♥ ◆✐❝✉

♦♥ t❤❡s❡ ♥❡❡❞s✱ ❛ ♥❡✇•② ♣r♦♣♦s❡❞ ❝♦♥tr♦• ♠❡t❤♦❞ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ✐♠♣•❡✲ ♠❡♥t❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ ■t ✐s ❛ s✐♠♣•❡ ❜✉t ❡✣❝✐❡♥t t❡❝❤♥✐q✉❡ ❢♦r ♥♦♥•✐♥❡❛r✱ ✉♥❦♥♦✇♥ ♦r ♣❛rt✐❛••② ❦♥♦✇♥ ❞②♥❛♠✐❝s✳ ■♥st❡❛❞ ♦❢ ❝♦♠♣✉t✐♥❣ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ s②st❡♠✱ t❤❡ ❞②♥❛♠✐❝s ✐s ❡✈❛•✉❛t❡❞ ❛s ❛ ✈❛r✐❛❜•❡ F ✐♥ r❡❛• t✐♠❡ ❜② t❤❡ s②st❡♠ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❝❛♥ ❝♦♠♣❡♥s❛t❡ ✇❡•• t❤❡ ♠♦❞❡•✐♥❣ ❡rr♦rs ❛♥❞ r❡❛•✲t✐♠❡ ❞✐st✉r❜❛♥❝❡s✳ ❚❤❡ ❛•❣♦r✐t❤♠ ♦❢
❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts✱ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t② ✭❈♦•✮✳ ❚❤❡ ❛①✐s x b ❛♥❞ y b ❛r❡ ❛•♦♥❣ t❤❡ t✇♦ ❛r♠s ♦❢ ❛ q✉❛❞r♦t♦r✱ ❛♥❞ t❤❡ ❛①✐s z b ✐s ✈❡rt✐❝❛• t♦ t❤❡ ❛r♠ ♣•❛♥❡✳ ❚❤❡ ❜♦❞② ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s ❛r❡ ♣r❡s❡♥t❡❞ ❛s p✱ q ❛♥❞ r✱ ✇❤✐❝❤ ❛r❡ ♣❤②s✐❝❛••② ♠❡❛s✉r❡❞ ❜② t❤r❡❡ ❣②r♦s❝♦♣❡s s❡♥s♦rs✳ ❚❤❡ ❢♦✉r r♦t♦rs ❛r❡ ♥✉♠❜❡r❡❞ ✶✱ ✷✱ ✸ ❛♥❞ ✹✳ ❚❤❡ ✜rst ♦♥❡ ✐s ❛•♦♥❣ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x b ❛①✐s✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❛•♦♥❣ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y b ❛①✐s✳ ❚❤❡ t❤✐r❞ ♦♥❡ ✐s ❛•♦♥❣ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x b ❛①✐s✱ ❛♥❞ t❤❡ ❢♦rt❤ ♦♥❡ ✐s ❛•♦♥❣ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y b ❛①✐s✳ ❚❤❡ r♦t♦r ✶ ❛♥❞ ✸ r♦t❛t❡ ❝•♦❝❦✇✐s❡ ✭s❡❡ ❢r♦♠ t❤❡ t♦♣ ♦❢ t❤❡ q✉❛❞r♦t♦r✮✳ ❚❤❡ r♦t♦r ✷ ❛♥❞ ✹ r♦t❛t❡ ❝♦✉♥t❡r❝•♦❝❦✇✐s❡✳ ■♥ t❤❡ ♥♦t❛t✐♦♥✱ t❤❡ s✉❜s❝r✐♣ts ✶ t♦ ✹ ♦❢ t❤❡ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ♠❡❛♥ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ r♦t♦r✳ ❋♦r ❡①❛♠♣•❡✱ t❤❡ ♣r♦♣✉•s✐♦♥ ❢♦r❝❡s✱ t❤❡ t❤r✉sts✱ ♦❢ ❡❛❝❤ ♠♦t♦r ❛r❡ ❞❡✜♥❡❞ T 1 ✱ T 2 ✱ T 3 ❛♥❞ T 4 ✇❤✐❝❤ ❛r❡ ❛•♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✶✳ ❋✐❣✉r❡ ✷✳✶✿ ❚❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ♦❢ ❛ q✉❛❞r♦t♦r✳ ❋♦r t❤❡ s✐♠♣•✐❝✐t② ♦❢ r❡❛❞✐♥❣✱ t❤❡ ♥♦t❛t✐♦♥s ✉s❡❞ ✐♥ t❤❡ ❦✐♥❡♠❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ♠♦❞❡•s ❛r❡ •✐st❡❞ ✐♥ ❚❛❜•❡ ✷✳✶✳ ✷✹ ✷✳✸✳ ❑✐♥❡♠❛t✐❝ ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ❚❛❜•❡ ✷✳✶✿ ❚❤❡ ♥♦t❛t✐♦♥s ✉s❡❞ ✐♥ t❤❡ ❦✐♥❡♠❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ♠♦❞❡•s P♦s✐t✐♦♥✱ ✈❡•♦❝✐t② ❛♥❞ ❛❝❝❡•❡r❛t✐♦♥ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ P w , V w , a w P♦s✐t✐♦♥✱ ✈❡•♦❝✐t② ❛♥❞ ❛❝❝❡•❡r❛t✐♦♥ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ P b , V b , a b ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s ✭❊✉•❡r ❛♥❣•❡s✮ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ φ✱ θ✱ ψ ❆♥❣✉•❛r ✈❡•♦❝✐t✐❡s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ p, q, r ❚❤r✉st T i ❍✉❜ ❢♦r❝❡ H i |♦••✐♥❣ ♠♦♠❡♥t M roll P✐t❝❤✐♥❣ ♠♦♠❡♥t M pitch |♦t♦r t♦rq✉❡ ✭②❛✇✐♥❣ ♠♦♠❡♥t✮ Q i ❇•❛❞❡ ✢❛♣♣✐♥❣ ♠♦♠❡♥t R i |♦t♦r ❣②r♦s❝♦♣✐❝ ❡✛❡❝t ♠♦♠❡♥ts M gx , M gy •r❛✈✐t② G ✷✳✸ ❑✐♥❡♠❛t✐❝ ▼♦❞❡• ■♥ t❤❡ ❦✐♥❡♠❛t✐❝ ❛♥❛•②s✐s✱ t❤❡ r❡•❛t✐♦♥s ♦❢ t❤❡ ♣♦s✐t✐♦♥s✱ ✈❡•♦❝✐t✐❡s ❛♥❞ ❛❝❝❡•❡r❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛♥❞ t❤❡ ❜♦❞② ❢r❛♠❡ ❛r❡ st✉❞✐❡❞✳ ❇❛s❡❞ ♦♥ t❤❡ ❢r❛♠❡s ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ t❤❡ tr❛♥s•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛♥❞ t❤❡ ❜♦❞② ❢r❛♠❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❡①♣•✐❝✐t•②✳ ✷✳✸✳✶ P♦s✐t✐♦♥✱ ▲✐♥❡❛r ❱❡•♦❝✐t② ❛♥❞ ❆❝❝❡•❡r❛t✐♦♥ ❚❤❡ ❊✉•❡r ❛♥❣•❡s φ✱ θ✱ ψ ❛r❡ t❤r❡❡ ❛♥❣•❡s ✐♥tr♦❞✉❝❡❞ ❜② ▲❡♦♥❤❛r❞ ❊✉•❡r t♦ ❞❡s❝r✐❜❡ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ ❛ r✐❣✐❞ ❜♦❞②✳ ❚❤❡② r❡♣r❡s❡♥t ❛ s❡q✉❡♥❝❡ ♦❢ t❤r❡❡ ❡•❡♠❡♥t❛• r♦t❛t✐♦♥s ❛❜♦✉t t❤❡ ❛①❡s ♦❢ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❆♥② ♦r✐❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ❝♦♠♣♦s✐♥❣ t❤r❡❡ ❡•❡♠❡♥t❛• r♦t❛t✐♦♥s✳ ❚❤❡r❡ ❛r❡ ♠❛♥② r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡s✱ s✉❝❤ ❛s ❳(❳✱ ❳()✱ ()(✳ ❋♦r ❡①❛♠♣•❡✱ ❛ s❡q✉❡♥❝❡ ❳(❳ ♠❡❛♥s ❛ ❜♦❞② ✜rst•② r♦t❛t❡s ❛❜♦✉t t❤❡ x ❛①✐s✱ t❤❡♥ r♦t❛t❡s ❛❜♦✉t t❤❡ y ❛①✐s✱ ❛♥❞ ❛t •❛st r♦t❛t❡s ❛❜♦✉t t❤❡ x ❛①✐s ❛❣❛✐♥✳ ❊✉•❡r ❛♥❣•❡s ✉s✉❛••② r❡♣r❡s❡♥t r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡s ❛❜♦✉t t✇♦ ❛①✐s✱ s✉❝❤ ❛s )❳)✱ ❳(❳✱ ()(✳ ❚❤✐s ♠❡❛♥s t❤❡r❡ ❛r❡ t✇♦ r♦t❛t✐♦♥s ❛❜♦✉t t❤❡ s❛♠❡ ❛①✐s✳ ❚❤❡ ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s r❡♣r❡s❡♥t r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡s ❛•♦♥❣ t❤r❡❡ ❞✐✛❡r❡♥t ❛①✐s✱ s✉❝❤ ❛s ❳()✱ ()❳✱ )(❳✳ •❡♥❡r❛••② ✇❡ ❝❛•• ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s ❛•s♦ ❊✉•❡r ❛♥❣•❡s✳ ❍❡r❡ ✇❡ ✉s❡ ❛ ❝♦♠♠♦♥ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ )(❳ ❢♦r t❤❡ ❊✉•❡r ❛♥❣•❡s✳ ■t ♠❡❛♥s ❛ ❜♦❞② ✜rst•② r♦t❛t❡s ❛❜♦✉t t❤❡ z ❛①✐s✱ t❤❡♥ r♦t❛t❡s ❛❜♦✉t t❤❡ y ❛①✐s✱ ❛♥❞ ❛t •❛st r♦t❛t❡s ❛❜♦✉t t❤❡ x ❛①✐s✳ ❚❤✐s r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ )(❳ ✐s ✉s❡❞ ✐♥ ♠❛♥② q✉❛❞r♦t♦r ♠♦❞❡•✐♥❣s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r♦t❛t✐♦♥ ♠❛tr✐① R b w ❢r♦♠ t❤❡ ✇♦r•❞ ❢r❛♠❡ t♦ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s✿ ✷✺ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✸✳ ❑✐♥❡♠❛t✐❝ ▼♦❞❡• R b w = R(x, φ)R(y, θ)R(z, ψ) =     1 0 0 0 ❝♦sφ -s✐♥φ 0 s✐♥φ ❝♦sφ         ❝♦sθ 0 s✐♥θ 0 1 0 -s✐♥θ 0 ❝♦sθ         ❝♦sψ -s✐♥ψ 0 s✐♥ψ ❝♦sψ 0 0 0 1     =     ❝♦sθ ❝♦sψ ❝♦sθ s✐♥ψ -s✐♥θ s✐♥φ s✐♥θ ❝♦sψ -❝♦sφ s✐♥ψ s✐♥φ s✐♥θ s✐♥ψ + ❝♦sφ ❝♦sψ s✐♥φ ❝♦sθ ❝♦sφ s✐♥θ ❝♦sψ + s✐♥φ s✐♥ψ ❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ ❝♦sφ ❝♦sθ     ✭✷✳✶✮ ❆s t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t② ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡✱ t❤❡ ♦r✐❣✐♥ P cb ❂✭✵✱✵✱✵✮✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡✇t❡❡♥ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡ ❛♥❞ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ✇♦r•❞ ❢r❛♠❡ ✐s ❞❡✜♥❡❞ ❛s r = P cb -P cw ✳ ❚❤❡r❡❢♦r❡✱ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ P b = (x b , y b , z b ) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛s✿ P w = P b + R b w r ✭✷✳✷✮ ❚❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛♥❞ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✿ V w = R w b V b = R b w T V b =     ❝♦sθ ❝♦sψ s✐♥φ s✐♥θ ❝♦sψ -❝♦sφ s✐♥ψ ❝♦sφ s✐♥θ ❝♦sψ + s✐♥φ s✐♥ψ ❝♦sθ s✐♥ψ s✐♥φ s✐♥θ s✐♥ψ + ❝♦sφ ❝♦sψ ❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ -s✐♥θ s✐♥φ ❝♦sθ ❝♦sφ ❝♦sθ     V b ✭✷✳✸✮ ♦r ❡①♣r❡ss❡❞ ✐♥ ❡❛❝❤ ❝♦♦r❞✐♥❛t❡ ❛s✿            V w x = (❝♦sθ ❝♦sψ)V b x + (s✐♥φ s✐♥θ ❝♦sψ -❝♦sφ s✐♥ψ)V b y + (❝♦sφ s✐♥θ ❝♦sψ + s✐♥φ s✐♥ψ)V b z V w y = (❝♦sθ s✐♥ψ)V b x + (s✐♥φ s✐♥θ s✐♥ψ + ❝♦sφ ❝♦sψ)V b y + (❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ)V b z V w z = (-s✐♥θ)V b x + (s✐♥φ ❝♦sθ)V b y + (❝♦sφ ❝♦sθ)V b z ✭✷✳✹✮ ❚❤❡♥✱ t❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ •✐♥❡❛r ❛❝❝❡•❡r❛t✐♦♥ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛♥❞ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✿ a w = R w b Vw ✭✷✳✺✮ ✷✻ ✷✳✸✳ ❑✐♥❡♠❛t✐❝ ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✸✳✷ ❆♥❣✉•❛r ❱❡•♦❝✐t② ❛♥❞ ❆❝❝❡•❡r❛t✐♦♥ ❆ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ )(❳ ✐s ❝❤♦s❡♥ ❢♦r t❤❡ ❊✉•❡r ❛♥❣•❡s✳ ■t ♠❡❛♥s t❤❡ ✇♦r•❞ ❢r❛♠❡ ✜rst•② r♦t❛t❡s ❛❜♦✉t ✐ts z ❛①✐s✱ t❤❡♥ r♦t❛t❡s ❛❜♦✉t ✐ts y ❛①✐s ❛♥❞ ❛t •❛st r♦t❛t❡s ❛❜♦✉t ✐ts x ❛①✐s t♦ ❣❡t t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② φ ✐s t❤❡ s❛♠❡ ✇✐t❤ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② p ❛❜♦✉t t❤❡ x ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ ❛s t❤❡ x ❛①✐s ✐s ♥♦t r♦t❛t❡❞ ❞✉r✐♥❣ t❤❡ •❛st st❡♣ ♦❢ t❤❡ r♦t❛t✐♦♥✳ ❚❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② θ ❛•♦♥❣ t❤❡ y ❛①✐s ✐s ✐♥✢✉❡♥❝❡❞ ♦♥•② ❜② t❤❡ •❛st st❡♣ r♦t❛t✐♦♥ ❛❜♦✉t t❤❡ x ❛①✐s✳ ❚❤❡r❡❢♦r❡✱ t❤✐s ✈❡•♦❝✐t② ♦♥•② ♥❡❡❞s t♦ ♠✉•t✐♣•② t❤❡ r♦t❛t✐♦♥❛• ♠❛tr✐① ❛❜♦✉t t❤❡ x ❛①✐s✳ ❆s t❤❡ z ❛①✐s ✐s ✜rst•② r♦t❛t❡❞✱ ✐ts ❛♥❣✉•❛r ✈❡•♦❝✐t② ψ ♥❡❡❞s t♦ ♠✉•t✐♣•② t✇♦ r♦t❛t✐♦♥❛• ♠❛tr✐❝❡ ❛❜♦✉t t❤❡ y ❛①✐s ✜rst•② t❤❡♥ ❛❜♦✉t t❤❡ x ❛①✐s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s p, q, r ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ❛♥❞ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s φ, θ, ψ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ✐s ♣r❡s❡♥t❡❞✿     p q r     = R ang     φ θ ψ    =     φ 0 0     + R T (x, φ)     0 θ 0     + R T (x, φ)R T (y, θ)     0 0 ψ    =     1 0 -s✐♥θ 0 ❝♦sθ s✐♥φ ❝♦sθ 0 -s✐♥φ ❝♦sφ ❝♦sθ         φ θ ψ    ✭✷✳✻✮     φ θ ψ    =     1 s✐♥φ t❛♥θ ❝♦sφ t❛♥θ 0 ❝♦sφ -s✐♥φ 0 s✐♥φ s❡❝θ ❝♦sφ s❡❝θ         p q r     ✭✷✳✼✮ ❚❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❛♥❣✉•❛r ❛❝❝❡•❡r❛t✐♦♥s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ❛♥❞ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❞✐✛❡r❡♥t✐❛t✐♥❣ ❊q✉❛t✐♦♥ ✭✷✳✻✮✳     ṗ q ṙ    = Ṙang     φ θ ψ    + R ang     φ θ ψ    ✭✷✳✽✮ ✷✼ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ✷✳✹ ❉②♥❛♠✐❝ ▼♦❞❡• ■♥ t❤❡ ❞②♥❛♠✐❝ ❛♥❛•②s✐s✱ ✇❡ ✜rst ✇r✐t❡ t❤❡ ◆❡✇t♦♥ ❊✉•❡r ❡q✉❛t✐♦♥s ❢♦r ❛ q✉❛❞r♦t♦r✳ ❚❤❡♥✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛♣♣•✐❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r ❛r❡ st✉❞✐❡❞✳ ❆t •❛st✱ ❛ s✐♠♣•✐✜❡❞ ❞②♥❛♠✐❝ ♠♦❞❡• ✐s ❢✉rt❤❡r ❞❡❞✉❝❡❞✳ ✷✳✹✳✶ ◆❡✇t♦♥ ❊✉•❡r ❋♦r♠❛•✐s♠ ■♥ ❝•❛ss✐❝❛• ♠❡❝❤❛♥✐❝s✱ t❤❡ ◆❡✇t♦♥ ❊✉•❡r ❡q✉❛t✐♦♥s s❤♦✇ t❤❡ tr❛♥s•❛t✐♦♥❛• ❛♥❞ r♦t❛t✐♦♥❛• ❞②♥❛♠✐❝s ♦❢ ❛ r✐❣✐❞ ❜♦❞②✳ ❚❤❡ ◆❡✇t♦♥ ❊✉•❡r ❢♦r♠❛•✐s♠ ✐s t❤❡ ❣r♦✉♣✐♥❣ ♦❢ ❊✉•❡r✬s t✇♦ ♠♦t✐♦♥ •❛✇s ❢♦r ❛ r✐❣✐❞ ❜♦❞② ✐♥t♦ ❛ s✐♥❣•❡ ❡q✉❛t✐♦♥ ✇✐t❤ s✐① ❝♦♠♣♦♥❡♥ts✳ ❚❤❡s❡ •❛✇s r❡•❛t❡ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t② ♦❢ ❛ r✐❣✐❞ ❜♦❞② ✇✐t❤ t❤❡ s✉♠ ♦❢ ❡①t❡r♥❛• ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛♣♣•✐❡❞ ♦♥ t❤❡ r✐❣✐❞ ❜♦❞②✳ ❲✐t❤♦✉t •♦ss ♦❢ ❣❡♥❡r❛•✐t②✱ ✇❡ ❞❡✜♥❡ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t②✳ ❙✉♣♣♦s✐♥❣ t❤❡r❡ ✐s ❛ s✐① ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✭❞✳♦✳❢✳✮ r✐❣✐❞ ❜♦❞② ✇❤✐❝❤ ❤❛s ❛ ♠❛ss m ❛♥❞ ❛♥ ✐♥❡rt✐❛ ♠❛tr✐① I ❛❜♦✉t t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t②✱ I ∈ R 3×3 ✳ ❚❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t② ✐s V b ✱ ❛♥❞ t❤❡ ❜♦❞② ❛♥❣✉•❛r ✈❡•♦❝✐t② ✐s ω b ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ V b , ω b ∈ R 3×1 ✳ ❚❤❡ ❡①t❡r♥❛• ❢♦r❝❡ ❛♥❞ t❤❡ ♠♦♠❡♥t ❛r❡ F b ❛♥❞ M b ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ F b , M b ∈ R 3×1 ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✈❡•♦❝✐t✐❡s ❛♥❞ t❤❡ ❡①t❡r♥❛• ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s✿ mI 0 3×3 0 3×3 I Vb ωb + ω b × (mV b ) ω b × (Iω b ) = F b M b ✭✷✳✾✮ ❚❤❡ s✉❜s❝r✐♣t 3×3 s❤♦✇s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ♠❛tr✐❝❡ ✐s ✸ ❜② ✸✳ ❚❤❡ s②♠❜♦• 0 3×3 ♠❡❛♥s ❛ 3×3 ❞✐♠❡♥s✐♦♥❛• ③❡r♦ ♠❛tr✐①✳ ■♥ ❛ q✉❛❞r♦t♦r s②st❡♠✱ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s ❝❤♦s❡♥ ❛s ✐♥ ❋✐❣✉r❡ ✷✳✶✳ ❆♣♣•②✐♥❣ t❤❡ ◆❡✇t♦♥ ❊✉•❡r ❢♦r♠❛•✐s♠ t♦ t❤❡ q✉❛❞r♦t♦r s②st❡♠✱ t❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❢♦r❝❡s ❛♥❞ t❤❡ •✐♥❡❛r ❛❝❝❡•❡r❛t✐♦♥s ✐s✿     ẍb ÿb zb     =     r ẏb -q żb p żb -r ẋb q ẋb -p ẏb     + 1 m     F b x F b y F b z     ✭✷✳✶✵✮ ✇❤❡r❡ ẋb ✱ ẏb ❛♥❞ żb ❛r❡ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t✐❡s ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ ❛♥❞ p✱ q ❛♥❞ r ❛r❡ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s ❛r♦✉♥❞ t❤❡ x✱ y ❛♥❞ z ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ✷✽ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ❆s t❤❡ q✉❛❞r♦t♦r ✐s s②♠♠❡tr✐❝ ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s✱ t❤❡ ❝r♦ss ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ ✐♥❡rt✐❛ ♠❛tr✐① ❛r❡ ❡q✉❛• t♦ ③❡r♦✱ I xy = I yz = I zx = 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥❡rt✐❛ ♠❛tr✐① ❝❛♥ ❜❡ ✇r✐tt❡♥✿ I =     I xx 0 0 0 I yy 0 0 0 I zz     ✭✷✳✶✶✮ ❚❤✉s✱ t❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♠♦♠❡♥ts ❛♥❞ t❤❡ ❛♥❣✉•❛r ❛❝❝❡•❡r❛t✐♦♥s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s✿     I xx ṗ I yy q I zz ṙ    =     (I yy -I zz )qr (I zz -I xx )pr (I xx -I yy )pq     +     M x M y M z     ✭✷✳✶✷✮ ◆♦✇✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❡①t❡r♥❛• ❢♦r❝❡s F b ❛♥❞ t❤❡ ♠♦♠❡♥t M b ❛♣♣•✐❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r✳ ✷✳✹✳✷ ❋♦r❝❡s ❚❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛♣♣•✐❡❞ ♦♥ ❛ q✉❛❞r♦t♦r ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❜② ❋❛② ❬✹✷(✳ ❯s✐♥❣ t❤❡ ❜•❛❞❡ ❡•❡♠❡♥t t❤❡♦r②✱ ❤❡ ❤❛s ❞❡❞✉❝❡❞ ❡①♣•✐❝✐t•② t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts✳ ❚❤❡s❡ ❡①♣r❡ss✐♦♥s ❛r❡ ✉s❡❞ ✐♥ t❤❡ ♠♦❞❡•✐♥❣✳ ❚❤❡ s✉♠♠❡r② ♦❢ t❤❡ ❡①t❡r♥❛• ❢♦r❝❡s ♠❡♥t✐♦♥❡❞ ✐♥ t❤✐s s✉❜s❡❝t✐♦♥ ❛♥❞ t❤❡✐r ❞✐r❡❝t✐♦♥s ❛r❡ •✐st❡❞ ✐♥ ❚❛❜•❡ ✷✳✷✳ ❆•• t❤❡ ♣❛r❛♠❡t❡rs ✉s❡❞ ✐♥ t❤✐s s✉❜s❡❝t✐♦♥ ❛r❡ •✐st❡❞ ✐♥ ❚❛❜•❡ ✷✳✸✳ ❆•• t❤❡ ❢♦r❝❡s ❛r❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❛❜•❡ ✷✳✷✿ ❚❤❡ ❡①t❡r♥❛• ❢♦r❝❡s ❛♣♣•✐❡❞ ♦♥ ❛ q✉❛❞r♦t♦r✳ ❋♦r❝❡ ❉✐r❡❝t✐♦♥ ❚❤r✉sts T i ❛•♦♥❣ t❤❡ z b ❛①✐s ❍✉❜ ❢♦r❝❡s H i ✐♥ t❤❡ x b , y b ♣•❛♥❡✱ ❛•♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② •r❛✈✐t② G ❛•♦♥❣ t❤❡ z w ❛①✐s ✶✳ ❚❤r✉st ✫ ❤✉❜ ❢♦r❝❡s ❚❤❡ ♣r♦♣✉•s✐♦♥ ♦❢ ❛ q✉❛❞r♦t♦r ✐s ✐♥❞✉❝❡❞ ❜② t❤❡ r♦t❛t✐♦♥s ♦❢ t❤❡ ❢♦✉r ♣r♦♣❡••❡rs✳ ❚❤❡ ✐♥✢♦✇ ✈❡•♦❝✐t② ✭❛•s♦ ❝❛••❡❞ ✐♥❞✉❝❡❞ ✈❡•♦❝✐t②✮ ✐s t❤❡ ✈❡•♦❝✐t② ♦❢ t❤❡ ❛✐r ✇❤❡♥ ✐t ❝r♦ss❡s t❤❡ ✷✾ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ❚❛❜•❡ ✷✳✸✿ ❚❤❡ ✈❛r✐❛❜•❡s ✐♥ t❤❡ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts✳ V H t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② ♦❢ t❤❡ r♦t♦r V 1 t❤❡ ✐♥✢♦✇ ✈❡•♦❝✐t② λ t❤❡ ✐♥✢♦✇ r❛t✐♦ µ t❤❡ ❛❞✈❛♥❝❡ r❛t✐♦ W t❤❡ q✉❛❞r♦t♦r ✇❡✐❣❤t ρ t❤❡ ❛✐r ❞❡♥s✐t② J r t❤❡ ♠♦♠❡♥ts ♦❢ ✐♥❡rt✐❛ ♦❢ t❤❡ r♦t♦rs A t❤❡ ♣r♦♣❡••❡r ❞✐s❦ ❛r❡❛ Ω t❤❡ r♦t❛t✐♦♥❛• ✈❡•♦❝✐t② ♦❢ t❤❡ r♦t♦r R rad t❤❡ ♣r♦♣❡••❡r r❛❞✐✉s a t❤❡ •✐❢t s•♦♣❡ θ 0 ✱ θ tw t❤❡ •✐♥❡❛r •✐❢t ♣❛r❛♠❡t❡rs σ t❤❡ s♦•✐❞✐t② r❛t✐♦ N t❤❡ ♥✉♠❜❡r ♦❢ ❜•❛❞❡s c t❤❡ ❜•❛❞❡ ❝❤♦r❞ Cd t❤❡ ❞r❛❣ ❝♦❡✣❝✐❡♥t m t❤❡ ♠❛ss ♦❢ t❤❡ q✉❛❞r♦t♦r g t❤❡ ❣r❛✈✐t② ❝♦❡✣❝✐❡♥t l t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❈♦• t♦ t❤❡ r♦t♦r V t❤❡ ✈♦•✉♠❡ ♦❢ t❤❡ q✉❛❞r♦t♦r ♣r♦♣❡••❡r ❜•❛❞❡✳ ■t ✐s ❛ss✉♠❡❞ t♦ ❜❡ ✉♥✐❢♦r♠ ❛♥❞ ✇✐t❤♦✉t ❞✐s❝♦♥t✐♥✉✐t②✳ ❚❤❡ ✐♥✢♦✇ ✈❡•♦❝✐t② ❝❛♥ ❜❡ ♠♦❞❡•❡❞ ❛s✿ V 1 = - V 2 H 2 + V 2 H 2 2 + W 2ρA 2 ✭✷✳✶✸✮ ✇❤❡r❡ W ✐s t❤❡ q✉❛❞r♦t♦r ✇❡✐❣❤t✱ ρ ✐s t❤❡ ❛✐r ❞❡♥s✐t②✱ A ✐s t❤❡ ♣r♦♣❡••❡r ❞✐s❦ ❛r❡❛✱ V H ✐s t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② ♦❢ t❤❡ r♦t♦r✱ ✇❤✐❝❤ ✐s t❤❡ q✉❛❞r♦t♦r •✐♥❡❛r ✈❡•♦❝✐t② ✐♥ t❤❡ x b ✱ y b ♣•❛♥❡ ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡✿ V H = ẋ2 b + ẏ2 b ✭✷✳✶✹✮ ❚❤❡ ✐♥✢♦✇ r❛t✐♦ λ ✐s ❛ ❞✐♠❡♥s✐♦♥•❡ss q✉❛♥t✐t② ✉s❡❞ ✐♥ t❤❡ ❤❡•✐❝♦♣t❡r •✐t❡r❛t✉r❡ t♦ r❡•❛t❡ t❤❡ ✐♥✢♦✇ ✈❡•♦❝✐t② t♦ t❤❡ r♦t♦r t✐♣ ✈❡•♦❝✐t②✳ ❚❤❡ ✐♥✢♦✇ r❛t✐♦ ✐s ❞❡✜♥❡❞ ❛s✿ λ = V 1 -żb ΩR rad ✭✷✳✶✺✮ ✇❤❡r❡ Ω ✐s t❤❡ r♦t❛t✐♦♥❛• ✈❡•♦❝✐t② ♦❢ t❤❡ r♦t♦r✱ R rad ✐s t❤❡ ♣r♦♣❡••❡r r❛❞✐✉s✳ ❆♥♦t❤❡r ❞✐♠❡♥s✐♦♥•❡ss q✉❛♥t✐t② ✐s t❤❡ ❛❞✈❛♥❝❡ r❛t✐♦✳ ■t r❡•❛t❡s t❤❡ ❤♦r✐③♦♥t❛• ✈❡•♦❝✐t② ✇✐t❤ t❤❡ t✐♣ ✈❡•♦❝✐t② ♦❢ ❛ r♦t♦r✳ ❚❤❡ ❛❞✈❛♥❝❡ r❛t✐♦ µ ✐s✿ µ = V H ΩR rad ✭✷✳✶✻✮ ✸✵ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ❲❤❡♥ t❤❡ ❛✐r ❝r♦ss❡s t❤❡ ♣r♦♣❡••❡rs✱ ✐t ✐♥❞✉❝❡s t✇♦ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s✿ t❤❡ •✐❢t ❢♦r❝❡ ❛♥❞ t❤❡ ❞r❛❣ ❢♦r❝❡✳ ❚❤❡ •✐❢t ❢♦r❝❡ ✐s ♣❡r♣❡♥❞✐❝✉•❛r t♦ t❤❡ •♦❝❛• ✈❡•♦❝✐t②✱ ✇❤✐•❡ t❤❡ ❞r❛❣ ❢♦r❝❡ ✐s ♣❛r❛••❡• t♦ ✐t✳ ❲❡ ❝❛♥ ♣r♦•❡❝t t❤❡s❡ t✇♦ ❢♦r❝❡s ♦♥t♦ t❤❡ t❤r✉st ❢♦r❝❡ ❛♥❞ t❤❡ ❤✉❜ ❢♦r❝❡✳ ❚❤❡ t❤r✉st ❢♦r❝❡ T ✐s ❛•♦♥❣ t❤❡ z b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡ r❡st ♦❢ t❤❡ ❢♦r❝❡ ✇❤✐❝❤ ✐s ✐♥ t❤❡ ♣•❛♥❡ x b ✱ y b ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡ ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ✐♥ t✇♦ ❞✐r❡❝t✐♦♥s✿ t❤❡ ❤✉❜ ❢♦r❝❡ H ✇❤✐❝❤ ✐s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② V H ✱ t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞✐♥❣ ✈❡•♦❝✐t② V ✐♥ t❤❡ x b ✱ y b ♣•❛♥❡❀ t❤❡ ♦t❤❡r ♦♥❡ ✐s t❤❡ s✐❞❡ ❢♦r❝❡ Y ✇❤✐❝❤ ✐s ✐♥ t❤❡ ♣❡r♣❡♥❞✐❝✉•❛r ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② ✐♥ t❤❡ s❛♠❡ ♣•❛♥❡✳ ❚❤❡s❡ ❢♦r❝❡s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✷✳ ❚❤❡ ❡••✐♣s❡ ✐s t❤❡ r♦t♦r ♣•❛t❡✳ ❋✐❣✉r❡ ✷✳✷✿ ❚❤❡ t❤r✉st ❛♥❞ ❤✉❜ ❢♦r❝❡s ♦♥ ❛ r♦t♦r✳ ❯s✐♥❣ t❤❡ ❜•❛❞❡ ❡•❡♠❡♥t t❤❡♦r②✱ ❋❛② ❤❛s ❞❡❞✉❝❡❞ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❬✹✷(✳ ❍❡ ❤❛s ♣r♦✈❡❞ t❤❡ s✐❞❡ ❢♦r❝❡ Y ✐s ❛•✇❛②s ③❡r♦✳ ❚❤✐s ♠❡❛♥s t❤❡ ❞r❛❣ ❢♦r❝❡ ✐s ♦♥•② ❛•♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ♦❢ t❤❡ q✉❛❞r♦t♦r✱ ✇❤✐❝❤ ✐s ♥❛♠❡❞ ❤✉❜ ❢♦r❝❡✳ ❚❤❡ t❤r✉st T ❛♥❞ t❤❡ ❤✉❜ ❢♦r❝❡ H ♦♥ ❛ r♦t♦r ❛r❡ ♠♦❞❡•❡❞ ❛s✿ T = C T ρA(ΩR rad ) 2 H = C H ρA(ΩR rad ) 2 ✭✷✳✶✼✮ ❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ t❤r✉st ❛♥❞ ❤✉❜ ❢♦r❝❡ ❛r❡✿ C T = σa 1 6 + 1 4 µ 2 θ 0 -1 + µ 2 θ tw 8 - 1 4 λ C H = σa 1 4a µ Cd + 1 4 λµ θ 0 - θ tw 2 ✭✷✳✶✽✮ ✇❤❡r❡ a ✐s t❤❡ •✐❢t s•♦♣❡✱ θ 0 ❛♥❞ θ tw ❛r❡ t❤❡ •✐♥❡❛r •✐❢t ♣❛r❛♠❡t❡rs✱ Cd ✐s t❤❡ ❞r❛❣ ❝♦❡✣❝✐❡♥t✱ σ ✐s t❤❡ s♦•✐❞✐t② r❛t✐♦ ✇❤✐❝❤ ❞❡✜♥❡s t❤❡ r❛t✐♦ ♦❢ t❤❡ ❜•❛❞❡ ❛r❡❛ t♦ t❤❡ ❞✐s❦ ❛r❡❛✳ ❚❤❡ s♦•✐❞✐t② ✸✶ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• r❛t✐♦ ✐s ❞❡✜♥❡❞ ❛s✿ σ = N c πR ✭✷✳✶✾✮ ✇❤❡r❡ N ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❜•❛❞❡s✱ c ✐s t❤❡ ❜•❛❞❡ ❝❤♦r❞✳ ❋✉rt❤❡r ❞❡t❛✐•s ❛❜♦✉t t❤❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✼✱ ✹✷(✳ ❚❤❡ ❡①♣r❡ss✐♦♥s ❛❜♦✈❡ ❛r❡ t❤❡ t❤r✉st ❛♥❞ ❤✉❜ ❢♦r❝❡ ♦♥ ❛ r♦t♦r✳ ❆s ❛ q✉❛❞r♦t♦r ❤❛s ❢♦✉r r♦t♦rs✱ t❤❡ t❤r✉sts ❛♥❞ ❤✉❜ ❢♦r❝❡s ❛r❡ ❞❡✜♥❡❞ ❛s T i ✱ H i ✱ i = 1,...,4✱ ✇❤✐❝❤ ❛r❡ r❡•❛t❡❞ t♦ ❡❛❝❤ r♦t♦r r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ t♦t❛• t❤r✉st ❛♥❞ ❤✉❜ ❢♦r❝❡ ♦♥ ❛ q✉❛❞r♦t♦r ❛r❡✿ - 4 i=1 H ix = -H 1x -H 2x -H 3x -H 4x - 4 i=1 H iy = -H 1y -H 2y -H 3y -H 4y 4 i=1 T i = T 1 + T 2 + T 3 + T 4 ✭✷✳✷✵✮ ❆s t❤❡ ❤✉❜ ❢♦r❝❡ H i ✐s ❛•♦♥❣ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t②✱ ✐t ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ✐♥t♦ t✇♦ ❢♦r❝❡s H ix ❛♥❞ H iy ✳ H ix ✐s ❛•♦♥❣ t❤❡ x b ❛①✐s ❛♥❞ H iy ✐s ❛•♦♥❣ t❤❡ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡ t❤r✉st T i ✐s ❛•✇❛②s ❛•♦♥❣ t❤❡ z b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ✷✳ •r❛✈✐t② ❚❤❡ ❣r❛✈✐t② ✐s ❛•♦♥❣ t❤❡ z w ❛①✐s ♦❢ t❤❡ ✇♦r•❞ ❢r❛♠❡✿ G = -mg ✭✷✳✷✶✮ ✇❤❡r❡ m ✐s t❤❡ ♠❛ss ♦❢ ❛ q✉❛❞r♦t♦r✱ g ✐s t❤❡ ❣r❛✈✐t② ❝♦❡✣❝✐❡♥t✳ ✸✳ ❚♦t❛• ❡①t❡r♥❛• ❢♦r❝❡s ❚❤❡ t❤r✉st ❛♥❞ ❤✉❜ ❢♦r❝❡s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡ ❢♦r❝❡s ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❝❛♥ ❜❡ tr❛♥s•❛t❡❞ ✐♥t♦ t❤❡ ❜♦❞② ❢r❛♠❡ ✉s✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ tr❛♥s•❛t✐♦♥✿ F b = R b w F w ✭✷✳✷✷✮ ✸✷ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ❚❤❡r❡❢♦r❡✱ t❤❡ t♦t❛• ❡①t❡r♥❛• ❢♦r❝❡ ✈❡❝t♦r ♦♥ ❛ q✉❛❞r♦t♦r ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s✿ F b =         - 4 i=1 H ix - 4 i=1 H iy 4 i=1 T i         + R b w     0 0 -G     ✭✷✳✷✸✮ ✷✳✹✳✸ ▼♦♠❡♥ts ❯s✐♥❣ t❤❡ ❜•❛❞❡ ❡•❡♠❡♥t t❤❡♦r②✱ ❋❛② ❤❛s ❛•s♦ ❞❡❞✉❝❡❞ ❡①♣•✐❝✐t•② t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ ❛❡r♦❞②♥❛♠✐❝ ♠♦♠❡♥ts ❬✹✷(✳ ❚❤❡s❡ ♠♦♠❡♥ts ❛r❡ ✉s❡❞ ✐♥ t❤❡ ♠♦❞❡•✐♥❣✳ ❚❤❡ s✉♠♠❡r② ♦❢ t❤❡ ❡①t❡r♥❛• ♠♦♠❡♥ts ♠❡♥t✐♦♥❡❞ ✐♥ t❤✐s s✉❜s❡❝t✐♦♥ ❛♥❞ t❤❡✐r ❞✐r❡❝t✐♦♥s ❛r❡ •✐st❡❞ ✐♥ ❚❛❜•❡ ✷✳✹✳ ❆•• t❤❡ ✈❛r✐❛❜•❡s ✉s❡❞ ✐♥ t❤✐s s✉❜s❡❝t✐♦♥ ❛r❡ •✐st❡❞ ✐♥ ❚❛❜•❡ ✷✳✸✳ ❆•• t❤❡ ♠♦♠❡♥ts ❛r❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❛❜•❡ ✷✳✹✿ ❚❤❡ ❡①t❡r♥❛• ♠♦♠❡♥ts ❛♣♣•✐❡❞ ♦♥ ❛ q✉❛❞r♦t♦r✳ ▼♦♠❡♥t ◆♦t❛t✐♦♥ ❉✐r❡❝t✐♦♥ |♦••✐♥❣ ♠♦♠❡♥t M roll ❛❜♦✉t t❤❡ x b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ P✐t❝❤✐♥❣ ♠♦♠❡♥t M pitch ❛❜♦✉t t❤❡ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ (❛✇✐♥❣ ♠♦♠❡♥ts Q i ❛❜♦✉t t❤❡ z b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❇•❛❞❡ ✢❛♣♣✐♥❣ ♠♦♠❡♥ts R ix ✱ R iy ❛❜♦✉t t❤❡ x b ✱ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ •②r♦s❝♦♣✐❝ ❡✛❡❝t ♠♦♠❡♥ts M gx ✱ M gy ❛❜♦✉t t❤❡ x b ✱ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ✶✳ |♦••✐♥❣ ♠♦♠❡♥t ✫ P✐t❝❤✐♥❣ ♠♦♠❡♥t ❆s ❡❛❝❤ r♦t♦r ❣❡♥❡r❛t❡s ❛ t❤r✉st✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❡❛❝❤ ♣❛✐r ♦❢ r♦t♦rs ❝❛✉s❡s t❤❡ ♠♦♠❡♥ts ❛r♦✉♥❞ t❤❡ x b ❛♥❞ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t❤r✉sts ❜❡t✇❡❡♥ t❤❡ r♦t♦r ✷ ❛♥❞ ✹ ❝❛✉s❡s t❤❡ r♦••✐♥❣ ♠♦♠❡♥t ❛❜♦✉t t❤❡ x b ❛①✐s✱ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t❤r✉sts ❜❡t✇❡❡♥ t❤❡ r♦t♦r ✶ ❛♥❞ ✸ ❝❛✉s❡s t❤❡ ♣✐t❝❤✐♥❣ ♠♦♠❡♥t ❛❜♦✉t t❤❡ y b ❛①✐s✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠♦♠❡♥ts ✐s ❞❡❝✐❞❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ r✐❣❤t ❤❛♥❞ r✉•❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r♦••✐♥❣ ♠♦♠❡♥t M roll ❛♥❞ t❤❡ ♣✐t❝❤✐♥❣ ♠♦♠❡♥t M pitch ❛r❡ ❞❡✜♥❡❞ ❛s✿ M roll = l(T 2 -T 4 ) M pitch = l(-T 1 + T 3 ) ✭✷✳✷✹✮ ✸✸ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ✇❤❡r❡ l ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t② ✭❈♦•✮ t♦ t❤❡ r♦t♦r✱ T i ✱ i = 1,...,4 ❛r❡ t❤❡ t❤r✉sts ♦❢ t❤❡ ❢♦✉r r♦t♦r✳ ❚❤❡ ♥✉♠❜❡rs ♦❢ t❤❡ r♦t♦rs ❛r❡ ❞❡✜♥❡❞ ✐♥ ❋✐❣✉r❡ ✷✳✶✳ ✷✳ (❛✇✐♥❣ ♠♦♠❡♥t ❚❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛•s♦ ❝❛✉s❡ ♠♦♠❡♥ts ❛❜♦✉t t❤❡ s❤❛❢t ❛♥❞ ❤✉❜✳ ❚❤❡ ♠♦♠❡♥t ❛❜♦✉t t❤❡ s❤❛❢t Q ♠❛❦❡s t❤❡ q✉❛❞r♦t♦r t✉r♥ ❛❜♦✉t t❤❡ z b ❛①✐s ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡✳ ■t ❞❡t❡r♠✐♥❡s t❤❡ ♣♦✇❡r r❡q✉✐r❡❞ ❢♦r t❤❡ r♦t♦r t♦ ❦❡❡♣ t❤❡ r♦t♦r s♣✐♥♥✐♥❣✳ ❚❤❡ ②❛✇✐♥❣ ♠♦♠❡♥t Q ✐s ♠♦❞❡•❡❞ ❛s✿ Q = C Q ρA(ΩR rad ) 2 R rad ✭✷✳✷✺✮ ❚❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ②❛✇✐♥❣ ♠♦♠❡♥t ✐s✿ C Q = σa 1 8a (1 + µ 2 ) Cd + λ 1 6 θ 0 - 1 8 θ tw - 1 4 λ ✭✷✳✷✻✮ ❆•• t❤❡ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❝♦❡✣❝✐❡♥t C Q ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✷✳✹✳✷✳ ❋✉rt❤❡r ❞❡t❛✐•s ❛❜♦✉t t❤❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✼✱ ✹✷(✳ ❚❤❡r❡ ❛r❡ ❢♦✉r r♦t♦rs ✐♥ ❛ q✉❛❞r♦t♦r✳ ❚❤❡r❡❢♦r❡ ❡❛❝❤ r♦t♦r ❤❛s ❛ ②❛✇✐♥❣ ♠♦♠❡♥t H i ✱ i = 1,...,4✳ ❚❤❡ ❢♦✉r r♦t♦rs ❛r❡ s❡♣❛r❛t❡❞ ✐♥t♦ t✇♦ ♣❛✐rs ✇✐t❤ ❞✐✛❡r❡♥t r♦t❛t✐♦♥❛• ❞✐r❡❝t✐♦♥s✳ ❚❤❡ r♦t♦r ✶ ❛♥❞ ✸ r♦t❛t❡ ❝•♦❝❦✇✐s❡ ✭s❡❡ ❢r♦♠ t❤❡ t♦♣ ♦❢ t❤❡ q✉❛❞r♦t♦r✮✳ ❚❤❡ r♦t♦r ✷ ❛♥❞ ✹ r♦t❛t❡ ❝♦✉♥t❡r❝•♦❝❦✇✐s❡✳ ❚❤❡s❡ t✇♦ ♣❛✐rs ♦❢ r♦t♦rs ✐♥❞✉❝❡ ♠♦♠❡♥ts ✐♥ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥✳ ❚❤✉s✱ t❤❡ t♦t❛• ②❛✇✐♥❣ ♠♦♠❡♥t ✐s✿ 4 i=1 Q i = Q 1 -Q 2 + Q 3 -Q 4 ✭✷✳✷✼✮ ✇❤❡r❡ t❤❡ s✉❜s❝r✐♣ts 1,...,4 ❛r❡ t❤❡ ♥✉♠❜❡rs ♦❢ t❤❡ r♦t♦rs ❞❡✜♥❡❞ ✐♥ ❋✐❣✉r❡ ✷✳✶✳ ✸✳ ❇•❛❞❡ ✢❛♣♣✐♥❣ ♠♦♠❡♥t ❉✉r✐♥❣ t❤❡ tr❛♥s•❛t✐♦♥❛• ✢✐❣❤t✱ t❤❡ ❛❞✈❛♥❝✐♥❣ ❜•❛❞❡ ❡①♣❡r✐❡♥❝❡s ❛ ❤✐❣❤❡r ✈❡•♦❝✐t② r❡•❛t✐✈❡ t♦ t❤❡ ❛✐r t❤❛♥ t❤❡ r❡tr❡❛t✐♥❣ ❜•❛❞❡✳ ❚❤✐s ❡✛❡❝t ✐s ❝❛••❡❞ t❤❡ ❜•❛❞❡ ✢❛♣♣✐♥❣✳ ❚❤✐s r❡s✉•ts ♠♦r❡ •✐❢t ♦♥ t❤❡ ❛❞✈❛♥❝✐♥❣ ❜•❛❞❡ t❤❛♥ t❤❡ r❡tr❡❛t✐♥❣ ❜•❛❞❡✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ •✐❢ts ❛♣♣•✐❡s ❛ ♠♦♠❡♥t t♦ t❤❡ r♦t♦r ❞✐s❦ ❛♥❞ •❡❛❞s t♦ ❛ ❣②r♦s❝♦♣❡ r❡s♣♦♥s❡ t❤❛t t✐•ts t❤❡ r♦t♦r ❞✐s❦ ❜❛❝❦ ✇✐t❤ ❛ ✢❛♣♣✐♥❣ ❛♥❣•❡ β✳ ❚❤❡ ❜•❛❞❡ ✢❛♣♣✐♥❣ ♠♦♠❡♥t R ✐s ♣❡r♣❡♥❞✐❝✉•❛r t♦ t❤❡ r♦t♦r s❤❛❢t ✸✹ ✷✳✹✳ ❉②♥❛♠✐❝ ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ❛♥❞ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ♦❢ t❤❡ q✉❛❞r♦t♦r✱ ✇❤✐❝❤ ✐s ♠♦❞❡•❡❞ ❛s✿ R = C R ρA(ΩR rad ) 2 R rad ✭✷✳✷✽✮ ✇✐t❤ t❤❡ ❝♦❡✣❝✐❡♥t ❛s✿ C R = -σaµ 1 6 θ 0 - 1 8 θ tw - 1 8 λ ✭✷✳✷✾✮ ❚❤❡ ♣❛r❛♠❡t❡rs ♦❢ C R ❛r❡ ♠❡♥t✐♦♥❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✷✳✹✳✷✳ ❋✉rt❤❡r ❞❡t❛✐•s ❛❜♦✉t t❤❡ ♣❛r❛♠✲ ❡t❡rs ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✼✱ ✹✷(✳ ❚❤❡ ❜•❛❞❡ ✢❛♣♣✐♥❣ ♠♦♠❡♥t ♦❢ ❡❛❝❤ r♦t♦r R i ✱ i = 1,...,4 ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ✐♥t♦ t✇♦ ♠♦♠❡♥ts R ix ❛♥❞ R iy ✳ ❚❤❡ t♦t❛• ❜•❛❞❡ ✢❛♣♣✐♥❣ ♠♦♠❡♥ts ♦❢ ❛ q✉❛❞r♦t♦r ❛r❡✿ 4 i=1 R ix = R 1x -R 2x + R 3x -R 4x 4 i=1 R iy = R 1y -R 2y + R 3y -R 4y
•♦♥❣ t❤❡ x b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ ❡❛❝❤ r♦t♦r ❛•s♦ ❤❛s ❛ ♠♦♠❡♥t M gy ❛•♦♥❣ t❤❡ y b ❛①✐s ✐♥❞✉❝❡❞ ❜② t❤❡ r♦t♦r ❣②r♦s❝♦♣✐❝ ❡✛❡❝t✳ ❲❤❡♥ ❛ q✉❛❞r♦t♦r r♦t❛t❡s ❛•♦♥❣ t❤❡ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ ❡❛❝❤ r♦t♦r ❛•s♦ ❤❛s ❛ ♠♦♠❡♥t ✸✺ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✺✳ ◗✉❛❞r♦t♦r ▼♦❞❡• M gx ❛•♦♥❣ t❤❡ x b ❛①✐s✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠♦♠❡♥t ✐s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ r✐❣❤t ❤❛♥❞ r✉•❡ ✭s❡❡ ❋✐❣✉r❡ ✷✳✸✮✳ ❚❤❡ ❣②r♦s❝♦♣✐❝ ❡✛❡❝t ♠♦♠❡♥ts ❛r❡ ♠♦❞❡•❡❞ ❛s✿ 4 i=1 M gix = J r q(Ω 1 -Ω 2 + Ω 3 -Ω 4 ) 4 i=1 M giy = -J r p(Ω 1 -Ω 2 + Ω 3 -Ω 4 ) ✭✷✳✸✶✮ ✇❤❡r❡ J r ✐s t❤❡ ♠♦♠❡♥ts ♦❢ ✐♥❡rt✐❛ ♦❢ t❤❡ r♦t♦rs✱ Ω i ✱ i = 1,...,4✱ ❛r❡ t❤❡ r♦t❛t✐♦♥❛• s♣❡❡❞s ♦❢ ❢♦✉r r♦t♦rs✳ ✺✳ ❚♦t❛• ♠♦♠❡♥ts ❚❤❡r❡❢♦r❡✱ t❤❡ t♦t❛• ❡①t❡r♥❛• ♠♦♠❡♥t ✈❡❝t♦r ♦♥ ❛ q✉❛❞r♦t♦r ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s✿ M b =         4 i=1 R ix + M roll + 4 i=1 M gix 4 i=1 R iy + M pitch + 4 i=1 M giy 4 i=1 Q i         ✭✷✳✸✷✮ ✷✳✺ ◗✉❛❞r♦t♦r ▼♦❞❡• ■♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥s✱ ✇❡ ❤❛✈❡ ❞❡❞✉❝❡❞ t❤❡ ◆❡✇t♦♥✲❊✉•❡r ❡q✉❛t✐♦♥s ❢♦r ❛ q✉❛❞r♦t♦r s②s✲ t❡♠✳ ❚❤❡♥ t❤❡ ✐♠♣♦rt❛♥t ❡①t❡r♥❛• ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ❛♥❛•②③❡❞✳ ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ❢✉•• ♠♦❞❡• ♦❢ ❛ q✉❛❞r♦t♦r s②st❡♠ ✇✐•• ❜❡ ✜rst•② ♣r♦♣♦s❡❞✳ ❇❛s❡❞ ♦♥ s♦♠❡ ❛♣♣•✐❝❛t✐♦♥s✱ t❤✐s ♠♦❞❡• ❝❛♥ ❜❡ ❢✉rt❤❡r s✐♠♣•✐✜❡❞✳ ✷✳✺✳✶ ❈♦♠♣•❡t❡ ▼♦❞❡• ■♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥s✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ♦♥ ❛ q✉❛❞r♦t♦r ❤❛✈❡ ❜❡❡♥ ❞❡❞✉❝❡❞✳ ❋r♦♠ ❊q✉❛t✐♦♥s ✭✷✳✸✮✱ ✭✷✳✼✮✱ ✭✷✳✶✵✮ ❛♥❞ ✭✷✳✶✷✮✱ ❛ ❢✉•• ♠♦❞❡• ❢♦r ❛ q✉❛❞r♦t♦r ✐s s❤♦✇♥ ✐♥ ❊q✉❛t✐♦♥ ✭✷✳✸✸✮✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ s②♠❜♦•s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❚❛❜•❡ ✷✳✶✳ ✸✻ ✷✳✺✳ ◗✉❛❞r♦t♦r ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣      ẍw ÿw zw      =     ❝♦sθ ❝♦sψ s✐♥φ s✐♥θ ❝♦sψ -❝♦sφ s✐♥ψ ❝♦sφ s✐♥θ ❝♦sψ +s✐♥φ s✐♥ψ ❝♦sθ s✐♥ψ s✐♥φ s✐♥θ s✐♥ψ +❝♦sφ ❝♦sψ ❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ -s✐♥θ s✐♥φ ❝♦sθ ❝♦sφ ❝♦sθ             r ẏb -q żb p żb -r ẋb q ẋb -p ẏb     + 1 m F b     ✭✷✳✸✸❛✮ F b =         - 4 i=1 H ix - 4 i=1 H iy 4 i=1 T i         + R b w     0 0 -G     ✭✷✳✸✸❜✮     φ θ ψ    =     1 s✐♥φ t❛♥θ ❝♦sφ t❛♥θ 0 ❝♦sφ -s✐♥φ 0 s✐♥φ s❡❝θ ❝♦sφ s❡❝θ         p q r     ✭✷✳✸✸❝✮     I x ṗ I y q I z ṙ    =     (I y -I z )qr (I z -I x )pr (I x -I y )pq     + M b , M b =     M b x M b y M b z     =         4 i=1 R ix + M roll + 4 i=1 M gix 4 i=1 R iy + M pitch + 4 i=1 M giy 4 i=1 Q i         ✭✷✳✸✸❞✮ ✇❤❡r❡ T = C T ρA(ΩR rad ) 2 , H = C H ρA(ΩR rad ) 2 Q = C Q ρA(ΩR rad ) 2 R rad , R = C R ρA(ΩR rad ) 2 R rad M roll = l(T 2 -T 4 ), M pitch = l(-T 1 + T 3 ) 4 i=1 M gix = J r q(Ω 1 -Ω 2 + Ω 3 -Ω 4 ), 4 i=1 M giy = -J r p(Ω 1 -Ω 2 + Ω 3 -Ω 4 ) ✭✷✳✸✹✮ ■♥ ❊q✉❛t✐♦♥ ✭✷✳✸✸✮✱ t❤❡ ✜rst t✇♦ ❡q✉❛t✐♦♥s ❛r❡ t❤❡ ♣♦s✐t✐♦♥ ❞②♥❛♠✐❝s✱ ❛♥❞ t❤❡ ♦t❤❡r t❤r❡❡ ❡q✉❛t✐♦♥s ❛r❡ t❤❡ ❛♥❣•❡ ❞②♥❛♠✐❝s✳ ❚❤❡ ❛♥❣•❡ ❞②♥❛♠✐❝s ✐s ♦♥•② r❡•❛t❡❞ ♦♥ t❤❡ ❛♥❣✉•❛r ✐♥❢♦r♠❛t✐♦♥✳ ❲❤✐•❡ ✐♥ t❤❡ ♣♦t✐♦♥ ❞②♥❛♠✐❝s✱ t❤❡ ♣♦s✐t✐♦♥s ❛r❡ ❛✛❡❝t❡❞ ❜② t❤❡ ❝❤❛♥❣❡s ♦❢ t❤❡ ❛♥❣•❡s✳ ✸✼ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✺✳ ◗✉❛❞r♦t♦r ▼♦❞❡• ❚❤✐s ♠♦❞❡• ❝❛♥ ❜❡ ❛ ❣♦♦❞ ✇♦r❦❜❡♥❝❤ ❢♦r t❡st✐♥❣ ❝♦♥tr♦• ♠❡t❤♦❞s✱ ❛♥❞ ❡s♣❡❝✐❛••② ❢♦r ❝♦♠♣❛r✐♥❣ ❞✐✛❡r❡♥t ♠❡t❤♦❞s✳ ■♥ t❤✐s ♠♦❞❡•✱ ❛ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ )(❳ ❢♦r ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s ✐s ❝❤♦s❡♥✱ ✇❤✐❝❤ ✐s ❣❡♥❡r❛••② ✉s❡❞ ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ▼♦st ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ✐♥ t❤❡ •✐t❡r❛t✉r❡ ❝❛♥ ❜❡ ❡❛s✐•② ✐♠♣•❡♠❡♥t❡❞ ♦♥ t❤✐s ♠♦❞❡• ✇✐t❤♦✉t ♠❛♥② ♠♦❞✐✜❝❛t✐♦♥s✳ ❆•s♦✱ ✐♠♣♦rt❛♥t ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ♠♦❞❡•✐♥❣✱ ✇❤✐•❡ ✐♥ t❤❡ •✐t❡r❛t✉r❡ ✉s✉❛••② t❤❡ t❤r✉st ❛♥❞ t❤❡ ②❛✇✐♥❣ ♠♦♠❡♥t ❛r❡ ✐♥❝•✉❞❡❞ ✐♥ t❤❡ ♠♦❞❡•s✳ ❚❤❡ ✐♠♣♦rt❛♥t ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s✱ t❤❡ t❤r✉sts ❛♥❞ ❤✉❜ ❢♦r❝❡s✱ ❛r❡ ♠♦❞❡•❡❞ ❜❛s❡❞ ♦♥ t❤❡ ❜•❛❞❡ ❡•❡♠❡♥t t❤❡♦r②✳ ❚❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② r❡•❛t❡❞ t♦ t❤❡ q✉❛❞r♦t♦r ❜♦❞② ❢r❛♠❡ ✐s ✉s❡❞ ✐♥ t❤❡ ♠♦❞❡•✐♥❣ ♦❢ t❤❡ ❢♦r❝❡s✱ s✐♥❝❡ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛r❡ ✐♥❞✉❝❡❞ ❜② t❤❡ •♦❝❛• ❛✐r✢♦✇ ❛r♦✉♥❞ t❤❡ r♦t♦rs✳ ❚❤❡ ♠♦❞❡•✐♥❣ ✐s ♠♦r❡ ♣r❡❝✐s❡ t❤❛♥ t❤❡ ♠♦❞❡• ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛••❛❤ ❬✷✵(✳ ■♥ ❤✐s t❤❡s✐s✱ ❤❡ ❤❛s ✉s❡❞ t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡✳ ❆s q✉❛❞r♦t♦rs t✐•t✱ t❤❡ s✐❞❡✇❛②s ✈❡•♦❝✐t② ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s ♥♦t ❛•✇❛②s ❡q✉❛• t♦ t❤❡ ♦♥❡ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❝•❡❛r•② ♣r♦♣♦s❡❞ ✐♥ t❤✐s t❤❡s✐s✳ ❚❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ r♦t♦rs✬ r♦t❛t✐♦♥✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ❝•❡❛r•② ❣✐✈❡♥✳ ■♥ ❇♦✉❛❜❞❛••❛❤✬s t❤❡s✐s✱ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦♥❡ ✉s❡❞ ✐♥ ❤✐s ♣r♦❣r❛♠✱ ❛♥❞ t❤✐s ❝❛✉s❡❞ s♦♠❡ ❞✐r❡❝t✐♦♥ ❡rr♦rs ✐♥ ❤✐s ♠♦❞❡•✐♥❣✳ ■♥ t❤❡ ✇♦r❦❜❡♥❝❤ ♣r♦♣♦s❡❞ ✐♥ t❤✐s t❤❡s✐s✱ ❛•• t❤❡ ♣r❡✈✐♦✉s•② ♠❡♥t✐♦♥❡❞ ❡rr♦rs ❤❛✈❡ ❜❡❡♥ ❝♦rr❡❝t❡❞✳ ❚❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛r❡ ♠♦❞❡•❡❞ ♠♦r❡ ♣r❡❝✐s❡•②✱ ❛♥❞ s♦♠❡ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♥❡❣•❡❝t❡❞ ❜❛s❡ ♦♥ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❡✛❡❝t✳ ✷✳✺✳✷ ❙✐♠♣•✐✜❡❞ ❙✐♠✉•❛t✐♦♥ ▼♦❞❡• ❆s ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✶✳✺✱ q✉❛❞r♦t♦rs ❤❛✈❡ ♠❛♥② ❛♣♣•✐❝❛t✐♦♥s✱ s✉❝❤ ❛s s✉r✈❡✐••❛♥❝❡✱ ❜♦r❞❡r ♣❛tr♦•✱ ❛❡r✐❛• ♣❤♦t♦❣r❛♣❤ ❛♥❞ ✈✐❞❡♦✱ ❡t❝✳ ■♥ t❤❡s❡ ❛♣♣•✐❝❛t✐♦♥s✱ q✉❛❞r♦t♦rs ✉s✉❛••② ♥❡❡❞ t♦ ❢♦••♦✇ ❛ tr❛•❡❝t♦r② ♦r ❤♦✈❡r ❛r♦✉♥❞ ❛ ♣♦✐♥t✳ ❚❤❡ ♠♦✈❡♠❡♥t ♦❢ q✉❛❞r♦t♦rs ✐s r❛t❤❡r s•♦✇ ❛♥❞ ✇✐t❤♦✉t ✈✐♦•❡♥t ❞✐s♣•❛❝❡♠❡♥t✳ ❚❤❡ P❤❛♥t♦♠ q✉❛❞r♦t♦r ✭s❡❡ ❋✐❣✉r❡ ✶✳✶✶ ✐♥ ❙❡❝t✐♦♥ ✶✳✹✮ ❤❛s ❛ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✹✺ ❞❡❣r❡❡s ❛♥❞ ❛ ♠❛①✐♠✉♠ ✢✐❣❤t ✈❡•♦❝✐t② ✶✵♠✴s✳ ❍♦✇❡✈❡r ✐♥ ❛♣♣•✐❝❛t✐♦♥s✱ t❤❡ t✐•t ❛♥❣•❡ ✐s ✉s✉❛••② ✐♥❢❡r✐♦r t♦ ✶✹ ❞❡❣r❡❡s ✭✵✳✷✺r❛❞✮✱ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② ✐s ✐♥❢❡r✐♦r t♦ ✵✳✷r❛❞✴s✱ ❛♥❞ t❤❡ ✢✐❣❤t ✈❡•♦❝✐t② ✐s ✐♥❢❡r✐♦r t♦ ✷♠✴s✳ φ, θ 0.25 r❛❞, φ, θ, ψ 0.2 r❛❞✴s p, q, r 0.2 r❛❞✴s, v b 2 ♠✴s ✭✷✳✸✺✮ ❚❤❡ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s φ✱ θ✱ ψ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛♥❞ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s p✱ q✱ r ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ✐s✿ ✸✽ ✷✳✺✳ ◗✉❛❞r♦t♦r ▼♦❞❡• ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣     p q r     =     1 0 -s✐♥θ 0 ❝♦sθ s✐♥φ ❝♦sθ 0 -s✐♥φ ❝♦sφ ❝♦sθ         φ θ ψ    ✭✷✳✸✻✮ ✇❤❡r❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛tr✐① ✐s •✐♠✐t❡❞✿     1 0 -s✐♥θ 0 ❝♦sθ s✐♥φ ❝♦sθ 0 -s✐♥φ ❝♦sφ ❝♦sθ         1 0 -0.247 0 0.9689 0.239 0 -0.247 0.939     ✭✷✳✸✼✮ ■♥ ❛♣♣•✐❝❛t✐♦♥s✱ ✐♥ ♦r❞❡r t♦ s❛✈❡ ❡♥❡r❣② ❛♥❞ s✐♠♣•✐❢② ❝♦♥tr♦• •❛✇s✱ ♦♥❡ ♦❢ t❤❡ ❛r♠s ♦❢ t❤❡ q✉❛❞r♦t♦r ✐s s❡t ❛•♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ ✈❡•♦❝✐t② t♦ r❡❞✉❝❡ t❤❡ r♦t❛t✐♦♥❛• ♠♦✈❡♠❡♥t✳ ❚❤✉s✱ ❡✈❡r② t✐♠❡ q✉❛❞r♦t♦rs t✐•t ♦♥•② ❛r♦✉♥❞ ♦♥❡ ❛①✐s x b ♦r y b ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s φ✱ θ ❛♥❞ ψ ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡ ❛r❡ ❡q✉❛• t♦ t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s p✱ q ❛♥❞ r ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ r❡s♣❡❝t✐✈❡•②✿ φ = p, θ = q, ψ = r ✭✷✳✸✽✮ ■♥ t❤❡ ❝♦♥tr♦• s②st❡♠✱ ❛ q✉❛❞r♦t♦r ✐s ❝♦♥tr♦••❡❞ ❜② t❤❡ r♦t❛t✐♦♥❛• s♣❡❡❞s ♦❢ ❢♦✉r r♦t♦rs Ω i ✳ ❆s t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ sq✉❛r❡❞ r♦t❛t✐♦♥❛• s♣❡❡❞s ♦❢ ❛ r♦t♦r✱ t❤❡ t❤r✉st ❛♥❞ ②❛✇✐♥❣ ♠♦♠❡♥t ❛r❡ ❝❤♦s❡♥ ❛s t❤❡ ❝♦♥tr♦• ✐♥♣✉ts ✐♥ t❤❡ ❝♦♥tr♦• s②st❡♠✿ u 1 = 4 i=1 T i , u 2 = l(T 2 -T 4 ), u 3 = l(-T 1 + T 3 ), u 4 = (-1) i+1 4 i=1 Q i . ✭✷✳✸✾✮ ❚❤❡♥✱ t❤❡ r♦t❛t✐♦♥❛• s♣❡❡❞s Ω i ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✉s✐♥❣ u i t❤r♦✉❣❤✿ u 1 = b 4 i=1 Ω 2 i , u 2 = bl(Ω 2 2 -Ω 2 4 ), ✭✷✳✹✵✮ u 3 = bl(-Ω 2 1 + Ω 2 3 ), u 4 = (-1) i+1 d 4 i=1 Ω 2 i . ✭✷✳✹✶✮ ✇❤❡r❡ b ❛♥❞ d ❛r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ t❤r✉st ❛♥❞ t❤❡ ②❛✇✐♥❣ ♠♦♠❡♥t✱ l ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t② ❛♥❞ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ r♦t♦r✳ ✸✾ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✻✳ ❈♦♥str❛✐♥ts ❚❤✉s✱ ❛ s✐♠♣•✐✜❡❞ s②st❡♠ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✳✸✸✮✿ I xx φ = θ ψ(I yy -I zz ) + J r θΩ r + u 2 + (-1) i+1 4 i=1 R mxi , ✭✷✳✹✷❛✮ I yy θ = φ ψ(I zz -I xx ) -J r φΩ r + u 3 + (-1) i+1 4 i=1 R myi , ✭✷✳✹✷❜✮ I zz ψ = θ φ(I xx -I yy ) + u 4 , ✭✷✳✹✷❝✮ mz w = s✐♥θ 4 i=1 H xi -(s✐♥φ ❝♦sθ) 4 i=1 H yi + (❝♦sφ ❝♦sθ)u 1 -mg + ρgV vol ✭✷✳✹✷❞✮ mẍ w = -(❝♦sθ ❝♦sψ) 4 i=1 H xi -(s✐♥φ s✐♥θ ❝♦sψ -❝♦sφ s✐♥ψ) 4 i=1 H yi + (❝♦sφ s✐♥θ ❝♦sψ + s✐♥φ s✐♥ψ)u 1 ✭✷✳✹✷❡✮ mÿ w = -(❝♦sθ s✐♥ψ) 4 i=1 H xi -(s✐♥φ s✐♥θ s✐♥ψ + ❝♦sφ ❝♦sψ) 4 i=1 H yi + (❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ)u 1 ✭✷✳✹✷❢✮ T = C T ρA(ΩR rad ) 2 , H = C H ρA(ΩR rad ) 2 ✭✷✳✹✷❣✮ Q = C Q ρA(ΩR rad ) 2 R rad , R = C R ρA(ΩR rad ) 2 R rad ✭✷✳✹✷❤✮ ✷✳✻ ❈♦♥str❛✐♥ts ◗✉❛❞r♦t♦rs ❤❛✈❡ ✈❛r✐♦✉s ❝♦♥str❛✐♥ts ✐♥ ❛♣♣•✐❝❛t✐♦♥s✳ ❚❤❡ ♠❡❝❤❛♥✐❝❛• str✉❝t✉r❡ ❤❛s ✐ts •✐♠✐ts✳ ◗✉❛❞r♦t♦rs ✉s✉❛••② ✉s❡ ❜r✉s❤•❡ss ❡•❡❝tr✐❝ ❉❈ r♦t♦rs ✇❤✐❝❤ ❤❛✈❡ •✐♠✐ts ✐♥ t❤❡✐r r♦t❛t✐♦♥❛• s♣❡❡❞s✳ ❚❤❡ ❍✉♠♠✐♥❣❜✐r❞ q✉❛❞r♦t♦r ♠❛❞❡ ❜② ❆s❝❡♥❞✐♥❣ ❚❡❝❤♥♦•♦❣✐❡s ✭s❡❡ ❋✐❣✉r❡ ✶✳✾ ✐♥ ❙❡❝t✐♦♥ ✶✳✹✮ ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❢♦✉r ❜r✉s❤•❡ss ❉❈ ♠♦t♦r✱ t②♣❡ ❍❆❈❑❊| ①✲❇▲ ✺✷s✳ ❚❤❡ r❛t❡❞ ♠♦t♦r s♣❡❡❞ ✇✐t❤♦✉t •♦❛❞ ✐s ✽✸✸✵|P▼ ✭|❡✈♦•✉t✐♦♥s P❡r ▼✐♥✉t❡✮✱ t❤❛t ✐s ✽✼✷ r❛❞✴s✳ ❚❤❡ r♦t❛t✐♦♥❛• s♣❡❡❞ ❞❡❝r❡❛s❡s t♦ t✇♦ t❤✐r❞ ✇✐t❤ •♦❛❞✳ ❍❡r❡✱ ✇❡ ❞❡✜♥❡ ❛ ♠❛①✐♠✉♠ r♦t❛t✐♦♥❛• s♣❡❡❞ ❞✉r✐♥❣ ✢✐❣❤ts ❛s✿ 10 r❛❞✴s Ω 300 r❛❞✴s ✭✷✳✹✸✮ ❋♦r t❤❡ s❛❦❡ ♦❢ s❛❢❡t②✱ ✇❡ s❤♦✉•❞ ❛•s♦ ✐♠♣♦s❡ s♦♠❡ ❝♦♥str❛✐♥ts ♦♥ t❤❡ s♣❡❡❞✳ ❚❤❡ P❤❛♥t♦♠ q✉❛❞r♦t♦r ✭s❡❡ ❋✐❣✉r❡ ✶✳✶✶ ✐♥ ❙❡❝t✐♦♥ ✶✳✹✮ ❤❛s ❛ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✹✺ ❞❡❣r❡❡s ✭✵✳✼✽ r❛❞✮✱ ❛ ♠❛①✐♠✉♠ ❛s❝❡♥t ♦r ❞❡s❝❡♥t s♣❡❡❞ ♦❢ ✻♠✴s ❛♥❞ ❛ ♠❛①✐♠✉♠ ✢✐❣❤t ✈❡•♦❝✐t② ♦❢ ✶✵♠✴s✳ ❲❡ ✹✵ ✷✳✼✳ ❈♦♥tr♦• ❖r✐❡♥t❡❞ ▼♦❞❡•s ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ s❡t ❛ ♠❛①✐♠✉♠ ❛s❝❡♥t ♦r ❞❡s❝❡♥❞✐♥❣ ❛❝❝❡•❡r❛t✐♦♥ ♦❢ ✺❣ ✭❣❂✾✳✽♠✴s 2 ✮ ❛♥❞ ❛ ♠❛①✐♠✉♠ t✐•t ❛❝❝❡•❡r❛t✐♦♥ ♦❢ ✹✽r❛❞✴s 2 ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ •✐♠✐ts ♦❢ t❤❡ ❝♦♥tr♦• ✐♥♣✉ts ❛r❡✿ m•g u 1 6 m•g = 31.2 ◆, |u 2 |, |u 3 | 48I xx = 0.3 ◆•♠ |u 4 | 48I zz = 0.5 ◆•♠ ✭✷✳✹✹✮ ✇❤❡r❡ m ✐s t❤❡ t♦t❛• ♠❛ss ✇❤✐❝❤ ✐s s❡t ❛s ✵✳✺✸❦❣ ✐♥ t❤❡ s✐♠✉•❛t✐♦♥✱ ❛♥❞ I xx ✱ I yy ✱ I zz ❛r❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦♠❡♥t ♦❢ ✐♥❡rt✐❛✱ I xx = I yy = 6.22×10 -3 ❦❣ • ♠ 2 ✱ I zz = 1.12×10 -2 ❦❣ • ♠ 2 ✳ ✷✳✼ ❈♦♥tr♦• ❖r✐❡♥t❡❞ ▼♦❞❡•s ■♥ ❙❡❝t✐♦♥ ✶✳✹✱ ♠❛♥② ❡①✐st✐♥❣ q✉❛❞r♦t♦rs ❛r❡ ♣r❡s❡♥t❡❞✳ ❚❤❡s❡ q✉❛❞r♦t♦rs ❛r❡ ❜✉✐•t ✉s✐♥❣ ♠✐❝r♦❝♦♥tr♦••❡rs✱ s✉❝❤ ❛s ❆|▼ ✾✳ ❊♠❜❡❞❞❡❞ s②st❡♠s ❤❛✈❡ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❞ ❡♥❡r❣② •✐♠✐ts✳ ❙♦♠❡ ♣❛rts ❛r❡ ♥❡❣•❡❝t❡❞ t♦ s✐♠♣•✐❢② t❤❡ ❝♦♥tr♦• •❛✇s✳ ❆ ❝♦♥tr♦• ♦r✐❡♥t❡❞ ♠♦❞❡• ❢♦r t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s✿ I xx φ = u 2 + θ ψ(I yy -I zz ) + J r θΩ

✭✷✳✹✺✮

❆ ❝♦♥tr♦• ♦r✐❡♥t❡❞ ♠♦❞❡• ❢♦r t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s✿ I xx φ = u 2 + F φ , I yy θ = u 3 + F θ , I zz ψ = u 4 + F ψ , mẍ = (sin ψ sin φ + cos ψ sin θ cos φ)u 1 + F x , mÿ = (-cos ψ sin φ + sin ψ sin θ cos φ)u 1 + F y , mz = (cos θ cos φ)u 1 + F z . ✭✷✳✹✻✮ ❆ ♣❛rt✐❛• ❦♥♦✇•❡❞❣❡ ♠♦❞❡• ✐s ✉s❡❞ ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ✐♥♣✉t u 1 ❛r❡ ❦❡♣t ✐♥ t❤❡ ❝♦♥tr♦• ♦r✐❡♥t❡❞ ♠♦❞❡• t♦ ❦❡❡♣ t❤❡ ❝♦♥tr♦• ♣r❡❝✐s✐♦♥✳ ❖t❤❡r ♣❛rts ✐♥ t❤❡ ✹✶ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✼✳ ❈♦♥tr♦• ❖r✐❡♥t❡❞ ▼♦❞❡•s s②st❡♠ ❛r❡ ✐♥❝•✉❞❡❞ ✐♥t♦ t❤❡ ✉♥❦♥♦✇♥ ♣❛rt F ✳ ❇❛s❡❞ ♦♥ t❤❡ t②♣❡ ♦❢ ❛ s②st❡♠✱ t❤❡ ❝♦♥tr♦• ♦r✐❡♥t❡❞ ♠♦❞❡• ❢♦r t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❝❛♥ ❜❡ ❢✉••② ♠♦❞❡•✲❢r❡❡ ♦r ✇✐t❤ ♣❛rt✐❛• ❦♥♦✇•❡❞❣❡✳ ■❢ ✇❡ s❡t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ✐♥♣✉t u 1 ❛s ❝♦♥st❛♥ts✱ t❤❡♥ ❛ ❢✉••② ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ✉s❡❞✳ ❚❤❡ P■❉ ❝♦♥tr♦• ❞♦❡s ♥♦t ♥❡❡❞ ♣❛rt✐❝✉•❛r•② t❤❡ ❦♥♦✇•❡❞❣❡ ♦❢ s②st❡♠✱ ❛♥❞ ✐t ❞♦❡s ♥♦t ♥❡❡❞ ❛ ❝♦♥tr♦• ♦r✐❡♥t❡❞ ♠♦❞❡•✳ ✹✷ ❈❤❛♣t❡r ✸ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✶ ■♥tr♦❞✉❝t✐♦♥ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐•• ✉s❡ t❤❡ ♥♦r♠❛• ❢♦r♠ t❤❡♦r② ❢♦r t❤❡ ❞②♥❛♠✐❝ ❛♥❛•②s✐s✳ ❚❤❡ ♥♦r♠❛• ❢♦r♠ t❤❡♦r② ✐s ❛ ✉s❡❢✉• ❛♣♣r♦❛❝❤ ✐♥ st✉❞②✐♥❣ t❤❡ ❞②♥❛♠✐❝ s②st❡♠ ♣r♦♣❡rt✐❡s ❬✺✽(✳ ❇② ❡♠♣•♦②✐♥❣ s✉❝❝❡ss✐✈❡ ❝♦♦r❞✐♥❛t❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❛ s②st❡♠ ✐s ❝❤❛♥❣❡❞ ✐♥t♦ ✐ts s✐♠♣•❡st ❢♦r♠✱ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ❚❤✐s ♠❛❦❡s t❤❡ ❛♥❛•②s✐s ♦❢ t❤❡ ❞②♥❛♠✐❝ s②st❡♠ ❡❛s✐❡r✳ ■♥ •✐♥❡❛r s②st❡♠s✱ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ t❤❡ ♥♦r♠❛• ❢♦r♠s ✇❡r❡ ❞❡❞✉❝❡❞ ❛♥❞ t❤❡ ❇r✉♥♦✈s❦② ❢♦r♠ ✐s ✉s❡❞ ❢♦r t❤❡ ❞❡s✐❣♥ ♦❢ t❤❡ ❝♦♥tr♦• •❛✇s✳ ▲✐♥❡❛r s②st❡♠s ❤❛✈❡ ❢♦✉r ♥♦r♠❛• ❢♦r♠s✿ ❝♦♥✲ tr♦••❛❜•❡✱ ♦❜s❡r✈❛❜•❡✱ ❝♦♥tr♦••❡r ❛♥❞ ♦❜s❡r✈❡r ❢♦r♠✳ ❚❤❡ ♥♦r♠❛• ❢♦r♠s ❢♦r ♥♦♥•✐♥❡❛r s②st❡♠s ❤❛✈❡ ❜❡❡♥ ❛❞❞r❡ss❡❞ ✜rst•② ❜② P♦✐♥❝❛ré ❛♥❞ t❤❡♥ ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❞✉r✐♥❣ t❤❡ •❛st ❝❡♥t✉r②✳ ❍✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ ✉s❡❞ ✐♥ t❤❡ ❛r❡❛ ♦❢ ♥♦♥•✐♥❡❛r ✈❡❝t♦r ✜❡•❞s✱ ❍❛♠✐•t♦♥ ❞②♥❛♠✐❝ s②st❡♠s✱ ♥♦♥•✐♥❡❛r ♠❛♣♣✐♥❣s ❛♥❞ ❜✐❢✉r❝❛t✐♦♥ ♣❤❡♥♦♠❡♥♦♥✱ ❡t❝✳ ■♥ t❤❡ •❛st ❞❡❝❛❞❡s✱ t❤❡ tr❛♥s❢♦r♠❛✲ t✐♦♥ ♦❢ ❛ ♥♦♥•✐♥❡❛r s②st❡♠ ✐♥t♦ ❛ ❇r✉♥♦✈s❦② ✉s✐♥❣ s✉❝❝❡ss✐✈❡ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛♥❞ st❛t❡ ❢❡❡❞❜❛❝❦ ✇❛s st✉❞✐❡❞ ❜② ♠❛♥② ❛✉t❤♦rs✳ ❚❤❡ ♥♦r♠❛• ❢♦r♠s ❢♦r •✐♥❡❛r ❛♥❞ ♥♦♥•✐♥❡❛r s②st❡♠s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ❜② ❑r❡♥❡r ✐♥ ✶✾✽✼ ❬✺✾(✳ ❍♦✇❡✈❡r✱ ♠♦st ♦❢ t❤❡ ♥♦♥•✐♥❡❛r s②st❡♠s ❞♦ ♥♦t ❛❞♠✐t ❛ ❇r✉♥♦✈s❦② ❢♦r♠ ♦r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ s②st❡♠ ✐♥t♦ t❤❡ ❇r✉♥♦✈s❦② ❢♦r♠ ✐s ♥✉♠❡r✐❝❛••② q✉✐t❡ ❞✐✣❝✉•t ✇❤✐❝❤ ♥❡❡❞s t♦ s♦•✈❡ t❤❡ ✜rst ♦r❞❡r •✐♥❡❛r ♣❛rt✐❛• ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥s✳ ❚❤❡ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥s ♦❢ t❤❡ ♥♦♥•✐♥❡❛r ❝♦♥tr♦••❡r ❛♥❞ ♦❜s❡r✈❡r ♥♦r♠❛• ❢♦r♠s ✇❡r❡ ♣r♦♣♦s❡❞ ❜② ❑r❡♥❡r ❡t ❛•✳ ✐♥ ✶✾✽✼ ❬✻✵(✳ ❆ s❡t ♦❢ ❡①t❡♥❞❡❞ q✉❛❞r❛t✐❝ ❝♦♥tr♦••❡r ♥♦r♠❛• ❢♦r♠s ♦❢ t❤❡ •✐♥❡❛r•② ❝♦♥tr♦••❛❜•❡ s②st❡♠s ✇✐t❤ s✐♥❣•❡ ✐♥♣✉t ✇❛s ❣✐✈❡♥ ❜② ❑❛♥❣ ❛♥❞ ❑r❡♥❡r ✐♥ ✶✾✾✷ ❬✻✷(✳ ❚❤❡② ❤❛✈❡ ❛•s♦ ♣r♦✈❡❞ t❤❡r❡ ❡①✐sts ❛ ❞②♥❛♠✐❝ st❛t❡ ❢❡❡❞❜❛❝❦ s♦ t❤❛t t❤❡ ❡①t❡♥❞❡❞ s②st❡♠ ❤❛s ❛ •✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✇❤✐❝❤ ✐s ❛❝❝✉r❛t❡ ❛t •❡❛st s❡❝♦♥❞ ❞❡❣r❡❡✳ ■♥ ✷✵✵✻✱ ❑❛♥❣ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✷✳ ◗✉❛❞r♦t♦r ▼♦❞❡• ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ♠❡t❤♦❞ t♦ ❞❡❞✉❝❡ t❤❡ ♥♦r♠❛• ❢♦r♠s ♦❢ ❛♥② ❞❡❣r❡❡ ❢♦r t❤❡ s②st❡♠s ✇✐t❤ ❛ s✐♥❣•❡ ✐♥♣✉t ❜② ✉s✐♥❣ s✉❝❝❡ss✐✈❡ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛♥❞ ❢❡❡❞❜❛❝❦ ❬✻✸(✳ ❋♦r ♠✉•t✐✲ ✐♥♣✉t s②st❡♠s✱ ❚❛•• ❡t ❛•✳ ❤❛✈❡ ❞❡❞✉❝❡❞ t❤❡ ♥♦r♠❛• ❢♦r♠s ❢♦r t❤❡ s②st❡♠s ✇✐t❤ t✇♦ ✐♥♣✉ts ✐♥ ✷✵✵✷ ❬✾✺(✳ ❇❛s❡❞ ♦♥ t❤❡ ♥♦r♠❛• ❢♦r♠s✱ t❤❡ ❜✐❢✉r❝❛t✐♦♥s ♦❢ ❛ s②st❡♠ ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❜② s❡✈❡r❛• ❛✉t❤♦rs✳ ❆ •♦❝❛• ❜✐❢✉r❝❛t✐♦♥ ♦❢ ❛ ♣❛r❛♠❡t❡r✐③❡❞ s②st❡♠ ♦❝❝✉rs ❛t ❛♥ ❡q✉✐•✐❜r✐✉♠ ✇❤❡r❡ t❤❡r❡ ✐s ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ st❛❜✐•✐t②✳ ❋♦r t❤❡ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ ❛ s✐♥❣•❡ ♣❛r❛♠❡t❡r✱ t❤❡ t②♣✐❝❛• ❜✐❢✉r❝❛t✐♦♥s ❛r❡ ❢✉••② ✉♥❞❡rst♦♦❞✿ t❤❡ ❢♦•❞ ✭♦r s❛❞❞•❡ ♥♦❞❡✮✱ t❤❡ tr❛♥s❝r✐t✐❝❛• ❛♥❞ t❤❡ ❍♦♣❢ ❜✐❢✉r❝❛t✐♦♥ ❬✺✽(✳ ❑r❡♥❡r ❡t ❛•✳ ❤❛✈❡ ♣r❡s❡♥t❡❞ t❤❡ q✉❛❞r❛t✐❝ ❛♥❞ ❝✉❜✐❝ ♥♦r♠❛• ❢♦r♠s ♦❢ ❛ s❝❛•❛r ✐♥♣✉t ♥♦♥•✐♥❡❛r ❝♦♥tr♦• s②st❡♠ ❛r♦✉♥❞ ❛♥ ❡q✉✐•✐❜r✐✉♠ ♣♦✐♥t✳ ❙♦♠❡ ✐♠♣♦rt❛♥t ❝♦♥tr♦• ❜✐❢✉r❝❛t✐♦♥s✱ t❤❡ ❛♥❛•♦❣✉❡s ♦❢ t❤❡ ❝•❛ss✐❝❛• ❢♦•❞✱ tr❛♥s❝r✐t✐❝❛• ❛♥❞ ❍♦♣❢ ❜✐❢✉r❝❛t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❬✻✶(✳ ❈❡♥t❡r ♠❛♥✐❢♦•❞ ✐s ✉s✉❛••② ❛♣♣•✐❡❞ ✇✐t❤ t❤❡ ♥♦r♠❛• ❢♦r♠s t❤❡♦r②✳ ■t r❡❞✉❝❡s t❤❡ s②st❡♠ t♦ ❛ ❝❡♥t❡r ♠❛♥✐❢♦•❞ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ♣❛rts ♦❢ t❤❡ s②st❡♠ ✇❤♦s❡ ❡✐❣❡♥✈❛•✉❡s ❤❛✈❡ ③❡r♦ r❡❛• ♣❛rts ❛t ❛ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ❬✸✽(✳ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇•❡❞❣❡✱ t❤❡ ♥♦r♠❛• ❢♦r♠s ❛♥❞ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r✐❡s ❤❛✈❡ ♥❡✈❡r ❜❡❡♥ ✉s❡❞ ✐♥ t❤❡ ❛♥❛•②s✐s ♦❢ ❛ q✉❛❞r♦t♦r✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ t❤❡ ♥♦r♠❛• ❢♦r♠ ♦❢ t❤❡ q✉❛❞r♦t♦r s②st❡♠ ✐s ✜rst•② ❝❛•❝✉•❛t❡❞✳ ❇② ✉s✐♥❣ s✉❝❤ ❛ ♠❡t❤♦❞♦•♦❣②✱ t❤❡ ❤✐❣❤•② ❝♦✉♣•❡❞ ♣❛rts ✐♥ t❤❡ q✉❛❞r♦t♦r s②st❡♠ ❛r❡ ❡•✐♠✐♥❛t❡❞✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥tr♦• •❛✇s✱ t❤❡ ♥♦r♠❛• ❢♦r♠ ✐s r❡❞✉❝❡❞ ✐♥t♦ ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛• s②st❡♠ ❛t t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ❜② ✉s✐♥❣ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r②✳ ❆ ❢♦r♠❡r r❡s❡❛r❝❤ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✵✶(✳ ✸✳✷ ◗✉❛❞r♦t♦r ▼♦❞❡• ❚❤❡ ❦✐♥❡♠❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ♠♦❞❡•s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷✳ ❆•• t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ✈❛r✐❛❜•❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❈❤❛♣t❡r ✷✳ ❆ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ )(❳ ✐s ❝❤♦s❡♥ ❢♦r t❤❡ ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s✳ ❚❤✉s✱ t❤❡ r♦t❛t✐♦♥❛• ♠❛tr✐① ❢r♦♠ t❤❡ ❜♦❞② ❢r❛♠❡ t♦ t❤❡ ✇♦r•❞ ❢r❛♠❡ R w b ✐s ♣r❡s❡♥t❡❞✿ R w b =     ❝♦sθ ❝♦sψ s✐♥φ s✐♥θ ❝♦sψ -❝♦sφ s✐♥ψ ❝♦sφ s✐♥θ ❝♦sψ + s✐♥φ s✐♥ψ ❝♦sθ s✐♥ψ s✐♥φ s✐♥θ s✐♥ψ + ❝♦sφ ❝♦sψ ❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ -s✐♥θ s✐♥φ ❝♦sθ ❝♦sφ ❝♦sθ     ✭✸✳✶✮ ■♥ t❤❡ ❞②♥❛♠✐❝ ❛♥❛•②s✐s✱ ♦♥•② t❤❡ t❤r✉st T ❛♥❞ t❤❡ ②❛✇✐♥❣ ♠♦♠❡♥t ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛♥❞ ❛•• ♦t❤❡r ❛❡r♦❞②♥❛♠✐❝❛• ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♥❡❣•❡❝t❡❞✳ ❆s t❤❡ t❤r✉sts ❛r❡ ❛•♦♥❣ t❤❡ z b ✹✹ ✸✳✸✳ ◆♦r♠❛• ❋♦r♠ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ t❤❡ s②st❡♠ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s✿ m     ẍ ÿ z    = R w b     0 0 T     +     0 0 -g     =     (❝♦sφ s✐♥θ ❝♦sψ + s✐♥φ s✐♥ψ) T (❝♦sφ s✐♥θ s✐♥ψ -s✐♥φ ❝♦sψ) T ❝♦sφ ❝♦sθ T -g     ✭✸✳✷✮ ■♥ ❊q✉❛t✐♦♥ ✭✸✳✶✮✱ t❤❡ t❤✐r❞ r♦✇ ✐s t❤❡ s✐♠♣•❡st t❤❛♥ ♦t❤❡r r♦✇s ❛♥❞ ❝♦•✉♠♥s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ ❳() ❢♦r t❤❡ ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s t♦ s✐♠♣•✐❢② t❤❡ s②st❡♠✳ ❚❤❡ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ ✐s ✐♥✈❡rs❡✱ ❛♥❞ t❤❡ ♠❛tr✐① R w b ✇✐•• ❜❡ t❤❡ ✐♥✈❡rs❡ ♠❛tr✐①✳ ❚❤❡ t❤✐r❞ r♦✇ ✐s ♥♦✇ ❝❤❛♥❣❡❞ t♦ t❤❡ t❤✐r❞ ❝♦•✉♠♥✱ ❛♥❞ ✇✐•• ❜❡ ♠✉•t✐♣•✐❡❞ ✇✐t❤ t❤❡ t❤r✉st✳ ◆♦t❡ t❤❛t t❤❡ r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ ✇✐•• ♥♦t ❝❤❛♥❣❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ s②st❡♠✳ ❚❤✉s✱ t❤❡ s②st❡♠ ✐s ❢✉rt❤❡r s✐♠♣•✐✜❡❞✱ ❛♥❞ t❤❡ ♠♦❞❡• ♦❢ ❛ q✉❛❞r♦t♦r ✐s s❤♦✇♥ ✐♥ ❊q✉❛t✐♦♥ ✭✸✳✸✮✳ ẍ = -w 1 sin θ, φ = w 2 , ÿ = w 1 cos θ sin φ, θ = w 3 , ✭✸✳✸✮ z = w 1 cos θ cos φ -g, ψ = w 4 .

❚❤❡ ✈❛r✐❛❜•❡s w

i (i = 1..4) ❛r❡ t❤❡ •✐♥❡❛r ❛♥❞ ❛♥❣✉•❛r ❛❝❝❡•❡r❛t✐♦♥s ✐♥❞✉❝❡❞ ❜② ❢♦✉r r♦t♦rs✱ ✇❤✐❝❤ ❛r❡ t❤❡ ✐♥♣✉ts ♦❢ t❤❡ s②st❡♠✳ ❚❤❡ r♦t❛t✐♦♥ ❛♥❣•❡s φ✱ θ ❛♥❞ ψ ❛r❡ ❛•♦♥❣ t❤❡ ✇♦r•❞ ❛①✐s x✱ y ❛♥❞ z r❡s♣❡❝t✐✈❡•②✱ ♥❛♠❡•② r♦••✱ ♣✐t❝❤ ❛♥❞ ②❛✇✳ g = 9.8♠✴s 2 ✐s t❤❡ ❣r❛✈✐t② ❝♦❡✣❝✐❡♥t✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ✈❛r✐❛❜•❡s ❛s x 1 = x✱ x 2 = ẋ✱ x 3 = y✱ x 4 = ẏ✱ x 5 = z✱ x 6 = ż✱ x 7 = φ✱ x 8 = φ✱ x 9 = θ✱ x 10 = θ✱ x 11 = ψ✱ x 12 = ψ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ✐s r❡✇r✐tt❡♥ ❛s✿ ẋ1 = x 2 , ẋ7 = x 8 , ẋ2 = -w 1 sin(x 9 ), ẋ8 = w 2 , ẋ3 = x 4 , ẋ9 = x 10 , ✭✸✳✹✮ ẋ4 = w 1 cos(x 9 ) sin(x 7 ), ẋ10 = w 3 , ẋ5 = x 6 , ẋ11 = x 12 , ẋ6 = w 1 cos(x 9 ) cos(x 7 ) -g, ẋ12 = w 4 . ✸✳✸ ◆♦r♠❛• ❋♦r♠ ■♥ t❤✐s s❡❝t✐♦♥✱ s♦♠❡ ♥♦♥•✐♥❡❛r ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛♥❞ ✐♥♣✉ts ❢♦r t❤❡ q✉❛❞r♦t♦r s②st❡♠ ✇✐•• ❜❡ ❞❡❞✉❝❡❞✳ ❚❤❡ ♥♦r♠❛• ❢♦r♠ ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠ ♦❢ ❛♥② ❞❡❣r❡❡ ✇✐•• ❜❡ ❢♦✉♥❞✳ ❆ ✹✺ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✸✳ ◆♦r♠❛• ❋♦r♠ ♥♦r♠❛• ❢♦r♠ ✐s ❛ s✐♠♣•❡st ❢♦r♠ ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✱ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ❚❤❡ ❡q✉✐•✐❜r✐❛ ♦❢ t❤❡ s②st❡♠ ✭✸✳✹✮ ❛r❡✿
x e = (c 1 , 0, c 2 , 0, c 3 , 0, kπ, 0, kπ, 0, c 4 , 0), w = (g, 0, 0, 0),

✇❤❡r❡ c i (i = 1,...,4) ∈ R ❛r❡ ❝♦♥st❛♥ts✱ k = 0, ±1, ±2, ... ✐s ✐♥t❡❣❡r ❛♥❞ g ✐s t❤❡ ❣r❛✈✐t②✳ ◆♦t❡ t❤❛t t❤❡ ❛♥❣•❡s ❛r❡ •✐♠✐t❡❞ ✐♥ t❤❡ r❡❛• ❝♦♥tr♦• s②st❡♠✱ t❤❡ t✐•t ❛♥❣•❡ ♦❢ ❛ q✉❛❞r♦t♦r ✐s •✐♠✐t❡❞ ❢r♦♠ ✲✾✵ ❞❡❣r❡❡s t♦ ✾✵ ❞❡❣r❡❡s✱ φ, θ ∈ (-π/2, π/2)✳ ❚❤❡ ②❛✇ ❛♥❣•❡ ✐s •✐♠✐t❡❞ t♦ ✸✻✵ ❞❡❣r❡❡s✱ ψ ∈ [0, π)✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✈❛r✐❛❜•❡ k ❤❡r❡ ✐s ❡q✉❛• t♦ ✵✳ ❲✐t❤♦✉t •♦s✐♥❣ ❣❡♥❡r❛•✐t②✱ t❤❡ ❝♦♥st❛♥ts c i (i = 1,...,4) ❝❛♥ ❜❡ s❡t ❛s ✵✳ ❚❤❡r❡❢♦r❡✱ ♦♥•② ♦♥❡ ❡q✉✐•✐❜r✐✉♠ ✐s ❝♦♥s✐❞❡r❡❞✿
x 0 = (x, w) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, g, 0, 0, 0)

❲❡ ♠♦✈❡ x 0 t♦ t❤❡ ♦r✐❣✐♥ ❜② ❝❤❛♥❣✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥♣✉ts✿ w 1 = u 1 + g, w 2 = u 2 w 3 = u 3 , w 4 = u 4 ✭✸✳✺✮ ❚❤❡♥✱ ✉s✐♥❣ t❤❡ ❚❛②•♦r s❡r✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s sin(x) ❛♥❞ cos(x) ❛t x = 0✱ t❤❡ s②st❡♠ ❝❛♥ ❜❡ ❢✉rt❤❡r ✇r✐tt❡♥ ✐♥ ♣♦•②♥♦♠✐❛• ❢♦r♠ ❛s ❢♦••♦✇s✿ ẋ1 = x 2 , ẋ2 = -gx 9 -u 1 x 9 + gx 3 9 6 + u 1 x 3 9 6 + O 5 , ẋ3 = x 4 , ẋ4 = gx 7 + u 1 x 7 - gx 3 7 6 - gx 2 9 x 7 2 - u 1 x 3 7 6 - u 1 x 2 9 x 7 2 + O 5 , ẋ5 = x 6 , ẋ6 = u 1 - gx 2 7 2 - gx 2 9 2 - u 1 x 2 7 2 - u 1 x 2 9 2 + gx 4 7 24 + gx 2 9 x 2 7 4 + gx 4 9 24 + O 5 , ẋ7 = x 8 , ẋ8 = u 2 , ẋ9 = x 10 , ẋ10 = u 3 , ẋ11 = x 12 , ẋ12 = u 4 . ✭✸✳✻✮ ❍❡r❡✱ O 5 ❛r❡ t❤❡ ♣♦•②♥♦♠✐❛•s ✇✐t❤ ✺t❤ ❛♥❞ ❤✐❣❤❡r ❞❡❣r❡❡✳ ✹✻ ✸✳✸✳ ◆♦r♠❛• ❋♦r♠ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ❯s✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ st❛t❡ ❛♥❞ ✐♥♣✉t tr❛♥s❢♦r♠❛t✐♦♥✿ y 1 = x 1 , y 2 = x 2 , y 3 = x 3 , y 4 = x 4 y 5 = x 5 , y 6 = x 6 , y 7 = gx 7 , y 8 = gx 8 y 9 = gx 9 , y 10 = gx 10 , y 11 = x 11 , y 12 = x 12 v 1 = u 1 , v 2 = gu 2 , v 3 = gu 3 , v 4 = u 4 ✭✸✳✼✮ ❚❤❡ s②st❡♠ ✭✸✳✻✮ ✐s t❤❡♥ ❝❤❛♥❣❡❞ ✐♥t♦ t❤❡ ❇r✉♥♦✈s❦② ❢♦r♠✿ ẏ1 = y 2 , ẏ2 = -y 9 - v 1 y 9 g + y 3 9 6g 2 + v 1 y 3 9 6g 3 + O 5 , ẏ3 = y 4 , ẏ4 = y 7 + v 1 y 7 g - y 7 y 2 9 2g 2 - y 3 7 6g 2 - v 1 y 3 7 6g 3 - v 1 y 7 y 2 9 2g 3 + O 5 , ẏ5 = y 6 , ẏ6 = v 1 - y 2 7 2g - y 2 9 2g - v 1 y 2 7 2g 2 - v 1 y 2 9 2g 2 + y 4 7 24g 3 + y 2 7 y 2 9 4g 3 + y 4 9 24g 3 + O 5 , ẏ7 = y 8 , ẏ8 = v 2 , ẏ9 = y 10 , ẏ10 = v 3 , ẏ11 = y 12 , ẏ12 = v 4 . ✭✸✳✽✮ ❚❤❡ s②st❡♠ ✭✸✳✽✮ ❝❛♥ ❛•s♦ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s✿ ẏ = f (y) + g(y)v = Ay + f (2) (y) + f (3) (y) + Bv + g (1) (y)v + g (2) (y)v + O 4 ✭✸✳✾✮ ✇❤❡r❡ A✱ B ❛r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ •✐♥❡❛r ♣❛rts✱ f (2) (y)✱ g (1) (y)v ❛r❡ t❤❡ s❡❝♦♥❞ ❞❡❣r❡❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦•②♥♦♠✐❛•s ♦❢ t❤❡ s②st❡♠✱ f (3) (y)✱ g (2) (y)v ❛r❡ t❤❡ t❤✐r❞ ❞❡❣r❡❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦•②♥♦♠✐❛•s✱ O 4 ❛r❡ t❤❡ ♣♦•②♥♦♠✐❛•s ✇✐t❤ ✹t❤ ❛♥❞ ❤✐❣❤❡r ❞❡❣r❡❡✳ ◆♦✇ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♦❜t❛✐♥ ❛ t❤✐r❞ ❞❡❣r❡❡ ♥♦r♠❛• ❢♦r♠ ♦❢ t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ ❋✐rst•②✱ ✇❡ t❛❦❡ ❛ t❤✐r❞✲❞❡❣r❡❡ ❤♦♠♦❣❡♥❡♦✉s tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r ❡①❛♠♣•❡ ❬✻✵(✿ y = z + φ (2) (z) + φ (3) (z) ✭✸✳✶✵✮ ✇❤✐❝❤ z ❛r❡ t❤❡ ♥❡✇ st❛t❡s ♦❢ t❤❡ s②st❡♠✳ φ (2) (z) ✐s ❛ s❡❝♦♥❞ ❞❡❣r❡❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦•②♥♦♠✐❛• ❛♥❞ φ (3) (z) ✐s ❛ t❤✐r❞ ❞❡❣r❡❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦•②♥♦♠✐❛• ♦❢ t❤❡ st❛t❡s z✱ ✇❤♦s❡ ❝♦❡✣❝✐❡♥ts ✇✐•• ❜❡ ❞❡✜♥❡❞ •❛t❡r✳ ✹✼ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✸✳ ◆♦r♠❛• ❋♦r♠ ❲❡ ❣❡t t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❊q✉❛t✐♦♥ ✭✸✳✶✵✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♥❡✇ st❛t❡s z ✐s✿ ż = (I + dφ (2) dz + dφ (3) dz ) -1 ẏ ✭✸✳✶✶✮ ✇❤❡r❡ (I + dφ (2) dz + dφ (3) dz ) -1 = I - dφ (2) dz - dφ (3) dz + ( dφ (2) dz ) 2 + ( dφ (3) dz ) 2 + 2 dφ (2) dz dφ (3) dz ... ■♥ ❊q✉❛t✐♦♥ ✭✸✳✾✮✱ ✇❡ r❡✇r✐t❡ t❤❡ f (y) ❛♥❞ g(y) ✉s✐♥❣ t❤❡ ♥❡✇ st❛t❡s z✳ f (y) = A(y) + f (2) (y) + f (3) (y) + O 4 = Az + Aφ (2) (z) + f (2) (z) + Aφ (3) (z) + f (3) (z)... ✭✸✳✶✷✮ g(y) = B + g (1) (y) + g (2) (y) + O 3 = B + g (1) (z) + g (1) (φ (2) (z)) + g (2) (z)... ❚❤❡r❡❢♦r❡✱ ✇✐t❤ t❤❡ ❤❡•♣ ♦❢ ❊q✉❛t✐♦♥s ✭✸✳✾✮✱ ✭✸✳✶✶✮✱ ✇❡ ❤❛✈❡ t❤❡ ♥❡✇ s②st❡♠✿ ż = Az + Bv + Aφ (2) (z) + f (2) (z) + g (1) (z)v - dφ (2) dz Az - dφ (2) dz Bv + Aφ (3) (z) +f (3) (z) + g (2) (z)v + g (1) (φ (2) (z))v - dφ (2) dz (Aφ (2) (z) + f (2) (z) + g (1) (z)v) - dφ (3) dz (Az + Bv) + ( dφ (2) dz ) 2 (Az + Bv) + O 4 ✭✸✳✶✸✮ ❚♦ s✐♠♣•✐❢② ♦❢ t❤❡ s②st❡♠✱ t❤❡ st❛t❡s z ❛♥❞ t❤❡ ✐♥♣✉ts v s❤♦✉•❞ ❜❡ s❡♣❛r❛t❡❞✱ ✇❤✐❝❤ ♠❡❛♥s t❤❡ ♣♦•②♥♦♠✐❛• g (1) (z)v✱ g (2) (z)v s❤♦✉•❞ ❜❡ ❝❛♥❝❡•❡❞ ✐♥ t❤❡ t❤✐r❞ ❞❡❣r❡❡ ♥♦r♠❛• ❢♦r♠✳ ❚❤✉s ✇❡ ❤❛✈❡✿ g (1) (z) - dφ (2) dz B = 0 g (2) (z) + g (1) (φ (2) (z)) - dφ (2) dz g (1) (z) - dφ (3) dz B + ( dφ (2) dz ) 2 B = 0 ❚❤❡r❡❢♦r❡✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐♥ ❊q✉❛t✐♦♥ ✭✸✳✶✵✮ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✿ φ (2) (z) = (0, - z 6 z 9 g , 0, z 6 z 7 g
, 0, 0, 0, 0, 0, 0, 0, 0),

φ (3) (z) = (0, 0, 0, 0, 0, - z 6 z 2 7 2g 2 - z 6 z 2 9
2g 2 , 0, 0, 0, 0, 0, 0).

✹✽ ✸✳✹✳ ▲✐♥❡❛r ❆♥❛•②s✐s ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ❯s✐♥❣ t❤❡ s❛♠❡ ♠❡t❤♦❞✱ ✇❡ ❝❛♥ ❝❛•❝✉•❛t❡ t❤❡ ♥♦r♠❛• ❢♦r♠ ♦❢ ❛♥② ❞❡❣r❡❡✿ ż1 = z 2 - z 6 z 9 g + 2z 6 z 3 9 3g 3 + z 6 z 2 7 z 9 2g 3 + O 5 , ż2 = -z 9 + z 6 z 10 g - z 3 9 3g 2 - z 2 7 z 9 2g 2 - z 6 z 2 7 z 10 2g 3 + z 6 z 7 z 8 z 9 g 3 + O 5 , ż3 = z 4 + z 6 z 7 g - z 6 z 7 z 2 9 g 3 - 2z 6 z 3 7 3g 3 + O 5 , ż4 = z 7 - z 6 z 8 g + z 3 7 3g 2 + z 6 z 8 z 2 9 g 3 + O 5 , ż5 = z 6 - z 6 z 2 7 2g 2 - z 6 z 2 9 2g 2 + O 5 , ż6 = v 1 - z 2 7 2g - z 2 9 2g + z 6 z 7 z 8 g 2 + z 6 z 9 z 10 g 2 + 5z 4 7 24g 3 - z 2 7 z 2 9 4g 3 - 5z 4 9 24g 3 + O 5 , ż7 = z 8 , ż8 = v 2 , ż9 = z 10 , ż10 = v 3 , ż11 = z 12 , ż12 = v 4 . ✭✸✳✶✹✮ ❍❡r❡✱ O 5 ❛r❡ t❤❡ ♣♦•②♥♦♠✐❛•s ✇✐t❤ ✺t❤ ❛♥❞ ❤✐❣❤❡r ❞❡❣r❡❡✳ ❆ ▼❛♣•❡ ♣❛❝❦❛❣❡ ❵◗✉❛•✐t❛t✐✈❡❖❉❊✬ ❤❛s ❜❡❡♥ ♠❛❞❡ ❢♦r ❝❛•❝✉•❛t✐♥❣ ❛ ♥♦r♠❛• ❢♦r♠ ♦❢ t❤❡ q✉❛❞r♦t♦r✳ ❚❤❡ ♣r♦❣r❛♠s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❆♣♣❡♥❞✐① ❇✳ ✸✳✹ ▲✐♥❡❛r ❆♥❛•②s✐s ❋r♦♠ t❤❡ ♥♦r♠❛• ❢♦r♠✱ ✇❡ ❝❛♥ ❡❛s✐•② ❣❡t t❤❡ •✐♥❡❛r✐③❡❞ q✉❛❞r♦t♦r s②st❡♠✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❛♥❛•②③❡ t❤❡ ❝♦♥tr♦••❛❜✐•✐t② ❛♥❞ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ •✐♥❡❛r✐③❡❞ s②st❡♠✳ ❆ •✐♥❡❛r st❛t✐❝ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦• ✐s ❛•s♦ ♣r♦♣♦s❡❞ t♦ s❤♦✇ ✐ts ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ s②st❡♠✬s st❛❜✐•✐t②✳ ✸✳✹✳✶ ❈♦♥tr♦••❛❜✐•✐t② ❚❤❡ •✐♥❡❛r✐③❡❞ q✉❛❞r♦t♦r s②st❡♠ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❢♦••♦✇✐♥❣ ❢♦r♠✿ ż = Az + Bv ✭✸✳✶✺✮ ✇❤❡r❡ ✹✾ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✹✳ ▲✐♥❡❛r ❆♥❛•②s✐s A =                       
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0                        , B =                        0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1                        ❚❤❡♥✱ t❤❡ r❛♥❦ ♦❢ t❤❡ ❝♦♥tr♦••❛❜✐•✐t② ♠❛tr✐① C ✐s✿ Rank(C) = Rank( B AB A 2 B • • • A n-1 B ) = 12 ✭✸✳✶✻✮ ❚❤❡r❡❢♦r❡✱ t❤❡ •✐♥❡❛r✐③❡❞ s②st❡♠ ✐s ❢✉••② ❝♦♥tr♦••❛❜•❡ ❛t ✐ts ❡q✉✐•✐❜r✐✉♠✱ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ♥♦r♠❛• ❢♦r♠✳ ■❢ t❤❡ •✐♥❡❛r✐③❡❞ ♥♦♥•✐♥❡❛r s②st❡♠ ✐s ❝♦♥tr♦••❛❜•❡ ❛t ✐ts ❡q✉✐•✐❜r✐✉♠✱ t❤❡♥ t❤❡ ♥♦♥•✐♥❡❛r s②st❡♠ ✐s ❛❝❝❡ss✐❜•❡ ❛t ✐ts ❡q✉✐•✐❜r✐✉♠✳ ✸✳✹✳✷ ❙t❛❜✐•✐t② ❚♦ ❝❤❡❝❦ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ •✐♥❡❛r✐③❡❞ s②st❡♠✱ ✇❡ ❝❛•❝✉•❛t❡ t❤❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ t❤❡ ♠❛tr✐① A ✐♥ ❊q✉❛t✐♦♥ ✭✸✳✶✺✮✳ ❚❤❡② t✉r♥ ♦✉t t♦ ❜❡ ❛•• ✵✳ ❚❤❡ •✐♥❡❛r✐③❡❞ s②st❡♠ ✐s ♠❛r❣✐♥❛••② st❛❜•❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ✉s❡ •✐♥❡❛r st❛t✐❝ ❢❡❡❞❜❛❝❦ t♦ ❝♦♥tr♦• t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠✳ ■❢ t❤❡ s②st❡♠ ✐s t✐♠❡ ✐♥✈❛r✐❛♥t✱ t❤❡ ✐♥❞✐r❡❝t ♠❡t❤♦❞ ♦❢ ▲②❛♣✉♥♦✈ s❛②s t❤❛t ✐❢ t❤❡ ❡✐❣❡♥✲ ✈❛•✉❡s ♦❢ ❏❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ t❤❡ s②st❡♠ ❛t t❤❡ ♦r✐❣✐♥ ❛r❡ ✐♥ t❤❡ ♦♣❡♥ •❡❢t ❤❛•❢ ❝♦♠♣•❡① ♣•❛♥❡✱ t❤❡♥ t❤❡ ♦r✐❣✐♥ ✐s ❛s②♠♣t♦t✐❝❛••② st❛❜•❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ st❛t❡ ❢❡❡❞❜❛❝❦ ❛s ❢♦•✲ •♦✇s t♦ ♠♦✈❡ ❛•• t❤❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ t❤❡ s②st❡♠ t♦ t❤❡ ♦♣❡♥ •❡❢t ❤❛•❢ ♣•❛♥❡✳ x r , y r , z r , ψ r ❛r❡ t❤❡ r❡❢❡r❡♥❝❡s✳ v 1 = -256(z 5 -z r ) -32z 6 , v 2 = -1700(z 3 -y r ) -1000z 4 -256z 7 -32z 8 v 4 = -256(z 11 -ψ r ) -32z 12 , ✭✸✳✶✼✮ v 3 = 1700(z 1 -x r ) + 1000z 2 -256z 9 -32z 10 ✺✵ ✸✳✹✳ ▲✐♥❡❛r ❆♥❛•②s✐s ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ❋✐❣✉r❡ ✸✳✶✿ ✭❛✮ |❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ❢♦r t❤❡ q✉❛❞r♦t♦r✳ ✭❜✮ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❋✐❣✉r❡ ✸✳✷✿ ❚❤❡ s✐♠✉•❛t✐♦♥ ✇✐t❤♦✉t ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✿ ✭❛✮ t❤❡ •✐♥❡❛r st❛t✐❝ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦•✳ ✭❜✮ ❛ st❛♥❞❛r❞ P■❉ ❝♦♥tr♦•✳ ■♥ ♦r❞❡r t♦ ✈❡r✐❢② t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ ❝♦♥tr♦• s②st❡♠✱ s♦♠❡ s✐♠✉•❛t✐♦♥s ❛r❡ ♠❛❞❡ ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥ ✇✐t❤♦✉t ❛♥❞ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ s✐♠✉•❛t✐♦♥ t❛s❦ ✐s t♦ •❡t ❛ q✉❛❞r♦t♦r ❢♦••♦✇ ❛ sq✉❛r❡ ♣❛t❤ ✇✐t❤ ❛ •❡♥❣t❤ ♦❢ ✷♠ ✇❤✐•❡ ❤♦✈❡r✐♥❣ ❛t ❛♥ ❛•t✐t✉❞❡ ♦❢ ✶✵♠✱ ✇❤✐❝❤ ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✸✳✶✳ ❚❤❡ ❞❡s✐r❡❞ r❡s♣♦♥s❡ t✐♠❡ ✐s ✶s✱ ✇❤✐❝❤ ♠❡❛♥s t❤❡ q✉❛❞r♦t♦r s❤♦✉•❞ ✢② ✷♠ ✇✐t❤✐♥ ✶ s❡❝♦♥❞✳ ❚❤❡ t♦t❛••② s❛♠♣•❡ t✐♠❡ ✐s 20s✳ ❋♦r ❝♦♠♣❛r✐s♦♥✱ t❤❡ s✐♠✉•❛t✐♦♥s ✉s✐♥❣ ❛ st❛♥❞❛r❞ P■❉ ❝♦♥tr♦• ❛r❡ ❛•s♦ ❣✐✈❡♥✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✇✐t❤♦✉t ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛r❡ ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✸✳✷✳ ❲❡ ❝❛♥ s❡❡ ✺✶ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✺✳ ◆♦♥•✐♥❡❛r ❆♥❛•②s✐s ❋✐❣✉r❡ ✸✳✸✿ ❚❤❡ s✐♠✉•❛t✐♦♥ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✿ ✭❛✮ t❤❡ •✐♥❡❛r st❛t✐❝ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦•✳ ✭❜✮ ❛ st❛♥❞❛r❞ P■❉ ❝♦♥tr♦•✳ t❤❛t t❤❡ •✐♥❡❛r st❛t✐❝ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦• ❤❛s ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ t❤❛♥ ❛ st❛♥❞❛r❞ P■❉ ❝♦♥tr♦•✳ ❉✉r✐♥❣ t❤❡ tr❛•❡❝t♦r②✱ t❤❡r❡ ♠❛② ❜❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ✇✐t❤ ✈❡•♦❝✐t② 1♠✴s ❛s ✐♥ ❋✐❣✉r❡ ✸✳✶✱ ✇❤✐❝❤ ♦❝❝✉rs ✐♥ ❛•• ①✱ ② ❛♥❞ ③ ❛①✐s✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✸✳✸✳ ❚❤❡ ❞❡s✐r❡❞ r❡s♣♦♥s❡ t✐♠❡ ✐s ✶s✳ ❚❤❡ •✐♥❡❛r st❛t✐❝ ❢❡❡❞❜❛❝❦ ❝❛♥ ❦❡❡♣ t❤❡ st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ ❛♥❞ ❤❛s ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ t❤❛♥ t❤❡ st❛♥❞❛r❞ P■❉ ❝♦♥tr♦•✳ ✸✳✺ ◆♦♥•✐♥❡❛r ❆♥❛•②s✐s ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ❜✐❢✉r❝❛t✐♦♥s ♦❢ t❤❡ q✉❛❞r♦t♦r s②st❡♠ ✐s ❛♥❛•②③❡❞✱ ❛♥❞ t❤❡ s②st❡♠ ❝❛♥ ❜❡ ❢✉rt❤❡r s✐♠♣•✐✜❡❞ ✉s✐♥❣ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r②✳ ✸✳✺✳✶ ❇✐❢✉r❝❛t✐♦♥s ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✐♥ t❤❡ •✐♥❡❛r ♣❛rt ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✸✳✶✹✮✱ z 1 ✐s r❡•❛t❡❞ ♦♥•② t♦ z 2 , z 9 , z 10 , v 3 ❀ z 3 ✐s r❡•❛t❡❞ t♦ z 4 , z 7 , z 8 , v 2 ❀ z 5 ✐s r❡•❛t❡❞ t♦ z 6 , v 1 ❀ z 11 ✐s r❡•❛t❡❞ t♦ z 12 , v 4 ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✺✷ ✸✳✺✳ ◆♦♥•✐♥❡❛r ❆♥❛•②s✐s ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ❋✐❣✉r❡ ✸✳✹✿ ❚❤❡ ❡✐❣❡♥✈❛•✉❡s ✇❤❡♥ K 12 ❝❤❛♥❣❡s ❢r♦♠ ✲✶✺✵ t♦ ✶✺✵✿ ✭❛✮ t❤❡ r❡❛• ♣❛rts✳ ✭❜✮ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts✳ ❝♦♥tr♦• •❛✇s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s✿ v 1 = K 11 z 5 + K 12 z 6 , v 2 = K 21 z 3 + K 22 z 4 + K 23 z 7 + K 24 z 8 , v 4 = K 41 z 11 + K 42 z 12 , ✭✸✳✶✽✮ v 3 = K 31 z 1 + K 32 z 2 + K 33 z 9 + K 34 z 10 . ■♥ t❤✐s ✇❛②✱ ✇❡ ❝❛♥ ♠♦✈❡ t❤❡ r❡•❛t❡❞ ❡✐❣❡♥✈❛•✉❡s ✐♥ ❡❛❝❤ ❣r♦✉♣ s❡♣❛r❛t❡•② ✇✐t❤♦✉t ❝❤❛♥❣✲ ✐♥❣ t❤❡ ❡✐❣❡♥✈❛•✉❡s ✐♥ ♦t❤❡r ❣r♦✉♣s✳ ❍❡r❡✱ ✇❡ ❞❡✜♥❡ v i (i = 1,...,4) ❛s✿ v 1 = -256z 5 + K 12 z 6 , v 2 = -100z 3 -308z 4 -256z 7 -32z 8 , v 4 = -1024z 11 + K 42 z 12 , ✭✸✳✶✾✮ v 3 = 100z 1 + 308z 2 -256z 9 -32z 10 .

❚❤❡ s②st❡♠ ❤❛s t❤r❡❡ ❡q✉✐•✐❜r✐❛✿

P e 1 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) P e 2 = (0, 0, 43.45, 0, -0.057, 0, -16.97, 0, 0, 0, 0, 0) P e 3 = (0, 0, -43.45, 0, -0.057, 0, 16.97, 0, 0, 0, 0, 0)

✭✸✳✷✵✮ ✺✸ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✺✳ ◆♦♥•✐♥❡❛r ❆♥❛•②s✐s ❍♦✇❡✈❡r✱ ♦♥•② t❤❡ ♦r✐❣✐♥ P e 1 ❝❛♥ ❜❡ st❛❜•❡ ✇❤❡♥ K 12 , K 42 ❝❤❛♥❣❡✳ ❆t t❤❡ ❡q✉✐•✐❜r✐✉♠ P e 1 ✱ ❢♦r s✐♠♣•✐❝✐t② K 12 = K 42 ✱ ✇❤❡♥ K 12 ❝❤❛♥❣❡s✱ t❤❡ r❡❛• ❛♥❞ ✐♠❛❣✲ ✐♥❛r② ♣❛rts ♦❢ t❤❡ ❡✐❣❡♥✈❛•✉❡s ❛r❡ ✐♥ ❋✐❣✉r❡ ✸✳✹✳ ❲❤❡♥ K 12 < 0✱ t❤❡ s②st❡♠ ❤❛s ❢♦✉r ❡✐❣❡♥✈❛•✉❡s ✇✐t❤ ♣♦s✐t✐✈❡ r❡❛• ♣❛rts✱ ❛♥❞ t❤❡ s②st❡♠ ❜❡❝♦♠❡s ✉♥st❛❜•❡✳ ❲❤❡♥ K 12 > 0✱ t❤❡ s②st❡♠ ❤❛s ❛•• ❡✐❣❡♥✈❛•✉❡s ✇✐t❤ ♥❡❣❛t✐✈❡ r❡❛• ♣❛rts✱ ❛♥❞ t❤❡ s②st❡♠ ✐s ❛s②♠♣t♦t✐❝❛••② st❛❜•❡✳ ❲❤❡♥ K 12 = 0✱ t❤❡ s②st❡♠ ❤❛s t✇♦ ♣❛✐rs ♦❢ ♣✉r❡ ✐♠❛❣✐♥❛r② ❡✐❣❡♥✈❛•✉❡s ±16i ❛♥❞ ±32i✱ ❛♥❞ ❛•• ♦t❤❡r ❡✐❣❡♥✈❛•✉❡s ❤❛✈❡ ♥❡❣❛t✐✈❡ r❡❛• ♣❛rts✱ ✇❤✐❝❤ ✐s ❛ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• ❝❡♥t❡r ♠❛♥✐❢♦•❞✳ ❚❤❡ st❛❜✐•✐t② ❝❛♥♥♦t ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ •✐♥❡❛r ♣❛rt ♦❢ t❤❡ s②st❡♠✳ ■t ❞❡♣❡♥❞s ♦♥ t❤❡ ♥♦♥•✐♥❡❛r✐t② ♦❢ t❤❡ s②st❡♠✳ ✸✳✺✳✷ ❈❡♥t❡r ▼❛♥✐❢♦•❞ ❚❤❡ ❛✐♠ ♦❢ t❤✐s ♣❛rt ✐s t♦ ❣❡t t❤❡ r❡❞✉❝❡❞ s②st❡♠ ✇❤✐❝❤ ❝❛♥ ❞❡t❡r♠✐♥❡ t❤❡ st❛❜✐•✐t② ❛♥❞ ♣♦ss✐❜•❡ •♦❝❛• ❜✐❢✉r❝❛t✐♦♥s ♦❢ t❤❡ s②st❡♠ ❛t ♦♥❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❆ s②st❡♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s✿ ẋ = A(b)x + F (x), x ∈ R n ✇❤❡r❡ ❜ ✐s ❛ ❢r❡❡ ♣❛r❛♠❡t❡r✱ b ∈ R✳ ❆t ✐ts ♦r✐❣✐♥ x = [0, ..., 0]✱ J(b) ✐s t❤❡ ❏♦r❞❛♥ ❢♦r♠ ♦❢ t❤❡ ♠❛tr✐① A(b) ❛♥❞ Q ✐s ❛ ♠❛tr✐① ✇❤✐❝❤ ❡♥❛❜•❡s Q(b)J(b)Q -1 (b) = A(b)✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ẋ = Q(b)J(b)Q -1 (b)x + F (x) Q -1 (b) ẋ = J(b)Q -1 (b)x + Q -1 (b)F (x) ✇❡ ❞❡✜♥❡ y = Q -1 (b)x✱ t❤❡♥ ẏ = J(b)y + Q -1 (b)F (Q(b)y) = J(b)y + F (y) ✭✸✳✷✶✮ ✇❡ ❝❛♥ s❡♣❛r❛t❡ t❤❡ ❏♦r❞❛♥ ♠❛tr✐① J ❛s ♠❛tr✐❝❡s B ❛♥❞ C ✇❤♦s❡ ❡✐❣❡♥✈❛•✉❡s ❤❛✈❡ ③❡r♦ r❡❛• ♣❛rts ❛♥❞ ♥❡❣❛t✐✈❡ r❡❛• ♣❛rts r❡s♣❡❝t✐✈❡•②✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤❡ s②st❡♠ ✭✸✳✷✶✮ ❛t t❤❡ ♦r✐❣✐♥ ✇✐t❤ x = [0, ..., 0]✳ ẏ0 = By 0 + f (y 0 , y -), ẏ-= Cy 0 + g(y 0 , y -). ✺✹ ✸✳✺✳ ◆♦♥•✐♥❡❛r ❆♥❛•②s✐s ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ❙✐♥❝❡ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ ✐s t❛♥❣❡♥t t♦ E c ✭t❤❡ y -= 0 s♣❛❝❡✮✱ ✇❡ ❞❡✜♥❡ y -= h(y 0 , b), h(0, 0) = Dh(0, 0) = 0, ḃ = 0. ✭✸✳✷✷✮ ❲❡ ❝❛♥ ❝❛•❝✉•❛t❡ t❤❡ ❢✉♥❝t✐♦♥ h(y 0 , b) ❜② ✉s✐♥❣ ẏ-= Dh(y 0 , b) ẏ0 = Dh(y 0 , b)[By 0 + f (y 0 , h(y 0 , b))] = Cy 0 + g(y 0 , h(y 0 , b)) ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❣❡t t❤❡ •♦❝❛• ❡✈♦•✉t✐♦♥ ❡q✉❛t✐♦♥s ♦❢ y 0 ✇❤✐❝❤ ❝❛♥ ❞❡t❡r♠✐♥❡ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ■♥ q✉❛❞r♦t♦r ❝❡♥t❡r ♠❛♥✐❢♦•❞ ❛♥❛•②s✐s✱ t❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡✜♥❡❞ ❛s✿ v 1 = -256z 5 -bz 6 -z 3 5 , v 2 = -100z 3 -308z 4 -256z 7 -32z 8 , v 4 = -10z 11 -24z 12 , v 3 = 100z 1 + 308z 2 -256z 9 -32z 10 . ❚❤❡ ❜✐❢✉r❝❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s •✐❦❡ ✐♥ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥✳ ❲❤❡♥ b < 0✱ t❤❡ s②st❡♠ ❤❛s t✇♦ ❡✐❣❡♥✈❛•✉❡s ✇✐t❤ ♣♦s✐t✐✈❡ r❡❛• ♣❛rts✳ ❲❤❡♥ b > 0✱ t❤❡ s②st❡♠ ❤❛s ❛•• ❡✐❣❡♥✈❛•✉❡s ✇✐t❤ ♥❡❣❛t✐✈❡ r❡❛• ♣❛rts✳ ❲❤❡♥ b = 0✱ t❤❡ s②st❡♠ ❤❛s t✇♦ ♣✉r❡ ✐♠❛❣✐♥❛r② ❡✐❣❡♥✈❛•✉❡s ±16i✱ ❛♥❞ ❛•• ♦t❤❡r ❡✐❣❡♥✈❛•✉❡s ❤❛✈❡ ♥❡❣❛t✐✈❡ r❡❛• ♣❛rts✳ ❚❤❡ st❛❜✐•✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ ♥♦♥•✐♥❡❛r ♣❛rts ♦❢ t❤❡ s②st❡♠✳ ❲❡ ❝❛♥ ✉s❡ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r② t♦ s✐♠♣•✐❢② t❤❡ s②st❡♠✱ ❛♥❞ ❢✉rt❤❡r s✐♠♣•✐❢② t❤❡ st✉❞② ♦❢ t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♦❢ t❤❡ s②st❡♠✳ ■♥ t❤✐s ❝♦♥tr♦• s②st❡♠✱ y 0 = [y 1 , y 2 ] T = [z 5 , z 6 ] T ❛♥❞ y -= [y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9 , y 10 , y 11 , y 12 ] T = [z 1 , z 2 , z 3 , z 4 , z 7 , z 8 , z 9 , z 10 , z 11 , z 12 ] T ✳ ❲❡ s❡❡❦ ❛ q✉❛❞r❛t✐❝ ❝❡♥t❡r ♠❛♥✐❢♦•❞ ✭a ❛r❡ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❞❡✜♥❡❞ •❛t❡r✮✿ y i = a i200 y 2 1 + a i020 y 2 2 + a i002 b 2 + a i110 y 1 y 2 + a i101 y 1 b + a i011 y 2 b, i = 3..12 ❯s✐♥❣ t❤❡ ♠❡t❤♦❞ ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ ✇❡ ❣❡t✿ h(y 0 , b) = [-0.62b 2 , -0.62b 2 , -0.76b 2 , -0.76b 2 , 0, 0, 0, 0, -0.42b 2 , -23.58b 2 ] ✭✸✳✷✸✮ ✺✺ ❈❤❛♣t❡r ✸✳ ❉②♥❛♠✐❝ ❆♥❛•②s✐s ✸✳✺✳ ◆♦♥•✐♥❡❛r ❆♥❛•②s✐s ❚❤❡r❡❢♦r❡✱ t❤❡ r❡❞✉❝❡❞ s②st❡♠ ♦♥ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ ❝❛♥ ❜❡ ✇r✐tt❡♥✿ ẏ1 = 16y 2 -0.41b 4 -0.011b 8 -(b + 0.057b 4 )y 1 + 0.00024y 3 2 ẏ2 = -16y 1 + 0.67b 4 y 1 ✭✸✳✷✹✮ ■♥ t❤❡ r❡❞✉❝❡❞ s②st❡♠✱ ✇❤❡♥ b ✐s ♣♦s✐t✐✈❡ ✭♥❡❣❛t✐✈❡✮✱ t❤❡ ♦r✐❣✐♥ ✐s ❛ st❛❜•❡ ✭✉♥st❛❜•❡✮ ❢♦❝✉s✳ ❲❤❡♥ b = 0✱ t❤❡ ♦r✐❣✐♥ ✐s ❛ ❝❡♥t❡r✳ ❚❤❡ ♣❤❛s❡ ♣♦rtr❛✐t ♦❢ ❊q✉❛t✐♦♥ ✭✸✳✷✹✮ ✇❤❡♥ b = -0.5✱ b = 0 ❛♥❞ b = 0.5 ❛r❡ ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✉r❡ ✸✳✺✳ ❋✐❣✉r❡ ✸✳✺✿ ❚❤❡ ♣❤❛s❡ ♣♦rtr❛✐t ♦❢ t❤❡ r❡❞✉❝❡❞ s②st❡♠✿ ✭❛✮ ❜❂✲✵✳✺✳ ✭❜✮ ❜❂✵✳ ✭❝✮ ❜❂✵✳✺✳ ✺✻ ❈❤❛♣t❡r ✹ ❙②st❡♠ ❈♦♥tr♦• ✹✳✶ ❙❝❡♥❛r✐♦s ❉✉❡ t♦ t❤❡ s✐♠♣•✐❝✐t② ✐♥ ❞❡s✐❣♥ ❛♥❞ ♠❛♥❡✉✈❡r✱ q✉❛❞r♦t♦rs ❤❛✈❡ ♠❛♥② ❛♣♣•✐❝❛t✐♦♥s✱ s✉❝❤ ❛s ❜♦r❞❡r ♣❛tr♦•✱ s✉r✈❡✐••❛♥❝❡✱ s❡❛r❝❤ ✫ r❡s❝✉❡✱ ❛❡r✐❛• ♣❤♦t♦❣r❛♣❤② ❛♥❞ ✈✐❞❡♦ ❡t❝✳ ❆s ❞✐s❝✉ss❡❞ ✐♥t❡♥s✐✈❡•② ❞✉r✐♥❣ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ♦❢ q✉❛❞r♦t♦rs ✐♥ ❈❤❛♣t❡r ✶✱ ✇❡ •✐st ❤❡r❡ t❤❡ ♣♦✐♥ts t❤❛t ❛ ❝♦♥tr♦• s②st❡♠ s❤♦✉•❞ ❤❛✈❡ ✐♥ ❛♣♣•✐❝❛t✐♦♥s✿ • ❈♦♠♣•❡①✐t② ♦❢ t❤❡ ❝♦♥tr♦• ❛•❣♦r✐t❤♠✿ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♣❡r❛t✐♦♥s ✐♥ t❤❡ ❝♦♥tr♦• •❛✇✳ • ▼❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✳ • ❱❛r✐❛♥❝❡ ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦r✳ • |♦❜✉st♥❡ss t♦ ♠♦❞❡• ✉♥❝❡rt❛✐♥t✐❡s✿ t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ tr❛❝❦✐♥❣ ❡rr♦r✳ • |♦❜✉st♥❡ss t♦ ❞✐st✉r❜❛♥❝❡s✿ t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ tr❛❝❦✐♥❣ ❡rr♦r✳ • ❆❞•✉st✐♥❣ t✐♠❡ t♦ ❞✐st✉r❜❛♥❝❡s✿ t❤❡ t✐♠❡ ♥❡❡❞❡❞ t♦ r❡t✉r♥ t♦ t❤❡ st❛❜•❡ st❛t❡✳ • ❊♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✿ t❤❡ ❡♥❡r❣② s♣❡♥t ♦♥ ♠♦t✐♦♥ ❝♦♥tr♦•✳ ■♥ t❤✐s t❤❡s✐s✱ ✇✐t❤♦✉t •♦s✐♥❣ ❣❡♥❡r❛•✐t②✱ ❛♥ ❛❡r✐❛• ♣❤♦t♦❣r❛♣❤② s❝❡♥❛r✐♦ ✐s s❡•❡❝t❡❞✳ ❋✐rst•②✱ ❛ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡ ✐s ♣r❡s❡♥t❡❞✳ ❍♦✇❡✈❡r✱ ✐♥ r❡❛• ❛♣♣•✐❝❛t✐♦♥s ❛s ❞✐s❝✉ss❡❞ ❛❜♦✈❡✱ ❝✐r❝✉♠st❛♥❝❡s ❝❛♥ ❜❡ r❛t❤❡r t✉r❜✉•❡♥t✱ ✇❤✐❝❤ r❡s✉•ts ❛ t♦t❛••② ❞✐✛❡r❡♥t ❡✛❡❝t ♦♥ q✉❛❞r♦t♦r s②st❡♠s ❝♦♠♣❛r✐♥❣ t♦ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❯s✉❛••② ✐♥ s❝✐❡♥t✐✜❝ r❡s❡❛r❝❤❡s✱ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ♦♥•② t❡st❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✶✳ ❙❝❡♥❛r✐♦s ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❝❛♥ ♥♦t ♣r♦✈❡ t❤❡ s❛♠❡ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❞✐st✉r❜❛♥❝❡✳ ❆ s✐♠♣•❡ ❡①❛♠♣•❡ ✐s ❣✐✈❡♥ ✐♥ ❈❤❛♣t❡r ✶✳ ❆ ❇❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❤❛s s♠❛•• ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥ ❛♥ ✐❞❡❛• ❝❛s❡✳ ❍♦✇❡✈❡r✱ ✐t •♦s❡s ✐ts st❛❜✐•✐t② ✈❡r② q✉✐❝❦•② ✇❤❡♥ t❤❡r❡ ✐s ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡r❡❢♦r❡✱ ✐♥ t❤✐s t❤❡s✐s✱ ✐♥ ♦r❞❡r t♦ ❣❡t ❝•♦s❡r t♦ t❤❡ r❡❛•✐st✐❝ ❛♣♣•✐❝❛t✐♦♥s ❛s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ s❡✈❡r❛• ♦t❤❡r s❝❡♥❛r✐♦s ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✱ s❡♥s♦r ♥♦✐s❡s ❛♥❞ ❛❝t✉❛t♦r ❢❛✉•ts ❛r❡ ❛•s♦ ♣r❡s❡♥t❡❞✳ ✹✳✶✳✶ ❇❛s✐❝ ❙❝❡♥❛r✐♦ ■♥ t❤✐s s❝❡♥❛r✐♦✱ ❛ s♠❛•• q✉❛❞r♦t♦r ✇✐t❤ ❛ ✹ ❝❡•• ✷✼✵✵♠❆❤ ▲✐t❤✐✉♠ P♦•②♠❡r ❜❛tt❡r② ❛♥❞ ❛♥ ♦♥❜♦❛r❞ ❝❛♠❡r❛ ✐s ✉s❡❞✳ ❚❤❡ t❛s❦ ✐s t❤❡ ❛❡r✐❛• ♣❤♦t♦ s❤♦♦t✐♥❣ ✐♥ ❛♥ ♦♣❡♥ s♣❛❝❡ ②❛r❞✳ ❚❤❡ q✉❛❞r♦t♦r ♥❡❡❞s t♦ ✢② t♦ ❢♦✉r ♣♦✐♥ts✱ ❛♥❞ ❛t ❡❛❝❤ ♣♦✐♥t t❤❡ q✉❛❞r♦t♦r ♥❡❡❞s t♦ t❛❦❡ s❡✈❡r❛• ♣❤♦t♦s✳ ❚❤❡s❡ ❢♦✉r ♣♦✐♥ts ❛r❡ t❤❡ ❢♦✉r ✈❡rt✐❝❡s ♦❢ ❛ sq✉❛r❡ ♣❛t❤ ✇✐t❤ t❤❡ •❡♥❣t❤ ♦❢ ✷♠✳ ❚❤❡ q✉❛❞r♦t♦r ❤♦✈❡rs ❛t t❤❡ ❛•t✐t✉❞❡ ♦❢ ✶✵♠✳ ❚❤❡ ✸❉ tr❛•❡❝t♦r② ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✹✳✶✳ ❋✐❣✉r❡ ✹✳✶✿ ✭❛✮ ✸❉ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ❢♦r t❤❡ q✉❛❞r♦t♦r✳ ✭❜✮ |❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ❛•♦♥❣ t❤❡ ① ❛①✐s ✭r❡❞✱ •✐♥❡✮ ❛♥❞ ② ❛①✐s ✭❜•✉❡✱ ❞❛s❤❡❞ •✐♥❡✮✳ ❲❤❡♥ ❛rr✐✈❡❞ ❛t ❡❛❝❤ ❝♦r♥❡r✱ t❤❡ q✉❛❞r♦t♦r ❤♦✈❡rs ❢♦r ❛❜♦✉t ✶✺s t♦ t❛❦❡ ♣✐❝t✉r❡s✱ t❤❡♥ ✢✐❡s t♦ t❤❡ ♥❡①t ❝♦r♥❡r✳ ❚❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ✐s ❡①♣r❡ss❡❞ ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✶✮✳ ❚❤❡ ♣❛r❛♠❡t❡r h d = 2♠ ✐s t❤❡ •❡♥❣t❤ ♦❢ t❤❡ tr❛•❡❝t♦r②✱ ❛♥❞ T f = 6s ✐s t❤❡ t✐♠❡ ♥❡❡❞❡❞ t♦ r❡❛❝❤ t❤❡ ❞❡s✐r❡❞ ❝♦r♥❡r✳ ❚❤❡ t♦t❛• s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s 150s✳ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ q✉❛❞r♦t♦r ✐s ✐♥ ❛♥ ✐❞❡❛• s✐t✉❛t✐♦♥ ✇✐t❤♦✉t ❡♥✈✐r♦♥♠❡♥t❛• ❞✐st✉r❜❛♥❝❡✱ ♠♦❞❡• ❡rr♦rs ❛♥❞ s❡♥s♦r ❞✐st✉r❜❛♥❝❡✱ ❡t❝✳ ✺✽ ✹✳✶✳ ❙❝❡♥❛r✐♦s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• σ(t) =                    0 ✇❤❡♥ 0 t t 1 , t 4 < t 150s✱ h d (t -t 1 ) 5 (t -t 1 ) 5 + (T f -t + t 1 ) 5 ✇❤❡♥ t 1 < t t 2 ✱ 2 ✇❤❡♥ t 2 < t t 3 ✱ h d -h d (t -t 3 ) 5 (t -t 3 ) 5 + (T f -t + t 3 ) 5 ✇❤❡♥ t 3 < t t 4 ✳ h d = 2♠, T f = 6s. x = σ(t), ✇✐t❤ t 1 = 10s, t 2 = 16s, t 3 = 90s, t 4 = 96s. y = σ(t), ✇✐t❤ t 1 = 40s, t 2 = 46s, t 3 = 120s, t 4 = 126s. z = 10♠. ✭✹✳✶✮ ✹✳✶✳✷ ❲✐♥❞ ❉✐st✉r❜❛♥❝❡ ❚❤✐s s❝❡♥❛r✐♦ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❡st t❤❡ st❛❜✐•✐t② ♦❢ ❝♦♥tr♦• s②st❡♠s ✇❤✐•❡ ❡♥❝♦✉♥t❡r✐♥❣ ❡♥✈✐r♦♥✲ ♠❡♥t ❞✐st✉r❜❛♥❝❡✱ ♥❛♠❡•② ✇✐♥❞✳ ◗✉❛❞r♦t♦rs ❛r❡ ❡❛s② t♦ ❜❡ ❛✛❡❝t❡❞ ❜② ❡①t❡r♥❛• ❞✐st✉r❜❛♥❝❡s ✇❤✐•❡ ✐ts ✢✐❣❤t✳ ❆s t❤❡ s❤♦♦t✐♥❣ ✐s t❛❦❡♥ ♣•❛❝❡ ❛t ❛♥ ♦♣❡♥ s♣❛❝❡ ②❛r❞✱ t❤❡ ♠♦st ❣❡♥❡r❛• ❞✐s✲ t✉r❜❛♥❝❡ ✐s t❤❡ ✇✐♥❞✳ ❙❡✈❡r❛• ✇✐♥❞ ♠♦❞❡•s ❛r❡ ♣r♦♣♦s❡❞ ✐♥ •✐t❡r❛t✉r❡ ❬✹✶✱ ✺✶(✳ ❚❤❡ ♠♦st ❣❡♥❡r❛• ♠♦❞❡• ✐s t❤❡ ✇✐♥❞ ❢♦r❝❡ ♠♦❞❡• ✇❤❡r❡ ✇✐♥❞ ✐♠♣♦s❡s ❡①tr❛ ❢♦r❝❡s ♦♥ t❤❡ s②st❡♠ ❬✺✶(✳ ❚❤❡ ❢♦r❝❡ ✐♥❞✉❝❡❞ ❜② ✇✐♥❞ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s✿ F w = k w A w v 2 w ✭✹✳✷✮ ✇❤❡r❡ v w ✐s t❤❡ ✇✐♥❞ ✈❡•♦❝✐t②✱ A w ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ s✉r❢❛❝❡ ❢❛❝✐♥❣ t❤❡ ✇✐♥❞✱ k w ✐s ❛ ❝♦❡✣❝✐❡♥t✱ k w = 0.0022 ◆•s 2 ✴♠ 2 ✳ ❆ ❛♥♦t❤❡r ♠♦❞❡• ✐s t❤❡ ✇✐♥❞ t♦rq✉❡ ♠♦❞❡• ❬✺✶(✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ❝❛♥ ❛•s♦ ✐♥❞✉❝❡ ❡①tr❛ t♦rq✉❡s ♦♥ t❤❡ s②st❡♠ ✇❤✐❝❤ ✐s ❡①♣r❡ss❡❞ ❛s✿ T w = c w A w Dα w v 2 w ✭✹✳✸✮ ✇❤❡r❡ v w ✐s t❤❡ ✇✐♥❞ ✈❡•♦❝✐t②✱ A w ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ s✉r❢❛❝❡ ❢❛❝✐♥❣ t❤❡ ✇✐♥❞✱ D ✐s t❤❡ ❞✐❛♠❡t❡r ♦❢ t❤❡ ❛r❡❛ ❢❛❝✐♥❣ t❤❡ ✇✐♥❞✱ α w ✐s t❤❡ st❛t✐❝ ❛✐r ❞❡♥s✐t②✱ c w ✐s ❛ t♦rq✉❡ ❝♦❡✣❝✐❡♥t✳ ❆s t❤❡ s✉r❢❛❝❡ ❛♥❞ ❞✐♠❡♥s✐♦♥ ♦❢ q✉❛❞r♦t♦rs ❛r❡ r❡•❛t✐✈❡•② s♠❛••✳ ❚❤❡ t♦rq✉❡ ✐♥❞✉❝❡❞ ❜② ✇✐♥❞ ❝❛♥ ❜❡ ♥❡❣•❡❝t❡❞✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ ♦♥•② t❤❡ ❢♦r❝❡ ✐♥❞✉❝❡❞ ❜② ✇✐♥❞ ✐s ❝♦♥s✐❞❡r❡❞✳ ❍❡r❡✱ ✇❡ s✉♣♣♦s❡ t❤❡ ✇✐♥❞ ❝❛✉s❡❞ ❛ s❛♠❡ ❛❝❝❡•❡r❛t✐♦♥ ✐♥t❡♥s✐t② ♦♥ ❛•• x✱ y ❛♥❞ z ❛①✐s✳ ❚❤❡s❡ ✺✾ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✶✳ ❙❝❡♥❛r✐♦s ❛❝❝❡•❡r❛t✐♦♥s ❛r❡ ♣❡rt✉r❜❛t✐♦♥s ✐♥ t❤❡ ❡q✉❛t✐♦♥s r❡•❛t❡❞ t♦ t❤❡ ❢♦r❝❡s ✐♥ t❤❡ q✉❛❞r♦t♦r ♠♦❞❡• ❊q✉❛t✐♦♥s ✭✷✳✹✷❡✮✱ ✭✷✳✹✷❢✮ ❛♥❞ ✭✷✳✹✷❞✮✳ ❚❤❡ ♠♦❞❡• ✐s ❡①♣r❡ss❡❞ ❛s ❢♦••♦✇s ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✹✮✳ x(t) = ẍ(t) + a(t) ỹ(t) = ÿ(t) + a(t) ✭✹✳✹✮ z(t) = z(t) + a(t) ❚❤✐s ❛❝❝❡•❡r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✉r❡ ✹✳✷✱ ❛♥❞ ❡①♣r❡ss❡❞ ❡①♣•✐❝✐t•② ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✺✮✳ ❚❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ a(t) ❤❛s ❛ ♠❛①✐♠✉♠ ❛❝❝❡•❡r❛t✐♦♥ ❛❜♦✉t 1.7♠✴s 2 ✳ a(t) =                                      0 ✇❤❡♥ 0s t 30s✱ 0.8 sin( π(t-30) 31 ) + 0.4 sin( π(t-30) 7 )
+0.08 sin( π(t-30)

2 ) + 0.056 sin( 24π(t-30)

11

) ✇❤❡♥ 30s < t 57s✱ 0 ✇❤❡♥ 57s< t 70s✱

♦•❡r❛♥❝❡ ♦❢ ❝♦♥tr♦• s②st❡♠s t♦ ♠♦❞❡•✐♥❣ ❡rr♦rs✳ ❉✉❡ t♦ t❤❡ ❝♦♠♣•❡①✐t② ♦❢ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❡✛❡❝ts✱ t❤❡ ❡①t❡r♥❛• ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ❞✐✣❝✉•t t♦ ❜❡ ✻✵ ✹✳✶✳ ❙❝❡♥❛r✐♦s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ❡①♣r❡ss❡❞ ❡①♣•✐❝✐t•② ✐♥ t❤❡ ♠♦❞❡•✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ❛ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ r❡❛• ❞②♥❛♠✐❝s ❛♥❞ t❤❡ ♠♦❞❡• ✉s❡❞ ✐♥ t❤❡ ❝♦♥tr♦• s②st❡♠✳ ❋✉rt❤❡r♠♦r❡✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❝♦♠♣•❡①✐t② ♦❢ t❤❡ ❝♦♥tr♦• ❛•❣♦r✐t❤♠ ❛♥❞ t❤❡ •✐♠✐t ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s ♦❢ t❤❡ ♦♥❜♦❛r❞ s②st❡♠✱ s♦♠❡ ✉♥✐♠♣♦rt❛♥t ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♥❡❣•❡❝t❡❞ ✐♥ t❤❡ ❝♦♥tr♦• s②st❡♠✱ s✉❝❤ ❛s ❢r✐❝t✐♦♥s✳ ❚❤✐s ❢✉rt❤❡r ✐♥❝r❡❛s❡s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ r❡❛• ❞②♥❛♠✐❝s ❛♥❞ t❤❡ ♠♦❞❡•✳ ■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ❤❛✈❡ ❞❡❞✉❝❡❞ ❛ ♠❛t❤❡♠❛t✐❝❛• ♠♦❞❡• ♦❢ q✉❛❞r♦t♦r ✭s❡❡ ❊q✉❛t✐♦♥ ✭✸✳✷✮✮✳ ❚❤❡r❡ ❛r❡ s❡r✈❛• ♣❛r❛♠❡t❡rs t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❛❝❝♦r❞✐♥❣ t♦ r❡❛• q✉❛❞r♦t♦rs✳ ❚❤❡ ♠❛ss m ❝❛♥ ❜❡ ❞✐r❡❝t•② ❡✈❛•✉❛t❡❞ ❜② ❛ ❜❛•❛♥❝❡✳ ❚❤❡ ✈❛•✉❡ ✇❡ ❣❡t ✐s ✉s✉❛••② ♣r❡❝✐s❡✳ ❚❤❡ ✐♥❡rt✐❛ ♠❛tr✐① ❡•❡♠❡♥ts I xx ✱ I yy ❛♥❞ I zz ❛r❡ ♦❢t❡♥ ❝♦♠♣✉t❡❞ ❜❛s❡❞ ♦♥ ♠❡❝❤❛♥✐❝❛• str✉❝t✉r❡s ✇❤✐❝❤ ❛r❡ ❞❡s✐❣♥❡❞ ♦♥ s♦♠❡ ♠❡❝❤❛♥✐❝❛• ❞❡s✐❣♥ s♦❢t✇❛r❡s✱ s✉❝❤ ❛s ❈❆❚■❆✱ Pr♦✴❊◆•■◆❊❊|✳ ❆s t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ♠❛t❡r✐❛• ✐s ♥♦t ♣r❡❝✐s❡ ❛♥❞ •❛t❡r ♦♥ ♠♦t♦rs ❛♥❞ ♦t❤❡r ❡•❡♠❡♥ts ❛r❡ ❛❞❞❡❞ ✐♥t♦ t❤❡ s②st❡♠✱ t❤❡ ✈❛•✉❡ ♦❢ t❤❡ ✐♥❡rt✐❛ ♠❛tr✐① ❝❛♥ ❜❡ ✐♥✢✉❡♥❝❡❞ ❛♥❞ ❝❛✉s❡ s♦♠❡ ♠♦❞❡• ✉♥❝❡rt❛✐♥t✐❡s✳ ■♥ ❊q✉❛t✐♦♥ ✭✸✳✷✮✱ t❤❡ ✐♥♣✉ts u i ♦❢ t❤❡ q✉❛❞r♦t♦r ♠♦❞❡• ❛r❡ ❞❡✜♥❡❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ t❤r✉sts ❛♥❞ t❤❡ ②❛✇✐♥❣ ♠♦♠❡♥ts✳ ❚❤❡♥✱ t❤❡ r♦t❛t✐♦♥❛• s♣❡❡❞s Ω i ♦❢ ❡❛❝❤ r♦t♦r ❛r❡ ❝❛•❝✉•❛t❡❞ ✉s✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥t ♣❛r❛♠❡t❡rs b ❛♥❞ d✳ u 1 = b 4 i=1 Ω 2 i , u 2 = bl(Ω 2 2 -Ω 2 4 ), u 3 = bl(-Ω 2 1 + Ω 2 3 ), u 4 = (-1) i+1 d 4 i=1 Ω 2 i . ✭✹✳✻✮ ❚❤❡ ♣❛r❛♠❡t❡rs b ❛♥❞ d ❛r❡ ✈❛r✐❛❜•❡s ❛♥❞ ❝❤❛♥❣❡ ❞✉r✐♥❣ t❤❡ ✢✐❣❤ts✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❡ ❝♦♥tr♦• s②st❡♠✱ t❤❡② ❛r❡ tr❡❛t❡❞ ❛s ❝♦♥st❛♥ts ✇❤✐❝❤ ❛r❡ ❝♦♠♣✉t❡❞ ❜❛s❡❞ ♦♥ s♦♠❡ st❛t✐❝ t❡sts✳ ❚❤❡s❡ ♣❛r❛♠❡t❡rs ❛•s♦ ❝❛✉s❡ s♦♠❡ ♠♦❞❡• ✉♥❝❡rt❛✐♥t✐❡s✳ ❚❤❡r❡❢♦r❡✱ ❝♦♥tr♦• s②st❡♠s s❤♦✉•❞ ❜❡ ❝❛♣❛❜•❡ t♦ t♦•❡r❛t❡ ♠♦❞❡• ✉♥❝❡rt❛✐♥t✐❡s ♦r ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✐♥ r❡❛• ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠ s❤♦✉•❞ ❛•✇❛②s ❜❡ ❡♥s✉r❡❞✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ s♦♠❡ r❛♥❞♦♠ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛r❡ ❣✐✈❡♥✳ ❙✉♣♣♦s✐♥❣ t❤❡ ✐♥❡rt✐❛ ♠❛tr✐① ❡•❡♠❡♥ts I xx ✱ I yy ❛♥❞ I zz ❛r❡ ✉♥❞❡rr❛t❡❞ ✺✵✪✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts b ❛♥❞ d ❛r❡ ❛•s♦ ✉♥❞❡rr❛t❡❞✱ ❛♥❞ t❤❡ ✈❛•✉❡ ✉s❡❞ ✐♥ t❤❡ ❝♦♥tr♦• s②st❡♠ ❛r❡ ♦♥•② ✽✵✪ ♦❢ t❤❡ r❡❛• ✈❛•✉❡s✳ Ĩxx = 0.5 I xx , Ĩyy = 0.5 I yy , Ĩzz = 0.5 I zz , b = 0.8 b, d = 0.8 d. ✭✹✳✼✮ ✹✳✶✳✹ ❙❡♥s♦r ◆♦✐s❡s ❚❤✐s s❝❡♥❛r✐♦ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❡st t❤❡ t♦•❡r❛♥❝❡ ♦❢ ❝♦♥tr♦• s②st❡♠s t♦ ♥♦✐s❡s ♦❢ t❤❡ s②st❡♠ st❛t❡s✳ ❆s t❤❡ ❝♦st ❛♥❞ s♣❛❝❡ ❛r❡ •✐♠✐t❡❞ ❢♦r q✉❛❞r♦t♦rs✱ ▼❊▼❙ ✭▼✐❝r♦✲❊•❡❝tr♦✲▼❡❝❤❛♥✐❝❛• ✻✶ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✶✳ ❙❝❡♥❛r✐♦s ❙②st❡♠s✮ s❡♥s♦rs ❛r❡ ✉s✉❛••② ✉s❡❞✳ ❚❤❡ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡s❡ s❡♥s♦r ❛r❡ •♦✇ ❛♥❞ t❤❡ ♦✉t♣✉t s✐❣♥❛•s ❛r❡ ❛•✇❛②s ♥♦✐s②✳ ▼♦r❡♦✈❡r✱ ✇❡❛t❤❡r ❝♦♥❞✐t✐♦♥ ❝❤❛♥❣❡s ✐♥ ❡❛❝❤ ❛♣♣•✐❝❛t✐♦♥✱ s✉❝❤ ❛s t❡♠♣❡r❛t✉r❡✱ ❤✉♠✐❞✐t②✱ ✇✐♥❞✳ ❚❤✐s ❝❛♥ ❛•s♦ ❝❛✉s❡ s❡♥s♦r ♥♦✐s❡s✳ ■♥ ♠❛♥② ❛♣♣•✐❝❛t✐♦♥s✱ s✉❝❤ ❛s ❛❡r✐❛• ♣❤♦t♦ ❛♥❞ ✈✐❞❡♦✱ ❛ •✐♠✐t ♦❢ ✈✐❜r❛t✐♦♥s ✐s ✐♠♣♦rt❛♥t ❢♦r t❤❡ q✉❛•✐t② ♦❢ ✐♠❛❣❡s✳ ❚❤❡ ♣♦s✐t✐♦♥ st❛t❡s x✱ y✱ z ❛♥❞ ❛♥❣✉•❛r st❛t❡s φ✱ θ✱ ψ ♦❢ t❤❡ q✉❛❞r♦t♦r s②st❡♠ ❛r❡ ♠❡❛s✉r❡❞ ❜② s❡♥s♦rs✳ ❚❤❡ ♣♦s✐t✐♦♥ st❛t❡s x✱ y✱ z ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② ❛❝❝❡•❡r♦♠❡t❡rs✳ ❆❝❝❡•❡r♦♠❡t❡rs ❣✐✈❡ ♦♥•② t❤❡ tr❛♥s✐❡♥t •✐♥❡❛r ❛❝❝❡•❡r❛t✐♦♥✳ ■♥ ♦r❞❡r t♦ ❣❡t t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥ ✐♥❢♦r♠❛t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ✐♥t❡❣r❛t❡ t❤❡ ❛❝❝❡•❡r❛t✐♦♥ ♦♥❝❡ ❛♥❞ t✇✐❝❡✳ ❚❤✐s ♠❛② ❝❛✉s❡ ❜✐❣ ❡rr♦rs ❛♥❞ ❞r✐❢ts✱ ❡s♣❡❝✐❛••② ✇❤❡♥ t❤❡ ❛❝❝❡•❡r♦♠❡t❡r ❤❛s ❛•r❡❛❞② ✐♥✐t✐❛• ❞❛t❛ ♥♦✐s❡s✳ ❚❤❡ ♣♦s✐t✐♦♥ st❛t❡s ❝❛♥ ❜❡ ❛•s♦ ♠❡❛s✉r❡❞ t❤r♦✉❣❤ ✈✐s✐♦♥ s②st❡♠s ❜② ✉s✐♥❣ tr✐❛♥❣✉•❛r ♣♦s✐t✐♦♥ ❞❡t❡r♠✐♥❛t✐♦♥ ♦r ♦t❤❡r ♣♦s✐t✐♦♥ ❞❡t❡r♠✐♥❛t✐♦♥ ♠❡t❤♦❞s✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ t❤❡ q✉❛❞r♦t♦r ✐s ❤♦✈❡r✐♥❣ ♦r ❡s♣❡❝✐❛••② ♠♦✈✐♥❣✱ t❤❡ ✈✐❜r❛t✐♦♥ ♦❢ q✉❛❞r♦t♦rs ❝❛♥ ❝❛✉s❡ ✉♥✇❛♥t❡❞ ♥♦✐s❡s✳ ❚❤❡ ♣♦s✐t✐♦♥ st❛t❡s ❝❛♥ ❜❡ ♠❡❛s✉r❡❞ t❤r♦✉❣❤ •P❙✱ s♦♥❛r✱ ♣r❡ss✉r❡ s❡♥s♦r ♦r ♦t❤❡r ❞✐st❛♥❝❡ ♠❡❛s✉r❡ s❡♥s♦rs✱ ✇❤❡r❡ s❡♥s♦r ♥♦✐s❡s ❛r❡ ❛•✇❛②s ❛ ♣r♦❜•❡♠✳ ❚❤❡ ❛♥❣✉•❛r st❛t❡s φ✱ θ✱ ψ ❛r❡ ✉s✉❛••② ❡st✐♠❛t❡❞ ❜② ❣②r♦s❝♦♣❡s✳ •②r♦s❝♦♣❡s ❣✐✈❡ t❤❡ r♦t❛t✐♦♥❛• ✈❡•♦❝✐t✐❡s ✐♥❢♦r♠❛t✐♦♥✳ ■♥ ♦r❞❡r t♦ ❣❡t t❤❡ r♦t❛t✐♦♥❛• ♣♦s✐t✐♦♥✱ ✇❡ ♥❡❡❞ t❤❡ ✐♥t❡❣r❛t❡ t❤❡ ❛❝❝❡•❡r❛t✐♦♥ ♦♥❝❡✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♣r❡❝✐s✐♦♥ ♦❢ ❛ ❣②r♦s❝♦♣❡✱ s❡♥s♦r ♥♦✐s❡s ❝❛♥ ❜❡ s♠❛•• ♦r ❜✐❣ ❡♥♦✉❣❤ t♦ ✐♥✢✉❡♥❝❡ ❝♦♥tr♦• s②st❡♠s✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ ✇❡ ❛❞❞ t❤❡ s❡♥s♦r ♥♦✐s❡s ♦♥ ❛•• ✶✷ st❛t❡s ♦❢ t❤❡ s②st❡♠✱ ♥❛♠❡•② t❤❡ ♣♦s✐t✐♦♥ st❛t❡s x✱ y✱ z✱ ẋ✱ ẏ✱ ż ❛♥❞ ❛♥❣✉•❛r st❛t❡s φ✱ θ✱ ψ✱ φ✱ θ✱ ψ✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ♥♦✐s❡ ❛❞❞❡❞ ✐♥ t❤❡ s②st❡♠ ✐s ❛s ✐♥ ❊q✉❛t✐♦♥ ✹✳✽✳ ❚❤❡ ❢✉♥❝t✐♦♥ |❛♥❞✭✮ ✐s ❛ ▼❆❚▲❆❇ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❣❡♥❡r❛t❡s ❛ r❛♥❞♦♠ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✵ t♦ ✶✳ ❚❤✉s✱ |❛♥❞✭✮✰✵✳✺ ❣❡♥❡r❛t❡s ❛ r❛♥❞♦♠ ✈❛•✉❡ ∈ [0.5, 1.5]✳ ❚❤❡r❡❢♦r❡✱ ❛•• t❤❡ st❛t❡s ❤❛✈❡ ±✺✵✪ ♥♦✐s❡s✳ x = x • |❛♥❞() + 0.5 ✭✹✳✽✮ ✹✳✶✳✺ ❆❝t✉❛t♦r ❋❛✉•ts ❚❤✐s s❝❡♥❛r✐♦ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❡st t❤❡ st❛❜✐•✐t② ♦❢ ❝♦♥tr♦• s②st❡♠s ✉♥❞❡r s✉❞❞❡♥ str♦♥❣ s✐❣♥❛• ❞✐st✉r❜❛♥❝❡s✳ ❆s ♠♦t♦rs ❛♥❞ s❡♥s♦rs ❛r❡ ✇♦r♥ ♦✉t✱ s♦♠❡ s❡✈❡r❡ s✐t✉❛t✐♦♥s ♠❛② ♦❝❝✉r ❞✉r✐♥❣ ✢✐❣❤ts✱ s✉❝❤ ❛s ❛❝t✉❛t♦r ❢❛✉•t✱ s❡♥s♦r ❢❛✐•✉r❡✱ ❡t❝✳ ■♥ ❛ ✈❡r② s❤♦rt t✐♠❡ ✐♥t❡r✈❛•✱ ❛ s✉❞❞❡♥ ❛♥❞ r❛t❤❡r str♦♥❣ ❛❜♥♦r♠❛• s✐❣♥❛• ♦❝❝✉rs✳ ❈♦♥tr♦• s②st❡♠s s❤♦✉•❞ ♥♦t ❜❡ ❣r❡❛t•② ✐♥✢✉❡♥❝❡❞ ❛♥❞ •♦♦s❡ ✐ts st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ ❢❛✐•✉r❡✳ ❙✉♣♣♦s❡ t❤❡ q✉❛❞r♦t♦r ✉s❡❞ ❢♦r t❤✐s s❤♦♦t✐♥❣ ✐s r❛t❤❡r ♦•❞✱ ❛♥❞ ❤❛s ❛•r❡❛❞② ❜❡❡♥ ✉s❡❞ ❢♦r s❡✈❡r❛• ②❡❛rs✳ ❉✉r✐♥❣ t❤❡ ✢✐❣❤ts✱ t❤❡ r♦t♦rs ♦❢ t❤❡ q✉❛❞r♦t♦r ❤❛✈❡ ❛❝t✉❛t♦r ❢❛✉•ts✳ ■♥ ❛ ✻✷ ✹✳✷✳ ❈♦♥tr♦• ❙❝❤❡♠❡s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• s❤♦rt t✐♠❡ ♣❡r✐♦❞✱ t❤❡ r♦t❛t✐♦♥❛• s♣❡❡❞ ♦❢ ♦♥❡ r♦t♦r ✐s ♥♦ •♦♥❣❡r ❛s ✇❤❛t ✐s r❡q✉✐r❡❞ ❜② t❤❡ ❝♦♥tr♦• s②st❡♠✳ ❆❢t❡r t❤✐s ♣❡r✐♦❞✱ t❤❡ r♦t♦rs r♦t❛t❡ ✇✐t❤♦✉t ❛♥② ♣r♦❜•❡♠ ❛s ✇❛♥t❡❞✳ ■❢ t❤✐s ❢❛✉•t ✐s ♥♦t ❞❡t❡❝t❡❞ ❞✉r✐♥❣ t❤❡ ✢✐❣❤t✱ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ q✉❛❞r♦t♦r ✇✐•• ❜❡ ✐♥✢✉❡♥❝❡❞✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ ✇❡ ❛❞❞ t❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ♦♥ t✇♦ r♦t♦rs ❞✉r✐♥❣ t❤❡ ✢✐❣❤t t❛s❦✱ r♦t♦r ✸ ❛♥❞ ✹ ✭s❡❡ ✐♥ ❋✐❣✉r❡ ✷✳✶ ♦♥ P❛❣❡ ✷✻✮✳ ❚❤❡s❡ ❢❛✉•ts ❛r❡ ♥♦t ❞❡t❡❝t❡❞ ❜② t❤❡ ♣r♦❣r❛♠✱ ❛♥❞ ✐♥✢✉❡♥❝❡ ❞✐r❡❝t•② t❤❡ ❝♦♥tr♦• s②st❡♠ ♣❡r❢♦r♠❛♥❝❡✳ ❚✇♦ r♦t♦rs ❛r❡ ♦♥ t✇♦ ❛r♠s ♦❢ t❤❡ q✉❛❞r♦t♦r ✇❤✐❝❤ ❡♥❛❜•❡s t♦ s❡❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ r♦•• ❛♥❞ ♣✐t❝❤ ❞②♥❛♠✐❝s✳ ❚❡st❡❞ ✐♥ s✐♠✉•❛t✐♦♥✱ ❛♥ ❛❝t✉❛t♦r ❢❛✉•t t✐♠❡ ✵✳✷s ❝❛♥ ❝❛✉s❡ ❛♥ ✉♥r❡❝♦✈❡r❛❜•❡ ❡rr♦r ❢♦r t❤❡ s②st❡♠ t♦ r❡t❛✐♥ ✐ts st❛❜✐•✐t②✳ ❚❤❡r❡❢♦r❡ ✐♥ t❤✐s s❝❡♥❛r✐♦✱ ❛ tr❛♥s✐❡♥t t✐♠❡ ♦❢ ❛❝t✉❛t♦r ❢❛✉•ts ✵✳✶s ✐s s❡•❡❝t❡❞✳ ❚✇♦ ❢❛✉•ts ❛r❡ ♣•❛♥♥❡❞✳ ❚❤❡ ✜rst ❢❛✉•t ❤❛♣♣❡♥s ❛t ✹✷s ❞✉r✐♥❣ t❤❡ r❡❢❡r❡♥❝❡ tr❛❝❦✐♥❣ ❛•♦♥❣ t❤❡ y ❛①✐s✳ ❚❤❡ ♦t❤❡r ❢❛✉•t ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ q✉❛❞r♦t♦r ✐s ❤♦✈❡r✐♥❣✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❝❤❡❝❦ ♦✉t t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠ ❞✉r✐♥❣ tr❛❝❦✐♥❣ ❛♥❞ ❤♦✈❡r✐♥❣✳ ❚❤❡ ♣r♦❣r❛♠ ✐s ✇r✐tt❡♥ ❛s ❢♦••♦✇✐♥❣✿ ✐❢ ✭t❃✹✷ ✫✫ t❁✹✷✳✶ ✮ ❖♠❞✭✸✮❂✷✷✵❀ ❡♥❞ ✐❢ ✭t❃✾✵ ✫✫ t❁✾✵✳✶ ✮ ❖♠❞✭✹✮❂✷✷✵❀ ❡♥❞ ✇❤❡r❡ ❖♠❞✭✮ ✐s t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② ♦❢ t❤❡ r♦t♦r✳ ❖♠❞✭✸✮❂✷✷✵ ♠❡❛♥s t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② ♦❢ t❤❡ r♦t♦r ✸ ✐s s❡t t♦ ❜❡ ✷✷✵r❛❞✴s✳ ✹✳✶✳✻ ❙❡♥s♦r ❋❛✉•ts ❆s s❡♥s♦rs ❛r❡ ✇♦r♥ ♦✉t✱ t❤❡② ❝❛♥ ❝❛✉s❡ s♦♠❡ s❡♥s♦r ❢❛✉•ts ✐♥ ❛♣♣•✐❝❛t✐♦♥s✳ ❚❤❡ s❡♥s♦r ❞❛t❛ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛•✇❛②s ❝♦rr❡❝t ✐♥ ❝♦♥tr♦• s②st❡♠s✳ ❚❤❡r❡❢♦r❡✱ s❡♥s♦r ❢❛✉•ts ❛r❡ ❞✐✣❝✉•t t♦ ❜❡ ❞❡t❡❝t❡❞ ❜② ♣r♦❣r❛♠✳ ❖t❤❡r s❡♥s♦rs ❛r❡ ♥❡❡❞❡❞ t♦ ❞❡t❡❝t ✇❤❡t❤❡r ♦♥❡ s❡♥s♦r ❤❛s ❢❛✉•ts ♦r ♥♦t✳ ❚❤✉s✱ s❡♥s♦r ❢❛✉•ts ❛r❡ ♥♦t s✐♠✉•❛t❡❞ ✐♥ t❤✐s t❤❡s✐s✳ ✹✳✷ ❈♦♥tr♦• ❙❝❤❡♠❡s ◆♦✇❛❞❛②s✱ t❤❡ ❝♦♥tr♦• •❛✇s ♦❢ ❝♦♥tr♦• s②st❡♠s ❛r❡ ♠❛✐♥•② ✐♠♣•❡♠❡♥t❡❞ ✐♥ ❞✐❣✐t❛• ♣•❛t❢♦r♠s s✐♥❝❡ ❝♦♠♣✉t❡r ❛♥❞ ♠✐❝r♦♣r♦❝❡ss♦rs ♦✛❡r ♠❛♥② ❛❞✈❛♥t❛❣❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥❛•♦❣ ♣•❛t❢♦r♠s✳ ❆s ❛ ❞✐❣✐t❛• ❝♦♠♣✉t❡r ✐s ❛ ❞✐s❝r❡t❡ s②st❡♠✱ ❆✴❉ ❝♦♥✈❡rt❡rs ❛r❡ ✉s❡❞ t♦ ❝❤❛♥❣❡ s❡♥s♦r ❞❛t❛ ✐♥t♦ ❞✐❣✐t❛• ✐♥♣✉ts ❢♦r t❤❡ ❝♦♥tr♦• s②st❡♠✱ ❛♥❞ ❉✴❆ ❝♦♥✈❡rt❡rs ❛r❡ ✉s❡❞ t♦ ❝❤❛♥❣❡ ❞✐❣✐t❛• ❝♦♥tr♦• s✐❣♥❛•s ✐♥t♦ ❝♦♥t✐♥✉♦✉s ❝♦♥tr♦• ♦✉t♣✉ts✳ ❚❤❡ ❉✴❆ ❝♦♥✈❡rt❡r ✐s ✉s✉❛••② ❛ ③❡r♦✲♦r❞❡r ❤♦•❞ ✭)❖❍✮✱ ✇❤✐❝❤ ❤♦•❞s ❡❛❝❤ s❛♠♣•❡ ✈❛•✉❡ ❢♦r ♦♥❡ s❛♠♣•✐♥❣ ✐♥t❡r✈❛•✳ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✻✸ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✷✳ ❈♦♥tr♦• ❙❝❤❡♠❡s s❝❤❡♠❡s t♦ s❝❤❡❞✉•❡ t❤❡ ❝♦♥tr♦• s✐❣♥❛•s ♦♥ ❞✐❣✐t❛• ♣•❛t❢♦r♠s✿ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡ ❛♥❞ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ✹✳✷✳✶ ❚✐♠❡ ❚r✐❣❣❡r❡❞ ❙❝❤❡♠❡ ❚❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡ ✐s ✇✐❞❡•② ✉s❡❞ ✐♥ ♠❛♥② ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❚❤❡ ❝♦♥tr♦• t❛s❦ ✐s ❡①❡❝✉t❡❞ ♣❡r✐♦❞✐❝❛••② ❡✈❡r② s❛♠♣•✐♥❣ ♣❡r✐♦❞ T t ✱ ✇❤✐❝❤ ✐s ♣r❡✈✐♦✉s•② ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r♦❣r❛♠ ❜❡❢♦r❡ t❤❡ r❡❛•✲t✐♠❡ ❝♦♥tr♦•✳ ❊✈❡r② T t ✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ❝♦♠♣✉t❡s t❤❡ ♥❡✇ ❝♦♥tr♦• s✐❣♥❛•s ❛♥❞ s❡♥❞s t❤❡♠ t♦ t❤❡ ❛❝t✉❛t♦rs✳ ■t ✐s t❤❡ ♠♦st ❝♦♠♠♦♥ ❛♥❞ s✐♠♣•❡st ✐♠♣•❡♠❡♥t❛t✐♦♥✳ ❚❤❡ ♠❛✐♥ ❞✐✣❝✉•t② ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❡ s❡•❡❝t✐♦♥ ♦❢ ❛ s❛♠♣•✐♥❣ ♣❡r✐♦❞ T t ✇❤✐❝❤ ❣✉❛r❛♥t❡❡s ❛ ❞❡s✐r❡❞ •❡✈❡• ♦❢ t❤❡ ❝♦♥tr♦• ♣❡r❢♦r♠❛♥❝❡✳ ❆ s♠❛•• s❛♠♣•✐♥❣ ♣❡r✐♦❞ ❝❛♥ ❦❡❡♣ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✉♥❞❡r ❛♥ ❛❝❝❡♣t❛❜•❡ •✐♠✐t ❞✉r✐♥❣ t✇♦ s❛♠♣•✐♥❣ t✐♠❡s✳ ❍♦✇❡✈❡r✱ ❛ s♠❛•• s❛♠♣•✐♥❣ ♣❡r✐♦❞ ♠❡❛♥s t❤❡ ❝♦♥tr♦• s②st❡♠ ♥❡❡❞s ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡ ❛♥❞ r❡s♦✉r❝❡s✱ ✇❤✐❝❤ ♥❡❡❞s t♦ t❛❦❡ ❢r♦♠ t❤❡ ✉♣♣❡r s②st❡♠✳ ❯s✉❛••② ✐♥ ♦r❞❡r t♦ ❦❡❡♣ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠✱ ❛ s❛♠♣•✐♥❣ ♣❡r✐♦❞ T t ✐s ❝❤♦s❡♥ ❜❛s❡❞ ♦♥ t❤❡ ✇♦rst ❝❛s❡ s❝❡♥❛r✐♦✳ ❋♦r •✐♥❡❛r s②st❡♠s✱ t❤❡ s❡•❡❝t✐♦♥ ♦❢ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ❡♥s✉r✐♥❣ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠ ✐s st✉❞✐❡❞ ❜❛s❡ ♦♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❡q✉✐✈❛•❡♥t ❞✐s❝r❡t❡ ♠♦❞❡•✳ ❋♦r ♥♦♥•✐♥❡❛r s②st❡♠s✱ ◆❡s✐❝ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ❝♦♠♠♦♥ ❛♣♣r♦❛❝❤ t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t❡ ❞✐s❝r❡t❡✲t✐♠❡ ♠♦❞❡•s✱ ❛♥❞ t❤❡♥ st✉❞✐❡❞ ❤♦✇ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✐♥✢✉❡♥❝❡ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠ ❬✼✼(✳ ❍♦✇❡✈❡r ✐♥ r❡❛•✲t✐♠❡ ❝♦♥tr♦• ❛♣♣•✐❝❛t✐♦♥s✱ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞s ❛r❡ s❡•❡❝t❡❞ ✉s✉❛••② ❜② t❤❡ ❡①♣❡r✐❡♥❝❡ ♦❢ t❤❡ ❡♥❣✐♥❡❡rs ❛♥❞ s♦♠❡ r♦✉❣❤ ♣r❡✈✐♦✉s s✐♠✉•❛t✐♦♥s✳ ✹✳✷✳✷ ❊✈❡♥t ❚r✐❣❣❡r❡❞ ❙❝❤❡♠❡ ❈♦♥tr❛r✐•② t♦ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• s❝❤❡♠❡ ✐♥ ✇❤✐❝❤ t❤❡ ❝♦♥tr♦• s✐❣♥❛•s ❛r❡ s❡♥t t♦ t❤❡ ❛❝t✉❛t♦rs ❡✈❡r② ✜①❡❞ s❛♠♣•✐♥❣ t✐♠❡✱ ✐♥ t❤❡ ❡✈❡♥t ❜❛s❡❞ s❝❤❡♠❡✱ t❤❡ ❝♦♥tr♦• s✐❣♥❛•s ❛r❡ s❡♥t ♦♥•② ✉♣♦♥ t❤❡ tr✐❣❣❡r✐♥❣ ♦❢ ❛♥ ❡✈❡♥t✳ ❚❤❡ ❡✈❡♥t ❞r✐✈❡♥ ❝♦♥tr♦• ✇❛s ✜rst•② ♣r♦♣♦s❡❞ ❜② ➴r③é♥ ❬✼(✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t✐♠❡ ❞r✐✈❡♥ ❛♥❞ ❡✈❡♥t ❞r✐✈❡♥ ❝♦♥tr♦• s❝❤❡♠❡ ❢♦r ✜rst ♦r❞❡r st♦❝❤❛st✐❝ ❛♥❞ ♥♦♥•✐♥❡❛r s②st❡♠s ❛r❡ ♣r♦♣♦s❡❞ ✐♥ ❬✽( ❛♥❞ ❬✹✵( r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• s❝❤❡♠❡ ❤❛s ♠❛♥② ❛❞✈❛♥t❛❣❡s✿ ✐♥ ♥❡t✇♦r❦❡❞ ❝♦♥tr♦• s②st❡♠s✱ ✐♥ ♦r❞❡r t♦ s❛✈❡ t❤❡ ❜❛♥❞✇✐❞t❤ ❛♥❞ ❡♥❡r❣②✱ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✐s ♦♥•② tr❛♥s♠✐tt❡❞ ✇❤❡♥ s♦♠❡ s✐❣♥✐✜❝❛♥t ❡✈❡♥ts ❤❛✈❡ ♦❝❝✉rr❡❞ ❬✾✾(❀ ✐♥ s♦♠❡ ❡♠❜❡❞❞❡❞ ❛♣♣•✐❝❛t✐♦♥s✱ r❡❞✉❝✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝♦♥tr♦• •❛✇ ❝♦♠♣✉t❛t✐♦♥s ❝❛♥ s❛✈❡ ❡♥❡r❣② ✭s✉❝❤ ❛s ✐♥ ✇✐r❡•❡ss ❝♦♠♠✉♥✐❝❛t✐♦♥✱ ✳✳✳✮ ❛♥❞ ❣✐✈❡ ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ t♦ ♦t❤❡r s✉❜✲s②st❡♠s ✭•♦❝❛•✐③❛t✐♦♥ s②st❡♠✱ ✈✐s✐♦♥ s②st❡♠✱ ❡t❝✮ ❬✾(✳ ❆ ❜❛s✐❝ ➴r③é♥✬s ❡✈❡♥t ❜❛s❡❞ ❝♦♥tr♦••❡r ❝♦♥s✐sts ♦❢ t✇♦ ♣❛rts✿ ❛ t✐♠❡ tr✐❣❣❡r❡❞ ❡✈❡♥t ✻✹ ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ❞❡t❡❝t♦r C t ❛♥❞ ❛♥ ❡✈❡♥t tr✐❣❣❡r❡❞ P■❉ ❝♦♥tr♦••❡r C e ❬✼(✳ ❚❤❡ •❛tt❡r ❝♦♠♣✉t❡s t❤❡ ❝♦♥tr♦• s✐❣♥❛• t♦ ❜❡ ❞❡•✐✈❡r❡❞ t♦ t❤❡ ❛❝t✉❛t♦rs✳ ❚❤❡ ❢♦r♠❡r C t r✉♥s ❛t ❛ ✜①❡❞ s❛♠♣•✐♥❣ ♣❡r✐♦❞ h e ✱ ❛♥❞ ✉♣♦♥ ❢✉•✜••♠❡♥t ♦❢ ❛ ❝❡rt❛✐♥ ❡✈❡♥t tr✐❣❣❡r✐♥❣ •❛✇ L e ✱ s❡♥❞s ❡✈❡♥ts t♦ C e ✳ ❯♣♦♥ r❡❝❡♣t✐♦♥ ♦❢ t❤❡ ❡✈❡♥t✱ C e ❝♦♠♣✉t❡s t❤❡ ❝♦♥tr♦• s✐❣♥❛• ❛♥❞ s❡♥❞s ✐t t♦ t❤❡ ❛❝t✉❛t♦rs✳ ❯s✉❛• ❡✈❡♥t tr✐❣❣❡r✐♥❣ •❛✇s L e ✐♥❝•✉❞❡✿ ✭✶✮ ❊rr♦r t❤r❡s❤♦•❞ •❛✇✿ |e(t k )| > e lim , ✭✹✳✾✮ ✇❤❡r❡ e = y r -y ✐s t❤❡ tr❛❝❦✐♥❣ ❡rr♦r✱ t k ✐s t❤❡ ❝✉rr❡♥t ❞✐s❝r❡t❡ s❡♥s✐♥❣ t✐♠❡ ❜② C e ✱ ❛♥❞ e lim ✐s ❛ ✜①❡❞ •✐♠✐t✳ ✭✷✮ ❊rr♦r ❞✐✛❡r❡♥❝❡ t❤r❡s❤♦•❞✿ |e(t k ) -e(t k-1 )| > e lim . ✭✹✳✶✵✮ ✭✸✮ ■❙❙ ❜❛s❡❞ •❛✇✿ e(t k ) = σ a b |y(t k )|, ✭✹✳✶✶✮ ❛ss✉♠✐♥❣ t❤❡ s②st❡♠ ❝❛♥ ❜❡ r❡♥❞❡r❡❞ ■❙❙ ✭■♥♣✉t t♦ ❙t❛t❡ ❙t❛❜•❡✮ t❤r♦✉❣❤ st❛t✐❝ ❢❡❡❞❜❛❝❦ ❬✾✶(✳ ❍❡r❡✱ σ ✐s ❝❤♦s❡♥ •❡ss t❤❛♥ ♦♥❡ t♦ ❡♥s✉r❡ ❛♥ ❛ss♦❝✐❛t❡❞ ▲②❛♣♦✉♥♦✈ ❢✉♥❝t✐♦♥ t♦ ❞❡❝r❡❛s❡✱ a ❛♥❞ b ❛r❡ ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥st❛♥ts ♦❢ K ∞ ✭❝♦♥s✐st✐♥❣ ♦❢ ❛•• ❢✉♥❝t✐♦♥s γ✿ R + → R + ✇❤✐❝❤ ❛r❡ ❝♦♥t✐♥✉♦✉s✱ str✐❝t•② ✐♥❝r❡❛s✐♥❣✱ s❛t✐s❢②✐♥❣ γ(0) = 0 ❛♥❞ lim ξ→∞ γ(ξ) = ∞✳ ❙❡❡ ❬✾✶(✮✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣ ♠♦t✐♦♥ ❝♦♥tr♦• s❝❤❡♠❡s✱ ✇❡ ✇✐•• t❛❦❡ t❤❡ ❢♦••♦✇✐♥❣ ❡✈❡♥t tr✐❣❣❡r✐♥❣ s❝❤❡♠❡s✿ |e(t k ) -e(t k-1 )| > e lim ✭✹✳✶✷✮ ✹✳✸ ❈♦♥tr♦• ▼❡t❤♦❞s ▼❛♥② ❝♦♥tr♦• ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ✐♠♣•❡♠❡♥t❡❞ ♦♥ q✉❛❞r♦t♦rs✳ ■♥ ❙❡❝t✐♦♥ ✶✳✼ ♦❢ ❈❤❛♣t❡r ✶✱ t❤❡s❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ❞✐s❝✉ss❡❞ ✐♥ ❞❡t❛✐•✳ ❆ •✐♥❡❛r ❙■❙❖ ❝♦♥tr♦• ✐s ✉s❡❞ ❜② P♦✉♥❞s ✐♥ t❤❡ ❞❡❝♦✉♣•❡❞ ❞②♥❛♠✐❝s ✐♥ •♦♥❣✐t✉❞✐♥❛• ✭♣✐t❝❤✴r♦••✮ ❛♥❞ ❛③✐♠✉t❤❛• ♠♦❞❡s ❬✽✸(✳ ❚❤❡ ❝♦♥tr♦•s ✉s✐♥❣ ▲②❛♣✉♥♦✈ t❤❡♦r② ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ❜② ❈❛st✐••♦ ❡t ❛•✳ ✉s✐♥❣ ❛ ♥❡st❡❞ ✻✺ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s s❛t✉r❛t✐♦♥ ❛•❣♦r✐t❤♠ ♦♥ ❉r❛❣❛♥✢②❡r ✐♥ ✷✵✵✹ ❬✸✷(✱ ❛♥❞ ❛•s♦ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛••❛❤ ❡t ❛•✳ ✉s✐♥❣ t❤❡ ❞❡❝♦✉♣•❡❞ ❞②♥❛♠✐❝s ✐♥ ✷✵✵✹ ❬✷✵(✳ ❚❤❡ ❇❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•s ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❜② ❆•t✉➜ ❡t ❛•✳ ✉s✐♥❣ ✈✐s✉❛• ❢❡❡❞❜❛❝❦ ❛s t❤❡ ♣r✐♠❛r② s❡♥s♦r ✐♥ ✷✵✵✷ ❬✹(✱ ❇♦✉❛❜❞❛••❛❤ ❡t ❛•✳ ✉s✐♥❣ t❤❡ ❞❡❝♦✉♣•❡❞ ❞②♥❛♠✐❝s ✐♥ ✷✵✵✺ ❬✷✵(✱ ❈❛st✐••♦ ❡t ❛•✳ ❛❞❞✐♥❣ s❛t✉r❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ✷✵✵✻ ❬✸✹(✱ ❛♥❞ ▼❛❞❛♥✐ ❡t ❛•✳ ❜② s❡♣❛r❛t✐♥❣ t❤❡ s②st❡♠ ✐♥t♦ t❤r❡❡ ✐♥t❡r❝♦♥♥❡❝t❡❞ s✉❜s②st❡♠s ✐♥ ✷✵✵✻ ❬✼✹(✳ ❚❤❡ ❙•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ❜② ❍♦✛♠❛♥♥ ❡t ❛•✳ ❢♦r t❤❡ ❛•t✐t✉❞❡ •♦♦♣ ❝♦♥tr♦• ✐♥ ✷✵✵✹ ❬✺✷(✱ ❲❛s•❛♥❞❡r ❡t ❛•✳ ✐♥ ✷✵✵✺ ❬✾✽(✱ ❛♥❞ ❇♦✉❛❜❞❛••❛❤ ❡t ❛•✳ ✉s✐♥❣ t❤❡ ❞❡❝♦✉♣•❡❞ ❞②♥❛♠✐❝s ✐♥ ✷✵✵✺ ❬✷✵(✳ ❆ ❞②♥❛♠✐❝ ❢❡❡❞❜❛❝❦ t❡❝❤♥✐q✉❡ t♦ •✐♥❡❛r✐③❡ t❤❡ s②st❡♠ ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❜② ▼✐st•❡r ❡t ❛•✳ ✐♥ ✷✵✵✶ ❬✻✾(✳ ❆ ♠✐①❡❞ r♦❜✉st ❢❡❡❞❜❛❝❦ •✐♥❡❛r✐③❛t✐♦♥ ✇✐t❤ •✐♥❡❛r •❍ ∞ ❝♦♥tr♦••❡r ❤❛s ❜❡❡♥ ❛♣♣•✐❡❞ ❜② ▼♦❦❤t❛r✐ ❡t ❛•✳ ✐♥ ✷✵✵✺ ❬✼✶(✳ ❆♥ ❛❞❛♣t✐✈❡ ❝♦♥tr♦• ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❜② •✉❡♥❛r❞ ❡t ❛•✳ ♦♥ t❤❡ q✉❛❞r♦t♦r ❳✹✲❋•②❡r ✐♥ ✷✵✵✻ ❬✹✾(✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r♦♣♦s❡ ❛ ♥❡✇•② ❞❡✈❡•♦♣❡❞ ❝♦♥tr♦• ♠❡t❤♦❞ ♥❛♠❡❞ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❚❤✐s ♠❡t❤♦❞ ✐s ✜rst•② ✐♠♣•❡♠❡♥t❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r ♠♦❞❡• ♣r♦♣♦s❡❞ ✐♥ ❈❤❛♣t❡r ✷✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ s❡✈❡r❛• ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ❛•s♦ ♣r♦♣♦s❡❞✱ s✉❝❤ ❛s ❛ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✱ ❛ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✱ ❡t❝✳ ❚❤❡ ❛✐♠ ♦❢ t❤✐s ♣❛rt ✐s t♦ ❝♦♠♣❛r❡ ❞✐✛❡r❡♥t ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ✈❛r✐♦✉s s❝❡♥❛r✐♦s ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥s✳ ❚❤❡ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s✲ ❛❞✈❛♥t❛❣❡s ♦❢ ❡❛❝❤ ❝♦♥tr♦• ♠❡t❤♦❞ ✇✐•• ❜❡ s❤♦✇♥ ❝•❡❛r•②✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ ♠♦st ❛♣♣r♦♣r✐❛t❡ ❝♦♥tr♦• ♠❡t❤♦❞ ❢♦r ❝♦♥tr♦••✐♥❣ t❤❡ q✉❛❞r♦t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❝❡♥❛r✐♦s✳ ✹✳✸✳✶ ▼♦❞❡• ❋r❡❡ ❈♦♥tr♦• ❚❤❡ r❡❝❡♥t•② ✐♥tr♦❞✉❝❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ♣r♦✈✐❞❡s ❛♥ ❡•❡❣❛♥t ❛♥❞ ❡✣❝✐❡♥t s♦•✉t✐♦♥ t♦ t❤❡ ♣r❡✈✐♦✉s ❝❤❛••❡♥❣❡s ❬✹✸✕✹✻(✳ ■t ✐s ❛ s✐♠♣•❡ ❜✉t ❡✣❝✐❡♥t t❡❝❤♥✐q✉❡ ❢♦r ♥♦♥•✐♥❡❛r✱ ✉♥❦♥♦✇♥ ♦r ♣❛rt✐❛••② ❦♥♦✇♥ ❞②♥❛♠✐❝s✳ ❚❤✐s ♠❡t❤♦❞ ✐s ❛•s♦ ❝❛••❡❞ ✐♥t❡••✐❣❡♥t P■❉ ✭♦r ✐✲P■❉✮✳ ❲❤✐•❡ r❡t❛✐♥✐♥❣ t❤❡ P■❉ r❡❞✉❝❡❞ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦st✱ ✐t ✐s ❛❜•❡ t♦ ❝♦♣❡ ✇✐t❤ ❣❡♥❡r❛• t②♣❡s ♦❢ ♥♦♥✲ •✐♥❡❛r✐t✐❡s✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ✐♥t❡••✐❣❡♥t P■❉ ❛♥❞ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦••❡rs ✐s ❣✐✈❡♥ ✐♥ ❞✬❆♥❞ré❛✲◆♦✈❡• ❡t ❛•✳ ❬✶✵(✳ ❚❤❡ ♣❛♣❡r ❡♠♣❤❛s✐③❡s t❤❡ ❡❛s② t✉♥✐♥❣ ♦❢ ✐✲P■❉ ❣❛✐♥s ❛♥❞ ❣✐✈❡s ❛ ❝•❡❛r✲❝✉t ❡①♣•❛♥❛t✐♦♥ ♦❢ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ✉s✉❛• P■❉s✳ ❆ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ❛ ❤✐❣❤✲♦r❞❡r s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦••❡rs ❛♥❞ ✐♥t❡••✐❣❡♥t P■❉ ❝♦♥tr♦••❡rs ✐s ♣r♦♣♦s❡❞ ❜② |✐❛❝❤② ❡t ❛•✳ ✐♥ ✷✵✶✶ ❬✾✵(✳ ▼♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s ❜❡❡♥ ✐♠♣•❡♠❡♥t❡❞ ✐♥ s♦♠❡ ❛❝❛❞❡♠✐❝ ❙■❙❖ s②st❡♠s ❬✹✸(✱ •♦✐♥✲ t ♠♦t✐♦♥ ❝♦♥tr♦• ✐♥ ❤✉♠❛♥♦✐❞ •♦❝♦♠♦t✐♦♥ ❬✾✻(✱ ❉❈ ♠♦t♦rs ✐♥ ✢❡①✐❜•❡ •♦✐♥ts ❬✾✼(✱ ♥♦♥✲ ♠✐♥✐♠✉♠ ♣❤❛s❡ s②st❡♠s ❬✶✶(✱ ❞❝✴❞❝ ❝♦♥✈❡rt❡rs ❬✼✸(✱ ❤②❞r♦❡•❡❝tr✐❝ ♣♦✇❡r ♣•❛♥ts ❬✺✻(✱ s♠❛•• ♣r♦❣r❛♠♠❛❜•❡ ❞❡✈✐❝❡s ❬✺✼(✱ ❛ q✉❛❞r♦t♦r ❬✶✵✵✱ ✶✵✷(✱ ❡t❝✳ ✻✻ ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ❆ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ❙■❙❖ s②st❡♠ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ✐♠♣•✐❝✐t•② ❛s E(y, ẏ, . . . , y (ι) , u, u, . . . , u (κ) ) = 0, ✭✹✳✶✸✮ ✇❤❡r❡ E : R ι+1 × R κ+1 → R ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥✳ ❆ss✉♠❡ t❤❛t ❢♦r ✐♥t❡❣❡r ν✱ 0 < ν ι✱ ∂E/∂y (ν) ≡ 0✳ ❚❤❡ ✐♠♣•✐❝✐t ❢✉♥❝t✐♦♥ t❤❡♦r❡♠ ❬✻✹( ❛••♦✇s t♦ ❡①♣r❡ss y (ν) •♦❝❛••②✿ y (ν) = E(t, y, ẏ, . . . , y (ν-1) , y (ν+1) , . . . , y (ι) , u, u, . . . , u (κ) ), ✭✹✳✶✹✮ ■♥ ❛ ✈❡r② s❤♦rt t✐♠❡ ✐♥t❡r✈❛•✱ t❤❡ s②st❡♠ ✭✹✳✶✹✮ ❝❛♥ ❜❡ ♠♦❞❡•❡❞ ❛s✿ y (ν) = αu + F, ✭✹✳✶✺✮ ❚❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ✐♥♣✉t α ∈ R ❝❛♥ ❜❡ ❝♦♥st❛♥t ♦r t✐♠❡✲✈❛r②✐♥❣✱ ✇❤✐•❡ ✐t ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛ ❝♦♥st❛♥t ✐♥ ❡❛❝❤ t✐♠❡ ✐♥t❡r✈❛•✳ ■ts ✈❛•✉❡ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❦♥♦✇•❡❞❣❡ ♦❢ ♠♦❞❡•s ♦r s②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥ ♠❡t❤♦❞s✳ ■❢ α ✐s ❝❤♦s❡♥ ❛s ❛ ❝♦♥st❛♥t✱ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞ ✐s ❛ ♣✉r❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤♦✉t ❛♥② ❦♥♦✇•❡❞❣❡ ♦❢ t❤❡ s②st❡♠✳ ■❢ α ✐s t✐♠❡✲✈❛r②✐♥❣✱ t❤❡ ♠❡t❤♦❞ ✐s ♣❛rt✐❛• ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ s♦♠❡ s②st❡♠ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡ t✐♠❡✲✈❛r②✐♥❣ ♣❛r❛♠❡t❡r F r❡♣r❡s❡♥ts t❤❡ r❡st ♦❢ t❤❡ ❞②♥❛♠✐❝s✱ ❛♥❞ ✐t ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛ ❝♦♥st❛♥t ✐♥ ❡❛❝❤ t✐♠❡ ✐♥t❡r✈❛•✳ ❚❤❡ ✈❛•✉❡ F ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② t❤❡ r❡❛•✲t✐♠❡ ✈❛•✉❡ ♦❢ y (ν) ❛♥❞ u ✐♥ t❤❡ ♣r❡✈✐♦✉s t✐♠❡ ✐♥t❡r✈❛•✿ F = ŷ(ν) -αû, ✭✹✳✶✻✮ ✇❤❡r❡ ŷ(ν) ✐s t❤❡ ν t❤ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♠❡❛s✉r❡ y ✐♥ t❤❡ ♣r❡✈✐♦✉s t✐♠❡ ✐♥t❡r✈❛•✱ ❛♥❞ û ✐s t❤❡ ❝♦♥tr♦• ✐♥♣✉t u ✐♥ t❤❡ ♣r❡✈✐♦✉s t✐♠❡ ✐♥t❡r✈❛•✳ ❆ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦••❡r ✐s ❞❡✜♥❡❞ ❛s✿ u = 1 α y (ν) r -F + Λ(e) , ✭✹✳✶✼✮ ✇❤❡r❡ y r ✐s ❛ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r②✱ ❛♥❞ e = y r -y ✐s t❤❡ tr❛❝❦✐♥❣ ❡rr♦r✳ Λ(e) ✐s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❝•♦s❡❞ •♦♦♣ ❡rr♦r ❞②♥❛♠✐❝s e (ν) = Λ(e) ❛s②♠♣t♦t✐❝❛••② st❛❜•❡✳ ❍❡r❡ ✇❡ ❝❤♦♦s❡ t❤❡ ❢✉♥❝t✐♦♥ Λ(e) ❛s✿ Λ(e) = K v-1 e (v-1) + K v-2 e (v-2) + ... + K 1 ė + K 0 e✱ ✇❤❡r❡ K i ✱ i = 0, ..., v -1 ❛r❡ t✉♥✐♥❣ ❣❛✐♥s✳ ✻✼ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈♦♠❜✐♥✐♥❣ ❊q✉❛t✐♦♥s ✭✹✳✶✻✮ ❛♥❞ ✭✹✳✶✼✮✱ ❛ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦••❡r ✐s ♦❜t❛✐♥❡❞✿ u = û + 1 α y (ν) r -ŷ(ν) + K v-1 e (v-1) + K v-2 e (v-2) + ... + K 1 ė + K 0 e . ✭✹✳✶✽✮ ✭✶✮ ❆❝❝✉r❛❝② ❜❡t✇❡❡♥ t❤❡ ♠♦❞❡•❡❞ s②st❡♠ ✭✹✳✶✺✮ ❛♥❞ t❤❡ ♦r✐❣✐♥❛• s②st❡♠ ✭✹✳✶✹✮✿ ❆t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ♦♥❡ t✐♠❡ ✐♥t❡r✈❛• t 0 ✱ y = ỹ, u = ũ✳ ❚❤❡ s②st❡♠ ✭✹✳✶✹✮ ✐s ❡①♣r❡ss❡❞ ❛s✿ ỹ(ν) = E(ỹ, ũ), ✭✹✳✶✾✮ ❚❤❡ ♠♦❞❡•❡❞ s②st❡♠ ✭✹✳✶✺✮ ❛t t = t 0 ✐s ♣r❡❝✐s❡✿ ỹ(ν) = αũ + F , ✭✹✳✷✵✮ ❆t t❤❡ ❡♥❞ ♦❢ t❤✐s t✐♠❡ ✐♥t❡r✈❛• t 0 + ∆t✱ t❤❡ s②st❡♠ ❜❡❝♦♠❡s y = ỹ + ∆y, u = ũ + ∆u, ✭✹✳✷✶✮ ❚❤❡r❡❢♦r❡✱ t❤❡ ♦✉t♣✉t y ❝❛♥ ❜❡ ❡①♣❛♥❞❡❞ ✐♥t♦ ❛ ❚❛②•♦r s❡r✐❡s ❛❜♦✉t (ỹ, ũ)✿ y (ν) =E(ỹ + ∆y, ũ + ∆u) =E(ỹ, ũ) + ∂E ∂y (ỹ, ũ)∆y + ∂E ∂u (ỹ, ũ)∆u ✭✹✳✷✷✮ + 1 2 ∂ 2 E ∂ 2 y (ỹ, ũ)∆y 2 + ∂ 2 E ∂u∂y (ỹ, ũ)∆u∆y + ∂ 2 E ∂ 2 u (ỹ, ũ)∆u 2 + ..., ❆♥❞ t❤❡ ♠♦❞❡•❡❞ s②st❡♠✬s ♦✉t♣✉t ✐s✿ y (ν) appr = α(ũ + ∆u) + F = E(ỹ, ũ) + α∆u = E(ỹ, ũ) + ∂E ∂u (ỹ, ũ)∆u, ✭✹✳✷✸✮ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❝❝✉r❛❝② ❡rr♦r ✐♥ ❡❛❝❤ t✐♠❡ ✐♥t❡r✈❛• ✐s✿ e appr = y (ν) -y (ν) appr ✭✹✳✷✹✮ = ∂E ∂y (ỹ, ũ)∆y + 1 2 ∂ 2 E ∂ 2 y(ỹ, ũ) ∆y 2 + ∂ 2 E ∂u∂y (ỹ, ũ)∆u∆y + ∂ 2 E ∂ 2 u (ỹ, ũ)∆u 2 + ..., ❚❤❡ ❛❝❝✉r❛❝② ❡rr♦r ✐s t❤❡ ❤✐❣❤ ♦r❞❡r ♣❛rts ♦❢ t❤❡ ❚❛②•♦r s❡r✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ❆s t❤❡ s❤♦rt t✐♠❡ ✐♥t❡r✈❛• ✐s ❞❡✜♥❡❞ r❡•❛t✐✈❡•② s♠❛•• ✇✐t❤ r❡s♣❡❝t t♦ s②st❡♠ ❞②♥❛♠✐❝s✱ ✇❡ ❝♦♥s✐❞❡r t❤✐s ❡rr♦r ✐s ♥❡❣•✐❣✐❜•❡✳ ✻✽ ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✭✷✮ ❙t❛❜✐•✐t② ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✿ ❈♦♠❜✐♥✐♥❣ ❊q✉❛t✐♦♥s ✭✹✳✶✺✮ ❛♥❞ ✭✹✳✶✼✮✱ t❤❡ ❡rr♦r ❞②♥❛♠✐❝s ♦❢ t❤❡ s②st❡♠ ✐s✿ y (ν) = y (ν) r + F -F + K v-1 e (v-1) + K v-2 e (v-2) + ... + K 1 ė + K 0 e, ✭✹✳✷✺✮ e (ν) + K v-1 e (v-1) + K v-2 e (v-2) + ... + K 1 ė + K 0 e + e F = 0 ✭✹✳✷✻✮ ✇❤❡r❡ e F = F -F ∈ R ✐s t✐♠❡✲✈❛r②✐♥❣✳ ■♥ ❡❛❝❤ s❤♦rt t✐♠❡ ✐♥t❡r✈❛• h✱ e F = y (ν) -ŷ(ν) -α(u -û) dŷ (ν) dt h + dû dt h . ✭✹✳✷✼✮ ■♥ ♣❤②s✐❝❛• s②st❡♠s✱ dŷ (ν) dt ❛♥❞ dû dt ❛r❡ ✉s✉❛••② ❜♦✉♥❞❡❞ ❛♥❞ ♥♦t ❡q✉❛• t♦ ✐♥✜♥✐t❡✱ dŷ (ν) dt < ǫ y (ν) ✱ dû dt < ǫ u ✱ ✇❤❡r❡ t❤❡ ♠❡❛s✉r❡ ŷ(ν) ✐s s✉♣♣♦s❡❞ ♥♦✐s❡ ❢r❡❡ ♦r ♣r❡✈✐♦✉s•② ✜•t❡r❡❞✳ ■❢ ❛ r❛t❤❡r s♠❛•• t✐♠❡ ✐♥t❡r✈❛• h ✇✐t❤ r❡s♣❡❝t t♦ s②st❡♠ ❞②♥❛♠✐❝s ✐s s❡•❡❝t❡❞✱ e F ✇✐•• ❜❡ ❜♦✉♥❞❡❞ ✐♥ ❛ s♠❛•• •✐♠✐t |e F | < ǫ, ǫ > 0✳ ❚❤✉s✱ ❜② ❝❤♦♦s✐♥❣ ❛♣♣r♦♣r✐❛t❡ ❣❛✐♥s K i ✱ t❤❡ tr❛❝❦✐♥❣ ❡rr♦r e ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✷✻✮ ✇✐•• ❝♦♥✈❡r❣❡ t♦ t❤✐s s♠❛•• •✐♠✐t ǫ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ✐s ♣r❛❝t✐❝❛••② st❛❜•❡✳ ❚♦ ❢✉rt❤❡r ♣r♦✈❡ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞✱ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❛ s♣❡✲ ❝✐✜❝ s②st❡♠ ✐s ♥❡❡❞❡❞✳ ✭✸✮ •❡♥❡r❛• s❡tt✐♥❣ ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ♦♥ t❤❡ ♠♦❞❡• ♦❢ q✉❛❞r♦t♦r✿ ❚❤❡ ❝♦♥tr♦• ♦r✐❡♥t❡❞ ♠♦❞❡• ✐s ♣r♦♣♦s❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✼✳ ❋✐rst•②✱ ✇❡ ❝♦♥tr♦• t❤❡ ❛•t✐t✉❞❡ ③✳ ❲❡ r❡✇r✐t❡ t❤❡ ✈❡rt✐❝❛• ❞②♥❛♠✐❝s ♦❢ t❤❡ q✉❛❞r♦t♦r ♠♦❞❡• ✐♥ ❊q✉❛t✐♦♥ ✭✷✳✹✷❞✮ ❛s✿ mz = (cos θ cos φ)u 1 + F z . ✭✹✳✷✽✮ ✇❤❡r❡ cos θ cos φ ✐s t❤❡ s②st❡♠ ✐♥❢♦r♠❛t✐♦♥ ✉s❡❞ ✐♥ ♣❛rt✐❛• ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❚❤❡ ✈❛•✉❡ F z ❞♦❡s ♥♦t ♦♥•② r❡♣r❡s❡♥t -mg✱ ❜✉t ❛•s♦ ✐♥❝•✉❞❡s ♥❡❣•❡❝t❡❞ ✈❡rt✐❝❛• ❞②♥❛♠✐❝s✳ ■♥ ❞✐s❝r❡t❡ t✐♠❡✱ t❤❡ ✉♥❦♥♦✇♥ ♣❛rt F z ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❢♦••♦✇✐♥❣✱ ✇❤❡r❡ ẑ(k) ✐s ❛♥ ❡st✐♠❛t❡ ♦❢ z(k)✿ Fz = m ẑ(t k ) -(cos θ cos φ)u 1 (t k-1 ). ✭✹✳✷✾✮ ❚❤✉s✱ t❤❡ ❝❤♦s❡♥ ❝♦♥tr♦• •❛✇ ✐s✿ u 1 (t k ) = u 1 (t k-1 ) + m cos θ cos φ êz 2d (t k ) + k z 1 e z d (t k ) + k z 0 e z (t k ) , ✭✹✳✸✵✮ ✻✾ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ✇✐t❤ êz 2d (t k ) = zr (t k ) -ẑ(t k ), e z d (t k ) = żr (t k ) -ẑ(t k ), e z (t k ) = z r (t k ) -z(t k ), ẑ(t k ) = T f T f + h ẑ(t k-1 ) + 1 T f + h ż(t k ) -ż(t k-1 ) , ẑ(t k ) = T f T f + h ẑ(t k-1 ) + 1 T f + h z(t k ) -z(t k-1 ) , ✭✹✳✸✶✮ ✇❤❡r❡ zr ✱ żr ✱ z r ❛r❡ t❤❡ r❡❢❡r❡♥❝❡ ❛❝❝❡•❡r❛t✐♦♥✱ ✈❡•♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥ ♦❢ z✳ ❚❤❡ ✈❛r✐❛❜•❡ h ✐s t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞✱ h = t k -t k-1 ✳ ❚❤❡♥ ✇❡ ❝♦♥tr♦• t❤❡ ♣♦s✐t✐♦♥ x ❛♥❞ y✳ ❆s t❤❡ ✐♥♣✉t u 1 ✐s ❛•r❡❛❞② ✉s❡❞ ✐♥ t❤❡ ❝♦♥tr♦• ♦❢ t❤❡ ❛•t✐t✉❞❡ z✱ ✇❡ ♥♦✇ ✉s❡ u 2 ❛♥❞ u 3 t♦ ❝♦♥tr♦• t❤❡ ♣♦s✐t✐♦♥s x ❛♥❞ y✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♥❡❡❞ t♦ ❞✐✛❡r❡♥t✐❛t❡ t✇✐❝❡ t❤❡ ❡q✉❛t✐♦♥s r❡•❛t❡❞ t♦ x ❛♥❞ y ✐♥ ❊q✉❛t✐♦♥s ✭✷✳✹✷❡✮ ❛♥❞ ✭✷✳✹✷❢✮ ✐♥ ♦r❞❡r t♦ ❣❡t t❤❡ ❝♦♥tr♦• ✐♥♣✉ts u 2 ❛♥❞ u 3 ✳ ❚❤❡♥✱ ✇❡ ♦❜t❛✐♥✿ x (4) = u 1 mI xx (sin ψ cos φ -cos ψ sin θ sin φ)u 2 + u 1 mI yy (cos ψ cos θ cos φ)u 3 + F x , y (4) = - u 1 mI xx (cos ψ cos φ + sin ψ sin θ sin φ)u 2 + u 1 mI yy (sin ψ cos θ cos φ)u 3 + F y , ✭✹✳✸✷✮ ✇❤❡r❡ t❤❡ ♣❛rt✐❛• ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ✉s❡❞ ❛♥❞ F x , F y ❛r❡ t❤❡ r❡♠❛✐♥✐♥❣ ♣❛rts ♦❢ t❤❡ ❤♦r✐✲ ③♦♥t❛• ❛♥❞ •❛t❡r❛• s②st❡♠✳ ❚♦ s✐♠♣•✐❢② t❤❡ ♥♦t❛t✐♦♥s✱ ✇❡ ❞❡✜♥❡ A = u 1 mI xx (sin ψ cos φ -cos ψ sin θ sin φ), B = u 1 mI yy (cos ψ cos θ cos φ), C = - u 1 mI xx (cos ψ cos φ + sin ψ sin θ sin φ), D = u 1 mI yy (sin ψ cos θ cos φ) ✭✹✳✸✸✮ ❲❡ ✐♠♣•❡♠❡♥t t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• s❝❤❡♠❡ ✐♥ ❛ s✐♠✐•❛r ♠❛♥♥❡r ❛s ♣r❡✈✐♦✉s✿ u 2 (t k ) u 3 (t k ) = u 2 (t k-1 ) u 3 (t k-1 ) + A B C D -1 êx 4d + k x 3 e x 3d + k x 2 e x 2d +
r ❛♥❞ t❤❡ ❡st✐♠❛t❡s ♦❢ x (4) , y (4)

✳ ✼✵ ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ❋♦r t❤❡ ②❛✇ ❛♥❣•❡ ❝♦♥tr♦•✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ r❡•❛t❡❞ ψ ❛s✿ I zz ψ = u 4 + F ψ . ✭✹✳✸✺✮ ❚❤❡♥ t❤❡ ❝♦♥tr♦• •❛✇ ✐s✿ u 4 (t k ) = u 4 (t k-1 ) + I zz êψ 2d (t k ) + k ψ 1 e ψ d (t k ) + k ψ 0 e ψ (t k ) , ✭✹✳✸✻✮ ✇❤❡r❡ êψ 2d ✐s t❤❡ ❡rr♦r ❜❡t✇❡❡♥ t❤❡ r❡❢❡r❡♥❝❡ ψr ❛♥❞ t❤❡ ❡st✐♠❛t❡ ♦❢ ψ✳ ✹✳✸✳✷ ❇❛❝❦st❡♣♣✐♥❣ ❈♦♥tr♦• ❋♦r t❤❡ ♣✉r♣♦s❡ ♦❢ ❝♦♠♣❛r✐s♦♥✱ ❛ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s ❛•s♦ ❛♣♣•✐❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ ❲❡ ❝❤♦♦s❡ t❤❡ ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛••❛❤ ❬✷✵(✳ ■♥ t❤❡ ❝♦♥tr♦• s②st❡♠✱ ✇❡ ♥❡❣•❡❝t s♦♠❡ ♣❛rts ✐♥ t❤❡ ♠♦❞❡• ❊q✉❛t✐♦♥s ✭✸✳✷✮ t♦ s✐♠♣•✐❢② t❤❡ ❝♦♥tr♦• •❛✇s✱ s✉❝❤ ❛s t❤❡ ❤✉❜ ❢♦r❝❡s✳ ❚❤❡ ❝♦♥tr♦• ♠♦❞❡• ✐s ❛s ❢♦••♦✇s✿

✭✹✳✸✼✮

◆♦t❡ t❤❛t t❤❡ ❛♥❣✉•❛r s✉❜s②st❡♠ ✐s ✐♥ ❝❛s❝❛❞❡ ✇✐t❤ t❤❡ ♣♦s✐t✐♦♥ s✉❜s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ✐♥t♦ t✇♦ s✉❜s②st❡♠s✿ t❤❡ ❛♥❣✉•❛r ❛♥❞ ♣♦s✐t✐♦♥ s✉❜s②st❡♠s✳ ❲❡ ✜rst•② ❝♦♥tr♦• t❤❡ ❛♥❣✉•❛r s✉❜s②st❡♠✱ ❛♥❞ t❤❡♥ ✉s❡ t❤❡ ❛♥❣•❡s t♦ ❝♦♥tr♦• t❤❡ ♣♦s✐t✐♦♥ s✉❜✲ s②st❡♠✳ ❲❡ r❡✇r✐t❡ t❤❡ s②st❡♠ ✐♥t♦ t❤❡ st❛t❡ s♣❛❝❡ ❢♦r♠ ✉s✐♥❣ t❤❡ st❛t❡ ✈❡❝t♦r (x 1 , . . . , x 12 ) ✇✐t❤ ✭✹✳✸✽✮ ✼✶ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s✿ ẋ1 = x 2 , ẋ2 = a 1 x 4 x 6 + a 2 x 4 Ω + b 1 u 2 , ẋ3 = x 4 , ẋ4 = a 3 x 2 x 6 + a 4 x 2 Ω + b 2 u 3 , ẋ5 = x 6 , ẋ6 = a 5 x 2 x 4 + b 3 u 4 , ẋ7 = x 8 , ẋ8 = -g + cos x 1 cos x 3 m u 1 , ẋ9 = x 10 , ẋ10 = u x m u 1 , ẋ11 = x 12 , ẋ12 = u y m u 1 ,
✭✹✳✸✾✮ ✇❤❡r❡ u x = sin ψ sin φ + cos ψ sin θ cos φ✱ u y = -cos ψ sin φ + sin ψ sin θ cos φ✱

a 1 = I yy -I zz I xx , a 2 = J r I xx , a 3 = I zz -I xx I yy , a 4 = - J r I yy , a 5 = I xx -I yy I zz , b 1 = 1 I xx , b 2 = 1 I yy , b 3 = 1 I zz . ✭✹✳✹✵✮ ❍❡r❡ ✇❡ ❞❡✜♥❡ ❛ ♥❡✇ st❛t❡ z 1 ✇❤✐❝❤ ❡q✉❛•s t❤❡ tr❛❝❦✐♥❣ ❡rr♦r ♦❢ t❤❡ st❛t❡ x 1 ✿ z 1 = x 1d -x 1 , ✭✹✳✹✶✮ ✇❤❡r❡ x 1d ✐s t❤❡ r❡❢❡r❡♥❝❡ x ♣♦s✐t✐♦♥✳ ❯s✐♥❣ t❤❡ ▲②❛♣✉♥♦✈ t❤❡♦r❡♠✱ ✇❡ ❜✉✐•❞ t❤❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ ♦❢ z 1 ❛s ❢♦••♦✇✐♥❣✿ V (z 1 ) = 1 2 z 2 1 , V (z 1 ) = z 1 ( ẋ1d -x 2 ). ✭✹✳✹✷✮ ❚❤❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ ♦❢ z 1 s❤♦✉•❞ ❜❡ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ s❤♦✉•❞ ❜❡ ♥❡❣❛t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡✳ ❍❡r❡ ✇❡ ❞❡✜♥❡ ❛ ✈✐rt✉❛• ❝♦♥tr♦• r❡❢❡r❡♥❝❡ ❢♦r x 2 ✿ x 2 = ẋ1d + α 1 z 1 , α 1 > 0 ✭✹✳✹✸✮ ✇❤✐❝❤ ❡♥s✉r❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ ♦❢ z 1 ♥❡❣❛t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡✿ V (z 1 ) = z 1 ( ẋ1d -x 2 ) = -α 1 z 2 1 < 0. ✭✹✳✹✹✮ ✼✷ ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ❍❡r❡ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ♥❡✇ st❛t❡ z 2 ✿ z 2 = x 2 -ẋ1d -α 1 z 1 . ✭✹✳✹✺✮ ❲❡ ❜✉✐•❞ t❤❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ ❢♦r z 2 ❛s ❢♦••♦✇✐♥❣✿ V (z 1 , z 2 ) = 1 2 (z 2 1 + z 2 2 )
✭✹✳✹✻✮ ❚❤❡♥✱ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s✿

V (z 1 , z 2 ) = z 2 (a 1 x 4 x 6 + a 2 x 4 ω + b 1 u 2 ) -z 2 (ẍ 1d -α 1 (z 2 + α 1 z 1 )) -z 1 z 2 -α 1 z 2 1 . ✭✹✳✹✼✮ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥tr♦• ✐♥♣✉t u 2 ✐s ❞❡✜♥❡❞ ❛s ❢♦••♦✇✐♥❣ t♦ ❡♥s✉r❡ V (z 1 , z 2 ) < 0✿ u 2 = 1 b 1 z 1 -a 1 x 4 x 6 -a 2 x 4 ω -α 1 (z 2 + α 1 z 1 ) -α 2 z 2 , α 2 > 0. ✭✹✳✹✽✮ ■♥ t❤❡ ❛♥❣✉•❛r s✉❜s②st❡♠✱ t❤❡ ♦t❤❡r ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡✜♥❡❞ ✉s✐♥❣ t❤❡ s❛♠❡ ♠❡t❤♦❞✿ u 3 = 1 b 2 z 3 -a 3 x 2 x 6 -a 4 x 2 ω -α 3 (z 4 + α 3 z 3 ) -α 4 z 4 , u 4 = 1 b 3 z 5 -a 5 x 2 x 4 -α 5 (z 6 + α 5 z 5 ) -α 6 z 6 , z 3 = x 3d -x 3 , z 4 = x 4 -ẋ3d -α 3 z 3 , z 5 = x 5d -x 5 , z 6 = x 6 -ẋ5d -α 5 z 5 . ✭✹✳✹✾✮ ❆•• t❤❡ α i ✱ i = 1,...,12. ❛r❡ t❤❡ ❝♦♥tr♦• ❣❛✐♥s✳ ■♥ t❤❡ ♣♦s✐t✐♦♥ s✉❜s②st❡♠✱ t❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡✜♥❡❞ ❛s✿ u 1 = m cos x 1 cos x 3 z 7 + g -α 7 (z 8 + α 7 z 7 ) -α 8 z 8 , u x = m u 1 z 9 -α 9 (z 10 + α 9 z 9 ) -α 10 z 10 , u y = m u 1 z 11 -α 11 (z 12 + α 11 z 11 ) -α 12 z 12 , z 7 = x 7d -x 7 , z 8 = x 8 -ẋ7d -α 7 z 7 , z 9 = x 9d -x 9 , z 10 = x 10 -ẋ9d -α 9 z 9 , z 11 = x 11d -x 11 , z 12 = x 12 -ẋ11d -α 11 z 11 . ✭✹✳✺✵✮ ❋r♦♠ u x ❛♥❞ u y ✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ♥❡❡❞❡❞ ❛♥❣•❡s φ✱ θ ❛♥❞ ψ ❢♦r t❤❡ ♣♦s✐t✐♦♥ ❝♦♥tr♦•✱ ✼✸ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s t❤❡♥ ✉s❡ t❤❡♠ ❛s t❤❡ ❛♥❣✉•❛r r❡❢❡r❡♥❝❡s ✐♥ t❤❡ ❛♥❣✉•❛r s✉❜s②st❡♠ ❝♦♥tr♦•✳ ▼♦r❡ ❞❡t❛✐•s ♦❢ t❤❡ ❝♦♥tr♦• ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✷(✳ ✹✳✸✳✸ ❙•✐❞✐♥❣ ▼♦❞❡ ❈♦♥tr♦• ❋♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣❛r✐s♦♥✱ ❛ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ❛•s♦ ♣r♦♣♦s❡❞ ♦♥ t❤❡ s❛♠❡ s❝❡♥❛r✐♦✳ ❲❡ ❝❤♦♦s❡ t❤❡ ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛••❛❤ ❬✷✵(✳ ❚❤❡ ♠♦❞❡• ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✸✼✮ ✐s ❛•s♦ ✉s❡❞ ❢♦r t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛•• t❤❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❙✉❜s❡❝t✐♦♥ ✹✳✸✳✷ ❇❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❚❤❡ q✉❛❞r♦t♦r s②st❡♠ ✐s ❛•s♦ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ s✉❜s②st❡♠s✿ t❤❡ ❛♥❣✉•❛r ❛♥❞ ♣♦s✐t✐♦♥ s✉❜s②st❡♠s✳ ❚❤✐s s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ❜❛s❡❞ ♦♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ♣r♦♣♦s❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥✳ ■♥ t❤❡ ❝♦♥tr♦• ♦❢ t❤❡ ❛♥❣•❡ r♦•• φ✱ t❤❡ ♥❡✇ st❛t❡ z 1 ✐s ❞❡✜♥❡❞ ❛s ✐♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ■♥ ❊q✉❛t✐♦♥ ✭✹✳✹✸✮✱ t❤❡ s•✐❞✐♥❣ s✉r❢❛❝❡ ✐s ❝❤♦s❡♥✿ S φ = z 2 = x 2 -ẋ1d -α 1 z 1 , z 1 = x 1d -x 1 . Ṡφ = ẋ2 -ẍ1d -α 1 ż1 = a 1 x 4 x 6 + a 2 x 4 ω + b 1 u 2 -φd + α 1 (z 2 + α 1 z 1 ). ✭✹✳✺✶✮ ❚❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛ttr❛❝t✐✈❡ s✉r❢❛❝❡ s❤♦✉•❞ s❛t✐s❢② S Ṡ < 0✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❞❡✜♥❡ Ṡφ ❛s Ṡφ = -k 1 ss✐❣♥(S φ ) -k 2 S φ . ✭✹✳✺✷✮ ✇❤❡r❡ k 1 , k 2 > 0 ❛r❡ t❤❡ ❝♦♥tr♦• ❣❛✐♥s✳ ss✐❣♥✭✮ ✐s ❛♥ ❡①t❡♥❞❡❞ s✐❣♥ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐s s❤♦✇♥ ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✺✸✮ ❛♥❞ ❋✐❣✉r❡ ✹✳✸✿ ss✐❣♥(x) =            -1 ✇❤❡♥ x -1✱ 2(x + 1) 5 (x + 1) 5 + (1 -x) 5 -1 ✇❤❡♥ -1 < x < 1✱ 1 ✇❤❡♥ x 1✳ ✭✹✳✺✸✮ ❚❤❡r❡❢♦r❡✱ t❤❡ ♦t❤❡r s•✐❞✐♥❣ s✉r❢❛❝❡s ✐♥ t❤❡ ❛♥❣✉•❛r s✉❜s②st❡♠ ❛r❡ ❝❤♦s❡♥ ❛s✿ S θ = z 4 = x 4 -ẋ3d -α 3 z 3 , z 3 = x 3d -x 3 , S ψ = z 6 = x 6 -ẋ5d -α 5 z 5 , z 5 = x 5d -x 5 .

✭✹✳✺✹✮

u 3 = 1 b 2 -k 3 ss✐❣♥(S θ ) -k 4 S θ -a 3 x 2 x 6 -a 4 x 2 ω + θd -α 2 2 z 3 , u 4 = 1 b 3 -k 5 ss✐❣♥(S ψ ) -k 6 S ψ -a 5 x 2 x 4 + ψd -α 2 3 z 5 . ✭✹✳✺✺✮ ■♥ t❤❡ ♣♦s✐t✐♦♥ s✉❜s②st❡♠✱ t❤❡ s•✐❞✐♥❣ s✉r❢❛❝❡s ❛r❡ ❝❤♦s❡♥ ❛s✿ S x = z 8 = x 8 -ẋ7d -α 7 z 7 , z 7 = x 7d -x 7 , S y = z 10 = x 10 -ẋ9d -α 9 z 9 , z 9 = x 9d -x 9 , S z = z 12 = x 11 -ẋ11d -α 11 z 11 , z 11 = x 11d -x 11 , ✭✹✳✺✻✮ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡✜♥❡❞✿ u 1 = m cos x 1 cos x 3 -k 7 ss✐❣♥(S z ) -k 8 S z + g + zd -α 2 7 z 7 , u x = m u 1 -k 9 ss✐❣♥(S z ) -k 10 S z + ẍd -α 2 9 z 9 , u y = m u 1 -k 11 ss✐❣♥(S z ) -k 12 S z + ÿd -α 2 11 z 11 . ✭✹✳✺✼✮ ✹✳✸✳✹ P■❉ ❈♦♥tr♦• ■♥ t❤❡ P■❉ ❝♦♥tr♦•✱ t❤❡ q✉❛❞r♦t♦r s②st❡♠ ✐s ❛•s♦ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ s✉❜s②st❡♠s✿ t❤❡ ❛♥❣✉•❛r ❛♥❞ ♣♦s✐t✐♦♥ s✉❜s②st❡♠✳ ❆ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦• •❛✇s ❛r❡ ✐♠♣•❡♠❡♥t❡❞✿ ✼✺ ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s u 1 = K z p e z + K z d ėz + K z i I(e z ), e z = z -z r u x = K x p e x + K x d ėx + K x i I(e x
), e x = x -x r u y = K y p e y + K y d ėy + K y i I(e y ), e y = y -y r

u 2 = K φ p e φ + K φ d ėφ + K φ i I(e φ ), e φ = φ -φ r u 3 = K θ p e θ + K θ d ėθ + K θ i I(e θ ), e θ = θ -θ r u 4 = K ψ p e ψ + K ψ d ėψ + K ψ i I(e ψ ), e ψ = ψ -ψ r ✭✹✳✺✽✮ ◆♦ ♠♦❞❡• ✐♥❢♦r♠❛t✐♦♥ ✐s t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✐♥ t❤✐s ❝♦♥tr♦• •❛✇s✳ ✼✻ ❈❤❛♣t❡r ✺ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✸ ✇✐•• ❜❡ ✐♠♣•❡♠❡♥t❡❞ ✐♥ ❞✐✛❡r✲ ❡♥t s❝❡♥❛r✐♦s✿ ❛ ❜❛s✐❝ s❝❡♥❛r✐♦ ❛♥❞ t❤❡ s❝❡♥❛r✐♦s ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✱ s❡♥s♦r ♥♦✐s❡s ❛♥❞ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡s❡ s❝❡♥❛r✐♦s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ✐s ❛ sq✉❛r❡ ♣❛t❤ ✇✐t❤ ❛ •❡♥❣t❤ ♦❢ ✷♠✳ ❚❤❡ ✸❉ tr❛•❡❝t♦r② ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✹✳✶✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ✐s ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✶✮✳ ❚❤❡ t♦t❛• s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s✳ ❆ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ♦❢ ✶✵♠s ✐s s❡•❡❝t❡❞ ✐♥ s✐♠✉•❛t✐♦♥✳ ❚❤❡ ❝♦♠♠❡r❝✐❛•✐③❡❞ q✉❛❞r♦t♦r ❆s❝❚❡❝ ❍✉♠♠✐♥❣❜✐r❞ ✐s ❞❡s✐❣♥❡❞ ❢♦r s❝✐❡♥t✐✜❝ r❡s❡❛r❝❤❡s ✭s❡❡ ❋✐❣✉r❡ ✶✳✾ ✐♥ ❈❤❛♣t❡r ✶✮✳ ❚❤❡ s②st❡♠ ♣r♦♣♦s❡s ❛ ✉s❡r ❢✉♥❝t✐♦♥ ✇❤✐❝❤ r✉♥s ❛t ✶❦❍③ ❬✹✽(✳ ■♥ t❤✐s ❢✉♥❝t✐♦♥✱ t❤❡ ✉s❡r ❝❛♥ ♣r♦❝❡ss t❤❡ s❡♥s♦r ❞❛t❛✱ ❝♦♠♣✉t❡ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ❛♥❞ ❡①❡❝✉t❡ ❢✉rt❤❡r ❛♣♣•✐❝❛t✐♦♥ ♣r♦❣r❛♠s✳ ❚❤❡ ❜❛s✐❝ ❝♦♥tr♦• st❛❜✐•✐③❛t✐♦♥ ♣r♦❣r❛♠ s❤♦✉•❞ ♥♦t ♦❝❝✉♣② t♦♦ ♠✉❝❤ ♣r♦❝❡ss✐♥❣ t✐♠❡ ✐♥ ♦r❞❡r t♦ ❛••♦✇ ♦t❤❡r ♣r♦❣r❛♠s ❤❛✈❡ ♠♦r❡ ♣r♦❝❡ss✐♥❣ r❡s♦✉r❝❡s✳ ❇❛s❡❞ ♦♥ t❤❡ r❡❛• s②st❡♠✱ ❛ ♣r♦❝❡ss✐♥❣ ❢r❡q✉❡♥❝② ♦❢ ✶✵✵❍③ ❢♦r t❤❡ ❝♦♥tr♦• t❛s❦ ❛♣♣❡❛rs t♦ ❜❡ r❡❛s♦♥❛❜•❡✳ ❚❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ♦❢ ❛ ❜❛s✐❝ ❝♦♥tr♦• s❤♦✉•❞ ♥♦t ❜❡ t♦♦ •♦♥❣ ❢♦r t❤❡ s❛❦❡ ♦❢ t❤❡ st❛❜✐•✐t②✳ ❚❤❡ •✐♥❡❛r ❛❝❝❡•❡r❛t✐♦♥ ♦❢ ❛ q✉❛❞r♦t♦r ✐s ✉s✉❛••② ✐♥❢❡r✐♦r t♦ g ✭g ❂ ✾✳✽♠✴s✮✳ ■♥ ♦♥❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ♦❢ ✶✵♠s✱ t❤❡ ♠❛①✐♠✉♠ ❝❤❛♥❣❡s ♦❢ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥ ❛r❡ ✵✳✵✾✽♠✴s ❛♥❞ ✵✳✹✾♠♠ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡r❡❢♦r❡✱ ❛ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ♦❢ ✶✵♠s ✐s s❡•❡❝t❡❞ ✇❤✐❝❤ ✐s r❛t❤❡r s♠❛•• ✇✐t❤ r❡s♣❡❝t t♦ s②st❡♠ ❞②♥❛♠✐❝s ❛♥❞ ❛t t❤❡ s❛♠❡ t✐♠❡ ❛❝❤✐❡✈❛❜•❡ ✐♥ t❤❡ r❡❛• ❝♦♥tr♦•✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ s❛♠♣•❡❞ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ ❛♥❞ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✐s s❡t t♦ ❜❡ ✶✵♠s✳ ❚❤❡r❡ ❛r❡ ✶✺✵✵✵ ❛❝t✉❛t✐♦♥ st❡♣s✳ ❊✈❡r② ✶✵♠s✱ t❤❡ s②st❡♠ r❡❛❞s t❤❡ ❞❛t❛ ❢r♦♠ s❡♥s♦rs✱ ❝♦♠♣✉t❡s ♥❡✇ ❝♦♥tr♦• s✐❣♥❛•s ❛♥❞ s❡♥❞s t❤❡♠ t♦ ❛❝t✉❛t♦rs✳ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ✺✳✶✳✶ ▼♦❞❡• ❋r❡❡ ❈♦♥tr♦• ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✐♠♣•❡♠❡♥t t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐♥ s❡✈❡r❛• ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✳ ❚❤❡ ❝♦♥tr♦• •❛✇ ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✸ ♦♥ P❛❣❡ ✼✸✳ ❚❤❡ t♦t❛• s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s ❛♥❞ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✐s ✶✵♠s✳ ❚❤❡r❡ ❛r❡ ✶✺✵✵✵ ❛❝t✉❛t✐♦♥ st❡♣s✳ ✶✳ ❇❛s✐❝ s❝❡♥❛r✐♦ ❚❤✐s ♠❡t❤♦❞ ✐s ✜rst•② s✐♠✉•❛t❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❝❤♦s❡♥ ❝♦♥✲ tr♦• ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡ ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✳ ❚❤❡ ✜rst ✜❣✉r❡ ✐s t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s✳ ❚❤❡ s❡❝♦♥❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣•❡s φ✱ θ ❛♥❞ ψ✳ ❚❤❡ t❤✐r❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s ♦❢ t❤❡ ❢♦✉r r♦t♦rs✳ ❚❤❡ ♥✉♠❜❡rs ♦❢ t❤❡ r♦t♦r ❛r❡ ❞❡✜♥❡❞ ✐♥ ❋✐❣✉r❡ ✷✳✶ ♦♥ P❛❣❡ ✷✻✳ ❚❛❜•❡ ✺✳✶✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❣❛✐♥s k x 3 ✺ k x 2 ✶✽ k x 1 ✶✺ k x 0 ✶✷ k z 1 ✷ k z 0 ✺ k y 3 ✺ k y 2 ✶✽ k y 1 ✶✺ k y 0 ✶✷ k ψ 1 ✺ k ψ 0 ✶✵ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ s②st❡♠ ✉s✐♥❣ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❦❡♣t t❤❡ st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ ✢✐❣❤t ❛♥❞ ❢♦••♦✇❡❞ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r②✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✹✷♠✱ ✇❤✐❝❤ ✐s ✷✳✶✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✳ ✷✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ✇✐♥❞ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶ ♦♥ P❛❣❡ ✻✺✳ ■♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥s✱ t❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ♣r❡✲s❡•❡❝t❡❞ ❛♥❞ ✉s✉❛••② ♥♦t ❝❤❛♥❣❡❞ ❛❝❝♦r❞✐♥❣ t♦ ✇❡❛t❤❡r ❝♦♥❞✐t✐♦♥s ❛♥❞ ❞✐st✉r❜❛♥❝❡s✳ ■♥ ♦r❞❡r t♦ s✐♠✉•❛t❡ t❤❡ r❡❛•✐st✐❝ ❝❛s❡✱ t❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞✱ ✇❤✐❝❤ ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✐♥ ❚❛❜•❡ ✺✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✳ ❚❤❡ ❝♦♥tr♦• s②st❡♠ s❤♦✇s ✐ts st❛❜✐•✐t② t♦ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ s②st❡♠ ❦❡♣t st❛❜•❡ ❞✉r✲ ✐♥❣ t❤❡ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✹✷♠✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ✇✐t❤ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❝♦♠♣❡♥s❛t❡❞ ✇❡•• t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ✸✳ P❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❚❤❡♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥✲ t✐❡s✳ ❚❤❡ ♣❛r❛♠
✹✳ ❙❡♥s♦r ♥♦✐s❡s ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ❚❤❡ s❡♥s♦r ♥♦✐s✲ ❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶ ♦♥ P❛❣❡ ✻✽✳ ❚❤❡ ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞ ❛s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✹✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✽♠✱ ✇❤✐•❡ ✵✳✵✹✷♠ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✺✳ ❆❝t✉❛t♦r ❢❛✉•ts ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶ ♦♥ P❛❣❡ ✻✾✳ ❚❤❡ ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞ ❛s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✺✳ ❚❤❡r❡ ✇❡r❡ t✇♦ ❛❝t✉❛t♦r ❢❛✉•ts✿ ♦♥❡ ✇❛s ❞✉r✐♥❣ t❤❡ ❤♦✈❡r✐♥❣ ❛♥❞ t❤❡ ♦t❤❡r ✐s ❞✉r✐♥❣ t❤❡ tr❛❝❦✐♥❣✳ ❚❤❡ s②st❡♠ ❦❡♣t st❛❜•❡ ❞✉r✐♥❣ t❤❡s❡ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✼✾ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❋✐❣✉r❡ ✺✳✷✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❋✐❣✉r❡ ✺✳✸✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ✽✵ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✹✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❋✐❣✉r❡ ✺✳✺✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ✽✶ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• r❡s♣♦♥❞s q✉✐❝❦•② t♦ t❤❡ s②st❡♠ ❢❛✉•t✱ ❛♥❞ s②st❡♠ ❛❞•✉sts t♦ ✐ts st❛❜•❡ st❛t❡s ✐♥ ✶s✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✼♠✱ ✇❤✐❝❤ ✐s ❤❛♣♣❡♥❡❞ ❛t t❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❞✉r✐♥❣ tr❛❝❦✐♥❣✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❞✉r✐♥❣ ❤♦✈❡r✐♥❣ ❝❛✉s❡❞ ✵✳✵✹♠ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❝❛✉s❡❞ t❤❡ q✉❛❞r♦t♦r t✐•t ❛❜♦✉t ✵✳✶ r❛❞ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣✳ ✺✳✶✳✷ ❇❛❝❦st❡♣♣✐♥❣ ❈♦♥tr♦• ■♥ t❤❡ s✉❜s❡❝t✐♦♥✱ ✇❡ ✐♠♣•❡♠❡♥t t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛•✲ •❛❤ ❬✷✵(✳ ❚❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✹✳✸✳✷ ♦♥ P❛❣❡ ✼✽✳ ❚❤❡ s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s✱ ❛♥❞ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✐s ✶✵♠s✳ ❚❤❡r❡ ❛r❡ t♦t❛• ✶✺✵✵✵ ❛❝t✉❛t✐♦♥ st❡♣s✳ ✶✳ ❇❛s✐❝ s❝❡♥❛r✐♦ ❚❤✐s ♠❡t❤♦❞s ✐s ✜rst•② s✐♠✉•❛t❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❝❤♦s❡♥ ❛s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐s✲ t✉r❜❛♥❝❡ ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✻✳ ❚❤❡ ✜rst ✜❣✉r❡ ✐s t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s✳ ❚❤❡ s❡❝♦♥❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣•❡s φ✱ θ ❛♥❞ ψ✳ ❚❤❡ t❤✐r❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s ♦❢ t❤❡ ❢♦✉r r♦t♦rs✳ ❚❤❡ ♥✉♠❜❡rs ♦❢ t❤❡ r♦t♦r ❛r❡ ❞❡✜♥❡❞ ✐♥ ❋✐❣✉r❡ ✷✳✶ ✐♥ ❈❤❛♣t❡r ✷✳ ❚❛❜•❡ ✺✳✷✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❣❛✐♥s α 1 ✹ α 2 ✶✵ α 3 ✹ α 4 ✶✵ α 5 ✶ α 6 ✺ α 7 ✺ α 8 ✸✵ α 9 ✷ α 10 ✸ α 11 ✷ α 12 ✸ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ s②st❡♠ ✉s✐♥❣ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❦❡♣t st❛❜•❡ ❛♥❞ ❢♦••♦✇❡❞ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r②✳ ❚❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✾♠✱ ✇❤✐❝❤ ✐s ✹✳✺✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✳ ■t ✐s ❜✐❣❣❡r t❤❛♥ ✵✳✵✹✷♠ ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ✷✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐s✲ t✉r❜❛♥❝❡✳ ❚❤❡ ✇✐♥❞ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✷✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✼✳ ❉✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ❤❛s ❜✐❣ tr❛❝❦✐♥❣ ❡rr♦rs✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✷✸♠✱ ✇❤✐❝❤ ✐s ✶✶✳✺✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✳ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ ♠❛①✐♠✉♠ ❡rr♦r ✐s ♦♥•② ✵✳✵✾♠✳ ■♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦
✹✳ ❙❡♥s♦r ♥♦✐s❡s ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ❚❤❡ s❡♥s♦r ♥♦✐s❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✷✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✾✳ ✺✳ ❆❝t✉❛t♦r ❢❛✉•ts ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✷✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✵✳ ❚✇♦ ❛❝t✉❛t♦r ❢❛✉•ts ❤❛♣♣❡♥❡❞ s❡♣❛r❛t❡•② ❞✉r✐♥❣ t❤❡ tr❛❝❦✐♥❣ ❛♥❞ ❤♦✈❡r✐♥❣✳ ■♥ ❜♦t❤ ❝❛s❡s✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❦❡♣t ✐ts st❛❜✐•✐t②✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉t ❞✉r✐♥❣ t❤❡ ♠♦✈✐♥❣ ❝❛✉s❡❞ ✵✳✶✺♠ ✽✸ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❋✐❣✉r❡ ✺✳✼✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❋✐❣✉r❡ ✺✳✽✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ✽✹ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✾✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❋✐❣✉r❡ ✺✳✶✵✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ✽✺ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r✱ ❝♦♠♣❛r✐♥❣ t♦ ✵✳✵✾♠ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❞✉r✐♥❣ ❤♦✈❡r✐♥❣ ❝❛✉s❡❞ ✵✳✵✻♠ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r✳ ❲❤✐•❡ ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ t❤❡ ❢❛✉•t ❝❛✉s❡❞ ✵✳✵✹♠ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ❞✉r✐♥❣ ❤♦✈❡r✐♥❣✳ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❝❛✉s❡❞ ❜✐❣ ✈✐❜r❛t✐♦♥s ❛❢t❡r t❤❡ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ❝❛✉s❡❞ t❤❡ q✉❛❞r♦t♦r t✐•t ❛❜♦✉t ✵✳✷✽r❛❞ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣ ✉s✐♥❣ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✱ ✇❤✐•❡ ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• t❤❡ t✐•t ❛♥❣•❡ ✐s ♦♥•② ✵✳✶ r❛❞✳ ❆❢t❡r t❤❡ ❢❛✉•ts✱ t❤❡ s②st❡♠ ❢♦✉♥❞ ✐ts st❛❜•❡ ♣♦s✐t✐♦♥ ✐♥ ❛❜♦✉t ✶✳✺s✱ ✇❤✐•❡ ✵✳✺s ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ✺✳✶✳✸ ❙•✐❞✐♥❣ ▼♦❞❡ ❈♦♥tr♦• ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✐♠♣•❡♠❡♥t t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛•✲ •❛❤ ❬✷✵(✳ ❚❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✸✳✸ ♦♥ P❛❣❡ ✽✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s✱ ❛♥❞ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✐s ✶✵♠s✳ ❚❤❡r❡ ❛r❡ t♦t❛• ✶✺✵✵✵ ❛❝t✉❛t✐♦♥ st❡♣s✳ ✶✳ ❇❛s✐❝ s❝❡♥❛r✐♦ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ✜rst•② ✐♠♣•❡♠❡♥t❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❝❤♦s❡♥ ❛s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡ ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✶✳ ❚❛❜•❡ ✺✳✸✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❣❛✐♥s k 1 ✷ k 2 ✶✵ α 1 ✻ k 3 ✷ k 4 ✶✵ α 3 ✻ k 5 ✶ k 6 ✷ α 5 ✷ k 7 ✺ k 8 ✶✺ α 7 ✽ k 9 ✶ k 10 ✹ α 9 ✸ k 11 ✶ k 12 ✹ α 11 ✸ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• s❤♦✇❡❞ ❛ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡✳ ❚❤❡ s②st❡♠ st❛②s st❛❜•❡ ❞✉r✐♥❣ t❤❡ t❛s❦✱ ❛♥❞ t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✼♠ ✇❤✐❝❤ ✐s ❛•♠♦st t❤❡ s❛♠❡ ✇✐t❤ ✵✳✵✾♠ ♦❢ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✱ ❜✐❣❣❡r t❤❛♥ ✵✳✵✹✷♠ ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ✷✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r✲ ❜❛♥❝❡✳ ❚❤❡ ✇✐♥❞ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❛•✇❛②s ❛s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✷✳ ❉✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ❤❛s ❜✐❣ tr❛❝❦✐♥❣ ❡rr♦rs✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✶✾♠✱ ✇❤✐❝❤ ✐s ✾✳✺✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✱ ✇❤✐•❡ ✐♥ t❤❡ ❜❛❝❦✲ ✽✻ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✶✶✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ st❡♣♣✐♥❣ ❝♦♥tr♦• t❤❡ ❡rr♦r ✐s ✵✳✷✸♠✳ ■♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ✐s ♦♥•② ✵✳✵✹✷♠✳ ✸✳ P❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ■♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✱ t❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✸✳ ❚❤❡ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✸✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✺✺♠✱ ✇❤✐•❡ ✵✳✵✽ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✹✳ ❙❡♥s♦r ♥♦✐s❡s ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ❚❤❡ s❡♥s♦r ♥♦✐s❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✹✳ ❚❤❡ s❡•❡❝t❡❞ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✸✳ ✺✳ ❆❝t✉❛t♦r ❢❛✉•ts ■♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✺✳ ❚❤❡ s❡•❡❝t❡❞ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✸✳ ✽✼ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❋✐❣✉r❡ ✺✳✶✷✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❋✐❣✉r❡ ✺✳✶✸✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ✽✽ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✶✹✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❋✐❣✉r❡ ✺✳✶✺✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ✽✾ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❚❤❡ ❢❛✉•t ❞✉r✐♥❣ t❤❡ ❤♦✈❡r✐♥❣ ❝❛✉s❡❞ ✵✳✷✶♠ tr❛❝❦✐♥❣ ❡rr♦r✳ ❉✉r✐♥❣ ❤♦✈❡r✐♥❣✱ t❤❡ s②st❡♠ ✇❛s ❜❛❝❦ t♦ ♥♦r♠❛• st❛❜•❡ st❛t❡ ✐♥ ❛❜♦✉t ✷s✱ ✇❤✐•❡ t❤❡ r❡s♣♦♥❞✐♥❣ t✐♠❡ ✐s ✶s ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❛♥❞ ✶✳✺s ✐♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❝❛✉s❡❞ t❤❡ q✉❛❞r♦t♦r t✐•t ❛❜♦✉t ✵✳✷✾r❛❞ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣✱ ✇❤✐•❡ ✵✳✶r❛❞ ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❛♥❞ ✵✳✷✽r❛❞ ✐♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ✺✳✶✳✹ ❉✐s❝✉ss✐♦♥ ■♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡s❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❡①♣•✐❝✐t•②✱ t❡♥ ✐♠♣♦rt❛♥t ❝r✐t❡r✐❛ ❛r❡ •✐st❡❞ ❛s ❢♦••♦✇s ❢♦r t❤❡ ❝♦♠♣❛r✐s♦♥✿ ✭✶✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥ t❤❡ ①✱ ② ❛♥❞ ③ ❛①✐s ✐♥ ❜❛s✐❝ s❝❡✲ ♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✿
3 i=1 t e 2 i ✳ ✭✻✮ ❆❞•✉st✐♥❣ t✐♠❡✿ t✳ ✭✼✮ ▼❛①✐♠✉♠ t✐•t ❛♥❣•❡✿ φ ♦r θ✳ ✭✽✮ ❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s✳ ✭✾✮ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t②✿ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♣❡r❛t✐♦♥s ✐♥ t❤❡ ❝♦♥tr♦• •❛✇✳ ✭✶✵✮ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✿ t❤❡ ❡♥❡r❣② s♣❡♥t ♦♥ ♠♦t✐♦♥ ❝♦♥tr♦•✱ t❤❡ s✉♠ ♦❢ t❤❡ ❢♦✉r ♠♦t♦rs✬ sq✉❛r❡❞ ❛♥❣✉•❛r s♣❡❡❞s✿ 4 i=1 t ω 2 i ✳ ■♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✱ t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛♥❞ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤r❡❡ ♠❡t❤♦❞s ✐♥ ✜✈❡ s❝❡♥❛r✐♦s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺✳✶✻✳ ❋r♦♠ t❤❡ ✜❣✉r❡✱ ✇❡ ❝❛♥ s❡❡ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s ❣❡♥❡r❛••② ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤❡s❡ ❝r✐t❡r✐❛✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤r❡❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✹✳ ❋r♦♠ t❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts✱ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❤❛✈❡ ❛•♠♦st t❤❡ s❛♠❡ ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❆•• t❤❡ ❝♦♥tr♦• s❤♦✇ ❛ ❣♦♦❞ st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ ✢✐❣❤t✱ ❛♥❞ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ •❡ss t❤❛♥ ✵✳✶♠✱ ✇❤✐❝❤ ✐s ✺✪ ♦❢ t❤❡ •❡♥❣t❤ ♦❢ ✷♠✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s s♠❛••❡st ✾✵ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✶✻✿ ❚✐♠❡ tr✐❣❣❡r❡❞ t❤r❡❡ ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✿ ✭❛✮ t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✳ ✭❜✮ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡✳ ✭❝✮ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥×10 -7 ✳ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✱ ❤♦✇❡✈❡r t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ ❡rr♦rs ✐s ♥♦t t❤❡ s♠❛••❡st✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤r❡❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✺✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ s❤♦✇s ✐ts ❛❞✈❛♥t❛❣❡ ❞✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥❝r❡❛s❡❞ ❜② ✻✪✳ ❚❤❡ ♠♦❞❡ ❢r❡❡ ❝♦♥tr♦• ❝♦♠♣❡♥s❛t❡s ✇❡•• t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤✐s ✐s ❞✉❡ t♦ t❤❡ r❡❛• t✐♠❡ ✉♣❞❛t❡❞ s②st❡♠ ❞②♥❛♠✐❝s ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❛t ❡❛❝❤ s❛♠♣•✐♥❣ ♣❡r✐♦❞✳ ❚❤❡ ✈❛•✉❡ F ✐s ❝♦♠♣✉t❡❞ ✉s✐♥❣ t❤❡ r❡❛• t✐♠❡ s②st❡♠ ♠❡❛s✉r❡♠❡♥t ŷ ❛♥❞ t❤❡ •❛st t✐♠❡ ❝♦♥tr♦• ✐♥♣✉t ũ✳ ❚❤❡ ✉♥❡①♣❡❝t❡❞ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ F ✐♥ t❤❡ r❡❛• t✐♠❡ ❝♦♥tr♦•✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❤❛s •✐tt•❡ ✐♥✢✉❡♥❝❡ ❢♦r t❤❡ ♠♦❞❡• ❢r❡❡ ✾✶ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❚❛❜•❡ ✺✳✹✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❜❛s✐❝ s❝❡♥❛r✐♦ ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✹✷ ✵✳✵✾ ✵✳✵✼ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✵✹✾ ✵✳✵✵✷✽ ✵✳✵✵✵✸ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✷✶ ✵✳✷✷ ✵✳✷✵✺ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✶✽❡✰✼ ✷✳✹✺✶✾❡✰✼ ✷✳✹✺✶✽❡✰✼ ❛❝t✉❛t✐♦♥ st❡♣s ✶✺✵✵✵ ✶✺✵✵✵ ✶✺✵✵✵ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❛❜•❡ ✺✳✺✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ✇✐♥❞ ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✹✷ ✵✳✷✸ ✵✳✶✾ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✵✺✷ ✷✳✹✼✷✸ ✵✳✽✹✼✻ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✹ ✵✳✹✶ ✵✳✹✶ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✸✺✾✽❡✰✼ ✷✳✸✻✵✵❡✰✼ ✷✳✸✺✾✽❡✰✼ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❝♦♥tr♦•✳ ❆s t❤❡ s❛♠❡ t✐♠❡✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❤❛✈❡ •❛r❣❡ tr❛❝❦✐♥❣ ❡rr♦rs ❞✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥❝r❡❛s❡❞ ❜② ✶✺✺✳✺✪ ❛♥❞ ✶✼✶✳✹✪ ✐♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ s✉♠s ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥❝r❡❛s❡❞ ✽✽✶✾✻✪ ❛♥❞ ✷✽✹✸✸✪✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✻✳ ❆•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s s❤♦✇ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤✐s s❝❡♥❛r✐♦✳ ❲❤❡♥ t❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ✉♥❞❡r❡st✐♠❛t❡❞ ❛♥❞ t❤❡ s②st❡♠ ✐s ✉♥❞❡r✲❛❝t✉❛t❡❞✱ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❦❡❡♣ t❤❡✐r st❛❜✐•✐t② ❛♥❞ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ r❛t❤❡r s♠❛••✳ ■♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ✐♥❝r❡❛s❡❞ ❜② ✼✽✪ ❛♥❞ ✸✸✸✪ r❡•❛t❡❞ t♦ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ■♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ✐♥❝r❡❛s❡❞ ❜② ✷✷✪ ❛♥❞ ✶✺✼✶✪✳ ■♥ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ✐♥❝r❡❛s❡❞ ❜② ✲✷✶✳✹✪ ❛♥❞ ✹✺✵✵✪✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✼✳ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❤❛s t❤❡ s♠❛••❡st ❡rr♦r ✈❛r✐❛♥❝❡✳ ❚❤❡ s❡♥s♦r ♥♦✐s❡s ❝❛✉s❡❞ ♦s❝✐••❛t✐♦♥s ✐♥ t❤❡ ♦t❤❡r t✇♦ ♠❡t❤♦❞s✱ ❡s♣❡❝✐❛••② ✐♥ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ■t ✐s ❞✉❡ t♦ t❤❡ s✇✐t❝❤✐♥❣ ❢❡❛t✉r❡ ♦❢ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ✾✷ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❚❛❜•❡ ✺✳✻✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡r✲ t❛✐♥t✐❡s✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✼✺ ✵✳✶✶ ✵✳✵✺✺ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✷✶✷ ✵✳✵✹✻✽ ✵✳✵✶✸✽ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✸✺ ✵✳✸✽ ✵✳✷✷ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✷✸❡✰✼ ✷✳✹✺✷✽❡✰✼ ✷✳✹✺✶✼❡✰✼ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❛❜•❡ ✺✳✼✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ s❡♥s♦r ♥♦✐s❡s ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✽ ✵✳✵✾ ✵✳✵✽ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✷✸✵✾ ✵✳✵✵✷✾ ✵✳✵✵✽✽ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✶✼❡✰✼ ✷✳✹✺✷✵❡✰✼ ✷✳✹✺✶✽❡✰✼ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✷✷ ✵✳✷✸ ✵✳✷✶ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❛❜•❡ ✺✳✽✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ❛❝t✉❛t♦r ❢❛✉•ts ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❡rr♦r ❝❛✉s❡❞ ❜② ❢❛✉•t ✵✳✵✹ ✵✳✵✻ ✵✳✷✶ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ❝❛✉s❡❞ ❜② ❢❛✉•t ✵✳✶ ✵✳✷✽ ✵✳✷✾ ❛❞•✉st✐♥❣ t✐♠❡ ❛❢t❡r t❤❡ ❢❛✉•t ✶ ✶✳✺ ✷✳✵ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✸✷✼ ✵✳✵✶✻✼ ✵✳✵✺✽✺ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✶✽❡✰✼ ✷✳✹✺✷✹❡✰✼ ✷✳✹✺✷✷❡✰✼ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✽✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s t❤❡ s♠❛••❡st ❛❞•✉st✐♥❣ t✐♠❡ ❛❢t❡r t❤❡ ❢❛✉•ts✳ ■t r❡s♣♦♥❞s q✉✐❝❦•② t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❞②♥❛♠✐❝s ❞✉❡ t♦ t❤❡ r❡❛• t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛•✉❡ F ✳ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❤❛s t❤❡ •❛r❣❡st ❛❞•✉st✐♥❣ t✐♠❡✱ ❛♥❞ t❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ❝❛✉s❡❞ ❛ ❜✐❣❣❡st ♦s❝✐••❛t✐♦♥ ✐♥ t❤❡ ❛❝t✉❛t♦rs ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ t✐•t ❛♥❣•❡s✳ ❚❤❡ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ♦❢ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❛r❡ ✵✳✷✽r❛❞ ❛♥❞ ✵✳✷✾r❛❞✱ ✇❤✐•❡ ✵✳✶ r❛❞ ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ✾✸ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ✺✳✷ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ◗✉❛❞r♦t♦rs ❤❛✈❡ ♠❛♥② ❛♣♣•✐❝❛t✐♦♥s✱ s✉❝❤ ❛s ❜♦r❞❡r ♣❛tr♦•✱ s✉r✈❡✐••❛♥❝❡✱ s❡❛r❝❤ ✫ r❡s❝✉❡✱ ❛❡r✐❛• ♣❤♦t♦❣r❛♣❤②✱ ❡t❝✳ ■♥ t❤❡s❡ ❛♣♣•✐❝❛t✐♦♥s✱ ❛ •♦♥❣ ✢✐❣❤t ❞✉r❛t✐♦♥ t✐♠❡ ✐s ♥❡❡❞❡❞ t♦ ✜♥✐s❤ t❤❡ t❛s❦s✳ ❚❤❡ ✢✐❣❤t ❞✉r❛t✐♦♥ t✐♠❡ ❞❡♣❡♥❞s ♦♥ t✇♦ ❢❛❝t♦rs✿ t❤❡ ❝❛♣❛❝✐t② ♦❢ t❤❡ ♦♥❜♦❛r❞ ❜❛tt❡r② ❛♥❞ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ ❚❤❡ ❝♦♠♠❡r❝✐❛•✐③❡❞ q✉❛❞r♦t♦rs ✉s✉❛••② ✉s❡ ▲✐t❤✐✉♠ P♦•②♠❡r ✭▲✐P♦✮ ❜❛tt❡r✐❡s✳ ❍♦✇❡✈❡r✱ ❛ ❜✐❣❣❡r ❜❛tt❡r② ❝❛♣❛❝✐t② ♠❡❛♥s ❛ ❤❡❛✈✐❡r •♦❛❞ ❛♥❞ ❛ ♠♦r❡ ❡①♣❡♥s✐✈❡ ♣r✐❝❡✳ ■♥ ♦r❞❡r t♦ ❡①t❡♥❞ t❤❡ ✢✐❣❤t ❞✉r❛t✐♦♥✱ ✇❡ ❝❛♥ ❛•s♦ r❡❞✉❝❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✳ ❆s ✐♥ t❤❡s❡ ❛♣♣•✐❝❛t✐♦♥s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ q✉❛❞r♦t♦rs ❤❛✈❡ ✉s✉❛••② ❛ s♠♦♦t❤ tr❛•❡❝t♦r② ❛♥❞ ✇✐t❤♦✉t ❛❣❣r❡ss✐✈❡ ♠♦✈❡♠❡♥t✳ ❲❤❡♥ q✉❛❞r♦t♦rs ❢♦••♦✇ ❛ tr❛•❡❝t♦r②✱ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ ♦❢t❡♥ ✐♥ ❛ •✐♠✐t❡❞ ❜♦✉♥❞✳ ❚❤❡r❡❢♦r❡✱ ❛♥ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• s❝❤❡♠❡ ✐s ♣r♦♣♦s❡❞ ❢♦r t❤❡s❡ ❛♣♣•✐❝❛t✐♦♥s✳ ▲✐♠✐t t♦•❡r❛♥❝❡s ❛r❡ s❡t ❢♦r ❝♦♥tr♦• s②st❡♠s✱ ❛♥❞ ❛ ❝♦♥tr♦• s②st❡♠ ✐s ❛❝t✉❛t❡❞ ♦♥•② ✇❤❡♥ t❤❡ •✐♠✐ts ❛r❡ ♣❛ss❡❞✳ ■♥ t❤✐s ✇❛②✱ ✉♥♥❡❝❡ss❛r② ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❛✈♦✐❞❡❞ ❛♥❞ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✐s r❡❞✉❝❡❞✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✸ ✇✐•• ❜❡ ✐♠♣•❡♠❡♥t❡❞ ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ❆s ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ❛•s♦ ✐♠♣•❡♠❡♥t❡❞ ✐♥ t❤❡ ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✿ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ s❝❡♥❛r✐♦s ✇✐t❤ ✇✐♥❞ ❞✐st✉r✲ ❜❛♥❝❡✱ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✱ s❡♥s♦r ♥♦✐s❡s ❛♥❞ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡s❡ s❝❡♥❛r✐♦s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ✐s ❛ sq✉❛r❡ ♣❛t❤ ✇✐t❤ ❛ •❡♥❣t❤ ♦❢ ✷♠✳ ❚❤❡ ✸❉ tr❛•❡❝t♦r② ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✹✳✶✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ✐s ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✶✮✳ ❚❤❡ t♦t❛• s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s✳ ❚❤❡ t❤r❡s❤♦•❞s ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ •❛✇s ❛r❡ s❡•❡❝t❡❞ ❜❛s❡❞ ♦♥ t❤❡ t❛s❦s ♦❢ q✉❛❞r♦t♦rs✳ ❆ t❤r❡s❤♦•❞ ✇❤✐❝❤ ❦❡❡♣s t❤❡ ♣r❡❝✐s❡♥❡ss ♦❢ tr❛❝❦✐♥❣ ❛♥❞ r❡❞✉❝❡s t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s s❤♦✉•❞ ❜❡ ✉s❡❞✳ ❚❤❡ s❡•❡❝t❡❞ s❝❡♥❛r✐♦ ✐s r❡❢❡r❡♥❝❡ tr❛❝❦✐♥❣✳ ❚❤❡ q✉❛❞r♦t♦r ♥❡❡❞s t♦ tr❛❝❦ ❛ sq✉❛r❡ tr❛•❡❝t♦r② ✇❤✐•❡ ❤♦✈❡r✐♥❣ ❛t ❛ ❝❡rt❛✐♥ ❛•t✐t✉❞❡✳ ❲❡ s❡t t❤❛t t❤❡ tr❛❝❦✐♥❣ ❛•♦♥❣ t❤❡ x ❛♥❞ y ❛①✐s s❤♦✉•❞ ❜❡ ♣r❡❝✐s❡ ❛♥❞ t❤❛t ❛•♦♥❣ t❤❡ z ❛①✐s ❝❛♥ ❤❛✈❡ ❛ ❜✐❣❣❡r tr❛❝❦✐♥❣ ❡rr♦r✳ ❚❤❡ t❤r❡s❤♦•❞s s❡t ❢♦r x✱ y ❛♥❞ z ❛r❡ ✵✳✵✵✶♠✱ ✵✳✵✵✶♠ ❛♥❞ ✵✳✵✷♠ r❡s♣❡❝t✐✈❡•②✳ ❆s ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✱ t❤❡ ♠❛①✐♠✉♠ ❝❤❛♥❣❡s ♦❢ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥ ❛r❡ ✵✳✵✾✽♠✴s ❛♥❞ ✵✳✹✾♠♠ r❡s♣❡❝t✐✈❡•② ✐♥ ♦♥❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✶✵♠s✳ ❆ t❤r❡s❤♦•❞ ♦❢ ✵✳✵✵✶♠ ✐s r❡❛s♦♥❛❜•❡✱ ✇❤✐❝❤ ❦❡❡♣s t❤❡ ♣r❡❝✐s❡♥❡ss ♦❢ tr❛❝❦✐♥❣ ❛♥❞ ❛t t❤❡ s❛♠❡ t✐♠❡ r❡❞✉❝❡s t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s✳ ❍❡r❡✱ t❤❡ ❡✈❡♥t tr✐❣❣❡r✐♥❣ •❛✇s ❛r❡ ❝❤♦s❡♥ t♦ ❜❡ t❤❡ ❡rr♦r ❞✐✛❡r❡♥❝❡ t❤r❡s❤♦•❞ ❛s ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✶✵✮✳ ❲❡ s❡t t❤❡ ❡rr♦r ❞✐✛❡r❡♥❝❡ t❤r❡s❤♦•❞ ♦❢ z ✵✳✵✷♠✱ ②❛✇ ❛♥❣•❡ ✵✳✶r❛❞✱ x ❛♥❞ ✾✹ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts y ✵✳✵✵✶♠✳ e z < 0.02♠, e x , e y < 0.001♠, e ψ < 0.1r❛❞. ✭✺✳✶✮ ❊✈❡r② s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✶✵♠s✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ✈❡r✐✜❡s t❤❡ tr❛❝❦✐♥❣ ❡rr♦r ❞✐✛❡r❡♥❝❡s✳ ■❢ ♦♥❡ ♦❢ t❤❡ ❡rr♦r ❞✐✛❡r❡♥❝❡s ✐s ❜✐❣❣❡r t❤❛♥ t❤❡ •✐♠✐t ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ t❤❡ s②st❡♠ ✇✐•• ❝♦♠♣✉t❡ ♥❡✇ ❝♦♥tr♦• s✐❣♥❛•s ❛♥❞ s❡♥❞ t❤❡♠ t♦ t❤❡ ❛❝t✉❛t♦r✳ ■❢ ❛•• ♦❢ t❤❡ ❡rr♦r ❞✐✛❡r❡♥❝❡s ❛r❡ s♠❛••❡r t❤❛♥ t❤❡ •✐♠✐ts✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ✇✐•• ♥♦t ❝♦♠♣✉t❡ t❤❡ ❝♦♥tr♦• s✐❣♥❛•s✱ ❛♥❞ s✐♠♣•② ❦❡❡♣ t❤❡ ♦•❞ ❝♦♥tr♦• s✐❣♥❛•s t♦ t❤❡ ♥❡①t s❛♠♣•✐♥❣✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ❝❛♥ s❛✈❡ ❝♦♠♣✉t❛t✐♦♥✲ ❛• r❡s♦✉r❝❡s ✇❤❡♥ q✉❛❞r♦t♦rs s♠❛•• tr❛❝❦✐♥❣ ❡rr♦rs✱ ❡s♣❡❝✐❛••② ✇❤❡♥ q✉❛❞r♦t♦r ♠♦✈❡ s♠♦♦t❤•② ♦r ❤♦✈❡r✐♥❣✳ ✺✳✷✳✶ ▼♦❞❡• ❋r❡❡ ❈♦♥tr♦• ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♠♣•❡♠❡♥t t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐♥ s❡✈❡r❛• ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✳ ❚❤❡ ❝♦♥✲ tr♦• •❛✇s ❛♥❞ t❤❡ s❝❡♥❛r✐♦s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❈❤❛♣t❡r ✹✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r✐♥❣ •❛✇ ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s ❝❤❛♣t❡r✳ ✶✳ ❇❛s✐❝ s❝❡♥❛r✐♦ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ✜rst•② s✐♠✉•❛t❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐s✲ t✉r❜❛♥❝❡✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❝❤♦s❡♥ ❛s ✐♥ ❚❛❜•❡ ✺✳✾✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡ ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✼✳ ❚❤❡ ✜rst ✜❣✉r❡ ✐s t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s✳ ❚❤❡ s❡❝♦♥❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣•❡s φ✱ θ ❛♥❞ ψ✳ ❚❤❡ t❤✐r❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s ♦❢ t❤❡ ❢♦✉r r♦t♦rs✳ ❚❤❡ ♥✉♠❜❡rs ♦❢ t❤❡ r♦t♦r ❛r❡ ❞❡✜♥❡❞ ✐♥ ❋✐❣✉r❡ ✷✳✶ ✐♥ ❈❤❛♣t❡r ✷✳ ❚❛❜•❡ ✺✳✾✿ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❣❛✐♥s k x 3 ✸ k x 2 ✾ k x 1 ✽ k x 0 ✷✳✺ k z 1 ✷ k z 0 ✺ k y 3 ✸ k y 2 ✽ k y 1 ✽ k y 0 ✸ k ψ 1 ✺ k ψ 0 ✶✵ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ s②st❡♠ ✉s✐♥❣ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❦❡♣t t❤❡ st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ ✢✐❣❤t ❛♥❞ ❢♦••♦✇❡❞ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r②✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✻♠✱ ✇❤✐•❡ ✵✳✵✹✷♠ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✾✵✵✽✱ ✇❤✐❝❤ ✐s ✻✵✪ ♦❢ ✶✺✵✵✵ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ✾✺ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ✷✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ✇✐♥❞ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ■♥ ♦r❞❡r t♦ s✐♠✉•❛t❡ t❤❡ r❡❛•✐st✐❝ ❝❛s❡✱ t❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞✱ ✇❤✐❝❤ ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✐♥ ❚❛❜•❡ ✺✳✾✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✽✳ ❚❤❡ ❝♦♥tr♦• s②st❡♠ s❤♦✇s ✐ts st❛❜✐•✐t② ✐♥ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ s②st❡♠ ❦❡♣t st❛❜•❡ ❞✉r✐♥❣ t❤❡ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✻♠✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ✇✐t❤ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✷✶✶✵✱ ✇❤✐•❡ ✾✵✵✽ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✸✳ P❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❚❤❡♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥✲ t✐❡s✳ ❚❤❡ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ❡st✐♠❛t❡❞ s♠❛••❡r t❤❛♥ t❤❡ r❡❛• ✈❛•✉❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ✐s ✉♥❞❡r✲❛❝t✉❛t❡❞✳ ❚❤❡ ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞ ❛s ✐♥ ❚❛❜•❡ ✺✳✾✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✶✾✳ ❚❤❡ s②st❡♠ st❛②s st❛❜•❡ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✶✾♠✱ ✇❤✐❝❤ ✐♥❝r❡❛s❡❞ ✷✶✼✪ r❡s♣❡❝t✐♥❣ t♦ ✵✳✵✻♠ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ■♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ t❤❡ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❝❛✉s❡❞ s♦♠❡ ♦s❝✐••❛t✐♦♥s ✐♥ t❤❡ s②st❡♠✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✵✽✵✸✱ ✇❤✐•❡ ✾✵✵✽ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✹✳ ❙❡♥s♦r ♥♦✐s❡s ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ❚❤❡ s❡♥s♦r ♥♦✐s❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞ ❛s ✐♥ ❚❛❜•❡ ✺✳✾✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✵✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✺✵✵✵✳ ❆❝t✉❛••② t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡ ✐s ❝❤❛♥❣❡❞ ✐♥t♦ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ ❛s t❤❡ ❛❝t✉❛t✐♦♥ •❛✇s ❛r❡ tr✐❣❣❡r❡❞ ❛t ❡❛❝❤ st❡♣s✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✽♠✱ ✇❤✐•❡ ✵✳✵✻♠ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✺✳ ❆❝t✉❛t♦r ❢❛✉•ts ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✾✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✶✳ ❚❤❡ s②st❡♠ ❦❡♣t st❛❜•❡ ❞✉r✐♥❣ t❤❡ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ s②st❡♠ ❛❞•✉sts t♦ ✐ts st❛❜•❡ st❛t❡s ✐♥ ✵✳✺s✳ ❚❤❡ ❛❞•✉st✐♥❣ t✐♠❡ ✐s t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✶✸♠✱ ✇❤✐❝❤ ✐s ❤❛♣♣❡♥❡❞ ❛t t❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❞✉r✐♥❣ tr❛❝❦✐♥❣✳ ✾✻ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✶✼✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❋✐❣✉r❡ ✺✳✶✽✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ✾✼ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❋✐❣✉r❡ ✺✳✶✾✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❋✐❣✉r❡ ✺✳✷✵✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ✾✽ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✷✶✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❞✉r✐♥❣ ❤♦✈❡r✐♥❣ ❝❛✉s❡❞ ✵✳✶✷♠ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❝❛✉s❡❞ t❤❡ q✉❛❞r♦t♦r t✐•t ❛❜♦✉t ✵✳✶ r❛❞ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✾✷✾✽✱ ✇❤✐•❡ ✾✵✵✽ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✺✳✷✳✷ ❇❛❝❦st❡♣♣✐♥❣ ❈♦♥tr♦• ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✐♠♣•❡♠❡♥t t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛•✲ •❛❤ ❬✷✵(✳ ❚❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✸✳✷✳ ❚❤❡ s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r✐♥❣ •❛✇ ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s ❝❤❛♣t❡r✳ ✶✳ ❇❛s✐❝ s❝❡♥❛r✐♦ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s ✜rst•② s✐♠✉•❛t❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❝❤♦s❡♥ ❛s ✐♥ ❚❛❜•❡ ✺✳✶✵✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡ ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✷✳ ❚❤❡ ✜rst ✜❣✉r❡ ✐s t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s✳ ❚❤❡ s❡❝♦♥❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣•❡s φ✱ θ ❛♥❞ ψ✳ ❚❤❡ t❤✐r❞ ✜❣✉r❡ ✐s t❤❡ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s ♦❢ t❤❡ ❢♦✉r r♦t♦rs✳ ❚❤❡ ♥✉♠❜❡rs ♦❢ t❤❡ r♦t♦r ❛r❡ ❞❡✜♥❡❞ ✐♥ ❋✐❣✉r❡ ✷✳✶ ✐♥ ❈❤❛♣t❡r ✷✳ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❢♦••♦✇❡❞ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② ✇❤✐•❡ ✾✾ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❚❛❜•❡ ✺✳✶✵✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❣❛✐♥s α 1 ✷ α 2 ✺ α 3 ✷ α 4 ✺ α 5 ✶ α 6 ✶ α 7 ✷ α 8 ✶✵ α 9 ✷ α 10 ✸ α 11 ✷ α 12 ✸ ❦❡❡♣✐♥❣ st❛❜•❡✳ ❚❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✶♠✱ ✇❤✐❝❤ ✐s ✹✳✺✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✳ ■♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✾♠✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✾✾✽✼✱ ✇❤✐•❡ ✾✵✵✽ ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✷✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r✲ ❜❛♥❝❡✳ ❚❤❡ ✇✐♥❞ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ❚❛❜•❡ ✺✳✶✵✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✸✳ ❉✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ❤❛❞ ❜✐❣ tr❛❝❦✐♥❣ ❡rr♦rs✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✷✷♠✱ ✇❤✐❝❤ ✐s ✶✶✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✳ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ ♠❛①✐♠✉♠ ❡rr♦r ✐s ♦♥•② ✵✳✵✾♠✳ ■♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✻♠✳ ❆❢t❡r t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ t❤❡ s②st❡♠ ❢♦••♦✇❡❞ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r②✱ ❛♥❞ ❦❡♣t t❤❡ tr❛❝❦✐♥❣ ❡rr♦r ✐♥ ❛ s♠❛•• •✐♠✐t ❛s ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✷✶✶✻✱ ✇❤✐•❡ ✾✾✽✼ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✸✳ P❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❚❤❡♥✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥✲ t✐❡s✳ ❚❤❡ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✵✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✹✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✷✷♠✱ ✇❤✐•❡ ✵✳✶✶♠ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ❚❤❡ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✐s ✵✳✹✾r❛❞✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✹✾✼✻✱ ✇❤✐•❡ ✾✾✽✼ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✹✳ ❙❡♥s♦r ♥♦✐s❡s ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ❚❤❡ s❡♥s♦r ♥♦✐s❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✵✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✺✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✺✵✵✵✳ ✺✳ ❆❝t✉❛t♦r ❢❛✉•ts ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❛❝t✉❛t♦r ✶✵✵ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✷✷✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❋✐❣✉r❡ ✺✳✷✸✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ✶✵✶ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❋✐❣✉r❡ ✺✳✷✹✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❋✐❣✉r❡ ✺✳✷✺✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ✶✵✷ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✷✻✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❢❛✉•ts ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✵✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✻✳ ❚✇♦ ❛❝t✉❛t♦r ❢❛✉•ts ❤❛♣♣❡♥❡❞ ❞✉r✐♥❣ t❤❡ ♠♦✈✐♥❣ ❛♥❞ ❤♦✈❡r✐♥❣✳ ■♥ ❜♦t❤ ❝❛s❡s✱ t❤❡ ❜❛❝❦✲ st❡♣♣✐♥❣ ❝♦♥tr♦• ❦❡♣t ✐ts st❛❜✐•✐t②✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉t ❞✉r✐♥❣ t❤❡ ♠♦✈✐♥❣ ❝❛✉s❡❞ ✵✳✷♠ ♠❛①✲ ✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r✱ ❝♦♠♣❛r✐♥❣ t♦ ✵✳✶♠ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❝❛✉s❡❞ t❤❡ q✉❛❞r♦t♦r t✐•t ❛❜♦✉t ✵✳✷✶r❛❞ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣ ✉s✐♥❣ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✱ ✇❤✐•❡ ✐♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• t❤❡ t✐•t ❛♥❣•❡ ✐s ♦♥•② ✵✳✶r❛❞✳ ❆❢t❡r t❤❡ ❢❛✉•ts✱ t❤❡ s②st❡♠ ❢♦✉♥❞ ✐ts st❛❜•❡ ♣♦s✐t✐♦♥ ✐♥ ❛❜♦✉t ✷✳✺s✳ ❚❤❡ r❡s♣♦♥❞✐♥❣ t✐♠❡ ✐s ❜✐❣❣❡r t❤❛♥ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✇❤✐❝❤ ✐s ❛❜♦✉t ✵✳✺s✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✵✵✷✵✱ ✇❤✐•❡ ✾✾✽✼ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✺✳✷✳✸ ❙•✐❞✐♥❣ ▼♦❞❡ ❈♦♥tr♦• ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✐♠♣•❡♠❡♥t t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛••❛❤ ❬✷✵(✳ ❚❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✸✳✸ ✐♥ ❈❤❛♣t❡r ✹✳ ❚❤❡ s✐♠✉•❛t✐♦♥ t✐♠❡ ✐s ✶✺✵s✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r✐♥❣ •❛✇s ❛r❡ ❞❡✜♥❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s s❡❝t✐♦♥✳ ✶✵✸ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ✶✳ ❇❛s✐❝ s❝❡♥❛r✐♦ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ✜rst•② ✐♠♣•❡♠❡♥t❡❞ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❝❤♦s❡♥ ❛s ✐♥ ❚❛❜•❡ ✺✳✶✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡ ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✼✳ ❚❛❜•❡ ✺✳✶✶✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❣❛✐♥s k 1 ✶ k 2 ✺ α 1 ✸ k 3 ✶ k 4 ✺ α 3 ✸ k 5 ✶ k 6 ✷ α 5 ✷ k 7 ✷ k 8 ✽ α 7 ✹ k 9 ✵✳✺ k 10 ✷ α 9 ✶✳✺ k 11 ✵✳✺ k 12 ✷ α 11 ✶✳✺ ■♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✱ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• s❤♦✇❡❞ ❛ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡✳ ❚❤❡ s②st❡♠ st❛②s st❛❜•❡ ❞✉r✐♥❣ t❤❡ t❛s❦✱ ❛♥❞ t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✶♠✱ ✇❤✐•❡ ✵✳✵✽♠ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❛♥❞ ✵✳✵✻♠ ❛♥❞ ✵✳✶♠ ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❛♥❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✽✹✾✸✱ ✇❤✐•❡ ✾✵✵✽ ❛♥❞ ✾✾✽✼ ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❛♥❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✷✳ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r✲ ❜❛♥❝❡✳ ❚❤❡ ✇✐♥❞ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❛•✇❛②s ❛s ✐♥ ❚❛❜•❡ ✺✳✶✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✽✳ ❉✉r✐♥❣ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ❤❛s ❜✐❣ tr❛❝❦✐♥❣ ❡rr♦rs✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✶✳✷✺♠✱ ✇❤✐❝❤ ✐s ✻✷✳✺✪ ♦❢ t❤❡ t♦t❛• •❡♥❣t❤ ✷♠✱ ✇❤✐•❡ ✵✳✷✸♠ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ■♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ t❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ✐s ♦♥•② ✵✳✵✻♠✳ ❆s t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✱ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ❛•s♦ ✈✉•♥❡r❛❜•❡ t♦ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❆❢t❡r t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ t❤❡ s②st❡♠ ❢♦••♦✇❡❞ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r②✱ ❛♥❞ ❦❡♣t t❤❡ tr❛❝❦✐♥❣ ❡rr♦r ✐♥ ❛ s♠❛•• •✐♠✐t ❛s ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤♦✉t ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❍♦✇❡✈❡r✱ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• s❤♦✉•❞ ♥♦t ❜❡ ✉s❡❞ ✐♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ ❜❡❝❛✉s❡ t❤❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s t♦♦ ❤✐❣❤✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✷✹✹✺✱ ✇❤✐•❡ ✽✹✾✸ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✸✳ P❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ■♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✱ t❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✷✾✳ ❚❤❡ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✶✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✹✶♠✱ ✇❤✐•❡ ✵✳✶✶♠ ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ✶✵✹ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✷✼✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❆•• t❤r❡❡ ♠❡t❤♦❞s s❤♦✇ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✹✾✽✸✱ ✇❤✐•❡ ✽✹✾✸ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✹✳ ❙❡♥s♦r ♥♦✐s❡s ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ❛•s♦ s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ❚❤❡ s❡♥s♦r ♥♦✐s❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✸✵✳ ❚❤❡ s❡•❡❝t❡❞ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✶✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ t❤❡ s❡♥s♦r ♥♦✐s❡s ❝❛✉s❡❞ ♠❛♥② ✈✐❜r❛t✐♦♥s ✐♥ t❤❡ ❛❝t✉❛t♦rs✱ ✇❤✐❝❤ ❛r❡ ♠✉❝❤ ♠♦r❡ t❤❛♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ♠✉❝❤ ♠♦r❡ s❡♥s✐❜•❡ t♦ t❤❡ s❡♥s♦r ♥♦✐s❡s t❤❛♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❚❤❡ r❡❛s♦♥ ✐s ❞✉❡ t♦ t❤❡ s✇✐t❝❤✐♥❣ ❢❡❛t✉r❡ ♦❢ t❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇❤✐❝❤ ✐s •❡ss st❛❜•❡ t❤❛♥ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✶✺✵✵✵✳ ✺✳ ❆❝t✉❛t♦r ❢❛✉•ts ■♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✸✶✳ ❚❤❡ s❡•❡❝t❡❞ ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✶✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•ts ❞✐❞ ♥♦t ❝❛✉s❡ ❜✐❣❣❡r ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ❞✉r✐♥❣ t❤❡ ✢✐❣❤t✳ ❚❤❡ ✶✵✺ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❋✐❣✉r❡ ✺✳✷✽✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❋✐❣✉r❡ ✺✳✷✾✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ✶✵✻ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✸✵✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ❋✐❣✉r❡ ✺✳✸✶✿ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✸✳ ✶✵✼ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❢❛✉•t ❞✉r✐♥❣ t❤❡ ❤♦✈❡r✐♥❣ ❝❛✉s❡❞ ✵✳✵✸✺♠ tr❛❝❦✐♥❣ ❡rr♦r✳ ❉✉r✐♥❣ ❤♦✈❡r✐♥❣✱ t❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❤❛s ❜❡❡♥ r❡s♣♦♥❞❡❞ q✉✐❝❦•②✱ ❛♥❞ ✐♥ ❛❜♦✉t ✶s t❤❡ s②st❡♠ ✇❛s ❜❛❝❦ t♦ ♥♦r♠❛• st❛❜•❡ st❛t❡✳ ❍♦✇❡✈❡r✱ t❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❞✉r✐♥❣ t❤❡ ♠♦✈✐♥❣ ❝❛✉s❡❞ ❛ •♦♥❣❡r ❛❞•✉st♠❡♥t✳ ❲❤✐•❡ t❤❡ r❡s♣♦♥❞✐♥❣ t✐♠❡ ✐s ✵✳✺s ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❛♥❞ ✶✳✺s ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❚❤❡ ❛❝t✉❛t♦r ❢❛✉•t ❝❛✉s❡❞ t❤❡ q✉❛❞r♦t♦r t✐•t ❛❜♦✉t ✵✳✶r❛❞ ❞✉r✐♥❣ ❤♦✈❡r✐♥❣✱ ✇❤✐•❡ ✵✳✶r❛❞ ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❛♥❞ ✵✳✷✶r❛❞ ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐s ✽✷✾✼✱ ✇❤✐•❡ ✽✹✾✸ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✺✳✷✳✹ ❉✐s❝✉ss✐♦♥ ❆s ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❝♦♥tr♦• s❝❤❡♠❡✱ ✐♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡s❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❡①♣•✐❝✐t•②✱ t❡♥ ✐♠♣♦rt❛♥t ❝r✐t❡r✐❛ ❛r❡ •✐st❡❞ ❛s ❢♦••♦✇s ❢♦r t❤❡ ❝♦♠♣❛r✐s♦♥✿ ✭✶✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥ t❤❡ ①✱ ② ❛♥❞ ③ ❛①✐s ✐♥ ❜❛s✐❝ s❝❡✲ ♥❛r✐♦ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡✿
3 i=1 t e 2 i ✳ ✭✻✮ ❆❞•✉st✐♥❣ t✐♠❡✿ t✳ ✭✼✮ ▼❛①✐♠✉♠ t✐•t ❛♥❣•❡✿ φ ♦r θ✳ ✭✽✮ ❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s✳ ✭✾✮ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t②✿ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♣❡r❛t✐♦♥s ✐♥ t❤❡ ❝♦♥tr♦• •❛✇✳ ✭✶✵✮ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✿ t❤❡ ❡♥❡r❣② s♣❡♥t ♦♥ ♠♦t✐♦♥ ❝♦♥tr♦•✱ t❤❡ s✉♠ ♦❢ t❤❡ ❢♦✉r ♠♦✲ t♦rs✬ sq✉❛r❡❞ ❛♥❣✉•❛r s♣❡❡❞s✿ 4 i=1 t ω 2 i ✳ ■♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✱ t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs✱ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ♦❢ t❤r❡❡ ♠❡t❤♦❞s ✐♥ ✜✈❡ s❝❡♥❛r✐♦s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❋✐❣✉r❡ ✺✳✸✷✳ ❋r♦♠ t❤❡ ✜❣✉r❡✱ ✇❡ ❝❛♥ s❡❡ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ ✐♥ ❣❡♥❡r❛•✳ ✶✵✽ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✸✷✿ ❊✈❡♥t tr✐❣❣❡r❡❞ t❤r❡❡ ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✿ ✭❛✮ t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✲ ✐♥❣ ❡rr♦r✳ ✭❜✮ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡✳ ✭❝✮ t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s✳ ✭❞✮ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥×10 -7 ✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✶✷✳ ❆•• t❤❡ t❤r❡❡ ♠❡t❤♦❞s ❛r❡ ❝❛♣❛❜•❡ t♦ ❦❡❡♣ t❤❡✐r st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ ✢✐❣❤t✳ ❚❤❡r❡ ❛r❡ ♥♦t ❛ ❜✐❣ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤r❡❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❤❛s t❤❡ ❜✐❣❣❡st ❛❝t✉❛t✐♦♥ st❡♣s ❛♥❞ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✶✸✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❝❛♥ st✐•• ✇❡•• ❝♦♠♣❡♥s❛t❡ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ ❤❛s t❤❡ s♠❛••❡st ❡rr♦r ✈❛r✐❛♥❝❡✱ ✇❤✐•❡ t❤❡ ♦t❤❡r t✇♦ ❝♦♥tr♦•s✱ ❡s♣❡❝✐❛••② s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✱ ❤❛✈❡ ❜✐❣ tr❛❝❦✐♥❣ ❡rr♦rs✳ ❚❤❡ ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s•✐❞✐♥❣ ♠♦❞❡ ✶✵✾ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❚❛❜•❡ ✺✳✶✷✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ❜❛s✐❝ s❝❡♥❛r✐♦ ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✻ ✵✳✶ ✵✳✵✼ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✷✶ ✵✳✷✶ ✵✳✷✶ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✷ ✵✳✵✷ ✵✳✵✶✶✻ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✶✼❡✰✼ ✷✳✹✺✷✷❡✰✼ ✷✳✹✸✻✺❡✰✼ ❛❝t✉❛t✐♦♥ st❡♣s ✾✵✵✽ ✾✾✽✼ ✼✻✵✸ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❛❜•❡ ✺✳✶✸✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ✇✐♥❞ ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✻ ✵✳✷✷ ✶✳✷✺ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✹ ✵✳✹ ✵✳✹✵✺ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✻✸✺ ✷✳✻✻✼✾ ✺✽✳✺✵✾✶ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✸✺✾✽❡✰✼ ✷✳✸✻✵✹❡✰✼ ✷✳✸✺✺✾❡✰✼ ❛❝t✉❛t✐♦♥ st❡♣s ✶✷✶✶✵ ✶✷✶✶✻ ✶✶✹✵✵ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❝♦♥tr♦• ✐s ✶✳✷✺♠✱ ✇❤✐❝❤ ✐s ✻✷✳✺✪ t❤❡ ❞❡s✐r❡❞ ♣❛t❤ •❡♥❣t❤ ✷♠✳ ❚❤❡ ❡rr♦r ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ✐s ✺✽✳✺✳✾✶♠✱ ✇❤✐•❡ ✵✳✵✻✸✺♠ ❛♥❞ ✷✳✻✻✼✾♠ ✐♥ ♦t❤❡r t✇♦ ♠❡t❤♦❞s r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✶✹✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s s♠❛••❡st tr❛❝❦✐♥❣ ❡rr♦rs ❛♥❞ ❛❝t✉❛t✐♦♥ st❡♣s✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• s❛✈❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s ❛♥❞ ❛t t❤❡ s❛♠❡ t✐♠❡ ❤❛s ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❤❛s t❤❡ ❜✐❣❣❡st ❡rr♦r ✈❛r✐❛♥❝❡ ✐♥ t❤❡ s②st❡♠✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✶✺✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ tr✐❣❣❡r❡❞ ❛•• t❤❡ t✐♠❡✱ ❛♥❞ t❤❡ r❡s✉•ts ❛r❡ ❛•♠♦st t❤❡ s❛♠❡ ✇✐t❤ t❤❡ ♠❡t❤♦❞s ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ■t s❤♦✇s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ■❢ t❤❡r❡ ❛r❡ ♠❛♥② ❞✐st✉r❜❛♥❝❡s✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ✇✐•• ♣❡r❢♦r♠ ❛s ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ■❢ t❤❡r❡ ❛r❡ ♥♦t ♠❛♥② ❞✐st✉r❜❛♥❝❡s✱ t❤❡ ❝♦♥tr♦• s②st❡♠ ✇✐•• s❡•❡❝t t❤❡ ❝♦♥tr♦• ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ tr✐❣❣❡r✐♥❣ •❛✇s✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts ✐s s❤♦✇♥ ✐♥ ❚❛❜•❡ ✺✳✶✻✳ ❚❤r❡❡ ❡✈❡♥t ✶✶✵ ✺✳✷✳ ❊✈❡♥t tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❚❛❜•❡ ✺✳✶✹✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡r✲ t❛✐♥t✐❡s✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✶✾ ✵✳✷✷ ✵✳✷✾ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✸✹ ✵✳✹✾ ✵✳✸ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✶✷✽✶ ✶✳✽✻✼✺ ✵✳✹✻✸✽ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✷✾✺❡✰✼ ✷✳✹✺✹✸❡✰✼ ✷✳✹✺✷✶❡✰✼ ❛❝t✉❛t✐♦♥ st❡♣s ✶✵✽✵✸ ✶✹✾✼✻ ✶✹✾✽✸ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❛❜•❡ ✺✳✶✺✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ s❡♥s♦r ♥♦✐s❡s ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ ❡rr♦r ✵✳✵✽ ✵✳✵✾ ✵✳✵✽ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ✵✳✷✷ ✵✳✷✸ ✵✳✷✶ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✷✸✵✾ ✵✳✵✵✷✾ ✵✳✵✵✽✽ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✶✼❡✰✼ ✷✳✹✺✷✵❡✰✼ ✷✳✹✺✶✽❡✰✼ ❛❝t✉❛t✐♦♥ st❡♣s ✶✺✵✵✵ ✶✺✵✵✵ ✶✺✵✵✵ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❚❛❜•❡ ✺✳✶✻✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ♥ ✐s ♥✲❞✐❣✐t ♥✉♠❜❡r✳ ❛❝t✉❛t♦r ❢❛✉•ts ♠♦❞❡• ❢r❡❡ ❜❛❝❦st❡♣♣✐♥❣ s•✐❞✐♥❣ ♠♦❞❡ ♠❛①✐♠✉♠ ❡rr♦r ❝❛✉s❡❞ ❜② ❢❛✉•t ✵✳✶✸ ✵✳✷ ✶✳✽ ♠❛①✐♠✉♠ t✐•t ❛♥❣•❡ ❝❛✉s❡❞ ❜② ❢❛✉•t ✵✳✶ ✵✳✷✶ ✵✳✺ ❛❞•✉st✐♥❣ t✐♠❡ ❛❢t❡r t❤❡ ❢❛✉•t ✵✳✺ ✷✳✺ ✷✳✺ ❡rr♦r ✈❛r✐❛♥❝❡ ✵✳✵✾✶✽ ✵✳✵✻✻✶ ✽✳✼✻✽ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✷✳✹✺✷✵❡✰✼ ✷✳✹✺✷✼❡✰✼ ✷✳✹✹✹✶❡✰✼ ❛❝t✉❛t✐♦♥ st❡♣s ✾✷✾✽ ✶✵✵✷✵ ✼✽✺✼ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ❖✭✺♥ 2 ✰✸♥✮ ❖✭✻♥ 2 ✰✹♥✮ ❖✭✻♥ 2 ✰✹♥✮ tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ❦❡❡♣ t❤❡ st❛❜✐•✐t② ❛❢t❡r t❤❡ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❤❛✈❡ ♦s❝✐••❛t✐♦♥s ✐♥ t❤❡ ❛❝✉t❛t♦r ❛❢t❡r t❤❡ ❢❛✉•ts✱ ❛♥❞ t❤❡ s②st❡♠ ♥❡❡❞ ❛❜♦✉t ✷✳✺s t♦ ❜❛❝❦ t♦ t❤❡ st❛❜•❡ st❛t❡✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❤❛s t❤❡ ❜✐❣❣❡st ♠❛①✐♠✉♠ tr❛❝❦✐♥❣ ❡rr♦r ❛♥❞ ❡rr♦r ✈❛r✐❛♥❝❡✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s t❤❡ s♠❛••❡st ❛❞•✉st✐♥❣ t✐♠❡ ❛♥❞ s♠❛•• tr❛❝❦✐♥❣ ❡rr♦rs✳ ✶✶✶ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✸✳ ❉✐s❝✉ss✐♦♥ ✺✳✸ ❉✐s❝✉ss✐♦♥ ■♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡s❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❡①♣•✐❝✐t•②✱ ✇❡ ❝❤♦♦s❡ s✐① ❝r✐t❡r✐❛ ❢♦r t❤❡ ❝♦♠♣❛r✐s♦♥✿ ✭✶✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ tr❛❝❦✐♥❣ ❡rr♦rs ✐♥ ①✱ ② ❛♥❞ ③ ❛①✐s ✇✐t❤♦✉t ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✿ 3 i=1 t e 2 i ✳ ✭✷✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✿ 3 i=1 t e 2 i ✳ ✭✸✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ r♦t❛t✐♦♥ ♠♦❞❡• ❡rr♦r✿ 3 i=1 t e 2 i ✳ ✭✹✮ ❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ✭✺✮ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t②✿ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♣❡r❛t✐♦♥s ✐♥ t❤❡ ❝♦♥tr♦• •❛✇✳ ✭✻✮ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✿ t❤❡ ❡♥❡r❣② s♣❡♥t ♦♥ ♠♦t✐♦♥ ❝♦♥tr♦•✱ t❤❡ s✉♠ ♦❢ t❤❡ ❢♦✉r ♠♦t♦rs✬ sq✉❛r❡❞ ❛♥❣✉•❛r s♣❡❡❞s✿ 4 i=1 t ω 2 i ✳ ❚❤❡ r❛❞❛r ❝❤❛rts ♦❢ t❤❡ ❝♦♠♣❛r✐s♦♥s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡s ✺✳✸✸ ❛♥❞ ✺✳✸✹✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦rs ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• t❤❡ s❝❡♥❛r✐♦s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❚❛❜•❡ ✺✳✶✼ ❛♥❞ ❋✐❣✉r❡ ✺✳✸✺✳ ❚❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❤❛s t❤❡ s♠❛••❡st ♠❛①✲ ✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✱ ❛♥❞ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❤❛s t❤❡ s♠❛••❡st ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r t❤❛♥ ♦t❤❡r ❡✈❡♥t tr✐❣❣❡r❡❞ ♠❡t❤♦❞s✳ ❚❤❡ t✐♠❡ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛✈❡ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡s ❡s♣❡❝✐❛••② ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ ♠✉❝❤ s♠❛••❡r t❤❛♥ ♦t❤❡r ♠❡t❤♦❞s✳ ❚❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❤❛s ❣r❡❛t ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ t✐♠❡ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✱ ✇❤✐•❡ t❤❡ t✐♠❡ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s ❛•♠♦st t❤❡ s❛♠❡ ♣❡r❢♦r♠❛♥❝❡ ❛s ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❞♦❡s ♥♦t ♠❛t❝❤ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✱ ❛♥❞ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ ♠✉❝❤ ♠♦r❡ ❜✐❣❣❡r t❤❛♥ ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❚❤❡ s✉♠ ♦❢ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡ ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• t❤❡ s❝❡♥❛r✐♦s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❚❛❜•❡ ✺✳✶✽ ❛♥❞ ❋✐❣✉r❡ ✺✳✸✻✳ ❚❤❡ t✐♠❡ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛✈❡ t❤❡ s♠❛••❡st ❡rr♦r ✈❛r✐❛♥❝❡ ✐♥ ❣❡♥❡r❛•✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❤❛s t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡ ❛•♠♦st t❤❡ s❛♠❡ ✇✐t❤ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❜❛❝❦st❡♣♣✐♥❣ ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•s ✐♥ ✜✈❡ s❝❡♥❛r✐♦s✱ ♦r ❡✈❡♥ ❜❡tt❡r ✐♥ t❤❡ ❝❛s❡ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• t❤❡ s❝❡♥❛r✐♦s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❚❛❜•❡ ✺✳✶✾ ❛♥❞ ❋✐❣✉r❡ ✺✳✸✼✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❤❛s r❡❞✉❝❡❞ ❛❜♦✉t ✸✵✪ ❛❝t✉❛t✐♦♥ st❡♣s ✐♥ ❣❡♥❡r❛• ✇❤✐•❡ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ ♥♦t ❣r❡❛t•② ✐♥❝r❡❛s❡❞✳ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ❛r❡ ♠✉❝❤ ♠♦r❡ r❡❞✉❝❡❞ t❤❛♥ ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✱ ✇❤✐•❡ ✶✶✷ ✺✳✸✳ ❉✐s❝✉ss✐♦♥ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✸✸✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ❋✐❣✉r❡ ✺✳✸✹✿ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡ ✐s ❛•♠♦st t❤❡ s❛♠❡ ✇✐t❤ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• t❤❡ s❝❡♥❛r✐♦s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❚❛❜•❡ ✺✳✷✵ ❛♥❞ ❋✐❣✉r❡ ✺✳✸✽✳ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞ t♦♦ ♠✉❝❤ ✐♥ ❞✐✛❡r❡♥t ♠❡t❤♦❞s✳ ❚❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦• ❤❛s s•✐❣❤t•② •❡ss ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✱ ❤♦✇❡✈❡r t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛r❡ t❤❡ ❜✐❣❣❡st ✐♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥s✳ ✶✶✸ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✸✳ ❉✐s❝✉ss✐♦♥ ❚❛❜•❡ ✺✳✶✼✿ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦rs ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ▼❋✿ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❇❙✿ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❙▼✿ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ t✐♠❡ tr✐❣❣❡r❡❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ▼❋ ❇❙ ❙▼ ▼❋ ❇❙ ❙▼ ❜❛s✐❝ s❝❡♥❛r✐♦ ✵✱✵✹✷ ✵✱✵✾ ✵✱✵✼ ✵✱✵✻ ✵✱✶ ✵✱✵✼ ✇✐♥❞ ✵✱✵✹✷ ✵✱✷✸ ✵✱✶✾ ✵✱✵✻ ✵✱✷✷ ✶✱✷✺ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✵✱✵✼✺ ✵✱✶✶ ✵✱✵✺✺ ✵✱✶✾ ✵✱✷✷ ✵✱✷✾ s❡♥s♦r ♥♦✐s❡ ✵✱✵✽ ✵✱✵✾ ✵✱✵✽ ✵✱✵✽ ✵✱✵✾ ✵✱✵✽ ❛❝t✉❛t♦r ❢❛✉•ts ✵✱✵✹ ✵✱✵✻ ✵✱✷✶ ✵✱✶✸ ✵✱✷ ✶✳✽ ❋✐❣✉r❡ ✺✳✸✺✿ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ❚❛❜•❡ ✺✳✶✽✿ ❚❤❡ s✉♠ ♦❢ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡s ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ▼❋✿ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❇❙✿ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❙▼✿ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ t✐♠❡ tr✐❣❣❡r❡❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ▼❋ ❇❙ ❙▼ ▼❋ ❇❙ ❙▼ ❜❛s✐❝ s❝❡♥❛r✐♦ ✵✱✵✵✹✾ ✵✱✵✵✷✽ ✵✱✵✵✵✸ ✵✱✷ ✵✱✷ ✵✱✵✶✶✻ ✇✐♥❞ ✵✱✵✵✺✷ ✷✱✹✼✷✸ ✵✳✽✹✼✻ ✵✱✵✻✸✺ ✷✱✻✻✼✾ ✺✽✳✺✵✾✶ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✵✱✵✷✶✷ ✵✱✵✹✻✽ ✵✱✵✶✸✽ ✵✱✶✷✽✶ ✶✱✽✻✼✺ ✵✳✹✻✸✽ s❡♥s♦r ♥♦✐s❡ ✵✱✷✸✵✾ ✵✱✵✵✷✾ ✵✱✵✵✽✽ ✵✱✷✸✵✾ ✵✱✵✵✷✾ ✵✱✵✵✽✽ ❛❝t✉❛t♦r ❢❛✉•ts ✵✱✵✸✷✼ ✵✱✵✶✻✼ ✵✱✵✺✽✺ ✵✱✵✾✶✽ ✵✱✵✻✻✶ ✽✳✼✻✽ ✶✶✹ ✺✳✸✳ ❉✐s❝✉ss✐♦♥ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✸✻✿ ❚❤❡ s✉♠ ♦❢ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡s ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ❚❛❜•❡ ✺✳✶✾✿ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ▼❋✿ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❇❙✿ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❙▼✿ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ t✐♠❡ tr✐❣❣❡r❡❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ▼❋ ❇❙ ❙▼ ▼❋ ❇❙ ❙▼ ❜❛s✐❝ s❝❡♥❛r✐♦ ✶✺✵✵✵ ✾✵✵✽ ✾✾✽✼ ✼✻✵✸ ✇✐♥❞ ✶✺✵✵✵ ✶✷✶✶✵ ✶✷✶✶✻ ✶✶✹✵✵ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✶✺✵✵✵ ✶✵✽✵✸ ✶✹✾✼✻ ✶✹✾✽✸ s❡♥s♦r ♥♦✐s❡ ✶✺✵✵✵ ✶✺✵✵✵ ✶✺✵✵✵ ✶✺✵✵✵ ❛❝t✉❛t♦r ❢❛✉•ts ✶✺✵✵✵ ✾✷✾✽ ✶✵✵✷✵ ✼✽✺✼ ❋✐❣✉r❡ ✺✳✸✼✿ ❚❤❡ ❛❝t✉❛t✐♦♥ st❡♣s ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ✶✶✺ ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ✺✳✸✳ ❉✐s❝✉ss✐♦♥ ❚❛❜•❡ ✺✳✷✵✿ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ▼❋✿ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❇❙✿ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦•✳ ❙▼✿ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ×10 7 t✐♠❡ tr✐❣❣❡r❡❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ▼❋ ❇❙ ❙▼ ▼❋ ❇❙ ❙▼ ❜❛s✐❝ s❝❡♥❛r✐♦ ✷✱✹✺✶✽ ✷✱✹✺✶✾ ✷✱✹✺✶✽ ✷✱✹✺✶✼ ✷✱✹✺✷✷ ✷✱✹✸✻✺ ✇✐♥❞ ✷✱✸✺✾✽ ✷✱✸✻✵✵ ✷✱✸✺✾✾ ✷✱✸✺✾✽ ✷✱✸✻✵✹ ✷✱✸✺✺✾ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ✷✱✹✺✷✸ ✷✱✹✺✷✽ ✷✱✹✺✶✼ ✷✱✹✷✾✺ ✷✱✹✺✹✸ ✷✱✹✺✷✶ s❡♥s♦r ♥♦✐s❡ ✷✱✹✺✶✼ ✷✱✹✺✷✵ ✷✱✹✺✶✽ ✷✱✹✺✶✼ ✷✱✹✺✷✵ ✷✱✹✺✶✽ ❛❝t✉❛t♦r ❢❛✉•ts ✷✱✹✺✶✽ ✷✱✹✺✷✹ ✷✱✹✺✷✷ ✷✱✹✺✷✵ ✷✱✹✺✷✼ ✷✱✹✹✹✶ ❋✐❣✉r❡ ✺✳✸✽✿ ❚❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥×10 -7 ♦❢ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❛•• s❝❡♥❛r✐♦s✳ ✶✶✻ ❈❤❛♣t❡r ✻ ❈♦♥❝•✉s✐♦♥s ✻✳✶ ❈♦♥❝•✉s✐♦♥ ■♥ t❤❡ •❛st ❞❡❝❛❞❡s✱ q✉❛❞r♦t♦rs ❜❡❝♦♠❡ ♣♦♣✉•❛r ✐♥ s❝✐❡♥t✐✜❝ r❡s❡❛r❝❤❡s✳ ❉✉❡ t♦ ✐ts s✐♠♣•✐❝✐t② ✐♥ ❞❡s✐❣♥ ❛♥❞ ♠❛♥❡✉✈❡r✱ q✉❛❞r♦t♦rs ❤❛✈❡ ♠❛♥② ❛♣♣•✐❝❛t✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♠❛♥② ❝❤❛••❡♥❣❡s ✐♥ t❤❡ ❝♦♥tr♦• ♦❢ q✉❛❞r♦t♦rs✳ ❚❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t❤❡r❡❢♦r❡ ❞❡❞✐❝❛t❡❞ t♦ t❤❡s❡ ♠❛✐♥ ❝❤❛••❡♥❣❡s✳ ❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ t✇♦ ✐♥s♣✐r✐♥❣ ❡①❛♠♣•❡s ❛r❡ ♠❛❞❡ t♦ s❤♦✇ t❤❡ •✐♠✐ts ♦❢ ❛ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦• ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ✐♥ ✐❞❡❛• s❝❡♥❛r✐♦s ❛♥❞ r❡❛•✐st✐❝ s❝❡♥❛r✐♦s✳ ❚❤❡♥✱ t❤❡ ❝❤❛••❡♥❣❡s ✐♥ q✉❛❞r♦t♦r r❡s❡❛r❝❤❡s ❛r❡ t❤♦r♦✉❣❤•② st✉❞✐❡❞✳ ❚❤r❡❡ ♠❛✐♥ ♣r♦❜•❡♠s ❛r❡ •✐st❡❞✳ ❋✐rst•②✱ ❛ ❞②♥❛♠✐❝ ♠♦❞❡• ✇❤✐❝❤ ✐s s✐♠♣•❡ ❢♦r ❝♦♠♣✉t❛t✐♦♥ ❛♥❞ ❝•♦s❡ t♦ q✉❛❞r♦✲ t♦r ❞②♥❛♠✐❝s ✐s ♥❡❡❞❡❞✳ ❙❡❝♦♥❞•②✱ ❛ ❝♦♥tr♦• ♠❡t❤♦❞ ✇❤✐❝❤ ❞❡♠❛♥❞s s♠❛•• ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s ❛♥❞ ❛t t❤❡ s❛♠❡ t✐♠❡ ✐s r❛t❤❡r st❛❜•❡ ❢♦r ❞✐st✉r❜❛♥❝❡s ❛♥❞ ♠♦❞❡• ✉♥❝❡rt❛✐♥t✐❡s ✐s r❡q✉✐r❡❞✳ ❆t •❛st✱ t❤❡ r❡s✉•ts ✐♥ r❡❛•✐st✐❝ s❝❡♥❛r✐♦s ❛r❡ ♥❡❡❞❡❞ t♦ s❤♦✇ t❤❡ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ s❡✈❡r❛• ♠❡t❤♦❞s ✐♥ t❤❡ s❛♠❡ s❝❡✲ ♥❛r✐♦s ❛r❡ ❛•s♦ ❞❡♠❛♥❞❡❞ t♦ ✈❡r✐❣② ✇❤✐❝❤ ♠❡t❤♦❞ ✐s ❜❡tt❡r ✐♥ ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ t❤❡s❡ ❝❤❛••❡♥❣❡s ❛r❡ st✉❞✐❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✳ ❚❤❡ ❡①✐st✐♥❣ q✉❛❞r♦t♦rs ❛r❡ ✐♥t❡♥s✐✈❡•② st✉❞✐❡❞ ❛♥❞ ♣r❡s❡♥t❡❞✳ ❚❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ q✉❛❞r♦t♦rs ❤❛✈❡ •✐♠✐t❛t✐♦♥s✿ ▲✐t❤✐✉♠ P♦•②♠❡r ❜❛tt❡r✐❡s ❤❛✈❡ •✐♠✐t❡❞ ❝❛♣❛❝✐t✐❡s❀ ❡♠❜❡❞❞❡❞ s②st❡♠s ❤❛✈❡ •✐♠✐t❡❞ ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s✱ ❡t❝✳ ❚❤❡♥✱ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ♦❢ q✉❛❞r♦t♦rs ❛r❡ t❤♦r♦✉❣❤•② st✉❞✐❡❞✱ ❛♥❞ ♥❡❝❡ss❛r② q✉❛•✐t✐❡s ♦❢ ❛ ❝♦♥tr♦• ♠❡t❤♦❞ ❛r❡ ❞✐s❝✉ss❡❞✳ ❚❤❡ ❡①✐st✐♥❣ q✉❛❞r♦t♦r ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ♣r❡s❡♥t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r r❡s❡❛r❝❤ ❣r♦✉♣s✳ ❙❡✈❡r❛• ❝♦♥tr♦• ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞✱ ❤♦✇❡✈❡r ✇❡ ❝❛♥ ♥♦t t❡•• ✇❤✐❝❤ ♦♥❡ ✐s ❜❡tt❡r ✐♥ ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ r❛r❡•② ❣✐✈❡♥ ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❚❤❡s❡ st✉❞✐❡s ♦❢ t❤❡ ❈❤❛♣t❡r ✻✳ ❈♦♥❝•✉s✐♦♥s ✻✳✶✳ ❈♦♥❝•✉s✐♦♥ •✐t❡r❛t✉r❡ s✉♣♣♦rt t❤❡ ❝❤♦s❡♥ ♠❛✐♥ ❝❤❛••❡♥❣❡s✳ ■♥ t❤❡ ♠♦❞❡•✐♥❣✱ t❤❡ ❦✐♥❡♠❛t✐❝ r❡•❛t✐♦♥s ❛♥❞ ❞②♥❛♠✐❝ ♠♦❞❡•s ❛r❡ ♣r♦♣♦s❡❞✳ ❚❤❡ ✇♦r•❞ ❢r❛♠❡✱ ❜♦❞② ❢r❛♠❡ ❛♥❞ t❤❡ ❊✉•❡r ❛♥❣•❡s r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ ❛r❡ s❡•❡❝t❡❞ ❜❛s❡❞ ♦♥ t❤❡ ♠♦st ❣❡♥❡r❛••② ✉s❡❞ r❡❢❡r❡♥❝❡s✳ ❚❤❡ r❡•❛t✐♦♥s ♦❢ t❤❡ ♣♦s✐t✐♦♥s✱ ✈❡•♦❝✐t✐❡s ❛♥❞ ❛❝❝❡•❡r❛t✐♦♥s ❛r❡ ❝•❡❛r•② ♣r❡s❡♥t❡❞✳ ❚❤❡♥✱ ❛ ❞②♥❛♠✐❝ ♠♦❞❡• ✉s✐♥❣ t❤❡ ◆❡✇t♦♥ ❊✉•❡r ❢♦r♠❛•✐s♠ ✐s ♣r♦♣♦s❡❞✳ ❚❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❜❛s❡❞ ♦♥ t❤❡ ❋❛②✬s ✇♦r❦ ❛r❡ st✉❞✐❡❞ ❛♥❞ ♣r❡s❡♥t❡❞✳ ❚❤❡ ♠♦❞❡•✐♥❣ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❇♦✉❛❜❞❛••❛❤✬s ♠♦❞❡•✐♥❣✳ ❚❤❡ ❇♦✉❛❜❞❛••❛❤✬s ♠♦❞❡• ✐s ❛ r❛t❤❡r r❡❛•✐st✐❝ ♠♦❞❡• ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ s♦♠❡ ❡rr♦rs ✐♥ ❤✐s ♠♦❞❡•✐♥❣✳ ❋♦r ❡①❛♠♣•❡✱ t❤❡ ❤✉❜ ❢♦r❝❡ ✐s ❝❛•❝✉•❛t❡❞ ✉s✐♥❣ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t✐❡s ❛•♦♥❣ t❤❡ x w ❛♥❞ y w ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡✳ ❚❤❡s❡ ❛r❡ ♥♦t ❡①❛❝t✳ ❚❤❡ ❤✉❜ ❢♦r❝❡ s❤♦✉•❞ ❜❡ ❝❛•❝✉•❛t❡❞ ✉s✐♥❣ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t✐❡s ❛•♦♥❣ t❤❡ x b ❛♥❞ y b ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✱ ❛s ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛r❡ ❝❛✉s❡❞ ❜② t❤❡ ❛✐r ✢♦✇ s♣❡❡❞ r❡•❛t❡❞ t♦ ❛ q✉❛❞r♦t♦r ✇❤✐❝❤ ❝❤❛♥❣❡s ✇❤❡♥ t❤❡ q✉❛❞r♦t♦r r♦t❛t❡s✳ ▼♦r❡♦✈❡r✱ s♦♠❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥t ❛r❡ ♥♦t ❝♦rr❡❝t✱ s✉❝❤ ❛s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ r♦••✐♥❣ ♠♦♠❡♥t✱ ❡t❝✳ ❚❤❡ ❢r❛♠❡s ❞❡✜♥❡❞ ✐♥ ❇♦✉❛❜❞❛••❛❤✬s t❤❡s✐s ❛r❡ ♥♦t t❤❡ s❛♠❡ ❛s ✐♥ ❤✐s ♣r♦❣r❛♠✱ ✇❤✐❝❤ ✐s ❛ •✐tt•❡ ❜✐t ❝♦♥❢✉s✐♥❣✳ ❚❤❡r❡❢♦r❡ ✐♥ t❤✐s t❤❡s✐s✱ t❤❡ ❝❤♦s❡♥ ✇♦r•❞ ❛♥❞ ❜♦❞② ❢r❛♠❡s✱ t❤❡ r❡•❛t✐♦♥s❤✐♣ ♦❢ •✐♥❡❛r ❛♥❞ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ❝•❡❛r•② ♣r❡s❡♥t❡❞✳ ❚❤❡ ❡rr♦rs ✐♥ ❇♦✉❛❜❞❛••❛❤✬s t❤❡s✐s ❛r❡ ❝♦rr❡❝t❡❞✳ ❙♦♠❡ ✉♥♥❡❝❡ss❛r② ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♥❡❣•❡❝t❡❞✳ ❚❤❡♥✱ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ q✉❛❞r♦t♦r ✐s ❛♥❛•②③❡❞✳ ❚❤❡ ❞②♥❛♠✐❝s ❤❛s ♥❡✈❡r ❜❡❡♥ st✉❞✐❡❞ ✐♥✲ t❡♥s✐✈❡•② ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❋✐rst•② ❜❛s❡❞ ♦♥ t❤❡ ♥♦r♠❛• ❢♦r♠ t❤❡♦r②✱ t❤❡ ♠♦❞❡• ♦❢ ❛ q✉❛❞r♦t♦r ✐s s✐♠♣•✐✜❡❞ t♦ ❛ s✐♠♣•❡st ❢♦r♠ ♥❛♠❡❞ t❤❡ ♥♦r♠❛• ❢♦r♠ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ■♥ t❤❡ ♦r✐❣✐♥❛• ♠♦❞❡•✱ t❤❡ ❝♦♥tr♦• ✐♥♣✉ts ❛r❡ ❝♦✉♣•❡❞ ✇✐t❤ t❤❡ st❛t❡s✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❛♥❛•②s✐s ♦❢ t❤❡ s②st❡♠ ❞✐✣❝✉•t✳ ❚❤❡ r❡•❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ s②st❡♠ st❛t❡s ❝❛♥ ♥♦t ❜❡ ❝•❡❛r•② s❡❡♥✳ ■♥ t❤❡ ♥♦r♠❛• ❢♦r♠✱ t❤❡ s②st❡♠ st❛t❡s ❛r❡ ♥♦ •♦♥❣❡r ❝♦✉♣•❡❞ ✇✐t❤ t❤❡ s②st❡♠ ✐♥♣✉ts✱ ❛♥❞ ✇❡ ❝❛♥ s❡❡ t❤❡ ♣✐t❝❤ ❛♥❣•❡ θ ❤❛s ♠♦r❡ ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ x ❛①✐s ❛♥❞ t❤❡ r♦•• ❛♥❣•❡ φ ❤❛s ♠♦r❡ ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ y ❛①✐s✳ ❚❤❡♥✱ t❤❡ ❜✐❢✉r❝❛t✐♦♥s ♦❢ ✐ts ♥♦r♠❛• ❢♦r♠ ❛r❡ st✉❞✐❡❞✳ ❚❤❡ s②st❡♠ ✇✐t❤♦✉t ✐♥♣✉t ❤❛s t✇❡•✈❡ ③❡r♦ ❡✐❣❡♥✈❛•✉❡s✱ ❛♥❞ t❤✐s ♠❡❛♥s t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠ ❞❡♣❡♥❞s ♦♥ t❤❡ ✐♥♣✉ts✳ •✐✈❡♥ s♦♠❡ ❣❡♥❡r❛• st❛t❡ ❢❡❡❞❜❛❝❦s ✇✐t❤ s♦♠❡ ♣❛✲ r❛♠❡t❡rs✱ t❤❡ ❝❤❛♥❣❡s ♦❢ t❤❡ ❡✐❣❡♥✈❛•✉❡s ✇❤✐❝❤ ❛r❡ ❝❛✉s❡❞ ❜② t❤❡ ❝❤❛♥❣❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ♣r♦♣♦s❡❞✳ ❯s✐♥❣ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r②✱ t❤❡ s②st❡♠ ❝❛♥ ❜❡ ❢✉rt❤❡r s✐♠♣•✐✜❡❞ ❛t ✐ts ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❚❤❡ t✇❡•✈❡ ❞✐♠❡♥s✐♦♥❛• s②st❡♠ ✐s t❤❡♥ s✐♠♣•✐✜❡❞ ✐♥t♦ ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛• s②st❡♠ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ■♥ ❈❤❛♣t❡r ✹✱ ✇✐t❤♦✉t •♦s✐♥❣ ❣❡♥❡r❛•✐t②✱ ❛♥ ❛❡r✐❛• ♣❤♦t♦❣r❛♣❤② s❝❡♥❛r✐♦ ✐♥ ❛♥ ♦♣❡♥ s♣❛❝❡ ✐s s❡•❡❝t❡❞✳ ❇❛s❡❞ ♦♥ t❤❡ ♥❡❝❡ss❛r② q✉❛•✐t✐❡s ✐♥ ❛♣♣•✐❝❛t✐♦♥s✱ ✜✈❡ r❡❛•✐st✐❝ s❝❡♥❛r✐♦s ❛r❡ ✶✶✽ ✻✳✷✳ ❋✉t✉r❡ ❲♦r❦ ❈❤❛♣t❡r ✻✳ ❈♦♥❝•✉s✐♦♥s ♣r❡s❡♥t❡❞✿ ❛♥ ✐❞❡❛• ❝❛s❡✱ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✱ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ❚❤❡s❡ r❡❛•✐st✐❝ ❝❛s❡s ❝❛♥ s❤♦✇ t❤❡ st❛❜✐•✐t② ❛♥❞ s❡♥s✐❜✐•✐t② ♦❢ ❛ ❝♦♥tr♦• ♠❡t❤♦❞ ✐♥ ❞✐✛❡r❡♥❝❡ ❛s♣❡❝ts✱ ❛♥❞ ❛ ❝♦♥tr♦• ♠❡t❤♦❞ ❝❛♥ ❤❛✈❡ ✈❛r②✐♥❣ ♣❡r❢♦r♠❛♥❝❡ ✐♥ ❡❛❝❤ ❛s♣❡❝t✳ ❚❤❡s❡ r❡s✉•ts ♠❛❦❡ ❣♦♦❞ r❡❢❡r❡♥❝❡s ❢♦r ❝❤♦♦s✐♥❣ ❛ ❝♦♥tr♦• ♠❡t❤♦❞ ❢♦r ❝❡rt❛✐♥ ❛♣♣•✐❝❛t✐♦♥✳ ■♥ ♦r❞❡r t♦ ❢✉rt❤❡r s❛✈❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s✱ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡ ✐s ♣r❡s❡♥t❡❞ ❛♥❞ ✉s❡❞ ♦♥ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❚❤❡♥✱ ❛ ♥❡✇•② ♣r♦♣♦s❡❞ ♠❡t❤♦❞ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ♣r❡s❡♥t❡❞✳ ■t ✐s ❛ s✐♠♣•❡ ❜✉t ❡✣❝✐❡♥t t❡❝❤♥✐q✉❡ ❢♦r ♥♦♥•✐♥❡❛r✱ ✉♥❦♥♦✇♥ ♦r ♣❛rt✐❛••② ❦♥♦✇♥ ❞②♥❛♠✐❝s✳ ■♥st❡❛❞ ♦❢ ❝♦♠♣✉t✐♥❣ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ s②st❡♠✱ t❤❡ ❞②♥❛♠✐❝s ✐s ❡✈❛•✉❛t❡❞ ❛s ❛ ✈❛r✐❛❜•❡ F ✐♥ r❡❛• t✐♠❡ ❜② t❤❡ s②st❡♠ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❝❛♥ ❝♦♠♣❡♥s❛t❡ ✇❡•• t❤❡ ♠♦❞❡•✐♥❣ ❡rr♦rs ❛♥❞ r❡❛•✲t✐♠❡ ❞✐st✉r❜❛♥❝❡s✳ ❚❤❡ ❛•❣♦r✐t❤♠ ♦❢ t❤✐s ❝♦♥tr♦• ♠❡t❤♦❞ ✐s r❛t❤❡r s✐♠♣•❡ ❛s ✐t ❞♦❡s ♥♦t ❢✉••② ❝♦♠♣✉t❡ t❤❡ ❞②♥❛♠✐❝ ♠♦❞❡•✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ t❤❡ st❛❜✐•✐t② ♦❢ t❤✐s ♠❡t❤♦❞ ✐s ❤✐❣❤ ❛s ✐t ❝♦♠♣❡♥s❛t❡s ✇❡•• t❤❡ ❞✐st✉r❜❛♥❝❡s✳ ■♥ ♦r❞❡r t♦ s❤♦✇ ✐ts ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s✱ s❡✈❡r❛• ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ❛•s♦ ♣r♦♣♦s❡❞✱ s✉❝❤ ❛s ❛ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ ❛ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ❚❤❡ t✐♠❡ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡s ❛r❡ ✐♠♣•❡♠❡♥t❡❞ ✐♥ ❛•• t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s✳ ■♥ ❈❤❛♣t❡r ✺✱ t❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ♦❢ t❤❡ t✐♠❡ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ ❛•• ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ✜✈❡ s❝❡♥❛r✐♦s ❛r❡ ❣✐✈❡♥✳ ■♥ ♦r❞❡r t♦ ❝•♦s❡ t♦ t❤❡ r❡❛•✐st✐❝ ❝❛s❡s✱ t❤❡ ❝♦♥tr♦• ❣❛✐♥s ♦❢ ❡❛❝❤ ❝♦♥tr♦• ♠❡t❤♦❞ ❛r❡ t❤❡ s❛♠❡ ✐♥ ✜✈❡ ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✳ ❯s✉❛••② ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ❝❤♦s❡♥ ❜❡❢♦r❡❤❛♥❞ ❛♥❞ ✇✐•• ♥♦t ❜❡ ❝❤❛♥❣❡❞ ❞✉r✐♥❣ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇❤❡♥ ❞✐st✉r❜❛♥❝❡s ♦❝❝✉r✳ ❚❤❡ r❡s✉•ts ❛r❡ ❛♥❛•②③❡❞ ❛♥❞ ❝♦♠♣❛r❡❞ t❤♦r♦✉❣❤•② ✐♥ ❞✐✛❡r❡♥t ❛s♣❡❝ts✳ ❚❡♥ ✐♠♣♦rt❛♥t ❝r✐t❡r✐❛ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✶ ❛r❡ •✐st❡❞✱ s✉❝❤ ❛s t❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r✱ t❤❡ ❡rr♦r ✈❛r✐❛♥❝❡✱ t❤❡ ❛❝t✉❛t✐♦♥ st❡♣s✱ t❤❡ ❡♥❡r❣② ❝♦♥s✉♠♣t✐♦♥✱ ❡t❝✳ ■♥ ❡❛❝❤ s❝❡♥❛r✐♦✱ t❤❡ ✈❛•✉❡s ♦❢ t❤❡s❡ ❝r✐t❡r✐❛ ❛r❡ ❝❛•❝✉•❛t❡❞ ❛♥❞ ♣r❡s❡♥t❡❞ ✐♥ t❛❜•❡s ❛♥❞ ✜❣✉r❡s✳ ❚❤❡ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s ♦❢ ❡❛❝❤ ❝♦♥tr♦• ♠❡t❤♦❞ ❝❛♥ ❜❡ s❡❡♥ ❝•❡❛r•②✳ ❚❤❡s❡ r❡s✉•ts ♠❛❦❡ ❛ ❣♦♦❞ r❡❢❡r❡♥❝❡ ❢♦r ❝❤♦♦s✐♥❣ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❝♦♥tr♦• ♠❡t❤♦❞ ❢♦r ❛♥ ❛♣♣•✐❝❛t✐♦♥✳ ✻✳✷ ❋✉t✉r❡ ❲♦r❦ ❙♦♠❡ ♦❢ t❤❡ ❝❤❛••❡♥❣❡s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ st✐•• s♦♠❡ ✇♦r❦ t♦ ✜♥✐s❤ ✐♥ t❤❡ ❢✉t✉r❡✳ ✭✶✮ ❆ s✐♠✉•❛t✐♦♥ ✇♦r❦❜❡♥❝❤✳ ❇❛s❡❞ ♦♥ t❤❡ ♣r♦❣r❛♠s ♦❢ ❇♦✉❛❜❞❛••❛❤✱ s♦♠❡ s✐♠✉•❛t✐♦♥ ♣r♦❣r❛♠s ❛r❡ ✇r✐tt❡♥ ✐♥ t❤✐s ✶✶✾ ❈❤❛♣t❡r ✻✳ ❈♦♥❝•✉s✐♦♥s ✻✳✷✳ ❋✉t✉r❡ ❲♦r❦ t❤❡s✐s✳ ■t ❝❛♥ ❜❡ ❛ ❣♦♦❞ s✐♠✉•❛t✐♦♥ ✇♦r❦❜❡♥❝❤ ❢♦r ❢✉t✉r❡ q✉❛❞r♦t♦r st✉❞✐❡s✳ ❍♦✇❡✈❡r ✐♥ t❤❡ ♠♦❞❡•✐♥❣✱ ♠❛♥② ❛❡r♦❞②♥❛♠✐❝ ❡✛❡❝ts ❛r❡ ♥♦t ❢✉••② ♣r❡s❡♥t❡❞ ♦r ♥❡❣•❡❝t❡❞✱ s✉❝❤ ❛s t❤❡ ❣r♦✉♥❞ ❡✛❡❝t✳ ❚❤❡ ❛❡r♦❞②♥❛♠✐❝ ❡✛❡❝ts ❛r❡ ❞❡✜♥✐t❡•② ♥❡❡❞❡❞ ♠♦r❡ ✐♥t❡♥s✐✈❡ st✉❞✐❡s✳ •❡tt✐♥❣ ❞❡✲ t❛✐•❡❞ ❛❡r♦❞②♥❛♠✐❝ ♠❛t❤❡♠❛t✐❝ ❡①♣r❡ss✐♦♥s ♠❛② ❜❡ ♥♦t ✉s❡❢✉• ❢♦r ❝♦♥tr♦• s②st❡♠ ❞❡s✐❣♥✱ ❛s t❤❡ ❡♠❜❡❞❞❡❞ s②st❡♠ ❤❛s •✐♠✐t ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s✳ ❍♦✇❡✈❡r✱ ✐t ✐s ✈❡r② ✉s❡❢✉• t♦ ❜✉✐•❞ ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ s✐♠✉•❛t✐♦♥ ✇♦r❦❜❡♥❝❤✳ ❆ st❛♥❞❛r❞ s✐♠✉•❛t✐♦♥ ✇♦r❦❜❡♥❝❤ ❛♥❞ s❝❡♥❛r✐♦s ❛r❡ ✉s❡❢✉• ❢♦r t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ ❝♦♥tr♦• ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ❜② ❞✐✛❡r❡♥t ❛✉t❤♦rs✳ ✭✷✮ ❋✉rt❤❡r r❡s❡❛r❝❤ ♦♥ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ♣r♦✈❡♥ ♣r❛❝t✐❝❛••② st❛❜•❡ ✐♥ t❤✐s t❤❡s✐s✳ ❚♦ ❢✉rt❤❡r ♣r♦✈❡ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ ♠❡t❤♦❞✱ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❛ s♣❡❝✐✜❝ s②st❡♠ ✐s ♥❡❡❞❡❞✳ ❆s t❤❡ ♣❛rt F ✐s ✉♣❞❛t❡❞ ❛t ❡❛❝❤ s❛♠♣•✐♥❣ t✐♠❡✱ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ✐s ✐♠♣♦rt❛♥t ❢♦r t❤❡ st❛❜✐•✐t②✳ ■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ❛ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ♦❢ ✶✵♠s ✐s s❡•❡❝t❡❞ ❜❛s❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r ❞②♥❛♠✐❝s ❛♥❞ r❡❛• ❝♦♥tr♦• ❞❡✈✐❝❡✳ ❚♦ s❡•❡❝t ❛ s❛♠♣•✐♥❣ ♣❡r✐♦❞ ❢♦r ❛ ❣❡♥❡r❛• s②st❡♠✱ ♠♦r❡ r❡s❡❛r❝❤❡s ❛r❡ ♥❡❡❞❡❞ ♦♥ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ✭✸✮ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦♥ ❛ r❡❛• q✉❛❞r♦t♦r✳ ▼❛♥② r❡❛•✐st✐❝ ❝❛s❡s ❛r❡ ♣r♦♣♦s❡❞ ✐♥ t❤✐s t❤❡s✐s✱ ❛♥❞ ♠❛♥② ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ s✐♠✉•❛t❡❞ ✐♥ ▼❆❚▲❆❇✳ ❋r♦♠ t❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts✱ t❤❡ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s ♦❢ ❡❛❝❤ ❝♦♥✲ tr♦• ♠❡t❤♦❞ ❝❛♥ ❜❡ s❡❡♥ ❝•❡❛r•②✳ ❍♦✇❡✈❡r ❞✉❡ t♦ t❤❡ •✐♠✐t ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥✱ t❤❡s❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ♥♦t ✐♠♣•❡♠❡♥t❡❞ ♦♥ ❛ r❡❛• q✉❛❞r♦t♦r✳ ❚❤❡r❡❢♦r❡✱ ♦♥❡ ♦❢ t❤❡ ❢✉t✉r❡ ✇♦r❦ ✐s t❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ♦♥ ❛ r❡❛• q✉❛❞r♦t♦r✳ ✶✷✵ ❆♣♣❡♥❞✐① ❆✳ ▼❆P▲❊ ❈♦❞❡ ❆♣♣❡♥❞✐① ❆ ▼❆P▲❊ ❈♦❞❡ > ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★ > ★★★★★★ ♥♦r♠❛• ❢♦r♠ ★★★★★★★★★★★★ > ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★ > ★★ t②•♦r ❡①♣❡ss✐♦♥ > r❡st❛rt✿ > ❞❡❣✿❂✼✿ ★★ ❞❡❣r❡❡ ❂ ❞❡❣✲✶ > s✐♥✭①✮✿❂s❡r✐❡s✭s✐♥✭①✮✱①✱❞❡❣✮✿ > ❝♦s✭①✮✿❂s❡r✐❡s✭❝♦s✭①✮✱①✱❞❡❣✮✿ > ★★ s✉❜st✐t✉❡ ✐♥ t❤❡ ♦r✐❣♥❛• s②st❡♠✳ ❡q ✸ ✐♥ t❤❡ ♣❛♣❡r ▼◆❚❙✳ > ❛rr❛②①✿❂s❡q✭①⑤⑤❦❂①❬❦(✱❦❂✶✳✳✶✷✮✿ > ①✶❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱①✷✮✮✿ > ①✷❞♦t✿❂♠t❛②•♦r✭s✉❜s✭❛rr❛②①✱❡①♣❛♥❞✭✲✭✉✶✰❣✮✯s✉❜s✭①❂①✾✱s✐♥✭①✮✮✮✮✱❬s❡q✭①❬❦(✱❦❂✶✳✳✶✷✮✱✉✶(✱❞❡❣✮✿ > ①✸❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱①✹✮✮✿ > ①✹❞♦t✿❂♠t❛②•♦r✭s✉❜s✭❛rr❛②①✱❡①♣❛♥❞✭✭✉✶✰❣✮✯s✉❜s✭①❂①✾✱❝♦s✭①✮✮✯s✉❜s✭①❂①✼✱s✐♥✭①✮✮✮✮✱❬s❡q✭①❬❦(✱ > ❦❂✶✳✳✶✷✮✱✉✶(✱❞❡❣✮✿ > ①✺❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱①✻✮✮✿ > ①✻❞♦t✿❂♠t❛②•♦r✭s✉❜s✭❛rr❛②①✱❡①♣❛♥❞✭✭✉✶✰❣✮✯s✉❜s✭①❂①✾✱❝♦s✭①✮✮✯s✉❜s✭①❂①✼✱❝♦s✭①✮✮✮✲❣✮✱❬s❡q✭①❬❦(✱ > ❦❂✶✳✳✶✷✮✱✉✶(✱❞❡❣✮✿ > ①✼❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱①✽✮✮✿ > ①✽❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱✉✷✮✮✿ > ①✾❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱①✶✵✮✮✿ > ①✶✵❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱✉✸✮✮✿ > ①✶✶❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱①✶✷✮✮✿ > ①✶✷❞♦t✿❂❡①♣❛♥❞✭s✉❜s✭❛rr❛②①✱✉✹✮✮✿ > ★★ ❝❤❛♥❣❡ t❤❡ ❡q✉✐•✐❜r✐✉♠ t♦ t❤❡ ♦r✐❣✐♥✳ > ①❬✶(
> ★★ s✉❜st✐t✉❡ ✐♥t♦ t❤❡ s②st❡♠✳ ❡q ✹ ✐♥ t❤❡ ♣❛♣❡r ▼◆❚❙✳ > ❞②✿❂▼❛tr✐①✭✶✷✱✶✮✿ > •✿❂❞♦t✿ > ❢♦r ✐ ❢r♦♠ ✶ t♦ ✶✷ ❞♦ > ❞②❬✐✱✶(✿❂♠t❛②•♦r✭①⑤⑤✐⑤⑤•✱❬s❡q✭②❬❦(✱❦❂✶✳✳✶✷✮✱✈❬✶(✱✈❬✷(✱✈❬✸(✱✈❬✹((✱❞❡❣✮✿ > ❡♥❞✿ > ❡✈❛•♠✭❞②✮❀ ✶✷✶ ❆♣♣❡♥❞✐① ❆✳ ▼❆P▲❊ ❈♦❞❡ > ★★ ❝❤❛♥❣❡ ✯ t♦ ✳ ❢♦r ♠❛tr✐①✳ > ♣r♦❞✉❝t❴t♦❴▼❛tr✐①❴♣r♦❞✉❝t✿❂♣r♦❝✭✐♥✈❤✮ > •♦❝❛• t✱❦✱♣✱♣❛rt❀ > ♣❛rt✿❂✵✿ > ❢♦r t ❢r♦♠ ✶ t♦ ♥♦♣s✭✐♥✈❤✮ ❞♦ > ✐❢ ♦♣✭✐♥✈❤✮❬t(❂✶✴✐❞❡♥ t❤❡♥ > ♣❛rt✿❂♣❛rt✰✶✴✐❞❡♥✿ > ❡•s❡ > ♣✿❂✶❀ > ❢♦r ❦ ❢r♦♠ ✶ t♦ ♥♦♣s✭♦♣✭✐♥✈❤✮❬t(✮ ❞♦ > ♣✿❂♣✳♦♣✭♦♣✭✐♥✈❤✮❬t(✮❬❦(❀ > ❡♥❞ ❞♦❀ > ♣❛rt✿❂♣❛rt✰♣❀ > ❡♥❞ ✐❢❀ > ❡♥❞ ❞♦❀ > ♣❛rt > ❡♥❞ ♣r♦❝✿ > ★★ ❡①♣r❡ss✐♦♥ ♦❢ ✭■ ✰ ❞◗✷ ✰✳✳✮)✭✲✶✮✳ ❡q ✻ ✐♥ t❤❡ ♣❛♣❡r ▼◆❚❙✳ > ✇✐t❤✭▲✐♥❡❛r❆•❣❡❜r❛✮✿ > ✐❞❡♥✿❂❡✈❛•♠✭■❞❡♥t✐t②▼❛tr✐①✭✶✷✮✮✿ > ✐♥✈❤s②♠❜♦•✿❂♠t❛②•♦r✭✶✴✭✐❞❡♥✰❞◗✷✰❞◗✸✰❞◗✹✰❞◗✺✰❞◗✻✮✱❬❞◗✷✱❞◗✸✱❞◗✹✱❞◗✺✱❞◗✻(✱❞❡❣✮✿ > ✐♥✈❤✿❂♣r♦❞✉❝t❴t♦❴▼❛tr✐①❴♣r♦❞✉❝t✭✐♥✈❤s②♠❜♦•✮✿ > ❞◗✸✿❂❡✈❛•♠✭▼❛tr✐①✭✶✷✱✶✷✮✮✿ > ❞◗✹✿❂❡✈❛•♠✭▼❛tr✐①✭✶✷✱✶✷✮✮✿ > ❞◗✺✿❂❡✈❛•♠✭▼❛tr✐①✭✶✷✱✶✷✮✮✿ > ❞◗✻✿❂❡✈❛•♠✭▼❛tr✐①✭✶✷✱✶✷✮✮✿❞◗✼✿❂❡✈❛•♠✭▼❛tr✐①✭✶✷✱✶✷✮✮✿ > ★★★★★★★ ✷t❤ ❞❡❣r❡❡ ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★ > ★★ ② ❞♦t ❜② st❛t❡s ③✳ > ❤③✶✿❂s❡q✭②❬✐(❂③❬✐(✱✐❂✶✳✳✶✷✮✿ > ❞②❜②③✶✿❂❡✈❛•♠✭s✉❜s✭❤③✶✱❞②✮✮❀ > ★★ ❝❛•✉•❛t❡ ✭■ ✰ ❞◗✷ ✰✳✳✮)✭✲✶✮✳ > ❞◗✷✿❂❡✈❛•♠✭▼❛tr✐①✭✶✷✱✭✐✱•✮✲❃❛❬✐✱•(✮✮✿ > ✐♥✈❤✶✿❂❡✈❛•♠✭✐♥✈❤✮✿ > ★★ ❡q ✻ ✐♥ t❤❡ ♣❛♣❡r ▼◆❚❙✳ > ❞③✶✿❂❡✈❛•♠✭✐♥✈❤✶✫✯❞②❜②③✶✮✿ > ★★ ❢✐♥❞ t❤❡ t❡r♠s ❢✭③✱❛✮✯✉✶✳ ✉✶ ❤❛s ❞❡❣r❡❡ ✶✳ t❤❡r❡❢♦r❡✱ t❤❡ ✶st ❞❡❣r❡❡ t❡r♠s ✐♥ ❢✭③✱❛✮ s❤♦✉•❞ ❜❡ ✵ t♦ ❡♥s✉r❡ t❤❡ ✷♥❞ ❞❡❣r❡❡ s②st❡♠ ❤❛s ♥♦ ❝♦✉♣•❡❞ t❡r♠s✳ t❤❡ ✶st ❞❡❣r❡❡ ✐♥ ❢✭③✱❛✮ ❛r❡ ❡✐t❤❡r ③ ♦r ❛✳ > ❢♦r ♠ ❢r♦♠ ✶ t♦ ✶✷ ❞♦ > ❡q❬♠(✿❂ > ♠t❛②•♦r✭♠t❛②•♦r✭❝♦❡❢❢✭❞③✶❬♠✱✶(✱✉✶✮✱❬s❡q✭s❡q✭❛❬✐✱•(✱✐❂✶✳✳✶✷✮✱•❂✶✳✳✶✷✮(✱✶✮✱❬s❡q✭③❬✐(✱✐❂✶✳✳✶✷✮(✱✷✮ > ✲♠t❛②•♦r✭♠t❛②•♦r✭❝♦❡❢❢✭❞③✶❬♠✱✶(✱✉✶✮✱❬s❡q✭s❡q✭❛❬✐✱•(✱✐❂✶✳✳✶✷✮✱•❂✶✳✳✶✷✮(✱✶✮✱❬s❡q✭③❬✐(✱✐❂✶✳✳✶✷✮(✱✶✮ > ✰♠t❛②•♦r✭♠t❛②•♦r✭❝♦❡❢❢✭❞③✶❬♠✱✶(✱✉✶✮✱❬s❡q✭③❬✐(✱✐❂✶✳✳✶✷✮(✱✶✮✱❬s❡q✭s❡q✭❛❬✐✱•(✱✐❂✶✳✳✶✷✮✱•❂✶✳✳✶✷✮(✱✷✮ > ✲♠t❛②•♦r✭♠t❛②•♦r✭❝♦❡❢❢✭❞③✶❬♠✱✶(✱✉✶✮✱❬s❡q✭③❬✐(✱✐❂✶✳✳✶✷✮(✱✶✮✱❬s❡q✭s❡q✭❛❬✐✱•(✱✐❂✶✳✳✶✷✮✱•❂✶✳✳✶✷✮(✱✶✮❂✵❀ > ❡♥❞❀ 0 = 0 > ■❞❡❛•✿❂❬s❡q✭❡q❬✐(✱✐❂✶✳✳✶✷✮(✿ > ❙✶✿❂s♦•✈❡✭■❞❡❛•✱s❡•❡❝t✭❤❛s✱✐♥❞❡ts✭■❞❡❛•✮✱❛✮✮✿ > ★★★★★★★★ ✸r❞ ❞❡❣r❡❡ ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★ > ★❡✈❛•♠✭❞②✮✿ > ◗✷✿❂❬s❡q✭✵✱✐❂✶✳✳✶✷✮(✿ ✶✷✷ ❆♣♣❡♥❞✐① ❆✳ ▼❆P▲❊ ❈♦❞❡ > ★★ ❝❛•❝✉•❛t❡ ◗✷ ❢r♦♠ ❞◗✷ ✭❙✶✮ > ✇✐t❤✭•✐♥❛•❣✮✿ > ❢♦r ❦ ❢r♦♠ ✶ t♦ ♥♦♣s✭❙✶✮ ❞♦ > ✐❢ •
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Event driven intelligent PID controllers with applications to motion control

Jing Wang * , * * * Hugues Mounier * Arben Cela * * Silviu-Iulian Niculescu *

INTRODUCTION

The trend to complex embedded control systems brings out a lot of new challenges. On one hand, the embedded character demands reduced complexity controllers. On the other hand, the complexity of the controlled systems enforces robustness of the proposed control schemes. Many constraints have to be taken into account, especially in distributed systems (see [START_REF] Murray | Future Directions in Control in an Information-Rich World[END_REF]). Low computational cost control schemes which are able to deal with nonlinear systems with robustness are needed.

Model free control has been proven to be a simple but very efficient nonlinear feedback technique for the unknown or partially known dynamics (see [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF], Choi et al. [2009]). We shall here use so-called intelligent PID (or i-PID). While retaining the PID reduced computational cost, it is able to cope with general types of nonlinearities. A precise relationship between i-PID and PIDs is given in d'Andréa-Novel et al. [2010]. It particularly emphasizes the ease of tuning of i-PID gains and gives a clearcut explanation of the performance of usual PIDs.

Contrarily to the time triggered control scheme which the control signals are sent to the actuator board every fixed sampling time, in the event based scheme, the control signals are sent only upon the triggering of an event (see Årzén [1999]). A typical event is that the tracking error goes beyond a specified limit. This type of scheme allows to go beyond the traditional Shannon sampling limit while still achieving asymptotic stability. We here propose an event based scheme for intelligent PID. The two techniques quoted above enable the efficiency and reduced complexity of the controller.

In the first section, the general setting of model free control and intelligent PID (i-PID) controllers are recalled. Then, event driven i-PID controllers are introduced. The simulations on the simplified models of longitudinal dynamics of a car and aerodynamics of quadrotor are then given.

MODEL FREE CONTROL

General setting

Model free control is a quite recent and very efficient technique for unknown and partially known systems (see [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF]). The input-output behavior of the system is approximatively governed within its operating range by a partially known or totally unknown finite-dimensional ordinary linear or non-linear differential equation. For the sake of simplicity, the input and output are assumed to be mono-variable. The system is described implicitly as E(y, ẏ, . . . , y (a) , u, u, . . . , u (b) ) = 0 (1) where E : R a+1 ×R b+1 → R is a sufficient smooth function of its arguments. Assume that for integer ν, 0 < ν ι, ∂E/∂y (ν) ≡ 0. The implicit function theorem (see [START_REF] Krantz | The Implicit Function Theorem: History, Theory, and Applications[END_REF]) allows to express y (ν) locally y (ν) = E(t, y, ẏ, . . . , y (ν-1) , y (ν+1) , . . . , y (ι) , u, u, . . . , u (κ) ) with the function E : R × R ι × R κ+1 → R.

Replace (1) by the following phenomenological model which is only valid in a very short time interval.

y (ν) = F + αu (2) where
• α ∈ R is a non-physical constant parameter, which is chosen by the engineer in such a way that F and αu are of the same magnitude.

• The derivation order ν is also an engineer's choice.

• F is determined thanks to the knowledge of u, α, and of the estimate of y (ν) .

An estimate of F is obtained as follows:

F = ŷ(ν) -αũ (3)
where ŷ(ν) is an estimate of the ν th derivative of the measure y which is assumed available, and ũ is an approximate value of u, in order to avoid algebraic loops in the controllers. Among the existing possibilities, ũ can be chosen as a past value of the control variable u. The resulting controller is then

u = 1 α y (ν) r -F + Λ(e -ξ,ζ
) where

• y r is a reference trajectory which is selected as in flatness-based control (see [START_REF] Fliess | Flatness and defect of nonlinear systems : introductory theory Fig. 4. Event triggered i-PID control of quadrotor and examples[END_REF]). • e = y r -y is the tracking error.

• e -ξ,ζ = ( ξ e, ξ-1 e, . . . , e, ė, e

(ζ) ), ξ, ζ ∈ [0, ν],
k is the k iterated integral, and Λ is an appropriate function R ξ+ζ+1 → R such that the closed loop error dynamics e (ν) = Λ(e -ξ,ζ ) is asymptotically stable. Remarks 2.1. a) The derivation order ν is not necessarily equal to the derivation order a of y in Equation ( 1). b) The derivation order ν, is often taken equal to 1 or 2, yielding so called intelligent PIDs or i-PID (see next subsection). c) A system may be partially unknown. It is straightforward to adapt the previous method. d) The estimate in (3) can be obtained for example through a simple first order filtering as

L ( ŷ) = s 1 + T f s L (y)
typically, 1/T f ranges from 8 to 20, and L denotes the transformation to the operational domain. It can also be given by efficient algebraic techniques (see [START_REF] Mboup | Numerical differentiation with annihiators in noisy environment[END_REF]) yielding for example the following estimate for the first derivative

ŷ = -3! T 3 T 0 (T -2τ )y(τ )dτ
with T an integration window size which order of magnitude is 20 times the sampling time in a time triggered setting.

Intelligent PIDs

The desired behavior is obtained by implementing, for instance ν = 2, the intelligent PID controller (i-PID) is

u = - F α + ÿr α + K P e + K I e + K D ė (4)
where K P , K I , K D are the usual tuning gains.

Let us consider the following special cases:

• If ν = 2, we may also employ an intelligent PD controller (i-PD)

u = - F α + ÿr α + K P e + K D ė (5)
• If ν = 1, we restrict ourselves to an intelligent PI controller (i-PI)

u = - F α + ẏr α + K P e + K I e (6)
or even to an intelligent P controller (i-P)

u = - F α + ẏr α + K P e (7)
Remarks 2.2. a) If ν = 2 (resp. 1), plugging Equations ( 4) or (5) (resp. ( 6) or ( 7)) in Equation ( 2) yields the control of a pure double (resp. simple) integrator. This is why tuning the gains of our intelligent controllers is quite straightforward. b) It should be emphasized, if ν = 2 (resp. 1), that Equation (5) (resp. ( 7)) is mathematically sufficient for ensuring stability around the reference trajectory. The integral term K I e in Equation (4) (resp. ( 6)) is however adding some well known robustness properties.

EVENT DRIVEN MODEL FREE CONTROL

The basic Årzén's event based controller consists of two parts: a time triggered event detector τ ed and an event triggered PID controller τ ec . See Årzén [1999]. The latter computes the control signal to be delivered to the actuator board. The former τ ed runs at a fixed sampling period h ed , and upon fulfillment of a certain event triggering law L et , sends events to τ ec . Upon reception of the event, τ ec computes the control signal and sends it to the actuator board.

Examples of event triggering laws L et are:

• Error threshold law: |e(t k )| > e lim (8
) where e = y r -y is the tracking error, t k is the current discrete sensing time by τ ed , and e lim is a fixed limit. [2007]). σ is chosen less than one to ensure an associated Lyapounov function decrease. a and b are chosen according to the Lipschitz constants of the K ∞ (consisting of all functions γ R + → R + which are continuous, strictly increasing, satisfying γ(0) = 0 and lim ξ→∞ = ∞. See, e.g., Sontag [2007]).

The present control goal is path tracking. We shall use geometric information on the reference trajectory y r . Namely, shall take the following event triggering scheme:

|e(t k ) -e(t k-1 )| > e lim ∧ t k -t k-1 > max(σ( ẏr )).h M σ( ẏr (t k )) (11
) where σ is a saturation function, and h M is the maximum sampling time ensuring stability in a time triggered scheme. We have chosen the following smooth saturating function

σ(ξ) = H -l 2(ξ H -ξ l ) φ(ξ) + ψ(ξ) + ξ H + ξ l 2 φ(ξ) = 1 ζ ln cosh(ζ(ξ -ξ l )) (12) ψ(ξ) = -1 ζ ln cosh(-ζ(ξ -ξ H ))
with l and H the low and high saturated values, ξ l and ξ H the beginning and ending abscissa of the linear part, and ζ is a stiffness value. The ln(cosh(ξ)) functions enable to have a linear part (when ξ ≪ 0 , cosh(ξ) ≈ exp(-ξ)/2, and ln(cosh(ξ)) ≈ -ξ/2; when ξ ≫ 0 , cosh(ξ) ≈ exp(ξ)/2, and ln(cosh(ξ)) ≈ ξ/2) with smooth transitions between the constant and linear parts.

APPLICATION TO VEHICLE LONGITUDINAL CONTROL

Model

We shall take a simplified model of longitudinal car dynamics as example. See [START_REF] Kiencke | Automotive Control Systems: For Engine, Driveline, And Vehicle[END_REF]. No attempt will be made to take longitudinal slip into account. Thus, the motor torque is supposed to be directly transmitted to the longitudinal dynamics.

The simplified model is as the following:

M Vx = C r -C ae (V x + V w ) V x + V w -M g sin(θ)- M gC rr sign(V x ) cos(θ) (13)
where M the vehicle's mass, V x the vehicle's longitudinal speed. C the traction torque which is taken as control input. r the wheel's mean radius. C ae the coefficient. V w the wind speed disturbance. g the gravity constant. θ the road slope. C rr the Rolling resistance coefficient.

The chosen values for the parameters are: M = 1200kg, V x = 0 to 36m/s, r = 0.025m, C ae = 0.015Ns 2 /m 2 , V w = 0 to 14m/s, θ = 0 to 0.52rad, C rr = 0.15. In the second member of equation ( 13): The first term is the traction force. The second term is the aerodynamics force. The third term is the slope effect force, and the fourth term is the rolling resistance force.

Model free setting

The model given in ( 13) can be expressed as

Vx = F + 1 M r C (14)
with

F = 1 M (-C ae (V x + V w ) V x + V w -M g sin(θ)-
M gC rr sign(V x ) cos(θ)) (15) which is of the form (2) with α = 1/M r. Thus, we have

C = M r Vxr -F -k p e -k i t 0 e(τ )dτ F = Vx - 1 M r C (16) e = V x -V xr
with V xr the reference speed, Vx an estimate of the derivative of V x , and C a past value of C (an approximation of C).

For instance, we can take the above form in discrete time

C(t k ) = C(t k-1 ) + M r ê(t k ) + k p e(t k ) + k i I(t k ) ê(t k ) = Vxr (t k ) -Vx (t k ) e(t k ) = V xr (t k ) -V x (t k ) I(t k ) = I(t k-1 ) + he(t k ) (17) Vx (t k ) = T f T f + h Vx (t k-1 ) + 1 T f + h V x (t k ) -V x (t k-1 ) h = t k -t k-1
For comparison, a usual PID takes the following form

C(t k ) = K p e(t k ) + K i I(t k ) e(t k ) = V xr (t k ) -V x (t k ) I(t k ) = I(t k-1 ) + he(t k ) (18) h = t k -t k-1

Simulations: continuous ideal flatness based control

Supposing we have the full knowledge of the dynamics, the ideal flatness based control is of the form:

C = M r V xr -F -k p e -k i t 0 e(τ )dτ F = -C ae (V x + V w ) V x + V w -M g sin(θ)- M gC rr sign(V x ) cos(θ) (19) e = V x -V xr
error in the case of flatness based control is depicted in figure 1.

Simulations: time triggered PI control and i-PID control

We first compare the cases of a time triggered PID and a time triggered i-PID.

Consider a fixed sampling time of h = 10ms (knowing that h = 35ms is the limit of stability). This yields 1976 actuation steps. We take a PI controller with gains k p = 17000 and k i = 100. The reference trajectory and the tracking error are depicted in figure 1.

Simulations: event triggered PI control and i-PID control

We now consider the event triggered controls. The event triggering scheme for PI control is the classical error difference of equation ( 9). The limit e lim in ( 9) is taken as e lim = max(y r ) -min(y r ) 200 It yields 291 actuation steps and the tracking error is given in figure 1. The i-PI controller is with gains K p = 60 and K i = 6.

Discussion

Note that the maximum absolute tracking error is 6.4.10 -2 m/s in the PI case, and 3.2.10 -3 m/s in the i-PI case which is 20 times less (2000%) than in the PI case. If we exclude the first second, the maximum absolute tracking error in the i-PI case is of 5.2.10 -4 , which is 123 times less than in the PI case.

Consider now an i-PI control. The event triggered scheme is the one given in equation (11), with l = 1, L = 20, ξ l = -2, ξ H = 2, and ζ = 6. The corresponding tracking error is given in figure 1, and was performed in 569 actuation steps. The gain in performance, when using an i-PI instead of a PI, is 68.18 and the loss in actuation steps is 1.95.

Model free control has better performance than PI control. Using the event triggered schemes, i-PID can further reduce the number of actuation loops, which is very useful for real time control systems.

APPLICATION TO QUADROTOR CONTROL

Model

The chosen model of quadrotor is depicted in equations (20). See Bouabdallah [2007]. The rotation angles φ, θ and ψ are along the world axis x, y and z respectively, namely, roll, pitch and yaw. Ω r (i = 1..4) are the angular velocities of each rotor, which are the real inputs of the quadrotor. The forces T i , H i (i = 1..4) are the thrust and hub forces of each motor. The moments R i , Q i (i = 1..4) are the drag and rolling moments of each rotor. The quantities ω1 ω2 (I ii -I ii ), J r ωΩ r (ω = φ, θ, ψ; i = x, y, z) are the body gyroscopic effects and propeller gyroscopic effects. The notations c and s represent cos and sin respectively. The values of all the parameters can be found in Bouabdallah [2007].

I xx φ = θ ψ(I yy -I zz ) + J r θΩ r + l(-T 2 + T 4 )- h( 4 i=1 H yi ) + (-1) i+1 4 i=1 R mxi I yy θ = φ ψ(I zz -I xx ) -J r φΩ r + l(T 1 -T 3 )- h( 4 i=1 H xi ) + (-1) i+1 4 i=1 R myi I zz ψ = θ φ(I xx -I yy ) + (-1) i 4 i=1 Q i + l(H x2 -H x4 ) + l(-H y1 + H y3 ) mz = -mg + (cθcφ) 4 i=1 T i mẍ = (sψsφ + cψsθcφ) 4 i=1 T i - 4 i=1 H xi - 1 2 C x A c ρ ẋ| ẋ| mÿ = (-cψsφ + sψsθcφ) 4 i=1 T i - 4 i=1 H yi - 1 2 C y A c ρ ẏ| ẏ| (20)
The most important forces and moments are the thrust T and the rolling moments Q. Therefore, we can take

u 1 = 4 i=1 T i u 2 = l(-T 2 + T 4 ) u 3 = l(T 1 -T 3 ) u 4 = (-1) i 4 i=1 Q i
as control inputs to compute the needed torques for each rotor, and then use them to control the altitude z, position x, y and direction ψ.

Altitude z control

The equation given in (20) related to z can be expressed as

mz = (cθcφ)u 1 + F z (21)
where F z can be considered as disturbances (e.g. the wind) or some parts of dynamics neglected in (20). In discrete time, the unknown part F z can be expressed as following. The estimate of z(k) is denoted as ẑ(k).

Fz = m ẑ(t k ) -(cθcφ)u 1 (t k-1 ) (22)
Therefore, the chosen control input is

u 1 (t k ) = u 1 (t k-1 ) + m cθcφ (ê z 2d (t k ) + k z 1 e z d (t k ) + k z 2 e z (t k )) (23) with êz 2d (t k ) = zr (t k ) -ẑ(t k ), e z d (t k ) = żr (t k ) -ż(t k ) e z (t k ) = z r (t k ) -z(t k ) ẑ(t k ) = T f T f + h ẑ(t k-1 ) + 1 T f + h z(t k ) -z(t k-1 )
zr , żr , z r are the reference acceleration, velocity and position of z. The variable sampling step is h = t k -t k-1 .

Position x,y control

We want to use u 2 and u 3 to control directly the position x, y. Therefore, we need to differentiate twice the equations related to x and y in (20) in order to appear the control inputs u 2 and u 3 . Since the equations in x and y are coupled, we get

x (4) = u 1 mI xx (sψcφ -cψsθsφ)u 2 + u 1 mI yy (cψcθcφ)u 3 + F x y (4) = - u 1 mI xx (cψcφ + sψsθsφ)u 2 + u 1 mI yy (sψcθcφ)u 3 + F y (24
) where F x , F y are considered as the badly known parts. For simplicity, we define A = u1 mIxx (sψcφ -cψsθsφ), B = u1 mIyy (cψcθcφ), C = -u1 mIxx (cψcφ + sψsθsφ) and D = u1 mIyy (sψcθcφ). Using the model free control scheme as before, we get

u 2 (t k ) u 3 (t k ) = u 2 (t k-1 ) u 3 (t k-1 ) + A B C D -1       êx 4d + 3 i=0 k x i e x id êy 4d + 3 i=0 k y i e y id       (25)
where êx 4d , êy 4d are the errors between the references x (4) r , y (4) r and the estimates of x (4) , y (4) .

Yaw control

For yaw control, we consider the equation of ψ as

I zz ψ = u 4 + F ψ (26) Then the control feedback is u 4 (t k ) = u 4 (t k-1 ) + I zz (ê ψ 2d (t k ) + k ψ 1 e ψ d (t k ) + k ψ 2 e ψ (t k )) (27)
where êψ 2d is the error between the reference ψr and the estimate of ψ.

Simulation: time triggered control

The task is to follow a rounded square path with length of 2m while hovering at the altitude of 10m, which is given in (28). The desired length is h d , and T f is the time needed to reach the desired length. Here we choose h d equals 2m, and T f equals 6s. The reference trajectory is in figure 2.

In the time triggered i-PID control, the sampling time is 10ms, and it yields 2785 actuation steps. The results are given in figure 3. The red lines are the desired trajectories. The maximum errors in x and y are both less than 0.05m, that is, less than 2, 5% of the desired length.

(t) =                0 0 t t 1 , t 4 < t 30 h d t 5 t 5 + (T f -t) 5 t 1 < t t 2 2 t 2 < t t 3 h d -h d t 5 t 5 + (T f -t) 5 t 3 < t t 4 h d = 2, T f = 6 x = σ(t) with t 1 = 3,

Simulation: event triggered control

In the event triggered i-PID control, the event triggering law is the absolute error limit. We set the error limit of z to be 0.1m. The error limit of yaw angle is 0.1rad. For x and y, we take the error limits both as 0.1m. The event triggered i-PID control yields 2389 actuation steps. The results are given in figure 4.

Discussion

The system mentioned in (20) is not complete. The aerodynamics of the system is complicated, and many more forces and moments will affect the system. Therefore, a control scheme which can adapt to the changes of the system is needed. The time triggered model free control performed nicely. It controls the system without the need of computing all the forces and moments in the system. In event triggered i-PID control, we set the the error limit to 5% of the reference. It has 396 steps less comparing to the time triggered model free scheme while still achieving stability.

CONCLUSION

Event triggered model free controllers which yields strong robustness while needing few computing resources is proposed in this paper. It is very efficient to control the nonlinear multi-input-output system which traditional PID is not able to. The i-PID control is also efficient to solve the partially known systems. From the simulation of a quadrotor model, we see that the i-PID control scheme avoids the heavy computations of the control laws, forces, moments and 4th derivatives of the variables. Moreover, the event triggered scheme enables to eliminate the small vibrations in the system while diminishing the number of actuation steps. 1.1) is a mini unmanned aerial vehicle (UVA) with four rotors, which has been widely studied in the last decades [1,2,3,4]. It is a system with four inputs, six outputs and highly coupled states. Due to its simplicity both in mechanical structure and maneuver, it is widely used in surveillance, search and rescue, mobile sensor networks [1]. Many methods have been proposed for controlling quadrotors. For example, Bouabdallah et al. [2] have proposed a backstepping control used separately in two subsystems. Besnard et al. [3] have proposed a sliding mode control driven by a disturbance observer. Wang et al. [4] have presented an event driven model free control which can avoid heavy computation. However, to the best of the authors' knowledge, the bifurcation of the dynamical system have never been studied. The method of normal forms is an useful approach in studying the dynamical system properties [5]. Its purpose is employing successive coordinate transformations 4.2. Center manifold. The aim of this part is to get the reduced system which can determine the stability and possible local bifurcations of the system at one bifurcation point [12]. A system can be written as: we can separate the Jordan matrix J as matrices B and C whose eigenvalues have zero real parts and negative real parts respectively. Therefore, we can rewrite the system (4.1) at the origin with x = [0, ..., 0]. ẏ0 = By 0 + f (y 0 , y -), ẏ-= Cy 0 + g(y 0 , y -).

Since the center manifold is tangent to E c (the y -= 0 space), we define Therefore, we can get the local evolution equations of y 0 which can determine the stability of the original system.

In quadrotor center manifold analysis, the control laws are defined as: The bifurcation of the system is like in previous subsection. When b < 0, the system has two eigenvalues with positive real parts. When b > 0, the system has all eigenvalues with negative real parts. When b = 0, the system has two pure imaginary eigenvalues ±16i, and all other eigenvalues have negative real parts. The stability depends on the nonlinear parts of the system. We can use the center manifold theory to simplify the system, and further simplify the study of the bifurcation of the system. In this control system, y 0 = [y 1 , y 2 ] T = [z 5 , z 6 ] T and y -= [y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9 , y 10 , y 11 , y 12 ] T = [z 1 , z 2 , z 3 , z 4 , z 7 , z 8 , z 9 , z 10 , z 11 , z 12 ] T . We seek a quadratic center manifold (a are parameters to be defined later):

y i = a i200 y 2 1 + a i020 y 2 2 + a i002 b 2 + a i110 y 1 y 2 + a i101 y 1 b + a i011 y 2 b, i = 3..12 Using the method mentioned before, we get h(y 0 , b) = [-0.62b 2 , -0.62b 2 , -0.76b 2 , -0.76b 2 , 0, 0, 0, 0, -0.42b 2 , -23.58b 2 ] in equation (4.2).

Therefore, the reduced system on the center manifold can be written:

Quadrotor control.

Here we propose a control method based on the normal form and Lyapunov theory. In equation (3.5), the Jacobian matrix of the system can be easily found. If the system is time invariant, the indirect method of Lyapunov says that if the eigenvalues of Jacobian matrix of the system at the origin are in the open left half complex plane, then the origin is asymptotically stable. Therefore, we can define the state feedback as follows to move all the eigenvalues of the system to the open left half plane. x r , y r , z r , ψ r are the references. The simulation task is to let quadrotor follow a square path with the length of 2m while hovering at the altitude of 10m, which is given in Figure 5.1. The totally sample time is 20s. For comparison, the simulations using a standard PID control are also given.

5.1. Simulation without wind disturbance. The simulation results are given in Figure 5.2. The desired response time is 1s. We can see that the proposed method has better performance than a standard PID control. 5.2. Simulation with wind disturbance. During the trajectory, there may have wind disturbance with velocity 1m/s as in Figure 5.1, which occurs in all x, y and z axis. The simulation results are given in Figure 5.3. The desired response time is 1s. The proposed method can keep the stability during the wind disturbance, and has better performance than a standard PID control.

Conclusion.

In this paper, the normal form of quadrotor is deduced. A Maple package 'QualitativeODE' [11] has been written for calculating the normal form of any degree of the system. From equation (3.5), we can see that the highly coupled parts in quadrotor system are eliminated. This makes the analysis of the dynamical system easier. Under certain control laws, the system can be further deduced using center manifold theorem. A two dimensional system is deduced which can determine the stability and possible local bifurcations of the control system at the origin. Based on the normal form and indirect method of Lyapunov, we proposed a state feedback control method with computational simplicity as well as practical implementation facility. This method achieved good results. In the simulations, the system can remain stable with small tracking errors even if there is wind disturbance. Also, this method has faster response time than a standard PID control.

I. INTRODUCTION

Quadrotors are vertical take-off and landing (VTOL) aircrafts with four rotors, which have embedded microprocessors, micro-electro-mechanical (MEMS) sensors and Lithium Polymer (LiPo) batteries. Due to the simplicity in the design and maneuver, quadrotors have many applications, such as border patrol, surveillance, aerial photography, etc. The quadrotor system is nonlinear, which has twelve states highly coupled with the inputs (see eq. ( 14)). Its aerodynamics is complex and difficult to include all the parts in the modeling, which makes it a partially known system. In the applications, quadrotors are often affected by disturbances, such as wind, weather conditions, etc. Therefore, the quadrotor control systems bring out many challenges: control algorithm complexity reduction; energy consumption reduction; robustness to perturbations; fast response to environmental and system changes, etc.

Many control methods are proposed in literature: Castillo et al. have proposed a Lyapunov controller using a nested saturation algorithm [1]; S. Bouabdallah has implemented a backstepping control and a sliding mode control [2]; Mistler et al. have used a dynamic feedback control [3]; Mokhtari et al. have applied a mixed feedback linearization with linear GH ∞ controller. However, to the best of authors' knowledge, these methods are tested in the ideal cases without disturbances, and the comparison of the control methods have never been proposed on quadrotor in realistic scenarios.

The recently introduced model free control is proposed for the challenges in the control of quadrotor. A preliminary work can be found in [4]. It is a simple but efficient technique for the nonlinear, unknown or partially known dynamics [5]. While retaining the PID reduced computational cost, it is able to cope with general types of nonlinearities. The comparison between the model free control and traditional PID controllers is given in d'Andréa-Novel et al. [6]. Model free control has been implemented in some academic SISO systems [5], joint motion control in humanoid locomotion [7], non-minimum phase systems [8], etc.

In order to further save the computational resources and energy consumption, the event triggered scheme is proposed on the model free control. Contrarily to the time triggered control scheme, in the event based scheme the control signals are sent only upon the triggering of an event. The event driven control was firstly proposed by Årzén [9]. The comparisons of the time driven and event driven control scheme for first order stochastic and nonlinear systems are proposed in [10] and [11] respectively.

The paper is organized as follows: In section II the event driven model free is presented; In section III the scenarios without and with wind disturbance are firstly presented. The model of quadrotor is then given. The model free control, backstepping control and sliding mode control laws are presented; In section IV the simulation results of the control laws in two scenarios are presented in both time and event driven schemes. The comparisons results in other realistic scenarios are also given.

II. EVENT DRIVEN MODEL FREE CONTROL

A finite dimensional SISO system can be described implicitly as E(y, ẏ, . . . , y (a) , u, u, . . . , u (b) ) = 0,

where E : R a+1 ×R b+1 → R is a sufficient smooth function of its arguments. Assume that for integer ν, 0 < ν ι, ∂E/∂y (ν) ≡ 0. The implicit function theorem allows to express y (ν) locally y (ν) = E(t, y, ẏ, . . . , y (ν-1) , y (ν+1) , . . . , y (ι) , u, u, . . . , u (κ) ),

(2) with the function E : R × R ι × R κ+1 → R. No matter the system is linear or not, we can rewrite the system (1) as following phenomenological model which is only valid in a very short time interval:

y (ν) = F + αu, (3)
where α ∈ R is a non-physical constant parameter, which is chosen by the engineer in such a way that F and αu are of the same order of magnitude. The derivation order ν is also an engineer's choice.

Here, F stands for the neglected parts of the system. It can be determined by the knowledge of u, α and y. An estimate of F is obtained as follows:

F = ŷ(ν) -αũ, (4)
where ŷ(ν) is an estimate of the ν th derivative of the measure y which is assumed available, and ũ is an approximate value of u. Among the existing possibilities, ũ can be chosen as a past value of the control variable u. The resulting controller is then defined as:

u = 1 α y (ν) r -F + Λ(e) , (5)
where y r is a reference trajectory. The variable e = y r -y is the tracking error and Λ is an appropriate function such that the closed loop error dynamics e (ν) = Λ(e) is asymptotically stable.

From above, we can see that the derivation order ν is not necessarily equal to the derivation order a of y in eq. ( 1). The derivation order ν is often taken equal to 1 or 2.

The estimate of y (v) in ( 4) can be obtained for example through a cascade of first order filter as:

L ( ŷ) = s 1 + T f s L (y). (6)
Typically, 1/T f ranges from 8 to 20, and L denotes the transformation to the operational domain.

Here, we choose the function Λ(e) as PID controller. The desired behavior is obtained by implementing the so-called intelligent PID controller (for instance ν = 2):

u = - F α + ÿr α + K P e + K I e + K D ė, (7)
where K P , K I , K D are the usual tuning gains. The basic Årzén's event based controller consists of two parts: a time triggered event detector C t and an event triggered PID controller C e [9]. The latter computes the control signal to be delivered to the actuators. The former C t runs at a fixed sampling period h e , and upon fulfillment of a certain event triggering law L e , sends events to C e . Upon reception of the event, C e computes the control signal and sends it to the actuators.

Usual event triggering laws L e include: (1) Error threshold law:

|e(t k )| > e lim , (8)
where e = y r -y is the tracking error, t k is the current discrete sensing time by C e , and e lim is a fixed limit.

(2) Error difference threshold:

assuming the system can be rendered ISS (Input to State Stable) through static feedback [12]. Here, σ is chosen less than one to ensure an associated Lyapounov function decreases, a and b are chosen according to the Lipschitz constants of K ∞ [12]).

In order to ensure the stability of the system, a maximum sampling period h M is defined in [10]. The time interval between two events must then be smaller than h M :

t k -t k-1 < h M . (11)
Other conditions ensuring the stability of the system have also proposed, such as the forgetting factor used in event driven PID control [11].

III. THE QUADROTOR MODEL: SCENARIO AND CONTROL

A. Scenario

The proposed scenario is a photo shooting at an outdoor garden. An autonomous quadrotor with limited energy is used. The quadrotor needs to follow a square path with length of 2m while hovering at the altitude of 10m, which is given in Figure 1. At each corner, the quadrotor will hover about 15s to take photos. The total simulation time is 150s. The reference trajectory is expressed as: x = σ(t), with t 1 = 10s, t 2 = 16s, t 3 = 90s, t 4 = 96s. y = σ(t), with t 1 = 40s, t 2 = 46s, t 3 = 120s, t 4 = 126s.

σ(t) =                0 0s t t 1 , h d (t-t1) 5 (t-t1) 5 +(T f -t+t1) 5 t 1 < t t
z = 10m. ( 12
)
As the shooting takes place at an outdoor garden, there may (

) 16
Further details about the model can be found in [START_REF] Wang | Analysis and control of quadrotor[END_REF].

C. Model free control

Firstly, we control the altitude z. We rewrite the vertical dynamics in (14) as:

mz = (cθcφ)u 1 + F z . ( 17
)
where F z includes the neglected vertical dynamics in eq. ( 14). In discrete time, the unknown part F z can be expressed as following, where ẑ(k) is an estimate of z(k):

Fz = m ẑ(t k ) -(cθcφ)u 1 (t k-1 ). (18)
Thus, the chosen control law is:

u 1 (t k ) =
) = z r (t k ) -z(t k ), ẑ(t k ) = T f T f + h ẑ(t k-1 ) + 1 T f + h ż(t k ) -ż(t k-1 ) , ẑ(t k ) = T f T f + h ẑ(t k-1 ) + 1 T f + h z(t k ) -z(t k-1 ) , (19)
where zr , żr , z r are the reference acceleration, velocity and position of z. The variable h is the sampling period, h = t k -t k-1 .

Then we control the position x and y. As the input u 1 is already used in the control of the altitude z, we now use u 2 and u 3 to control the positions x and y. Therefore, we need to differentiate twice the equations related to x and y in eq. ( 14) in order to get the control inputs u 2 and u 3 . Then, we obtain:

x (4) =
where F x , F y are the remaining parts of the horizontal and lateral system. For further simplicity, here we define

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  ♥ t❤✐s s❡❝t✐♦♥✱ t✇♦ ❡①❛♠♣•❡s ❛r❡ ❡①♣♦s❡❞✳ ■♥ t❤❡ ✜rst ❡①❛♠♣•❡✱ t❤❡ •✐♠✐t❛t✐♦♥ ♦❢ t❤❡ tr❛✲ ❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦• ✐s ♣r❡s❡♥t❡❞✳ ❆s ✇✐❞❡•② ❦♥♦✇♥✱ ❝•❛ss✐❝❛• P■❉ ❝♦♥tr♦••❡rs ❤❛✈❡ s✐♠♣•❡ ❝♦♥tr♦• •❛✇s ❛♥❞ ❞♦ ♥♦t ♥❡❝❡ss❛r✐•② ♥❡❡❞ s♣❡❝✐✜❝ ❦♥♦✇•❡❞❣❡ ♦❢ t❤❡ s②st❡♠✳ ❚❤✐s ✐s ♦♥❡ ♦❢ t❤❡ r❡❛s♦♥s ♦❢ ✐ts ♣♦♣✉•❛r✐t② ✐♥ ♣r♦❝❡ss ❝♦♥tr♦• ❬✶✱ ✼✽(✳ ❆•t❤♦✉❣❤ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦••❡rs ❛r❡ ✇✐❞❡•② ✉s❡❞ ✐♥ ✐♥❞✉str✐❛• ❝♦♥tr♦•✱ t❤❡② s❤♦✇ ♣♦♦r ♣❡r❢♦r♠❛♥❝❡ ❢♦r s❡✈❡r❛• ❝•❛ss❡s ♦❢ ♥♦♥•✐♥❡❛r s②st❡♠s✱ s✉❝❤ ❛s t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ t❤❡ st✉❞✐❡s ♦♥ s✉♣❡r✐♦r ❝♦♥tr♦• ♠❡t❤♦❞s ❢♦r t❤❡ q✉❛❞r♦t♦r s②st❡♠ ❛r❡ ♥❡❡❞❡❞✳■♥ t❤❡ s❡❝♦♥❞ ❡①❛♠♣•❡✱ t❤❡ ❞✐✛❡r❡♥t ♣❡r❢♦r♠❛♥❝❡s ♦❢ ❛ ❝♦♥tr♦• s②st❡♠ ✐♥ ❛♥ ✐❞❡❛• ❝❛s❡ ❛♥❞ ✐♥ ❛ r❡❛•✐st✐❝ ❝❛s❡ ❛r❡ ♣♦✐♥t❡❞ ♦✉t✳ ❯s✉❛••②✱ ✇❤❡♥ ❛ ♥❡✇ ❝♦♥tr♦• ♠❡t❤♦❞ ✐s ♣r♦♣♦s❡❞✱ s♦♠❡ s✐♠✉•❛t✐♦♥s ❛r❡ ♠❛❞❡ ✐♥ ❛♥ ✐❞❡❛• ❝❛s❡ t♦ ♣r♦✈❡ t❤❡ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤✐s ♠❡t❤♦❞✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✉s✉❛••② ❤✐❣❤ t♦ ❣❡t ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡✳ ■♥ ✐❞❡❛• ❝❛s❡s✱ t❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ❣♦♦❞ ✇✐t❤ s♠❛•• tr❛❝❦✐♥❣ ❡rr♦rs✳ ❍♦✇❡✈❡r✱ t❤❡ ❝•♦s❡ •♦♦♣ s②st❡♠s ✇✐t❤ ❤✐❣❤ ❣❛✐♥s •♦♦s❡ ✐ts st❛❜✐•✐t② q✉✐t❡ q✉✐❝❦•② ✐♥ r❡❛•✐st✐❝ ❝❛s❡s ✇✐t❤ ❞✐st✉r❜❛♥❝❡s✳ ❚❤❡r❡❢♦r❡✱ ✐t ❛♣♣❡❛rs t❤❛t t❤❡ ❝♦♠♣❛r✐s♦♥s ♦❢ ❞✐✛❡r❡♥t ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ✐❞❡❛• ❝❛s❡s ❛r❡ ♥♦t ❡♥♦✉❣❤✳ ❲❡ ❝❛♥ ♥♦t s❛② ♦♥❡ ❝♦♥tr♦• ♠❡t❤♦❞ ❤❛s ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ t❤❛♥ ❛♥♦t❤❡r s✐♠♣•② ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣❡r❢♦r♠❛♥❝❡s ✐♥ ✐❞❡❛• ❝❛s❡s✳ ✷ ✶✳✷✳ ❚✇♦ ■♥s♣✐r✐♥❣ ❊①❛♠♣•❡s ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✶✳✷✳✶ ▲✐♠✐t❛t✐♦♥ ♦❢ t❤❡ ❚r❛❞✐t✐♦♥❛• P■❉ ❈♦♥tr♦• ❚r❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦••❡rs ❛r❡ ✇✐❞❡•② ✉s❡❞ ✐♥ ✐♥❞✉str✐❛• ❝♦♥tr♦•s✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ t❤❡ s②s✲ t❡♠ ✐s ❤✐❣❤•② ♥♦♥•✐♥❡❛r✱ t✐♠❡ ✈❛r②✐♥❣ ✇✐t❤ ♠✉•t✐♣•❡ st❛t❡s✱ P■❉ ❝♦♥tr♦••❡rs s❤♦✇ ♣♦♦r ♣❡r✲ ❢♦r♠❛♥❝❡s✳ ❍❡r❡✱ ✇❡ ✐♠♣•❡♠❡♥t ❛ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦• ♦♥ t❤❡ q✉❛❞r♦t♦r ♠♦❞❡• ✇❤✐❝❤ ✐♥ ♣r♦♣♦s❡❞ ✐♥ ❈❤❛♣t❡r ✷ ✭s❡❡ ❊q✉❛t✐♦♥ ✭✸✳✷✮✮✳ ❚❤❡ ❝♦♥tr♦• •❛✇s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙✉❜s❡❝t✐♦♥ ✹✳✸✳✹✳ ❚❤❡ s❡•❡❝t❡❞ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜•❡ ✶✳✶✳ ❚❛❜•❡ ✶✳✶✿ ❚❤❡ P■❉ ❝♦♥tr♦• ❣❛✐♥s P■❉ ❝♦♥tr♦• ♦❢ ❛ q✉❛❞r♦t♦r✳ ❚❤❡ tr❛•❡❝t♦r② ❛•♦♥❣ t❤❡ x ❛①✐s ✭❛✮✱ ❛•♦♥❣ t❤❡ y ❛①✐s ✭❜✮✳ x r ✱ y r t❤❡ r❡❢❡r❡♥❝❡s✳ ✭❝✮ ❚❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛•♦♥❣ t❤❡ x ❛♥❞ y ❛①✐s✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛•♦♥❣ t❤❡ ❛①✐s ① ❛♥❞ ② ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✶✳✶ ✭❛✮ ❛♥❞ ✭❜✮✳ ❚❤❡ ✈❛•✉❡s x r ❛♥❞ y r ❛r❡ t❤❡ r❡❢❡r❡♥❝❡s✳ ❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦• ❤❛s ♣♦♦r ♣❡r❢♦r♠❛♥❝❡ ♦♥ t❤✐s ♥♦♥•✐♥❡❛r ♠♦❞❡•✳ ■t ❤❛s ❛ s•♦✇ r❡s♣♦♥s❡ t✐♠❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞❡s✐r❡❞ ♣❡r❢♦r♠❛♥❝❡✱ ✇❤✐❝❤ ❣✐✈❡s ♥♦t✐❝❡❛❜•❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✶✳✶ ✭❝✮✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✶✳✺♠✱ ✇❤✐❝❤ ✐s ✼✺✪ ♦❢ t❤❡ ✢② ❞✐st❛♥❝❡ ✷♠✳ ❲❡ ❝❛♥ s❡❡ t❤✐s tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦••❡r ❞♦❡s ♥♦t s✉✣❝❡ ✐♥ t❡r♠s ♦❢ r❡s♣♦♥s❡ t✐♠❡ ❢♦r t❤✐s ❝•❛ss ♦❢ ♥♦♥•✐♥❡❛r s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ t❤❡ st✉❞✐❡s ♦♥ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❢♦r t❤❡ q✉❛❞r♦t♦r s②st❡♠ ❛r❡ ♥❡❝❡ss❛r②✳ ✶✳✷✳✷ ❉✐✛❡r❡♥❝❡ ❇❡t✇❡❡♥ ■❞❡❛• ❛♥❞ |❡❛•✐st✐❝ ❈❛s❡s ■♥ t❤❡ •✐t❡r❛t✉r❡✱ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ✉s✉❛••② s✐♠✉•❛t❡❞ ♦r t❡st❡❞ ✐♥ ✐❞❡❛• ❝❛s❡s ✇✐t❤♦✉t ❛♥② ❞✐st✉r❜❛♥❝❡✳ ■♥ ♦r❞❡r t♦ ❣❡t ❛ ❜❡tt❡r r❡s✉•t✱ ❤✐❣❤ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✉s✉❛••② ❝❤♦s❡♥✳ ■♥ r❡❛•✐st✐❝ ✸ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✶✳✷✳ ❚✇♦ ■♥s♣✐r✐♥❣ ❊①❛♠♣•❡s ❝❛s❡s✱ t❤❡r❡ ❛r❡ ♠❛♥② ❞✐st✉r❜❛♥❝❡s ❛♥❞ ✉♥❝❡rt❛✐♥t✐❡s✳ ❆ ❝♦♥tr♦• s②st❡♠ ✇✐t❤ ❤✐❣❤ ❝♦♥tr♦• ❣❛✐♥s✱ ✇❤✐❝❤ ✐s ♠♦r❡ ✈✉•♥❡r❛❜•❡ t♦ ❝❤❛♥❣❡s✱ ♠❛② •♦s❡ ✐ts st❛❜✐•✐t② ✐♥ r❡❛•✐st✐❝ ❝❛s❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ✐♥ ✐❞❡❛• ❝❛s❡s ❝❛♥ ♥♦t ❢✉••② s❤♦✇ ✐ts ❢❡❛t✉r❡s ♦❢ ❛ ❝♦♥tr♦• ♠❡t❤♦❞✳ ■t s❤♦✉•❞ ❜❡ t❡st❡❞ ✐♥ ❞✐✛❡r❡♥t r❡❛•✐st✐❝ ❝❛s❡s t♦ s❤♦✇ ✐ts ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s✳ ❍❡r❡✱ ❛ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ♣r♦♣♦s❡❞ ✐♥ ❈❤❛♣t❡r ✹ ✐s ✜rst•② s✐♠✉•❛t❡❞ ✐♥ ❛♥ ✐❞❡❛• ❝❛s❡ ✇✐t❤♦✉t ❞✐st✉r❜❛♥❝❡s ❛♥❞ ✉♥❝❡rt❛✐♥t✐❡s✳ ❚❤❡ ❝❤♦s❡♥ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✶✳✷✱ ✇❤✐❝❤ ❛r❡ s❡•❡❝t❡❞ t♦ ♦❜t❛✐♥ t❤❡ s♠❛••❡st ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐♥ t❤❡ ✐❞❡❛• ❝❛s❡✳ ❚❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ❛•♦♥❣ t❤❡ x✱ y ❛♥❞ z ❛①✐s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✶✳✷ ✭❛✮✳ ■♥ t❤❡ ✐❞❡❛• ❝❛s❡✱ t❤❡ s②st❡♠ s❤♦✇s ✐ts st❛❜✐•✐t② ❞✉r✐♥❣ t❤❡ s✐♠✉•❛t✐♦♥✳ ❯s❡❞ t❤❡s❡ ❝♦♥tr♦• ❣❛✐♥s✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❤❛s t❤❡ s♠❛••❡st ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✵✳✵✵✾♠✱ ✇❤✐❝❤ ✐s ♦♥•② ✵✳✹✺✪ ♦❢ t❤❡ ✢② ❞✐st❛♥❝❡ ✷♠✳ ❚❛❜•❡ ✶✳✷✿ ❚❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❣❛✐♥s ✐♥ t❤❡ ✐❞❡❛• ❝❛s❡ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✶✳✼✳ ❊①✐st✐♥❣ ◗✉❛❞r♦t♦r ❈♦♥tr♦• ▼❡t❤♦❞s •✐♥❡❛r ❙■❙❖ s②st❡♠ ❜② ✉s✐♥❣ ❛ st❛t✐❝ st❛t❡ ❢❡❡❞❜❛❝❦✳ ❚❤❡② ❤❛✈❡ ✉s❡❞ ❛ ❞②♥❛♠✐❝ ❢❡❡❞❜❛❝❦ t❡❝❤♥✐q✉❡ t♦ •✐♥❡❛r✐③❡ t❤❡ s②st❡♠✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❝•♦s❡❞ •♦♦♣ s②st❡♠ •✐♥❡❛r ❛♥❞ ❝♦♥tr♦••❛❜•❡ ✐♥ ✷✵✵✶ ❬✻✾(✳ ❚❤❡ st❛❜✐•✐t② ❛♥❞ t❤❡ r♦❜✉st♥❡ss ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❝♦♥tr♦• ✉♥❞❡r ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ❛♥❞ ♣❛r❛♠❡tr✐❝ ✉♥❝❡rt❛✐♥t✐❡s ✇❡r❡ st✉❞✐❡❞✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✇❛s ♥♦ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s✳ ▼♦❦❤t❛r✐ ❡t ❛•✳ ❤❛✈❡ ❛♣♣•✐❡❞ ❛ ❞②♥❛♠✐❝ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦••❡r ♦❢ ❊✉•❡r ❛♥❣•❡s ✐♥ ✷✵✵✹ ❬✼✵(✱ ❛ ♠✐①❡❞ r♦❜✉st ❢❡❡❞❜❛❝❦ •✐♥❡❛r✐③❛t✐♦♥ ✇✐t❤ •✐♥❡❛r •❍ ∞ ❝♦♥tr♦••❡r ✐♥ ✷✵✵✺ ❬✼✶( ❛♥❞ ✇✐t❤ ❛ s•✐❞✐♥❣ ♠♦❞❡ ♦❜s❡r✈❡r ✐♥ ✷✵✵✻ ❬✼✷(✳ ■♥ ✷✵✵✻✱ ▼❛❞❛♥✐ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ❢✉•• st❛t❡ ❜❛❝❦st❡♣♣✐♥❣ t❡❝❤♥✐q✉❡ ♦♥ q✉❛❞r♦t♦r ❜② s❡♣❛r❛t✐♥❣ t❤❡ s②st❡♠ ✐♥t♦ t❤r❡❡ ✐♥t❡r❝♦♥♥❡❝t✲ ❡❞ s✉❜s②st❡♠s ❬✼✹(✳ ■♥ ✷✵✵✻✱ ❇❡♥❛••❡❣✉❡ ❡t ❛•✳ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ ❢❡❡❞❜❛❝❦ •✐♥❡❛r✐③❛t✐♦♥✲❜❛s❡❞ ❝♦♥tr♦••❡r ✇✐t❤ ❛ ❤✐❣❤✲♦r❞❡r s•✐❞✐♥❣ ♠♦❞❡ ♦❜s❡r✈❡r✳ ❚❤❡ s•✐❞✐♥❣ ♠♦❞❡ ♦❜s❡r✈❡r ✇♦r❦❡❞ ❛s ❛♥ ♦❜s❡r✈❡r ❛♥❞ ❡st✐♠❛t♦r ♦❢ ❡①t❡r♥❛• ❞✐st✉r❜❛♥❝❡s s✉❝❤ ❛s ✇✐♥❞ ❛♥❞ ♥♦✐s❡s ❬✷✸✱ ✷✹(✳ ❆t ❈❊❆ ❋r❛♥❝❡✱ •✉❡♥❛r❞ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛♥ ❛❞❛♣t✐✈❡ ❝♦♥tr♦• ♠❡t❤♦❞ ♦♥ ❛ q✉❛❞r♦t♦r ❳✹✲❋•②❡r ✐♥ ✷✵✵✻ ❬✹✾(✳ ❆♥ ❛tt✐t✉❞❡ ❝♦♥tr♦• ❜❛s❡❞ ♦♥ t❤❡ ♠♦❞❡• ♦❢ t❤❡ r♦t♦r✲❝r❛❢t ✇❛s ❞❡s✐❣♥❡❞ ❢♦r st❛t✐♦♥❛r② ❛♥❞ q✉❛s✐ st❛t✐♦♥❛r② ✢✐❣❤ts✳ ❆♥ ❛❞❛♣t✐✈❡ ❝♦♥tr♦••❡r ✇❛s ❞❡✈❡•♦♣❡❞ ❜❛s❡❞ ♦♥ ✉•tr❛s♦♥✐❝ ♠❡❛s✉r❡♠❡♥ts ✐♥ ♦r❞❡r t♦ •✐♠✐t t❤❡ ❣r♦✉♥❞ ❡✛❡❝ts ❢♦r t❤❡ ❛•t✐t✉❞❡ st❛❜✐•✐③❛t✐♦♥✳ •✉❡♥❛r❞ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ♣r❛❝t✐❝❛• ✈✐s✉❛• s❡r✈♦ ❝♦♥tr♦• ❢♦r ❛♥ ✉♥♠❛♥♥❡❞ ❛❡r✐❛• ✈❡❤✐❝•❡ ✐♥ ✷✵✵✽ ❬✺✵(✳ ❇❡rtr❛♥❞ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ❤✐❡r❛r❝❤✐❝❛• ❝♦♥tr♦••❡r ❢♦r ♠✐♥✐❛t✉r❡ ❱❚❖▲ ❯❆❱s ✉s✐♥❣ s✐♥❣✉•❛r ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ✐♥ ✷✵✶✶ ❬✷✽(✳ ❆t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❆•❛❜❛♠❛ ✭❯❙❆✮✱ ❇❡s♥❛r❞ ❡t ❛•✳ ❤❛✈❡ ❞❡✈❡•♦♣❡❞ ❛ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥✲ tr♦• ❞r✐✈❡♥ ❜② ❛ s•✐❞✐♥❣ ♠♦❞❡ ❞✐st✉r❜❛♥❝❡ ♦❜s❡r✈❡r ✐♥ ✷✵✵✼ ❬✷✺(✳ ❚❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞ ✇❛s r♦❜✉st t♦ ❡①t❡r♥❛• ❞✐st✉r❜❛♥❝❡ ✭✐♥❝•✉❞✐♥❣ ✇✐♥❞✱ ❝♦••✐s✐♦♥✱ ❛❝t✉❛t♦r ❢❛✐•✉r❡✮ ❛♥❞ ♠♦❞❡• ✉♥❝❡r✲ t❛✐♥t✐❡s ✇✐t❤♦✉t ✉s✐♥❣ ❤✐❣❤ ❝♦♥tr♦• ❣❛✐♥s ♦r ❡①t❡♥s✐✈❡ ❝♦♠♣✉t❛t✐♦♥❛• ♣♦✇❡r ❬✷✻(✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✇❛s ♥♦ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❆t t❤❡ ❈r❛♥✜❡•❞ ❯♥✐✈❡rs✐t② ✭❯❑✮✱ ❈♦✇•✐♥❣ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ▲◗| ❝♦♥tr♦••❡r ❢♦r ♣❛t❤ ❢♦••♦✇✐♥❣ ✐♥ ✷✵✵✼ ❬✸✻(✳ ❚❤❡② ❛•s♦ ♣r❡s❡♥t❡❞ ❛ tr❛•❡❝t♦r② ♣•❛♥♥✐♥❣ ♠❡t❤♦❞ ✉s✐♥❣ ❛ ❞✐✛❡r❡♥t✐❛• ✢❛t♥❡ss ♣r♦♣❡rt② ❛s ❛ ❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜•❡♠ ✐♥ t❤❡ ♦✉t♣✉t s♣❛❝❡ ✭❛s ♦♣♣♦s❡❞ t♦ t❤❡ ❝♦♥tr♦• s♣❛❝❡✮✳ ❚❤❡ ❝♦♥tr♦• ♠❡t❤♦❞ ❤❛s ❜❡❡♥ s✐♠✉•❛t❡❞ ✇✐t❤ ♥♦✐s❡ ❛♥❞ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡ ♠♦❞❡•❡❞ ❛s ❛ ❞r✐❢t ✐♥ t❤❡ tr❛♥s•❛t✐♦♥❛• ♣♦s✐t✐♦♥ ❬✸✼(✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✇❛s ♥♦ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s✳ ❆t t❤❡ ▲❛❦❡❤❡❛❞ ❯♥✐✈❡rs✐t② ✭❈❛♥❛❞❛✮✱ ❚❛②❡❜✐ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ P❉ 2 ❝♦♥tr♦••❡r✱ ✇❤❡r❡ t❤❡ ♣r♦♣♦rt✐♦♥❛• ❛❝t✐♦♥ ✐s ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡❝t♦r q✉❛t❡r♥✐♦♥ ❛♥❞ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡ ❛❝t✐♦♥s ❛r❡ ✐♥ t❡r♠s ♦❢ t❤❡ ❛✐r❢r❛♠❡ ❛♥❣✉•❛r ✈❡•♦❝✐t② ❛♥❞ t❤❡ ✈❡❝t♦r q✉❛t❡r♥✐♦♥ ✈❡•♦❝✐t② s✐♥❝❡ ✷✵✵✹ ❬✾✷✕✾✹(✳ ❚❤❡ ♠❡t❤♦❞ ❤❛✈❡ ♦♥•② t❡st❡❞ ✐♥ ✐❞❡❛• ❝❛s❡s✳ ❆t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❙❡✈✐••❛ ✭❙♣❛✐♥✮✱ |❛✛♦ ❡t ❛•✳ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛♥ ✐♥t❡❣r❛• ♣r❡❞✐❝t✐✈❡ ❛♥❞ ✶✽ ✶✳✽✳ ❈♦♥tr✐❜✉t✐♦♥s ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ♥♦♥•✐♥❡❛r ❍ ∞ ❝♦♥tr♦••❡r ✐♥ ✷✵✶✵ ❬✽✽✱ ✽✾(✳ ■t ✇❛s ❛ ❤✐❡r❛r❝❤✐❝❛• s❝❤❡♠❡ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ♠♦❞❡• ♣r❡❞✐❝t✐✈❡ ❝♦♥tr♦••❡r ✭▼P❈✮ t♦ tr❛❝❦ t❤❡ r❡❢❡r❡♥❝❡ tr❛•❡❝t♦r② t♦❣❡t❤❡r ✇✐t❤ ❛ ♥♦♥•✐♥❡❛r ❍ ∞ ❝♦♥tr♦••❡r t♦ st❛❜✐•✐③❡ t❤❡ r♦t❛t✐♦♥❛• ♠♦✈❡♠❡♥ts✳ ❚❤❡ ♠❡t❤♦❞ ✇❛s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦s ✇✐t❤ ♣❛r❛♠❡tr✐❝ ✉♥❝❡rt❛✐♥t✐❡s ✐♥ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ♣r♦♣♦s❡❞ ❜② ❇♦✉❛❜❞❛••❛❤ ❬✷✵(✳ ❆t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ P❛tr❛s ✭•r❡❡❝❡✮✱ ❆•❡①✐s ❡t ❛•✳ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ s✇✐t❝❤✐♥❣ ♠♦❞❡• ♣r❡❞✐❝t✐✈❡ ❛tt✐t✉❞❡ ❝♦♥tr♦••❡r ❜❛s❡❞ ♦♥ ❛ ♣✐❡❝❡✇✐s❡ ❛✣♥❡ ✭P❲❆✮ ♠♦❞❡• ✐♥ ✷✵✶✶ ❬✻(✳ ❚❤❡ ❝♦♥tr♦• ♠❡t❤♦❞ ✐s t❡st❡❞ ✐♥ t❤❡ ❝❛s❡ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✳ ✶✳✽ ❈♦♥tr✐❜✉t✐♦♥s ❚❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❝❤❛••❡♥❣❡s ✐♥ t❤❡ ♠♦❞❡•✐♥❣ ❛♥❞ ❝♦♥tr♦• ♦❢ ❛ q✉❛❞r♦t♦r ♠❡♥t✐♦♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✶✳✸✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤✐s t❤❡s✐s ❝❛♥ ❜❡ s✉♠♠❛r✐③❡❞ ❛s ❢♦••♦✇s✿ • ❆ ♥❡✇ ❞②♥❛♠✐❝ ♠♦❞❡• ✐s ♣r♦♣♦s❡❞ ❛♥❞ ❛ ❞②♥❛♠✐❝ ❛♥❛•②s✐s ❜❛s❡❞ ♦♥ t❤❡ ♥♦r♠❛• ❢♦r♠ t❤❡♦r② ✐s ❣✐✈❡♥✳ • ❆ ♥❡✇ ❝♦♥tr♦• ♠❡t❤♦❞✱ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✱ ✐s ♣r♦♣♦s❡❞ ❢♦r t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ • ❈♦♠♣❛r✐s♦♥s ♦❢ s❡✈❡r❛• ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ✜✈❡ r❡❛•✐st✐❝ s❝❡♥❛r✐♦s ❛r❡ ❣✐✈❡♥✳ ✭✶✮ ▼♦❞❡•✐♥❣✳ ■♥ t❤✐s t❤❡s✐s✱ t❤❡ r❡•❛t✐♦♥s❤✐♣ ♦❢ t❤❡ ❦✐♥❡♠❛t✐❝ ✈❛r✐❛❜•❡s ❜❡t✇❡❡♥ t❤❡ ❜♦❞② ❛♥❞ ✇♦r•❞ ❢r❛♠❡s ❛r❡ ✜rst•② s❤♦✇♥✳ ❚❤❡♥✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♣r❡s❡♥t❡❞✳ ❚❤❡ ✇♦r•❞ ❛♥❞ ❜♦❞② ❢r❛♠❡s ❛♥❞ t❤❡ ❊✉•❡r ❛♥❣•❡s r♦t❛t✐♦♥❛• s❡q✉❡♥❝❡ ❛r❡ s❡•❡❝t❡❞ ❜❛s❡❞ ♦♥ t❤❡ ♠♦st ❣❡♥❡r❛••② ✉s❡❞ r❡❢❡r❡♥❝❡s✳ ❚❤❡ ♠♦❞❡•✐♥❣ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❇♦✉❛❜s❛••❛❤✬s ♠♦❞❡•✐♥❣✳ ❚❤❡ ❇♦✉❛❜s❛••❛❤✬s ♠♦❞❡• ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ❝♦♠♣•❡t❡ ♠♦❞❡•✐♥❣ ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ s♦♠❡ ❡rr♦rs ✐♥ ❤✐s ♠♦❞❡•✳ ❚❤❡ ❤✉❜ ❢♦r❝❡ ✐s ❝❛•❝✉•❛t❡❞ ✉s✐♥❣ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t✐❡s ❛•♦♥❣ t❤❡ x w ❛♥❞ y w ✐♥ t❤❡ ✇♦r•❞ ❢r❛♠❡✳ ❚❤❡s❡ ❛r❡ ♥♦t ❡①❛❝t✳ ❚❤❡ ❤✉❜ ❢♦r❝❡ s❤♦✉•❞ ❜❡ ❝❛•❝✉•❛t❡❞ ✉s✐♥❣ t❤❡ •✐♥❡❛r ✈❡•♦❝✐t✐❡s ❛•♦♥❣ t❤❡ x b ❛♥❞ y b ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡ ♦♥ t❤❡ q✉❛❞r♦t♦r ♣•❛♥❡✱ ❛s t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡ ✐s ❝❛✉s❡❞ ❜② t❤❡ ❛✐r✢♦✇ s♣❡❡❞ r❡•❛t❡❞ t♦ t❤❡ q✉❛❞r♦t♦r ✇❤✐❝❤ ✐s ❝❤❛♥❣❡❞ ❞✉❡ t♦ t❤❡ q✉❛❞r♦t♦r ♣•❛♥❡✳ ▼♦r❡♦✈❡r✱ s♦♠❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥t ❛r❡ ♥♦t ❝♦rr❡❝t✱ s✉❝❤ ❛s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ r♦••✐♥❣ ♠♦♠❡♥t✱ ❡t❝✳ ❚❤❡ ❢r❛♠❡s ✉s❡❞ ✐♥ ❇♦✉❛❜s❛••❛❤✬s t❤❡s✐s ❛r❡ ♥♦t t❤❡ s❛♠❡ ❛s ✐♥ ❤✐s ♣r♦❣r❛♠✱ ✇❤✐❝❤ ✐s ❛ •✐tt•❡ ❜✐t ❝♦♥❢✉s✐♥❣✳ ❚❤❡r❡❢♦r❡ ✐♥ t❤✐s ✶✾ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✶✳✽✳ ❈♦♥tr✐❜✉t✐♦♥s t❤❡s✐s✱ t❤❡ ❝❤♦s❡♥ ✇♦r•❞ ❛♥❞ ❜♦❞② ❢r❛♠❡s✱ t❤❡ r❡•❛t✐♦♥s❤✐♣ ♦❢ t❤❡ •✐♥❡❛r ❛♥❞ ❛♥❣✉•❛r ✈❡•♦❝✐t✐❡s✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ❝•❡❛r•② ♣r❡s❡♥t❡❞✳ ❚❤❡ ❡rr♦rs ✐♥ t❤❡ ❇♦✉❛❜s❛••❛❤✬s t❤❡s✐s ❛r❡ ❝♦rr❡❝t❡❞✳ ❙♦♠❡ ✉♥♥❡❝❡ss❛r② ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ ♥❡❣•❡❝t❡❞✳ ❚❤❡♥ ❜❛s❡❞ ♦♥ t❤❡ ♥♦r♠❛• ❢♦r♠ t❤❡♦r②✱ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ q✉❛❞r♦t♦r ✐s ❛♥❛•②③❡❞✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ q✉❛❞r♦t♦r ❤❛s ♥❡✈❡r ❜❡❡♥ st✉❞✐❡❞ ✐♥t❡♥s✐✈❡•② ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❚❤❡ ♠♦❞❡• ♦❢ ❛ q✉❛❞r♦t♦r ✐s s✐♠♣•✐✜❡❞ ✐♥t♦ ❛ s✐♠♣•❡st ❢♦r♠ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ■♥ t❤❡ ♦r✐❣✐♥❛• ♠♦❞❡•✱ t❤❡ ❝♦♥tr♦• ✐♥♣✉ts ❛r❡ ❝♦✉♣•❡❞ ✇✐t❤ t❤❡ st❛t❡s✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❛♥❛•②s✐s ♦❢ t❤❡ s②st❡♠ ❞✐✣❝✉•t✳ ■♥ t❤❡ ♥♦r♠❛• ❢♦r♠✱ t❤❡ s②st❡♠ st❛t❡s ❛r❡ ♥♦ •♦♥❣❡r ❝♦✉♣•❡❞✇✐t❤ t❤❡ s②st❡♠ ✐♥♣✉ts✱ ❛♥❞ ✇❡ ❝❛♥ s❡❡ t❤❡ ♣✐t❝❤ ❛♥❣•❡ θ ❤❛s ♠♦r❡ ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ x ❛①✐s ❛♥❞ t❤❡ r♦•• ❛♥❣•❡ φ ❤❛s ♠♦r❡ ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ y ❛①✐s✳ ❚❤❡ ❜✐❢✉r❝❛t✐♦♥s ♦❢ t❤❡ s✐♠♣•✐✜❡❞ s②st❡♠ ❛r❡ st✉❞✐❡❞✳ ❯s✐♥❣ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r②✱ t❤❡ s②st❡♠ ❝❛♥ ❜❡ ❢✉rt❤❡r s✐♠♣•✐✜❡❞ ❛t ✐ts ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❚❤❡ t✇❡•✈❡ ❞✐♠❡♥s✐♦♥❛• s②st❡♠ ✐s s✐♠♣•✐✜❡❞ ✐♥t♦ ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛• s②st❡♠ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ✭✷✮ ❈♦♥tr♦• ♠❡t❤♦❞s✳ ■♥ t❤❡ ✜rst ✐♥s♣✐r✐♥❣ ❡①❛♠♣•❡✱ t❤❡ tr❛❞✐t✐♦♥❛• P■❉ ❝♦♥tr♦••❡r ❤❛s ♣♦♦r ♣❡r❢♦r♠❛♥❝❡ ✐♥ t❤❡ ❝♦♥tr♦• ♦❢ ❛ q✉❛❞r♦t♦r✳ ❆s ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥ ✶✳✸✱ q✉❛❞r♦t♦r s②st❡♠s ❜r✐♥❣ ♦✉t ♠❛♥② ❝❤❛••❡♥❣❡s✿ t❤❡ ❝♦♠♣•❡①✐t② ♦❢ t❤❡ ❝♦♥tr♦• ❛•❣♦r✐t❤♠ s❤♦✉•❞ ❜❡ s✐♠♣•❡❀ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞ s❤♦✉•❞ ❜❡ st❛❜•❡ ❢♦r ♠♦❞❡• ❛♥❞ ♣❛r❛♠❡t❡r ❡rr♦rs❀ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞ s❤♦✉•❞ ❜❡ ❛❜•❡ t♦ ❝♦♠♣❡♥s❛t❡ ❞✐st✉r❜❛♥❝❡s ❛♥❞ ❦❡❡♣ t❤❡ st❛❜✐•✐t② ♦❢ t❤❡ s②st❡♠ ✇✐t❤ •❡ss tr❛❝❦✐♥❣ ❡rr♦rs ❛♥❞ ✈✐❜r❛t✐♦♥s ♣♦ss✐❜•❡❀ ❡t❝✳ ❇❛s❡❞

  t❤✐s ❝♦♥tr♦• ♠❡t❤♦❞ ✐s r❛t❤❡r s✐♠♣•❡ ❛s ✐t ❞♦❡s ♥♦t ❢✉••② ❝♦♠♣✉t❡ t❤❡ ❞②♥❛♠✐❝ ♠♦❞❡•✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ t❤❡ st❛❜✐•✐t② ♦❢ t❤✐s ♠❡t❤♦❞ ✐s ❤✐❣❤ ❛s ✐t ❝♦♠♣❡♥s❛t❡s ✇❡•• t❤❡ ❞✐st✉r❜❛♥❝❡s✳ ❚♦ ❢✉rt❤❡r s❛✈❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s✱ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡ ✐s ♣r♦♣♦s❡❞ ♦♥ t❤✐s ❝♦♥tr♦• ♠❡t❤♦❞✳ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❛❞❛♣ts ✇❡•• t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡✳ ❲❤✐•❡ s❛✈✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• r❡s♦✉r❝❡s✱ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐s ♥♦t ❤❡❛✈✐•② ❝❤❛♥❣❡❞ ✐♥ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦•✳ ✷✵ ✶✳✾✳ ❖✉t•✐♥❡ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ■♥ ♦r❞❡r t♦ s❤♦✇ ✐ts ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s✱ s❡✈❡r❛• ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ❛•s♦ ♣r♦♣♦s❡❞✱ s✉❝❤ ❛s ❛ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ❛♥❞ ❛ s•✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦•✳ ❋✐✈❡ ❞✐✛❡r❡♥t r❡❛•✐st✐❝ s❝❡♥❛r✐♦s ❛r❡ ♣r♦♣♦s❡❞ t♦ ❣❡t ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ ❡✈❛•✉❛t✐♦♥✳ ✭✸✮ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♥ ❞✐✛❡r❡♥t r❡❛•✐st✐❝ s❝❡♥❛r✐♦s✳ ❆s s❤♦✇♥ ✐♥ t❤❡ s❡❝♦♥❞ ✐♥s♣✐r✐♥❣ ❡①❛♠♣•❡✱ t❤❡ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ ❝♦♥tr♦• ♠❡t❤♦❞ ✐♥ ❛♥ ✐❞❡❛• ❝❛s❡ ❝❛♥ ♥♦t ❡♥s✉r❡ t❤❡ s❛♠❡ ❣♦♦❞ ♣❡r❢♦r♠❛♥❝❡ ✐♥ ❛ r❡❛•✐st✐❝ ❝❛s❡✳ ❚❤❡r❡❢♦r❡ ✐♥ t❤✐s t❤❡s✐s✱ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ♦❢ q✉❛❞r♦t♦rs ❤❛✈❡ ❜❡❡♥ ❝❛r❡❢✉••② st✉❞✐❡❞✱ ❛♥❞ ✜✈❡ t②♣✐❝❛• r❡❛•✐st✐❝ s❝❡♥❛r✐♦s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞✿ ❛ ❜❛s✐❝ ✐❞❡❛• ❝❛s❡✱ ❛ ❝❛s❡ ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✱ ❛ ❝❛s❡ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✱ ❛ ❝❛s❡ ✇✐t❤ s❡♥s♦r ♥♦✐s❡s ❛♥❞ ❛ ❝❛s❡ ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✳ ■♥ ♦r❞❡r t♦ ❝•♦s❡ t♦ t❤❡ r❡❛•✐st✐❝ ❝❛s❡s✱ t❤❡ ❝♦♥tr♦• ❣❛✐♥s ♦❢ ❡❛❝❤ ❝♦♥tr♦• ♠❡t❤♦❞ ❛r❡ t❤❡ s❛♠❡ ✐♥ ✜✈❡ ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✳ ❚❤❡ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✉s✉❛••② ❝❤♦s❡♥ ❜❡❢♦r❡❤❛♥❞ ❛♥❞ ✇✐•• ♥♦t ❜❡ ❝❤❛♥❣❡❞ ❞✉r✐♥❣ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇❤❡♥ t❤❡ ❞✐st✉r❜❛♥❝❡s ♦❝❝✉r✳ ❋r♦♠ t❤❡ r❡s✉•ts ✐♥ t❤❡s❡ r❡❛•✐st✐❝ s❝❡♥❛r✐♦s✱ ✇❡ ❝❛♥ ❣❡t ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ ❡✈❛•✉❛t✐♦♥ ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s✱ ✇❤✐❝❤ ✐s ♠♦r❡ ♦❜•❡❝t✐✈❡ t❤❛♥ ✉s✐♥❣ •✉st t❤❡ r❡s✉•t ✐♥ ✐❞❡❛• ❝❛s❡s✳ ■♥ t❤❡ •✐t❡r❛t✉r❡✱ t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ r❛r❡•② ♣r❡s❡♥t❡❞✳ ▲❛❝❦ ♦❢ ❛ st❛♥❞❛r❞ s❝❡♥❛r✐♦✱ ✇❡ ❝❛♥ ❤❛r❞•② s❛② ✇❤✐❝❤ ♠❡t❤♦❞ ❤❛s ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ s❡✈❡r❛• ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ❛•s♦ ✐♠♣•❡♠❡♥t❡❞ ✐♥ t❤❡s❡ ✜✈❡ s❝❡♥❛r✐♦s✳ ❋r♦♠ t❤❡ r❡s✉•ts ✐♥ ❞✐✛❡r❡♥t ❝❛s❡s✱ t❤❡ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞✐s❛❞✈❛♥t❛❣❡s ♦❢ ❡❛❝❤ ♠❡t❤♦❞ ❝❛♥ ❜❡ ❡❛s✐•② ♦❜s❡r✈❡❞✳ ❚❤❡r❡❢♦r❡✱ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❝♦♥tr♦• ♠❡t❤♦❞ ❝❛♥ ❜❡ s❡•❡❝t❡❞ ❜❛s❡❞ ♦♥ t❤❡ ♥❡❡❞s ♦❢ ❛ ❝❡rt❛✐♥ ❛♣♣•✐❝❛t✐♦♥s✳ ✶✳✾ ❖✉t•✐♥❡ ❚❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦••♦✇s✿ ■♥ ❈❤❛♣t❡r ✷✱ t❤❡ ❦✐♥❡♠❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ♠♦❞❡•s ♦❢ q✉❛❞r♦t♦r ❛r❡ ♣r❡s❡♥t❡❞✳ ❚❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❛r❡ st✉❞✐❡❞✳ ❚❤❡ ❢✉•• ♠♦❞❡• ✐s ♣r♦♣♦s❡❞ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝❤❛♣t❡r✱ ❛♥❞ ❛ s✐♠♣•✐✜❡❞ ♠♦❞❡• ✐s ❛•s♦ ♣r❡s❡♥t❡❞ ❜❛s❡❞ ♦♥ s♦♠❡ ❛♣♣•✐❝❛t✐♦♥s✳ ■♥ ❈❤❛♣t❡r ✸✱ t❤❡ ♥♦r♠❛• ❢♦r♠ t❤❡♦r② ✐s ✐♠♣•❡♠❡♥t❡❞ ♦♥ t❤❡ q✉❛❞r♦t♦r ♠♦❞❡•✱ ❛♥❞ ❛ s✐♠♣•✐✜❡❞ ♠♦❞❡• ✐s ♦❜t❛✐♥❡❞ ✇❤✐❝❤ ❡①❤✐❜✐ts ❛•• ♣♦ss✐❜•❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛• s②st❡♠✳ ❚❤❡♥ t❤❡ ❜✐❢✉r❝❛t✐♦♥s ❜❛s❡❞ ♦♥ t❤❡ s✐♠♣•✐✜❡❞ ♠♦❞❡• ❛r❡ st✉❞✐❡❞✳ ❯s✐♥❣ t❤❡ ❝❡♥t❡r ♠❛♥✐❢♦•❞ t❤❡♦r②✱ t❤❡ ♠♦❞❡• ✐s ❢✉rt❤❡r s✐♠♣•✐✜❡❞ ❛t ✐ts ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ■♥ ❈❤❛♣t❡r ✹✱ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ♦❢ q✉❛❞r♦t♦rs ❛r❡ st✉❞✐❡❞✱ ❛♥❞ ✜✈❡ t②♣✐❝❛• r❡❛•✐st✐❝ s❝❡♥❛r✐♦s ❛r❡ ♣r♦♣♦s❡❞✳ ❚❤❡♥✱ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❛♥❞ ❡✈❡♥t tr✐❣❣❡r❡❞ s❝❤❡♠❡s ❛r❡ ♣r♦♣♦s❡❞✳ ❆t t❤❡ ❡♥❞✱ t❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ❛♥❞ ♦t❤❡r ❝♦♥tr♦• ♠❡t❤♦❞s ❛r❡ ♣r❡s❡♥t❡❞✳ ■♥ ❈❤❛♣t❡r ✺✱ t❤❡ t✐♠❡ tr✐❣❣❡r❡❞ ❛♥❞ ✷✶ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✶✳✾✳ ❖✉t•✐♥❡ ❡✈❡♥t tr✐❣❣❡r❡❞ ❝♦♥tr♦• ♠❡t❤♦❞s ✐♠♣•❡♠❡♥t❡❞ ✐♥ ✜✈❡ s❝❡♥❛r✐♦s ❛r❡ ♣r♦♣♦s❡❞ ✐♥ ❈❤❛♣t❡r ✹✳ ■♥ ❈❤❛♣t❡r ✻✱ t❤❡ ❝♦♥❝•✉s✐♦♥ ✐s ❣✐✈❡♥ ❛♥❞ ❢✉t✉r❡ ✇♦r❦ ✐s ❞✐s❝✉ss❡❞✳ ✷✷ ❈❤❛♣t❡r ✷ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✶ ■♥tr♦❞✉❝t✐♦♥ ▼❛♥② ❞②♥❛♠✐❝ ♠♦❞❡•s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ■♥ ❙❡❝t✐♦♥ ✶✳✻✱ t❤❡s❡ ♠♦❞❡•s ✇❡r❡ ❞✐s❝✉ss❡❞ ✐♥ ❞❡t❛✐•✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ♥♦ st❛♥❞❛r❞ ♠♦❞❡•✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ❛ ❦✐♥❡♠❛t✐❝ ♠♦❞❡• ❛♥❞ ❛ ❞②♥❛♠✐❝ ♠♦❞❡• ♦❢ q✉❛❞r♦t♦r ✇✐•• ❜❡ ❝♦♥str✉❝t❡❞✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ •✐t❡r❛t✉r❡✱ t❤❡ ♠♦st ❝♦♠♠♦♥ ❊✉•❡r ❛♥❣•❡ s❡q✉❡♥❝❡ )(❳ ✐s s❡•❡❝t❡❞✳ ■♥ t❤❡ ❞②♥❛♠✐❝ ♠♦❞❡•✐♥❣✱ t❤❡ ❛❡r♦❞②♥❛♠✐❝ ❢♦r❝❡s ❛♥❞ ♠♦♠❡♥ts ❞❡❞✉❝❡❞ ❜② ❋❛② ❛r❡ ✉s❡❞ ❬✹✷(✳ ❚❤❡ ❞②♥❛♠✐❝ ♠♦❞❡• ✐s ❜❛s❡❞ ♦♥ ❝♦♥st❛♥t t❤r✉st ❛♥❞ ♠♦♠❡♥t ❝♦❡✣❝✐❡♥ts✳ ❖t❤❡r ✐♠♣♦rt❛♥t ♠♦♠❡♥ts •✐❦❡ t❤❡ ❣②r♦s❝♦♣✐❝ ❡✛❡❝t ♠♦♠❡♥ts ❛r❡ ❛•s♦ ❝♦♥s✐❞❡r❡❞✳ ❆ s✐♠♣•✐✜❡❞ ♠♦❞❡• ✐s ❛•s♦ ♣r♦♣♦s❡❞ ❜❛s❡❞ ♦♥ s♦♠❡ s❝❡♥❛r✐♦s✳ ❆t t❤❡ ❡♥❞✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ •✐♠✐ts ♦❢ t❤❡ ♠♦t♦r ❛♥❞ t❤❡ s❛❢❡t② ♦❢ t❤❡ ✢✐❣❤t✱ s♦♠❡ ❝♦♥str❛✐♥ts ❛r❡ ❣✐✈❡♥ ❢♦r t❤❡ q✉❛❞r♦t♦r s②st❡♠✳ ✷✳✷ Pr❡•✐♠✐♥❛r② ❑♥♦✇•❡❞❣❡ ❇❡❢♦r❡ ❛♥❛•②③❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ q✉❛❞r♦t♦r✱ s❡✈❡r❛• ❛ss✉♠♣t✐♦♥s ❛r❡ ♠❛❞❡✿ • ❚❤❡ q✉❛❞r♦t♦r ✐s ❛ r✐❣✐❞ ❜♦❞②✳ • ❚❤❡ ♣r♦♣❡••❡rs ♦❢ t❤❡ ♠♦t♦rs ❛r❡ ❛•s♦ r✐❣✐❞✳ • ❚❤❡ q✉❛❞r♦t♦r ✐s s②♠♠❡tr✐❝❛• ❛•♦♥❣ t❤❡ x ❛♥❞ y ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❜♦❞② ❢r❛♠❡ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❝❡♥t❡r ♦❢ ❣r❛✈✐t②✳ ❚❤❡r❡❢♦r❡✱ I xx = I yy ✱ I xy = I yz = I zx = 0✳ • ❚❤❡ ♠♦t♦r ✐♥❡rt✐❛ ✐s ♥❡❣•✐❣✐❜•❡✳ ❈❤❛♣t❡r ✷✳ ❙②st❡♠ ▼♦❞❡•✐♥❣ ✷✳✷✳ Pr❡•✐♠✐♥❛r② ❑♥♦✇•❡❞❣❡ ❆s t❤❡ q✉❛❞r♦t♦r ✐s ❛ r✐❣✐❞ ❜♦❞②✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ◆❡✇t♦♥✲❊✉•❡r ❢♦r♠❛•✐s♠ ❢♦r t❤❡ ❞②♥❛♠✐❝ ♠♦❞❡•✐♥❣✳ ■♥ t❤❡ ❛♥❛•②s✐s✱ t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s s❡•❡❝t❡❞ ❛s ✐♥ ❋✐❣✉r❡ ✷✳✶✳ ❚❤❡ ❢r❛♠❡ W = {x w , y w , z w } ✐s t❤❡ ✇♦r•❞ ❢r❛♠❡✳ ❚❤❡ ❛①✐s z w ✐s ✈❡rt✐❝❛• t♦ t❤❡ ❣r♦✉♥❞✱ ❛♥❞ t❤❡ ❛①✐s x w ❛♥❞ y w •♦❝❛t❡ ✐♥ t❤❡ ❣r♦✉♥❞ ♣•❛♥❡✳ ❚❤❡ ❛♥❣•❡s φ, θ ❛♥❞ ψ ❛r❡ ❚❛✐t✲❇r②❛♥ ❛♥❣•❡s ✭❛•s♦ ❝❛••❡❞ ❈❛r❞❛♥ ♦r ❊✉•❡r ❛♥❣•❡s✮✱ ✇❤✐❝❤ r♦t❛t❡ ❛❜♦✉t t❤❡ x w ✱ y w ❛♥❞ z w ❛①✐s r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ ❢r❛♠❡ B = {x b , y b , z b } ✐s t❤❡ ❜♦❞② ❢r❛♠❡✳ ❚♦ s✐♠♣•✐❢② t❤❡ ♠❛t❤❡♠❛t✐❝❛• ❡①♣r❡ss✐♦♥s ♦❢ ♣❤②s✐❝❛•

  ✭✷✳✸✵✮ ❚❤❡s❡ t✇♦ ♠♦♠❡♥ts r❡s✉•t t❤❡ t✉r♥✐♥❣ ❛❜♦✉t t❤❡ x b ❛♥❞ y b ❛①✐s ✐♥ t❤❡ ❜♦❞② ❢r❛♠❡✳ ✹✳ |♦t♦r ❣②r♦s❝♦♣✐❝ ❡✛❡❝t ❋✐❣✉r❡ ✷✳✸✿ |♦t♦r ❣②r♦s❝♦♣✐❝ ❡✛❡❝t✳ ❲❤❡♥ t❤❡ r♦t♦r r♦t❛t❡s ❛r♦✉♥❞ ✐ts ♦✇♥ ❛①✐s ❛♥❞ ❛ s❡❝♦♥❞ ❛①✐s✱ ✐t ❛•s♦ ❜r✐♥❣s ✐♥t♦ r♦t❛t✐♦♥ ❛r♦✉♥❞ ❛ t❤✐r❞ ❛①✐s ✇❤✐❝❤ ✐s ♣❡r♣❡♥❞✐❝✉•❛r t♦ t❤❡ ❢♦r♠❡r t✇♦ ❛①✐s✳ ❚❤✐s ❡✛❡❝t ✐s ❝❛••❡❞ t❤❡ ❣②r♦s❝♦♣✐❝ ❡✛❡❝t✳ ❚❤❡r❡❢♦r❡✱ ✇❤❡♥ ❛ q✉❛❞r♦t♦r r♦t❛t❡s ❛

  r , I yy θ = u 3 + φ ψ(I zz -I xx ) -J r φΩ r , I zz ψ = u 4 + θ φ(I xx -I yy ), mẍ = (sin ψ sin φ + cos ψ sin θ cos φ)u 1 , mÿ = (-cos ψ sin φ + sin ψ sin θ cos φ)u 1 , mz = -mg + (cos θ cos φ)u 1 .

1 . 2 ) 5 ) 11 )

 12511 35 sin( π(t-70) 55 ) + 0.15 sin( π(t-70) +0.225 sin( π(t-70) + 0.105 sin( 24π(t-70) ✇❤❡♥ 70s < t 124s✱ 0 ✇❤❡♥ 124s t 150s✳ ✭✹✳✺✮ ❋✐❣✉r❡ ✹✳✷✿ ❲✐♥❞ ❞✐st✉r❜❛♥❝❡ ✇✳r✳t t✐♠❡✳ ✹✳✶✳✸ ▼♦❞❡• P❛r❛♠❡t❡r ❯♥❝❡rt❛✐♥t✐❡s ❚❤✐s s❝❡♥❛r✐♦ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❡st t❤❡ t

I

  xx φ = θ ψ(I yy -I zz ) + J r θΩ r + u 2 , I yy θ = φ ψ(I zz -I xx ) -J r φΩ r + u 3 , I zz ψ = θ φ(I xx -I yy ) + u 4 , mẍ = (sin ψ sin φ + cos ψ sin θ cos φ)u 1 , mÿ = (-cos ψ sin φ + sin ψ sin θ cos φ)u 1 , mz = -mg + (cos θ cos φ)u 1 .

  ✼✹ ✹✳✸✳ ❈♦♥tr♦• ▼❡t❤♦❞s ❈❤❛♣t❡r ✹✳ ❙②st❡♠ ❈♦♥tr♦• ❋✐❣✉r❡ ✹✳✸✿ ❚❤❡ ♣•♦t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ss✐❣♥✭①✮✳ ✇❤❡r❡ ẋid (i = 1, 3, 4, 5, 7, 9, 11.) ❛r❡ t❤❡ r❡❢❡r❡♥❝❡s✳ ❚❤❡ ❝♦♥tr♦• •❛✇s ❛r❡✿

  ❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶ ♦♥ P❛❣❡ ✻✻✳ ❚❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ❡st✐♠❛t❡❞ s♠❛••❡r t❤❛♥ t❤❡ r❡❛• ✈❛•✉❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s②st❡♠ ✐s ✉♥❞❡r✲❛❝t✉❛t❡❞✳ ❚❤❡ ✼✽ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✶✿ ❚❤❡ ♠♦❞❡• ❢r❡❡ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✶✳ ❝♦♥tr♦• ❣❛✐♥s ❛r❡ ✐♥ ❚❛❜•❡ ✺✳✶✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✸✳ ❚❤❡ s②st❡♠ st❛②s st❛❜•❡ ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✵✼✺♠✱ ✇❤✐•❡ ✵✳✵✹✷♠ ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦✳

  r ✐s ✵✳✵✹✷♠✳ ✽✷ ✺✳✶✳ ❚✐♠❡ tr✐❣❣❡r❡❞ ❈♦♥tr♦• ❈❤❛♣t❡r ✺✳ ❙✐♠✉•❛t✐♦♥ |❡s✉•ts ❋✐❣✉r❡ ✺✳✻✿ ❚❤❡ ❇❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐♥ t❤❡ ❜❛s✐❝ s❝❡♥❛r✐♦ ✇✐t❤ t❤❡ ❣❛✐♥s ✐♥ ❚❛❜•❡ ✺✳✷✳ ✸✳ P❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❚❤❡♥✱ t❤❡ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦• ✐s s✐♠✉•❛t❡❞ ✐♥ t❤❡ s❝❡♥❛r✐♦ ✇✐t❤ ♠♦❞❡• ♣❛r❛♠❡t❡r ✉♥❝❡r✲ t❛✐♥t✐❡s✳ ❚❤❡ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✳ ❚❤❡ ❣❛✐♥s ❛r❡ ♥♦t ❝❤❛♥❣❡❞ ❛s ✐♥ ❚❛❜•❡ ✺✳✷✳ ❚❤❡ s✐♠✉•❛t✐♦♥ r❡s✉•ts ❛r❡ ✐♥ ❋✐❣✉r❡ ✺✳✽✳ ❚❤❡ ♠❛①✐♠✉♠ ❛❜s♦•✉t❡ tr❛❝❦✐♥❣ ❡rr♦r ✐s ✵✳✶✶♠✳

3 i=1 t e 2 i 3 i=1 t e 2 i 3 i=1 t e 2 i

 323232 ♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✿ ✳ ✭✸✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✿ ✳ ✭✹✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✿ ✳ ✭✺✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✿

3 i=1 t e 2 i 3 i=1 t e 2 i 3 i=1 t e 2 i

 323232 s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ t❤❡ ✇✐♥❞ ❞✐st✉r❜❛♥❝❡✿ ✳ ✭✸✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ ♣❛r❛♠❡t❡r ✉♥❝❡rt❛✐♥t✐❡s✿ ✳ ✭✹✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ s❡♥s♦r ♥♦✐s❡s✿ ✳ ✭✺✮ ❚❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s ♦❢ t❤❡ tr❛❝❦✐♥❣ ❡rr♦rs ✇✐t❤ ❛❝t✉❛t♦r ❢❛✉•ts✿

  ✿❂②❬✶(❀ ①❬✷(✿❂②❬✷(❀ ①❬✸(✿❂②❬✸(❀ ①❬✹(✿❂②❬✹(❀ ①❬✺(✿❂②❬✺(❀ > ①❬✻(✿❂②❬✻(❀ ①❬✶✶(✿❂②❬✶✶(❀ ①❬✶✷(✿❂②❬✶✷(❀ > ①❬✼(✿❂②❬✼(✴❣❀ ①❬✽(✿❂②❬✽(✴❣❀ ①❬✾(✿❂②❬✾(✴❣❀ ①❬✶✵(✿❂②❬✶✵(✴❣❀ > ①✼❞♦t✿❂①✼❞♦t✯❣❀①✽❞♦t✿❂①✽❞♦t✯❣❀①✾❞♦t✿❂①✾❞♦t✯❣❀①✶✵❞♦t✿❂①✶✵❞♦t✯❣❀ > ✉✶✿❂✈❬✶(❀ ✉✷✿❂✈❬✷(✴❣❀ ✉✸✿❂✈❬✸(✴❣❀ ✉✹✿❂✈❬✹(❀

Fig. 1 .

 1 Fig. 1. (1) Reference trajectory. (2) Tracking errors of ideal flatness based control. (3) Tracking errors of time triggered PI control. (4) Tracking errors of time triggered i-PID control. (5) Tracking errors of event triggered PI control. (6) Tracking errors of event triggered i-PID control.

Fig. 2 .

 2 Fig. 2. Reference trajectory for the quadrotor.

  σ

  t 2 = 9, t 3 = 15, t 4 = 21. y = σ(t) with t 1 = 9, t 2 = 15, t 3 = 21, t 4 = 27. z = 10 (28)

Fig. 3 .

 3 Fig. 3. Time triggered i-PID control of quadrotor

Fig. 1 . 1 .

 11 Fig. 1.1. The quadrotor(right) from Ascending technologies(Available at ESIEE).

Fig. 4 . 1 .

 41 Fig. 4.1. The eigenvalues when K 12 changes from -150 to 150: (a) the real parts. (b) the imaginary parts.

  ẋ = A(b)x + F (x), x ∈ R n where b is a free parameter, b ∈ R. At its origin x = [0, ..., 0], J(b) is the Jordan form of the matrix A(b) and Q is a matrix which enables Q(b)J(b)Q -1 (b) = A(b). Therefore, we have: ẋ = Q(b)J(b)Q -1 (b)x + F (x) ⇒ Q -1 (b) ẋ = J(b)Q -1 (b)x + Q -1 (b)F (x) we define y = Q -1 (b)x, then ẏ = J(b)y + Q -1(b)F (Q(b)y) = J(b)y + F (y) (4.1)

y

  -= h(y 0 , b), h(0, 0) = Dh(0, 0) = 0, ḃ = 0. (4.2) We can calculate the function h(y 0 , b) by using ẏ-= Dh(y 0 , b) ẏ0 = Dh(y 0 , b)[By 0 + f (y 0 , h(y 0 , b))] = Cy 0 + g(y 0 , h(y 0 , b))

v 1 = 5 , v 2 =

 152 -256z 5 -bz 6 -z 3 -100z 3 -308z 4 -256z 7 -32z 8 , v 4 = -10 11 -24z 12 , v 3 = 100z 1 + 308z 2 -256z 9 -32z 10 .

ẏ1 = 16y 2 -

 2 0.41b 4 -0.011b 8 -(b + 0.057b 4 )y 1 + 0.00024y 3 2 ẏ2 = -16y 1 + 0.67b 4 y 1 (4.3) In the reduced system, when b is positive (negative), the origin is a stable (unstable) focus. When b = 0, the origin is a center. The phase portrait of equation (4.3) when b = -0.5, b = 0 and b = 0.5 are depicted in Figure 4.2.

Fig. 4 . 2 .

 42 Fig. 4.2. The phase portrait of the reduced system: (a) b=-0.5. (b) b=0. (c) b=0.5.

v 1 = 8 v 4 =

 184 -256(z 5 -z r ) -32z 6 , v 2 = -1700(z 3 -y r ) -1000z 4 -256z 7 -32z -256(z 11 -ψ r ) -32z 12 , v 3 = 1700(z 1 -x r ) + 1000z 2 -256z 9 -32z 10

Fig. 5 . 1 .

 51 Fig. 5.1. (a) Reference trajectory for the quadrotor. (b) Wind disturbance.

Fig. 5 . 2 .

 52 Fig. 5.2. The simulation without wind disturbance: (a) the proposed method. (b) a standard PID.

Fig. 5 . 3 .

 53 Fig. 5.3. The simulation with wind disturbance: (a) the proposed method. (b) a standard PID.

  |e(t k ) -e(t k-1 )| > e lim .(9)(3) ISS based law:e(t k ) = σ a b |y(t k )|,

2

 2

2 t 2 < t t 3 h 5 t 3 < t t 4 0 t 4

 235344 d -h d (t-t3) 5 (t-t3) 5 +(T f -t+t3) < t 150s h d = 2m, T f = 6s.

Fig. 1 .

 1 Fig. 1. (a) and (b) Reference trajectory for the quadrotor. (c) The wind disturbance w.r.t time.

x 1 = φ, x 2 = φ, x 3 = θ, x 4 = θ, x 5 = ψ, x 6 = ψ, x 7 = z, x 8 = ż, x 9 = x, x 10 = ẋ, x 11 = y, x 12 = ẏ.

ACKNOWLEDGEMENTS

Jing Wang is financially supported by IPSA (Institut Polytechnique des Sciences Avancées).

2

J. WANG, I. BOUSSAADA, H. MOUNIER, A. CELA AND S.I. NICULESCU to construct the simplest form of the system. The normal form exhibits all possible properties of the original system. The normal forms of any degree with a single input were obtained by using change of coordinates and feedback [6]. For multi-input systems, the normal forms are deduced from the system with two inputs [7]. Based on the normal forms, the bifurcations and its control were studied by several authors [6,8]. Center manifold is usually applied with the normal forms. It reduces the system to a center manifold associated with parts of the system with the eigenvalues with zero real parts at a bifurcation point [9].

To the best of our knowledge, the normal form and center manifold theories have never been used in the analysis and control of quadrotor. In this paper, the normal form of the quadrotor system is firstly calculated. By using such a methodology, the highly coupled parts in quadrotor system are eliminated. Under certain control laws, the normal form is reduced into a two dimensional system at the bifurcation point by using center manifold theory. Also, a simple control method based on the normal form using state feedback is proposed. The control laws are proposed to ensure the asymptotical stability of the system by moving all the eigenvalues of the system to the open left half plane. Comparing to a standard PID control, the proposed method has faster response time and less tracking errors especially when there is wind disturbance, as illustrated at the end of the paper. The interest of considering such control laws lies in the simplicity of the controller as well as in its practical implementation facility.

The paper is organized as follows: In Section 2, the model of quadrotor is given. In Section 3, the normal form of quadrotor is deduced. In Section 4, the bifurcation of the system under certain control laws is analyzed. In Section 5, simulations with and without wind disturbance using the proposed method and PID control are given.

2. The quadrotor model. The chosen model of quadrotor is depicted in equations (2.1). The rotation angles φ, θ and ψ are along the world axis x, y and z respectively, namely roll, pitch and yaw. w i (i = 1..4) are the accelerations caused by four rotors, which are the inputs of the system. (g = 9.8m/s 2 the gravity).

We introduce the variables as x 1 = x, x 2 = ẋ, x 3 = y, x 4 = ẏ, x 5 = z, x 6 = ż, x 7 = φ, x 8 = φ, x 9 = θ, x 10 = θ, x 11 = ψ, x 12 = ψ. Therefore, we can rewrite the system as:

3. Normal form of the system. It is easy to see that the equilibria of the system (2.2) are x e = (c 1 , 0, c 2 , 0, c 3 , 0, kπ, 0, kπ, 0, c 4 , 0), w = (g, 0, 0, 0), where k = 0, ±1, ±2, ..., c i (i = 1..4) ∈ R are constants and g is the gravity. Note in the real control system, φ, θ ∈ (-π/2, π/2) and ψ ∈ [0, π). Therefore, without losing generality, only the equilibrium x 0 = (x, w) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, g, 0, 0, 0) is considered. We move x 0 to the origin by changing the coordinates of the inputs w 1 = u 1 + g, w 2 = u 2 , w 3 = u 3 , w 4 = u 4 . Then, using the Taylor series of function sin(x) and cos(x) at x = 0. The system (2.2) can be written in polynomial form as follows. Here, O 5 are the polynomials with 5th and higher degree:

Using the state and input transformation y 1 = x 1 , y 2 = x 2 , y 3 = x 3 , y 4 = x 4 , y 5 = x 5 , y 6 = x 6 , y 7 = gx 7 , y 8 = gx 8 , y 9 = gx 9 , y 10 = gx 10 , y 11 = x 11 ,

, we change the system (3.1) into Brunovsky form:

The system (3.1) can be written as:

where A, B are the coefficients of the linear parts, f (2) (y), g (1) (y)v are the second degree homogeneous polynomials of the system, f (3) (y), g (2) (y)v are the third degree homogeneous polynomials.

We take a third-degree homogeneous transformation for example [10]:

which z are the new states of the system. φ (2) (z) is a second degree homogeneous polynomial and φ (3) (z) is a third degree homogeneous polynomial of the states z, whose coefficients will be defined later.

We get the derivative of equation (3.3). Therefore, the derivative of the new states z are:

where, (I + dφ (2) dz 2) dz dφ (3) dz ...

2), we rewrite the f (y) and g(y) using the new states z.

Therefore, with the help of the equations (3.2), (3.4), by now we have the new system:

For the simplicity of the system, the states z and the inputs v should be separated. In the third degree normal form, the polynomial g (1) (z)v, g (2) (z)v should be canceled.

g (1) (z) -dφ (2) dz B = 0

Therefore, the transformation in equation ( 3.3) should be:

, 0, 0, 0, 0, 0, 0, 0, 0),

2g 2 , 0, 0, 0, 0, 0, 0). Using the same method, we can calculate the normal form of any degree. A Maple package 'QualitativeODE' [11] has been made for calculating the normal form of quadrotor. Using this programme, we get the third degree normal form of the system (3.1) as:

4. Bifurcation and simplification of the control system.

4.1. Bifurcation of the roots. It is easy to see that in the linear part of the equation (3.5), z 1 is related only to z 2 , z 9 , z 10 , v 3 ; z 3 is related to z 4 , z 7 , z 8 , v 2 ; z 5 is related to z 6 , v 1 ; z 11 is related to z 12 , v 4 . Therefore, the control laws can be defined as:

In this way, we can move the related eigenvalues in each group separately without changing the eigenvalues in other groups. Here, we define v i (i = 1..4) as:

The system has three equilibria P e 1 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), P e 2 = (0, 0, 43.45, 0, -0.057, 0, -16.97, 0, 0, 0, 0, 0) and P e 3 = (0, 0, -43.45, 0, -0.057, 0, 16.97, 0, 0, 0, 0, 0). However, only the origin P e 1 can be stable when K 12 , K 42 change. At the equilibrium P e 1 , for simplicity K 12 = K 42 , when K 12 changes, the real and imaginary parts of the eigenvalues are in Figure 4.1. When K 12 < 0, the system has four eigenvalues with positive real parts, and the system becomes unstable. When K 12 > 0, the system has all eigenvalues with negative real parts, and the system is asymptotically stable. When K 12 = 0, the system has two pairs of pure imaginary eigenvalues ±16i and ±32i, and all other eigenvalues have negative real parts, which is a four dimensional center manifold. The stability cannot be determined by the linear part of the system. It depends on the nonlinearity of the system.

Event driven model free control of quadrotor

Jing Wang, Marcel-Stefan Geamanu, Arben Cela, Hugues Mounier and Silviu-Iulian Niculescu Abstract-In this paper we propose a new control approach, event driven model free control, which deals with the "tradeoff" between computational cost and system performance. The model free control scheme demands low computational resources and has high robustness, which is especially suitable for systems with complex dynamics and/or affected by disturbances. Particularly for the embedded systems, the event driven model free control demands even fewer computational resources, since the actuation is allowed only when an event is triggered. The proposed method is implemented on a quadrotor model in different realistic scenarios with disturbances and uncertainties. Under the time and event triggered schemes, the model free control is compared with the backstepping and sliding mode controls in these scenarios.

be wind during the shooting. The wind is represented as the extra acceleration and affects all x, y and z axis, which is depicted in Figure 1.

B. Model

The model of the quadrotor is a six d.o.f. system with twelve states and four inputs which is depicted in eq. ( 14). The notations c and s represent cos and sin respectively. The rotation angles φ, θ and ψ are along the x, y and z axis respectively, namely roll, pitch and yaw. The parameters in the system can be found in the footnote. 1 Ixx φ = θ ψ(Iyy -Izz)+Jr θΩr +l(-T2 +T4)+(-1) i+1 (14) Here, T i and H i are the thrusts and hub forces of each motor; Q i and R i are the drag and rolling moments; ω i are the rotational speeds of the four motors.

In eq. ( 14), φ ψ(I zz -I xx ), θ ψ(I yy -I zz ), θ φ(I xx -I yy ) are the body gyro effect moments; J r θΩ r , J r φΩ r (Ω r = ω 1 -ω 2 +ω 3 -ω 4 ) are the propeller gyro effect moments; (-1) i+1

The system is controlled by the rotational speeds of the four motors ω i . In the control system, we define

Then, the rotational speeds ω i can be computed using u i through:

mIxx (cψcφ + sψsθsφ) and D = u1 mIyy (sψcθcφ). We implement the model free control scheme in a similar manner as previous:

where êx 4d , êy 4d are the errors between the references x (4) r , y

and the estimates of x (4) , y (4) , i = 0, ..., 3.

For the yaw control, we consider the equation of ψ as:

Then the needed control law is:

where êψ 2d is the error between the reference ψr and the estimate of ψ.

D. Backstepping control

For the purpose of comparison, a backstepping control proposed by S. Bouabdallah et al. [2] is also used on the quadrotor system. In order to simplify the control laws, some parts in the model in eq. ( 14) are neglected, such as the rolling moments and the hub forces. The system is written into the state space form using the state vector (x 1 , . . . , x 12 )

The system is separated into the angular and position subsystems. The angular subsystem is thus firstly controlled, and then the position subsystem is controlled by using the angles from the angular subsystem.

In the angle subsystem, the control laws are defined as:

with In the position subsystem, the control laws are defined as:

with z 7 = x 7d -x 7 , z 8 = x 8 -ẋ7d -α 7 z 7 , z 9 = x 9d -x 9 , z 10 = x 10 -ẋ9d -α 9 z 9 , z 11 = x 11d -x 11 , z 12 = x 12 -ẋ11dα 11 z 11 and u x = sψsφ + cψsθcφ, u y = -cψsφ + sψsθcφ.

E. Sliding mode control

For comparison, a sliding mode control proposed by S. Bouabdallah et al. [2] is also proposed. The state variables are defined in the backstepping control. The sliding surfaces are chosen as:

where ẋid (i = 1, 3, 4, 5, 7, 9, 11.) are the references. The control laws are:

The definition of all the parameters can be found in the previous backstepping control section III-D.

IV. COMPARISON OF THE CONTROL LAWS

A. Basic scenario: time and event triggered schemes

In the time triggered scheme, the sampling period is 10ms, and it yields 15000 actuation steps. The simulation results of the three control methods are given in Figure 2. Three methods have followed the reference trajectory nicely. The maximum absolute tracking errors are 0.042m, 0.09m and 0.08m for the model free, backstepping and sliding mode control respectively, which are 2.1%, 4.5% and 4% of the desired length 2m.

In the event triggered scheme, the chosen event triggering law is the error difference threshold as in eq. ( 9). Here, we set the error difference threshold of z 0.02m, yaw angle 0.1rad, x and y 0.001m. Every 10ms, the control system verifies the tracking error differences, and decides the corresponding actions. The simulation results are shown in Figure 2. The event triggered model free control, backstepping control and sliding mode control yield 9008, 9987 and 8493 actuation steps, which are 60%, 66.6% and 56.6% of 15000 actuation steps in the time driven scheme. The maximum absolute tracking errors are 0.06m, 0.1m and 0.1m in the three control methods, which increased 0.43%, 0.11% and 0.25% with respect to their results in the time triggered scheme.

B. Scenario with wind disturbance: time and event triggered schemes

All the control methods are simulated in the scenario with wind disturbance. The simulation results are in Figure 3. The chosen event triggering law is the same as in the basic scenario.

In the time triggered scheme, the maximum absolute tracking errors are 0.042m, 0.23m and 0.21m in three control methods, which increased 0%, 155.6% and 162.5% with respect to the time triggered controls in the basic scenario. The wind does not have great influence in the model free control, however it has highly affected the backstepping and sliding mode controls.

In the event triggered scheme, the model free, backstepping and sliding mode controls yield 12110, 12116 and 12445 actuation steps, which are 80.7%, 80.8% and 83.0% of 15000 actuation steps. The maximum absolute tracking errors are 0.11m, 0.22m and 1.25m respectively in three control methods, which increased 83.3%, 120% and 1150% with respect to the event triggered controls in the basic scenario.

C. Discussion

In order to get a comprehensive evaluation, all the methods are implemented in other realistic scenarios with parameter uncertainties, with sensor noises and with actuator faults. Further details can be found in [START_REF] Wang | Analysis and control of quadrotor[END_REF]. The maximum absolute tracking errors, the sum of the error variances and the actuation steps of all the control methods in all scenarios can be found in Table I and Figure 4.

In different realistic scenarios, the model free control has the smallest tracking errors than other control methods, and the event triggered model free control has the smallest errors than other event triggered methods. In the model free control, the part F is evaluated at each actuation step using the system measurement ŷ and the last time input ũ. The model free control avoids the time-consuming computation of the full control model and has an algorithm complexity O(5n 2 +3n), while the backstepping and sliding mode controls have to compute the full model and have an algorithm complexity O(6n 2 +4n). The disturbances and uncertainties are also considered into F in the real time control, and that is why the model free control compensate them well.

The event triggered scheme matches the model free control. While the maximum tracking errors are bounded in desirable limits, the actuation steps are reduced more than one third. The event triggered model free control works well in the realistic scenarios with disturbances. The sliding mode control does not match with the event triggered scheme. The tracking errors are too high to be accepted in the scenarios with disturbances. basic scenario 0,042 0,09 0,08 0,06 0,1 0,1 wind 0,042 0,23 0,21 0,11 0,22 1,25 parameter uncertainties 0,075 0,11 0,11 0,19 0,22 0,41 sensor noise 0,08 0,09 0,08 0,08 0,09 0,08 actuator faults 0,04 0,06 0,07 0,13 0,2 0,35 error variance basic scenario 0,0049 0,0028 0,0058 0,2 0,2 0,077 wind 0,0052 2,4723 3,5238 0,0635 2,6679 65.4849 parameter uncertainties 0,0212 0,0468 0,2805 0,1281 1,8675 3,5517 sensor noise 0,2309 0,0029 0,0088 0,2309 0,0029 0,0088 actuator faults 0,0327 0,0167 0,0188 0,0918 0,0661 0,554 V. CONCLUSION

In this paper, event driven model free controllers have been proposed and applied on quadrotor system in different realistic scenarios. The proposed method avoids the timeconsuming computation of the full control model by evaluating the part F at each actuation step using the system measurement ŷ and the last time input ũ. The part F can also take the disturbances and uncertainties into account in the real time control, therefore the model free control has a better performance in the realistic scenarios with disturbances.

Under the event driven scheme, the model free control has smaller tracking errors and lower actuation steps than backstepping and sliding mode control. In the realistic scenarios with disturbances and uncertainties, the event driven model free control has achieved almost the same results as in the scenarios without disturbance, which proves its robustness to perturbations. In the same scenarios, the backstepping and sliding mode controls have higher tracking errors.

The proposed method gives promising results in terms of control algorithm complexity reduction, computational resources reduction and robustness to perturbations, which is appropriate for implementation in embedded systems.