Edoardo Idà
email: edoardo.ida2@unibo.it

Marco Carricato
email: marco.carricato@unibo.it

Natural Oscillations of Underactuated Cable-Driven Parallel Robots

Keywords:

Underactuated Cable-Driven Parallel Robots (CDPR) employ a number of cables smaller than the degrees of freedom (DoFs) of the end-effector (EE) that they control. As a consequence, the EE is underconstrained and preserves some freedoms even when all actuators are locked, which may lead to undesirable oscillations. This paper proposes a methodology for the computation of the EE natural oscillation frequencies, whose knowledge has proven to be convenient for control purposes. This procedure, based on the linearization of the system internal dynamics about equilibrium configurations, can be applied to a generic robot suspended by any number of cables comprised between 2 and 5. The kinematics, dynamics, stability and stiffness of the robot free motion are investigated in detail. The validity of the proposed method is demonstrated by experiments on 6-DoF prototypes actuated by 2, 3, and 4 cables. Additionally, in order to highlight the interest in a robotic context, this modelling strategy is applied to the trajectory planning of a 6-DoF 4-cable CDPR by means of a frequency-based method (multi-mode input shaping), and the latter is experimentally compared with traditional non-frequency-based motion planners.

I. INTRODUCTION

C ABLE-driven parallel robots (CDPRs) control the end-effector (EE) pose by means of extendable cables. A CDPR is underactuated if the number n of actuated cables is smaller than the number of the EE degrees of freedom (DoFs). As a consequence, only a sub-set of the EE coordinates can be directly controlled, with the remaining ones being determined by the system mechanical equilibrium. An underactuated CDPR (UACDPR in short) is always underconstrained, thus its EE preserves some DoFs once actuators are locked. Accordingly, in case the EE is not in a static equilibrium configuration when actuators cease to move, the UACDPR exhibits (possibly dangerous) oscillatory motions. This oscillatory behaviour is naturally expected to occur at the end-point of a trajectory, if suitable motion-planning and control techniques are not employed [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF], or it may result from an emergency stop or an actuator failure.

Despite these drawbacks, the use of CDPRs with a limited number of cables may be favorable in several applications, in which a limitation of mobility is acceptable in order to enhance workspace accessibility or decrease mechanical complexity and robot cost [START_REF] Fink | Planning and control for cooperative manipulation and transportation with aerial robots[END_REF]- [START_REF] Scalera | Design and experimental validation of a 3-dof underactuated pendulum-like robot[END_REF]. Thus, the study of UACDPRs is attracting the interest of more and more researchers, who have dealt with their geometricostatic problems [START_REF] Abbasnejad | Direct geometrico-static problem of underconstrained cable-driven parallel robots with n cables[END_REF], [START_REF] Berti | Solving the direct geometricostatic problem of underconstrained cable-driven parallel robots by interval analysis[END_REF], equilibrium stability analysis [START_REF] Carricato | Stability Analysis of Underconstrained Cable-Driven Parallel Robots[END_REF], [START_REF] Surdilovic | Practical stability of under-constrained cable-suspended parallel robots[END_REF], trajectory planning [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF], [START_REF] Cunningham | The winch-bot: A cable-suspended, under-actuated robot utilizing parametric self-excitation[END_REF]- [START_REF] Idà | Robust trajectory planning of underactuated cable-driven parallel robot with 3 cables[END_REF], system parameter identification [START_REF] Idá | Automatic self-calibration of suspended under-actuated cable-driven parallel robot using incremental measurements[END_REF], and control [START_REF] De Rijk | Out-of-plane vibration control of a planar cable-driven parallel robot[END_REF]- [START_REF] Jafari Harandi | Point-to-point motion control of an underactuated planar cable driven robot[END_REF].

The knowledge of natural oscillation frequencies of UACDPRs can be used in order to derive frequency-based trajectory planners based on periodic excitation [START_REF] Zoso | Point-to-point motion planning of a parallel 3dof underactuated cable-suspended robot[END_REF] or input-shaping [START_REF] Park | Anti-sway trajectory generation of incompletely restrained wire-suspended system[END_REF], [START_REF] Hwang | Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots[END_REF]- [START_REF] Idà | Robust trajectory planning of underactuated cable-driven parallel robot with 3 cables[END_REF]: these planners limit oscillations, and are real-time capable, as opposed to restto-rest trajectory planners [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF], which can completely stop a UACDPR EE after a point-to-point motion, but needs to be computed off-line (and are not frequency-based). Additionally, natural oscillation frequencies may also be exploited for optimal robot design [START_REF] Briot | On the Optimal Design of Parallel Robots Taking Into Account Their Deformations and Natural Frequencies[END_REF].

In order to compute UACDPRs natural oscillation frequencies, the EE internal dynamics [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF] needs to be derived, and expressed in terms of a minimal set of EE residual DoFs. Natural frequencies are determined by linearizing, about an equilibrium configuration, the EE internal dynamics with respect to (w.r.t.) the EE residual DoFs, and by solving the resulting eigenproblem. The authors of [START_REF] Zoso | Point-to-point motion planning of a parallel 3dof underactuated cable-suspended robot[END_REF], [START_REF] Park | Anti-sway trajectory generation of incompletely restrained wire-suspended system[END_REF] derived the single configuration-dependent natural oscillation frequency of a planar 3-DoF 2-cable robot, by intuitively selecting the platform orientation as the EE residual DoF, whereas in [START_REF] Hwang | Oscillation reduction and frequency analysis of under-constrained cable-driven parallel robot with three cables[END_REF] the same technique was employed for a spatial 6-DoF 3-cable system, where ZYX Tait-Bryan angles were chosen as residual DoFs. Due to the specific UACDPR architectures reported in [START_REF] Zoso | Point-to-point motion planning of a parallel 3dof underactuated cable-suspended robot[END_REF], [START_REF] Park | Anti-sway trajectory generation of incompletely restrained wire-suspended system[END_REF], [START_REF] Hwang | Oscillation reduction and frequency analysis of under-constrained cable-driven parallel robot with three cables[END_REF], the translational and rotational mechanical equilibria of the EE could be decoupled, which resulted in a mathematically simpler internal dynamics formulation and linearization. This was not the case, instead, for the 6-DoF 4-cable manipulator considered in [START_REF] Hwang | Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots[END_REF], where the authors determined the system natural frequencies with a method similar to the one developed in [START_REF] Park | Anti-sway trajectory generation of incompletely restrained wire-suspended system[END_REF], by approximating the 6-DoF robot with two 3-DoF planar sub-systems and selecting the orientations of these subsystems' platforms as residual DoFs. In fact, because of the intrinsic coupling of rotational and translational equilibria of 6-DoF UACDPRs with more than 3 cables, it is not straightforward to select the corresponding residual DoFs (1 DoF for 5-cable robots and 2 DoFs for 4-cable robots), and to derive and linearize the manipulator internal-dynamic equations: singularities may arise in the computation, which results in the failure of natural frequency determination.

The contributions presented in this paper are the following. 1) A novel unified technique is proposed for the computation of the natural oscillation frequencies of UACDPRs with a generic number n of cables (1 < n < 6), a generic geometry, and subject to a generic external wrench. Previous works only analyzed specific architectures, such as 2-cable 3-DoF UACDPRs [START_REF] Zoso | Point-to-point motion planning of a parallel 3dof underactuated cable-suspended robot[END_REF], [START_REF] Park | Anti-sway trajectory generation of incompletely restrained wire-suspended system[END_REF] and 3-cable 6-DoF UACDPRs [START_REF] Hwang | Oscillation reduction and frequency analysis of under-constrained cable-driven parallel robot with three cables[END_REF], [START_REF] Idà | Robust trajectory planning of underactuated cable-driven parallel robot with 3 cables[END_REF], or an approximation of a 4-cable 6-DoF UACDPR with two 2-cable 3-DoF UACDPRs [START_REF] Hwang | Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots[END_REF]. Our approach allows an easy selection of the EE residual DoFs and the opportunity of easily switching between a selection and another, so that representation singularities may always be avoided in the formulation of the internal dynamics; the subsequent natural frequency computation is performed with a well-known tool, namely linearizing the internal dynamics about an equilibrium configuration. Additionally, the proposed modelling method has the merit of determining out-of-the-plane oscillation frequencies of planar systems with 2 cables, which were not previously considered [START_REF] Zoso | Point-to-point motion planning of a parallel 3dof underactuated cable-suspended robot[END_REF], [START_REF] Park | Anti-sway trajectory generation of incompletely restrained wire-suspended system[END_REF]: the determination of these frequencies proved to be useful in [START_REF] De Rijk | Out-of-plane vibration control of a planar cable-driven parallel robot[END_REF], where the authors determined them experimentally, and used them in the design of a stabilizing controller for the robot EE.

2) The natural oscillation frequencies of generic mediumscale UACDPRs with 2, 3 and 4 cables are experimentally determined and compared with the ones computed by means of the new technique, thus showing that the latter is adequate for real-world applications.

3) The stiffness and equilibrium stability of UACDPR are refined w.r.t. the state of the art [START_REF] Carricato | Stability Analysis of Underconstrained Cable-Driven Parallel Robots[END_REF], [START_REF] Surdilovic | Practical stability of under-constrained cable-suspended parallel robots[END_REF] by taking into account swivel pulleys in the kinematic model and a generic wrench (not necessarily a pure force) acting on the EE. Shaper [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF]. The results are experimentally compared with those that can be achieved by non-frequency-based motion planners. The structure of the paper is as follows. Section II reports the kinematic model. Section III investigates the behaviour of an UACDPR when it is in free motion, that is, when cable lengths are constant. The computation of natural oscillation frequencies is then carried out in Section IV, while Section V experimentally verifies that the frequencies computed by the proposed approach closely match the ones of physical 2-, 3-and 4-cable UACDPR prototypes. Section VI shows an application example: the experimental comparison between an Input-Shaped trajectory of a 4-cable UACDPR and other motion planners. Conclusions are drawn in Section VII.

II. KINEMATIC MODEL

The kinematic model of an UACDPR is derived by considering the geometric constraints imposed by n (taut) cables, with 1 < n < 6, on the robot 6-DoF EE, as in [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF]. The pose ζ = [p T T ] T of the EE is described in the inertial frame Ox y z by the position vector p of P , the EE reference point, and a minimal set of angles (Fig. 1a).

Cables are assumed to be massless (which is reasonable for small-to medium-scale CDPRs), infinitely rigid and always subject to non-zero tensile loads. Additionally, each cable is guided into the workspace by a swivel pulley, which can rotate about its swivel axis. Such a pulley has radius r i and center C i , and is mounted on an hinged support, whose swivel axis is tangent to the pulley in point D i (Figs. 1b,1c). The cable enters the pulley groove in D i , exits from it at point B i , and it is attached to the platform at point A i . d i and a i are the position vectors of D i and A i , whereas a i is a vector pointing from P to A i . All position vectors, except d i , are functions of the EE pose ζ in Ox y z.

The coordinates of position vector b i of point B i , in the inertial frame, depend on ζ and also on the pulley model. k i , w i , u i , n i , and t i are additional unit vectors associated with the pulley geometry. As shown in Figs. 1b,1c: k i is directed along the swivel axis, u i points from D i to C i , w i = k i × u i is normal to the plane defined by the swivel axis and the i-th cable, n i points from C i to B i , t i = w i ×n i is directed as the i-th cable; σ i and ψ i are swivel and tangency angles. All these variables depend on the EE pose ζ and, in case ζ is known, they can be computed in closed form, as shown in [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF]. Accordingly:

b i = d i + r i (u i + n i ) (1)
The constraint imposed by each cable onto the EE is:

ρ T i ρ i -l i -r i (π -ψ i ) | 2 = 0 (2)
where ρ i ∆ = a ib i , and l i is the total cable length, comprising the rectilinear part ρ i and the arc B i D i wrapped onto the pulley.

A. DIFFERENTIAL KINEMATICS

If ω is the angular velocity of the EE, the EE twist is v = [ ṗT ω T ] T and its linear relationship with ζ is given by:

v = D( ) ζ, D( ) ∆ = I 3×3 0 3×3 0 3×3 H( ) (3)
where I 3×3 ∈ R 3×3 and 0 3×3 ∈ R 3×3 are identity and null matrices, and H( ) depends on the parametrization used to describe the orientation [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF].

The rate of change of l i , li , can be computed as the projection of the velocity of point A i along the i-th cable direction t i [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF], [START_REF] Pott | Influence of pulley kinematics on cable-driven parallel robots[END_REF], [START_REF] Surdilovic | Geometric Stiffness Analysis of Wire Robots: A Mechanical Approach[END_REF], that is:

ȧ i = ṗ + ω × a i (4) li = t i • ȧ i = ξ i • v, ξ i ∆ = t i a i × t i (5)
where ξ i is a zero-pitch screw directed as t i and passing through A i , and the symbols • and × denote the scalar VOLUME 4, 2016

and vector products, respectively. The relationship between the EE twist v and the derivatives of the system actuated variables, l = [l 1 . . . , l n ] T , is thus given by:

Ξv = l, Ξ ∆ = [ξ 1 • • • ξ n ] T (6)
where matrix Ξ ∈ R n×6 is the kinematic Jacobian of the manipulator. In general, rank (Ξ) = n, but, if a directkinematics singularity is encountered [START_REF] Conconi | A new assessment of singularities of parallel kinematic chains[END_REF], rank (Ξ) < n.

By substituting (3) in [START_REF] Abbasnejad | Direct geometrico-static problem of underconstrained cable-driven parallel robots with n cables[END_REF], the relationship between l and the derivative of the EE pose is obtained as:

ΞD ζ = J ζ = l, J ∆ = ΞD (7)
Matrix J ∈ R n×6 is the analytic Jacobian of the manipulator, thus it is a proper gradient, which may also be obtained by differentiating (2) w.r.t.

ζ for i = 1, • • • , n [26]
.

III. FREE MOTION

The aim of this section is to highlight some kinematic and dynamic properties of the UACDPR when actuators are locked, namely l = l 0 , l = l = 0 n×1 , and thus the EE is in free motion.

A. FREE-MOTION KINEMATICS

The free-motion first-order kinematics can be described by setting l = 0 n×1 in ( 6):

Ξv = 0 n×1 (8)
namely (cf. ( 3) and ( 7)):

J ζ = 0 n×1 (9)
Once the lengths of the n cables are set and all cables are taut, that is, all kinematic constraints are active, and Ξ has full column rank, the EE can still move on a variety of dimension λ = 6n in SE(3), thus preserving λ DoFs. Consequently, λ components of ζ are free to vary, and are called free pose components ζ f . The remaining n components of ζ are called dependent pose components ζ d , since they depend on the value of cable lengths l 0 and free pose components ζ f . When actuators are locked, the free twist v of the EE can be expressed as a function of the EE λ residual DoFs. This can be done by computing the right nullspace Ξ ⊥ of matrix Ξ. By definition, the right nullspace of the full-rank (n × 6) matrix Ξ is spanned by the (independent) columns of a (6 × λ) matrix Ξ ⊥ such that ΞΞ ⊥ = 0 n×λ , so that its columns define a basis for the free twist v:

v = Ξ ⊥ c for some c ∈ R λ (10)
If J ⊥ is the right nullspace of matrix J, then:

v = D ζ = DJ ⊥ c for some c ∈ R λ (11)
Comparing ( 10) and ( 11) and choosing c = c yields:

Ξ ⊥ = DJ ⊥ (12)
Equation ( 12) has great significance in the computation of natural oscillation frequencies of UACDPR, as Section IV will highlight. Since most orientation parametrizations of SO(3) allow rank (D) ≥ 5 (even in case of representation singularities), and 5 ≥ n for any UACDPR, one can always assume rank (D) ≥ n. This allows us to find an expression of J ⊥ , and thus of Ξ ⊥ , so that the parameter array c can be chosen as a subset of ζ. Since Ξ has full column rank and rank (D) ≥ n, a permutation matrix 1 P ∈ R 6×6 can be determined so that: ζP

∆ = P ζ = ζd ζf , ζd ∈ R n , ζf ∈ R λ ( 13
)
J P ∆ = JP T = ΞDP T = Ξ D d D f = J d J f ( 14
)
D d ∈ R 6×n , J d ∆ = ΞD d ∈ R n×n ( 15
)
D f ∈ R 6×λ , J f ∆ = ΞD f ∈ R n×λ ( 16
)
J ⊥ P ∆ = PJ ⊥ , J ⊥ P , J ⊥ ∈ R 6×λ (17)
where:

JJ ⊥ = JP T PJ ⊥ = J P J ⊥ P = 0 n×λ ( 18
)
The permutation matrix P must always be chosen so that rank(D d ) = n and, since rank(Ξ) = n, this also means rank(J d ) = n. This allows us to express ζ, and thus v, as a function of ζf in free motion. Indeed, since:

J ζ = J P ζP = J d ζd + J f ζf = 0 n×1 (19) then: ζd = -J -1 d J f ζf (20) ζP = J ⊥ P ζf , J ⊥ P = -J -1 d J f I λ×λ (21) ζ = P T ζP = P T J ⊥ P ζf = J ⊥ ζf , J ⊥ = P T J ⊥ P (22) v = D ζ = DJ ⊥ ζf = Ξ ⊥ ζf , Ξ ⊥ = DP T J ⊥ P = DJ ⊥ (23)
Matrix J ⊥ P in (21) always satisfies [START_REF] De Rijk | Out-of-plane vibration control of a planar cable-driven parallel robot[END_REF]. Basically, matrix P allows us to group n dependent pose-derivative components in ζd and λ free pose-derivative components in ζf , so that it is possible to express the free twist of the EE as a linear combination of the columns of Ξ ⊥ , with the combination coefficients c being the components ζf . The choice of the permutation matrix P, and subsequently of the free pose components, is not arbitrary, since it must ensure rank(D d ) = n. However, this choice does not need to be unique throughout the robot workspace, but it can be changed locally in order to avoid representation singularities: this is always possible, because rank (D) ≥ 5. In addition, in case a direct-kinematics singularity is encountered in the workspace, so that rank(Ξ) = n , with n < n, the method proposed in this Section for the 1 A permutation matrix is an orthogonal matrix that has exactly one entry of 1 in each row and each column, and has 0's elsewhere [START_REF] Golub | Matrix computations[END_REF].

description of the free motion may still be employed: the DoFs preserved by the EE would be λ = 6n , and λ free pose coordinates and n dependent pose coordinates could be chosen.

B. FREE-MOTION DYNAMICS

The non-linear dynamic model of an UACDPR emerges from the EE mechanical equilibrium, subject to cable constraints, inertial actions, and an external wrench [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF]:

Mv + Cv = -Ξ T τ + f (24)
M ∆ = mI 3×3 -ms ms I P , I P ∆ = I G -ms s C ∆ = 0 3×3 -m ωs 0 3×3 ωI P , f ∆ = φ q φ + µ (25)
where m is the EE mass, I G is the EE inertia tensor about its center of mass G expressed in the inertial frame, the symbol ∼ over a vector denotes its skew-symmetric representation, and τ ∈ R n is an array containing the cable tension magnitudes. f ∈ R 6 is a generic external wrench, resulting from a force φ applied in point Q and a moment µ directed along φ (Fig. 1a). Vectors s and q point from P to G and Q, respectively.

Since the natural oscillation frequencies characterize the free motion of the EE about equilibria (l = l 0 , l = l = 0 n×1 ), it is useful to express [START_REF] Surdilovic | Geometric Stiffness Analysis of Wire Robots: A Mechanical Approach[END_REF] in terms of the λ DoFs that the EE preserves and their derivatives, ζ f , ζf , ζf . This is achieved by considering (2) for i = 1, • • • , n and a fixed l 0 , as well as [START_REF] Pott | Influence of pulley kinematics on cable-driven parallel robots[END_REF] and its time derivative:

ζ = ζ(l 0 , ζ f ), v = Ξ ⊥ ζf , v = Ξ⊥ ζf + Ξ ⊥ ζf ( 26
)
where Ξ⊥ , as any first-order time derivative, is linearly dependent from ζf and can be symbolically computed by differentiating the right-hand side of (23) w.r.t. time. The free motion internal dynamics of the EE can be obtained by substituting [START_REF] Taghirad | Parallel robots: mechanics and control[END_REF] in [START_REF] Surdilovic | Geometric Stiffness Analysis of Wire Robots: A Mechanical Approach[END_REF] and left-multiplying by Ξ ⊥ T . Since Ξ ⊥ T Ξ T = 0 λ×n , then:

M ⊥ ζ f ζf + C ⊥ ζ f , ζf ζf -f ⊥ ζ f = 0 λ×1 ( 27
)
where:

M ⊥ ∆ = Ξ ⊥ T MΞ ⊥ , C ⊥ ∆ = Ξ ⊥ T M Ξ⊥ + CΞ ⊥ , f ⊥ ∆ = Ξ ⊥ T f (28)

C. FREE MOTION STIFFNESS

An equilibrium configuration is a set (ζ, l) = ζ 0 , l 0 such that (2), for i = 1, • • • , n, and ( 27) are satisfied for ζf = ζf = 0 λ×1 , and τ is element-wise stricly positive [START_REF] Abbasnejad | Direct geometrico-static problem of underconstrained cable-driven parallel robots with n cables[END_REF].

After equilibrium is altered, the restoring action that pushes the system back towards the equilibrium is due to the external wrench and the cable constraint forces. These restoring actions generate the Free-Motion Stiffness (FMS)

K ⊥ ∆ = -∂f ⊥ /∂ζ f ∈ R λ×λ of the UACDPR 2 .
2 Please refer to the Appendix for additional details about the tensor notation used in this paper.

The FMS was implicitly formulated for UACDPRs in [START_REF] Carricato | Stability Analysis of Underconstrained Cable-Driven Parallel Robots[END_REF], under the assumptions that cables exit the frame through eyelets, and the platform is subject to the gravitational action only, and explicitely formulated accounting for pulley kinematics in [START_REF] Surdilovic | Practical stability of under-constrained cable-suspended parallel robots[END_REF]. Here, K ⊥ is formulated as in [START_REF] Surdilovic | Practical stability of under-constrained cable-suspended parallel robots[END_REF], but the application of a generic external wrench f on the platform is also considered.

According to [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF] ( ζ = J ⊥ ζf ), one can infer:

J ⊥ = ∂ζ/∂ζ f (29)
Thus, accounting for [START_REF] Cui | Research on controllable stiffness of redundant cable-driven parallel robots[END_REF]:

K ⊥ = - ∂f ⊥ ∂ζ f = - ∂f ⊥ ∂ζ J ⊥ = - ∂Ξ ⊥ T ∂ζ f + Ξ ⊥ T ∂f ∂ζ J ⊥ (30)
Since the restoring actions under examination are those around equilibrium configurations, f = Ξ T τ (see [START_REF] Surdilovic | Geometric Stiffness Analysis of Wire Robots: A Mechanical Approach[END_REF]), and thus:

K ⊥ 0 = - ∂Ξ ⊥ T ∂ζ Ξ T τ + Ξ ⊥ T ∂f ∂ζ J ⊥ (31)
where the subscript 0 denotes that K ⊥ 0 is calculated in the equilibrium configuration. Differentiating Ξ ⊥ T Ξ T = 0 λ×n and substituting in (31) yields:

K ⊥ 0 = Ξ ⊥ T ∂Ξ T ∂ζ τ - ∂f ∂ζ J ⊥ (32)
The first term in the parentheses at the right-hand side of (32) may be calculated by considering the right-hand side of (5):

∂Ξ T ∂ζ τ = n i =1 τ i ∂ξ i ∂ζ = n i =1 τ i ∂t i ∂ζ ã i ∂t i ∂ζ -ti ∂a i ∂ζ ( 33
)
According to [START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF], it can be shown by computation that:

∂t i ∂ζ = T i -T i ã i D, T i ∆ = sin ψ i w i w T i u i • (a i -d i ) + n i n T i ρ i ( 34
)
∂a i ∂ζ = 0 3×3 -ã i D (35)
thus obtaining:

∂Ξ T ∂ζ τ = KD (36)
with:

K ∆ = n i =1 τ i T i -T i ã i ã i T i -ã i T i ã i + n i =1 τ i 0 3×3 0 3×3 0 3×3 ti ã i ( 37
)
In the literature, the (6 × 6) matrix K is referred to as Geometric [START_REF] Surdilovic | Geometric Stiffness Analysis of Wire Robots: A Mechanical Approach[END_REF], Controllable [START_REF] Cui | Research on controllable stiffness of redundant cable-driven parallel robots[END_REF] or Active [START_REF] Behzadipour | Stiffness of Cable-based Parallel Manipulators With Application to Stability Analysis[END_REF] Stiffness of the CDPR, because it is geometry dependent and, in overconstrained CDPRs, τ can be actively controlled independently from the EE configuration. It should be noted that its definition is fundamentally different from the socalled Passive Stiffness generated by cable deformations (not considered in this paper, since cables are modelled as rigid). However, in UACDPRs, K cannot be actively VOLUME 4, 2016 controlled, because τ depends on the equilibrium configuration.

The second term in the parentheses at the right-hand side of ( 32) is calculated from [START_REF] Conconi | A new assessment of singularities of parallel kinematic chains[END_REF]. Since q = R P q , with R being the rotation matrix between the moving and the inertial frame, and P q being the coordinates of q expressed in P x y z , one has:

- ∂f ∂ζ = QD -F (38)
Q ∆ = 0 3×3 0 3×3 0 3×3 -φi q , F ∆ = ∂φ ∂ζ R ∂ P q ∂ζ + q ∂φ ∂ζ + ∂µ ∂ζ (39) since ∂q /∂ζ = [0 3×3 -q i D + R ∂ P q ∂ζ ]
. Finally, substituting (36) and ( 38) in [START_REF] Idà | Natural oscillations of underactuated cable-driven parallel robots -complete experiments data[END_REF], yields:

K ⊥ 0 = Ξ ⊥ T [(K + Q) D -F] J ⊥ (40)
where:

K + Q = n i =1 τ i T i -T i ã i ã i T i -ã i T i ã i + + n i =1 τ i 0 3×3 0 3×3 0 3×3 -ti ã i + 0 3×3 0 3×3 0 3×3 φi q i ( 41
)
Notice that matrix K + Q is generally non-symmetric, since, while the first summation in (41) is always symmetric, the other terms are not. In fact, at the static equilibrium:

n i =1 τ i a i × t i = q i × φ + µ (42)
which, in skew-symmetric representation, is equivalent to:

n i =1 τ i ã i ti -ti ã i = q φ -φq + μ (43)
Equation ( 43) shows that the summation of the second and third term in (41), namely, φq -

n i =1 τ i ti ã i = q φ - n i =1 τ i ã i ti + μ = = φq - n i =1 τ i ti ã i T + μ (44)
is symmetric if and only if µ = 0 3×1 . If, furthermore, the force φ is constant, then F = 0 6×6 and the FMS is symmetric for any choice of EE pose parameters (both p and ):

K ⊥ 0 = Ξ ⊥ T (K + Q) Ξ ⊥ (45)

IV. LINEARIZED FREE-MOTION DYNAMICS AND NATURAL OSCILLATIONS

The UACDPR natural oscillation frequencies can be computed from the eigenvalue problem arising from the EE free-motion dynamics, after its linearization about an equilibrium configuration. A linearized form of ( 27) can be obtained by expanding it in Taylor series and truncating the expansion at the first order (an example of application to the linearization of the dynamic model of fully-actuated parallel manipulators can be found in [START_REF] Kozak | Locally Linearized Dynamic Analysis of Parallel Manipulators and Application of Input Shaping to Reduce Vibrations[END_REF]).

In the following, an approach similar to [START_REF] Kozak | Locally Linearized Dynamic Analysis of Parallel Manipulators and Application of Input Shaping to Reduce Vibrations[END_REF] is followed, but the EE coordinates are not considered independent from each other (cf. ( 22)), which is a distinctive feature of underactuated mechanisms.

If the left-hand side of ( 27) is denoted by h ζ f , ζf , ζf , the Taylor-series expansion of ( 27) about an equilibrium configuration (ζ f = ζ f 0 , ζf = 0 λ×1 , ζf = 0 λ×1 ) truncated at the first order yields:

h ζ f , ζf , ζf h ζ f 0 , 0, 0 + ∂h ∂ ζf ζ f 0 ,0,0 ζf + + ∂h ∂ ζf ζ f 0 ,0,0 ζf + ∂h ∂ζ f ζ f 0 ,0,0 ζ f -ζ f 0 = 0 λ×1 (46)
At equilibrium, h ζ f 0 , 0, 0 = f ⊥ = 0 λ×1 . The partial derivatives are readily obtained as:

∂h ∂ ζf ζ f 0 ,0,0 = M ⊥ ζ f 0 ,0,0 = M ⊥ 0 (47) ∂h ∂ ζf ζ f 0 ,0,0 = C ⊥ + ∂C ⊥ ∂ ζf ζf ζ f 0 ,0,0 = 0 λ×λ (48) ∂h ∂ζ f ζ f 0 ,0,0 = ∂M ⊥ ∂ζ f ζf + ∂C ⊥ ∂ζ f ζf - ∂f ⊥ ∂ζ f ζ f 0 ,0,0 = = - ∂f ⊥ ∂ζ f ζ f 0 ,0,0 = K ⊥ ζ f 0 ,0,0 = K ⊥ 0 ( 49
)
where many elements vanishing in (47), ( 48) and (49) are linearly dependent on ζf and ζf (and thus are naught), and matrices M ⊥ 0 and K ⊥ 0 , given in ( 28) and (40), are reported below for the sake of convenience:

M ⊥ 0 = Ξ ⊥ T MΞ ⊥ (50)
K ⊥ 0 = Ξ ⊥ T [(K + Q) D -F] J ⊥ (51)
All quantities at the right-hand sides of (50) and ( 51) are intended to be computed in the equilibrium configuration.

Finally, (46) can be rewritten as:

M ⊥ 0 ζf + K ⊥ 0 ζ f -ζ f 0 = M ⊥ 0 ∆ ζf 0 + K ⊥ 0 ∆ζ f 0 = 0 λ×1 (52)
where

∆ ζf 0 ∆ = ζf -0 λ×1 and ∆ζ f 0 ∆ = ζ f -ζ f 0 .
This formulation leads to a generalized eigenvalue problem, whose solution allows for the determination of the system natural oscillation frequencies in the equilibrium configuration under investigation. By considering a solution of (52) in the form ∆ζ f 0 (t ) = γe Λt , with Λ ∈ C, so that:

Λ 2 M ⊥ 0 + K ⊥ 0 γ = 0 λ×1 (53)
    x i j -i -j i y i -k -k -k -k z i -i -j i j
det Λ 2 M ⊥ 0 + K ⊥ 0 = 0 (54)
Based on (54), it is possible to define the stability conditions of a UACDPR equilibrium configuration: if and only if Λ 2 1 , . . . , Λ 2 λ are real negative numbers, the equilibrium configuration is stable, otherwise it is unstable [START_REF] Nayfeh | Nonlinear oscillations[END_REF]. When K ⊥ 0 is symmetric (and only in this case), the latter assertion is equivalent to requiring K ⊥ 0 to be positive-definite [START_REF] Carricato | Stability Analysis of Underconstrained Cable-Driven Parallel Robots[END_REF]. Finally, natural oscillation frequencies are computed (in [Hz]) as:

f j = ℑ Λ j 2π ( 55
)
where ℑ (•) denotes the imaginary part of a complex number. Additionally, eigenvectors γ j can be determined by solving (53) for each j and normalized according to γ T j M ⊥ 0 γ j = 1.

V. EXPERIMENTAL VALIDATION

In order to validate the methodology proposed in this paper, a series of experiments were conducted on the 6-DoF UACDPR prototype of the University of Bologna (Fig. 2). Geometrical and inertial properties of the prototype are deduced from the prototype CAD models, and are summarized in Tables 1 and2, where i = [1; 0; 0] T , j = [0; 1; 0] T , and k = [0; 0; 1] T , and the only external load applied to the robot EE is gravity, thus q = s , φ = -mg k and µ = 0 3×3 . The coordinates of a i , s and I G are constant

A. EXPERIMENTAL METHODOLOGY

The procedure described in this Section was applied to, respectively: 30 equilibrium configurations in which the platform was constrained by 4 cables (cables 1 to 4, Fig. 3a), 30 configurations in which only cables 1 through 3 were attached to the platform (Fig. 3b), and 30 configurations in which the platform was only suspended by cables 1 and 3 (Fig. 3c). Each equilibrium configuration was reached by quasistatically varying robot cable lengths; once the assigned set-point was reached, actuators were controlled to hold their angular positions so that cable lengths could not vary any longer, and motor torques were checked to ensure that their values were compatible with cables being taut. The EE was then manually slightly displaced w.r.t. its equilibrium configuration, and swiftly released next: this operation was equivalent to impose non-equilibrium initial conditions to the free-motion dynamics of the platform. The positions p k , k = 1, . . . , 5, of 5 optical markers mounted on the robot platform (2 of which can be seen in Fig. 2) were tracked by 8 cameras of a VICON Motion Capture System (measurement accuracy was ±0.2 mm for each marker's Cartesian component, at a 100 Hz sampling rate) for a total duration of 10 s for each experiment, thus acquiring n s = 1001 samples per marker coordinate.

These coordinates were then filtered by using a zerophase finite-impulse response low-pass digital filter with a stop-band frequency of 10 Hz. No natural oscillation frequency above 4 Hz was expected from the model, thus measurement noise and unmodelled oscillatory phenomena at higher frequencies, such as cable elastic axial vibrations, were accordingly removed.

For each experiment, the n s EE poses recorded during oscillations were reconstructed from the position of the 5 markers, and the corresponding cable lengths were calculated by the inverse geometric model (see ( 2)). The mean value over the n s samples of each cable length differed from its maximum and minimum value by less than 1 mm and thus it was considered as the constant experimental value of the variable l 0 . Alternatively, cable lengths could be computed as the result of the inverse model applied to the rest pose of the EE that is eventually reached. On the other hand, the employed procedure is considered to be more robust, because static friction may lead the EE to stop in a configuration different from the theoretical one.

The natural-oscillation-frequency computation method proposed is summarized as follows:

• given the experimental value l 0 of cable lengths, compute the corresponding EE static equilibrium pose ζ 0 according to the direct geometric-static model [START_REF] Abbasnejad | Direct geometrico-static problem of underconstrained cable-driven parallel robots with n cables[END_REF] ( is expressed by x y z Tait-Bryan angles); since the problem has possibly multiple solutions, keep only the stable pose that is closer to the initial one; • once the equilibrium configuration ζ 0 , l 0 is known, compute M ⊥ 0 and K ⊥ 0 according to (50) and ( 51); • solve the generalized eigenvalue problem in (53) and compute the natural oscillation frequencies f j , for j = 1, . . . , λ, according to (55).

The experimental value f j of each EE natural oscillation frequency was then identified, for j = 1, . . . , λ, from the marker recorded positions, so that a comparison with the corresponding modelled value f j could be performed. The oscillation of each marker w.r.t. its equilibrium position was experimentally computed as:

∆p k (t ) = p k (t ) -p k (56)
where (•) denotes the mean value operator. The signal of any coordinate of ∆p k (t ) contains, in general, the system natural frequencies since, if p k is chosen as the platform reference point, ∆p k (t ) can be modelled as:

∆p k (t ) = J ⊥ k ∆ζ f 0 (t ) = J ⊥ k γe Λt = γ k e Λt , γ k ∆ = J ⊥ k γ (57)
where J ⊥ k groups the first 3 rows of J ⊥ as in the lefthand side of [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF]. Then, the Fast Fourier Transformation (FFT) of each coordinate of ∆p k (t ), for k = 1, . . . , 5, was performed. This operation was deemed necessary since: (i) depending to the actual value of γ k , some modes may be absent in some coordinate, (ii) depending on the manually imposed initial condition of the EE oscillation, some modes may have an experimentally negligible amplitude in the frequency spectra of a certain coordinate FFT, and (iii) high data redundancy, which is achieved by considering 15 signals theoretically possessing frequency spectra peaks corresponding to the same frequency values, robustifies the experimental investigation. Figure 4 shows, as an example, the FFTs produced while analyzing experiment 77 on the 2-cable UACDPR. Several small-amplitude peaks can be noticed surrounding high amplitude-peaks: they are not present in the original signals, but artificially introduced because of an FFT resolution upscaling process. In fact, n s = 1001 samples recorded at 100 Hz would produce an FFT with 0.1 Hz frequency resolution. This resolution was upscaled to 0.01 Hz in order to better isolate nearby peaks of the signal FFTs. This operation was performed by adding, at the end of the n s recorded samples, 9n s additional zerovalue samples, for a total of 10n s = 10001 samples.

B. DISCUSSION OF RESULTS

For each experiment, the experimental natural frequency f j is determined as the weighted mean of the frequencies f j ,kc corresponding to FFT peaks of a single coordinate, with the oscillation amplitude A kc used as weight (k = 1, . . . , 5, c = x, y, z):

f j = 5 k=1 z c=x A kc f j ,kc 5 k=1 z c=x A kc (58)
The results 3 of all experiments are summarized in Fig. 5, where each integer between 1 and 90 on the abscissa axis represents one of the experimental configurations portrayed in Fig. 3, with the ordinate representing the corresponding values of f j (in black) and f j (in red), for j = 1, . . . , λ. 3 Complete experimental data, and the associated descriptive statics, can be found in [START_REF] Idà | Natural oscillations of underactuated cable-driven parallel robots -complete experiments data[END_REF].

In order to evaluate how different FFT peaks corresponding to the same natural frequency are dispersed w.r.t. their mean value, the percentage standard deviation of the weighted mean was analyzed:

σ j % = 100 5 k=1 z c=x A kc f j ,kc -f j 2 5 k=1 z c=x A kc
(59) The smaller σ j % is, the better the natural frequency is experimentally identified. A descriptive statistics of σ j % for each mode j = 1, . . . , λ, organized by architecture (4-, 3-, 2-cable UACDPRs), is given in Table 3. Mean values are less than 3% across all modes and architectures, minimum values are below 1% and the largest value is roughly 6%: the proposed frequencyidentification method is deemed well performing, especially considering the prototype nature of the robot used in the experiments (most structural components, except for the winches, are made of 3D-printed plastic).

In order to asses how well the experimental natural frequency f j matches the modelled one f j , the (absolute) VOLUME 4, 2016 j mean(σ j %) max(σ j %) min(σ j %)

∆ f j % = 100 f j -f j f j (60)
A descriptive statistics of ∆ f j % for each mode j = 1, . . . , λ, organized by architecture (4-, 3-, 2-cable UACDPRs), is given in Table 4, where the experiment number matching maximum and minimum statistical indicators is reported within parentheses. Mean values are less than 2.5% across all modes and architectures, minimum errors are below 0.5% and the largest error is less than 6%: it can be ultimately concluded that the proposed method for natural-frequency computation is accurate, in practice.

In fact, from an engineering perspective, a mean error of the order of 1 -2% is negligible in most applications.

In addition, various error sources, such as an imperfect knowledge of robot geometry and inertial parameters, which were estimated from CAD drawings, could also have had a negative impact on the evaluation of the modelled natural frequencies f j .

It should be noted that, while experimentally studying UACDPRs natural frequencies, additional FFT smallamplitude peaks, which did not match any phenomena modelled in this paper, were occasionally detected. These additional vibratory/oscillatory phenomena were expected, since cables may vibrate axially or flexurally, or they may oscillate out of the pulley planes. On the other hand, the amplitude of these phenomena (i.e. tenths of a millimeter) is negligible w.r.t. the amplitude of natural oscillations (up to dozens of millimeters, see Fig. 4), for the prototype used in the experimentation. These additional effects will be addressed in our future work on a larger-scale prototype.

VI. APPLICATION EXAMPLE

In order to highlight the interest of the modelling strategy proposed in this paper in a robotic context, a potential application example is presented in the following: the trajectory planning of a 6-DoF 4-cable UACDPR by a frequency-based method (i.e. Multi-Mode Zero-Vibration Input Shaping [START_REF] Idà | Robust trajectory planning of underactuated cable-driven parallel robot with 3 cables[END_REF], [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF]), experimentally compared with traditional methods. The robot under investigation has λ = 2 free pose components, and thus the trajectory of 4 EE dependent coordinates can be assigned for planning purposes 4 . The natural frequencies of the robot are determined by taking into account the exact 4-cable 6-DoF architecture of the robot (an approximated method is reported in [START_REF] Hwang | Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots[END_REF], which, according its authors, may have limitations when motion is performed near workspace edges).

Three trajectories are compared, in the form:

ζ d (t ) = ζ d ,s + (ζ d , f -ζ d ,s )u(t ) (61)
with ζ d ,s and ζ d , f being start and final values of dependent pose coordinates respectively, and u(0) = 0, u(T ) = 1, 0 ≤ u(t ) ≤ 1 ∀t . All trajectories have equal start and final configurations, but they differ in the choice of the motion law u(t ), as follows:

• the first motion law, called STD T , is a standard trapezoidal velocity profile, with total transition time T , and αT acceleration and deceleration duration (0 ≤ α ≤ 0.5):

u STD T (t ) =        (t /T ) 2 2α(1-α) , t < αT (-α+2t /T ) 2(1-α) , αT ≤ t ≤ (1 -α)T -2α 2 +2α-1+t /T -(t /T ) 2 2α(1-α) , t > (1 -α)T (62)
Figure 6: Modelled oscillation frequencies along u(t ).

• the second motion law, called STD T -IS, is the convolution of STD T with a multi-mode zero-vibration input shaper [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF]:

u STD T -IS (t ) = u STD T (t ) * S(t ) (63) S(t ) = k i =1 A i δ i (t -t i ) (64)
where * denotes the convolution operation,

δ i (t = t i ) = 1, δ i (t = t i ) = 0, A i is the impulse amplitude, t i
is the time at which the impulse occurs, and k is the number of impulses; the convolution with an inputshaper delays the total duration of the trajectory by ∆T = t k ;

• the last motion law, called STD T +∆T , is a standard trapezoidal velocity profile, with total transition time T +∆T , and α(T +∆T ) acceleration and deceleration duration. Once a trajectory for the dependent coordinates is assigned, the evolution of the free coordinates when the system is following the prescribed trajectory must be evaluated. Free coordinates are computed by numerically integrating the system internal dynamics, which is obtained by pre-multiplying (24) by Ξ ⊥ T and substituting the lefthand side of (3) and ( 13), and their time derivatives 5 :

M P ζP + C P ζP = f ⊥ (65) M P ∆ = Ξ ⊥ T MDP T = M d M f , M d ∈ R λ×n , M f ∈ R λ×λ (66) C P ∆ = Ξ ⊥ T M Ḋ + CD P T (67)
After algebraic manipulation, the time-derivative of vec-

tor x ∆ = ζ T f , ζT f T can be expressed as: ẋ = ζT f M -1 f -M d ζd -C P ζP + f ⊥ (68)
5 While performing a trajectory, cable lengths change and the elements of ζ P and its time derivatives are independent, as opposite to when the EE is in free-motion and only ζ f and its time derivatives are independent.

and (68) can be numerically integrated for assigned initial rest condition x 0 (ζ f 0 is the static equilibrium value for an assigned ζ d 0 and ζf 0 = 0 λ ). Finally, cable lengths can be computed according to the inverse geometric model in [START_REF] Fink | Planning and control for cooperative manipulation and transportation with aerial robots[END_REF], and fed to low-level motor drivers for manipulator control. While servo-motor angular positions are closedloop controlled, there is no feedback on the platform pose, and thus its configuration is only feed-forward controlled.

Start and end configurations are selected near the UACDPR static workspace edges [START_REF] Kumar | Workspace analysis of a 4 cable-driven spatial parallel robot[END_REF], in order to stress the importance of careful trajectory planning so as to avoid potentially dangerous situations, such as cable loss of tension due to platform large oscillatory motions. is expressed by x y z Tait-Bryan angles, since no representation singularities are expected throughout the manipulator static workspace: 6) and can be computed by the method described in Sections III and IV. Since the ratio between the maximum and minimum frequency is almost 2, a convoluted multi-mode zero-vibration Input Shaper with 3 modes is designed: the 4 pairs (A i , t i ), i = 1, . . . , 4 are determined by setting to zero both summations inside the parentheses of the residual-amplitude equation [START_REF] Idà | Robust trajectory planning of underactuated cable-driven parallel robot with 3 cables[END_REF], [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF]:

ζ s = [0.
A % ( f ) = k i =1 A i cos(2π f t i ) 2 + k i =1 A i sin(2π f t i ) 2 (69)
for f = 1.19, 1.7, 2.21 Hz (the minimum, mean and maximum frequencies in the range), by considering t 1 = 0 s, and imposing k i =1 A i = 1; this procedure results in:

I S : A 1 = A 4 = 0.1575, A 2 = A 3 = 0.3425 t 1 = 0 s, t 1 = 0.294 s, t 3 = 0.588 s, t 4 = 0.882 s (70)
with ∆T = t 4 . Trapezoidal motion law parameters are selected as α = 0.2 and T = 1.5 s. Finally, dependent components are selected as p and 3 . While the choice of p as part of the dependent coordinates is natural if a positioning task has to be performed, no particular strategy is readily available for the choice of orientation parameters as dependent coordinates. For our demonstrative purpose, any choice is suitable: 3 is chosen due to its limited variation between the start and final configurations.

Complete experiments can be visualized in the media material attached to this paper, and free components 1 and 2 time evolution during experiment is shown in Fig. 7 as recorded by the Vicon Camera system described in Sec. V. When comparing trajectories with the same total duration, namely STD T -IS and STD T +∆T , it is evident that the former allows for smaller amplitude oscillations, which are rapidly damped by unmodelled frictional effects, once the target destination is reached. On the other hand, when comparing unshaped and shaped trajectories, namely STD T and STD T -IS, the advantage in employing the latter is even more evident, since the former results in large platform oscillations not only at the final destination, but also during the transition: this fact could easily lead to platform instability and cable loss of tension, thus ultimately robot loss of control. In the attached video material, it can be easily appreciated a complete loss of tension in cable number 2 during the STD T trajectory, as well as an overall better tracking performance of the STD T -IS trajectory.

VII. CONCLUSIONS

This paper presented a methodology for the computation of the natural oscillation frequencies of underactuated cable-driven parallel robots. The approach was experimentally validated on 2-,3-, and 4-cable prototypes. According to the experimental data, the method was shown to be effective, because recorded oscillation frequencies deviated from the model less than 6%. As a possible application example, the proposed approach was employed for comparing standard trajectory-planning methods and a frequency-based input-shaping planning of a 4-cable UACDPR, resulting in remarkably reduced EE residual oscillations even at workspace edges. In the future, this method will be considered for robust calibration and dynamic parameter identification of UACDPRs. In addition, cable deformation and sagging will be modelled in order to account for vibrational effects, which may play a role in large-scale UACDPRs. Finally, another line of investigation will be the use of different techniques to approximate non-linear systems compared to linearization, such as the method of multiple scales [START_REF] Daqaq | Nonlinear input-shaping controller for quay-side container cranes[END_REF]: these techniques may better describe the system behaviour over large oscillations.

APPENDIX: TENSOR NOTATION

Let:

A ∆ = a 1 • • • a h ∈ R k×h , a i ∆ =    a 1i . . .
c l    ∈ R l ×1 (72)
The derivative of a scalar w.r.t. a vector is:

∂b i ∂c ∆ = ∂b i ∂c 1 • • • ∂b i ∂c l ∈ R 1×l (73)
and the derivative of a vector w.r.t. a vector is:

∂a i ∂c ∆ =     ∂a 1i ∂c 1 • • • ∂a 1i ∂c l . . . . . . . . . ∂a ki ∂c 1 • • • ∂a ki ∂c l     ∈ R k×l (74)
The derivative of a matrix w.r.t. a vector can be obtained as follows. Let:

Ab = a 1 • • • a h    b 1 . . . b h    = h i =1 a i b i ∈ R k×1 (75)
Then:

∂ (Ab) ∂c = h i =1 ∂a i ∂c b i + h i =1 a i ∂b i ∂c (76)
Formally, (76) may also be written as:

4 )Figure 1 :

 41 Figure 1: CDPR geometric model

6 VOLUME 4

 64 , 2016 Idà et al.: Natural Oscillations of Underactuated Cable-Driven Parallel Robots

Figure 2 :

 2 Figure 2: UACDPR Prototype

Figure 3 :

 3 Figure 3: Layout of experimental configurations

Figure 4 :Figure 5 :

 45 Figure 4: Example of experimental FFTs: experiment 77 on the 2-cable UACDPR

  36, -0.82, -0.37, -0.35, 0.51, 0.12] T [m, rad] ζ e = [1.82, 0.55, -0.37, 0.38, -0.25, 0] T [m, rad] Natural oscillation frequencies along the path defined by ζ s and ζ e vary in the range [1.19, 2.21] Hz (see Fig.

Figure 7 :

 7 Figure 7: Free components of the 4-cable UACDPR prototype

  EDOARDO IDÀ received the M.Sc. degree (with honors) in Mechanical Engineering in from the University of Bologna, and is currently a Ph.D. student in Mechanics and Advanced Engineering Sciences at the same university. His research interests focus on cabledriven robotic systems. VOLUME 4, 2016

Table 1 :

 1

		 0.1338 0.0059	0.0021			0.002	
	8	   0.0059 0.1814 -0.0055	  	  	-0.002   
		0.0021 -0.0055 0.2602			0.200

Actuators' properties m [Kg] P I G [Kg • m 2 ] P s [m]

Table 2 :

 2

Platform inertial properties the eigenvalues Λ 2 1 , . . . , Λ 2 λ are found by solving the characteristic equation, namely:

Table 3 :

 3 σ j % descriptive statistics.

			4-cable UACDPR
	1	0.96%	3.18% (Exp. 26)	0.04% (Exp. 7)
	2	1.56%	4.21% (Exp. 29)	0.26% (Exp. 9)
			3-cable UACDPR
	1	2.36%	6.06% (Exp. 46) 0.21% (Exp. 33)
	2	1.79%	5.52% (Exp. 42) 0.19% (Exp. 56)
	3	1.78%	3.50% (Exp. 35) 0.12% (Exp. 50)
			2-cable UACDPR
	1	0.70%	2.70% (Exp. 76) 0.09% (Exp. 65)
	2	1.34%	3.94% (Exp. 63) 0.25% (Exp. 75)
	3	2.28%	4.30% (Exp. 68) 0.53% (Exp. 77)
	4	2.87%	4.91% (Exp. 87) 0.29% (Exp. 63)
	j mean(∆ f j %)	max(∆ f j %)	min(∆ f j %)
			4-cable UACDPR
	1	1.07%	2.66% (Exp. 6)	0.09% (Exp. 13)
	2	0.70%	1.87% (Exp. 28) 0.04% (Exp. 21)
			3-cable UACDPR
	1	1.41%	4.45% (Exp. 43) 0.08% (Exp. 42)
	2	1.61%	5.05% (Exp. 60) 0.06% (Exp. 49)
	3	1.38%	3.10% (Exp. 52) 0.25% (Exp. 32)
			2-cable UACDPR
	1	1.86%	3.87% (Exp. 80) 0.05% (Exp. 65)
	2	2.14%	4.92% (Exp. 73) 0.03% (Exp. 64)
	3	1.06%	3.08% (Exp. 78) 0.16% (Exp. 74)
	4	1.88%	5.38% (Exp. 77) 0.16% (Exp. 61)

Table 4 :

 4 ∆ f j % descriptive statistics.

	percentage estimation error is additionally analyzed:

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 VOLUME 4, 2016

 VOLUME 4, 2016

In the context of trajectory planning, dependent coordinates are also called controlled coordinates, or actuated coordinates[START_REF] Idà | Rest-to-rest trajectory planning for underactuated cable-driven parallel robots[END_REF]. 10 VOLUME 4, 2016

 VOLUME 4, 2016

Comparing (76) and (77) yields:


J M García-Ruiz

Al Jorge
email: jorge.vago@esa.int

L Vago

Frances Westall

Similarities between terrestrial planets at the time life appeared on Earth Comment on "Mineral self-organization on a lifeless planet" by

Venus, Earth, and Mars differ in size, distance from the Sun, atmospheric composition, and ambient conditions. However, during the first 500 Ma after their formation-approximately 4.56 Ga ago-they probably had comparable surface environments.

Although each terrestrial planet formed mainly from locally sourced materials, the final stages of accretion softened chemical differences by incorporating water and other volatiles from the outer Solar System-shepherded inwards by the wanderings of Jupiter and Saturn [START_REF] Walsh | A low mass for Mars from Jupiter's early gas-driven migration[END_REF]. As the three planets cooled, their mantles degassed atmospheres, a large part of whose water later condensed, giving rise to global oceans, or at least, large expanses of water many hundreds of metres deep [START_REF] Elkins-Tanton | Formation of early water oceans on rocky planets[END_REF]. We may expect this process to have taken place initially on Mars, then on Earth, and (perhaps) finally on Venus. We include this qualifying "perhaps" because high-altitude water is readily dissociated by ultraviolet light with subsequent loss of hydrogen to space. Some authors believe that Venus' atmosphere may have dried out before its water had a chance to condense [START_REF] Genda | Origin of Earth's oceans: an assessment of the total amount, history and supply of water[END_REF]; many others disagree [START_REF] Way | Was Venus the first habitable world of our solar system?[END_REF].

Fuelled by internal heat, rapid material turnover coupled with vigorous hydrothermal circulation brought about the sequestration of most atmospheric CO 2 into carbonate minerals. Because the luminosity of the young Sun was 70% that of today, in the absence of strong greenhouse forcing, surface temperatures would have rapidly plunged to subfreezing values-on Earth and Mars, not on Venus [START_REF] Vago | Habitability on early Mars and the search for biosignatures with the ExoMars Rover[END_REF]. However, a gradual buildup of CH 4 , assisted by a generally low planetary albedo (water is dark), alleviated the effect of the rapidly thinning CO 2 presence. As water diffused downward through fractured ocean crust, it reacted with mantle rocks at high temperatures to transform into a hydrothermal fluid able to promote serpentinization [START_REF] Holm | Serpentinization and the formation of H2 and CH 4 on celestial bodies (planets, moons, comets)[END_REF]. Serpentinization produces H 2 that can react with oxidized carbon compounds under reducing conditions to emit CH 4 and other simple organic molecules in the form of ubiquitous thick, smoke-like underwater plumes [START_REF] Nisbet | The habitat and nature of early life[END_REF][START_REF] Martin | On the origins of cells: a hypothesis for the evolutionary transitions from abiotic geochemistry to chemoautotrophic prokaryotes, and from prokaryotes to nucleated cells[END_REF]. And here we connect with the García-Ruiz et al. [START_REF] García-Ruiz | Mineral self-organization on a lifeless planet[END_REF] article.

The paper starts by explaining how, under alkaline conditions, silica can induce the formation of biomimetic structures. The emphasis of this work is not placed on warning investigators about the perils of attempting to identify

potential microorganisms on the basis of morphology alone. In fact, those familiar with the analysis of microbial colonies' palaeorelics would likely not be "taken in" by the biomorphs. Rather, the message is another: that the mineral world can, under certain conditions, approximate some of the behaviours we associate with living things, and that this distinction may have once been very much blurred-see also Saladino et al. [7].

The authors propose that, while CH 4 levels were important, the terrestrial ocean's pH would have remained high. We are not so sure. CO 2 will readily dissolve in water to give a slightly acidic ∼pH 6 sea, as supported by our analysis of 3.5 Ga samples from the Pilbara and Barberton [11]. However, most of their assertions regarding the mineral scenario's ability to support interesting prebiotic reactions would hold true even for the case where these alkaline environments would be geographically restricted, or even local, or circumscribed to the plumes.

The work mentions the upcoming ExoMars rover and its objective to search for biosignatures in Noachian (4.0+ Ga) Mars deposits [8]. We would really welcome an extension, or a discussion, of the paper's thesis to the Venus and Mars case. For the former, the assumption should be a global ocean with water temperatures in the range 20-60 • during the faint, young Sun period. For the latter, the waters' surface temperatures would have hovered around the freezing point, or below. In both cases, the internal heat of the young planets would have been the engine driving rock-water interactions releasing their warm, organics-rich exhalation in a submarine environment.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.