B G B Hunnekens

R H B Fey

A Shukla

H Nijmeijer

Vibrational self-alignment of a rigid object exploiting friction

Keywords: Self-alignment, Friction, Stick-slip vibrations, Accumulation position, Positioning

In this paper, one-dimensional self-alignment of a rigid object via stick-slip vibrations is studied. The object is situated on a table, which has a prescribed periodic motion. Friction is exploited as the mechanism to move the object in a desired direction and to stop and self-align the mass at a desired end position with the smallest possible positioning error. In the modeling and analysis of the system, theory of discontinuous dynamical systems is used. Analytic solutions can be derived for a model based on Coulomb friction and an intuitively chosen table acceleration profile, which allows for a classification of different possible types of motion. Local stability and convergence is proven for the solutions of the system, if a constant Coulomb friction coefficient is used. Next, near the desired end position, the Coulomb friction coefficient is increased (e.g. by changing the roughness of the table surface) in order to stop the object. In the transition region from low friction to high friction coefficient, it is shown that, under certain conditions, accumulation of the object to a unique end position occurs. This behavior can be studied analytically and a mapping is given for subsequent stick positions.

Introduction

Accurate positioning of objects is very important in industrial applications, e.g. in printer heads, CD-players, pick-and-place machines, welding robots etc. Usually, the positioning and tracking problem is tackled using closed-loop control using feedback to asymptotically stabilize the error dynamics. In this paper, the self-aligning positioning problem will be addressed. A rigid object is placed on a table, which will be submitted to a periodic motion profile. Friction based stick-slip vibrations will be used as a mechanism, by which the rigid object (in this paper also referred to as: the mass) will move in a desired direction. By locally increasing the friction the mass will self-align at a desired end position. Some (simplified) friction models for dynamics and control applications have been studied in for example [START_REF] Armstrong-Hélouvry | A survey of models, analysis tools and compensation methods for the control of machines with friction[END_REF]. An extensive treatment of friction modeling can be found in [START_REF] Berger | Friction modeling for dynamic system simulation[END_REF]. In literature, the dry-friction oscillator with Coulomb friction has received much attention. In this system, a mass experiencing Coulomb friction is connected to the world by a spring and a viscous damper, while an external periodic force acts on the mass, see [START_REF] Den | Forced vibrations with combined Coulomb and viscous damping[END_REF], [START_REF] Luo | Stick and non-stick periodic motions in periodically forced oscillators with dry friction[END_REF], and [START_REF] Stein | On dry friction modelling and simulation in kinematically excited oscillatory systems[END_REF]. As early as 1930, Den Hartog gave the exact solution of a single degree of freedom harmonic oscillator with dry friction [START_REF] Den | Forced vibrations with combined Coulomb and viscous damping[END_REF]. Cases where the static and dynamic friction coefficient are not equal to each other have been studied in [START_REF] Shaw | On the dynamical response of a system with dry friction[END_REF]. Also describing function approaches have been used to analyse the behavior of this type of system, see [START_REF] Duarte | Fractional describing function of systems with Coulomb friction[END_REF] for example. In [START_REF] Fidlin | Predicting vibration-induced displacement for a resonant friction slider[END_REF], the method of averaging is used to analyse highfrequency vibration-induced movements of a mass between two layers of different friction, while in [START_REF] Thomsen | Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening[END_REF] the method of direct separation of motion, see [START_REF] Blekhman | Vibrational Mechanics -Nonlinear Dynamic Effects, General Approach, Applications[END_REF], is used to calculate equilibrium speeds (including zero speed) of a mass on a table vibrating at high frequency. In [START_REF] Thomsen | Analytical approximations for stick-slip vibration amplitudes[END_REF], analytical approximations for stick-slip vibration amplitudes are derived for the classical mass-on-moving-belt system. For the same system, in [START_REF] Galvanetto | Dynamics of a simple damped oscillator undergoing stick-slip vibrations[END_REF], bifurcations are studied, which are non-standard due to the discontinuous behavior of the dry friction force. In [START_REF] Dankowicz | On the origin and bifurcations of stick-slip oscillations[END_REF], the bifurcations associated with the appearance of stick-slip vibrations are studied using a state variable friction law for a mass on a fixed horizontal surface, moving under the influence of a horizontal force transmitted by a linear spring, whose other end is moving with constant velocity. A method of calculating exact analytical solutions for parts of the oscillation cycles, which are 'stitched' together, has been used in [START_REF] Chernous'ko | Analysis and optimization of the motion of a body controlled by means of a movable internal mass[END_REF]; vibration-induced motions of a mass on a friction plane are studied, and optimal parameter values, for which maximum mean velocity is reached, have been determined.

Self-alignment of microparts is studied in [START_REF] Sato | Self-alignment of microparts using liquid surface tension -behavior of micropart and alignment characteristics[END_REF], where the self-alignment is achieved using liquid surface tension. In [START_REF] Frutiger | Small, fast, and under control: Wireless resonant magnetic micro-agents[END_REF], resonant magnetic micro-agents are used to move micro-robots using magnetic fields and visionfeedback for positioning applications. Modeling and closed-loop control of a 2-DOF micropositioning device, using stick-slip motion based on piezoelectric materials, is treated in [START_REF] Rakotondrabe | Development, modeling, and control of a micro-/nanopositioning 2-dof stick-slip device[END_REF]. The work in [START_REF] Reznik | Leaving on a plane jet[END_REF] studies the use of a planar manipulator to position multiple objects on a vibrating surface using the frictional forces along with feedback control.

An important motivation for using self-alignment compared to conventional closed-loop control is to reduce cost by eliminating the need for feedback control equipment. Possible practical applications could include parts feeding, automated assembly, or inspection of parts. To the best knowledge of the authors, there is no literature available, in which friction is exploited to move a mass in a desired direction and in which a mass self-aligns using friction, which motivates this study.

The paper is structured as follows. In Section 2, the two major objectives of this paper will be discussed. The focus in Section 3 will be on the dynamical model. The stopping regime will be discussed in detail. The table trajectory design will be studied in Section 4. In Section 5, the classification of different types of motion will be introduced. Local stability and convergence of solutions will be proven in Section 6. The positioning accuracy obtained, when the mass stops, will be investigated in Section 7. Finally, in Section 8, conclusions and some recommendations for future work will be given.

Problem formulation

The basic objective of this paper is to design a selfpositioning method, in which friction is used to move a mass on a table in a desired direction, using dry friction and periodic motion of the table. For the table, a suitable periodic motion profile must be designed that achieves this as fast as possible given some actuator constraints. Furthermore, the mass needs to be positioned/stopped at a desired end position accurately. Summarizing, the two main objectives of this research are:

-Design a periodic In this paper, a ∼ above a quantity means that the quantity is in SI units, whereas for dimensionless quantities the ∼ will be omitted.

Modeling the system dynamics

Dynamic model

Consider the system depicted in Fig. 1. For a practical application, the table displacement signal ỹ( t) will have a zero average displacement. Therefore, it is assumed that ỹ( t) will be periodic. The position of the mass relative to the table is denoted by x( t). Note that an arbitrary position on the table can be chosen as the origin of x( t). Between the table and the mass m, a Coulomb friction force Ff with friction coefficient µ exists, which mathematically can be described as follows, see e.g. [START_REF] Leine | Dynamics and Bifuractions of Non-Smooth Mechanical Systems[END_REF]:

Ff ∈ µ mgSign( ẋ) ( 1
)
where g is the acceleration due to gravity, and Sign( ẋ) is the set-valued sign function, defined by:

Sign( ẋ) =    {1} if ẋ > 0 [-1, 1] if ẋ = 0 {-1} if ẋ < 0 (2)
Note that in case the mass sticks to the table (so ẋ = 0) the friction force can take values in the range [-µ mg, µ mg]. This allows the mass to stick to the table as long as the external force acting on the mass is small enough. Using Newton's second law it is straightforward to derive the equation of motion for the mass:

-Ff = m( ẍ + ÿ) (3)
Mass m ɶ

Sign f F mg x µ = ɺ ɶ ɶ ɶ ɶ g ɶ Fig. 1 Schematic of the system.
If the relative velocity of the mass is zero ( ẋ = 0) and the external force acting on the mass is small enough (i.e. | mÿ | < µ mg), the friction force will balance this external force (-Ff = mÿ ) resulting in ẍ = 0, i.e. in sticking of the mass. If the external force is larger than the maximum friction force (i.e. | mÿ | > µ mg), the mass will start to slip, resulting in ẋ = 0 and Ff = µ mgSign( ẋ).

Dimensionless dynamic model

The equation of motion derived in the previous section will be made dimensionless to obtain a model, which incorporates a minimum number of parameters. A characteristic length scale L and a characteristic time scale T need to be introduced. The dimensionless length scales x and y and dimensionless time scale t are defined in the following way:

x = x/ L, y = ỹ/ L, t = t/ T (4)
Also a dimensionless differential operator is defined as = d/dt. If we choose T = L/µg, the equation of motion (3) can be written in dimensionless form as:

-F f = x + y (5)
where F f = Ff /µ mg ∈ Sign(x ), using [START_REF] Armstrong-Hélouvry | A survey of models, analysis tools and compensation methods for the control of machines with friction[END_REF]. Note that (5) can now be written as:

-Sign(x ) x + y (6)
The dimensionless condition for sticking of the mass (x = 0) now reduces to |y | < 1.

Stopping region

One of the objectives is to accurately stop the mass at a desired end location. To achieve this, an increase in friction coefficient is used near the desired end location x = x d of the mass. In essence, the table will consist of two regions, a low friction region and a high friction region. The high friction region will start at position x = x µ , see Fig. 2. The low friction region has a friction coefficient µ 1 = µ and the high friction region has a friction coefficient µ 2 = c µ µ, with c µ > 1.

Mass

z 2 µ x x µ w 1 µ
Fig. 2 Schematic of the table with two regions, one region with low friction coefficient µ 1 and one region with high friction coefficient µ 2 .

Because the mass has a finite dimensionless width w = w/ L ( w is the width in SI units), it will not experience a discontinuous step in the friction coefficient, when it enters the high friction region. More realistically, in the transition region, the mass will experience an effective friction coefficient μ, depending on the distribution of the weight of the mass over the low friction region and the high friction region:

μ(z) = w -z w µ 1 + z w µ 2 = w + (c µ -1)z w µ ( 7
)
where z is the portion of the mass on the high friction region, see Fig. 2. Note that z is bounded by 0 ≤ z ≤ w. This constraint can conveniently be formulated using a min-max formulation:

z = min(max(x -x µ , 0), w) (8)
The friction coefficient μ that the mass experiences, thus depends on the position x of the mass. Note that in the transition region, using [START_REF] Fidlin | Predicting vibration-induced displacement for a resonant friction slider[END_REF] and ( 8), the friction coefficient can be written as follows:

μ(x) = µ 1 + c µ -1 w (x -x µ ) if x µ ≤ x ≤ x µ + w (9)
In all simulation results presented throughout this paper, it is assumed that the mass does not tip over, i.e. it is assumed that the mass will remain in full contact with the table, so that the equation of motion [START_REF] Duarte | Fractional describing function of systems with Coulomb friction[END_REF] remains valid. In general, this will be valid for an object with a height, which is relatively small with respect to its width w.

Table trajectory design

The design of the table motion profile will be discussed in this section. First, in Section 4.1, a simulation method called the time-stepping method will be discussed. In Section 4.2, an example of a stick-slip motion of the mass, calculated using the time-stepping method, will be shown. An objective function will be introduced in Section 4.3, which is used in Section 4.4 to design a suitable table excitation signal.

Simulation method

The system under study exhibits dry friction, which makes it a nonlinear system with unilateral constraints [START_REF] Leine | Stability and Convergence of Mechanical Systems with Unilateral Constraints[END_REF]. There are different methods that can be used to simulate these kind of systems. Time-stepping is an efficient method for numerically solving the equations describing the dynamics of systems with unilateral constraints. A thorough mathematical description of timestepping method can be found in [START_REF] Leine | Dynamics and Bifuractions of Non-Smooth Mechanical Systems[END_REF] and [START_REF] Studer | Numerics of Unilateral Contacts and Friction -Modeling and Numerical Time Integration in Non-Smooth Dynamics[END_REF]. In contrast to, for example, event-driven techniques, in timestepping it is not necessary to detect every event (e.g. a stick-slip transition). The solution is computed using fixed time-steps forward in time. The time-stepping method of Moreau [START_REF] Moreau | Unilateral contacts and dry friction in finite freedom dynamics[END_REF] is used as the integration routine in this paper. A fixed-point iteration is carried out to solve for the unknown velocity in the system at every time-step. This velocity is used to estimate the position at the end of the time-step.

Example of stick-slip motion

An example of a simulated response using time-stepping is shown in Fig. 3. Note that the time-stepping routine does not explicitly solve for the acceleration x (t) of the mass. A periodic excitation signal consisting of a sequence of second order polynomials was initially used for the table displacement signal y(t). Intuitively, the table should move slowly in the desired direction dragging the mass along, and fast in the opposite direction, resulting in slipping of the mass. Fig. 3 shows that the principle of moving a mass in a desired direction using stick-slip vibrations works. However, the shape of the prescribed table displacement signal y(t) may be far from optimal. Therefore, the focus will be on the design and optimization of the actuation signal based on an objective function, which will be defined in Section 4.3. Fig. 3 Simulation of the system using the time-stepping approach (x: object, y: table).

Objective function

In order to study the effectiveness of a certain prescribed periodic table trajectory, a sensible choice has to be made for an objective function. In conformity with the first objective from Section 2, the mass should move in one direction as fast as possible, meeting certain actuator constraints related to stroke and excitation frequency. An appropriate quantity to use in an objective function will therefore be the average velocity v of the mass in steady-state (in the µ 1 region). Consequently, transient behavior is not considered. This average velocity v can be written as:

v = lim t→∞ x(t + ∆t) -x(t) ∆t ( 10
)
where ∆t is the period time of the periodic relative velocity signal of the mass. Obviously, in the case of pure stick, which is undesirable in the motion phase, v will be zero. The following objective function f is introduced:

f = 1 cv 2 (11)
where the positive constant c = 10000 is used for normalizing the objective function values throughout this paper. Obviously, this constant in principle has no influence on the optimization process itself. Minimizing f is equivalent to maximizing the average steady-state velocity v and to minimizing the process time T p = Tp / T (neglecting transients effects).

Design of the table motion profile

From ( 6) it is clear that the trajectory of the table directly influences the dynamics of the mass through y (t). Because the excitation of the table will be periodic, the stroke of the table ∆y = max(y) -min(y) and the excitation frequency 1/∆t will be two important parameters that influence the dynamics. In the following, the influence of these two parameters ∆y and ∆t is studied, and the design of the shape of the trajectory profile of the table (which completely specifies y (t)) is examined.

Preliminary study of influence of excitation frequency and stroke

The excitation frequency 1/∆t and stroke ∆y both have a big influence on the average steady-state velocity of the mass. Here, again, a sequence of second order polynomials similar to y(t) in Fig. 3 is used for the table motion profile along with the time-stepping simulation routine. In Figs. 4 and 5 the objective function value f defined by ( 11) is plotted as a function of ∆t (for ∆y = 0.02) and ∆y (for ∆t = 0.05) respectively. It is clear that for this type of table trajectories and the values given above, a large excitation frequency and a large stroke are both beneficial in obtaining a large average steady-state velocity v of the mass. Of course, suitable values for ∆t and ∆y will depend on the application in mind and on actuator constraints.

Preliminary analysis of a Fourier based motion profile

Once the values of excitation frequency and stroke are fixed, there is still freedom in designing the precise shape of the table trajectory. The shape of a table motion profile parameterized using a Fourier series of twelfth order has been optimized by minimizing the objective function value f , see [START_REF] Leine | Dynamics and Bifuractions of Non-Smooth Mechanical Systems[END_REF], using constrained From Fig. 6 it is clear that the optimization procedure converged to a table motion profile, which in approximation has the following nature:

a time interval with zero acceleration and constant nonzero velocity a time interval with constant negative acceleration values a time interval with constant positive acceleration values It seems that the largest part of the time interval with approximately zero-acceleration in approximation results in sticking of the mass to the table. This effect is clearly visible in Fig. 6: in steady-state the mass (almost) does not slide back, i.e. it does not move in the undesired direction. These observations are the motivation for the design of a table trajectory generator incorporating these three distinct time intervals with constant acceleration values. This will be discussed in detail in the next subsection.

Prescribed table motion with three constant acceleration time intervals

The prescribed table motion with three constant acceleration time intervals will consist of: a time interval with zero acceleration, a time interval with constant negative acceleration and a time interval with constant positive acceleration. This is, in approximation, the signal, which was obtained after optimization in the previous subsection. As stated before, for practical reasons, the table should return to the same position after each excitation period. As indicated in Fig. 7, the following 9 parameters determine the shape of the table trajectory: y 1 , y 2 , y 3 , t 0 , t 1 , t 2 , t 3 , y ini , and y ini . There are some constraints on the table motion profile that need to be satisfied:

1. Minimal table displacement should be 0 (this is a choice rather than a constraint) 2. t 0 = 0 (this is also a choice rather than a constraint) 3. Fix the period time ∆t 4. Fix the stroke ∆y 5. The start position should be equal to the end position (periodicity requirement, in the second subplot of Fig. 7 the shaded area below the zero axis should be equal to the shaded area above the zero axis) 6. The start velocity should be equal to the end velocity (periodicity requirement, in the third subplot of Fig. 7 the shaded area below the zero axis should be equal to the shaded area above the zero axis) 7. y 1 = 0 Thus, 2 parameters can be chosen freely, for example y 2 and y 3 . Subsequently, the trajectory generator needs to determine 5 unknowns: the times t 1 , t 2 , t 3 , the initial position y ini , and the initial velocity y ini . The mathematical description of the constraints and the trajectory generator algorithm can be found in appendix A. In general, parameter values will be chosen such that certain phenomena can be clearly observed and explained.

Analysis of steady-state response

The analytical steady-state solutions of the mass and the classification of these solutions, using the prescribed table motion with three acceleration parts, will be addressed in Section 5.1. The results of some parameter studies will be discussed in Section 5.2. 7 Sketch of the table motion profile, which consists of three time intervals with constant acceleration.

' 0,1 y t = y t ' y ' ini y '' 1 0 y = '' 2 y - '' 3 y 0 t 1 t 2 t 3 t ' 0,2 y t = ini y t t '' y y ∆ t ∆ Fig.

Analytical solutions and classification

Given a prescribed motion profile of the table, obtained in section 4.4.3, the equation of motion ( 6) can be solved analytically by splitting the solution into different time intervals and combining them in one general solution by requiring continuity of x(t). This allows for a classification of different types of mass motions. The analytic solution will be discussed in detail here for one type of motion only.

For the case considered here, y 2 > 1 and y 3 < 1, which means that the mass cannot stick in [t 1 , t 2 ], but can stick in [t 2 , t 3 ]. Note that the accelerations are defined in such a way that y 2 > 0 and y 3 > 0, see Fig. 7. Every time the table acceleration values change (so at times t 1 , t 2 , t 3 ), and when the relative velocity x becomes zero, a discontinuity occurs in the mass acceleration x , such that the solution needs to be determined for all these time-spans separately. Starting with x 0 = x 0s = 0 will yield a steady-state trajectory immediately, as will be shown. In the remainder of this paper, if the initial conditions are chosen such that these yield a steady-state trajectory from the start (corresponding to the correct phase of the table motion), this will be denoted by the subscript 's'. First, consider the timespan [t 0 , t 1 ]. The table acceleration is zero in this timespan, so the mass will stick to the table. The relative velocity and acceleration of the mass will thus be zero. Assuming that the initial position of the mass is also zero gives for t 0 ≤ t ≤ t 1 :

x (t) = 0 (12)
x (t) = 0 (13)

x(t) = 0 (14)
In the second time interval [t 1 , t 2 ], the mass will start to slip because y 2 > 1. Note that from (6) it follows that the acceleration of the mass becomes x = -1 + y 2 , so for t 1 ≤ t ≤ t 2 :

x (t) = -1 + y 2 (15)
x (t) = (-1 + y 2 )(t -t 1 ) ( 16)

x(t) = (-1 + y 2 )(t -t 1 ) 2 /2 (17)
At time instance t 2 , the acceleration changes from -y 2 to y 3 . Therefore, the acceleration of the mass changes to x = -1 -y 3 . For t 2 ≤ t ≤ t x =0 :

x (t) = -1 -y 3 (18)
x (t) = x (t 2 ) + (-1 -y 3 )(t -t 2 ) ( 19
)
x(t) = x(t 2 ) + x (t 2 )(t -t 2 ) + (20) (-1 -y 3 )(t -t 2 ) 2 /2
where t x =0 is the time, at which the relative velocity of the mass x becomes zero because y 3 < 1. This time can be calculated analytically:

t x =0 = t 2 + -x (t 2 ) -1 -y 3 (21)
The mass will stick until t = t 3 . Indeed, the initial condition x 0s = 0 leads to a steady-state trajectory. Note that in principle only the velocity and acceleration of the mass are in steady-state in contrast to the position, because the mass has a net non-zero displacement every period time ∆t = t 3 -t 0 . For t x =0 ≤ t ≤ t 3 :

x (t) = 0 (22)
x (t) = 0 (23)

x(t) = x(t x =0 ) (24)
Simulations and solving different cases analytically point out that in the µ 1 region 5 different types of steady-state responses can occur, depending on the magnitudes of y 2 and y 3 , see table 1. Note that the analytic solution for case 1 has been derived above. For cases 1-4, the period of the relative velocity response is equal to the excitation period ∆t.

The analytic solutions of cases 2-5 can be derived in an equivalent manner as was done above for the case 1 type of motion. Determination of the resulting case can be done a priori by following the flowchart displayed in Fig. 8. Fig. 9 gives an illustrative graphical overview of the 5 cases that can occur, when the table motion with three constant acceleration time intervals is used. In addition, a non-physical, complex table trajectory may be obtained if, for example, a very large stroke ∆y is requested in a very short time span ∆t using very small acceleration values y 2 and y 3 . This non-physical table trajectory will be denoted by case 0.

Parameter studies and objective function evaluation

The analytic solutions of the mass motion resulting from the prescribed periodic table motion, can be used for very fast and exact parameter studies. In this way, it is possible to make surface plots of the objective function value f , see [START_REF] Leine | Dynamics and Bifuractions of Non-Smooth Mechanical Systems[END_REF], as a function of y 2 and y 3 . There is a strong analogy between the case of motion that is occurring and the objective function value. In Fig. 10, the average steady-state velocity v of the mass drops dramatically (f dramatically increases), if the type of motion changes from a general stick-slip motion to a slip-slip type of motion. As a design rule, this would mean that slip-slip trajectories should be avoided. Objective function values are limited to f = 1.6 • 10 -3 for clarity. Fig. 11 considers a range of acceleration values that are much smaller, such that stick can occur in time intervals of nonzero acceleration (case 1 or case 2 solutions). Also here, there is a clear relation between the type of motion and the average steady-state velocity. Case 1 solutions seem preferable. Objective function values f ≥ 0.8 are set to f = 0.8 for clarity.

The design rules for making a trajectory, for the current system and studied parameter ranges, can be summarized as follows:

-Increase the frequency of excitation (subject to practical constraints) -Increase the stroke (subject to practical constraints) -Use surface plots to make a sensible choice for the acceleration values y 2 and y 3 such that the chosen settings are not too sensitive. The response should be a case 1 (stick-slip, x ≥ 0), a case 3 (general stick-slip), or a case 4 (slip-slip) response; the latter two cases are only acceptable if v > 0. Case 2 (stickslip, x ≤ 0) and case 5 (just stick) responses are prohibited.

6 Local stability and convergence

Local stability analysis

Using Floquet theory, it can be proven that all steadystate solutions from Section 5.1 are locally stable. Three different approaches have been used to calculate the Floquet multipliers: 1) an analytical method based on linearization of the effects of the perturbations of the analytical steady-state solutions over a complete period, 2) a second analytical method, in which the Monodromy matrix is calculated by coupling smooth parts of the Fundamental solution matrix using Saltation matrices [START_REF] Leine | Dynamics and Bifuractions of Non-Smooth Mechanical Systems[END_REF], and 3) a numerical method, which estimates the Monodromy matrix using a sensitivity analysis. Here, the first method will be treated. Local stability is studied using the analytical expressions from Section 5.1, and following the same approach as in [START_REF] Natsiavas | Stability of piecewise linear oscillators with viscous and dry friction damping[END_REF]. Infinitesimally small perturbations ∆x 0 and ∆x 0 on the initial position respectively the initial velocity are considered. After linearization, perturbations in the position and velocity of the mass at the end of one period, i.e. at t = t 3 , ∆x 3 and ∆x 3 are linearly related to the perturbations in initial position ∆x 0 and velocity ∆x 0 as follows:

∆x 3 ∆x 3 = Φ(t 3 , t 0 ) ∆x 0 ∆x 0 (25)
where Φ(t 3 , t 0 ) is the so-called Monodromy matrix. Note that the times t i correspond to the notation of Section 4.4.3, see also Fig. 7. For a case 1 response (stick-slip with x (t) ≥ 0, ∀t) the analytic local stability analysis will be carried out here. An illustrative case 1 response is shown in Fig. 12. The mass will be in steady-state at t = t 0 , if the initial velocity is chosen as x 0s = 0. Using the analytic expressions derived in Section 5.1 and applying the initial conditions x(t 0 ) = x 0s and x (t 0 ) = x 0s , it can be shown that the position and velocity at t = t 3 can be written as:

x(t 3 ) = x 0s + x 2 0s /2 + c 1 (26) x (t 3 ) = 0 (27)
where c 1 is a constant. Using x 0s +∆x 0 and x 0s +∆x 0 as initial conditions at t = t 0 , will result in position x(t 3 )+ ∆x 3 and velocity x (t 3 ) + ∆x 3 at t = t 3 . Substitution of these quantities in (26) and (27) gives:

x(t 3 ) + ∆x 3 = x 0s + ∆x 0 + (x 0s + ∆x 0 ) 2 /2 + c 1 (28)
x (t 3 ) + ∆x 3 = 0 (29)

By substituting (26)-( 27) in ( 28)-( 29), keeping only the linear terms in ∆x 0 and ∆x 0 , and applying the initial condition x 0s = 0 (to assess the local stability of the steady-state solution), the following relation between perturbations at t = t 0 and t = t 3 can be derived:

∆x 3 ∆x 3 = 1 0 0 0 ∆x 0 ∆x 0 (30)
The Floquet multipliers, i.e. the eigenvalues of the Monodromy matrix, are λ 1 = 1 and λ 2 = 0. The eigenvalue equal to 1 is expected due to the fact that a perturbation in the initial position only, will only shift the end position by the same amount. The corresponding eigenvector of

λ 1 = 1 is v 1 = [1, 0]
T , which confirms that this eigenvalue is purely related to the freedom in position. The second eigenvalue λ 2 = 0 and the corresponding eigenvector v 2 = [0, 1] T , which illustrates the fact that a small change in initial velocity will not change the end velocity at all, because the mass will get into stick anyway, despite the small perturbation at t = t 0 .

For the cases 2 and 3 a completely equivalent reasoning can be used. In both cases, this leads to the same eigenvalues as for case 1. Case 4, the slip-slip case, is different, because the mass never sticks. This means that the end velocity after one period can be influenced by the initial velocity. It can be shown that the position and velocity after one period can be written as:

x(t 3 ) = x 0s + c 1 x 2 0s + c 2 x 0s + c 3 (31) x (t 3 ) = c 4 x 0s + c 5 ( 32
)
where c i for i = 1, 2, ..., 5 are all constants. The following (linearized) relation describes the evolution of small perturbations ∆x 0 , ∆x 0 on the initial conditions:

∆x 3 ∆x 3 = 1 2c 1 x 0s + c 2 0 c 4 ∆x 0 ∆x 0 (33)
The Floquet multipliers can be shown to be λ 1 = 1, which again represents the freedom in position, and λ 2 = c 4 = ((y 2 -1)(y 3 -1))/((y 2 + 1)(y 3 + 1)). Note that this eigenvalue is always positive and smaller than 1 (so within the unit circle) because y i > 1 for i = 2, 3 in the slip-slip case.

The local stability analysis presented in this section has shown that, besides the fact that the response can be shifted in position, the remaining dynamics are all locally asymptotically stable. The analysis using Saltation matrices to compute the Monodromy matrix gives identical results. The method using numerical estimation using a sensitivity analysis gives a good approximation of the exact analytical results.

Convergence

In this subsection, convergence of the relative velocity solutions x will be proven, which also proves that the solution converges to a unique velocity solution in steady-state, independent of the initial conditions. The (dimensionless) relative velocity is here denoted by v = x . In terms of the relative velocity, the dynamics is described by, see [START_REF] Duarte | Fractional describing function of systems with Coulomb friction[END_REF]:

v ∈ -Sign(v) -y (t) (34)
Now consider two trajectories, v 1 (t, t 0 , v 10 ) and v 2 (t, t 0 , v 20 ), with different initial conditions v 10 and v 20 at t = t 0 , and consider the following Lyapunov function:

V (v 1 , v 2 ) = (v 1 -v 2 ) 2 2 (35)
The time derivative of this Lyapunov function along solutions of the system can be written as follows [START_REF] Yu | On solution concepts and well-posedness of linear relay systems[END_REF]:

V (v 1 , v 2 ) = (v 1 -v 2 )(v 1 -v 2 ) ∈ -(v 1 -v 2 )(Sign(v 1 ) -Sign(v 2 )) (36)
≤ 0

If v 1 and v 2 are both positive or both negative, V = 0. Therefore, V is only negative semi-definite and convergence is not proven yet. Let v1 ∈ Sign(v 1 ) and v2 ∈ Sign(v 2 ), see [START_REF] Berger | Friction modeling for dynamic system simulation[END_REF]. Now, it can be shown that if v1 = v2 and v 1 (t A ) = v 2 (t A ) at a certain time t = t A (otherwise convergence has already occurred), there will always be a time t B > t A , for which v1 = v2 , resulting in further decrease of V . Consider the situation, in which v1 = v2 . For both trajectories the acceleration v i will be equal, see (34). Both trajectories will eventually reach zero velocity v i = 0. To see this, consider (34). The excitation signal y (t) is periodic and over a single period ∆t it has an average acceleration equal to zero, such that its net effect on the velocity x of the mass over one period time ∆t is zero. This is shown by integrating (34) over one period time ∆t, for v = 0:

v(t + ∆t) -v(t) = - t+∆t t (Sign(v) + y (t)) dt (37) = - t+∆t t

Sign(v)dt

The term -Sign(v) will force the relative velocity of the mass to v = 0, which will occur at some point in time. After the trajectory reaches v = 0, the mass can stick or slip, depending on the value of the acceleration y of the table. At the moment one of the trajectories reaches v = 0, v1 = v2 , and V < 0, which means that v 1 and v 2 will approach each other. This will last until a point in time is reached, at which v1 = v2 again, and the whole process will repeat itself. Eventually, for case 1-3 solutions, in finite time a point will be reached, at which v 1 = v 2 = 0. From this time onwards, both velocity signals are fully converged, i.e. are equal to each other. In case of a case 4 solution, the mass will never stick, and v 1 = v 2 = 0 will be reached in the limit t → ∞. As an example to illustrate this convergence of the velocity solutions, consider Fig. 13. The upper plot of Fig. 13 shows two transients leading to a stick-slip trajectory (case 1). The lower plot of Fig. 13 shows the corresponding Lyapunov function V as a function of time. In Fig. 13, time intervals, in which v1 = v2 , are indicated by '='-signs. Time intervals, in which v1 = v2 , are indicated by ' ='-signs. Clearly, in the time intervals, where v1 = v2 , the solutions converge to each other and V < 0. In the time intervals, where v1 = v2 , the solutions 'run in parallel' and V = 0. The velocity solutions become identical in the last time interval in Fig. 13. The Lyapunov function V therefore becomes zero.

Summarizing, although the time derivative of the Lyapunov function is only negative semi-definite, it has been shown that in the time intervals, in which V = 0, the solutions evolve in such a way that always a time interval, in which V < 0, will be reached. This proves that the two solutions v 1 and v 2 will converge to each other. Therefore, the steady-state solution in terms of velocity is unique and is independent of the initial velocity of the mass. In other words, depending on the system parameters, different types of steady-state motion occur as discussed in Section 5.1. However, each steady-state solution is unique. There are no coexisting steady-state solutions.

7 Positioning accuracy in the stopping phase For stopping and accurate positioning of the mass an increased friction coefficient µ 2 , as described earlier in Section 3.3, is used. Depending on the weight distribution of the mass over the two table regions with two different Coulomb friction coefficients µ 1 and µ 2 , the effective friction coefficient μ experienced by the mass can be calculated using [START_REF] Galvanetto | Dynamics of a simple damped oscillator undergoing stick-slip vibrations[END_REF]. Due to the existence of a low friction and high friction region, the system dynamics in the friction transition region, see ( 5) and ( 9), is given by:

-1 + c µ -1 w (x -x µ ) Sign(x ) x + y (38)
As long as y i < 1 + ((c µ -1)(x -x µ )/w) for i = 2, 3, the mass will be able to stick in the high friction region.

Definition of positioning accuracy

Before defining and studying the accuracy of the end position of the mass, first a discussion on the derivation of the end position itself will be given. Consider the situation sketches in Figs. 14 and15. It is assumed that the mass will enter the high friction region in a steady-state motion. The end position of the mass will depend on the state of the mass [x, x ] T (corresponding to a certain phase of the periodic table motion) when it reaches x = x µ , see Fig. 2, for the first time. This state will obviously depend on the initial state [x 0s , x 0s ] T (the subscript 's' stresses the fact that the initial conditions are chosen corresponding to a steady-state solution) at t = t 0 , the phase of the periodic excitation signal at t = t 0 , and on x µ -x 0s .

The position where the mass stops, is defined by x µ + ∆x overshoot . The effective displacement in the low friction region, during one excitation period in steadystate, can be written as v∆t (v is the average velocity during one excitation period). The maximum displacement in one period is defined as ∆x. To include all possible ways of entering the high friction region in steadystate, define x µ = ∆x and vary the initial positions corresponding to steady-state x 0s in the range -v∆t ≤ x 0s ≤ 0. Corresponding initial velocities x 0s are used to assure steady-state motion from the start. Now, collecting all overshoots ∆x overshoot ∈ [∆x overshoot,min ,

x µ x t x ∆ v t -∆ overshoot x ∆ t ∆ v t ∆ 0 t ∆ v t ∆ 1 µ 2 µ
Fig. 14 Situation sketch of when the mass is stopped. ∆x overshoot,max ] for all initial conditions corresponding to steady-state responses, will provide a range of the possible end positions of the mass. The positioning error is now defined as:

∆x d = ∆x overshoot,max -∆x overshoot,min 2 (39)
The desired end position can now be defined as x d = x µ + ∆x overshoot,min + ∆x d , see Fig. 15.

Results

The positioning accuracy will, next to system properties, obviously depend on the prescribed table motion. In Section 5, the stroke and frequency of the table were fixed and the accelerations y 2 and y 3 were varied. For different combinations of y 2 and y 3 , the positioning error is calculated and stored. In this section and in Section 7.3, the stroke and the excitation period of the table are set to respectively ∆y = 0.084 and ∆t = 1. The dimensionless width is chosen to be w = 0.05. It is assumed that c µ = 2.5 such that the friction coefficient in the high friction region is µ 2 = 2.5µ. Hence, the interesting region of accelerations is limited to y i < 1 + (c µ -1)(x -x µ )/w, so to accelerations y i < 2.5 (enabling the mass to permanently stick to the table). Remarkably, as will be shown below, a design space for y 2 and y 3 can be identified, where the positioning error ∆x d = 0.

Consider the following two table acceleration settings: (y 2 , y 3 ) = (2.3, 1.3) and (y 2 , y 3 ) = (1.8, 1.1). In Fig. 16, the two corresponding phase portraits are shown. For each setting, two responses corresponding to two different initial conditions are shown. The two dashed vertical lines respectively indicate the positions, where the high friction region is starting (x µ ), and from where it is possible for the mass to permanently stick to the table (x stick,min ). The position on the table, from where the mass is able to permanently stick to the table, is determined by the following condition:

max(y i ) = 1 + c µ -1 w (x -x µ ) (40)
Using this condition, the position x stick,min can be calculated to be: In the lower plot of Fig. 16, the end positions are clearly different. In the upper plot of Fig. 16, however, the mass stops at the same position for both trajectories, although the initial conditions are different: an accumulation position can be identified. A detailed analysis of this accumulation position will be presented in Section 7.3. There, among others, it will be shown that depending on the parameter values, sometimes for all initial conditions, but also sometimes for only a part of the initial conditions, accumulation results. When accumulation occurs, the same type of trajectory repeats itself on smaller and smaller scales up to the accumulation position. Note that the accumulation position in theory only will be reached for t → ∞. Interestingly, when accumulation occurs, it will be shown that the trajectories accumulate to the position, from where it is possible for the mass to permanently stick, i.e. x stick,min . Fig. 17 combines several analysis results showing a surface plot (A) of the objective function value f , see [START_REF] Leine | Dynamics and Bifuractions of Non-Smooth Mechanical Systems[END_REF], a case-plot (B) indicating the type of motion, an accumulation identification plot (C), and a surface plot with the positioning accuracy (D). From the function value plot (A), it is clear that a high value for y 2 results in fast movement of the mass, but y 2 should be low enough to allow sticking in the high friction region. Simultaneously, a low value for y 3 increases the chance of a case 1 type of motion, see plot (B). Fortunately, this is also the parameter space, in which the lowest positioning error can be achieved, see plot (D). When plot (C) is compared to the positioning error plot (D), it becomes clear that, for the current parameter settings and used initial conditions, the occurrence of accumulation results in extreme positioning accuracy of the mass (∆x d = 0).

x stick,min = x µ + w max(y i ) -1 c µ -1 (41)

Analysis of the accumulation position

The previous section showed that for some parameter settings, the mass stops at a unique end position, even though the initial conditions are different, leading to ultimate positioning accuracy of the mass. It is now assumed that the mass enters the high friction region with a case 1 steady-state motion without any negative relative velocity. This corresponds to the lowest values of f in the region with low acceleration values, see subplot (A) of Fig. 17. This case 1 motion guarantees that the mass will stay in the high friction region once it entered it.

In this section, analytic expressions will be derived for the subsequent positions, where relative velocities x = 0 are found, in the region x > x µ . Simulations based on these expressions indicate that, in case of accumulation, the distance between these positions will become shorter and shorter and that accumulation to a fixed position will occur.

In the transition region (x µ ≤ x ≤ x µ + w), when the mass is moving, so in forward slip, the equation of motion (38) can be written as:

x (t) + b 1 x(t) = b 3 (t) (42)
where b

1 = (c µ -1)/w, b 3 (t) = b 2 -y (t)
, and b 2 = ((c µ -1)x µ /w) -1. Thus, in essence, the system behaves as a forced mass-spring system in the friction transition region. The general analytic solution to differential equation (42) can be written as follows:

x(t) = b 3 b 1 + d 1 cos( b 1 (t -t * )) + (43) d 2 sin( b 1 (t -t * )) x (t) = -d 1 b 1 sin( b 1 (t -t * )) + (44) d 2 b 1 cos( b 1 (t -t * ))
where the constants d 1 and d 2 are determined by the conditions at the time t = t * , at which the mass enters the transition region, i.e. x(t * ) = x µ :

d 1 = x(t * ) - b 3 b 1 (45) d 2 = x (t * ) √ b 1 (46)
The time, at which the velocity reaches zero for the first time for t > t * can be calculated analytically to be:

t x =0 = t * + 1 √ b 1 arcsin d 2 d 2 1 + d 2 2 (47)
Now, the goal is to find expressions for the subsequent positions, where the mass sticks to the table. By studying these subsequent stick positions, or actually the mapping between them, it is possible to explain the accumulation to a unique end position. The position, at which stick occurs for the n th time, is denoted by x n . For n = 1, x 1 can be calculated by substituting (47) in (43). To make the notation more compact, a new time scale τ is introduced, which is zero at the moment, when the mass gets out of the first stick phase in the transition region. The mass will stick at the y 3 part of the trajectory. The mass therefore starts to slip at τ = 0 when the acceleration changes from y 1 = 0 to -y 2 . The following values apply at τ = 0:

b 3,1 = b 2 + y 2 (48) d 1,1 = x n - b 3,1 b 1 (49) d 2,1 = 0 √ b 1 = 0 (50)
Note that b 1 and b 2 are constant in the transition region. The first part of the analytic solution can therefore be written as follows, for 0 ≤ τ ≤ t 2 -t 1 :

x(τ ) = b 3,1 b 1 + d 1,1 cos( b 1 τ ) (51) x (τ ) = -d 1,1 b 1 sin( b 1 τ ) (52)
At τ = t 2 -t 1 , the acceleration changes from -y 2 to y 3 . At τ = t 2 -t 1 , the following values apply:

b 3,2 = b 2 -y 3 (53) d 1,2 = x(t 2 -t 1 ) - b 3,1 b 1 (54) d 2,2 = x (t 2 -t 1 ) √ b 1 (55)
which yields the following analytic solution for time interval t 2 -t 1 ≤ τ ≤ τ x =0 , where τ x =0 is the time, at which the relative velocity x becomes zero again: The displacement at τ x =0 will be denoted by x n+1 = x(τ x =0 ). After algebraic manipulation the following mapping between subsequent positions x n and x n+1 can be found:

x(τ ) = b 3,2 b 1 + d 1,2 cos( b 1 (τ -(t 2 -t 1 ))) (56)
x n+1 = b 2 -y 3 b 1 + ( 59
)
y 2 + y 3 b 1 + x n - b 2 + y 2 b 1 cos( b 1 (t 2 -t 1 )) 2 + x n - b 2 + y 2 b 1 sin( b 1 (t 2 -t 1 )) 2 1 2
The fixed point of this mapping can be calculated to be:

x n = x n+1 = b 2 + y 2 b 1 = x stick,min (60)
Recall that b 1 = (c µ -1)/w and b 2 = (c µ -1)x µ /w -1, so that in fact the fixed point of the mapping is the position x stick,min , from where permanent stick is possible, see (41)! In Fig. 18, mapping (59) is shown for settings y 2 = 2.2, y 3 = 0.9, ∆t = 1.05, and ∆y = 0.084. Note that this also specifies t 2 -t 1 appearing in (59), see appendix A. The left and the right dashed vertical lines in Fig. 18 indicate respectively the starting position of the µ 2 friction region (x = x µ ), and from where the mass is able to permanently stick (x = x stick,min ). Indeed, the fixed point of the mapping is located at the same position as x stick,min .

Note again that in fact the end position, in case of accumulation to this fixed point, is reached for t → ∞. In a practical situation, however, the end position will be reached in finite time. Consider a dimensionless scalar p, where 0 ≤ p ≤ 1. For the cases, where accumulation occurs in Fig. 17, the position x µ + p(x stick,minx µ ) is reached in at most 5 period times for p = 0.99, and in at most 7 period times for p = 0.999. This has been verified by time-stepping simulations.

The initial conditions, which determine the first stick position x 1 , can influence the occurrence of accumulation. Consider again Fig. 18, which by means of the middle dashed vertical line shows that, depending on the initial conditions, the mass can stick to the table at a position x > x stick,min , or can exhibit the accumulating behavior and end up at x = x stick,min . Note that accumulation to an end position will always occur if x n = x n+1 Fig. 18 Depending on the initial conditions, accumulation to a fixed end position is possible.

x n+1 (x µ ) < x stick,min , and is then independent of the initial conditions.

Conclusions and recommendations

In this paper, one-dimensional self-alignment of a mass via stick-slip vibrations has been studied. Using a suitable periodic table trajectory, it is possible to move the mass into a desired direction and using an increase in the friction coefficient it is possible to stop the mass at a certain end position.

The first objective of this paper is to design a trajectory for the table, such that the mass moves to a desired end position in the least possible amount of time, given some table motion constraints. The design of a table motion profile, consisting of three time intervals with different constant accelerations has been discussed. For the mass, analytical steady-state solutions have been derived, and a classification of the types of steady-state responses of the mass has been given.

Using Floquet theory for discontinuous systems, the local stability of the solutions has been proven. Due to the freedom in position, one Floquet multiplier will always be equal to one. Furthermore, using a Lyapunov analysis, convergence of the velocity response has been proven. In other words, different velocity solutions will converge to each other, independent of the initial velocities. This convergence implies uniqueness of the velocity response of the mass.

The second objective of this paper is to stop the mass at a desired end position with the best possible positioning accuracy. For certain parameter settings, an interesting phenomenon is observed in the transition

Fig. 4

 4 Fig. 4 Influence of excitation frequency, objective function value f vs ∆t.

Fig. 5 Fig. 6

 56 Fig. 5 Influence of stroke, objective function value f vs ∆y.

Fig. 9

 9 Fig. 9 Examples of time histories of the 5 different cases that can occur.

Fig. 10

 10 Fig. 10 Surface plot and case plot for the settings ∆y = 0.021, ∆t = 0.072.

Fig. 11

 11 Fig. 11 Surface plot and case plot for the settings ∆y = 0.064, ∆t = 1.38.

Fig. 12

 12 Fig. 12 Example of an illustrative case 1 trajectory.

Fig. 13

 13 Fig. 13 Illustration of convergence of velocity solutions.

Fig. 15

 15 Fig. 15 Region of end positions where the mass can stop.

Fig. 16

 16 Fig.[START_REF] Yu | On solution concepts and well-posedness of linear relay systems[END_REF] Phase portraits of two different settings when stopping the mass.

Fig. 17

 17 Fig. 17Objective function value f (A), case-plot (B), accumulation identification plot (C), positioning error ∆x d (D).

  Fig. 17Objective function value f (A), case-plot (B), accumulation identification plot (C), positioning error ∆x d (D).

+d 2 , 2

 22 sin( b 1 (τ -(t 2 -t 1 ))) x (τ ) = -d 1,2 b 1 sin( b 1 (τ -(t 2 -t 1 ))) + (57) d 2,2 b 1 cos( b 1 (τ -(t 2 -t 1 )))Using (47), the mass gets into stick in the transition region for the second time at time:

τ x =0 = (t 2 -t 1

 21

  table displacement signal ỹ( t) such that the mass travels to a desired end location xd in the smallest possible process time Tp , where Tp = min( W ) and W ∈ { t ≥ 0|x( t) = xd }. -Minimize the positioning error |∆x d | of the end position of the mass, in other words, minimize the interval [x d -∆x d , xd +∆x d ], in which the mass stops.

Table 1

 1

  Table showing possible types of steady-state responses.

	Case	Description	(Un)desirable?
	1	Stick-slip with	Desirable
		x (t) ≥ 0, ∀t
	2	Stick-slip with	Undesirable
		x (t) ≤ 0, ∀t
	3	General Stick-slip	Acceptable if v > 0
	4	Slip-slip	Acceptable if v > 0
	5	Pure stick	Only desirable
		(no relative motion)	in stopping region

  Flowchart to calculate a priori which case will occur.

	Values for	y	'' 2,dim	and	'' 3,dim y
		Physical table			no	Case 0
		solution?		yes			α	'' 3,dim y	>	1	get into stick? Enough time to
	α	y	'' 2,dim	>	1			yes		yes	yes	Case 3
							no			no	no
	α	'' 3,dim y	>	1		yes	Case 2	Case 1	Case 4
							no
						Case 5
	Fig. 8

Acknowledgements

This work was supported in part by a grant from Miami University's School of Engineering and Applied Sciences for International collaborations.

region from low friction to high friction, namely accumulation of the mass position to a unique end position. This accumulation behavior benefits the positioning accuracy in an ultimate manner. In essence, the system behaves as a forced mass-spring system in this region, which allows for an analytical study of the accumulation behavior. Subsequent stick-positions have been related to each other resulting in a discrete mapping. It is shown that the fixed point of this mapping is exactly the position, from where permanent stick becomes possible. Under certain conditions, accumulation will always occur, independent of the initial conditions.

Recommendations for future research are: 1) experimental verification of the analyzed behavior, 2) analysis of models with resembling, but more realistic table motion profiles, i.e. finite values for the jerk should be introduced at times, when the table acceleration values change, 3) use of enhanced friction models, 4) extension to two-dimensional models describing general planar motion of the mass, and 5) extension to selfalignment of multiple masses.

A Details of the prescribed table motion generation using three acceleration parts

In order to calculate a table trajectory that satisfies the 7 constraints formulated in Section 4.4.3, it is necessary to find explicit expressions for the following 5 variables: t 1 , t 2 , t 3 , y ini , and y ini . Note that the following two variables are already explicitly known: y 1 = 0 and t 0 = 0. It is therefore only needed to solve 5 constraint equations, which can be derived from the constraints formulated in Section 4.4.3. Using Fig. 7 the following 5 equations can be derived:

1.

.

.

For convenience, in these equations two additional unknowns are introduced, namely t y =0,1 and t y =0,2 , which are the times, at which the velocity y of the table becomes zero. Therefore, to solve the set of equations, two additional equations are needed, which define these two additional unknowns: 6.

y ini -y 3 (t 3 -t y =0,2 ) = 0 (67) Now 7 equations are available in the 7 unknowns t 1 , t 2 , t 3 , y ini , y ini , t y =0,1 , and t y =0,2 , This set of equations is solved by the symbolic toolbox of Matlab. The explicit solutions are not given here, because the expressions are too long to display.

Please note that actually, there is a second case that needs to be considered. If the initial velocity y ini appears to be smaller than 0, then the previously discussed mathematical description of the constraints is not valid anymore. However, a completely equivalent reasoning as for the case y ini > 0 can be followed. Therefore, the second case is not discussed in detail here. Moreover, note that it is possible that no physical solutions exist for the above specified equations if, for example, a large displacement ∆y is demanded in a short amount of time ∆t, using small acceleration values y 2 and y 3 . In that case the solutions of some variables will be complex valued, lacking a physical interpretation.


Anna Moniuszko
email: annamoniuszko@op.pl

Wojciech Łuczaj

Magdalena Rusak

Sławomir Pancewicz

Joanna Zajkowska

Elżbieta Skrzydlewska

"Moreover the analysis of correlation showed that in the group with Lyme arthritis the correlations between all analyzed lipid peroxidation products are significantly weaker than in the control." Lipid peroxidation products as potential bioindicators of Lyme arthritis

Keywords: Lyme disease, malondialdehyde, 4-hydroxynonenal, 8-isoPGF2α Lyme disease, malondialdehyde, 4-hydroxynonenal, 8-isoPGF 2

Aim: Lipid peroxidation products -malondialdehyde, 4-hydroxynonenal and 8-isoPGF2α were determined in the plasma and urine of patients with Lyme arthritis and healthy people. Methods: The group consisted of 19 patients with Lyme arthritis (mean age -47 years). The control group consisted of 16 healthy individuals (mean age -38 years). Diagnosis of Lyme disease was confirmed by epidemiological anamnesis, clinical manifestation of arthritis and serological examinations. Lipid peroxidation was estimated by the measurement of aldehydes (malondialdehyde (MDA) and 4-hydroxynonenal (4-HNE) determined by HPLC) and prostaglandin derivatives (8-isoPGF2α -determined by LCMS). Results: Malondialdehyde and 4-hydroxynonenal levels were increased about 2-4-fold in the plasma, while in the urine about 2-fold. More significant increase, which referred to the 8-isoPGF2α total plasma level, was enhanced over 4-fold, while the urine 8-isoPGF2α level was increased over 8-fold. The 8-isoPGF2α total plasma level consists of free and esterified form. During infection the ratio of free to esterified form is significantly smaller compared to healthy people. Conclusions: The ratio of free to esterfied form of 8-isoPGF2α may be a useful indicator of Lyme arthritis. Moreover the complementarities of three lipid peroxidation products levels  may be helpful in the diagnosis of Lyme arthritis.

2.The discussion has been significantly shortened.

3.The problem with control we tried to solve as follows; *In the Materials and methods section the following information was added "In the history of the control group no other diseases, which could be responsible for increased lipid peroxidation (e.g. arthritis of any etiology) were observed." *In the Discussion after statement "Therefore 8-isoPGF2α can be implicated as mediators of Lyme arthritis development. " the following sentences have been added [with citation of Basu et al paper] "However it has been shown that an increase in 8-isoPGF2α and its major metabolite in the course of other type of arthritis including rheumatoid arthritis, reactive arthritis, psoriatic arthritis and osteoarthritis may be observed, which in consequence may cause diagnostic problems [START_REF] Basu | Raised levels of F2-isoprostanes and prostaglandin F2α in different rheumatic diseases[END_REF]. In case of Lyme disease, both immunological data and lipid peroxidation products levels should allow to establish the diagnosis."

Answers for comments of Reviewer 2:

1.In order not to stress only one lipid peroxidation product as the most important and to indicate that the used technique for MDA and HNE determination is HPLC and for 8-isoPGF2α is LCMS the following changes wave been done in the abstract and discussion: in abstract: "Lipid peroxidation was estimated by the measurement of aldehydes (malondialdehyde (MDA) and 4hydroxynonenal (4-HNE) determined by HPLC) and prostaglandin derivatives (8-isoPGF2αdetermined by LCMS) in the plasma and urine." "Moreover the complementarities of three lipid peroxidation products (MDA, 4-HNE and 8-isoPGF2α levels may be very helpful in the diagnosis of Lyme arthritis." in discussion: In conclusion, it should be emphasized that during Lyme arthritis lipid peroxidation is enhanced. In the course of infection the ratio of free to esterified form of 8-isoPGF2α is significantly smaller compared to the level in healthy people. This founding indicates that the ratio of free to esterfied form of 8-isoPGF2α may be a useful indicator of Lyme arthritis. Moreover the complementarities of the three lipid peroxidation products (MDA, 4-HNE and 8-isoPGF2α levels may be very helpful in the diagnosis of Lyme arthritis.

2. Information that HNE level reflects only free aldehydes level has been added however the comparison of advantages of ELISA and HPLC technique has not been done, because this paper is not methodological paper. We want to prepare the methodological paper showing 8-isoPGF2 determinations and there we will include information about comparison of the LC and ELISA techniques.

3 . The discussion has been significantly shortened and significance of MDA and HNE has been emphasized. [START_REF] Uesugi | Inflammatory Properties of IgG Modified by Oxygen Radical and Peroxynitrite[END_REF].Conclusion indicates for complementarities of three lipid peroxidation products as above (point 1).

5.Errors have been corrected.

Introduction

The Ixodes ricinus ticks belonging to the arachnids are responsible for transferring many bacteriae, viruses and riccetsiae pathogenic to humans and animals. In Central Europe TBEV (Tick-Borne Encephalitis Virus) and spirochete Borrelia burgdorferi cause the most wide spread zoonoses, namely Lyme disease and TBE [START_REF] Steere | Lyme disease[END_REF]. In the last two decades a dramatic increase of incidence of both diseases was reported. 100 000 new cases in Europe and 15.000-20.000 in North America are diagnosed annually. Considering all the tick-borne diseases, several European countries such as Poland, Estonia, Sweden, Austria, Czech Republic or Slovenia are characterized by the highest incidence in the northern hemisphere.

In the pathomechanism of Lyme disease, a significant role is played by the inflammatory and immunological mechanisms. B. burgdorferi causes damage of various organs and systems including skin, nervous system, osteoarticular system, heart, or eyes. Clinical outcome of Lyme disease depends on the spirocheates genotype and the immunological system of patients. This infection may lead to various pathologies, most commonly inflammatory arthritis characterized by cellular infiltration in the joints, predominantly with neutrophils and macrophages [START_REF] Asch | Lyme disease: an infectious and postinfectious syndrome[END_REF]. It predisposes to the generation of reactive oxygen species, which cause changes in antioxidant abilities of the organism [START_REF] Pancewicz | Role of reactive oxygen species (ROS) in patients with erythema migrans, an early manifestation of Lyme borreliosis[END_REF][START_REF] Uesugi | Inflammatory Properties of IgG Modified by Oxygen Radical and Peroxynitrite[END_REF]. In consequence, cellular components such as lipids, DNA and proteins are oxidatively modified, which leads to changes in their structure and functions, and significantly contributes to the pathophysiology of oxidative stress.

Lipids are readily attacked by reactive oxygen species, which results in the formation of numerous peroxidation products. One group of oxidation products that may exert toxic effects on the cells is very reactive carbonyls, such as malondialdehyde and 4-hydroxynonenal [START_REF] Esterbauer | Chemistry and biochemistry of 4-hydroxynonenal, malonaldehyde and related aldehydes[END_REF]. Unlike reactive oxygen species, aldehydes are rather long-lived and can therefore diffuse from the site of their origin reaching and attacking the intracellular and extracellular targets [START_REF] Esterbauer | Chemistry and biochemistry of 4-hydroxynonenal, malonaldehyde and related aldehydes[END_REF]. They disrupt various important structural and protective functions associated with biomembranes in various pathological events in vivo [START_REF] Barclay | Quantitative studies of peroxidation, antioxidant action, partitioning and oxidative stress[END_REF]. Moreover, the generated aldehydes are highly reactive and may act as "secondary toxic messengers" of the primary free radical event. These biogenic aldehydes possess strong electrophilic property which is reflected in their ability to form adducts with nucelophilic proteins amino acids residues as well as with DNA bases accompanied by alterations in their structure and functions.

Another class of lipid oxidation products formed in abundance in vitro and in vivo is the prostaglandin F2-like compounds groupisoprostanes, mainly 8-isoPGF 2 [START_REF] Montuschi | Isoprostanes: markers and mediators of oxidative stress[END_REF]. 8-isoPGF 2 are produced in vivo independently of cyclooxygenases, primarily by free radical induced peroxidation of arachidonic acid [START_REF] Morrow | A series of prostaglandin F2like compounds are produced in vivo in humans by a non-cyclooxygenase, free radical-catalyzed mechanism[END_REF]. It has been shown that levels of 8-isoPGF 2 are increased in a number of human diseases. Therefore, the measurement of these molecules is currently recognised as the most accurate analytical method to assess the oxidative injury in vivo.

Moreover they are more stable than aldehydes generated during lipid peroxidation. In addition, products of the 8-isoPGF 2 pathway have been found to exert potent biological actions and thus may be pathophysiologic mediators of diseases [START_REF] Montuschi | Isoprostanes: markers and mediators of oxidative stress[END_REF][START_REF] Milne | Human biochemistry of the isoprostane pathway[END_REF].

The clinical symptoms caused by B. burgdorferi, especially in the early stage of infection, may not be characteristic and may not allow to base a diagnosis on them. Specific IgM and IgG antibodies titer in the serum of the Lyme disease patients enable etiological diagnosis. The results of serological examinations do not correlate with the severity of the disease and are not useful in the treatment monitoring and potential complications prediction.

Therefore, the objective of our study was the measurement of the serum lipids peroxidation products concentration and potential designation of their role in the pathomechanism of the Lyme disease.

Material and methods

The group consisted of 19 patients with Lyme arthritis (7 females and 12 males) aged 25 to 67 years (mean age-47 years). The control group consisted of 16 healthy individuals (12 females and burgdorferi antibodies titer was 18 BBU/ml (min-11 BBU/ml, max-88 BBU/ml) and the mean IgG anti/B. burgdorferi antibodies titer was 38 BBU/ml (min-11 BBU/ml, max-87 BBU/ml). In all cases results of ELISA test were confirmed by Western blot testing. In the history of control group no other diseases, which could be responsible for increased lipid peroxidation (e.g. arthritis of any etiology) were observed.

In epidemiological anamnesis 90% of patients reported previous tick bites. In all cases pain limitating movement of big joints (especially knee, elbow, shoulder) and muscles, especially of tights and shoulders was observed.

From patients with diagnosis of Lyme disease as well as from healthy people the blood was taken into EDTA tubes. The blood was centrifuged at 3 000 x g (4C) to obtain the plasma. From the same people urine was collected.

Lipid peroxidation was estimated by measuring malondialdehyde (MDA), 4-hydroxynonenal (4-HNE) and 8-isoPGF 2 [START_REF] Li | Quantitative high performance liquid chromatography/tandem mass spectrometric analysis of the four classes of F 2isoprostanes in human urine[END_REF][START_REF] Londero | Automated HPLC separation with spectrofluorometric detection of malondialdehyde-thiobarbituric acid adduct in plasma[END_REF][START_REF] Yoshino | Fluorometric liquid chromatographic of aliphatic aldehydes arising from lipid peroxides[END_REF] in the plasma as well as in the urine. Malondialdehyde was determined as a malondialdehyde-tiobarbituric acid adducts separated by HPLC with spectrofluorometric quantification at 532 nm exitation and 553 nm emission. 4-hydroxynonenal was determined as a one of the fluorescent decahydroacridine derivatives formed in reaction with 1,3-cyclohexanodione. It was determined by HPLC with fluorescence detector with excitation at 380 nm and emission at 445 nm.

The plasma 8-isoPGF 2 were determined as free 8-isoPGF 2 (hydrolyzed in the organism), notesterified in phospholipids and as total 8-isoPGF 2  contained free and esterified 8-isoPGF 2  To determine total 8-isoPGF 2 1.0 ml of plasma was thawed and homogenised with 30l internal standard (300 pg 8-isoPGF 2 ), 80l of a solution containing BHT (10mM) and triphenylphosphine (1 mM) in ethanol, and the mixture was vortexed thoroughly. Then 400 l of a solution containing 25% (w/v) KOH was added and the mixture was heated on a water bath for 45 minutes at 40C in order to hydrolyse esterified isoprostanes. After hydrolysis the sample was diluted with 3ml water (in order to prevent protein precipitation) followed by vortexing. Acetic acid was added to pH 3 to prevent dissociation of the isoprostanes. A solid phase column (tC18 Sep-pak Vac, 3ml, 200mg; Waters Chromatography) was conditioned with 3ml methanol, followed by 3ml water that was adjusted to pH 3 with glacial acetic acid. The sample was vortexed again and loaded onto the SPE column. The test-tube was washed with 3mL water (pH 3, corrected with glacial acetic acid) and the washing solution was also loaded onto the column and cartridge was dried. The cartridge was then washed with 3ml heptane to remove lipids. The cartridges were dried again under vacuum after which the sample was eluted with 7ml of eluent consisting of ethyl acetate/heptane/methanol (50/40/10%). The resulting solution was evaporated to dryness under a gentle stream of nitrogen. The residue was dissolved in 60 l acetonitrile/water (60/40% with 0.05% (v/v) acetic acid).

 determine free 8-isoPGF 2 1.0 ml of plasma was thawed and homogenised with 30l internal standard (300 pg 8-isoPGF 2 ), 80l of a solution containing BHT (10mM) and triphenylphosphine (1 mM) in ethanol, and the mixture was vortexed thoroughly. Then 3400 l of water was added. Acetic acid was added to pH 3 to prevent dissociation of the isoprostanes. A solid phase column (tC18 Sep-pak Vac, 3ml, 200mg; Waters Chromatography) was conditioned with 3ml methanol, followed by 3ml water that was adjusted to pH 3 with glacial acetic acid. The sample was vortexed again and loaded onto the SPE column. The test-tube was washed with 3mL water (pH 3, corrected with glacial acetic acid) and the washing solution was also loaded onto the column and cartridge was dried. The cartridge was then washed with 3ml heptane to remove lipids. The cartridges were dried again under vacuum after which the sample was eluted with 7ml of eluent consisting of ethyl acetate/heptane/methanol (50/40/10%). The resulting solution was evaporated to dryness under a gentle stream of nitrogen. The residue was dissolved in 60 l acetonitrile/water (60/40% with 0.05% (v/v) acetic acid).

To urine 8-isoPGF 2 determination samples (1 ml) were added to 0.5 ml of 15% KOH and mixed and then allowed to stand for 30 min. The pH was then adjusted with 6 M HCl to ~2.5. The sample was then subjected to solid phase extraction (SPE). BondElut LMS polymer 100-mg SPE cartridges were used (purchased from Varian). The SPE cartridge was conditioned with 1 ml of ethanol and 1 ml of Sorensen buffer. The sample was applied to the cartridge, which was then sequentially washed with 1 ml each of Sorensen buffer and 5% ethanol in water. The analyte and internal standards were eluted from the cartridge by using 1 ml of ethanol. The eluate was collected and dried under a gentle stream of nitrogen. The resulting residue was then reconstituted with 100 ml of 20% acetonitrile.

For chromatographic analysis Phenomenex RP C18 column (150x2 mm, 3m) was used. A gradient separation was performed starting with a solvent that consisted of 60% water (pH 3.85) and 40% acetonitrile running for 4 minutes. From 4 to 5 minutes a gradient was run to 100% acetonitrile which ran till 8 minutes.

Between 8th and 9th minute a gradient was used to return to the original eluent composition and the system was equilibrated till 15 minutes. The flow rate was 200 l/min, the temperature column was set at 30C and the injection volume was 10 l. The following parameters were employed throughout all MS experiments for ESI with negative ion polarity the capillary voltage was set to 3.5 kV, the drying temperature to 350C, the nebulizer pressure to 45 psi, and drying gas flow to 10 l/min. The maximum accumulation time was 400 msec, the scan speed was ultra scan mode and the fragmentation time was 40 ms. The target ions m/z 353, EIC (extracted ion chromatogram): m/z 193 and 255, the fragments detected over scan range of m/z 150-400.

Statistical analysis

Data obtained in the current study are expressed as mean  SD. These data were analyzed by standard statistical analyses, one-way analysis of variance (ANOVA) with Tukey's test, to determine significant differences between different groups. A p value of < 0.05 was considered statistically significant.

Results

In the pathogenesis of Lyme arthritis lipid peroxidation is significantly enhanced. The levels of lipid peroxidation products are enhanced in the plasma as well as in the urine of patients. In the plasma the MDA level is enhanced by about 2 fold, while the HNE level is increased by about 4 fold ( p<0.001) (Table 1). In the urine of the Lyme arthritis patients, the level of MDA as well as HNE is about 2 fold higher than in the control group (p<0.01) (Table 1). Significantly important differences are observed in prostanoids derivatives -8-isoPGF 2nonenzymatic lipid oxidation products level. In patients plasma the total 8-isoPGF 2 level is about 3 fold higher than in the control while in the urine it increased tenfold (p<0.001) (Table 1). However the level of free 8-isoPGF 2 in the control patients plasma gives 32 percent of the total level while the level of free 8-isoPGF 2 in the Lyme arthritis patients plasma gives 4 percent of the total level (p<0.001) (Table 1,Fig 1,Fig 2).

Moreover the analysis of correlation showed that in group with Lyme arthritis the correlations between all analyzed lipid peroxidation products are significantly weaker than in the control group.

Discussion

In this study, a significant increase in the level of membrane phospholipids peroxidation products in the course of Lyme arthritis has been revealed. The enhanced lipid peroxidation results from ROS generated by stimulated phagocytes located near the joints during B.burgdorferi infection [START_REF] Pancewicz | Role of reactive oxygen species (ROS) in patients with erythema migrans, an early manifestation of Lyme borreliosis[END_REF]. The lipid peroxidation process is additionally intensified by a decrease in the antioxidant abilities observed during inflammation especially by the decrease in the level of vitamin E, which acts as an efficient antioxidant in the lipid phase of the biomembrane [START_REF] Bradford | The effect of vitamin E on the structure of membrane lipid assemblies[END_REF]. ROS reacting with phospholipids polyunsaturated fatty acids produce mainly superoxides and low molecular aldehydes [START_REF] Esterbauer | Chemistry and biochemistry of 4-hydroxynonenal, malonaldehyde and related aldehydes[END_REF]. The augmentation of the malondialdehyde (MDA) and 4-hydroxynonenal (HNE) level in the plasma as well as in urine of patients with B.burgdorferi infection has been presented in this paper. These aldehydes have longer lifetime than ROS and can diffuse from the formation site and cause greater cell damages. In addition, aldehydes may be considered as a secondary transmitter of lipid peroxidation, because they are very reactive electrophilic compounds which react e.g. with antioxidants participating in diminishing lipid peroxidation, such as GSH and glutathione peroxidase [START_REF] Kinter | Glutathione consumption and glutathione peroxidase inactivation in fibroblast cell lines by 4HNE[END_REF]. Aldehydes also form adducts with nucleophilic sulfhydryl, amino and histydyl groups of proteins, which causes changes in the membrane proteins structure and functions [START_REF] Dotan | Lipid peroxydation cannot be used as a universal criterion of oxidative stress[END_REF]. High reactivity of aldehydes generated during lipid peroxidation contributes to the fact that estimation of their total level is difficult. In consequence, the level of their free form in the plasma as well as in urine during Lyme arthritis is only about 2 fold higher in comparison to the control group. However, the level of another, more stable, lipid peroxidation product -8-isoPGF 2 that was determined in the plasma and urine of patients with Lyme arthritis was significantly higher than the level of aldehydes and its level in plasma was over 3 fold higher, while in urine was over 8 fold higher than in healthy people.

8-isoPGF 2 belongs to the eicosanoids family that comprise a diverse class of over one hundred bioactive lipid mediators derived from the metabolism of polyunsaturated fatty acids by enzymatic pathways connected with the action of cyclooxygenase, lipoxygenase and cytochrom P450 as well as non-enzymatic pathways [START_REF] Buczynski | An integrated omics analysis of eicosanoid biology[END_REF]. The non-enzymatic mechanism generation of eicosanoids derivatives results from enhanced ROS generation and their reactions with nonsaturated fatty acids from phospholipids. However, very little is known about the role of eicosanoids in infectious arthritis. Analysis of joints from Lyme arthritis-resistant andsusceptible mouse strains revealed e.g. production of PGE 2 and PGD 2 [START_REF] Blaho | Lipidomic analysis of dynamic eicosanoid responses during the induction and resolution of Lyme arthritis[END_REF]. In such a situation, the nonenzymatic mechanisms of arachidonic acid oxidation may play important role. This acid can be autooxidized in vivo by free radicals to yield a series of PGF 2 -like compounds termed 8-isoPGF 2 . The formation of 8-isoPGF 2 results in the generation of four regioisomers that contain 16 stereoisomers within each regioisomeric group [START_REF] Yin | Identification of a Novel Class of Endoperoxides from Arachidonate Autoxidation[END_REF]. It is important to note that unlike COX, which generates stereoisomerally pure eicosanoids, 8-isoPGF 2 are formed as racemic diastereomers [START_REF] Yin | Mechanisms for the formation of isoprostane endoperoxides from arachidonic acid. "Dioxetane" intermediate versus beta-fragmentation of peroxyl radicals[END_REF]. These compounds possess potent bioactivity and their quantification represents the most reliable index of oxidative stress in vivo [START_REF] Morrow | A series of prostaglandin F2like compounds are produced in vivo in humans by a non-cyclooxygenase, free radical-catalyzed mechanism[END_REF].

A novel aspect of the 8-isoPGF 2 formation is that they are generated in situ esterified to phospholipids and realized by phospholipases, mainly by phospholipase A 2enzyme hydrolyzing bond in sn-2 position of the glycerin skeleton of phospholipids, the place of the location of unsaturated fatty acids including arachidonic acid [START_REF] Morrow | The Isoprostanes: unique prostaglandinlike products of free radical initiated lipid peroxidation[END_REF][START_REF] Nigam | Phospholipase A 2 s and lipid peroxidation[END_REF]. It was shown that B.burgdorferi induced cytosolic phospolipase A 2 mRNA level in the joints of infected mice [START_REF] Blaho | Arthritis Develops but Fails to Resolve During Inhibition of Cyclooxygenase 2 in a Murine Model of Lyme Disease[END_REF]. Phospholipase A 2 that separates arachidonic acid, indirectly contributes to increased ROS generation because the further arachidonic acid metabolism is partially connected with COX and LOX action in which ROS generation is observed [START_REF] Katsuki | Arachidonic acid as a neurotoxic and neurotrophic substance[END_REF].

It was proved that over 60% of 8-isoPGF 2 present in the plasma was carried as lipid esters, with the remainder circulating as free 8-isoPGF 2 [START_REF] Coolen | Kinetics of biomarkers: biological and technical validity of isoprostanes in plasma[END_REF], what is confirmed in this study. Measurement of free 8-isoPGF 2 in the plasma provides an index of production of 8-isoPGF 2 representing the equilibrium between the formation of esterified 8-isoprostanes and further released as the free acid, which indicates for metabolism. The concentration of free 8-isoPGF 2 seems to change more dynamically during the course of the disease. Free 8-isoPGF 2 are formed by oxidation of the tissues or plasma lipids and then released as free acids by the action of phospholipases. Because it was proved that B.burgdorferi induce the cytosolic phospolipase A 2 mRNA level in the joints of infected mice [START_REF] Blaho | Lipidomic analysis of dynamic eicosanoid responses during the induction and resolution of Lyme arthritis[END_REF], the level of the free 8-isoPGF 2 should be increased. However significant decrease in the level of free 8-isoPGF 2 was observed in the plasma of patients with Lyme arthritis. It may be connected with the fact that oxidatively modified phospholipids, such as esterified 8-isoPGF 2 are good substrates for plasma platelet-activating factor acetylhydrolase (PAF-AH) [START_REF] Foulks | PAF-acetylhydrolase expressed during megakaryocyte differentiation inactivates PAF-like lipids[END_REF], which also hydrolyzes the long chain polyunsaturated fatty acyl groups attached to the sn-2 position of the membrane phospholipids [START_REF] Mcintyre | The emerging roles of PAF acetylhydrolase[END_REF]. It was also shown that during some diseases such as asthma, atopy or stroke [START_REF] Ito | Evidence for an association between plasma platelet-activating factor acetylhydrolase deficiency and increased risk of childhood atopic asthma[END_REF][START_REF] Satoh | Plasma Platelet-activating Factor Acetylhydrolase Deficiency in Japanese Patients with Asthma[END_REF], the plasma PAF-AH was decreased [START_REF] Stafforini | Biology of platelet-activating factor acetylhydrolase (PAF-AH, lipoprotein associated phospholipase A2)[END_REF]. Because 8-isoprostanes are initially formed in situ from esterified arachidonic acids in phospholipids, we estimated the content of esterified 8-isoPGF 2 in phospholipids by subtracting the amount of free 8-isoPGF 2 from the total amount of 8-isoPGF 2 . The present study shows that the total 8-isoPGF 2 levels in the plasma from Lyme arthritis patients were significantly higher than those of the control group, whereas the free 8-isoPGF 2 levels stayed at the similar level as in controls. These results indicate that the esterified 8-isoPGF 2 levels were increased in patients with Lyme arthritis and also imply that accumulation of oxidized phospholipids was more pronounced in the plasma of patients with Lyme arthritis than that of the control probably because of the impaired oxidized phospholipids-hydrolyzing activity. Because most isoprostanes, which are produced under oxidative stress, are in the esterified form, it is possible that their release from phospholipids is a key step for the biological action.

Independently of the significant increase in the plasma level of 8-isoPGF 2  a higher than 8-fold increase in the urine 8-isoPGF 2 from patients with Lyme arthritis in comparison with the control group was observed. Measurement of the urinary 8-isoPGF 2 does not allow the COX to influence on the level of 8-isoPGF 2 existing only as a free form [START_REF] Yin | Urinary prostaglandin F 2 is generated from the isoprostane pathway and not the cyclooxygenase in humans[END_REF]. It was proven that the majority of 8-isoprostanes present in the human urine is derived from the free radical-autooxidation of arachidonic acid rather than via COX [START_REF] Frölich | Urinary prostaglandins. Identification and origin[END_REF], because COX inhibitors do not suppress the urinary levels of 8-isoPGF 2 and the major urinary metabolites of eicosanoids such as PGE 2 are significantly reduced [START_REF] Yin | Urinary prostaglandin F 2 is generated from the isoprostane pathway and not the cyclooxygenase in humans[END_REF]. The source of the urinary PGF is presumed to be from the tissue phospholipids-esterified PGF that is hydrolyzed by the action of phospholipase A 2 and excreted into the urine [START_REF] Stafforini | Biology of platelet-activating factor acetylhydrolase (PAF-AH, lipoprotein associated phospholipase A2)[END_REF]. The evidence that the levels of 8-isoPGF 2 are significantly increased in human disorders associated with oxidative stress provides additional support for the mechanism of formation of 8-isoPGF 2 via free-radicals mechanism. A number of important physiological and pathophysiological issues have emerged from this study. This is related with the fact that formation of bioactive 8-isoPGF 2 occurs in vivo to a significant extent via free-radical mechanisms at the baseline and is markedly increased in oxidative stress e.g. by B.burgdorferi infection. Therefore 8-isoPGF 2 can be implicated as mediators of Lyme arthritis development.

However it has been shown that increased in 8-isoPGF 2 and its major metabolite in the course of other type of arthritis including rheumatoid arthritis, reactive arthritis, psoriatic arthritis and osteoarthritis may be observed, which in consequence may cause diagnostic problems [START_REF] Basu | Raised levels of F2-isoprostanes and prostaglandin F2α in different rheumatic diseases[END_REF]. Therefore the comparison of the level of different lipid peroxidation products (MDA, HNE and 8-isoPGF 2  may be helpful in diagnosis. Additionally in case of Lyme disease, using both immunological data and lipid peroxidation products levels should allow to establish the correct diagnosis.

In conclusion, it should be emphasized that during Lyme arthritis lipid peroxidation is enhanced.

During infection the ratio of free to esterified form of 8-isoPGF 2 is significantly smaller compared to the level in healthy people. This founding indicates that the ratio of free to esterified form of 8-isoPGF 2 may be a useful indicator of Lyme arthritis. Moreover the complementarities of three lipid peroxidation products (MDA, 4-HNE and 8-isoPGF 2 levels may be very helpful in the diagnosis of Lyme arthritis.

  4 males) aged 23 to 62 years (mean age-38 years). Diagnosis of Lyme disease was confirmed by epidemiological anamnesis, clinical manifestationarthritis caused by B. burgdorferi and serological examinations results -detection of anti/ B. burgdorferi IgM and IgG antibodies in ELISA method (Borrelia recombinant IgG and IgM High sensitivity of Biomedica Company, Austria) and at least 4-time increase of its concentration in IgG class. The mean IgM anti/B.

Fig. 1 .FigureFig. 2 .

 12 Fig. 1. The level of total (a), esterified (b) and free 8-isoPGF 2 (c) in the plasma and the level of 8-isoPGF 2 in the urine (d) of patients with Lyme arthritis (N=19) and control subjects (N=16)

Table 1 .

 1 Comparison of means and standard deviations from lipid peroxidation products level in plasma and urine of patients with Lyme arthritis[START_REF] Yin | Mechanisms for the formation of isoprostane endoperoxides from arachidonic acid. "Dioxetane" intermediate versus beta-fragmentation of peroxyl radicals[END_REF] and control subjects[START_REF] Buczynski | An integrated omics analysis of eicosanoid biology[END_REF]

	Parameter	Group	N	Mean	SD	p<
	plasma total 8-isoPGF 2  level	Lyme arthritis	19	76.55	37.38
						0.001
	[pg/ml]	control	16	19.94	5.53
	plasma free 8-isoPGF 2  level	Lyme arthritis	19	3.13	1.10
						0.001
	[pg/ml]	control	16	6.30	1.38
	plasma esterified 8-isoPGF 2  level	Lyme arthritis	19	73.42	36.73
						0.001
	[pg/ml]	control	16	13.58	4.42
	plasma MDA level	Lyme arthritis	19	3.13	0.48
						0.001
	[nmol/ml]	control	16	1.39	0.17
	plasma HNE level	Lyme arthritis	19	45.31	16.86
						0.001
	[pmol/ml]	control	16	11.24	2.19
	urine 8-isoPGF 2 level	Lyme arthritis	19	328.21	154.76
						0.001
	[pg/mg creatinine]	control	16	36.88	12.15
	urine MDA level	Lyme arthritis	19	528.47	222.51
						0.008
	[pmol/mg creatinine]	control	16	344.69	72.96
	urine HNE level	Lyme arthritis	19	1 124.16	482.52
						0.002
	[pmol/mg creatinine]	control	16	675.13	169.16

Table