Samer Riachy

Thierry Floquet

Jean-Pierre Richard

Nonlinear control for linear motors with friction -Application to an inverted pendulum system

A linear motor is used here as an actuator for a cart-pendulum system. The global, upward stabilization of the inverted position is aimed at. In such an under-actuated situation, (i) constraints on the motor motion (limited length) have to be taken into account and (ii) friction effects may have a strong influence (limit cycles). A two-step path planningplus-tracking strategy allows for dealing with constraint (i). Regarding point (ii), and since friction effects are hardly modeled, a second-order sliding mode algorithm is chosen. The resulting controller is designed without any knowledge of neither the electromagnetic circuits of the linear motor (the model is a simple gain), nor the friction models (only upper bounds for the friction forces acting on both the linear motor and pendulum are required). Experimental results show good performances in tracking and regulation, both for the swing-up and stabilization phases of the inverted pendulum.

I. INTRODUCTION

Control of electric drives constitutes a large domain of research. When coupled to a mechanical system, as in the case of a cart-pendulum system, the modeling may result in a high order model. The resulting system becomes difficult to investigate in control. In this paper, the linear drive will be modeled in a very simplified way (i.e., a linear gain) and this lack of modeling information will be compensated by a robust control design.

The cart-pendulum system has played a long-standing test-bed role in control laboratories. An interesting problem (and probably the most impressive for demonstration) is the swing-up and upward stabilization of the inverted pendulum around its unstable, top equilibrium position (say: angle θ = 0). It constitutes an under-actuated situation, in which two outputs (translatory motion of the cart, say: x variable, and rotation angle θ of the rod) have to be controlled by means of a single actuator. Besides, such an electromechanical process is nonlinear, since the global motion presents several equilibrium points, and a simple linear model cannot represent it. Lastly, it refers to several concrete situations 1 {samer.riachy, thierry.floquet, jean-pierre.richard}@ec-lille.fr. This work is supported by the regional council Nord-Pas de Calais and the FEDER (European Funds of Regional Development) under the grant ARCir RoboCoop, and program TAT31. 1 The damping of a mass hanged to a travelling crane refers to the bottom position (local) control [7]; The top position (local) control is encountered when balancing new two-wheel vehicles as B2 [8] or Segway [9]; The bottom-to-top -then, global-stabilization arises in particular wheel-chairs [START_REF] Ding | Electric powered wheelchairs[END_REF], [10] that can raise from a 4-wheels to a 2-wheels situation, so as to make the sited person reach a higher position.

Considering the cart-pendulum system, two main problems are encountered:

1) Constraints on the motor motion, which must move within a limited length, have to be taken into account.

Here, it is chosen to combine a linear motor (whose high acceleration performance allows for generating high speeds within short lengths) with a two-step path planning + tracking strategy. Such a control ensures the tracking of a planned (then, admissible) reference trajectory in x, which is computed so as to be admissible with regard to the constraints, and to make the angle θ behave according to a tunable, Van der Pol-like equation. An additional closed-loop control guarantees the tracking of this reference whatever the perturbations are. 2) Friction effects may have a strong influence. This will be shown in the paper by considering control laws from the literature [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF], the design of which was relying on a simplified model without friction. The resulting concrete application makes limit cycles appear. Here, we take it explicitly into account and, since friction effects are hardly modeled, a second-order sliding mode algorithm is chosen (for a detailed discussion on Higher Order Sliding Modes see [START_REF]Sliding mode control in engineering[END_REF] and references therein). The resulting controller is designed without any knowledge of neither the electromagnetic circuits of the linear motor, nor the friction models (only upper bounds for the friction forces acting on both the linear motor and pendulum are required).

The developed second order sliding mode controller is shown to be robust to unmodeled nonlinear phenomena and disturbances. In addition good performances are obtained in stabilization, trajectory tracking, swing-up and balancing of an inverted pendulum coupled to the linear motor.

II. TRACKING OF A MODIFIED VAN DER POL REFERENCE

SIGNAL

In the next section a second order sliding mode controller is developed for the linear motor to swing-up and balance an inverted pendulum. In order to swing the inverted pendulum, the linear motor needs to follow a modified Van der Pol reference signal [START_REF] Orlov | Model orbit robust stabilization (MORS) of Pendubot with application to swing up control[END_REF].

The pendulum coupled to the linear motor is shown in Fig. 1 and is governed by ( 1)-( 3)

(M + m) ẍ + ml sin θ θ 2 -ml cos θ θ = τ + w 1 (t) -ψ( ẋ), (1) 4 3 ml 2 θ -ml cos θ ẍ -mgl sin θ = w 2 (t) -ϕ( θ ) (2) τ = KV (3)
where x is the cart position, θ is the angular deviation of the pendulum from the vertical, M is the cart mass, m is the rod mass, l is the distance to the center of mass of the pendulum, g is the gravitational acceleration, τ is the force generated by the linear motor, w 1 (t), w 2 (t) are external disturbances, ψ( ẋ) and ϕ( θ ) are friction forces, affecting the linear motor and the pendulum, respectively. Note that the linear motor is modeled in (3) as a linear gain K acting on the input voltage V.

The general idea of the swing-up control is inspired from [START_REF] Orlov | Model orbit robust stabilization (MORS) of Pendubot with application to swing up control[END_REF], while the local control relies on [START_REF] Riachy | Stabilization of the cart-pendulum system via quasi-homogeneous switched control. 9-th International Workshop on Variable Structure Systems[END_REF]. For lack of space, only general ideas are given here. The modified Van der Pol oscillator is given by: ..

z +ε (z 2 + ż2 µ 2 ) -ρ 2 ż + µ 2 z = 0, (4)
This modification possess a stable limit cycle (a sinusoidal one), the advantage of which is to be expressible in the explicit form

z 2 + ż2 µ 2 = ρ 2 (5)
Our objective is to design a controller that causes the actuated part of the cart-pendulum system, i.e. the linear motor, to track a trajectory generated by the modified Van der Pol equation (4), i.e:

lim t→∞ [z(t) + x(t)] = 0, (6)
while attenuating the effect of the friction forces, external disturbances, and unmodeled dynamics. Consider the sliding variable:

y(t) = z(t) + x(t), (7)
that combines the actuated position x(t) of the system and the reference variable z(t) governed by the modified Van der Pol equation ( 4). The control problem is to drive the system output (7) onto the surface y = 0 in finite time and maintain it there in spite of the friction forces, external disturbances, and unmodeled dynamics affecting the system. By differentiating (7) twice, one obtains:

..

y = u + 3 cos θ 4lJ w 2 (t) -ϕ( θ ) + 1 J [w 1 (t) -ψ( ẋ)] -ε (z 2 + ż2 µ 2 ) -ρ 2 ż -µ 2 z. ( 8
)
Relating the quasihomogeneous synthesis from [START_REF] Riachy | Stabilization of the cart-pendulum system via quasi-homogeneous switched control. 9-th International Workshop on Variable Structure Systems[END_REF], the following second order sliding mode control law

u = 3ϕ v cos θ 4lJ θ + ψ v J ẋ + ε (z 2 + ż2 µ 2 ) -ρ 2 ż + µ 2 z -αsign(y) -β sign( ẏ) -hy -p ẏ (9)
with the parameters such that

h, p ≥ 0, α -β > 3(ϕ c + N 2 ) 4lJ + ψ c + N 1 J ( 10
)
is proposed. It was demonstrated in [START_REF] Orlov | Model orbit robust stabilization (MORS) of Pendubot with application to swing up control[END_REF] that the resulting quasihomogeneous system (8), (9) with the parameter subordination (10) is finite time stable regardless of unmodeled dynamics, friction forces and uniformly bounded external disturbances affecting the system. So, after a finite time, the cart-pendulum system evolves on the second order sliding manifold i.e. on the zero dynamics surface y = 0. The control objective is thus achieved in finite time (non asymptotic).

III. SWING UP CONTROL AND STABILIZATION

To swing the pendulum up from the bottom position to the top position, the orbitally stabilizing synthesis (4), (8), ( 9) is applied to pump into the system as much energy as required to approach the basin of attraction of a locally stabilizing controller. Switching to a local second order sliding mode controller (a detailed discussion can be found in [START_REF] Riachy | Stabilization of the cart-pendulum system via quasi-homogeneous switched control. 9-th International Workshop on Variable Structure Systems[END_REF]) completes the objective of swing-up and balancing of the inverted pendulum.

A. Cart-pendulum prototype

In order to study the performance of the proposed synthesis experimental tests were made. The real parameters of the laboratory cart-pendulum system are given in table I.

B. Swinging controller design

To swing the pendulum up from the downward position to the upright position, the orbitally stabilizing synthesis ( 4), (8), ( 9) is applied to pump into the system as much energy as required to approach a homoclinic orbit with the same energy level as the one corresponding to the desired equilibrium point. The idea is inspired from [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF] which consists of driving a frictionless cart-pendulum to its homoclinic orbit obtained by zeroing both the total energy of the system and the cart velocity. The model used in this paper is that of a real life cart-pendulum, having friction both on the cart and the pendulum axis. So, we speak of a quasi-homoclinic orbit which still gives large oscillations of the pendulum from the downward position to the upright one which is the main interest for swing-up. In experiments, the total energy of the cart-pendulum

E(q, q) = 1 2 (M + m) ẋ2 -ml ẋ θ cos θ + 2 3 ml 2 θ 2 +mgl(cos θ -1) (11)
is calculated at each instant when the cart is at its maximal displacement (i.e. zero cart velocity) on the modified Van der Pol trajectory. Starting from a negative value E 1 = -2mgl when the system is at rest on the downward pendulum position, the energy increases due to (9) until it comes slightly greater than zero (to take into account friction forces)

E 0 0. ( 12
)
The swinging controller is then switched off, and the pendulum is left to travel on its quasi-homoclinic orbit toward the upright position. Being crucial to a successful swing up, this is achieved by tuning both the controller parameters α, β , h, p and the reference parameters ε, ρ, µ of the Van der Pol modification (4). Appropriate values of the parameters to be tuned are carried out in successive numerical experiments. In our experimental study, the controller gains in (9) were set to α = 20rad/sec 2 , β = 5rad/sec 2 , h = 0, p = 0 whereas the reference parameters were tuned to

ε = 10 [rad] -2 s -1 , ρ = 0.2 rad, µ = 0.7 × 2π s -1 .

C. Locally stabilizing controller design

The locally stabilizing controller developed in [START_REF] Riachy | Stabilization of the cart-pendulum system via quasi-homogeneous switched control. 9-th International Workshop on Variable Structure Systems[END_REF] is defined as:

τ = D cos θ 3 + 8lλ 2 θ sin θ × -µ(θ , θ ) -α 1 sign(s) -β 1 sign( ṡ) -h 1 s -p 1 ṡ , α 1 , β 1 , h 1 , p 1 ≥ 0 (13) with µ(θ , θ ) = 2 tan θ cos 2 θ θ 2 + 3+8lλ 2 θ sin θ cos θ (M+m)g-ml cos θ θ 2 D × tan θ + λ 1 g + 4 3 l θ 2 cos θ tan θ + λ 2 g + 4 3 l θ 2 (1+sin 2 θ ) cos θ θ cos 2 θ , (14)
D = l(4M +m+3m sin 2 θ ), s = tan θ -λ 1 ω -λ 2 ω, λ 1 , λ 2 > 0 (15) ω = x - 4l 3 ln 1 + sin θ cos θ , |θ | < π 2 , (16)
is tested in an experimental study. Quite impressive experimental results are obtained for the local stabilization of the pendulum about the upright position with the controller parameters α 1 = 30 m/s 2 , β 1 = 7.5 m/s 2 , h 1 = 0, p 1 = 0, λ 1 = 2 1/m, λ 2 = 1 s/m. The results can be observed in Subsection III-E where the proposed controller is introduced into a hybrid synthesis of swinging the Cart-Pendulum up and balancing it about the vertical.

D. Hybrid Controller Design

In order to accompany swinging the pendulum up by the subsequent stabilization around the upright position, the swinging controller, presented in Subsection III-B, is turned off once the system reaches the corresponding homoclinic orbit, and then the locally stabilizing controller from III-C is turned on whenever the pendulum enters the basin of attraction, numerically found for the latter controller. The resulting hybrid controller, thus constructed, moves the inverted pendulum, located on the cart, from its downward position to the upright position and stabilizes it about the vertical whereas the cart is stabilized at the desired endpoint. While being not studied in details here, the capability of the closed-loop system to reach the homoclinic orbit and entering the attraction basin of the locally stabilizing controller is supported by experimental results.

E. Experimental results

Experimental results are plotted in Figs. 34. They show that the pendulum starts from the "stable" position π, oscillates until it reaches the unstable equilibrium position 0 and remains there.

Other approaches have been implemented and compared (see Fig. 5 corresponding to [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF]). It can be seen that the swing-up controller was not able to drive the pendulum close to its top position. This is mainly due to the neglected friction effects. A linear static state feedback has also been implemented for local upright stabilization (see Fig. 6). Large amplitude oscillations (limit cycles) occur.

A drawback of sliding mode control could be the high frequency oscillations in the control signal (see Fig. 4) called chattering. There is a huge literature on how to attenuate this problem. A simple solution, which we used here, is to replace the sign function in (9) by a continuous one (arctan).

IV. CONCLUSIONS

Orbital stabilization of a cart-pendulum system, presenting a simple underactuated (two degrees-of-freedom, one actuator) manipulator with friction, was under study. The system was constrained to track a modified Van der Pol oscillator that possesses a stable limit cycle, governed by a standard tunable linear oscillator equation. A quasihomogeneous second order sliding mode based control synthesis was utilized to design a variable structure controller that drives the pendulum to a desired zero dynamics manifold in finite time and maintains it there in sliding mode in spite of the presence of external disturbances. Once the cart-pendulum reached a sufficient level of energy, the control was switched to a locally stabilizing quasihomogeneous controller, solving the problem of moving the pendulum from its stable downward position to the unstable inverted position and stabilizing it about the vertical.

Capabilities of the quasihomogeneous synthesis and its robustness against friction forces and external disturbances were successfully illustrated on an underactuated cartpendulum testbed and it is hoped to suggest a practical framework for orbital stabilization of underactuated manipulators.

Fig. 1 .

 1 Fig. 1. Inverted pendulum located at the LAGIS laboratory.

Fig. 2 .

 2 Fig. 2. Experimental tracking of the Van der Pol reference signal (reference and ouput).

Fig. 2

 2 Fig.2shows experimental results of the linear motor (green) tracking the Van der Pol reference (blue). It can be seen that the linear motor position starts from the Van der Pol reference signal, this initial condition is chosen for practical reason (short cart length).

Fig. 3 .

 3 Fig. 3. Cart position and rod angular position versus time.

Fig. 4 .

 4 Fig. 4. Control input versus time.

Fig. 5 .

 5 Fig. 5. Experimental results for the control designed in [3] (cart position and pendulum angular position).

Fig. 6 .

 6 Fig. 6. Experimental results for linear static state feedback local stabilization [1] (cart position and pendulum angular position).

  .

	S. Riachy, T. Floquet and J.P. Richard are with ALIEN (INRIA FUTURS)
	and LAGIS (CNRS), Ecole Centrale de Lille, BP 48, 59651 Villeneuve
	d'Ascq,	France.

TABLE I PARAMETERS

 I OF THE CART-PENDULUM.

	Notation	Value	Units
	M	3.4	kg
	m	0.147	kg
	l	0.175	m
	ψ v	8.5	N • s/m
	ϕ v	0.0015	N • m • s/rad
	ψ c	6.5	N
	ϕ c	0.00115	N • m


Damien Chablat
email: damien.chablat@cnrs.fr

Chengwei Zhang

Kinematics, Workspace and Singularity Analysis of a Parallel Robot with Five Operation Modes *

Most multi-mode parallel robots can change operation modes by passing through constraint singularities. This paper deals with a comprehensive kinematic study of a 3-DOF multi-mode 3-PRPiR parallel robot developed at Heriot-watt University. This robot is able to reach several operation modes without crossing any constraint singularity by using lockable Pi and R joints . Here a Pi joint may act as a 1-DOF planar parallelogram if its lockable P (prismatic) joint is locked or a 2-DOF RR serial chain if its lockable P joint is released. The operation modes of the robot include a 3T operation mode and four 2T1R operation modes with two different directions of the rotation axis of the moving platform. The inverse kinematics and forward kinematics of the robot in each operation mode are dealt with in detail. The joint space and workspace analysis of the robot allow us to know the regions of the workspace that the robot can reach in each operation mode. It is shown that the robot is able to change assembly mode in one operation mode by passing through another operation mode.

Introduction

Reconfigurable Parallel Manipulators (PMs), such as multi-mode PMs [START_REF] Fanghella | Parallel robots that change their group of motion[END_REF][START_REF] Refaat | Two-mode overconstrained three-DOFs rotational-translational[END_REF][START_REF] Kong | Type synthesis of parallel mechanisms with multiple operation modes[END_REF][START_REF] Li | Parallel mechanisms with bifurcation of Schoenflies motion[END_REF][START_REF] Gogu | Maximally regular T2R1-type parallel manipulators with bifurcated spatial motion[END_REF][START_REF] Ruggiu | Mobility and kinematic analysis of a parallel mechanism with both PPR and planar operation modes[END_REF][START_REF] Kong | Type synthesis of variable degrees-of-freedom parallel manipulators with both planar and 3T1R operation modes[END_REF][START_REF] Kong | Type synthesis of 3-DOF parallel manipulators with both a planar operation mode and a spatial translational operation mode[END_REF][START_REF] Kong | Type synthesis of two-degrees-of-freedom 3-4R parallel mechanisms with both spherical translation mode and sphere-on-sphere rolling mode[END_REF][START_REF] Gan | Geometry constraint and branch motion evolution of 3-PUP parallel mechanisms with bifurcated motion[END_REF][START_REF] Zeng | Design of parallel hybrid-loop manipulators with kinematotropic property and deployability[END_REF][START_REF] Ye | A new family of reconfigurable parallel mechanisms with diamond kinematotropic chain[END_REF], have received much attention from a number of researchers in the past decade [START_REF] Ding | Advances in Reconfigurable Mechanisms and Robots II[END_REF]. The main characteristics of multi-mode PMs include [START_REF] Kong | Type synthesis of variable degrees-of-freedom parallel manipulators with both planar and 3T1R operation modes[END_REF][START_REF] Kong | Type synthesis of 3-DOF parallel manipulators with both a planar operation mode and a spatial translational operation mode[END_REF][START_REF] Kong | Type synthesis of two-degrees-of-freedom 3-4R parallel mechanisms with both spherical translation mode and sphere-on-sphere rolling mode[END_REF]: a) fewer actuators are needed for the moving platform to realize several specified motion patterns; and b) less time is needed in reconfiguring the PM since there is no need to disassemble the PM in the process of reconfiguration.

A number of interesting multi-mode PMs have been proposed, such as multi-mode PMs with a 3-DOF (degrees-offreedom) translational mode and a 3-DOF planar mode or a 4-DOF 3T1R mode [START_REF] Kong | Type synthesis of variable degrees-of-freedom parallel manipulators with both planar and 3T1R operation modes[END_REF][START_REF] Kong | Type synthesis of 3-DOF parallel manipulators with both a planar operation mode and a spatial translational operation mode[END_REF], PMs with two 3T1R (also Schönflies motion) operation mode which has three translational DOF and one rotational DOF operation modes [START_REF] Li | Parallel mechanisms with bifurcation of Schoenflies motion[END_REF], PMs with two 2T1R operation modes [START_REF] Gogu | Maximally regular T2R1-type parallel manipulators with bifurcated spatial motion[END_REF][START_REF] Ruggiu | Mobility and kinematic analysis of a parallel mechanism with both PPR and planar operation modes[END_REF], and 2-DOF 3-4R PMs with both spherical translation mode and sphere-on-sphere rolling mode [START_REF] Kong | Type synthesis of two-degrees-of-freedom 3-4R parallel mechanisms with both spherical translation mode and sphere-on-sphere rolling mode[END_REF]. References [14,15] show 4-DOF 3T1R PMs that have an extra 2-DOF or 3-DOF motion mode in addition to the required 4-DOF 3T1R motion mode. It is noted the metamorphic PMs based on reconfigurable U (universal) or S (spherical) joints [16,17] and the multi-mode PMs with lockable joints [18,19] are in fact kinematically redundant PMs in nature and do not belong to multi-mode PMs that this paper focuses on. In addition, several methods have been proposed [14,15,[19][START_REF] Walter | Chapter A: Complete kinematic analysis of the SNU 3-UPU parallel manipulator[END_REF][START_REF] Kong | Reconfiguration analysis of a 3-DOF parallel mechanism using Euler parameter quaternions and algebraic geometry method[END_REF][START_REF] Kong | Reconfiguration analysis of a 4-DOF 3-RER parallel manipulator with equilateral triangular base and moving platform[END_REF][23][START_REF] Chablat | Non-singular assembly mode changing trajectories in the workspace for the 3-RPS parallel robot[END_REF][START_REF] Chablat | Workspace and joint space analysis of the 3-RPS parallel robot[END_REF] to the reconfiguration analysis of multi-mode PMs for identifying all the operation modes and the transition configurations of a PM.

Operation modes Conditions

No Translation Rotation Joints Joints along about locked released 1

x, y, z --

P 1 , P 2 R 1 , R 2 , R 3 2 x, y z(α) R 1 , R 2 , P 2 R 3 , P 1 3 x, y x(β) R 1 , R 2 , P 1 R 3 , P 2 4 y, z x(β) R 3 , P 1 R 1 , R 2 , P 2 5 y, z z(α) R 3 , P 2 R 1 , R 2 , P 1
However, only a few multi-mode PMs, such as the PM with two 1T1R operation modes used as a novel swivel head for machine tools [START_REF] Refaat | Two-mode overconstrained three-DOFs rotational-translational[END_REF], have been put into use in practice. As pointed out in [START_REF] Kong | Reconfiguration analysis of a 4-DOF 3-RER parallel manipulator with equilateral triangular base and moving platform[END_REF]23], one of the causes is the lack of practical methods for switching a PM from one operation mode to another.

In [START_REF] Zhang | Improvement of a disassembly-free reconfigurable parallel manipulator[END_REF], a multi-mode PM was proposed, which is a revised version of the multi-mode PM developed at Heriot-Watt University as shown in Fig. 1 of [18]. This multi-mode PM can switch from one operation mode to another securely by using reconfigurable planar parallelograms and lockable R (revolute) joints. Unlike robot architectures that cross a constraint singularity to change operation mode, no external force or torque is required to change operation modes. This paper will perform a systematic study on the inverse kinematics, forward kinematics, workspace and singularity analysis of this multimode PM.

In the literature, trajectories for changing assembly modes has been investigated either by using dynamic proper trajectories [START_REF] Nenchez | Singularity-consistent path tracking: A null space based approach[END_REF] or the properties of the cuspidal robots [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF][START_REF] Zein | Non-singular assembly-mode changing motions for 3-RPR parallel manipulators[END_REF]. In this paper, we will present a new way to change assembly modes, which are associated with different solutions of the forward kinematics, by switching operation modes.

In Section 2, the description of a 3-PRPiR PM is given. Section 3 deals with the inverse and direct kinematic problem. In Section 4, the singularity analysis is presented. Section 5 presents the workspace analysis to find out the connected regions reachable for a given working mode, i.e. a given solution of the inverse kinematics [30], in each operation mode. Finally, conclusions are drawn on the advantages of this robot.

Description of a 3-PRPiR parallel robot

The 3-DOF 3-PRPiR1 multi-mode parallel robot developed by the second author's team at Heriot-Watt University [START_REF] Zhang | Improvement of a disassembly-free reconfigurable parallel manipulator[END_REF] is composed of a base and moving platform connected by three RPiR legs. Here R and Pi represent revolute and reconfigurable parallelogram joint respectively (Fig. 1). The Pi joint may act as a 1-DOF planar parallelogram if its lockable P (prismatic) joint is locked or a 2-DOF RR serial chain if its lockable P joint is released. In total, there are three lockable revolute joints (R 1 , R 2 and R 3 ) and two lockable prismatic joints (P 1 and P 2 ) which can be locked/released to change the operation modes of the robot. With this prototype, it was possible to validate the movements of the robot according to the operation mode. The change of operating mode is carried out in the home pose shown in Fig. 1 or whenever the robot is under the xy or yz planes passing through this pose with α = 0 and β = 0. A practical home configuration would be one does not encounter parallel singularity. Currently, the change of operation mode is done by inserting pins manually to lock or unlock the joints. In the future, magnetic devices will be used to make this change. Compared to the use of motors, this solution is simpler because a logic output of the controller is used and there is no optical encoder and no drive.

In the home pose, the axes of all the revolute joints, including those within the Pi joint, are parallel to two directions. The input joint variables are the three prismatic actuated joints ρ 1 , ρ 2 and ρ 3 along the y-axis. The motion of the moving platform depends on the operation mode of the robot (Fig. 1). Table 1 describes the robot's mobility in its five operation modes.

In a fixed reference frame, the positions of the fixed points A i (i = 1, 2, 3) as well as the points B i controlled by the

ρ 1 ρ 3 R 1 A 1 A 5 B 1 B 5 C 1 C 5 C 1 B 3 C 4 x y z ρ 2 P 1 P 2 R 2 R 3 A 4 B 4 C 2 P C 3 B 2
Fig. 1. A 3-PRPiR multi-mode parallel robot with the three actuators in red, the passive joints in white and the lockable joints in other colors and its prototype actuated prismatic joints are defined as

A 1 = [-d 1 , 0, 0] T A 2 = [d 6 , 0, 0] T A 3 = [0, 0, d 2 ] T A 4 = [d 6 + d 7 , 0, 0] T A 5 = [0, 0, d 2 + d 5 ] T
Points A 2 and A 3 are virtual points only to simplify the model.

B 1 = [-d 1 , ρ 1 , 0] T B 2 = [d 6 , ρ 2 , 0] T B 3 = [0, ρ 3 , d 2 ] T B 4 = [d 6 + d 7 , ρ 2 , 0] T B 5 = [0, ρ 3 , d 2 + d 5 ] T
The motion of the end-effector P depends on the operation modes of the robot. The coordinates of point P will be given in details in the next subsections on the constraint equations for each operation mode. The rotations around x-axis and z-axis are denoted by β and α respectively.

Based on the prototype built at Heriot-Watt University, the design parameters are defined in millimeters as (Fig. 2):

d 1 = 55, d 2 = 113, d 3 = 385/10, d 4 = 95, d 5 = 25, d 6 = 33, d 7 = 405/10, l 1 = 124, l 2 =
165, and l 3 = 170. For all the operation mode, the constraint equations are obtained by calculating the distances between the points B i and C i .

||B i -C i || = l i for i = 1, 2, 3
In the following equations, we will introduce the position and orientation of the end-effector P as a function of the operation modes of the robot.

Operation Mode 1

Operation mode 1 is a 3-DOF pure translation mode. That is to say the moving platform cannot rotate. This architecture is identical to the linear Delta robot found in the Renault Automation UraneSX [START_REF] Company | Modelling and design issues of a 3-axis parallel machine-tool[END_REF] or the IRCCyN Orthoglide [START_REF] Chablat | Architecture optimization of a 3-DOF parallel mechanism for machining applications, the Orthoglide[END_REF][START_REF] Chablat | An interval analysis based study for the design and the comparison of 3-DOF parallel kinematic machines[END_REF].

C 1 = [x -d 3 , y, z] T C 2 = P = [x, y, z] T C 4 = [x + d 7 , y, z] T C 3 = [x, y, z + d 4 ] T C 5 = [x, y, z + d 4 + d 5 ] T 2 DESCRIPTION OF A 3-PRPIR PARALLEL ROBOT B 1 C 3 P d 4 A 1 A 4 P C 1 C 4 C 2 d 3 l 1 l 2 l 3 B 2 B 3 B 1 d 2 d 1 d 1 C 2 C 5 B 5 d 5 B 4 d 6 d 7 d 1 A 2 A 3 A 5
Fig. 2. Link parameters of the 3-PRPiR multi-mode parallel robot.

The constraint equations are

   (x -d 3 + d 1 ) 2 + (y -ρ 1 ) 2 + z 2 = l 1 2 (x -d 6 ) 2 + (y -ρ 2 ) 2 + z 2 = l 2 2 x 2 + (y -ρ 3 ) 2 + (z + d 4 -d 2 ) 2 = l 3 2 (1)
Operation Mode 2 For this operation mode, the moving platform undergoes a 3-DOF planar motion along the O-XY plane: it translates in the plane (x, y) and rotate around the z-axis. We can write the positions of C i and P as

C 1 = [x -d 3 cos α, y -d 3 sin α, 0] T C 2 = P = [x, y, 0] T C 3 = [x, y, d 4 ] T C 4 = [x + d 7 cos α, y + d 7 sin α, 0] T C 5 = [x, y, d 4 + d 5 ] T
The constraint equations are

   (x -d 3 cos α + d 1 ) 2 + (y -d 3 sin α -ρ 1 ) 2 = l 2 1 (x -d 6 ) 2 + (y -ρ 2 ) 2 = l 2 2 x 2 + (y -ρ 3 ) 2 + (d 4 -d 2 ) 2 = l 2 3 ( 2
)
Operation Mode 3 For this operation mode, the translations of the moving platform are in the plane (x, y) and the rotation around the x-axis. Such motion is called PPR equivalent motion in [START_REF] Kong | Type synthesis of 3-DOF PPR-equivalent parallel manipulators based on screw theory and the concept of virtual chain[END_REF]. We can write the positions of C i and P as

C 1 = [x -d 3 , y, 0] T C 2 = P = [x, y, 0] T C 4 = [x + d 7 , y, 0] T C 3 = [x, y + d 4 sin β, d 4 cos β] T C5 = [x, y + (d 4 + d 5 ) sin β, (d 4 + d 5 ) cos β] T
The constraint equations are

   (x -d 3 + d 1 ) 2 + (y -ρ 1 ) 2 = l 2 1 (x -d 6 ) 2 + (y -ρ 2 ) 2 = l 2 2 x 2 + (y + d 4 sin β -ρ 3 ) 2 + (d 4 cos β -d 2 ) 2 = l 2 3 (3)

INVERSE AND DIRECT KINEMATIC ANALYSIS OF THE PARALLEL ROBOT IN DIFFERENT OPERATION MODES

Operation Mode 4 For operation mode 4, the moving platform undergoes a 3-DOF planar motion along the O-YZ plane. The translations of the moving platform are in the plane (y, z) and the rotation around the x-axis. The positions of C i and P are

C 1 = [0 -d 3 , y, z] T C 2 = P = [0, y, z] T C 4 = [0 + d 7 , y, z] T C 3 = [0, y + d 4 sin β, z + d 4 cos β] T C 5 = [0, y + (d 4 + d 5 ) sin β, z + (d 4 + d 5 ) cos β]] T
The constraint equations are

   (d 1 -d 3 ) 2 + (y -ρ 1 ) 2 + z 2 = l 2 1 d 2 6 + (y -ρ 2 ) 2 + z 2 = l 2 2 (y + d 4 sin β -ρ 3 ) 2 + (z + d 4 cos β -d 2 ) 2 = l 2 3 ( 4
)
This operation mode was not presented in [START_REF] Chablat | Kinematics, workspace and singularity analysis of a multi-mode parallel robot[END_REF] because for the design parameters in [START_REF] Chablat | Kinematics, workspace and singularity analysis of a multi-mode parallel robot[END_REF], the robot was singular throughout its workspace. The constraint equations are

   (x -d 3 + d 1 ) 2 + (y -ρ 1 ) 2 = l 1 2 (x -d 6 ) 2 + (y -ρ 2 ) 2 = l 2 2 x 2 + (y + d 4 sin β -ρ 3 ) 2 + (d 4 cos β -d 2 ) 2 = l 3 2 ( 5
)
3 Inverse and direct kinematic analysis of the parallel robot in different operation modes 3.1 Inverse kinematic analysis For each of the five operation modes, there are 8 solutions to the inverse kinematic model or working modes [30]. From the constraint equations [Eqs. (1)-( 5)], it is quite straightforward to solve quadratic equations to find the inputs of the actuators. These solutions are given below without detailed derivation. Figure 3 shows the 8 working modes of the robot for the home pose. Operation Mode 1 Operation Mode 2

ρ 1 = y ± 2(d 3 d 1 + xd 3 -xd 1 ) -d 2 1 -d 2 3 + l 2 1 -x 2 -z 2 ρ 2 = y ± 2xd 6 -d 2 6 + l 2 2 -x 2 -z 2 ρ 3 = y ± 2(d 4 d 2 + zd 2 -zd 4 ) -d 2 2 -d 2 4 + l 2 3 -x 2 -z 2
ρ 1 = -d 3 sin α + y ± d 2 3 sin 2 α + 2(d 3 d 1 + xd 3 ) cos α -(d 1 + x) 2 -d 2 3 + l 2 1 ρ 2 = y ± l 2 2 -(d 6 -x) 2 ρ 3 = y ± l 2 3 -x 2 -(d 2 -d 4 ) 2
Operation Mode 3

ρ 1 = y ± -d 2 1 + 2d 3 d 1 -2xd 1 -d 2 3 + 2 xd 3 + l 2 1 -x 2 ρ 2 = y ± -d 2 6 + 2xd 6 + l 2 2 -x 2 ρ 3 = d 4 sin β + y ± d 2 4 sin 2 β + 2d 4 cos βd 2 -d 2 2 -d 2 4 + l 2 3 -x 2
Operation Mode 4 Operation Mode 5

ρ 1 = y + l 2 1 -z 2 -(d 3 -d 1 ) 2 ρ 2 = y + l 2 2 -z 2 -d 2 6 ρ = d sin β + y + d 2 sin 2 β + 2d cos β(d -z) -(d -z) 2 -d 2 + l 2

Direct kinematic analysis 4 SINGULARITY ANALYSIS OF THE PARALLEL ROBOT IN DIFFERENT OPERATION MODES

ρ 1 = -d 3 sin α + y ± d 2 3 sin 2 α + 2d 3 cosαd 1 -d 2 1 -d 2 3 + l 2 1 -z 2 ρ 2 = y ± l 2 2 -d 2 6 -z 2 ρ 3 = y ± 2zd 2 -(d 2 -d 4 ) 2 -2d 4 + l 2 3 -z 2

Direct kinematic analysis

Solving the constraint equations associated with each operation mode [Eqs. (1)-( 5)], one can find the locations of the moving platform for a given set of inputs.

For operation mode 1, the resolution of the direct kinematic analysis amounts to calculating the intersection of three spheres. An efficient geometric method is presented by Pashkevich in [START_REF] Pashkevich | Kinematics and workspace analysis of a three-axis parallel manipulator: the Orthoglide[END_REF]. For the other operation modes, we obtain two values for the position according to x,y or z. For a given orientation of the moving platform, we obtain a quadratic equation as a function of the position parameters. Therefore, we usually have four real solutions for the orientation of the moving platform.

Since the equations for all the five operation modes are too large to be displayed in an article and the solutions are well-documented in the literature, we will focus on the variation of the number of solutions to the direct kinematic model in the joint space. For clarity, we make a slice in the joint space by setting ρ 1 = 0. This is equivalent to the change of variable ρ ′ 2 = ρ 2ρ 1 and ρ ′ 3 = ρ 3ρ 1 . Figures 45678represent the joint space in blue, the regions where the direct kinematic model admits two real solutions and in red, the regions where it admits four real solutions.

In operation mode 1, there is two separate regions where the direct kinematic problem admits two solutions (Fig. 4). In operation mode 2, there are eight regions where the direct kinematic problem admits either two or four solutions (Fig. 5) that we can merge in two connected regions. However, there is one hole in each one. This property can cause problems if the robot is to be protected by introducing limits on the active joints.

In operation mode 3, there are three regions that we can merge in a single connected regions (Fig. 6). The region with four direct kinematic solutions connects two regions where there are only two solutions.

In operation mode 4, there are two connected regions (Fig. 7). In each one, we have four regions with either two or four solutions.

In operation mode 5, there are four connected regions with two holes (Fig. 8). The regions with four direct kinematic solutions are inside a region with two solutions.

It is noted that except operation mode 3, the regions in the joint space are not connected. Figures 910111213shows several postures of the robot when we have the two or four solutions for the direct kinematics in these five operation modes. det(A) and det(B) respectively [30]. In each operation mode, the serial singularities are located at the boundaries of the workspace while the parallel singularities divide it.

As the robots studied have several solutions to the direct and inverse kinematic problem, the singularity conditions are defined in the cross product of the joint space and workspace. Their projections into the workspace are not requested because we cannot distinguish the curves associated with a given working mode. The equations of the singularities are written according to the design parameters of the prototype because it is not possible to write these conditions in this paper without affecting their values. However, the Siropa library makes it possible to do all the calculations in the general case with all the design parameters (d i for i = 1..7 and l i for i = 1, 2, 3 [START_REF] Siropa | A library for manipulator singularities analysis[END_REF]. For example, for the operation mode 1, without assigning the design parameters, the number of operands of the parallel singularity locus in the Cartesian space is 2929 and the degree

Operation mode 1

The singular configurations in operation mode 1 of this robot are the same as the conventional parallel robot presented legs are in the same plane (flat position) or all parallel (bar position).

Serial singularities occur if and only if

ρ 3 -y = 0, ρ 2 -y = 0 or ρ 1 -y = 0
Parallel Singularities occur if and only if

(-12 x -22 z + 396)ρ 1 + (12 x -11 z + 198)ρ 2 + 33 zρ 3 -594 y = 0
In this operation mode, the determinant of the matrix A cannot be factorized. In a single equation, we have the singularity locus associated with the eight working modes.

Operation mode 2

Serial singularities occur if and only if

77 sin α -2y + 2ρ 1 = 0, ρ 2 -y = 0 or ρ 3 -y = 0
Only the serial singularities associated with the first leg ρ 1 depend on the orientation of the moving platform.

Parallel Singularities occur if and only if

x sin α + cosαρ 1 -y cosα + 55 sin α = 0 or xρ 2 -ρ 3 x + 33 ρ 3 -33 y = 0
The determinant of the matrix A is factorized into two components. This properties means that the workspace is divided into at least four regions. We notice that the first component does not depend on the orientation of the moving platform. The workspace analysis of a multi-mode parallel robot requires analyzing the workspace of the robot in each operation mode. In each operation mode, one needs to separate the postures of the robot according to the current working modes. An aspect was defined in [30] as the maximal sets in the product of the workspace, W , by the joint space, Q. However, we can only represent its projection in the workspace that we will call parallel aspects, for a given working mode.

Since that it is impossible to change the working modes without disassembling this multi-mode parallel robot, we will study only the working modes depicted in Fig. 1 for simplicity reasons. With the Siropa library, we can make an algebraic cylindrical decomposition (CAD) [START_REF] Collins | Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition[END_REF][START_REF] Manubens | Cusp points in the parameter space of degenerate 3-RPR planar parallel manipulators[END_REF] in the space which includes the joint space and the workspace taking into account the serial and parallel singularities. In this paper, we assume there is no limitation on the range of motion of the joints. Therefore, we can study the workspace by considering only its section at y = 0.

In each cross-section of the workspace, we have represented the parallel aspects [30], i.e., the largest regions without singularity in the Cartesian space. The boundaries of these regions are the parallel singularities. Curves may exist that do not divide the workspace. These curves are associated with the other operation modes. These curves are obtained by the discriminant varieties of the constraint equations with the serial and parallel singularities [START_REF] Lazard | Solving parametric polynomial systems[END_REF].

Workspace analysis

In operation mode 1, the workspace is a divided in two regions due to the parallel singularities. In Fig. 14, the curves correspond to the parallel singularities associated with the 8 working modes and only a part of these curves divides the workspace in to two regions for the working mode under study. Unlike parallel robots with equal lengths and three parallel actuators [START_REF] Chablat | Kinematics, workspace and singularity analysis of a multi-mode parallel robot[END_REF], one singularity curve is associated to the working mode of the Fig. 1. In operation mode 2, the workspace is divided in two regions (Fig. 15). Please note the continuity of certain regions due to the cyclicity of the angles. The workspace in operation mode 3 consists of only two regions (Fig. 16). The workspace in operation mode 4 has four aspects (Fig. 17). The brown region is connected due to the cyclicity of the angles. For z = 0, and for any orientation, the robot is in singularity.

The workspace in operation mode 5 has two aspects (Fig. 18). The green region is connected due to the cyclicity of the one operation mode to another have been identified. The aspects that the robot can achieve in a given operation mode without going through a parallel singularity have also been obtained. Two non-singular assembly mode changing trajectories have been introduced by changing operation mode. This feature can justify the additional brakes requested to change operation mode by increasing the reachable workspace of one operation mode. This work, together with the literature on the reconfiguration analysis of multi-mode parallel mechanisms [15,[START_REF] Walter | Chapter A: Complete kinematic analysis of the SNU 3-UPU parallel manipulator[END_REF][START_REF] Kong | Reconfiguration analysis of a 3-DOF parallel mechanism using Euler parameter quaternions and algebraic geometry method[END_REF][START_REF] Kong | Reconfiguration analysis of a 4-DOF 3-RER parallel manipulator with equilateral triangular base and moving platform[END_REF][23], provide a comprehensive framework for the analysis of multi-mode parallel robots.

Future works will be done on the prototype by adding some brakes to automatically change operation mode and optimize the values of the design parameters to optimize the accuracy of the robot as well as its stiffness.

Operation Mode 5

 5 For operation mode 5, the moving platform undergoes a 3-DOF planar motion along the O-YZ plane. The translations of the moving platform are in the plane (y, z) and the rotation around the z-axis. The positions of C i and P are C 1 = [0d 3 cos α, yd 3 sin α, z] T C 2 = P = [0, y, z] T C 4 = [0 + d 7 cos α, y + d 7 sin α, z] T C 3 = [0, y, z + d 4 ] T C 5 = [0, y, z + d 4 + d 5 ] T

3. 1 Fig. 3 .

 13 Fig. 3. The eight working modes of the robot associated with the home pose.

Fig. 4 .

 4 Fig. 4. Joint space of operation mode 1 for ρ 1 = 0.

Fig. 5 .

 5 Fig. 5. Joint space of operation mode 2 for ρ 1 = 0.Fig. 6. Joint space of operation mode 3 for ρ 1 = 0.

Fig. 6 .

 6 Fig. 5. Joint space of operation mode 2 for ρ 1 = 0.Fig. 6. Joint space of operation mode 3 for ρ 1 = 0.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Joint space of operation mode 4 for ρ 1 = 0.Fig. 8. Joint space of operation mode 5 for ρ 1 = 0.

Fig. 9 .

 9 Fig. 9. Example of joint configuration with two direct kinematic solutions in operation mode 1 with ρ 1 = -50, ρ 2 = 0, and ρ 3 = 0.

Fig. 10 .

 10 Fig. 10. Example of joint configuration with four direct kinematic solutions in operation mode 2 with ρ 1 = 0, ρ 2 = 272, and ρ 3 = -53.

Fig. 11 .

 11 Fig. 11. Example of joint configuration with four direct kinematic solutions in operation mode 3 with ρ1 = 0, ρ 2 = 200, and ρ 3 = 200.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Example of joint configuration with four direct kinematic solutions in operation mode 4 with ρ1 = 0, ρ 2 = 200, and ρ 3 = 200.

4. 3 0 4. 4 0 4. 5 0 5

 304050 Operation mode 3 Serial singularities occur if and only if ρ 1y = 0 or ρ 2y = 0 or y + 95 sin βρ 3 = 0 As the axis of rotation of the moving platform is parallel to the x-axis, only the serial singularities related to the third leg depend on the orientation of the moving platform. Parallel Singularities occur if and only if cos β(ρ 3y) -113 sinβ = 0 or 2x(ρ 1 xρ 2 ) -66ρ 1 -33ρ 2 + 99y = Operation mode 4 Serial singularities occur if and only if ρ 1y = 0 or ρ 2y = 0 or y + 95 sinβρ 3 = 0 As the axis of rotation of the moving platform is parallel to the x-axis, only the serial singularities related to the first leg depend on the orientation of the moving platform. Parallel Singularities occur if and only if cos βρ 3y cosβ + z sinβ -113 sinβ = 0 or z = 0 or ρ 1ρ 2 = Operation mode 5 5 WORKSPACE ANALYSIS OF THE PARALLEL ROBOT IN DIFFERENT OPERATION MODES 4.5 Operation mode 5 Serial singularities occur if and only if 77 sin α + 2(ρ 1y) = 0 or ρ 2y = 0 or ρ 3y = 0 As the axis of rotation of the moving platform is parallel to the z-axis, only the serial singularities related to the first leg depend on the orientation of the moving platform. Parallel Singularities occur if and only if cos αρ 1y cosα + 55 sinα or ρ 2 zzρ 3 -18ρ 2 + 18y = Workspace analysis of the parallel robot in different operation modes 5.1 Definition of the aspects

Fig. 14 .

 14 Fig. 14. Workspace in operation mode 1 for y = 0.

Fig. 15 .

 15 Fig. 15. Workspace in operation mode 2 for y = 0. Fig. 16. Workspace in operation mode 3 for y = 0.

Fig. 17 .

 17 Fig. 17. Workspace in operation mode 4 for y = 0. Fig. 18. Workspace in operation mode 5 for y = 0.

Fig. 19 .

 19 Fig. 19. Transition among the five operation modes.

Fig. 20 .Fig. 21 .Fig. 22 .

 202122 Fig. 20. Workspace (a) and connected regions (b) for y = 0 for translation motion and the rotations around the z-axis

Fig. 23 .

 23 Fig. 23. Non-singular and singular assembly mode changing trajectory for β = 0 (a) and α = 0 (b) between T 1 = [-100, 0] and T 2 = [100, 0].

Table 1 .

 1 Five operation modes of the multi-mode parallel robot.

One of the Pi joint is replaced with an RR serial chain to simplify the structure without affecting the function of the robot.

Singularity analysis of the parallel robot in different operation modes The singularity analysis of conventional parallel robots has been well-documented in the literature. The singularity analysis of a multi-operation mode parallel robot involves the singularity analysis of the robot in each operation mode. In each operation mode, we first find the parallel and serial Jacobian matrix, named A and B respectively[START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Wenger Ph | Definition sets for the direct kinematics of parallel manipulators[END_REF], by differentiating

Acknowledgments

Dr. Xianwen Kong would like to thank the Engineering and Physical Sciences Research Council (EPSRC), United Kingdom, for the support under grant No.EP/K018345/1.

Connectivity

The multi-mode parallel robot can switch among its five operation modes in its home in which x = 0, z = 0, α = 0 and β = 0. In Fig. 19, starting from the home pose, the red arrows show the corresponding postures for each operation mode. The grey regions are the regions of the workspace that the robot cannot reach without crossing a parallel singularity.

Based on this analysis, Figs. 20 and 21 depict, at the left side, the workspace for the operations modes 1, 2 and 5 and the operation modes 1, 3 and 4, respectively, and at the right side, the remaining connected regions.

In operation mode 1, only the yellow region can be reached if we do not switch to operation modes 4 or 5. For example, in the operation mode 1, the trajectory T 1 to T 2 defined in Fig. 22(a) is not feasible because the point T 3 is singular. So, along the negative z-axis, we can reach at most z = -70.456 for x = 0 starting from the home pose. In operation mode 4, the robot is singular at any configuration with z = 0 (Fig. 22(c)). The pink and green regions can be reached only if we apply a trajectory either with operation mode 1 or operation mode 5 as shown Fig. 21. Conversely, in the operation mode 5 (Fig. 22(b)), the trajectory T 1 to T 2 is fully included inside a single region.

Based on the above results, we conclude that it is possible to change the parallel aspects for a given operation mode by passing through operation mode 1 or 4 or 5 (Figs. 22(a)-(c)). This change only happens at configurations with x = 0 and any values of y (Fig. 23). This property increases the workspace of the robot theoretically.

We notice that the ranges of translation in operation modes 2, 3, 4 and 5 are within the ranges defined by the intersections of the workspace with the x-and z-axes in operation mode 1. The above analysis has shown that the asymmetric 3-PRPiR multi-mode PM is better than the symmetric case of the multi-mode PM detailed in [START_REF] Chablat | Kinematics, workspace and singularity analysis of a multi-mode parallel robot[END_REF].

Conclusions

In this paper, we have presented the kinematics of a multi-mode parallel robot that can change operation modes by using reconfigurable Pi joints and lockable R joints without the problem of the constraint singularities. Changes in operation modes are realized by locking/releasing certain lockable joints. This feature allow us to know the operation mode without external or internal sensors. We have investigated the singularities of the parallel robot in different operation modes from the study of the determinant of the Jacobian matrices. For an operation mode resulting from the "home" configuration of the robot, we have

LIST OF FIGURES LIST OF TABLES