OPTIMAL MOTION PLANNING OF A PLANAR PARALLEL MANIPULATOR WITH KINEMATICALLY REDUNDANT DEGREES OF FREEDOM

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1 OPTIMAL MOTION PLANNING OF A PLANAR PARALLEL MANIPULATOR WITH KINEMATICALLY REDUNDANT DEGREES OF FREEDOM Bahman Nouri Rahmat Abadi, Sajjad Taghvaei and Ramin Vatankhah School of Mechanical Engineering, Shiraz University, Shiraz, Iran bahman.nouri@shirazu.ac.ir Received August 2015, Accepted February 2016 No. 15-CSME-102, E.I.C. Accession 3855 ABSTRACT In this paper, an optimal motion planning algorithm and dynamic modeling of a planar kinematically redundant manipulator are considered. Kinematics of the manipulator is studied, Jacobian matrix is obtained and the dynamic equations are derived using D Alembert s principle. Also, a novel actuation method is introduced and applied to the 3-PRPR planar redundant manipulator. In this approach, the velocity of actuators is determined in such a way to minimize the 2-norm of the velocity vector, subjected to the derived kinematic relations as constraints. Having the optimal motion planning, the motion is controlled via a feedback linearization controller. The motion of the manipulator is simulated and the effectiveness of the proposed actuation strategy and the designed controller is investigated. Keywords: parallel manipulators; kinematic redundancy; optimal motion planning; control. PLANIFICATION OPTIMALE DE MOUVEMENT D UN PLAN MANIPULATEUR PARALLÈLE AVEC DEGRÉS DE LIBERTÉ CINÉMATIQUEMENT REDONDANTS RÉSUMÉ Cet article considère un algorithme de planification optimale de mouvement et la modélisation dynamique d un plan manipulateur cinématiquement redondant. La cinématique du manipulateur est étudiée, la matrice jacobienne est obtenue et les équations dynamiques sont calculées en utilisant le principe de D Alembert. En outre, un nouveau procédé d actionnement est introduit et appliqué au manipulateur avec plans redondants 3-PRPR. Dans cette approche la vitesse des actionneurs est déterminée de manière à réduire au minimum la norme-2 du vecteur de vitesse, soumis à des relations cinématiques dérivées comme des contraintes. Avec la planification optimale de mouvement, le mouvement est contrôlé par l intermédiaire d un contrôleur de linéarisation de rétroaction. Le mouvement du manipulateur est simulé et l efficacité de la stratégie d actionnement proposée et le contrôleur conçu sont étudiés. Mots-clés : contrôle. manipulateurs parallèles; redondance cinématique; planification optimale de mouvement; Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

2 1. INTRODUCTION Parallel manipulators have attracted the attention of researchers for their potential advantages. High stiffness, large load-carrying capacity, good dynamic performance and position accuracy are among the advantages of parallel manipulators in comparison with serial manipulators 1. Many researchers have extensively studied kinematic and dynamics of parallel manipulators 2 5, as well as workspace and singularity analyses 6, 7. In spite of the promising features of parallel robots, there are some disadvantages such as a small workspace, reduced dexterity and complexity in the forward kinematic equations. Furthermore, the workspace of parallel manipulators contains singularities 8. In the vicinity of these configurations, large actuator forces are needed which can cause serious damage to the mechanism. For solving the mentioned drawback, redundancy has been proposed in the literature 9, 10. Two types of redundancy have been introduced for parallel manipulators; kinematic redundancy and actuator redundancy 9, 10. Actuator redundancy can be obtained by actuating a passive joint which causes internal preload. Also, a redundant manipulator can be obtained by adding an extra kinematic chain but this may reduce the manipulator s reachable workspace. Due to these drawbacks, kinematic redundancy is considered in this research. This type of redundancy can be obtained by adding at least one actuated joint to one kinematic chain. Such a parallel mechanism has a larger singularity-free workspace in comparison with non-redundant manipulators. Most of the previous works have focused on inverse kinematics, workspace and singularities avoidance of these mechanisms (e.g ). Wong and Gosselin 11 have introduced three new types of kinematically redundant manipulators by adding one degree of freedom in one of the kinematic chains. In 11, the Jacobian matrices are obtained and the analytic expressions describing the singularities are presented. The results show that singularity configurations are significantly reduced. Ebrahimi et al. 12 have introduced a novel 3-PRRR planar mechanism. The workspace of the mechanism is compared to that of a non-redundant one. It was shown that the proposed mechanism has a larger reachable and dexterous workspace. Where, the dexterous workspace is considered as a region of the reachable workspace within which every point can be reached by the end-effector in all possible orientations 12. Kotlarski et al. 15 have demonstrated the influence of kinematic redundancy on the singularity-free workspace of some planar and spatial parallel mechanisms. In 16, a novel family of singularity-free kinematically redundant planar parallel mechanisms that have unlimited rotational capabilities is introduced. Assal 17 has developed a planar parallel manipulator for a hybrid machine tool. The proposed architecture has some advantages such as a large workspace, unlimited orientation capability, base-mounted actuators and no singularities. On the other hand, some researchers have focused on the force capacity of kinematically redundant mechanisms. Boudreau and Nokleby 18 presented an optimization-based methodology for resolving the joint generalized forces of a 3-PRPR manipulator following a desired trajectory. Moreover, the force capability of a 3-RPRR was elaborated by Weihmann et al. 19. Jiang et al. 20 have proposed a novel planar 2-DOF parallel kinematic machine with kinematic redundancy and presented a method for redundant force optimization to improve the precision of the machine. In the context of parallel robots, some researchers have looked into the actuation scheme of redundant manipulators. Ruggiu and Carretero 21 have presented a new actuation strategy which is based on minimization of the actuated joints acceleration. In 22, two methods are proposed to determine the actuation scheme; using the condition number and a geometrical interpretation. It is illustrated that the mechanism is capable of following a path while avoiding singularities. The authors have also introduced a motion planning algorithm for kinematically redundant manipulators 23. The main implication emerging from this research is to present a novel optimization based approach for motion planning and actuation strategy, and to design a controller for kinematically redundant parallel manipulators. Due to lower computational cost, the algorithm gives the optimal trajectories in real-time. The 384 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

3 Fig PRPR kinematically redundant planar parallel manipulator 21. proposed method is implemented on a 3-PRPR redundant mechanism but also applicable for other similar redundant parallel mechanisms. In Section 2, a general geometric description of the 3-PRPR redundant mechanism is provided. In sequence, kinematic analysis of the redundant manipulators is considered and the Jacobian matrix is derived. Section 4 presents the dynamic equation of the manipulator. Then, in Section 5, the actuation algorithm for trajectory tracking is described. Section 6 deals with designing a tracking controller based on feedback linearization. Finally, the motion of the redundant manipulators is simulated and the performance of the proposed method for actuation of the considered manipulator is considered. 2. STRUCTURE OF THE MANIPULATOR The kinematically redundant manipulator studied here is generated by adding an actuator to each kinematic chain of a 2D parallel robot. A diagram of the mechanism, introduced in 21, is shown in Fig. 1. The manipulator consists of a fixed base and the moving plate connected by several limbs. Each limb consists of an actuated prismatic joint fixed to the based followed by a passive revolute joint, an actuated prismatic joint, and a passive revolute joint attached to the end effector. Three proximal actuators (redundant actuators) on the base of the manipulator and three distal prismatic actuators (non-redundant actuators) control the position and orientation of the equilateral triangle moving plate. 3. KINEMATIC MODELING In contrast to non-redundant mechanisms, kinematic analysis of kinematically redundant manipulators is more challenging. Due to the presence of additional actuated joint, the inverse kinematics has an infinite number of solutions. A complete kinematic analysis of the mechanism is presented in 21. In this section, the inverse kinematic, velocity and acceleration of the manipulator are presented and finally the Jacobian matrix is obtained Position Analyses For the purpose of kinematic analysis, two sets of coordinate systems are used. The first one is a moving coordinate system: the M-frame is attached to the geometric center of the moving platform where the y-axis Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

4 passes through node B 1 and the z-axis is perpendicular to the plane of the revolute joints B i (henceforth i = 1,2,3) as shown in Fig. 1. In the second coordinate system, the R-frame X-Y is fixed to the intersection of the proximal actuators as shown in Fig. 1. The position of the geometric center of the moving platform is depicted by the vector p = x y T and its orientation is described by θ. The rotation matrix R, represents the transformation matrix from moving frame to the reference frame and is defined as R = cθ sθ sθ cθ, (1) where cθ and sθ indicate cosθ and sinθ, respectively. Three proximal prismatic actuators on the base are symmetrically distributed and their lengths are designated by a i. As illustrated in Fig. 1, the ith prismatic joint changes the position of point A i. The coordinate of point A i with respect to the R-frame and the position of the revolute joints on the moving plate, B i presented in the M-frame are given as r Ai = Z ai 0 (2) ū Bi = Z b 0 where b represents the lengths of B 1 B 1+1 and Z is defined as Z = cγi sγ i sγ i cγ i (3) (4) in which γ 1 = π/2, γ 2 = 7π/6, γ 3 = 11π/6 (5) The position vector of revolute joints on the moving plate, presented in the reference coordinate system, is given by r Bi = p + u Bi (6) In the above equation, u Bi is the position of point B i presented in the reference coordinate frame u Bi = Rū Bi (7) Also, the vector r Bi can be expressed in terms of the distal prismatic actuator as given by r Bi = r Ai + l i s i (8) where s i is the unit vector from revolute joint A i to the revolute joint B i and l i is the length of the ith distal actuator. As mentioned, the mobility of the mechanism is more than the number of degrees of freedom. Thus, for a given pose of the moving platform, an infinite number of solutions for the lengths of the prismatic actuators can be selected to satisfy Eq. (8) and the stroke limitations. The configuration of the mechanism can be selected such as to avoid singularities 18. In conclusion, the proposed mechanism has a larger singularity-free workspace in comparison with the 3-RPR platform. 386 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

5 3.2. Velocity Analysis The velocity of point B i is found by taking the derivate of Eq. (6) with respect to time. Here the angular velocity of the moving plate is depicted by ω. ṙ Bi = ṗ + ω u Bi (9) ω = 0 0 θ T. (10) Equation (9) can be rewritten more compactly as ṙ Bi = I 2 2 R θ ū Bi ṗ θ (11) where sθ cθ R θ = cθ sθ On the other hand, the velocity of point B i can be obtained by the time derivate of Eq. (8) (12) ṙ Bi = ṙ Ai + i i s i + l i Ω i s i (13) in which Ω i is the angular velocity of distal prismatic limbs. In the remainder of this paper, the component of Ω i along vector s i is neglected, i.e. Ω i s i = 0. Using this simplified assumption, an expression for Ω i and l i is developed. Cross multiplication of both sides of Eq. (13) by s i yields s i ṙ Bi = s i ṙ Ai + l i s i (Ω i s i ) (14) Using the above assumption, Eq. (14) is rewritten and an analytical expression for the angular velocity of the lateral limbs is developed as Dot multiplication of both sides of Eq. (13) by s i gives Ω i = 1 l i s i (ṙ Bi ṙ Ai ) (15) s i ṙ Bi = s i ṙ Ai + i i s i s i + l i s i (Ω i s i ) (16) By neglecting the component of Ω i along vector s i, the third term in the right side will vanish. Thus, the following relation for i i can be developed: i i = s i (ṙ Bi ṙ Ai ) (17) Also, the velocity of point A i can be calculated as ṙ Ai = Z ȧi 0 (18) Finally, applying Eqs. (11) and (18) to Eq. (17) determines the relation between the linear velocity of the actuators and the velocity of the moving platform expressed as ṗ l i = s T i I 2 2 R θ ū Bi s θ T cγi i ȧ i (19) sγ i Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

6 3.3. Acceleration Analysis For the acceleration analysis an analytical expression for l i is obtained by taking the time derivate of Eq. (17). in which the acceleration of the ith joint B i can be obtained as l i = s i ( r Bi r Ai ) + (Ω i s i ) (ṙ Bi r Ai ) (20) r Bi = p θ 2 Rū Bi + θr θ ū Bi (21) Additionally, the angular acceleration of the ith non redundant actuator is obtained by the time derivative of Eq. (15). Ω i = 1 l i (Ω i s i ) (ṙ Bi ṙ Ai ) + s i ( r Bi r Ai ) i i l i Ω i (22) 3.4. Jacobian Matrix formulation Now, let q = q 1,q 2 T be the vector of input (actuated joints) and x = x,y,θ T be the output vector (moving platform pose). The vectors q 1 and q 2 are defined as Using this notation, Eq. (19) can be rewritten in the matrix form as q 1 = l 1,l 2,l 3 T, q 2 = a 1,a 2,a 3 T (23) Aẋ = B q (24) where matrixes A and B are the 3 3 and 3 6 Jacobian matrices, respectively, given by e J 1 B = e 2 0, A = J 2 (25) e 3 J 3 in which e i = s T i cγi sγ i (26) J i = s T i I 2 2 R θ ū Bi (27) Gosselin and Angeles 8 have defined three types of singularities for parallel manipulators. The most problematic singularity, referred to as the second type, is located inside the workspace of the manipulator. In such a pose, the moving platform is able to perform local motion, even though all of the actuated joints are locked. For the proposed mechanism, this type of singularity can occur when the determinant of A is zero. In such a case, the singular configuration can be avoided by adjusting the lengths of prismatic actuators which directly change the Jacobians elements. In another words, thanks to the redundancy, it is always possible to select a solution for the inverse position problem such as to avoid singularities. 4. DYNAMIC MODELING In this section, dynamic equations of the manipulator are derived using D Alembert s principle. For this purpose the following assumptions are considered: 1. A lumped mass of magnitude m a is assumed in each joint a i. 2. Friction is not incorporated. 388 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

7 Moreover, in order to determine the variation of expressions, the kinematic approach is used. This approach states that taking the variation of the position vectors is similar to taking time derivative of the position vectors 24. Let u ri and u nri be the force in the redundant and non-redundant prismatic actuators, respectively. The dynamic equations can be obtained as follows: m p p δp + J p θδθ + 3 i=1 m a ä i δa i 3 i=1 (u ri δa i + u nri δl i ) = 0 (28) where m p and J p are mass and the mass moment of inertia of the moving platform. In the above equation, δl i designates the virtual change in l i obtained via the kinematic approach. By considering Eqs (26), (27) and (19), the following relation for δl i can be derived: δl i = J i δp δθ e i δa i (29) Using Eqs. (28) and (29), the principle of virtual work can be rewritten as 3 δp 3 m p p δp + J p θδθ u nri J i + δθ m a ä i u ri + e i u nri δa i = 0 (30) In sequence, Eq. (30) is written in a simpler form given as T m p 0 0 T δp p 3 δp 0 m p 0 δθ θ u nri J T i=1 δθ 0 0 J p i=1 i=1 3 i + i=1 m a ä i u ri + e i u nri δa i = 0 (31) Dynamic equations are obtained by setting the coefficient of δp δθ T and δa i equal to zero. Finally, the equations of motion are depicted in a compact form as M1 0 ẍ P u1 = (32) 0 M 2 q 2 P 2 I 3 3 u 2 The components of mass matrix are given as follows: m p 0 0 M 1 = 0 m p 0, M 2 = 0 0 J p m a m a m a (33) The matrices P 1 and P 2 are obtained as s 1 s 2 P 1 = ū T B 1 R T θ s 1 ū T B 2 R T θ s 2 s 3 ū T B 1 R T θ s 3, P 2 = e e e 3 (34) Also, the vector of control parameters u 1 and u 2 are defined by u 1 = u nr1 u nr2 u nr3 T, u 2 = u r1 u r2 u r3 T (35) Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

8 5. MOTION PLANNING Having obtained the dynamic modeling of the system, a novel optimal motion planning strategy which is based on minimizing the velocity of the actuators is proposed and used to actuate the 3-PRPR manipulator. It is worth mentioning that the introduced procedure can be applied for a large class of parallel manipulators with kinematic redundancy. Obviously, the control of kinematically redundant manipulators is more challenging in comparison with similar non-redundant manipulators. For a given trajectory of the moving platform, the desired pose of the end-effector is specified. But, there is no idea for the desired length of the redundant actuators. In the introduced method, the length of the redundant actuators is obtained so that the velocity of the actuators is minimized. Therefore, the actuation strategy is presented as follows: 1. Firstly, the desired trajectory of the moving platform and the initial position of the redundant actuators are determined. For the considered manipulator, the initial lengths of the proximal actuators are specified. Then, using Eq. (8), the lengths of the distal actuators are obtained. 2. By considering Eq. (19) as a constraint,the velocity of the distal and proximal prismatic actuators are optimized to minimize the objective function F given as F = 3 i=1 ( l 2 i + ȧ 2 i ) (36) Also, to consider the stroke and the velocity limitations of the distal actuators, the following inequality constraints are incorporated to minimize F. l min l i l max (37) In this step, FMINCON, included in MATLAB s optimization toolbox, is used to find the minimum of a constrained linear multivariable function. 3. Using the optimized velocity of the actuators at each moment, the desired length and acceleration of the proximal actuators are obtained by numerical methods. 4. Finally, since the desired configuration of the mechanism is specified in the previous steps, a controller is designed to control the motion of the manipulator. In the following, some discussions and remarks on the introduced motion planning method are given as: Remark 1. The introduced motion planning strategy has less computational cost in comparison with the methods proposed in the literature. The velocities of the actuators are considered as the search variables which lead to optimization of a multivariable function with linear constraints. Whereas in other approaches 21, 22 the displacements of the redundant actuators are considered as the search variables which requires solving the inverse kinematics at each step. For instance in 21, the displacements of the non-redundant actuators are obtained from the inverse kinematics for trial values of the search variables. In sequence, the acceleration of the search variables are estimated using time history and the accelerations of the non-redundant actuators are obtained using the acceleration equation. The optimization problem involves solving a set of nonlinear equations by imposing constraints on the velocity and acceleration of the actuators. Whereas, in the method presented in this paper, the inverse kinematic is not solved and the computational calculations are reduced. Thus, the desired length, velocity and acceleration of actuators can be obtained on line. 390 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

9 Remark 2. The introduced path planning strategy is based on minimizing the 2-norm of the velocity vector and does not guarantee that large changes in acceleration do not occur. The objective function, F in Eq. (36), is a function of velocity and displacement, not the acceleration. Therefore, a constraint cannot be imposed on the acceleration of the joints in the defined optimization problem. However, the bounds on the acceleration are more practical than velocity limit. Imposing the bounds on the joint accelerations will be considered as an extension of this work in the future. This would require the definition of a new objective function which is explicitly dependent to acceleration. Remark 3. It is worth mentioning that, the optimatization problem involves the minimatization of a quadratic function subjected to three linear constraints. Thus, the obtained solution in each step is the unique global solution CONTROL STRATEGY For the purpose of tracking a given trajectory, two steps are considered in this paper. In the first step, using the above mentioned algorithm, for a given trajectory of the moving plate, the desired length, velocity and acceleration of the actuators are obtained which is based on minimizing the 2-norm of velocity vector. In the next step, using the desired kinematics of the actuators and a feedback linearization control algorithm, the actuator forces are obtained in such a way to minimize the tracking error. In this section we are going to explain the second step, namely the feedback linearization control 26. To achive tracking the control task, the following control law is used and the nonlinear terms are canceled: u1 u 2 = P P 2 I M1 0 0 M 2 v1 v 2 (38) which applied to Eq. (32) yields a set of six decoupled linear equations ẍ v1 q 2 = v 2 (39) The typical choice for the auxiliary control input, v = v 1 v 2 T is v1 q1d KD1 0 q1d ẋ KP1 0 q1d x v 2 = vq 2d + 0 K D2 q 2d q K P2 q 2d q 2 (40) leading to the error equation q 1 q 2 KD K D2 q 1 q 2 KP K P2 q1 q 2 = O O (41) where q 1 = q 1d x, q 2 = q 2d q 2 (42) The error equation is exponentially stable by a suitable choice of the matrixes K D1,K D2,K P1 and K P2. In what follows, the mentioned algorithm is used to design a controller for tracking a desired trajectory. Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

10 Table 1. Geometric parameters of the redundant manipulator. m p 0.4 kg m a 0.1 kg PB l 0.1 m l max 0.9 m/s 0.9 m/s l min Fig. 2. Velocity of proximal actuators. 7. SIMULATION AND RESULTS In this section, some numerical eamples are presented. In this regards, the limitation on the velocity of the distal and other geometric parameters used in the simulations are listed in Table 1. The chosen trajectory is an elliptical path described by X = 0.1sin(2πt) + 0.2cos(2πt) 0.1cos(2πt) + 0.2sin(2πt) π/6 T (43) Using the obtained velocity of the acuators, the position and acceleration of the proximal actuators can be determined in each step as depicted in Figs. 4 and 5, respectively. To investigate the ability of the considered control procedure, some simulations are performed. As shown in Fig. 6, the manipulator with the designed controller can follow the elliptical path, perfectly. The controller performance is scrutinized via 2 cycles. In Fig. 7, the forces in the distal actuators of the redundant mechanism and the forces in the actuators of the non-redundant mechanism are depicted. Generalized forces in the proximal actuators are depicted in Fig. 8. It is worth mentioning that the structure of the redundant and non-redundant mechanism is different and this should be considered in any comparison between these two mechanisms. For instance the dimensions of the base platforms of the non-redundant manipulator greatly affect the manipulator s performance. However, for similar size of the base platform it is seen that the forces required by distal actuators of the kinematically 392 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

11 Fig. 3. The velocities of the actuators of the non-redundant and the velocities of the distal actuators of the redundant mechanism. Fig. 4. Position of the nodes in the base of the manipulator. Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

12 Fig. 5. Acceleration of the distal prismatic actuators. Fig. 6. Implementation results for a trajectory. redundant manipulators were lower than those of the non-redundant mechanisms as illustrate in Fig. 7. Thus, it can be seen that the redundant mechanism has improved force capability in comparison with nonredundant mechanisms. 394 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

13 Fig. 7. The forces in the actuators of the non-redundant mechanism and the forces in the distal actuators of the redundant mechanism. Fig. 8. Force in the proximal prismatic actuators. Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,

14 8. CONCLUSION This paper addresses the issue of kinematic redundancy in parallel manipulators by proposing an optimal motion planning algorithm. By considering the velocity of the actuators as a cost function subjected to the kinematic relations constraints, the optimal trajectory is determined. The computational cost of the algorithm is comparatively low such that the motion planning is obtained in real-time. To implement the method, the kinematic and dynamic modeling of a 3-PRPR manipulator is considered. In sequence, the actuation strategy is performed to the manipulator. It is shown that the velocity and acceleration of the distal actuators can be decreased by choosing different solutions for the inverse kinematics. Using the proposed actuation strategy, a controller based on feedback linearization is implemented and the performance of the controller is investigated by numerical simulation of a tracking problem. The proposed method can be utilized for other kinematically redundant parallel mechanisms. REFERENCES 1. Merlet, J.-P., Parallel Robots (2nd edition), Springer, Merlet, J.-P., Direct kinematics of parallel manipulators, IEEE Transactions. Robotics and Automation. Vol. 9, No. 6, pp , Tahmasebi, F., and Tsai, L.W., Closed-form direct kinematics solution of a new parallel minimanipulator, ASME Journal of Mechanical Design, Vol. 116, No 4, pp , Gosselin, C.M., Parallel computational algorithms for the kinematics and dynamics of planar and spatial parallel manipulators, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 118, No. 1, pp , Tsai, L.W., Solving the inverse dynamics of a Stewart-Gough manipulator by the principle of virtual work, ASME Journal of Mechanical Design, Vol. 122, No. 1, pp. 3 9, Lee, M.K. and Park, K.W., Workspace and singularity analysis of a double parallel manipulator, Mechatronics, IEEE/ASME Transactions, Vol. 5, No. 4, pp , Bonev, I.A., Zlatanov, D. and Gosselin, C.M., Singularity analysis of 3-dof planar parallel mechanisms via screw theory, ASME Journal of Mechanical Design, Vol. 125, No. 3, pp , Gosselin, C.M. and Angeles, J., Singularity analysis of closed-loop kinematic chains, IEEE Transactions on Robotics and Automation, Vol. 6, No. 3, pp , Zanganeh, K. and Angeles, J., Instantaneous kinematics and design of a novel redundant parallel manipulator, in Proceedings of IEEE Conference on Robotics and Automation, pp , Merlet, J.-P., Redundant parallel manipulators, Journal of Laboratory Robotic and Automation, Vol. 8, No. 1, pp , Wang, J. and Gosselin, C.M., Kinematic analysis and design of kinematically redundant parallel mechanisms, ASME Journal of Mechanical Design, Vol. 126, No. 1, pp , Ebrahimi, I. Carretero, J.A. and Boudreau, R., 3-PRRR redundant planar parallel manipulator: inverse displacement, workspace and singularity analyses, Journal of Mechanism and Machine Theory, Vol. 42, No. 8, pp , Ebrahimi, I., Carretero, J.A. and Boudreau, R., A family of kinematically redundant planar parallel manipulators, ASME Journal of Mechanical Design, Vol. 130, No. 6, pp , Gallant, A., Boudreau, R. and Gallant, M., Dexterous workspace of a 3-PRRR Kinematically Redundant Planar Parallel Manipulator, Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 4, Kotlarski, J. Heimann, B. and Ortmaier, T., Influence of kinematic redundancy on the singularity-free workspace of parallel kinematic machines, Frontiers of Mechanical Engineering, Vol. 7, No. 2, pp , Gosselin, C.M., Singularity-free kinematically redundant planar parallel mechanisms with unlimited rotational capability, IEEE Transactions on Robotics, Vol. 31, No. 2, pp , Assal, A., Piecewise kinematically redundant planar parallel manipulator for a hybrid machine tool, in Proceedings IEEE International Conference on Industrial Technology, Seville, Spain, pp , Boudreau, R. and Nokleby, S., Force optimization of kinematically-redundant planar parallel manipulators following a desired trajectory, Mechanism and Machine Theory, Vol. 56, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016

15 19. Weihmann, L., Martins, D. and Coelho, L.S., Force capabilities of kinematically redundant planar parallel manipulators, in Proceedings 13th World Congress in Mechanism and Machine Science, Guanajuato, México, June, Jiang, Y., Li, T.M. and Wang, L.P., Dynamic modelling and redundant force optimization of a 2-DOF parallel kinematic machine with kinematic redundancy, Robotics and Computer-Integrated Manufacturing, Vol. 32, pp. 1 10, Ruggiu, M. and Carretero, J.A., Actuation strategy based on the acceleration model for the 3-PRPR redundant planar parallel manipulator, in Advances in Robot Kinematics: Analysis and Design, J. Lenarcic and M.M. Stanisic (Eds.), Springer, Ebrahimi, I. Carretero, J.A. and Boudreau, R., Kinematic analysis and path planning of a new kinematically redundant planar parallel manipulator, Robotica, Vol. 26, No. 3, pp , Carretero, J.A. Ebrahimi I. and Boudreau, R., Comparison between two motion planning strategies for kinematically redundant parallel manipulators, Advances in Robot Kinematics: Analysis and Design, Springer, the Netherlands, pp. A243 A252, Baruh, H., Analytical Dynamics, McGraw-Hill, New York, Wright, S. and Nocedal, J., Numerical Optimization, Springer, New York, Canudas de Witt, C., Siciliano, B. and Bastin, G., Theory of Robot Control, Springer, Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3,