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OPTIMAL MOTION PLANNING OF A PLANAR PARALLEL MANIPULATOR WITHKINEMATICALLY REDUNDANT DEGREES OF FREEDOM

Bahman Nouri Rahmat Abadi, Sajjad Taghvaei and Ramin VatankhahSchool of Mechanical Engineering, Shiraz University, Shiraz, Iran

E-mail: [email protected]

Received August 2015, Accepted February 2016No. 15-CSME-102, E.I.C. Accession 3855

ABSTRACTIn this paper, an optimal motion planning algorithm and dynamic modeling of a planar kinematically redun-dant manipulator are considered. Kinematics of the manipulator is studied, Jacobian matrix is obtained andthe dynamic equations are derived using D’Alembert’s principle. Also, a novel actuation method is intro-duced and applied to the 3-PRPR planar redundant manipulator. In this approach, the velocity of actuatorsis determined in such a way to minimize the 2-norm of the velocity vector, subjected to the derived kine-matic relations as constraints. Having the optimal motion planning, the motion is controlled via a feedbacklinearization controller. The motion of the manipulator is simulated and the effectiveness of the proposedactuation strategy and the designed controller is investigated.

Keywords: parallel manipulators; kinematic redundancy; optimal motion planning; control.

PLANIFICATION OPTIMALE DE MOUVEMENT D’UN PLAN MANIPULATEUR PARALLÈLEAVEC DEGRÉS DE LIBERTÉ CINÉMATIQUEMENT REDONDANTS

RÉSUMÉCet article considère un algorithme de planification optimale de mouvement et la modélisation dynamiqued’un plan manipulateur cinématiquement redondant. La cinématique du manipulateur est étudiée, la matricejacobienne est obtenue et les équations dynamiques sont calculées en utilisant le principe de D’Alembert. Enoutre, un nouveau procédé d’actionnement est introduit et appliqué au manipulateur avec plans redondants3-PRPR. Dans cette approche la vitesse des actionneurs est déterminée de manière à réduire au minimum lanorme-2 du vecteur de vitesse, soumis à des relations cinématiques dérivées comme des contraintes. Avecla planification optimale de mouvement, le mouvement est contrôlé par l’intermédiaire d’un contrôleur delinéarisation de rétroaction. Le mouvement du manipulateur est simulé et l’efficacité de la stratégie d’ac-tionnement proposée et le contrôleur conçu sont étudiés.

Mots-clés : manipulateurs parallèles; redondance cinématique; planification optimale de mouvement;contrôle.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 3, 2016 383

1. INTRODUCTION

Parallel manipulators have attracted the attention of researchers for their potential advantages. High stiff-ness, large load-carrying capacity, good dynamic performance and position accuracy are among the advan-tages of parallel manipulators in comparison with serial manipulators [1]. Many researchers have extensivelystudied kinematic and dynamics of parallel manipulators [2–5], as well as workspace and singularity anal-yses [6, 7]. In spite of the promising features of parallel robots, there are some disadvantages such as asmall workspace, reduced dexterity and complexity in the forward kinematic equations. Furthermore, theworkspace of parallel manipulators contains singularities [8]. In the vicinity of these configurations, largeactuator 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 ofredundancy 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 themanipulator’s reachable workspace. Due to these drawbacks, kinematic redundancy is considered in thisresearch. This type of redundancy can be obtained by adding at least one actuated joint to one kinematicchain. Such a parallel mechanism has a larger singularity-free workspace in comparison with non-redundantmanipulators.

Most of the previous works have focused on inverse kinematics, workspace and singularities avoidanceof these mechanisms (e.g. [11–17]). Wong and Gosselin [11] have introduced three new types of kinemati-cally redundant manipulators by adding one degree of freedom in one of the kinematic chains. In [11], theJacobian matrices are obtained and the analytic expressions describing the singularities are presented. Theresults show that singularity configurations are significantly reduced. Ebrahimi et al. [12] have introduced anovel 3-PRRR planar mechanism. The workspace of the mechanism is compared to that of a non-redundantone. 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 canbe reached by the end-effector in all possible orientations [12]. Kotlarski et al. [15] have demonstratedthe influence of kinematic redundancy on the singularity-free workspace of some planar and spatial parallelmechanisms. In [16], a novel family of singularity-free kinematically redundant planar parallel mechanismsthat have unlimited rotational capabilities is introduced. Assal [17] has developed a planar parallel manipu-lator 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 mech-anisms. Boudreau and Nokleby [18] presented an optimization-based methodology for resolving the jointgeneralized forces of a 3-PRPR manipulator following a desired trajectory. Moreover, the force capabilityof a 3-RPRR was elaborated by Weihmann et al. [19]. Jiang et al. [20] have proposed a novel planar2-DOF parallel kinematic machine with kinematic redundancy and presented a method for redundant forceoptimization to improve the precision of the machine.

In the context of parallel robots, some researchers have looked into the actuation scheme of redundantmanipulators. Ruggiu and Carretero [21] have presented a new actuation strategy which is based on mini-mization of the actuated joints’ acceleration. In [22], two methods are proposed to determine the actuationscheme; using the condition number and a geometrical interpretation. It is illustrated that the mechanism iscapable of following a path while avoiding singularities. The authors have also introduced a motion planningalgorithm for kinematically redundant manipulators [23].

The main implication emerging from this research is to present a novel optimization based approachfor motion planning and actuation strategy, and to design a controller for kinematically redundant parallelmanipulators. 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

Fig. 1. 3-PRPR kinematically redundant planar parallel manipulator [21].

proposed method is implemented on a 3-PRPR redundant mechanism but also applicable for other similarredundant parallel mechanisms.

In Section 2, a general geometric description of the 3-PRPR redundant mechanism is provided. In se-quence, 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 algorithmfor trajectory tracking is described. Section 6 deals with designing a tracking controller based on feedbacklinearization. Finally, the motion of the redundant manipulators is simulated and the performance of theproposed 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 kinematicchain of a 2D parallel robot. A diagram of the mechanism, introduced in [21], is shown in Fig. 1. Themanipulator consists of a fixed base and the moving plate connected by several limbs. Each limb consists ofan 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) onthe base of the manipulator and three distal prismatic actuators (non-redundant actuators) control the positionand orientation of the equilateral triangle moving plate.

3. KINEMATIC MODELING

In contrast to non-redundant mechanisms, kinematic analysis of kinematically redundant manipulators ismore challenging. Due to the presence of additional actuated joint, the inverse kinematics has an infinitenumber 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 Jacobianmatrix is obtained.

3.1. Position AnalysesFor the purpose of kinematic analysis, two sets of coordinate systems are used. The first one is a movingcoordinate system: the M-frame is attached to the geometric center of the moving platform where the y-axis


passes through node B1 and the z-axis is perpendicular to the plane of the revolute joints Bi (henceforthi = 1,2,3) as shown in Fig. 1. In the second coordinate system, the R-frame X-Y is fixed to the intersectionof the proximal actuators as shown in Fig. 1. The position of the geometric center of the moving platformis depicted by the vector p = [x y]T and its orientation is described by θ . The rotation matrix R, representsthe 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 aresymmetrically distributed and their lengths are designated by ai. As illustrated in Fig. 1, the ith prismaticjoint changes the position of point Ai. The coordinate of point Ai with respect to the R-frame and the positionof the revolute joints on the moving plate, Bi presented in the M-frame are given as

rAi = Z

[ai

0

](2)

uBi = Z

[b0

](3)

where b represents the lengths of B1B1+1 and Z is defined as

Z =

[cγi −sγi

sγi cγi

](4)

in whichγ1 = π/2, γ2 = 7π/6, γ3 = 11π/6 (5)

The position vector of revolute joints on the moving plate, presented in the reference coordinate system,is given by

rBi = p+uBi (6)

In the above equation, uBi is the position of point Bi presented in the reference coordinate frame

uBi = RuBi (7)

Also, the vector rBi can be expressed in terms of the distal prismatic actuator as given by

rBi = rAi + lisi (8)

where si is the unit vector from revolute joint Ai to the revolute joint Bi and li is the length of the ith distalactuator.

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 prismaticactuators can be selected to satisfy Eq. (8) and the stroke limitations. The configuration of the mechanismcan be selected such as to avoid singularities [18]. In conclusion, the proposed mechanism has a largersingularity-free workspace in comparison with the 3-RPR platform.


3.2. Velocity AnalysisThe velocity of point Bi is found by taking the derivate of Eq. (6) with respect to time.

rBi = p+ω ×uBi (9)

Here the angular velocity of the moving plate is depicted by ω .

ω = [0 0 θ ]T . (10)

Equation (9) can be rewritten more compactly as

rBi = [I2×2 Rθ uBi ]

[pθ

](11)

where

Rθ =

[−sθ −cθ

cθ −sθ

](12)

On the other hand, the velocity of point Bi can be obtained by the time derivate of Eq. (8)

rBi = rAi + iisi + liΩi × si (13)

in which Ωi is the angular velocity of distal prismatic limbs. In the remainder of this paper, the componentof Ωi along vector si is neglected, i.e. Ωi · si = 0. Using this simplified assumption, an expression for Ωi andli is developed. Cross multiplication of both sides of Eq. (13) by si yields

si × rBi = si × rAi + lisi × (Ωi × si) (14)

Using the above assumption, Eq. (14) is rewritten and an analytical expression for the angular velocity ofthe lateral limbs is developed as

Ωi =1li

si × (rBi − rAi) (15)

Dot multiplication of both sides of Eq. (13) by si gives

si · rBi = si · rAi + iisi · si + lisi · (Ωi × si) (16)

By neglecting the component of Ωi along vector si, the third term in the right side will vanish. Thus, thefollowing relation for ii can be developed:

ii = si · (rBi − rAi) (17)

Also, the velocity of point Ai can be calculated as

rAi = Z

[ai

0

](18)

Finally, applying Eqs. (11) and (18) to Eq. (17) determines the relation between the linear velocity of theactuators and the velocity of the moving platform expressed as

li = sTi [I2×2 Rθ uBi ]

[pθ

]− sT

i

[cγi

sγi

]ai (19)


3.3. Acceleration AnalysisFor the acceleration analysis an analytical expression for li is obtained by taking the time derivate of Eq. (17).

li = si · (rBi − rAi)+(Ωi × si) · (rBi − rAi) (20)

in which the acceleration of the ith joint Bi can be obtained as

rBi = p− θ2RuBi + θRθ uBi (21)

Additionally, the angular acceleration of the ith non redundant actuator is obtained by the time derivativeof Eq. (15).

Ωi =1li[(Ωi × si)× (rBi − rAi)+ si × (rBi − rAi)]−

iili

Ωi (22)

3.4. Jacobian Matrix formulationNow, let q = [q1,q2]

T be the vector of input (actuated joints) and x = [x,y,θ ]T be the output vector (movingplatform pose). The vectors q1 and q2 are defined as

q1 = [l1, l2, l3]T , q2 = [a1,a2,a3]T (23)

Using this notation, Eq. (19) can be rewritten in the matrix form as

Ax = Bq (24)

where matrixes A and B are the 3×3 and 3×6 Jacobian matrices, respectively, given by

B =

1 0 0 e1 0 00 1 0 0 e2 00 0 1 0 0 e3

, A =

J1

J2

J3

(25)

in which

ei = sTi

[cγi

sγi

](26)

Ji = sTi [I2×2 Rθ uBi ] (27)

Gosselin and Angeles [8] have defined three types of singularities for parallel manipulators. The mostproblematic singularity, referred to as the second type, is located inside the workspace of the manipulator. Insuch a pose, the moving platform is able to perform local motion, even though all of the actuated joints arelocked. 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 actuatorswhich directly change the Jacobians’ elements. In another words, thanks to the redundancy, it is alwayspossible 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 thispurpose the following assumptions are considered:

1. A lumped mass of magnitude ma is assumed in each joint ai.

2. Friction is not incorporated.


Moreover, in order to determine the variation of expressions, the kinematic approach is used. This ap-proach states that taking the variation of the position vectors is similar to taking time derivative of theposition vectors [24].

Let uri and unri be the force in the redundant and non-redundant prismatic actuators, respectively. Thedynamic equations can be obtained as follows:

mpp ·δp+ Jpθ δθ +3

∑i=1

maaiδai −3

∑i=1

(uri δai +unri δ li) = 0 (28)

where mp and Jp are mass and the mass moment of inertia of the moving platform. In the above equation,δ li designates the virtual change in li obtained via the kinematic approach. By considering Eqs (26), (27)and (19), the following relation for δ li can be derived:

δ li = Ji

[δpδθ

]− eiδai (29)

Using Eqs. (28) and (29), the principle of virtual work can be rewritten as

mpp ·δp+ Jpθ δθ −3

∑i=1

unri Ji

[δpδθ

]+

3

∑i=1

[maai −uri + eiunri]δai = 0 (30)

In sequence, Eq. (30) is written in a simpler form given as

[δpδθ

]T mp 0 0

0 mp 00 0 Jp

[ pθ

]−

3

∑i=1

unri

[δpδθ

]T

JTi +

3

∑i=1

[maai −uri + eiunri]δai = 0 (31)

Dynamic equations are obtained by setting the coefficient of [δp δθ ]T and δai equal to zero. Finally, theequations of motion are depicted in a compact form as[

M1 00 M2

][xq2

]=

[P1 03×3

P2 I3×3

][u1

u2

](32)

The components of mass matrix are given as follows:

M1 =

mp 0 00 mp 00 0 Jp

, M2 =

ma 0 00 ma 00 0 ma

(33)

The matrices P1 and P2 are obtained as

P1 =

[[s1

uTB1

RTθ

s1

][s2

uTB2

RTθ

s2

][s3

uTB1

RTθ

s3

]], P2 =

e1 0 00 e2 00 0 e3

(34)

Also, the vector of control parameters u1 and u2 are defined by

u1 = [unr1 unr2 unr3]T , u2 = [ur1 ur2 ur3]

T (35)


5. MOTION PLANNING

Having obtained the dynamic modeling of the system, a novel optimal motion planning strategy which isbased 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 manipulatorswith kinematic redundancy.

Obviously, the control of kinematically redundant manipulators is more challenging in comparison withsimilar non-redundant manipulators. For a given trajectory of the moving platform, the desired pose of theend-effector is specified. But, there is no idea for the desired length of the redundant actuators. In theintroduced method, the length of the redundant actuators is obtained so that the velocity of the actuators isminimized. Therefore, the actuation strategy is presented as follows:

1. Firstly, the desired trajectory of the moving platform and the initial position of the redundant actuatorsare determined. For the considered manipulator, the initial lengths of the proximal actuators arespecified. 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 areoptimized to minimize the objective function F given as

F =3

∑i=1

(l2i + a2

i ) (36)

Also, to consider the stroke and the velocity limitations of the distal actuators, the following inequalityconstraints are incorporated to minimize F.

lmin ≤ li ≤ lmax (37)

In this step, FMINCON, included in MATLAB’s optimization toolbox, is used to find the minimumof a constrained linear multivariable function.

3. Using the optimized velocity of the actuators at each moment, the desired length and acceleration ofthe proximal actuators are obtained by numerical methods.

4. Finally, since the desired configuration of the mechanism is specified in the previous steps, a controlleris 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 themethods proposed in the literature. The velocities of the actuators are considered as the search variableswhich 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 requiressolving the inverse kinematics at each step.

For instance in [21], the displacements of the non-redundant actuators are obtained from the inversekinematics for trial values of the search variables. In sequence, the acceleration of the search variables areestimated using time history and the accelerations of the non-redundant actuators are obtained using theacceleration equation. The optimization problem involves solving a set of nonlinear equations by imposingconstraints 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.


Remark 2. The introduced path planning strategy is based on minimizing the 2-norm of the velocity vectorand 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 onthe acceleration of the joints in the defined optimization problem. However, the bounds on the accelerationare more practical than velocity limit.

Imposing the bounds on the joint accelerations will be considered as an extension of this work in thefuture. This would require the definition of a new objective function which is explicitly dependent toacceleration.

Remark 3. It is worth mentioning that, the optimatization problem involves the minimatization of a quadraticfunction subjected to three linear constraints. Thus, the obtained solution in each step is the unique globalsolution [25].

6. CONTROL STRATEGY

For the purpose of tracking a given trajectory, two steps are considered in this paper. In the first step, usingthe above mentioned algorithm, for a given trajectory of the moving plate, the desired length, velocity andacceleration of the actuators are obtained which is based on minimizing the 2-norm of velocity vector. Inthe 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 toexplain 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

u2

]=

[P1 03×3

−P2 I3×3

]−1[M1 00 M2

][v1

v2

](38)

which applied to Eq. (32) yields a set of six decoupled linear equations[xq2

]=

[v1

v2

](39)

The typical choice for the auxiliary control input, v = [v1 v2]T is[

v1

v2

]=

[q1d

vq2d

]+

[KD1 0

0 KD2

][q1d − xq2d − q2

]+

[KP1 0

0 KP2

][q1d −xq2d −q2

](40)

leading to the error equation[¨q1¨q2

]+

[KD1 0

0 KD2

][˙q1˙q2

]+

[KP1 0

0 KP2

][q1

q2

]=

[OO

](41)

whereq1 = q1d −x, q2 = q2d −q2 (42)

The error equation is exponentially stable by a suitable choice of the matrixes KD1,KD2,KP1 and KP2. Inwhat follows, the mentioned algorithm is used to design a controller for tracking a desired trajectory.


Table 1. Geometric parameters of the redundant manipulator.mp 0.4 kgma 0.1 kgPBl 0.1 mlmax 0.9 m/slmin –0.9 m/s

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 thedistal 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 bedetermined 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 shownin Fig. 6, the manipulator with the designed controller can follow the elliptical path, perfectly. The controllerperformance 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 thenon-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 thisshould be considered in any comparison between these two mechanisms. For instance the dimensions of thebase 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


Fig. 3. The velocities of the actuators of the non-redundant and the velocities of the distal actuators of the redundantmechanism.

Fig. 4. Position of the nodes in the base of the manipulator.


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 non-redundant mechanisms.


Fig. 7. The forces in the actuators of the non-redundant mechanism and the forces in the distal actuators of theredundant mechanism.

Fig. 8. Force in the proximal prismatic actuators.


8. CONCLUSION

This paper addresses the issue of kinematic redundancy in parallel manipulators by proposing an optimalmotion planning algorithm. By considering the velocity of the actuators as a cost function subjected tothe kinematic relations constraints, the optimal trajectory is determined. The computational cost of thealgorithm 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 accel-eration 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 theperformance of the controller is investigated by numerical simulation of a tracking problem. The proposedmethod can be utilized for other kinematically redundant parallel mechanisms.

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