On the dynamic performance of parallel kinematic manipulators with actuation and kinematic redundancies

Mechanism and Machine Theory 103 (2016) 148–166

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On the dynamic performance of parallel kinematic manipulators with actuation and kinematic redundancies João Vitor Fontes* , Maíra Martins da Silva University of São Paulo, São Carlos School of Engineering, Department of Mechanical Engineering, Av. Trabalhador Sãocarlense, 400, 13566-590 São Carlos, Brazil

A R T I C L E

I N F O

Article history: Received 12 November 2015 Received in revised form 12 May 2016 Accepted 13 May 2016 Available online xxxx Keywords: Parallel kinematic manipulators Actuation redundancy Kinematic Redundancy Dynamic performance

A B S T R A C T The performance of parallel kinematic manipulators may be enhanced by the use of actuation and kinematic redundancies since they promote a significant reduction in the singularities and homogenization on the actuation forces. Considering these facts, this manuscript proposes a strategy to assess if the use of one or more actuation/kinematic redundancies can be good alternatives to improve the dynamic performance of a planar parallel kinematic manipulator. This may not be a trivial task since their dynamic performance can be highly dependent on which trajectory is performed and how the redundancy is treated. To overcome the former issue, a dynamic index depicted in the system workspace is proposed. To overcome the latter issue, different optimization strategies to treat actuation and kinematic redundancies are compared. From numerical results, one can conclude that the dynamic performance of the manipulator can only be enhanced if the redundancies are treated accordingly. Moreover, the benefits of redundancy are highly dependent on the chosen trajectory, the end effector’s position and orientation. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction It is well-known that parallel kinematic machines (PKMs) are capable of providing precision, high stiffness, good dynamic performance and load capability. Although being a promising alternative to industrial manipulators, PKMs may suffer from the presence of singularities in their workspace [1,2]. As a result, the ratio between the useful working space and physical space occupied by the equipment is rather low, which is an important drawback. Some authors, e.g. [3–7], have suggested that the use of redundancy can be a good alternative for minimizing the presence of singularities enhancing the workspace usability. Moreover, the addition of redundancy might also provide higher accuracy [3], better movements and forces’ transmissibility [8], improved dynamic performance [9–12], enhanced energy efficiency [13], among others. In this manuscript, two distinct kinds of redundancies are exploited: actuation and kinematic redundancies. Actuation redundancy can be implemented by the actuation of passive joints or by the inclusion of active kinematic chains [14]. The degrees-of-freedom (DOFs) of the end effector’s manipulator are kept constant in both strategies. In spite of possessing more actuators than DOFs, the kinematic description of a manipulator with actuation redundancy is unique. In order words, the inverse kinematic problem presents a single solution. On the other hand, kinematic redundancy can be implemented by the introduction of extra active joints in a kinematic chain. The inverse kinematic problem presents infinite solutions, i.e. there

* Corresponding author. E-mail address: [email protected] (J. de Carvalho Fontes).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.05.004 0094-114X/© 2016 Elsevier Ltd. All rights reserved.

J. de Carvalho Fontes, M. da Silva / Mechanism and Machine Theory 103 (2016) 148–166

(a)

(b)

3RRR

4RRR

(d)

(e)

(P)RRR+2RRR

Passive Revolute Joint

Active Revolute Joint

2(P)RRR+RRR

End effector

149

(c)

6RRR

(f)

3(P)RRR

Link

Active Prismatic Joint

Fig. 1. Non-redundant and redundant manipulators: (a) 3RRR, (b) 4RRR, (c) 6RRR, (d) PRRR + 2RRR, (e) 2PRRR + RRR and (f) 3PRRR.

are infinite possible joint parameters for a single end effector’s pose (rotation and position). According to [14], kinematically redundant PKMs provide higher mobility than required for the task allowing for the avoidance of singularities and obstacles. Regarding actuation redundancy, several authors have claimed that this strategy may enhance the performance of PKMs. In [15], the contribution of actuation redundancies to enhance the dynamic performance of pick-and-place robotic systems have been exploited. In order to do that, two dynamic indexes based on the maximum end effector’s acceleration have been proposed. The authors have concluded that the considered redundancy is capable of homogenizing the required torques/forces enhancing the dynamic performance of such manipulators. In [11,16], kinematic and dynamic models of the 2RRR, 3RRR and 4RRR PKMs (see Fig. 1) have been described. It could be concluded that the redundant manipulators have presented not only better conditioning, payload and stiffness performances but also better dynamic response. In an industrial application, the authors of [17] have concluded that the performance of a redundant parallel tool head leads to notable advantages over the non-redundant one, including enlarged singularity-free workspace, improved dexterity performance, and higher stiffness. Recently, a measure for evaluating the dynamic performance of parallel manipulators with a single actuation redundancy has been proposed by [18]. Regarding kinematic redundancy, reports exploiting the reconfiguration capabilities of this strategy can be found. Among them, [3,19–21] have compared numerically and experimentally the performance of non-redundant and redundant manipulators. The main objective of these works has been to obtain enlarged workspace with an acceptable precision and free of singularities. The results have shown that the redundant manipulator is capable to reconfigure itself avoiding singularities and increasing the useful workspace. The work of [5] has presented a manipulator with three levels of kinematic redundancy called 3PRRR. An optimization strategy has been proposed and applied to avoid the singularities of the manipulator. The comparison between the redundant and non-redundant manipulator has shown that redundancy is capable of avoiding the singularities and increasing the workspace. Recently, [22] has described optimization strategies to deal with kinematic redundancies. Different trajectories have been exploited demonstrating that the redundant manipulator is capable of overcoming the issues with singularities. The comparison between the dynamic performance of non-redundant and redundant manipulators is highly dependent on the selected trajectory. This is an important drawback since a fully understanding of the redundancy capability cannot be assessed. The authors of [23] have proposed a novel dynamic performance index that combines the acceleration, velocity, and gravity terms of the dynamic equations. This index can be directly employed to the performance assessment of manipulators with actuation redundancy. Due to the reconfiguration capabilities of the kinematically redundant manipulators, the same index cannot be directly exploited. Considering these facts, this manuscript proposes a strategy to assess if the use of one or more

150

J. de Carvalho Fontes, M. da Silva / Mechanism and Machine Theory 103 (2016) 148–166 Table 1 Number of kinematic chains and kinematically redundant actuators for the considered combinations. Manipulator

N

M

3RRR 4RRR 6RRR PRRR + 2RRR 2PRRR + RRR 3PRRR

3 4 6 3 3 3

0 0 0 1 2 3

actuation/kinematic redundancies can be good alternatives to improve the dynamic performance of a planar parallel kinematic manipulator. This may not be a trivial task since their dynamic performance can be highly dependent on which trajectory is performed and how the redundancy is treated. To overcome the former issue, a dynamic index depicted in the system workspace is proposed. To overcome the latter issue, different optimization strategies to treat kinematic redundancies are compared. In order to infer about this quest, kinematic and dynamic models of the planar parallel manipulator 3RRR, the redundantly actuated planar parallel manipulators 4RRR and 6RRR and the kinematically redundant planar parallel manipulators PRRR + 2RRR, 2PRRR + RRR and 3PRRR, are exploited. These manipulators are illustrated in Fig. 1. As commonly used, the letter R corresponds to the revolute joint, the letter P corresponds to the prismatic joint, the underline letter to the actuated joint and the number in front of the name refers to the number of kinematic chains. The end effector of these planar manipulators is considered to be a rigid body that can move in plane, yielding systems with three DOFs. The kinematic and dynamic models are revised in Section 2. Section 3 presents both dynamic performance evaluation strategies addressed in this manuscript. In this section, the proposed dynamic index is also fully described. Numerical results exploit these performance evaluation strategies in Section 4. Finally, conclusions are drawn in Section 5. 2. Short review on kinematic and dynamic modeling methodology In spite of having different quantities of moving parts and kinematic chains, the understudied manipulators present a rather similar structure. In other words, the main differences between the manipulators are the number of the kinematic chains, N, and the number of kinematically redundant actuators, M. These aspects modify the number of moving parts in each kinematic chain. For instance, regarding the number of kinematic chains, the difference between the non-redundant manipulator 3RRR (N = 3, M = 0) and redundant manipulator 4RRR (N = 4, M = 0) is a single kinematic chain, while the same amount of kinematic chains is found on the 3RRR (N = 3, M = 0) and the redundant manipulator 3PRRR (N = 3, M = 3). Regarding the number of moving parts, the kinematically redundant manipulators present one or more extra moving parts, the kinematically redundant actuator that can move on the linear guide. Table 1 summarizes the number of kinematic chains and the number of kinematically redundant actuators for all possible combinations. These aspects are exploited in order to derive comprehensive kinematic and dynamic models for the redundant and non-redundant manipulators. The manipulators may have three kinds of moving parts: the kinematically redundant actuators, the links, and the end effector. The subscript j is employed to address the actuators and links, while the subscript e is used to address the end effector. In this manuscript, it has been adopted that the j = 0 represents the mass of kinematically redundant actuator while j = 1 and j = 2 represent the links 1 and 2, respectively. On the other hand, the manipulators may have three, four, or six kinematic chains according to Table 1. The subscript i is employed to address the kinematic chain under evaluation. In this way, i can be equal to 1, 2, or 3 for the 3RRR, while 1, 2, 3, 4, 5, or 6 for the 6RRR. The generic representation of a single chain i including the moving parts is depicted in Fig. 2. For instance, using the proposed representation, the manipulator PRRR + 2RRR has one kinematic chain with three moving parts and two chains with two moving parts. So, in this case, i can be equal to 1, 2 or 3. If i = 1 (the kinematic chain with three moving parts), j can be equal to 0, 1 or 2. If i = 2 or i = 3 (the kinematic chains with two moving parts), j can be equal to 1 or 2. Table 2 shows the representation of each moving part j in each kinematic chain i considering the different manipulators. This table summarizes the subscripts for all possible combinations. 2.1. Kinematic analysis The following description of a single kinematic chain is employed in order to derive comprehensive models for the understudied manipulators. The kinematic chain i is illustrated in Fig. 2. The base coordinate system O-xo yo is located in the center of the workspace of the manipulator and the point D is the end effector’s center. The inverse kinematic modelscan be solved by evaluating geometrical constraint equations. As it can be seen in Fig. 2, the  −→       geometrical constraint equation  Bi Ci  = rCi − rBi = l2 can be imposed, where rCi and rBi are the position vectors of the passive joints Bi and Ci , respectively. This constraint equation can be described by evaluating the length of the link 2:        li − l1 cos(hi )    = l2 cos(bi )  = l2 ,    q − l1 sin(h )   sin(b ) i i i

(1)

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151

yO xO

O y ai x D l2

Ci

βi

Bi γi l1 Ai

y

θi

x α

δi hi

λi

ηi

Fig. 2. Illustration of the kinematic chain.

where l i and qi are defined by: 

li qi



 =

x y



 − hi

cos(a + gi ) sin(a + gi )



 − ai

cos(ki ) sin(ki )



 − di

 cos(ci ) . sin(ci )

(2)

In these definitions, a is the end effector rotational angle, hi is the rotational angle of the active joint Ai , di is the position of the redundant actuator on the linear guide, ai is the shortest distance between the linear guide and the point O, hi is the distance between the center of the end effector and the point Ci , and x and y are the linear positions of the end effector’s center relative to the base coordinate system. Only the manipulators with kinematic redundancy have the position of the redundant actuator di non-zero. Eq. (1) can be rewritten by using the following notation: ei1 sin hi + ei2 cos hi + ei3 = 0,

(3)

where ei1 = −2l1i qi , ei2 = −2l1i l i and ei3 = l 2i + q2i + (l1i )2 − (l2i )2 . The angles hi can be calculated using [16]: ⎛ ⎜ −ei1 ± hi = 2tan−1 ⎝

⎞  e2i1 + e2i2 − e2i3 ⎟ ⎠. ei3 − ei2

(4)

In the same way, the angles bi can be calculated using the vectorial equation obtained from Eq. (1): bi = tan−1



qi − l1 sin(hi ) . li − li cos(hi )

(5)

Table 2 Number of kinematic chains and moving parts for each manipulator. Manipulator

Kinematic chain i

Moving part j

3RRR 4RRR 6RRR PRRR + 2RRR 2PRRR + RRR 3PRRR

1, 2, 3 1, 2, 3, 4 1, 2, 3, 4, 5, 6 1, 2, 3 1, 2, 3 1, 2, 3

1, 2 1, 2 1, 2 i = 1, j = 0, 1, 2; i = 2, 3, j = 1, 2 i = 1, 2, j = 0, 1, 2; i = 3, j = 1, 2 0, 1, 2

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In order to evaluate the velocities and the accelerations of the passive and active joints of the manipulators, the velocities and accelerations of the end effector should be known. One can express the end effector’s velocities of the link i by taking the time derivative of the Eq. (1): ˙ 2 hi sin(bi − gi − a)] = h˙i [l1 l2 sin(bi − hi )] + d˙i [l2 cos(bi − ci )]. x˙ [l2 cos(bi )] + y˙ [l2 sin(bi )] − a[l

(6)

Eq. (6) can be rewritten in a matrix form: ˙ ˙ = BH, AX

(7)

˙ are the velocities of the active joints. For the non-redundant ˙ = [x˙ y˙ a] ˙ T are the velocities of the end effector and H where X manipulator and the manipulators with actuation redundancy, the quantity of velocities varies with the number of kinematic ˙ = [h˙1 h˙2 . . . h˙N ]T . The velocities of the active joints of the kinematically redundant manipuchains of the manipulator, H ˙ = [h˙1 h˙2 h˙3 d˙1 ]T , for the lators vary according to the number of redundant actuators. In this way, for the PRRR + 2RRR, H ˙ = [h˙1 h˙2 h˙3 d˙1 d˙2 ]T and for the 3PRRR, H ˙ = [h˙1 h˙2 h˙3 d˙1 d˙2 d˙3 ]T . 2PRRR + RRR, H For the non-redundant and redundant manipulators, the Jacobian matrix A can be described by: ⎡

an1 an2 an3

⎢ . A=⎢ ⎣ ..

⎤

.. ⎥ ⎥ . ⎦,

.. .

(8)

aN1 aN2 aN3 where an1 = l2 cos(bn ), an2 = l2 sin(bn ) and an3 = −l2 h sin(bn − gn − a). The subscript n can vary according to the number of kinematic chains N, n = 1 . . . N (see Table 1). In this way, the dimension of the Jacobian matrix A of the models of the non-redundant and redundant manipulators is N × 3. For the non-redundant manipulator and the manipulators with actuation redundancies (M = 0), the Jacobian matrix B is defined by B = diag(b11 , b22 , . . . , bNN ),

(9)

where bkk = l1 l2 sin(bk − hk ) and k = 1 . . . N. For the kinematically redundant manipulators (M = 0), the Jacobian matrix B is defined by ⎡

b11 0

0 b1(m+3) . . .

0

.. .

0

⎢ B=⎢ ⎣ 0 b22 0 0 0 b33

0

..

.

⎤ ⎥ ⎥, ⎦

(10)

0 b3(M+3)

where bkk = l1 l2 sin(bk − hk ) and bk(m+3) = l2 cos(bk − ck ), where k = 1 . . . 3 and m = 1 . . . M, according to the number of kinematically redundant actuators. In this way, the dimension of the Jacobian matrix B is 3 × (M + 3). So, the dimension of the matrix B is 3 × 3 for the non-redundant manipulator, 4 × 4 for the manipulator 4RRR, 6 × 6 for 6RRR, 3 × 4 for the PRRR + 2RRR, 3 × 5 for 2PRRR + RRR and 3 × 6 for 3PRRR. The velocities and accelerations of each moving part of the manipulator should be calculated to perform the dynamic analysis. ˙ and X, ¨ can be calculated by taking the time derivatives of the position The velocities and accelerations of the end effector, X vector: ˙ ˙ = JH X

and

(11)

¨ + J˙H, ˙ ¨ = JH X

(12)

where J is the Jacobian matrix. This Jacobian matrix can be decomposed into two matrices:  J = A+ B =

He Ge

 ,

(13)

˙ with the vector where A + is the pseudo inverse matrix of A. The matrix He is defined as the matrix that relates the vector H ˙ with [x˙ y˙ ]T which is the vector of the translational velocities of the end effector. Similarly, the vector Ge relates the vector H ˙ the angular velocity of the end effector a. Following the aforementioned notation (see Table 1), the kinematically redundant actuator Ai receives value j = 0, the link Ai Bi receives j = 1 with pivotal point in Ai , and the link Bi Ci j = 2 with pivotal point in Bi .

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In order to find the velocities and accelerations of the redundant actuator and links, one can define the position vector dij of each of these moving parts as:  dij =

rij 0ij

 ,

(14)

where rij is the linear position vector and 0ij is the rotational position of the moving part j of the kinematic chain i. In this way, the rotation position of the actuator Ai is not defined (concentrated mass), the rotational position of the link Ai Bi is 0i1 = hi and the rotational position of the link Bi Ci is 0i2 = bi (see Fig. 2). The velocities and accelerations of the redundant actuators and links can be calculated by taking the time derivative of the position vector: ˙ d˙ ij = Kij H

and

(15)

¨ +K ˙ ˙ ij H, d¨ ij = Kij H

(16)

where the Jacobian matrices Kij can be calculated using the same methodology used to define J (see Eq. (13)). These Jacobian matrices can also be decomposed in two major matrices:  Kij =

Hij Gij



.

(17)

˙ with the vector r˙ ij which comprises the linear velocities of The matrix Hij is defined as the matrix that relates the vector H ˙ with the angular velocity 0˙ij of the body j of the chain i. the body j of the chain i. Similarly, the vector Gij relates the vector H These matrices are usually sparse matrices. For the actuator Ai in the kinematic chain i, the linear velocities are defined as: r˙ Ai = r˙ i0 = d˙i



cos(ci ) sin(ci )



 =

0 · · · cos(ci ) · · · 0 0 · · · sin(ci ) · · · 0



˙ H,

(18)

2×6

where the non-zero terms are located in column i + 3. In this way, Hi0 can be defined as:  Hi0 =

0 · · · cos(ci ) · · · 0 0 · · · sin(ci ) · · · 0



.

(19)

2×6

Since the angular velocity of this moving part is always zero,   Gi0 = 0 · · · 0 1×6 .

(20)

For the link Ai Bi (link 1 in Fig. 2), the linear velocities are the same as previously defined to the point Ai and the angular velocity is h˙ i . Consequently, Hi1 = Hi0

and

(21)

  Gi1 = 0 · · · 1 · · · 0 1×6 .

(22)

For the link Bi Ci (link 2 in Fig. 2), the linear velocities are defined by the velocities of the point Bi , which can be defined by: r˙ Bi = r˙ i2 = h˙i l1



− sin(hi ) cos(hi )



+ d˙i



 cos(ci ) . sin(ci )

(23)

In this way, Hi2 can be defined as:  Hi2 =

0 · · · l1 (− sin(hi )) · · · 0 · · · cos(ci ) · · · 0 0 · · · l1 cos(hi ) · · · 0 · · · sin(ci ) · · · 0

 ,

(24)

2×6

where non-zero values are in columns i and i + 3. Gi2 can be determined by the relation between r˙ Ci and r˙ Bi , r˙ Ci = r˙ Bi + b˙i l2



 − sin(bi ) . cos(bi )

(25)

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J. de Carvalho Fontes, M. da Silva / Mechanism and Machine Theory 103 (2016) 148–166

˙ and a˙ = Ge H ˙ (see Eq. (13)): Also, r˙ Ci can be calculated from the velocities of the end effector, using [x˙ y˙ ]T = He H  r˙ Ci = [He + hi

 − sin(gi + a) ˙. Ge ]H cos(gi + a)

(26)

Using these relations, Gi2 can be expressed as:  Gi2 =

     1  cos(gi + a) Ge + Hi2 . − sin(bi ) cos(bi ) He − h sin(gi + a) l2 1×6

(27)

It is important to note that the values with index i + 3 only exist if the manipulator has kinematic redundancy. In other words, these columns are null for the non-redundant manipulator and for the manipulators with actuation redundancies. 2.2. Dynamic analysis Using the aforementioned approach, the velocities and accelerations of the moving parts can be calculated. Using these quantities, one can employ the Newton–Euler formulation to find the equations of motion. In this analysis, the mass at the actuator Ai is m0 , the mass and inertia of the link Ai Bi is m1 and I1 , the mass and inertia of the link Bi Ci is m2 and I2 and the mass and inertia of the end effector is me and Ie . Considering the body j of the chain i, the vector pij composed by the combination of forces and moment applied on this body can be described as:  pij =

Fij Mij



.

(28)

Using the Newton–Euler formulation, Fij and Mij can be calculated:  Fij =

¨ ij sj sin(0ij ) − 0 ˙ 2 sj cos(0ij )) mj (r¨xij − 0 ij ¨ ij sj cos(0ij ) − 0 ˙ 2 sj sin(0ij )) mj (r¨yij + 0 ij



¨ ij Mij = mj sj (r¨xij (− sin(0ij )) + r¨yij cos(0ij )) + Ij 0

(29) (30)

where sj is the distance between of the mass center and the pivotal point of the body j. The position vector rij = [rxij ryij ]T and angular position 0ij of the moving body j are defined in the vector dij in Eq. (14). Similarly, considering the end effector, the vector pe composed by the forces and moment applied on the end effector can be described as:  pe =

Fe Me



.

(31)

Using the Newton–Euler formulation, Fe and Me can be calculated:  Fe =

me x¨ me y¨

 (32)

Me = Ie a¨ .

(33)

The relation between the generalized forces tg applied by the actuators and the forces and moments on the system can be expressed using the Principle of Virtual Work [16]. This strategy yields the following relation: tg T dH = pTe dX +

N 2  

pTij d dij ,

(34)

i=1 j=0 or 1

where N and M are defined in Table 1 and the lower limit of the sum related to the subscript j varies according to the presence of kinematically redundant actuators. If there is a redundant actuator in the considered kinematic chain i, j starts from 0, otherwise j starts from 1. Using the definitions of the Jacobian matrices, Eqs. (13) and (15), Eq. (34) can be rewritten as: tg T dH = pTe JdH +

N 2   i=1 j=0 or 1

pTij Kij dH.

(35)

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155

Since the virtual displacements are non-zero, Eq. (35) can be completely defined by:

tg = JT pe +

2 N  

KTij pij .

(36)

i=1 j=0 or 1

In order to describe Eq. (36) in function of the position vector H, the matrices Zij , Nij and Ze can be introduced by rewriting the vectors pij and pe (Eqs. (29) and (30)): ⎡

⎤ ⎡ ⎤ r¨x r˙x ⎢ ¨ ij ⎥ ⎢ ˙ ij ⎥ pij = Zij ⎣ ryij ⎦ + N ij ⎣ ryij ⎦ ¨ ij ˙ ij 0 0

and

(37)

⎡

⎤ x¨ pe = Ze ⎣ y¨ ⎦ , a¨

(38)

where: ⎡

⎤ mj 0 −mj sij sin 0ij 0 mj mj sij cos 0ij ⎦ , −mj sij sin 0ij mj sij cos 0ij Ij

Zij = ⎣

(39)

⎡

˙ ij sij cos 0ij ⎤ 0 0 −mj 0 ˙ ij sij sin 0ij ⎦ Nij = ⎣ 0 0 −mj 0 0 0 0

and

(40)

⎡

⎤ me 0 0 ⎣ Ze = 0 me 0 ⎦ . 0 0 Ie

(41)

Substituting Eqs. (37) and (38) in Eq. (36), the generalized forces, tg , can be expressed in function of the time derivatives of the position vector H: ¨ + VH, ˙ tg = MH

(42)

where: ⎛ M = ⎝JT Ze J +

⎞

2 N  

KTij Zij Kij ⎠

and

(43)

i=1 j=0 or 1

⎛ V = ⎝J Ze J˙ + T

N 2   i=1 j=0 or 1

˙ ij KTij Zij K

+

N 2  

⎞ KTij Nij Kij ⎠

.

(44)

i=1 j=0 or 1

It is important to note that indexes i and j of the Eqs. (43) and (44) follow the suggested nomenclature in Table 2. Hereafter, the proposed performance indexes aim to compare the required torques to perform a predefined kinematic situation (predefined positions, velocities and accelerations). Aiming that, the generalized forces generated by the redundant actuators have to be also expressed in terms of the actual required torques. This is accomplished by using a conversion factor Tm = ddm /dhredundantm , where m = 1 . . . M according to the number of kinematic redundancy. The transformation can be mathematically expressed by the diagonal matrix T = diag(1, 1, 1, Tm , . . . , TM ). Using this transformation, one can assess the required torques delivered by the actuators: ¨ + TVH, ˙ t = TMH

(45)

where t is the vector of the actual required torques. The dimension of T varies according to the number of redundant actuators. The dimension of T is 3 × 3 when M = 0 and (3 + M) × (3 + M) when M = 0.

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2.3. Workspace analysis A dynamic evaluation index is defined in Section 3.2 and its value is depicted on the manipulator’s workspace. For sake of completeness, the methodology to find the workspace of a planar PKM is briefly described hereafter. In order to draw the workspace, all reachable positions of each kinematic chain i have to be assessed. This region has a ring-shape format and is denoted as RSi hereafter [1,18]. The outer and the inner radius of this ring-shape region are defined by the maximum and minimum reachable distances of this kinematic chain, i.e., (|Ai D|max = max|Ai D|) and (|Ai D|min = min|Ai D|), respectively. The −→

−→

−→



−→

−→

vector Ai D = Ai Bi + Bi Ci + Ci D can be described by the following equation:

Ai D = l 1

cos (hi ) sin (hi )



 + l2

cos (bi ) sin (bi )



 +h

 cos (gi + a) . sin (gi + a)

(46)

The total workspace, denoted as WS, considering a variable end effector’s pose can be given by the intersection of the ringshape regions of each kinematic chain. In this way, WS = RS1 ∩ RS2 ∩ . . . RSN ,

(47)

where N is the number of kinematic chains of the manipulator. For the kinematically redundant manipulators, one should consider that the position of the redundant actuator Ai can vary (dmin < di < dmax ). 3. Dynamic performance evaluation strategies In this section, two dynamic performance evaluation strategies are presented. The first approach is the calculation and comparison of the required torques to perform preselected trajectories. From this evaluation, one can conclude that the evaluation of the dynamic performance of manipulators are highly dependent on the specified trajectory. To overcome this issue, the second approach is the proposal of a dynamic index depicted in the system workspace which results in the dynamic maps. Both strategies are treated in the following subsections. 3.1. Dynamic performance evaluation based on the inverse kinematic and dynamic models — maximum required torque The inverse kinematic and dynamic models of the non-redundant manipulator and the manipulators with actuation redundancy present a single solution. In this way, once a trajectory is selected, the calculation of the required torques to be employed by the actuators is straightforward (Eq. (42)). This is not true to the manipulators with kinematic redundancy. Since manipulators with kinematic redundancy have infinity kinematic configurations for a constant end effector configuration, the solutions of the inverse kinematic and dynamic models are also infinity. In this way, a choice among the possibilities should be made in order to calculate the required torques to perform a selected trajectory. This choice can be done through an optimization problem. The definitions of this cost function and constraints of the optimization problem are not unique and depend on the designer expertise. In this work, the absolute value of the maximum required torque during the execution of a preselected trajectory of the end effector should be minimized, t(di )  from Eq. (42). The variables of this optimization problem are the positions di (t) of the redundant actuators which are limited by the length of the linear guides. The optimal positions of the redundant actuators, di,opt , are given by:

di,opt (t) = arg subject

to :

min

di (t)

t

dmin ≤ di (t) ≤ dmax .

(48)

The required torques are dependent on the positions of the active rotational joints and the positions of the redundant actuators H. From a predefined path of the end effector and the presented modeling strategy, the positions hi can be calculated while the positions di are used as optimization variables. During the trajectory, the positions di are dependent on the time. To overcome this issue, Kotlarski et. al [3] proposed the offline and online strategies for a manipulator with a single kinematically redundant actuator. This nomenclature may be misleading due to issues related to real time control of the manipulators. In this way, in spite of exploiting rather similar strategies, a new nomenclature is proposed by the authors: the prepositioning and the ongoing positioning approaches. Moreover, the hereafter described strategies deal with multiple redundant actuators and different optimization variables compared to the strategies proposed by [3]. The prepositioning approach consists in determining which are the best positions of kinematically redundant actuators, the positions of the linear guides di , before moving the end effector. Note that the values di are the same throughout the entire execution of the end effector’s trajectory. So, in this case, the optimization problem has just one optimization variable for each

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kinematic chain with a redundant actuator, a fixed dfixedi . The optimal values of this variable, dfixedi ,opt , can be found by the following optimization problem: dfixedi ,opt = arg subject

to :

t

min

dfixed

i

dmin ≤ dfixedi ≤ dmax .

(49)

In the ongoing positioning approach, the redundant actuators are allowed to move on the linear guides while the end effector performs the desired task. In order to calculate appropriate movements for the redundant actuators, a parametric trajectory is predefined for the actuators’ position on the linear guide. This trajectory is selected in such a way that only the initial and final positions are considered as optimization variables. This parametric trajectory is defined as a polynomial of degree five that describes the movement of di from d0i to dfi in 2 s with initial and final velocities/acceleration equal to zero, according to: di (t) = (dfi − d0i )



      3t5 /16 − 15t4 /16 + 5t3 /4 + d0i .

(50)

In this case, the optimization problem has, for each level of kinematic redundancy, two variables, d0i and dfi . The optimal values of these variables, d0i ,opt and dfi ,opt , can be found by: [d0i ,opt , dfi ,opt ] = arg

min ] ]d0 ,df i

subject

t

(51)

i

to :

dmin ≤ d0i ≤ dmax dmin ≤ dfi ≤ dmax dmin ≤ di ≤ dmax . This evaluation strategy requires the selection of a predefined trajectory. For sake of comparison, the chosen trajectory is a parametric straight line. The trajectory is calculated using the same polynomial of degree five described in Eq. (50). In this way, for each end effector’s degree-of-freedom, starting values (x0 , y0 , a 0 ) and final values (xf , yf , a f ) should be selected a priori. During the trajectory, these values are modified according to the polynomial. Two sets of starting and final points are exploited: (i) (x0 , y0 , a 0 ) = (−0.05, −0.05, 0) and (xf , yf , a f ) = (0.05, 0.05, 0) and : (ii) (x0 , y0 , a 0 ) = (−0.05, −0.05, −p/8) and (xf , yf , a f ) = (0.05, 0.05, p/8). As one can realize, the difference between the sets (i) and (ii) is the evaluation of the rotation of end effector during the movement. The prepositioning and the ongoing positioning approaches, aim to seek trajectories that minimizes the required torques. There is no guarantee that the end effector will not pass by a singularity region. However, since the torques requires by the manipulator in these regions are relatively higher, these regions are usually avoided. These optimization problems have been solved using an iterative method for nonlinear optimization, the Sequential Quadratic Programming [24]. This method is implemented in Matlab. 3.2. Dynamic maps The above described evaluation strategy is highly dependent on the selected trajectory. A fair comparison between the dynamic performances of different manipulators can only be accomplished by adopting a metric depicted in the manipulator’s workspace. Recently, [23] proposed a dynamic index that consists of calculating the required torque to obtain fixed values of velocities and accelerations in specific points. In a similar way, the dynamic index metric proposed hereafter aims to evaluate the required torque to ensure predefined values of acceleration of the end effector, W, all over the workspace considering that ˙ = 0. The formulation of the index is described below. ˙ = 0 and H the manipulator is at rest, X ˙ = 0, Eq. (42) can be rewritten as: Considering the presented dynamic formulation and the velocities H ¨. t = TMH

(52)

Taking the time derivative of Eq. (12), the accelerations of the active joints can be written as a function of the accelerations of the end effector using the Jacobian matrix: ¨ = J˙−1 X ˙ + J−1 X ¨. H

(53)

¨ = J−1 X. ¨ Moreover, using the definition of the ˙ = 0, the accelerations can be calculated as H As the manipulator is at rest, X Jacobian matrix, J −1 = B −1 A (Eq. (7)), the torques t can be calculated as: ¨. t = TMB−1 AX

(54)

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Obtaining δi,opt by minimizing of DIpen

Recalculating DI using δi,opt

Ploting DI in the workspace: Dynamic Maps

Fig. 3. Procedure to plot the Dynamic Maps using DIpen for the kinematically redundant manipulators.

¨ are equal to predefined values X ¨ = W = [Wx Considering that the accelerations X t = TMB−1 AW,

Wy

Walpha ]T : (55)

where W is a column vector of weighting values selected by the designer to evaluate the dynamic performance. These weighting values may account for distinct requirements on the degrees-of-freedom, homogenization and for unit conversion issues. For instance, manipulators that may require to have higher dynamic performance in the x-direction than the y-direction may be compared with different weighting values in these directions. One can conclude that the proposed approach is rather similar to a multi-objective optimization problem where different weights are employed to construct a single objective function. This objective function can take into account diverse functions with different units. In this way, these weighting values are not dimensionless [24]. The dynamic index DI is defined as the maximum required torque in the proposed conditions, which corresponds to the infinity norm of the matrix TMB −1 AW:     DI = max {t} = TMB−1 AW . ∞

(56)

This index DI can be calculated across the workspace since it is a function of the pose of the end effector and the actuators. In the case of the manipulators with kinematic redundancy, the matrix B is not square so it was used the pseudoinverse of the matrix B. Moreover, as aforementioned, the inverse kinematic and dynamic models of these manipulators present infinity solutions. So, the optimal DI can be found by solving an optimization problem. In to this optimization problem, the minimum value of DI has be found and the optimization variables are the positions of the redundant actuators on the linear guides di . In the following section, these positions are depicted on the workspace (see Fig. 10). In this figure, one can observe some important discontinuities. They represent areas where the optimal positions of the redundant actuators on the linear guides may abruptly change (see Fig. 11). These abrupt changes should be avoided since they are not practicable. One can add constraints to the optimization problem in order to avoid this issue. A common strategy to deal with constraints in optimization problems is the use of penalty function methods [24]. In these methods, a weighted penalty term is added to the objective function modifying the original problem to an unconstrained optimization problem [24]. This penalty term presents the same unit of the original objective function, so its weight is not dimensionless. In this way, in order to avoid the abrupt modifications on the position of the redundant actuators, a penalty term has been added to the dynamic index, yielding the DIpen . This penalty term ensures that the optimization problem searches for small values of DI while modifications on the positions of the redundant actuators on the linear guides are penalized. The proposed dynamic index with the penalty term is described by:     DIpen = TMB−1 AW

∞

  + 4d2 ,

(57)

where d = [d1 . . . dM ]T , according to the number of kinematically redundant actuators, M. In the same way, the minimum value of DIpen should be assessed at each position of the end effector. The variables of this optimization problem are the positions of the linear guides di . Once DIpen has no physical meaning because of the penalty term, it is necessary to recalculate DI using the optimized positions of the linear guides di,opt . Fig. 3 illustrates this procedure. The minimal values of the DI and the DIpen have been found using extensive search [24]. In order to do that, the total length of the linear guides have been equally divided in 20, yielding 21 possible linear guide’s positions for each kinematic chain, di . The indexes DI and DIpen have been calculated for all possible positions of the redundant actuator on the linear guide. It is necessary to highlight that since singularities should be avoided, the possible positions must satisfy the condition (det A > 0). Finally, the workspace have been discretized in several possible positions of the end effector. The discretization of the workspace has been evaluated until the results have been converged. In order words, finer discretization would lead to similar plots. The plot of DI in the workspace is denoted as Dynamic Maps in this work. 4. Results and discussion As discussed previously, two different strategies have been exploited in order to evaluate the dynamic performance of the understudy manipulators: (i) the comparison of the required torque to perform a preselected trajectory and (ii) the comparison of the dynamic maps which allows for a more comprehensive understanding on the manipulator’s performance regardless of the chosen trajectory. These strategies are treated in the following subsections.

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Table 3 Fixed angles of each manipulator. Manipulator

ki

ci

3RRR 4RRR 6RRR PRRR + 2RRR 2PRRR + RRR 3PRRR

p/2, −p/6, 7p/6 0, p/2, p, 3p/2 0, p/3, 2p/3, p, 4p/3, 5p/3 p/2, −p/6, 7p/6 p/2, −p/6, 7p/6 p/2, −p/6, 7p/6

––– ––– ––– 0–– 0, −2p/3 – 0, −2p/3, 2p/3

In order to perform a suitable comparison, not only each link has the same geometry and inertia properties but also the geometries of manipulators have been selected in such way that the workspaces have a similar area. The length of each link is 0.2 m. The limits of the linear guide are dmin = −0.3 m and dmax = 0.3 m. The weight of the motor in the point Ai is 0.4 kg. The weights/moments of inertia of the links 1 and 2, represented by l1 and l2 in Fig. 2, are 0.2 kg/0.00667 kg • m2 . The weight/moment of inertia of the end effector is 0.4 kg/0.008 kg • m2 . The values have been selected arbitrarily. The distances ai and hi are the same for all kinematic chains and are equal to 0.2 m and 0.05 m, respectively. In order to evaluate the torques on the redundant actuators, responsible for the translational movements on the linear guides, the factors of T1 = T2 = T3 = 0.1 have been employed (see Eq. (45)). This factor accounts for the transmission path from the actuators’ torques to the linear guides’ forces. In this way, the required torques are compared hereafter. Some geometrical characteristics of the manipulators are described in the Table 3. These characteristics lead to symmetrical designs such that gi = ki + p. 4.1. Maximum required torque In this subsection, the required torques to perform the preselected trajectory are exploited for the understudy six manipulators. Fig. 4 depicts the calculated torques for the selected trajectory with no rotation modification. The maximum values of

Fig. 4. Required torques when no rotation of the end effector is requested: (a) 3RRR, (b) 4RRR, (c) 6RRR, (d) PRRR + 2RRR, (e) 2PRRR + RRR, (f) 3PRRR, (g) PRRR + 2RRR, (h) 2PRRR + RRR and (i) 3PRRR.

160

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0,02

0,0175

0,0171

0,0162

0,015

0,0158

0,0129

0,0117

0,0109 0,0105 0,01 0,005 0 (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Peak of torque

Fig. 5. Maximum required torques to perform the task when no rotation of the end effector is requested: (a) 3RRR, (b) 4RRR, (c) 6RRR, (d) PRRR + 2RRR, (e) 2PRRR + RRR, (f) 3PRRR, (g) PRRR + 2RRR, (h) 2PRRR + RRR and (i) 3PRRR.

these quantities are shown in Fig. 5. The same plots are depicted in Figs. 6 and 7 considering the preselected trajectory considering rotation of the end effector. In these figures, (a) refers to the non-redundant manipulator 3RRR; (b) and (c) refer to the manipulators with actuation redundancy 4RRR and 6RRR, respectively; (d), (e), and (f) refer to the kinematically redundant manipulators PRRR + 2RRR, 2PRRR + RRR and 3PRRR, respectively, using the prepositioning approach; and (g), (h), and (i) refer to the kinematically redundant manipulators PRRR + 2RRR, 2PRRR + RRR and 3PRRR, respectively, using the ongoing positioning approach. From Figs. 4 and 5, one can conclude that the addition of actuation or kinematic redundancies may promote a reduction on the required torques to perform the preselected trajectory when no rotation of the end effector is imposed.

Fig. 6. Required torques considering rotation on the end effector is requested: (a) 3RRR, (b) 4RRR, (c) 6RRR, (d) PRRR + 2RRR, (e) 2PRRR + RRR, (f) 3PRRR, (g) PRRR + 2RRR, (h) 2PRRR + RRR and (i) 3PRRR.

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0,04 0,035

0,0344 0,0315 0,0314

Torque (N.m)

0,03

0,0305

0,0263

0,025

0,0262 0,0232

0,0208

0,0228

0,02 0,015 0,01 0,005 0 (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Peak of torque

Fig. 7. Maximum required torques considering rotation on the end effector is requested: (a) 3RRR, (b) 4RRR, (c) 6RRR, (d) PRRR + 2RRR, (e) 2PRRR + RRR, (f) 3PRRR, (g) PRRR + 2RRR, (h) 2PRRR + RRR and (i) 3PRRR.

Regarding the manipulators with actuation redundancy, the 4RRR and the 6RRR have similar performances when no rotation of the end effector is considered. These manipulators required almost 50.0% less peak of torque than the non-redundant manipulator. This result may indicate that adding more than one level of actuation redundancy does not promote any improvement on the dynamic performance of the manipulator. Comparing the kinematically redundant manipulators, one can conclude that the manipulator 3PRRR presents the best performance when the ongoing positioning approach is executed when no rotation of the end effector is considered. The reduction of peak torque reaches 43.2% in this case. One can see that the ongoing positioning approach presents better results than the prepositioning approach. Nevertheless, both approaches present reasonable results (see Figs. 4 and 5). The results are quite different when rotation on the end effector is considered, as illustrated in Figs. 6 and 7. In this case, when compared to the results of the non-redundant manipulators, the reduction of the peak of torque reaches 23.5% for a single level of actuation redundancy and 39.5% for three levels of redundancies. In this way, one may conclude that extra levels of actuation redundancy can indeed promote significant dynamic performance improvement. Regarding the kinematically redundant 3PRRR manipulator, the prepositioning and ongoing positioning approaches promote a reduction of the peak of torque of 31% and 41%, respectively. From Figs. 6 and 7, one can conclude that the addition of actuation or kinematic redundancies may promote a reduction on the required torques to perform the preselected trajectory when rotation of the end effector is requested. However, this improvement is not so evident as previously verified (no rotation). In a general way, it can be verified that if redundancies are exploited, smaller and cheaper actuators can be selected. Nevertheless, this conclusion is limited to the dynamic performance related to the preselected trajectory and it is not necessary related to the rotation of the end effector. In the next subsection, the Dynamic Maps are presented in order to improve the understanding of the dynamic performance of manipulators with redundancies. 4.2. Dynamic Maps The previous analyses are highly case-dependent and the outcome may not be valid for the entire workspace. In this subsection, the dynamic index, proposed in Section 3.2, is depicted on the workspace enabling the designer to assess the manipulator’s dynamic performance over the entire workspace. These plots, denoted as Dynamic Maps, are found considering the angular position of the end effector equal to zero. Similar analyses can be performed for different angular positions. In these figures, the workspace is shown as a limiting blue line. Moreover, values higher than 0.8 are considered unacceptable since they might

(b)

(a)

(c)

DI (N.m)

(m)

(m)

(m)

0.25

0 (m)

(m)

(m)

Fig. 8. Dynamic Maps of the Weighting PRRR + 2RRR considering: (a) W = [100]T , (b) W = [010]T and (c) W = [001]T .

162

(a)

(b)

(c)

(f )

(m)

(m)

(e)

DI (N.m)

(m)

(m)

(m)

(d)

(m)

(m)

(m)

(m)

0 (m)

(m)

(m)

Fig. 9. Dynamic Maps (DI): (a) 3RRR, (b) 4RRR, (c) 6RRR, (d) PRRR + 2RRR, (e) 2PRRR + RRR and (f) 3PRRR.

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163

(m)

(a)

(m)

(b)

(m)

(m)

(c)

(m)

(m)

(e)

(f) (m)

(m)

(m)

(d)

(m)

(m)

-0.3

δ

(m)

0.3

(m)

Fig. 10. Optimal redundant actuators’ positions of the linear guides di,opt : (a) PRRR + 2RRR — linear guide 1, (b) 2PRRR + RRR — linear guide 1, (c) 2PRRR + RRR — linear guide 2, (d) 3PRRR — linear guide 1, (e) 3PRRR — linear guide 2 and (f) 3PRRR — linear guide 3.

(m)

(m)

(m)

represent a singularity region or regions that cannot be reached by the limitation imposed on the pose of the end effector. Alternatively, one can find these regions by evaluating the sign of the Jacobian determinant. These regions are in red in the Dynamic Maps. The rest of the Dynamic Map is depicted using contour plot. In order to select reasonable weighting values, W = [Wx Wy Walpha ]T , a first study on their impact on the Dynamic Maps has been performed using the kinematically redundant manipulator PRRR + 2RRR. Fig. 8 shows the Dynamic Maps for: (a) W = [100]T , (a) W = [010]T and (c) W = [001]T . Higher values of DI are found when translational DoFs are evaluated. In this way, one can conclude that if the designer selects W = [111]T , more importance is given to the translational DoFs. This is the choice made in this section. Similar choices are made when multi-objective optimization problems are solved using a weighted objective function [24]. Fig. 9 depicts the Dynamic Maps of the understudied manipulators using the DI proposed by Eq. (56), considering W = [111]T . Fig. 9a presents the Dynamic Map of the non-redundant manipulator 3RRR while Fig. 9b and 9c shows the Dynamic Maps of the manipulators 4RRR and 6RRR, respectively. These maps show that adding extra actuation redundancies to a manipulator can improve the dynamic performance throughout the workspace. However, one can realize that higher values of DI can be

(m)

(m)

Fig. 11. Two optimal configurations of the PRRR + 2RRR.

(m)

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(m)

(a)

(m)

(b) (m)

(m)

(c)

(m)

(m)

(d)

(f) (m)

(m)

(m)

(e)

(m)

(m)

-0.3

δ

(m)

0.3

(m)

Fig. 12. Optimal redundant actuators’ positions of the linear guides considering the proposed penalty term: (a) PRRR + 2RRR — linear guide 1, (b) 2PRRR + RRR — linear guide 1, (c) 2PRRR + RRR — linear guide 2, (d) 3PRRR — linear guide 1, (e) 3PRRR — linear guide 2 and (f) 3PRRR — linear guide 3.

found near to the position of the actuators, limiting the workspace. This assessment is in line with what has been discussed previously in [9–12]. Regarding the manipulators with kinematic redundancy, one can recognize that the inclusion of a singular redundancy may not guarantee the elimination of singularity regions according to Fig. 9d (Dynamic Map of the manipulator PRRR + 2RRR). However, the inclusion of two or three kinematic redundancy not only has been capable of reducing the singularity regions but also has presented a notable improvement of the dynamic performance according to Fig. 9e and 9f (Dynamic Maps of manipulators 2PRRR + RRR and 3PRRR, respectively). During the calculation of the DI, optimal linear guide’s positions have been calculated di,opt . Fig. 10a shows the optimal position of the linear guide of the manipulator PRRR + 2RRR. Fig. 10b and c depicts the optimal positions of the two linear guides of the manipulator 2PRRR + RRR. Finally, Fig. 10d, e and f illustrates the optimal positions of the three linear guides of the manipulator 3PRRR. This figure shows that the redundancies should be fully exploited in order to improve the manipulator’s dynamic performance since the position of the redundant actuators, di,opt , should be modified according to the position of the end

(a)

(b)

(c)

DI (N.m)

(m)

(m)

(m)

0.8

0 (m)

(m)

(m)

Fig. 13. Dynamic Maps (DI) according to the optimal linear guides’ position considering the penalty term: (a) PRRR + 2RRR, (b) 2PRRR + RRR and (c) 3PRRR.

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effector. Their values may vary from −0.3 m to 0.3 m. Abrupt modifications on the optimal position of the redundant actuator may cause discontinuities in these figures. In order to illustrate this issue, Fig. 11 shows two optimal configurations of the kinematically redundant manipulator PRRR + 2RRR. One can realize that both configurations are completely different from each other in spite of being calculated for rather close end effector’s poses. This issue may impose non-realistic movements to the manipulators. To overcome this important matter, a penalty term has been added to DI penalizing large modifications on the position of the redundant actuators on the linear guides, di,opt . The proposed dynamic index DIpen is described in Eq. (57). Fig. 12 shows these optimal positions on the workspace using the DIpen . The proposed penalty term has been capable of avoiding such discontinuities. This procedure is illustrated in Fig. 3. For sake of comparison, Fig. 13 illustrates the Dynamic Maps using the DI for the optimal positions of the redundant actuators on the linear guides calculated using DIpen . As it can be verified, no expressive loss on the dynamic performance of the manipulators has been caused by imposing the penalty factor. 5. Conclusions In this manuscript, the impact of kinematic and actuation redundancies on the dynamic performance of planar parallel kinematic manipulators were assessed. Aiming this, two strategies have been considered: (i) the comparison of the required torque to perform a preselected trajectory and (ii) the Dynamic Maps which allow for a more comprehensive understanding on the manipulator’s performance regardless of the chosen trajectory. Based on the strategy (i), one can conclude that a single level of actuation redundancy may present limited performance compared to the addition of extra levels of actuation redundancies when variations on the end effector’s rotation are imposed. These variations may lead to larger singularity regions and extra redundancies levels can be able to avoid them properly. Moreover, it can be concluded that how kinematic redundancies are treated may have an important impact on the performance of the manipulator. In general, the ongoing positioning strategy has shown to be superior to the prepositioning approach. Moreover, this evaluation strategy may lead to the misleading conclusion that manipulators with actuation redundancies present better dynamic performance that the kinematically redundant manipulators. Based on the strategy (ii), one can conclude that kinematically redundant manipulators have better dynamic capacities when compared to the non-redundant and redundant actuated manipulators. This important conclusion is based on the fact that this kind of redundancy promotes an enlargement of the usable workspace with an improved dynamic performance as indicated by the Dynamic Maps. However, the kinematic redundancy should be carefully exploited since the movement of the linear guides must not present abrupt modifications. The manipulators with actuation redundancies suffered from reduced workspace since more chains draw the kinematics constrains of the manipulator. The fully exploitation of the kinematically redundant manipulators depends on how the redundancies are treated. In this way, more comprehensive strategies should be proposed overcoming the limitation of imposing parametric movements on the linear guides. Moreover, some of the results shown in this manuscript are highly dependent on weights and penalty terms. One may argue that the use of such weights to assure a single performance index for linear and rotational motions may not be appropriate due to unit conversion and homogenization issues. The fully understanding of such terms should be further exploited. Nevertheless, using the proposed performance index, the Dynamic Maps, the designer can assess the dynamic performance of the system regarding linear and rotational motions separately, using the appropriate weights. Finally, strategies proposed in the manuscript clearly are configuration dependent metrics. A Global Dynamic Index capable of rate a manipulator regardless its configuration is still to be proposed. When dealing with redundant manipulators, the designer should deal with the compromise of deciding on the number of redundant actuators and their size. It seems that redundant manipulators may require smaller actuators. In this way, the initial cost of the product may be higher, but its energy consumption may be smaller. This topic is still an open research field and should be investigate carefully. Acknowledgments This research is supported by FAPESP2014/01809-0 and FP7-EMVeM (Energy Efficiency Management for Vehicles and Machines) (EMVeM FP7 ITN GA# 315967). Moreover, João V. de C. Fontes is thankful for his grant, CNPq140561/2015-3. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2016.05.004.

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