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Singularity Robust Inverse Dynamics of Parallel Manipulators

Written By

S. Kemal Ider

Published: April 1st, 2008

DOI: 10.5772/5378

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1. Introduction

Parallel manipulators have received wide attention in recent years. Their parallel structures offer better load carrying capacity and more precise positioning capability of the end-effector compared to open chain manipulators. In addition, since the actuators can be placed closer to the base or on the base itself the structure can be built lightweight leading to faster systems (Gunawardana & Ghorbel, 1997; Merlet, 1999; Gao et al., 2002 ).

It is known that at kinematic singular positions of serial manipulators and parallel manipulators, arbitrarily assigned end-effector motion cannot in general be reached by the manipulator and consequently at those configurations the manipulator loses one or more degrees of freedom. In addition, the closed loop structure of parallel manipulators gives rise to another type of degeneracy, which can be called drive singularity, where the actuators cannot influence the end-effector accelerations instantaneously in certain directions and the actuators lose the control of one or more degrees of freedom. The necessary actuator forces become unboundedly large unless consistency of the dynamic equations are guaranteed by the specified trajectory.

The previous studies related to the drive singularities mostly aim at finding only the locations of the singular positions for the purpose of avoiding them in the motion planning stage (Sefrioui & Gosselin, 1995; Daniali et al, 1995; Alici, 2000; Ji, 2003; DiGregorio, 2001; St-Onge & Gosselin, 2000). However unlike the kinematic singularities that occur at workspace boundaries, drive singularities occur inside the workspace and avoiding them limits the motion in the workspace. Therefore, methods by which the manipulator can move through the drive singular positions in a stable fashion are necessary.

This chapter deals with developing a methodology for the inverse dynamics of parallel manipulators in the presence of drive singularities. To this end, the conditions that should be satisfied for the consistency of the dynamic equations at the singular positions are derived. For the trajectory of the end-effector to be realizable by the actuators it should be designed to satisfy the consistency conditions. Furthermore, for finding the appropriate actuator forces when drive singularities take place, the dynamic equations are modified by using higher order derivative information. The linearly dependent equations are replaced by the modified equations in the neighborhoods of the singularities. Since the locations of the drive singularities and the corresponding modified equations are known (as derived in Section 3), in a practical scenario the actuator forces are found using the modified equations in the vicinity of the singular positions and using the regular inverse dynamic equations elsewhere. Deployment motions of 2 and 3 dof planar manipulators are analyzed to illustrate the proposed approach (Ider, 2004; Ider, 2005).

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2. Inverse dynamics and singular positions

Consider an n degree of freedom parallel robot. Let the system be converted into an open-tree structure by disconnecting a sufficient number of unactuated joints. Let the degree of freedom of the open-tree system be m, i.e. the number of the independent loop closure constraints in the parallel manipulator be m-n. Let η=[η1,...,ηm]T denote the joint variables of the open-tree system and q=[q1,...,qn]T the joint variables of the actuated joints. The m-n loop closure equations, obtained by reconnecting the disconnected joints, can be written as

ϕi(η1,...,ηm)=0           i=1,...,m-nE1

and can be expressed at velocity level as

ΓijGη˙j=0           i=1,...,m-n           j=1,...,mE2

whereΓijG=ϕiηj. A repeated subscript index in a term implies summation over its range. The prescribed end-effector Cartesian variables xi(t), i=1,...,n represent the tasks of the non-redundant manipulator. The relations between the joint variables due to the tasks are

fi(η1,...,ηm)=xi           i=1,...,nE3

Equation (3) can be written at velocity level as

ΓijPη˙j=x˙i           i=1,...,n           j=1,...,mE4

whereΓijP=fiηj. Equations (2) and (4) can be written in combined form,

Γη˙=hE5

where ΓT=[ΓGTΓPT] which is an m×m matrix andhT=[0x˙T]. The derivative of equation (5) gives the acceleration level relations,

Γη¨=Γ˙η˙+h˙E6

The dynamic equations of the parallel manipulator can be written as

Mη¨ΓGTλZTT=RE7

where M is the m×m generalized mass matrix and R is the vector of the generalized Coriolis, centrifugal and gravity forces of the open-tree system, λis the (m-n)×1 vector of the joint forces at the loop closure joints, T is the n×1 vector of the actuator forces, and each row of Z is the direction of one actuator force in the generalized space. If the variable of the joint which is actuated by the i th actuator isηk, then for the i th row of Z, Zik=1and Zij=0 for j=1,...,m(jk).

Combining the terms involving the unknown forces λ and T, one can write equation (7) as

ATτ=Mη¨RE8

where the m×m matrix AT and the m×1 vector τ are

AT=[ΓGTZT]E9
and
τT=[λTTT]E10

The inverse dynamic solution of the system involves first findingη¨, η˙and η from the kinematic equations and then finding τ (and hence T) from equation (8).

For the prescribed x(t), η¨can be found from equation (6), η˙from equation (5) and η can be found either from the position equations (1,3) or by numerical integration. However during the inverse kinematic solution, singularities occur when|Γ|=0. At these configurations, the assigned x˙ cannot in general be reached by the manipulator since, in equation (3), a vector h lying outside the space spanned by the columns of Γ cannot be produced and consequently the manipulator loses one or more degrees of freedom.

Singularities may also occur while solving for the actuator forces in the dynamic equation (8), when|A|=0. For each different set of actuators, Z hence the singular positions are different. Because this type of singularity is associated with the locations of the actuators, it is called drive singularity (or actuation singularity). At a drive singularity the assigned η¨ cannot in general be realized by the actuators since, in equation (8), a right hand side vector lying outside the space spanned by the columns of AT cannot be produced, i.e. the actuators cannot influence the end-effector accelerations instantaneously in certain directions and the actuators lose the control of one or more degrees of freedom. (The system cannot resist forces or moments in certain directions even if all actuators are locked.) The actuator forces become unboundedly large unless consistency of the dynamic equations are guaranteed by the specified trajectory.

Let ΓGu be the (m-n)×(m-n) matrix which is composed of the columns of ΓG that correspond to the variables of the unactuated joints. Since Zik=1 and Zij=0 forjk, the drive singularity condition |A|=0 can be equivalently written as|ΓGu|=0.

In the literature the singular positions of parallel manipulators are mostly determined using the kinematic expression between q˙ and x˙ which is obtained by eliminating the variables of the unactuated joints (Sefrioui & Gosselin, 1995; Daniali et al, 1995; Alici, 2000; Ji, 2003; DiGregorio, 2001; St-Onge & Gosselin, 2000),

Jq˙+Kx˙=0E11

References (Sefrioui & Gosselin, 1995 ; Daniali et al, 1995; Ji, 2003) name the condition |J|=0 as “Type I singularity” and the condition |K|=0 “Type II singularity”. And in reference (DiGregorio, 2001) they are called “inverse problem singularity” and “direct problem singularity”, respectively. Since it shows the lost Cartesian degrees of freedom, the condition |Γ|=0 shown above corresponds to|J|=0. For the drive singularity, equation (2) can be written as

ΓGuη˙u=ΓGaq˙E12

where ηu is the vector of the joint variables of the unactuated joints and ΓGa is the matrix composed of the columns of ΓG associated with the actuated joints. Since after finding η˙u from eqn (12) one can find h and hence x˙ from eqn (5) directly, the drive singularity condition |A|=0 (i.e.|ΓGu|=0) given above is equivalent to|K|=0. It should be noted that the identification of the singular configurations as shown here is easier since elimination of the variables of the passive joints is not necessary.

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3. Consistency conditions and modified equations

At the motion planning stage one usually tries to avoid singular positions. This is not difficult as far as inverse kinematic singularities are concerned because they usually occur at the workspace boundaries (DiGregorio, 2001). In this paper it is assumed that Γ always has full rank, i.e. the desired motion is chosen such that the system never comes to an inverse kinematic singular position. On the other hand, drive singularities usually occur inside the workspace and avoiding them restricts the functional workspace. It is therefore important to devise techniques for passing through the singular positions while the stability of the control forces is maintained. To this end, equation (8) must be made consistent at the singular position. In other words, since the rows of AT become linearly dependent, the same relation must also be present between the rows of the right hand side vector (Mη¨R), so that it lies in the vector space spanned by the columns ofAT.

3.1. Consistency conditions and modified equations when rank(A) becomes m-1

At a drive singularity, usually rank of A becomes m-1. Let at the singular position the s th row of AT become a linear combination of the other rows ofAT.

AsjT=αpApjT           p=1,...,m  (ps),     j=1,...,mE13

where αp are the linear combination coefficients (which may depend also onηi). Notice that only those rows of AT which are associated with the unactuated joints can become linearly dependent, hence αp corresponding to the actuated joints are zero. Then for the rows of equation (6) one must have

AsjTτjαpApjTτj=Msjη¨jRsαp(Mpjη¨jRp)E14

Substitution of equation (13) into equation (14) yields

Msjη¨jRs=αp(Mpjη¨jRp)E15

Equation (15) represents the consistency condition that η¨j should satisfy at the singular position. Since η¨j are obtained from the inverse kinematic equations (6), the trajectory x¨ must be planned in such a way to satisfy equation (15) at the drive singularity. Otherwise an inconsistent trajectory cannot be realized and the actuator forces grow without bounds as the drive singularity is approached. Time derivative of equation (14) is(AsjTαpApjT)τ˙j+(A˙sjTαpA˙pjTα˙pApjT)τj=(MsjαpMpj)ηj

+(M˙sjαpM˙pjα˙pMpj)η¨jR˙s+αpR˙p+α˙pRpE16

Now, because equation (13) holds at the singular position, there exists a neighborhood in which the first term in equation (16) is negligible compared to the other terms. Therefore in that neighborhood this term can be dropped to yield

(A˙sjTαpA˙pjTα˙pApjT)τj=(MsjαpMpj)ηj+(M˙sjαpM˙pjα˙pMpj)η¨jR˙s+αpR˙p+α˙pRpE17

Equation (17) is the modified equation that can be used to replace the s th row of equation (8) or any other equation in the linearly dependent set.

3.2. Consistency conditions and modified equations when rank(A) becomes r<m

In the general case where the rank of AT becomes r<m at the singular position, let rowssk, k=1,...,m-rof AT become linear combinations of the other r rows ofAT,

AskjT=αkpApjT           p=1,...,m  (psk)  j=1,...,m     k=1,...,m-rE18

where αkp are the linear combination coefficients. Then the following relations must be present among the rows of equation (8)

AskjTτjαkpApjTτj=Mskjη¨jRskαkp(Mpjη¨jRp)           k=1,...,m-rE19

The consistency relations are obtained as below

Mskjη¨jRsk=αkp(Mpjη¨jRp)           k=1,...,m-rE20

Substitution of equation (18) into the derivative of equation (19) yields the modified equations,

(A˙skjTαkpA˙pjTα˙kpApjT)τj=(MskjαkpMpj)ηj+(M˙skjαkpM˙pjα˙kpMpj)η¨j
R˙sk+αkpR˙p+α˙kpRp           k=1,...,m-rE21

3.3. Inverse dynamics algorithm in the presence of drive singularities

When the linearly dependent dynamic equations in equation (8) are replaced by the modified equations, equation (8) takes the following form, which is valid in the vicinity of the singular configurations.

DTτ=SE22

where in the case the s th row of AT becomes a linear combination of the other rows,

DijT={AijTA˙ijTαpA˙pjTαpA˙pjTisi=sE23
and
Si={Mijη˙jRi(MijαpMpj)ηj+(M˙ijαpM˙pjα˙pMpj)η¨jR˙i+αpR˙p+α˙pRpisi=sE24

In the general case when the rank of AT becomes r, DTand S take the following form.

DijT={AijTA˙ijTαkpA˙pjTα˙kpApjTisk,k=1,...,mri=sk,k=1,...,mrE25
and
Si={Mijη˙jRi(MijαkpMpj)ηj+(M˙ijαkpM˙pjα˙kpMpj)η¨jR˙i+αkpR˙p+α˙kpRpisk,k=1,...,mri=sk,k=1,...,mrE26

Notice that η in the modified equation should be found from the derivative of equation (6),

Γη=2Γ˙η¨Γ¨η˙+h¨E27
ηobtained from equation (27) corresponds to the prescribed end-effector jerks x (inh¨). Also the coefficients of the forces in the modified equations (17,21) depend on velocities. Therefore, if at the singularity the system is in motion, then by the modified equations the driving forces affect the end-effector jerk instantaneously in the singular directions.

The inverse dynamics algorithm in the presence of drive singularities is given below.

  1. Find the loci of the positions where the actuation singularities occur and find the linear dependency coefficients associated with the singular positions.

  2. If the assigned path of the end-effector passes through singular positions, design the trajectory so as to satisfy the consistency conditions at the singular positions.

  3. Set timet=0.

  4. Calculateη, η˙and η¨ from kinematic equations.

  5. If the manipulator is in the vicinity of a singular position, i.e. |g(η1,...,ηm)|<εwhere g(η1,...,ηm)=0 is the singularity condition and ε is a specified small number, calculate η from eqn (27) and then find τ (hence T) from equation (22).

  6. If the manipulator is not in the vicinity of a singular position, i.e.|g(η1,...,ηm)|>ε, find τ (hence T) from equation (8).

  7. Sett=t+Δt. If the final time is reached, stop. Otherwise continue from step 3.

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4. Case studies

4.1. Two degree of freedom 2-RRR planar parallel manipulator

The planar parallel manipulator shown in Figure 1 has 2 degrees of freedom (n=2). Considering disconnection of the revolute joint at P, the joint variable vector of the open-chain system isη=[θ1θ2θ3θ4]T. The joints at A and C are actuated, i.e.q=[θ1θ2]T. The end point P is desired to make a deployment motion s(t) along a straight line whose angle with x-axis isγ=330o, starting from initial positionxPo=0.431m,yPo=1.385m. The time of the motion is T=1s and its length is L=2.3min the positive s sense.

Figure 1.

Two degree of freedom 2-RRR planar parallel manipulator.

The moving links are uniform bars. The fixed dimensions are labelled asro=AC, r1=AB, r2=CD, r3=BPandr4=DP. The numerical data arero=1.75m, r1=r2=r3=r4=1.4m, m1=m2=6kgandm3=m4=4kg.

The loop closure constraint equations at velocity level are ΓGη˙=0 where

ΓG=[r1s1r3s13r2s2+r4s24r3s13r4s24r1c1+r3c13r2c2r4c24r3c13r4c24]E28
Heresi=sinθi, ci=cosθi, sij=sin(θi+θj),cij=cos(θi+θj). The prescribed Cartesian motion of the end point P, x can be written as
x=[xP(t)yP(t)]=[xPo+s(t)sinγyPo+s(t)cosγ]E29

Then the task equations at velocity level areΓPη˙=x˙, where

ΓP=[r1s1r3s130r3s130r1c1+r3c130r3c130]E30

The mass matrix M and the vector of the Coriolis, centrifugal and gravitational forces R are

M=[M110M1300M220M24M130M3300M240M44]E31

where

M11=m1r123+m3(r12+r323+r1r3c3), M13=m3(r323+r1r3c32), M33=m3r323
M22=m2r223+m4(r22+r423+r2r4c4),  M24=m4(r423+r2r4c42),  M44=m4r423E32
and
R=[R1R2R3R4]=[m3r1r3s3θ˙3(θ˙112θ˙3)+12m1gr1c1+m3g(r1c1+12r3c13)12m3r1r3s3θ˙12+12m3gr3c13m4r2r4s4θ˙4(θ˙212θ˙4)+12m2gr2c2+m4g(r2c2+12r4c24)12m4r2r4s4θ˙22+12m4gr4c24]E33

Since the variables of the actuated joints are θ1 andθ2, the matrix Z composed of the actuator direction vectors is

Z=[10000100]E34

Then the coefficient matrix of the constraint and actuator forces, ATis

AT=[r1s1r3s13r1c1+r3c1310r2s2+r4s24r2c2r4c2401r3s13r3c1300r4s24r4c2400]E35

The drive singularities are found from |A|=0 assin(θ1+θ3θ2θ4)=0, i.e. as the positions when points A, B and D become collinear. Hence, drive singularities occur inside the workspace and avoiding them limits the motion in the workspace. Defining a path for the operational point P which does not involve a singular position would restrict the motion to a portion of the workspace where point D remains on one side of the line joining A and D. In fact, in order to reach the rest of the workspace (corresponding to the other closure of the closed chain system) the manipulator has to pass through a singular position.

When the end point comes tos=Ld=0.80m, θ1+θ3becomes equal toπ+θ2+θ4, hence a drive singularity occurs. At this position the third row of AT becomes r3/r4 times the fourth row. Then, for consistency of equation (8), the third row of the right hand side of equation (8) should also be r3/r4 times the fourth row. The resulting consistency condition that the generalized accelerations must satisfy is obtained from equation (15) as

M31θ¨1r3r4M24θ¨2+M33θ¨3r3r4M44θ¨4=R3r3r4R4E36

Hence the time trajectory s(t) of the deployment motion should be selected such that at the drive singularity the generalized accelerations satisfy equation (36).

An arbitrary trajectory that does not satisfy the consistency condition is not realizable. This is illustrated by considering an arbitrary third order polynomial for s(t) having zero initial and final velocities, i.e.s(t)=3Lt2T22Lt3T3. The singularity position is reached whent=0.48s. The actuator torques are shown in Figure 2. The torques grow without bounds as the singularity is approached and become infinitely large at the singular position. (In Figure 2 the torques are out of range around the singular position.)

For the time function s(t), a polynomial is chosen which satisfies the consistency condition at the drive singularity in addition to having zero initial and final velocities. The time Td when the singular position is reached and the velocity of the end point P atTd, vP(Td)can be arbitrarily chosen. The loop closure relations, the specified angle of the acceleration of P and the consistency condition constitute four independent equations for a unique solution of θ¨i,i=1,...,4 at the singular position. Hence, using θi and θ˙i atTd, the acceleration of P atTd, aP(Td)is uniquely determined. Consequently a sixth order polynomial is selected where s(0)=0, s˙(0)=0, s(T)=L, s˙(T)=0, s(Td)=Ld, s˙(Td)=vP(Td) ands¨(Td)=aP(Td). Tdand vP(Td) are chosen by trial and error to prevent any overshoot in s ors˙. The values used are Td=0.55s andvP(Td)=3.0m/s, yieldingaP(Td)=18.2m/s2. s(t) so obtained is given by equation (37) and shown in Figure 3.

s(t)=30.496t2154.909t3+311.148t4265.753t5+81.318t6E37

Figure 2.

Motor torques for the trajectory not satisfying the consistency condition: 1.T1, 2.T2

Furthermore, even when the consistency condition is satisfied, ATis ill-conditioned in the vicinity of the singular position, hence τ cannot be found correctly from equation (8). Deletion of a linearly dependent equation in that neighborhood would cause task violations due to the removal of a task. For this reason the modified equation (17) is used to replace the dependent equation in the neighborhood of the singular position. The modified equation, which relates the actuator forces to the system jerks, takes the following form.(A˙31Tr3r4A˙41T)τ1+(A˙32Tr3r4A˙42T)τ2=M31θ1r3r4M24θ2+M33θ3r3r4M44θ4

+M˙31θ¨1r3r4M˙24θ¨2+M˙33θ¨3r3r4M˙44θ¨4R˙3+r3r4R˙4E38

The coefficients of the constraint forces in eqn (38) are

A˙31Tr3r4A˙41T=r3(θ˙1+θ˙3)c13r3(θ˙2+θ˙4)c24E39a
A˙32Tr3r4A˙42T=r3(θ˙1+θ˙3)s13r3(θ˙2+θ˙4)s24E39b

which in general do not vanish at the singular position if the system is in motion.

Once the trajectory is chosen as above such that it renders the dynamic equations to be consistent at the singular position, the correspondingθi, θ˙iand θ¨i are obtained from inverse kinematics, and when there is no actuation singularity, the actuator torques T1 and T2 (along with the constraint forces λ1andλ2) are obtained from equation (8). However in the neighborhood of the singular position, equation (22) is used in which the third row of equation (8) is replaced by the modified equation (38). The neighborhood of the singularity where equation (22) is utilized is taken as|θ1+θ3θ2θ4180o|<ε=1o. The motor torques necessary to realize the task are shown in Figure 4. At the singular position the motor torques are found as T1=138.07Nm andT2=30.66Nm. To test the validity of the modified equations, when the simulations are repeated with ε=0.5o andε=1.5o, no significant changes occur and the task violations remain less than 104m.

Figure 3.

Time function satisfying the consistency condition.

4.2. Three degree of freedom 2-RPR planar parallel manipulator

The 2-RPR manipulator shown in Figure 5 has 3 degrees of freedom (n=3). Choosing the revolute joint at D for disconnection (among the passive joints) the joint variable vector of the open chain system isη=[θ1ζ1θ2ζ2θ3]T, where ζ1=AB andζ2=CD. The link dimensions of the manipulator are labelled asa=AC, b=BD, c=BPandα=PBD. The position and orientation of the moving platform is x=[xPyPθ3]T wherexP, yPare the coordinates of the operational point of interest P in the moving platform.

The velocity level loop closure constraint equations areΓGη˙=0, where

ΓG=[ζ1sinθ1cosθ1ζ2sinθ2cosθ2bsinθ3ζ1cosθ1sinθ1ζ2cosθ2sinθ2bcosθ3]E40

The prescribed position and orientation of the moving platform, x(t)represent the tasks of the manipulator. The task equations at velocity level are ΓPη˙=x˙ where

ΓP=[ζ1sinθ1cosθ100csin(θ3+α)ζ1cosθ1sinθ100ccos(θ3+α)00001]E41

Figure 4.

Motor torques for the trajectory satisfying the consistency condition: 1.T1, 2.T2.

Figure 5.

RPR planar parallel manipulator.

Let the joints whose variables are θ1,ζ1andζ2 be the actuated joints. The actuator force vector can be written as T=[T1F1F2]T where T1 is the motor torque corresponding toθ1, and F1 and F2 are the translational actuator forces corresponding to ζ1 andζ2, respectively. Consider a deployment motion where the platform moves with a constant orientation given as θ3=320o and with point P having a trajectory s(t) along a straight line whose angle with x-axis isγ=200o, starting from initial positionxPo=0.800m, yPo=0.916m(Figure 5). The time of the deployment motion is T=1s and its length isL=1.5m. Hence the prescribed Cartesian motion of the platform can be written as

x=[xP(t)yP(t)θ3(t)]=[xPo+s(t)sinγyPo+s(t)cosγ320o]E42

The link dimensions and mass properties are arbitrarily chosen as follows. The link lengths areAC=a=1.0m, BD=b=0.4m, BP=c=0.2m,PBD=α=0. The masses and the centroidal moments of inertia arem1=2kg, m2=1.5kg, m3=2kg, m4=1.5kg, m5=1.0kg, I1=0.05kgm2, I2=0.03kgm2, I3=0.05kgm2, I4=0.03kgm2andI5=0.02kgm2. The mass center locations are given byAG1=g1=0.15m, BG2=g2=0.15m, CG3=g3=0.15m, DG4=g4=0.15m, BG5=g5=0.2mandG5BD=β=0.

The generalized mass matrix M and the generalized inertia forces involving the second order velocity terms R are

M=[M11000M150M2200M2500M3300000M440M51M5200M55],    R=[R1R2R3R4R5]E43

where Mij and Ri are given in the Appendix.

For the set of actuators considered, the actuator direction matrix Z is

Z=[100000100000010]E44

Hence, ATbecomes

AT=[ζ1sinθ1ζ1cosθ1100cosθ1sinθ1010ζ2sinθ2ζ2cosθ2000cosθ2sinθ2001bsinθ3bcosθ3000]E45

Since|A|=bζ2sin(θ2θ3), drive singularities occur when ζ2=0 orsin(θ2θ3)=0. Noting that ζ2 does not become zero in practice, the singular positions are those positions where points B, D and C become collinear.

Hence, drive singularities occur inside the workspace and avoiding them limits the motion in the workspace. Avoiding singular positions where θ2θ3=±nπ (n=0,1,2,...) would restrict the motion to a portion of the workspace where point D is always on the same side of the line BC. This means that in order to reach the rest of the workspace (corresponding to the other closure of the closed chain system) the manipulator has to pass through a singular position.

When point P comes tos=Ld=0.662m, a drive singularity occurs since θ2 becomes equal toθ3+π. At this position the third and fifth rows of AT become linearly dependent asA3jTζ2bA5jT=0,j=1,...,5. The consistency condition is obtained as below

M33θ¨2ζ2b(M51θ¨1+M52ζ¨1+M55θ¨3)=R3ζ2bR5E46

The desired trajectory should be chosen in such a way that at the singular position the generalized accelerations should satisfy the consistency condition.

If an arbitrary trajectory that does not satisfy the consistency condition is specified, then such a trajectory is not realizable. The actuator forces grow without bounds as the singular position is approached and become infinitely large at the singular position. This is illustrated by using an arbitrary third order polynomial for s(t) having zero initial and final velocities, i.e.s(t)=3Lt2T22Lt3T3. The singularity occurs whent=0.46s. The actuator forces are shown in Figures 6 and 7. (In the figures the forces are out of range around the singular position.)

Figure 6.

Motor torque for the trajectory not satisfying the consistency condition.

Figure 7.

Actuator forces for the trajectory not satisfying the consistency cond.: 1.F1, 2.F2.

For the time function s(t) a polynomial is chosen that renders the dynamic equations to be consistent at the singular position in addition to having zero initial and final velocities. The time Td when singularity occurs and the velocity of the end point whent=Td, vP(Td)can be arbitrarily chosen. The acceleration level loop closure relations, the specified angle of the acceleration of P (γ=200o), the specified angular acceleration of the platform (θ¨3=0) and the consistency condition constitute five independent equations for a unique solution of η¨i,i=1,...,5 at the singular position. Hence, using η and η˙ atTd, the acceleration of P atTd, aP(Td)is uniquely determined. Consequently a sixth order polynomial is selected where s(0)=0, s˙(0)=0, s(T)=L, s˙(T)=0, s(Td)=Ld, s˙(Td)=vP(Td) ands¨(Td)=aP(Td). The values used for Td and vP(Td) are 0.62s and 1.7m/s respectively, yieldingaP(Td)=10.6m/s2. s(t) so obtained is shown in Figure 8 and given by equation (47).

s(t)=20.733t287.818t3+146.596t4103.669t5+25.658t6E47

Bad choices for Td and vP(Td) would cause local peaks in s(t) implying back and forth motion of point P during deployment along its straight line path.

However, even when the equations are consistent, in the neighborhood of the singular positions AT is ill-conditioned, hence τ cannot be found correctly from equation (8). This problem is eliminated by utilizing the modified equation valid in the neighborhood of the singular position. The modified equation (17) takes the following form

Bjτj=Q           j=1,2E48

where

B1=A˙31Tζ2bA˙51Tζ˙2bA51T,          B2=A˙32Tζ2bA˙52Tζ˙2bA52TE49a
Q=M33θ2ζ2b(M51θ1+M52ζ1+M55θ3)+M˙33θ¨2ζ2b(M˙51θ¨1+M˙52ζ¨1
+M˙55θ¨3)ζ˙2b(M51θ¨1+M52ζ¨1+M55θ¨3)R˙3+ζ2bR˙5+ζ˙2bR5E49b

Figure 8.

A time function that satisfies the consistency condition.

Once the trajectory is specified, the correspondingη, η˙and η¨ are obtained from inverse kinematics, and when there is no actuation singularity, the actuator forcesT1, F1and F2 (and the constraint forces λ1 andλ2) are obtained from equation (8). However in the neighborhood of the singularity, A is ill-conditioned. So the unknown forces are obtained from equation (22) which is obtained by replacing the third row of equation (8) by the modified equation (48). The neighborhood of the singular position where equation (22) is utilized is taken as|θ2θ3+180o|<ε=0.5o. The motor torques and the translational actuator forces necessary to realize the task are shown in Figures 9 and 10, respectively. At the singular position the actuator forces areT1=30.31Nm, F1=26.3NandF2=1.61N. The joint displacements under the effects of the actuator forces are given in Figures 11 and 12. To test the validity of the modified equations in a larger neighborhood, when the simulations are repeated withε=1o, no significant changes are observed, the task violations remaining less than 105m.

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5. Conclusions

A general method for the inverse dynamic solution of parallel manipulators in the presence of drive singularities is developed. It is shown that at the drive singularities, the actuator forces cannot influence the end-effector accelerations instantaneously in certain directions. Hence the end-effector trajectory should be chosen to satisfy the consistency of the dynamic equations when the coefficient matrix of the drive and constraint forces, A becomes singular. The satisfaction of the consistency conditions makes the trajectory to be realizable by the actuators of the manipulator, hence avoids the divergence of the actuator forces.

Figure 9.

Motor torque for the trajectory satisfying the consistency condition.

Figure 10.

Actuator forces for the trajectory satisfying the consistency condition: 1.F1, 2.F2

To avoid the problems related to the ill-condition of the force coefficient matrix, A in the neighborhood of the drive singularities, a modification of the dynamic equations is made using higher order derivative information. Deletion of the linearly dependent equation in that neighborhood would cause task violations due to the removal of a task. For this reason the modified equation is used to replace the dependent equation yielding a full rank force coefficient matrix.

Figure 11.

Rotational joint displacements: 1.θ1, 2.θ2.

Figure 12.

Translational joint displacements: 1.ζ1, 2.ζ2.

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6. Appendix

The elements of M and R of the 2-RPR parallel manipulator shown in equation (41) are given below, wheremi, i=1,...,5are the masses of the links, Ii, i=1,...,5are the centroidal moments of inertia of the links and the locations of the mass centersGi, i=1,...,5are indicated byg1=AG1, g2=BG2, g3=CG3, g4=DG4, g5=BG5andβ=G5BD.

M11=m1g12+I1+m2(ζ1g2)2+I2+m5ζ12EA1
M15=m5ζ1g5cos(θ1θ3β)EA2
M22=m2+m3EA3
M25=m5g5sin(θ1θ3β)EA4
M33=m3g32+I3+m4(ζ2g4)2+I4EA5
M44=m4EA6
M51=m5ζ1g5cos(θ1θ3β)EA7
M52=m5g5sin(θ1θ3β)EA8
M55=m5g52+I5EA9
R1=2m2(ζ1g2)ζ˙1θ˙1+m5ζ1g5θ˙32sin(θ1θ3β)+[m1g1+m2(ζ1g2)+m5ζ1]gcosθ1EA10
R2=m5g5θ˙32cos(θ1θ3β)m2(ζ1g2)θ˙12m5ζ1θ˙12+(m2+m5)gsinθ1EA11
R3=2m4(ζ2g4)ζ˙2θ˙2+[m3g3+m4(ζ2g4)]gcosθ2EA12
R4=m4(ζ2g4)θ˙22+m4gsinθ2EA13
R5=m5g5[2ζ˙1θ˙1cos(θ1θ3β)ζ1θ˙12sin(θ1θ3β)+gcos(θ3+β)]EA14

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Written By

S. Kemal Ider

Published: April 1st, 2008