Singularity Robust Inverse Dynamics of Parallel Manipulators 381 11 313 11 313 22 424 22 424 T 313 313 424 424 10 01 00 00 rs rs rc rc rs rs rc rc rs rc rs rc −− + ⎡ ⎤ ⎢ ⎥ +−− ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ A (35) The drive singularities are found from 0 = A as 1324 sin( ) 0 θ θθθ + −− =, 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 to 0.80m d sL = = , 13 θ θ + becomes equal to 24 π θθ + + , hence a drive singularity occurs. At this position the third row of T A becomes 34 /rr times the fourth row. Then, for consistency of equation (8), the third row of the right hand side of equation (8) should also be 34 /rr times the fourth row. The resulting consistency condition that the generalized accelerations must satisfy is obtained from equation (15) as 333 31 1 24 2 33 3 44 4 3 4 444 rrr M MM MRR rrr θθθθ −+−=−     (36) 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 ()st having zero initial and final velocities, i.e. 23 23 32 () Lt Lt TT st =−. The singularity position is reached when 0.48st = . 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 d T when the singular position is reached and the velocity of the end point P at d T , () Pd vT 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 ,1, ,4 i i θ =  at the singular position. Hence, using i θ and i θ  at d T , the acceleration of P at d T , () Pd aT is uniquely determined. Consequently a sixth order polynomial is selected where (0) 0,s = (0) 0,s =  () ,sT L = () 0,sT =  () , dd sT L = () () dPd sT v T =  and () () dPd sT a T=  . d T and () Pd vT are chosen by trial and error to prevent any overshoot in s or s  . The values used are 0.55 s d T = and ()3.0m/s Pd vT = , yielding 2 ( ) 18.2 m/s Pd aT= . s(t) so obtained is given by equation (37) and shown in Figure 3. Parallel Manipulators, New Developments 382 23456 ( ) 30.496 154.909 311.148 265.753 81.318stttttt=− + − + (37) Figure 2. Motor torques for the trajectory not satisfying the consistency condition: 1. T 1 , 2. T 2 Furthermore, even when the consistency condition is satisfied, T A is 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. rr r r AA AA M MM M rr r r τ τθ θθ θ −+−=−+−       TT TT 33 3 3 31 41 1 32 42 2 31 1 24 2 33 3 44 4 44 4 4 ()() 333 31 1 24 2 33 3 44 4 3 4 444 rrr M MM MRR rrr θθθθ +− +− −+       (38) The coefficients of the constraint forces in eqn (38) are TT 3 31 41 3 1 3 13 3 2 4 24 4 ()() r AAr cr c r θθ θθ −=−+−+    (39a) TT 3 32 42 3 1 3 13 3 2 4 24 4 ()() r AAr sr s r θθ θθ −=−+−+    (39b) 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 corresponding i θ , i θ  and i θ  are obtained from inverse kinematics, and when there is no actuation singularity, the actuator torques 1 T and Singularity Robust Inverse Dynamics of Parallel Manipulators 383 2 T (along with the constraint forces 1 λ and 2 λ ) 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 1324 180 1 oo θθθθ ε + −−− <=. The motor torques necessary to realize the task are shown in Figure 4. At the singular position the motor torques are found as 1 138.07NmT = − and 2 30.66NmT = − . To test the validity of the modified equations, when the simulations are repeated with 0.5 o ε = and 1.5 o ε = , no significant changes occur and the task violations remain less than 4 10 − m. 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 is [] T 11223 θ ζθζθ =η , where 1 AB ζ = and 2 CD ζ = . The link dimensions of the manipulator are labelled as aAC = , bBD = , cBP = and PBD α =∠ . The position and orientation of the moving platform is [] T 3 PP xy θ =x where P x , P y are the coordinates of the operational point of interest P in the moving platform. The velocity level loop closure constraint equations are G = Γη 0  , where 11 122 2 3 11 122 2 3 sin cos sin cos sin cos sin cos sin cos G b b ζ θθζθ θ θ ζ θθζθ θ θ −−− ⎡ ⎤ = ⎢ ⎥ −− ⎣ ⎦ Γ (40) The prescribed position and orientation of the moving platform, ()tx represent the tasks of the manipulator. The task equations at velocity level are P = Γη x  where Parallel Manipulators, New Developments 384 11 1 3 11 1 3 sin cos 0 0 sin( ) cos sin 0 0 cos( ) 00001 P c c ζ θθ θα ζ θθ θα −−+ ⎡ ⎤ ⎢ ⎥ =+ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Γ (41) Figure 4. Motor torques for the trajectory satisfying the consistency condition: 1. T 1 , 2.T 2 . Figure 5. 2-RPR planar parallel manipulator. Let the joints whose variables are 11 2 ,and θ ζζ be the actuated joints. The actuator force vector can be written as [] T 112 TFF=T where 1 T is the motor torque corresponding to s 5 4 G 2 G 5 G 1 G 1 3 x y o P γ 1 θ 2 θ A C Singularity Robust Inverse Dynamics of Parallel Manipulators 385 1 θ , and 1 F and 2 F are the translational actuator forces corresponding to 1 ζ and 2 ζ , respectively. Consider a deployment motion where the platform moves with a constant orientation given as o 3 320 θ = and with point P having a trajectory s(t) along a straight line whose angle with x-axis is o 200 γ = , starting from initial position m0.800 o P x = , m0.916 o P y = (Figure 5). The time of the deployment motion is s1T = and its length is m1.5L = . Hence the prescribed Cartesian motion of the platform can be written as o 3 () ()sin () ()cos () 320 o o PP PP xt x st yt y st t γ γ θ ⎡ ⎤ + ⎡⎤ ⎢ ⎥ ⎢⎥ ==+ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎣⎦ ⎣ ⎦ x (42) The link dimensions and mass properties are arbitrarily chosen as follows. The link lengths are m1.0AC a== , m0.4BD b = = , m0.2BP c = = , 0PBD α ∠ ==. The masses and the centroidal moments of inertia are kg 1 2m = , kg 2 1.5m = , kg 3 2m = , kg 4 1.5m = , kg 5 1.0m = , kg m 2 1 0.05I = , kgm 2 2 0.03I = , kg m 2 3 0.05I = , kgm 2 4 0.03I = and kg m 2 5 0.02I = . The mass center locations are given by m 11 0.15AG g = = , m 22 0.15BG g = = , m 33 0.15CG g== , m 44 0.15DG g== , m 55 0.2BG g = = and 5 0GBD β ∠ ==. The generalized mass matrix M and the generalized inertia forces involving the second order velocity terms R are 11 15 22 25 33 44 51 52 55 000 000 00 00 000 0 00 MM MM M M MM M ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ M , 1 2 3 4 5 R R R R R ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ R (43) where ij M and i R are given in the Appendix. For the set of actuators considered, the actuator direction matrix Z is 10000 01000 00010 ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Z (44) Hence, T A becomes 111 1 11 T 22 2 2 22 33 sin cos 1 0 0 cos sin 0 1 0 sin cos 000 cos sin 0 0 1 sin cos 000bb ζθζθ θθ ζθ ζθ θθ θθ − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − = ⎢ ⎥ −− ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ A (45) Parallel Manipulators, New Developments 386 Since 223 sin( )b ζ θθ =−A , drive singularities occur when 2 0 ζ = or 23 sin( ) 0 θθ − = . Noting that 2 ζ 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 23 n θ θπ − =± (0,1,2, )n = 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 to m0.662 d sL = = , a drive singularity occurs since 2 θ becomes equal to 3 θ +π. At this position the third and fifth rows of T A become linearly dependent as 2 35 0 TT jj AA b ζ −= , 1, ,5j = . The consistency condition is obtained as below 22 33 2 51 1 52 1 55 3 3 5 () M MMM RR bb ζζ θθζθ −++=−     (46) 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 ()st having zero initial and final velocities, i.e. 23 23 32 () Lt Lt TT st =−. The singularity occurs when s0.46t = . 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. Singularity Robust Inverse Dynamics of Parallel Manipulators 387 Figure 7. Actuator forces for the trajectory not satisfying the consistency cond.: 1. F 1 , 2. F 2 . 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 d T when singularity occurs and the velocity of the end point when d tT = , () Pd vT can be arbitrarily chosen. The acceleration level loop closure relations, the specified angle of the acceleration of P ( o 200 γ = ), the specified angular acceleration of the platform 3 (0) θ =  and the consistency condition constitute five independent equations for a unique solution of ,1, ,5 i i η =  at the singular position. Hence, using η and η  at d T , the acceleration of P at d T , () Pd aT is uniquely determined. Consequently a sixth order polynomial is selected where (0) 0,s = (0) 0,s =  () ,sT L = () 0,sT =  () , dd sT L= () () dPd sT v T=  and () () dPd sT a T=  . The values used for d T and () Pd vT are s0.62 and ms1.7 / respectively, yielding ms 2 ( ) 10.6 / Pd aT= . s(t) so obtained is shown in Figure 8 and given by equation (47). 23 4 56 ( ) 20.733 87.818 146.596 103.669 25.658stttttt=−+ − + (47) Bad choices for d T and () Pd vT 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 T A 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 jj BQ τ = 1,2j = (48) where 22 1315151 TTT BAAA bb ζζ =− −   , 22 2325252 TTT BAAA bb ζζ =− −   (49a) Parallel Manipulators, New Developments 388 22 33 2 51 1 52 1 55 3 33 2 51 1 52 1 ()(QM MMM M MM bb ζζ θ θζθ θ θζ =− ++ +− +           2 55 3 51 1 52 1 55 3 )( )MMMM b ζ θ θζθ +− ++      22 355 RRR bb ζζ −+ +   (49b) Figure 8. A time function that satisfies the consistency condition. Once the trajectory is specified, the corresponding η , η  and η  are obtained from inverse kinematics, and when there is no actuation singularity, the actuator forces 1 T , 1 F and 2 F (and the constraint forces 1 λ and 2 λ ) 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 oo 23 180 0.5 θθ ε −+ <= . 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 are Nm 1 30.31T = , N 1 26.3F = and N 2 1.61F = . 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 with o 1 ε = , no significant changes are observed, the task violations remaining less than 5 10 − m. 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 Singularity Robust Inverse Dynamics of Parallel Manipulators 389 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. F 1 , 2. F 2 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. Parallel Manipulators, New Developments 390 Figure 11. Rotational joint displacements: 1. θ 1 , 2. θ 2 . Figure 12. Translational joint displacements: 1. ζ 1 , 2. ζ 2 . 6. References Alıcı, G. (2000). Determination of singularity contours for five-bar planar parallel manipulators, Robotica, Vol. 18, No. 5, (September 2000) 569-575. Daniali, H.R.M.; Zsombor-Murray, P.J. & Angeles, J. (1995). Singularity analysis of planar parallel manipulators, Mechanism and Machine Theory, Vol. 30, No. 5, (July 1995) 665-678. [...]... planar parallel manipulator Since both the input shaping technique and the two-time scale control scheme, applied to parallel manipulators, 406 Parallel Manipulators, New Developments can only control command inputs to joint actuators of a manipulator, their performance on vibration reduction of flexible linkage are limited This chapter introduces a methodology for the dynamic analysis of a planar parallel. .. 11 Tip-position feedback with the PC Parallel Manipulators, New Developments 21 Dynamic Modelling and Vibration Control of a Planar Parallel Manipulator with Structurally Flexible Linkages Bongsoo Kang1 and James K Mills2 Hannam University1, University of Toronto2 South Korea1 Canada2 1 Introduction A parallel manipulator provides an alternative design to serial manipulators, and can be found in many... Dynamics of Parallel Manipulators 391 Di Gregorio, R (2001) Analytic formulation of the 6-3 fully -parallel manipulator’s singularity determination, Robotica, Vol 19, No 6, (September 2001) 663-667 Gao, F.; Li, W.; Zhao, X.; Jin, Z & Zhao, H (2002) New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs, Mechanism and Machine Theory, Vol 37, No 11, (November 2002) 1395 -141 1 Gunawardana,... Nantes, France, September 1997, Cambridge University Press, New York Ider, S.K (2004) Singularity robust inverse dynamics of planar 2-RPR parallel manipulators, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol 218, No 7, (July 2004) 721-730 Ider, S.K (2005) Inverse dynamics of parallel manipulators in the presence of drive singularities, Mechanism... (2003) Study of planar three-degree-of-freedom 2-RRR parallel manipulators, Mechanism and Machine Theory, Vol 38, No 5, (May 2003) 409-416 Kong, X & Gosselin, C.M (2001) Forward displacement analysis of third-class analytic 3RPR planar parallel manipulators, Mechanism and Machine Theory, Vol 36, No 9, (September 2001) 1009-1018 Merlet, J.-P (1999) Parallel Robotics: Open Problems, Proceedings of Ninth... consecutive revolute joints In contrast to well-known RPR type parallel manipulators, actuators of the proposed PRR configuration remain stationary that results in low inertia of moving parts Workspace analysis of a planar parallel manipulator has been addressed (Gosselin et al., 1996, Heerah et al., 2002) and singularity analysis of a planar parallel manipulator has been studied (Gosselin & Angeles, 1990,... n ) 2 ⎪ − (W (n − 1) + f (n )v (n )) ⎪ 3 3 if case 4.1 α 2 ( n) = ⎨ v3 ( n) 2 ⎪ ⎪ 0 if case 1, 2, 4.2 ⎩ where each case is as follows: Case 1: energy does not flow out W (n ) ≥ 0 (4) 398 Parallel Manipulators, New Developments Case 2: energy flows out from the left port W (n ) < 0, f 2 (n )v2 (n ) < 0, f 3 (n )v3 (n ) ≥ 0 Case 3: energy flows out from the right port W (n ) < 0, f 2 (n )v2 (n ) ≥ 0,... (passive) system and a dynamics from the collocated output (joint velocity) to the noncollocated output (end-point velocity) As a result, if it is possible, and it generally is, to use the 400 Parallel Manipulators, New Developments velocity information of the actuating position, we can construct the network model (controller and passive plant) that is suitable to our framework as in Fig 8 by including the... 11a), and the PO is constrained to positive values (Fig 11c) The PC at the both side is active only when these are required, and dissipate the just amount of energy generation (Fig 11d) 402 Parallel Manipulators, New Developments 5 Conclusions In this paper, we propose a stability guaranteed control scheme of noncollocated feedback control systems without any model information The main contribution of... cos(θ1 − θ 3 − β ) (A2) M 22 = m2 + m3 (A3) M 25 = m5 g5 sin(θ 1 − θ 3 − β ) (A4) 2 M 33 = m3 g3 + I 3 + m4 (ζ 2 − g4 )2 + I 4 (A5) M 44 = m4 (A6) M 51 = m5ζ 1 g5 cos(θ1 − θ 3 − β ) (A7) 392 Parallel Manipulators, New Developments M 52 = m5 g5 sin(θ1 − θ 3 − β ) (A8) 2 M 55 = m5 g5 + I 5 (A9) 2 R1 = 2 m2 (ζ 1 − g2 )ζ 1θ1 + m5ζ 1 g5θ 3 sin(θ1 − θ 3 − β ) + [m1 g1 + m2 (ζ 1 − g2 ) + m5ζ 1 ] g cosθ 1 (A10) . obtained is given by equation (37) and shown in Figure 3. Parallel Manipulators, New Developments 382 23456 ( ) 30.496 154.909 311 .148 265.753 81.318stttttt=− + − + (37) Figure 2. Motor. of the manipulator. The task equations at velocity level are P = Γη x  where Parallel Manipulators, New Developments 384 11 1 3 11 1 3 sin cos 0 0 sin( ) cos sin 0 0 cos( ) 00001 P c c ζ θθ. cos 000bb ζθζθ θθ ζθ ζθ θθ θθ − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − = ⎢ ⎥ −− ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ A (45) Parallel Manipulators, New Developments 386 Since 223 sin( )b ζ θθ =−A , drive singularities occur when