x Condition Number Condition Number 0.5

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1 Redundant Actuation for Improving Kinematic Manipulability John F. O'Brien and John T. Wen Center for Advanced Technology in Automation, Robotics & Manufacturing Department of Electrical, Computer, & Systems Engineering Rensselaer Polytechnic Institute, Troy, NY 28 Abstract Parallel mechanisms frequently contain an unstable type of singularity that has no counterpart in serial mechanisms. When the mechanism is at or near this type of singularity, it loses the ability to counteract external forces in certain directions. By altering the mechanism through, for example, additional kinematic linkages, the singularity can be modied or even removed. Another approach is to actuate certain unactuated degrees of freedom. The manipulability is guaranteed to improve over the original mechanism, but the mechanism is now overactuated. By studying a recently proposed parallel machining center, we examine the eectiveness of singularity modication through redundant actuation. As a metric, we use the condition number of the recently developed manipulability ellipsoid for general parallel mechanisms. Introduction Parallel robots provide a sti connection between the payload and the base structure, with pose accuracies that are superior to serial chain manipulators. The principal drawbacks concerning parallel robots are their limited workspace, and the complexity of singularity analysis [4]. In contrast to serial chain manipulators, singularities in parallel mechanisms have dierent manifestations. This issue has been studied in the multi-nger grasping context in [, 3] and more recently for general parallel mechanisms in [5, 2, 7]. In [5], the singularities are separated into two broad classications: end-eector and actuator singularities. The former is comparable to the serial arm case, where the end-eector losses a degree-offreedom in the Cartesian task space. The latter is dened when a certain task wrench cannot be resisted by active joint torques. Or equivalently, the task frame can move even when all the active joints are locked. These are called the unstable congurations in [7] which correspond to unstable grasps in the multi-nger grasp literature. The unstable type of singularity isobviously unattractive, as unpredictable task motion could result. A systematic approach to determine the set of singular poses for a given parallel robot short of an exhaustive search ofitsworkspace has yet to be developed. The inability to a priori determine the entire singular pose set, coupled with the particularly catastrophic nature of singularity in parallel mechanisms, prompts the development of techniques to reduce the size of this set. In [6], a new parallel machining center is proposed. It was discovered, after the mechanism was built, that the workspace contains unstable type of singularities { severely compromising the usefulness of the mechanism. Asolu- tion proposed in [6] is to introduce additional linkages (degrees of freedom and constraints) to provide required forces at or near singular congurations. This approach, however, may compromise manipulability in other portions of the workspace. Another approach is to actuate some unactuated degrees of freedom in the original mechanism. This paper considers both approaches and their eectiveness on the mechanism described in [6]. We use the manipulability ellipsoid concept developed in [7] and consider the eect of redundant actuation on the condition number of the ellipsoid. Terminology and Notation: We shall use the term \spatial velocity" at a given frame to mean where! is the angular velocity of the frame and v is the linear velocity of the origin of the frame. Given a matrix G, we use G e to either denote the annihilator of G ( GG e = ) or the transpose of the annihilator of G T (GG e = ). The distinction between the two cases will be clear from the context. 2 Dierential Kinematics This section considers the dierential kinematics of general rigid multibody systems. Consider a general mechanism subject to kinematic constraints. The generalized coordinate (with the constraints re-! v

2 moved) is denoted by. The active joints' angles are denoted by a and passive onesby p. We order the angles so that T =[ T a T p ]. Consider a general constraint (written in terms of the joint velocity vector) J C () _ =: () Let the spatial velocity of the task frame be v T = J T () _ : (2) Partition J C and J T according to the dimension of a and p : J C = J Ca J Cp Then () can be used to solve for _ p : J T = J Ta J Tp : _ p = ;J y C p J Ca _ a + e J Cp (3) where col( e J Cp ) spans the null space of J Cp,and is arbitrary. Substituting into (2), we have v T =(J Ta ; J Tp J y Cp J Ca) _ a + J Tp e J Cp : (4) Dene the manipulability Jacobian as J T = J Ta ; J Tp J y Cp J Ca : (5) There are two cases of singularities:. Unmanipulable Singularity: This corresponds to congurations at which J T loses rank. 2. Unstable Singularity: This corresponds to con- gurations at which J Tp e J Cp 6=. It may happen that J e Tp J Cp =butj e Cp 6=. This corresponds to the existence of self motion involving only passive joints in the mechanism. We can now dene the manipulability ellipsoid as the ellipsoid corresponding to J T. Additional weighting matrices for active joint velocities and task velocities can also be included. Manipulability ellipsoids provide a geometric visualization for singular congurations. At an unmanipulable singularity, the ellipsoid becomes degenerate (the length of one or more axes become zero, implying that the ellipsoid has zero volume). At an unstable singularity, the ellipsoid becomes innite (the length of one or more axes become innite, implying that arbitrary task velocity is possible even when active joint velocities are constrained). When the mechanism is at a conguration close to an unstable singularity, the ellipsoid would become badly conditioned as one or more axes would be very large. When the mechanism is close to an unmanipulable conguration, the ellipsoid would also be badly conditioned, since the length of one or more of the axes will be close to zero. Hence, a measure of the \closeness" to singularity maybechosen to be the condition number of J T. However, this measure should be used in conjunction with the minimum singular value of J T to distinguish between the two types of singularities. There is also the dual force perspective of the above. The active torque is related to the task frame force f T and internal force f C as follows [7]: J T C f C + J T T f T = Expand this corresponding to active and passive joints: J T C a f C + J T C a f T = J T C p f C + J T T p f T = : Premultiply the second equation by J et,we obtain: Cp ej T Cp J T Tp f T = for all f T. Clearly,ifJ et J T 6= (unstable singularity), there exists some f T that the condition cannot Cp Tp be satised. This corresponds to the task frame spatial forces that cannot be resisted by active torque and internal constraint forces alone. Hence, the mechanism is unstable. If the mechanism is away from the unstable singularities, then can be solved in terms of external forces as well as \free" internal forces: =(J T Ta ; J T Ca (J T Cp )y J T Tp )f T + J T Ca : ^f C (6) where ^f C 2N(J T ). Cp In terms of the interpretation using ellipsoids, the force manipulability ellipsoid can be considered T to be the ellipsoid corresponding to (J T ) ; (mapping from to f T ). In this case, the ellipsoid would have the same principal axes as the velocity manipulability ellipsoid except that the lengths are reciprocal. Consequently, whenthemechanism is at an unmanipulable singularity, the force manipulability ellipsoid is innite in certain directions, and when the mechanism is at an unstable singularity, the ellipsoid is degenerate. In [7], the stable conguration condition is stated in terms of J e Ca being of full column rank (or, equivalently, N ( J e Ca )=fg). This is in fact equivalent to the analysis here since N ( J e Ca )=fg if and only if N (J Cp )=fg. This can be shown as follows. Let 2

3 2N( e J Ca ). Then =J C e J C = J C " ej Ca ej Cp # = J e Cp J Cp which implies that e J Cp 2 N (J Cp ). Suppose N (J Cp ) = fg. Then e J Cp = and therefore ej C = (since e J Ca = ). By construction e J C is of full column rank, hence, =. In the reverse direction, suppose there exists _ p 2N(J Cp ). Then there exists such that _ p = e J Cp. Now, = J Ca J Cp Hence 2N( e J Ca ). " ej Ca ej Cp # = ej Ca 3 Redundant Actuation to Increase Manipulability Qualitatively, a parallel robot at a singular pose has \run out" of actuation, in that it can either not provide an arbitrary instantaneous change in task velocity, or that it cannot resist arbitrary task space force/torques. An obvious approach is to increase the dimension of the active joint space beyond what is strictly necessary for the desired task velocity dimension in an eort to reduce the set of (near) singular poses. This approach consists of two categories: the activation of passive degrees-offreedom in the joint space (i.e. move rows of e J Ca and e J Cp ), and the introduction of new, active degrees of freedom in the joint space (i.e. increasing the dimension of the generalized coordinate ). 3. Activation of Passive Joints For a given mechanism, the activation of passive joints (e.g. attaching a motor to a U-joint at the base end of a Stewart Platform prismatic actuator) is usually not feasible. It does, however, provide a computationally tractable means of determining the eect of redundant actuation, as no additional absolute kinematic analysis is needed. Additionally, the optimization process involves a relatively small number of computations for a given cost function. The activation of passive joints involves the division of J Cp into two groups: J Cp S and J Cp S where S is a selection matrix that extracts rows from J Cp corresponding to activated joints. The matrix S retains the remaining rows of J Cp corresponding to passive joints. The constraint matrix J C now becomes J C = J Ca J Cp S J Cp S : where the new active/passive partition is J C a = J Ca J Cp S J C p = J Cp S: Foragiven selection matrix, S, the velocity ellipsoid can be calculated by (5), using the augmented J and reduced J in place of J C Ca Cp a and J Cp. The eectiveness of the redundant actuation can be assessed by choosing a performance metric associated with how well the Jacobian is conditioned (e.g. maximum singular value, condition number, etc), and a cost function can be found to nd the optimal S (i.e. determine which joint(s) should be activated). The cost function could include a term that weights the number of passive joints activated. The minimum singular value of J T should also be computed to distinguish between unmanipulable and unstable singularities. 3.2 Increasing Manipulator Joint Space A more involved approach to redundant actuation is the introduction of entirely new actuators to the mechanism. As opposed to mapping to new active joint space velocities through a given shape matrix J e Cp, this approach introduces additional joint space velocities using a new shape matrix. Using the Stewart Platform again as an example, this approachmightinvolve a seventh prismatic actuator between the base and the payload plate. This approach is more computationally intensive, as the optimization problem includes the absolute kinematics of the augmented manipulator. The approach, however, has greater mechanical feasibility and is likely more eective in reducing the singular pose set. While the issue of design optimization for kinematic manipulability has been pursued [8], the work has focused on the geometric properties of a given parallel mechanism and the development of a suitable metric. The inclusion of design modications in the form of additional actuators and serial arm bracing is still an open topic. 4 An Example This section presents the results of redundant actuation using passive joint activation and joint space augmentation for a 6-DOF parallel mechanism used for accurate machining [6]. The Eclipse is a 6-DOF parallel mechanism designed for rapid, 5-face machining. The active joints are the three coupled circular prismatic joints of the rotating base (joints -3), and the three linear prismatic joints on the guides (joints 4-6). The passive joints are three revolute joints attaching the payload plate bars to 3

4 the guide prismatic actuators (joints 7-9). The absolute and dierential kinematics of the Eclipse are programmed in Matlab with arbitrary geometric values. To gain insight into the global singularity structure we rst investigate the manipulability over a subset of the workspace while activating passive joints. For this analysis, the payload plate orientation is swept in tip/tilt (no yawing) in. radian increments. The position of the payload frame remains xed. The condition number of the Jacobian is calculated at each orientation. Figure show the condition numbers for the nominal manipulator. Figures 2{6 show the same metric for the passive joint activations of joint 7, joint 8, joint9, joints 7 and 9, and joints 8 and 9, respectively. Figure shows unmanipulable singularites in the neighborhood of 4 degrees tilt. The maximum condition number in this set is The activation of passive joints 7, 8 and 9 reduces the maximum condition number to 4493, 227 and 344, respectively. Activation of these joints has the eect of reducing the average condition number ofthejaco- bian from a nominal value of 48, to 79, 8, and 83, respectively. The activation of multiple passive joints can have a more dramatic eect on manipulability. For example, activation of joints 8 and 9 reduces the maximum and mean values of Jacobian condition number to 249 and 5. It is interesting to note that the activation of joints 7 and 9 does not yield signicant improvement over the activation of 7 alone. Figure 7 shows the condition numbers over the same pose set for a modied Eclipse manipulator. The modication is the addition of a fourth serial subchain, thus increasing the joint space dimension to 2x (active joint space dimension increased to 8x). The geometric properties of the subchains are identical to the nominal values. Figure 7 shows three regions of (near) unmanipulable singularity, where the maximum condition number is 953. It is obvious in comparison with the nominal case that redundant actuation improves the manipulability when using the presence of singularity as a metric. It is interesting to note that the minimum condition number for the modied manipulator over this pose set is 27, in comparison to the nominal value of 3. This suggests that from the perspective of kinematic isotropy, this design modication might have little eect. It should be noted that the pose set selected for this analysis is a small subset of the workspace of the Eclipse manipulator, a fact that could signicantly change the results of the analysis should a larger pose set be investigated. It should also be noted that the Jacobian was not dimensionally homogonized to account for the presence of prismatic and revolute joints in the mechanism. This coupled with the arbitrary selection of geometric parameters for the analytical model could have a signicant effect on the results x s over Pose Range Figure : Orientation Sweep (nominal actuation) over Pose Range (Joint 7 Activated) Figure 2: Orientation Sweeps (Joint 7 activated) 5 Conclusion This paper provides a general velocity ellipsoid expression for manipulability analysis of constrained, multibody systems. It further proposes a computationally tractable means of singular pose set reduction via activation of passive joints in the.5.5 4

5 over Pose Range (Joint 8 Activated) over Pose Range (Joints 7 and 8 Activated) Figure 3: Orientation Sweeps (Joint 8 activated) Figure 5: Orientation Sweeps (redundant actuation, revolute joints 7 and 8 activated) over Pose Range (Joint 9 Activated) Figure 4: Orientation Sweeps (Joint 9 activated) mechanism. It is suggested that augmenting the joint space of a mechanism provides a mechanically feasible means of increasing kinematic manipulability. It is shown that the average and maximum Jacobian condition numbers can be reduced over asetof poses that include singularities by passive joint activation or joint space augmentation for constrained mechanisms such as the Eclipse proposed in [6]. Future work will include the development ofan optimization method for improving kinematic manipulability by increasing the joint space dimension for a given mechanism. One aspect of overactuation is the need to regulate internal forces (actuators may ghteach other to produce unacceptably large.5 internal force). This aspect will be investigated in the future. Acknowledgment This work is supported in part by the Center for Advanced Technology in Automation, Robotics & Manufacturing under a block grant from the New York State Science and Technology Foundation, and a U.S. Department of Energy Integrated Manufacturing Predoctoral Fellowship. References [] A. Bicchi, C. Melchiorri, and D. Balluchi, \On the mobility and manipulability of general multiple limb robots," IEEE Transactions on Robotics and Automation, vol., pp. 25{228, Apr [2] A. Bicchi and D. Prattichizzo, \Manipulability of cooperating robots with passive joints," in Proc. 998 IEEE International Conference on Robotics &Automation, (Leuven, Belgium), pp. 38{44, May 998. [3] P. Chiacchio, S. Chiaverini, L. Sciavicco, and B. Siciliano, \Global task space manipulability ellipsoids for multiple-arm systems," IEEE Transactions on Robotics and Automation, vol. 7, pp. 678{685, Oct. 99. [4] J.-P. Merlet, \Parallel manipulators: state of the art and perspective," in Robotics, Mechatronics and Manufacturing Systems (T. Takamori and K. Tsuchiya, eds.), Elsevier, 993. [5] F. Park and J. Kim, \Manipulability and singularity analysis of multiple robot systems: A geometric approach," in Proc. 998 IEEE International Confer- 5

6 over Pose Range (Joints 8 and 9 Activated) Figure 6: Orientation Sweeps (redundant actuation, revolute joints 7 and 9) s over Pose Range ence on Robotics & Automation, (Leuven, Belgium), pp. 32{37, May 998. [6] S. Ryu, C. Park, J. Kim, J. Hwang, J. Kim, and F. Park, \Deisgn and performance analysis of a parallel mechanism-based universal machining center," technical report, Seoul National University, 998. [7] J. Wen and L. Wilnger, \Kinematic manipulability of general constrained rigid multibody systems," in Proc. 998 IEEE International Conference on Robotics & Automation, (Leuven, Belgium), pp. 2{25, May 998. [8] K. Zanganeh and J. Angeles, \Kinematic isotropy and the optimum design of parallel manipulators," International Journal of Robotics Research, vol. 6, Apr Figure 7: Orientation Sweeps (redundant actuation, 4th kinematic subchain added).5 6