Force-Moment Capabilities of Redundantly-Actuated Planar-Parallel Architectures S. B. Nokleby F. Firmani A. Zibil R. P. Podhorodeski UOIT University of Victoria University of Victoria University of Victoria Oshawa, Canada Victoria, Canada Victoria, Canada Victoria, Canada Abstract The force-moment capabilities of redundantlyactuated planar-parallel manipulator (PPM) architectures are investigated. A previously developed explicit methodology for generating the force-moment capabilities of redundantly-actuated PPMs is used on three different architectures. The results for the 3-RRR, 3-RPR, 3-PRR layouts, where the underline denotes the actuated joints, are presented and discussed. Keywords: planar parallel manipulators, redundantly-actuated, force-moment capabilities I. Introduction The force-moment capability of a manipulator is defined as the maximum wrench that can be applied (or sustained) by a manipulator. This capability is for a given pose and is based on the limits of its actuators. Redundancy in parallel manipulators (PMs) can be divided into four categories. The first category features actuating some of the passive joints within the branches of a PM. For example, consider the three-branch planar parallel manipulator (PPM) with three-revolute joints per branch, the 3-RRR device [9]. For a non-redundant device, one joint per branch must be actuated. By actuating an additional joint in one or more branches, the PPM is redundantly actuated. The second category of redundant manipulators are those that feature additional branches beyond the minimum necessary to actuate the device. Again considering the 3-RRR device, if an additional branch (featuring an actuated joint within the branch) is added to the device, the device would be a redundantly actuated 4-RRR manipulator. The third category features branches that are joint redundant, i.e., there are additional actuated joints added to one or more of the branches to form redundant serial chains. The final category of redundant PMs are devices that are a hybrid of the first three categories. This paper focuses on the force-moment capabilities of redundant PPM architectures that fall within the first category, i.e., those that feature actuating additional passive joints within the branches. Recently, there has been a growing interest in the re- E-mail: scott.nokleby@uoit.ca E-mail: ffirmani@me.uvic.ca E-mail: azibil@me.uvic.ca E-mail: podhoro@me.uvic.ca UOIT: University of Ontario Institute of Technology search community into redundantly-actuated PMs (see for example [1 8, 10 18]). This growing interest is due to the benefits associated with redundant actuation. With redundant actuation, the solution to the inverse force problem (given the desired wrench to be applied by the platform, what are the required joint torques/forces) no longer has a unique solution. An infinity of possible solutions exists to the inverse force problem. This infinity of possible solutions allows the joint torques/forces to be optimized. In addition, redundant actuation eliminates force-unconstrained configurations 1. For the three-branch PPM layouts, there are seven possible architectures: RRR, RPR, PRR, RRP, PPR, RPP, and PRP, where R and P denote a revolute and prismatic joint, respectively. Of these seven architectures, PRR and the RRP are kinematically equivalent. Likewise, the PPR and RPP are kinematically equivalent. Considering only one of the kinematically equivalent architectures leaves only five unique architectures. For this work, those architectures with two prismatic joints are not considered as they are not convenient for implementation as PPMs. As such only the RRR, RPR, and PRR architectures are studied. The force-moment capabilities are critical in the design and development of PMs. For a given application, a designer needs to know the capabilities of the manipulator that is being developed. By being able to graphically visualize the force-moment capabilities, comparisons between different design parameters, including actuator output and link lengths, can easily be explored. The designer can then optimize the design according to the force-moment capabilities required for a specific task. The goal of this paper is to investigate the force-moment capabilities of the 3- RRR, 3-RPR, and 3-PRR, where the underline denotes the joints that are actuated within each branch. To determine the force-moment capabilities, a methodology developed by Zibil et al. [19,20] of determining the force-moment capabilities of redundantly-actuated PMs in an explicit manner is employed. The paper first briefly summarizes the methodology of Zibil et al. [19, 20]. Next, the force-moment capabilities for 1 A force-unconstrained configuration is a configuration in which the platform of a parallel manipulator cannot sustain or apply an arbitrary force and instantaneously gains an uncontrollable degree-of-freedom (DOF) of motion.
the 3-RRR, 3-RPR, and 3-PRR PPMs are presented. This is followed by a discussion of the results. The paper finishes with proposals for future work and conclusions. II. Explicit Methodology for the Determination of Force-Moment Capabilities In previous work (see [17]) it has been observed that greater output wrenches are obtained when the individual actuators are close to their maximum capabilities. In [19, 20] a solution is developed to explicitly set the largest number of actuators to their maximum capabilities while ensuring the correct output direction and prescribed moment. Assume that the PPM has k actuated joints. The forward force solution yields: [$ D] 3 k τ k 1 = F 3 1 (1) where τ is a vector of the actuated joint torques or forces and [$ ] is the matrix of associated reciprocal screw quantities. In equation (1), [D] is a diagonal matrix which converts the vector of joint torques/forces to a vector of wrench intensities. For the purposes of this work, F can be expressed as [f cos(α), f sin(α); m z ] T ; with f being the wrench intensity, α being the direction and m z being the moment of the output wrench. All the elements of [$ D] are known through the geometry and the pose of the manipulator. Thus, the elements of τ and F are unknown, i.e., there are k + 3 variables. With three equations to satisfy, k variables can be set arbitrarily. By properly adjusting the six actuator outputs for different scenarios, Zibil, et al. [20] formulated four studies of the force-moment capabilities of the 3-RRR PPM, namely: i) the maximum/minimum pure forces, ii) the maximum pure moments, iii) the maximum/minimum applicable forces with an associated moment; and iv) the maximum applicable moment with an associated force. The details of these formulations are presented in [20] and will not be covered here due to space limitations. For case (i), by varying α through 360 the unknowns of equation (1) are f and two actuator outputs (torques for revolute joints or forces for prismatic joints). Thus there are four actuator outputs that are acting at their maximum capabilities (either positive or negative). For case (ii), the unknowns of equation (1) are m z and two actuator outputs. Thus there are four actuator outputs that are acting at their maximum capabilities. For case (iii), by varying α through 360 the unknowns of equation (1) are f, m z and one actuator output. Thus there are five actuator outputs that are acting at their maximum capabilities. For case (iv), the unknowns of equation (1) are f, α, and m z. Thus all actuator outputs are acting at their maximum capabilities. The developed explicit methodology of [19, 20] was shown to be significantly more efficient than optimizationbased methods [17, 18] for resolving the inverse force problem for redundantly-actuated PPMs. Hence, it was used to perform the analysis presented in this paper. III. Force-Moment Capabilities Figure 1 shows the layouts for the 3-RRR, 3-RPR, 3- PRR, respectively. For the 3-RRR, the base triangle edge length was 0.5 m, the end-effector triangle edge length was 0.2 m, and the lengths of the first and second links of each branch were 0.2 m. The torque limits of the base revolute joints were ±4.2 Nm, and the torque limits of the elbow revolute joints were ±2.1 Nm. For the 3-RPR, the base triangle edge length was 0.5 m, the end-effector triangle edge length was 0.2 m, the prismatic joints extension limits were 0.4 m and the force limits were ±10 N, and the torque limits of the base revolute joints were ±4.2 Nm. For the 3-PRR, the base triangle edge length was 0.5 m, the end-effector triangle edge length was 0.2 m, the prismatic joints orientations were 0, 120, 240 (see the right plot of Figure 1), the prismatic joints extension limits were 1 m and the force limits were ±10 N, the lengths of the second links were 0.23 m, and the torque limits of the elbow revolute joints were ±2.1 Nm. For this work, the platform orientations are held constant at 0 (as shown in Figure 1). Also, for the eight possible inverse displacement solutions for both the 3-RRR and the 3-PRR, only the assembly modes shown in Figure 1 are used. Figures 2, 3, and 4 show the force-capabilities of the 3- RRR, 3-RPR, 3-PRR, respectively. For each figure, each column of plots are associated. From top to bottom, for the first column the plots are: i) the minimum of the maximum resultant forces among all directions throughout the workspace of the manipulator; ii) the minimum of the maximum applicable forces with an associated moment that can be applied or sustained in any direction; and iii) the associated moments of the minimum of the maximum applicable forces shown in (ii). From top to bottom, for the second column the plots are: iv) the maximum of the maximum resultant forces among all directions throughout the manipulator s workspace; v) the maximum of the maximum applicable forces with an associated moment among all directions; and vi) the associated moments of the maximum of the maximum applicable forces shown in (v). From top to bottom, for the third column the plots are: vii) the maximum pure moments that can applied or sustained throughout the manipulator s workspace; viii) the maximum applicable moments with an associated force; and ix) the associated forces of the maximum applicable moments shown in (viii). In Figures 2, 3, and 4, due to the fact that for a few poses, the values obtained are very large compared to the rest of the poses in the workspace, therefore, the values for some of
12th IFToMM World Congress, Besanc on, June 18-21, 2007 Fig. 1. The PPM Layouts: 3-RRR (left), 3-RPR (middle), 3-PRR (right) Fig. 2. The force-moment capabilities of the 3-RRR. the plots have been capped to give a better overall greyscale gradient. The maximum and minimum values achieved for each analysis are presented in Tables I, II, and III.
12th IFToMM World Congress, Besanc on, June 18-21, 2007 Fig. 3. The force-moment capabilities of the 3-RPR. i) iv) vii) Max: 67.00 N Min: 33.56 N Min: 42.17 N ii) v) viii) Min: 61.33 N Min: 77.05 N iii) vi) ix) Max: 8.40 Nm Min: 3.73 Nm Min: 5.55 Nm Min: 0.11 N TABLE I. The maximum and minimum values for the nine sub-plots of Figure 2 for the 3-RRR. i) iv) vii) Max: 53.69 N Max: 62.00 N Max: 6.28 Nm Min: 33.19 N Min: 35.34 N ii) v) viii) Min: 45.97 N Min: 54.59 N iii) vi) ix) Max: 8.40 Nm Min: 3.63 Nm Min: 4.47 Nm Min: 0 N TABLE II. The maximum and minimum values for the nine sub-plots of Figure 3 for the 3-RPR. IV. Discussion Referring to Figures 2, 3, and 4, a number of observations can be made. The most obvious difference is in the shape of the workspace of the 3-PRR, compared with the other two PPMs. This is completely due to the geometric properties
Fig. 4. The force-moment capabilities of the 3-PRR. i) Max: 66.27 N Min: 27.48 N ii) Min: 40.86 N iii) Max: 13.34 Nm Min: 4.40 Nm iv) Min: 28.36 N v) Min: 41.20 N vi) Min: 5.91 Nm vii) viii) ix) Min: 0 N TABLE III. The maximum and minimum values for the nine sub-plots of Figure 4 for the 3-PRR. of the 3-PRR and not a result of the redundant actuation. In terms of the force-moment capabilities, the 3-RRR (Figure 2) and 3-RPR (Figure 3) have very similar plots with only minor differences. Looking closely at the plots between these two manipulators, the plots for the 3-RRR (Figure 2) appear to be distortions of the plots of the 3- RPR (Figure 3). The distortions are due to the force capabilities of the second revolute joints in the branches of the 3-RRR being moment-arm (and therefore pose) dependent. The maximum forces and moments achieved by these two layouts are comparable. It is worthy of mention that the non-redundantly actuated 3-RRR and 3-RPR are kinematically equivalent. Thus elements of symmetry are expected to happen.
Referring to Figure 2 for the 3-RRR, the locations with large force-moment capabilities correspond to poses where the manipulator is at or near singular configurations. Infinite values at the limits of the workspace correspond to configurations where the manipulator has a fully extended arm. Infinite values within the workspace occur when the manipulator has one arm folded back on itself. Referring to Figure 3 for the 3-RPR, the locations with large force-moment capabilities correspond to poses where the manipulator has one prismatic joint at or close to zero, i.e., a singular configuration. Note that this singularity occurs within the workspace. Referring to Figure 4 for the 3-PRR, the singular configurations, i.e., locations with large force-moment capabilities, correspond to poses where the second link of a branch is perpendicular to the fixed prismatic joint. Note that this singularity occurs at the boundary of the workspace. The plots generated are a useful way to visualize the complete force-moment capabilities of a given PPM layout. As a design tool they allow for easy visualization of the differences in force-moment capabilities. V. Proposals for Future Work The results presented here are based on symmetric actuation schemes. Studies into the effects of non-symmetric actuating schemes could be undertaken. This work focused on redundant PPMs featuring the actuation of some of the passive joints within the branches of the PPM. Another avenue of future study would be to investigate the force-moment capabilities of redundant PPMs that feature additional branches to achieve redundancy. Explorations of the effects on the force-moment capabilities of modifying the geometric parameters, the assembly modes, and the platform orientation of the three PPMs considered could also be investigated. VI. Conclusion A previously developed explicit methodology for generating the force-moment capabilities of redundantlyactuated PPM was used to determine the force-moment capabilities of the three PPM architectures. The results for the 3-RRR, 3-RPR, 3-PRR layouts were presented. It was shown that the force-moment capabilities for the 3-RRR and 3-RPR were very similar. An analysis was carried out on the effects of singular configurations on the forcemoment capabilities. The plots presented are a useful way to visualize the complete force-moment capabilities of a given PPM architecture. References [1] Beiner, L. Redundant Actuation of a Closed-Chain Device. Advanced Robotics, 11(3):233-245, 1997. [2] Buttolo, P. and Hannaford, B. Advantages of Actuation Redundancy for the Design of Haptic Displays. In Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition - Part 2, pp. 623-630, San Francisco, USA, November 12-17, 1995. [3] Cheng, H., Liu, G. F., Yiu, Y. K., Xiong, Z. H., and Li, Z. X. Advantages and Dynamics of Parallel Manipulators with Redundant Actuation. In Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 171-176, Maui, USA, October 29 - November 3, 2001. [4] Cheng, H., Yiu, Y.-K., and Li., Z. Dynamics and Control of Redundantly Actuated Parallel Manipulators. IEEE Transactions on Mechatronics, 8(4):483-491, 2003. [5] Dasgupta, B. and Mruthyunjaya, T. S. Force Redundancy in Parallel Manipulators: Theoretical and Practical Issues. Mechanism and Machine Theory, 33(6):727-742, 1998. [6] Firmani, F. and Podhorodeski, R. P. Force-Unconstrained Poses for a Redundantly-Actuated Planar Parallel Manipulator. Mechanism and Machine Theory, 39(5):459-476, 2004. [7] Ganovski, L. L., Fisette, P., and Samin, J.-C. Modeling of Overactuated Closed-Loop Mechanisms with Singularities: Simulation and Control. In Proceedings of the 2001 ASME Design Engineering Technical Conferences and the Computers and Information in Engineering Conference, 8 pages, Pittsburgh, USA, September 9-12, 2001. [8] Gonzalez, L. J. and Sreenivasan, S. V. Representational Singularities in the Torque Optimization Problem of an Active Closed Loop Mechanism. Mechanism and Machine Theory, 35(6):871-886, 2000. [9] Gosselin, C. and Angeles, J. Singularity Analysis of Closed-Loop Kinematic Chains. IEEE Transactions on Robotics and Automation, 6(3):281-290, 1990. [10] Kerr, D. R., Griffis, M., Sanger, D. J., and Duffy, J. Redundant Grasps, Redundant Manipulators, and Their Dual Relationship. Journal of Robotic Systems, 9(7):973-1000, 1992. [11] Kim, H. S. and Choi, Y. J. The Kinetostatic Capability Analysis of Robotic Manipulators. In Proceedings of the 1999 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1241-1246, Kyongju, South Korea, October 17-21, 1999. [12] Kurtz, R. and Hayward, V. Multiple-Goal Kinematic Optimization of a Parallel Spherical Mechanism with Actuator Redundancy. IEEE Transactions on Robotics and Automation, 8(5):644-651, 1992. [13] Lee, S. and Kim, S. Kinematic Feature Analysis of Parallel Manipulator Systems. In Proceedings of the 1994 IEEE International Conference on Robotics and Automation, pp. 77-82, San Diego, USA, May 8-13, 1994. [14] Lee, S. H., Yi, B.-J., Kim, S. H., and Kwak, Y. K. Control of Impact Disturbance by Redundantly Actuated Mechanism. In Proceedings of the 2001 IEEE International Conference on Robotics and Automation, pp. 3734-3741, Seoul, South Korea, May 21-26, 2001. [15] Liu, G. F. and Wu, Y. L. and Wu, X. Z. and Kuen, Y. Y. and Li, Z. X. Analysis and Control of Redundant Parallel Manipulators. In Proceedings of the 2001 IEEE International Conference on Robotics and Automation, pp. 3748-3754, Seoul, South Korea, May 21-26, 2001. [16] Merlet, J.-P. Redundant Parallel Manipulators. Laboratory Robotics and Automation, 8(1):17-24, 1996. [17] Nokleby, S. B., Fisher, R., Podhorodeski, R. P., and Firmani, F. Force Capabilities of Redundantly-Actuated Parallel Manipulators. Mechanism and Machine Theory, 40(5):578-599, 2005. [18] Nokleby S. B. Force-Moment Capabilities of Planar-Parallel Manipulators Using Different Redundant-Actuation Configurations. Transactions of the CSME, 29(4):669-678, 2005. [19] Zibil, A., Firmani, F., Nokleby, S. B., and Podhorodeski, R. P. An Analytic Method for Determining the Force-Moment Capabilities of Redundantly-Actuated Planar Parallel Manipulators. In Proceedings of the 2006 CSME Forum, 12 pages, Kananaskis, Canada, May 21-24, 2006. [20] Zibil, A., Firmani, F., Nokleby, S. B., and Podhorodeski, R. P. An Analytic Method for Determining the Force Moment Capabilities of Redundantly Actuated Planar Parallel Manipulators. To appear in Transactions of the ASME, Journal of Mechanical Design.