Research Papers

Directional Stiffness Modulation of Parallel Robots With Kinematic Redundancy and Variable Stiffness Joints

[+] Author and Article Information
Andrew L. Orekhov

Department of Mechanical Engineering,
Vanderbilt University,
Nashville, TN 37235
e-mail: andrew.orekhov@vanderbilt.edu

Nabil Simaan

Department of Mechanical Engineering,
Vanderbilt University,
Nashville, TN 37235
e-mail: nabil.simaan@vanderbilt.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received July 2, 2018; final manuscript received April 26, 2019; published online July 8, 2019. Assoc. Editor: Shaoping Bai.

J. Mechanisms Robotics 11(5), 051003 (Jul 08, 2019) (9 pages) Paper No: JMR-18-1197; doi: 10.1115/1.4043685 History: Received July 02, 2018; Accepted April 26, 2019

Parallel robots have been primarily investigated as potential mechanisms with stiffness modulation capabilities through the use of actuation redundancy to change internal preload. This paper investigates real-time stiffness modulation through the combined use of kinematic redundancy and variable stiffness actuators. A known notion of directional stiffness is used to guide the real-time geometric reconfiguration of a parallel robot and command changes in joint-level stiffness. A weighted gradient-projection redundancy resolution approach is demonstrated for resolving kinematic redundancy while satisfying the desired directional stiffness and avoiding singularity and collision between the legs of a Gough/Stewart parallel robot with movable anchor points at its base and with variable stiffness actuators. A simulation study is carried out to delineate the effects of using kinematic redundancy with or without the use of variable stiffness actuators. In addition, modulation of the entire stiffness matrix is demonstrated as an extension of the approach for modulating directional stiffness.

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Luces, M., Mills, J. K., and Benhabib, B., 2017, “A Review of Redundant Parallel Kinematic Mechanisms,” J. Intell. Robot. Syst., 86(2), pp. 175–198. [CrossRef]
Gosselin, C., and Schreiber, L.-T., 2018, “Redundancy in Parallel Mechanisms: A Review,” ASME Appl. Mech. Rev., 70(1), p. 010802. [CrossRef]
Kim, S., 1997, “Operational Quality Analysis of Parallel Manipulators With Actuation Redundancy,” Proceedings of the 1997 IEEE International Conference on Robotics and Automation, Vol. 3, Albuquerque, NM, Apr. 20, pp. 2651–2656, IEEE.
Radojicic, J., Surdilovic, D., and Schreck, G., 2009, “Modular Hybrid Robots for Safe Human–Robot Interaction,” Int. J. Mech.. Mechatron. Eng., 3(12), pp. 1601–1607.
Salisbury, J. K., 1980, “Active Stiffness Control of a Manipulator in Cartesian Coordinates,” 19th IEEE Conference on Decision and Control Including the Symposium on Adaptive Processes, Albuquerque, NM, Dec. 10, pp. 95–100.
Hogan, N., 1985, “Impedance Control: An Approach to Manipulation, Parts I, II, III,” ASME J. Dyn. Syst. Meas. Control, 107(1), pp. 1–24. [CrossRef]
Yi, B.-J., and Freeman, R. A., 1993, “Geometric Analysis of Antagonistic Stiffness in Redundantly Actuated Parallel Mechanisms,” J. Robot. Syst., 10(5), pp. 581–603. [CrossRef]
Chen, S.-F., and Kao, I., 2000, “Conservative Congruence Transformation for Joint and Cartesian Stiffness Matrices of Robotic Hands and Fingers,” Int. J. Robot. Res., 19(9), pp. 835–847. [CrossRef]
Simaan, N., and Shoham, M., 2003, “Geometric Interpretation of the Derivatives of Parallel Robots Jacobian Matrix With Application to Stiffness Control,” ASME J. Mech. Des., 125(1), pp. 33–42. [CrossRef]
Simaan, N., and Shoham, M., 2003, “Stiffness Synthesis of a Variable Geometry Six-Degrees-of-Freedom Double Planar Parallel Robot,” Int. J. Robot. Res., 22(9), pp. 757–775. [CrossRef]
Abdolshah, S., Zanotto, D., Rosati, G., and Agrawal, S. K., 2017, “Optimizing Stiffness and Dexterity of Planar Adaptive Cable-Driven Parallel Robots,” ASME J. Mech. Robot., 9(3), p. 031004. [CrossRef]
Anson, M., Alamdari, A., and Krovi, V., 2017, “Orientation Workspace and Stiffness Optimization of Cable-Driven Parallel Manipulators With Base Mobility,” ASME J. Mech. Robot., 9(3), p. 031011. [CrossRef]
Kim, W.-K., Lee, J.-Y., and Yi, B. J., 1997, “Analysis for a Planar 3 Degree-of-Freedom Parallel Mechanism With Actively Adjustable Stiffness Characteristics,” KSME Int. J., 11(4), p. 408. [CrossRef]
Zhou, X., Jun, S.-K., and Krovi, V., 2016, “Planar Cable Robot With Variable Stiffness,” Experimental Robotics (No. 109 in Springer Tracts in Advanced Robotics), M. A. Hsieh, O. Khatib, and V. Kumar, eds., Springer International Publishing, New York, pp. 391–403.
Loncaric, J., 1987, “Normal Forms of Stiffness and Compliance Matrices,” IEEE J. Robot. Autom., 3(6), pp. 567–572. [CrossRef]
Huang, S., and Schimmels, J. M., 1998, “The Bounds and Realization of Spatial Stiffnesses Achieved With Simple Springs Connected in Parallel,” IEEE Trans. Robot. Autom., 14(3), pp. 466–475. [CrossRef]
Jamshidifar, H., Khajepour, A., Fidan, B., and Rushton, M., 2017, “Kinematically-Constrained Redundant Cable-Driven Parallel Robots: Modeling, Redundancy Analysis, and Stiffness Optimization,” IEEE/ASME Trans. Mech., 22(2), pp. 921–930. [CrossRef]
Kock, S., and Schumacher, W., 1998, “A Parallel X–Y Manipulator With Actuation Redundancy for High-Speed and Active-Stiffness Applications,” Proceedings of the 1998 IEEE International Conference on Robotics and Automation, Vol. 3, Leuven, Belgium, May 16, pp. 2295–2300, IEEE.
Chakarov, D., 2004, “Study of the Antagonistic Stiffness of Parallel Manipulators With Actuation Redundancy,” Mech. Mach. Theory, 39(6), pp. 583–601. [CrossRef]
Muller, A., 2006, “Stiffness Control of Redundantly Actuated Parallel Manipulators,” Proceedings of the 2006 IEEE International Conference on Robotics and Automation, 2006 (ICRA 2006), Orlando, FL, May 15, pp. 1153–1158, IEEE.
Yu, K., Lee, L. F., Tang, C. P., and Krovi, V. N., 2010, “Enhanced Trajectory Tracking Control With Active Lower Bounded Stiffness Control for Cable Robot,” 2010 IEEE International Conference on Robotics and Automation, Anchorage, AK, May 3, pp. 669–674.
Pitt, E. B., Simaan, N., and Barth, E. J., 2015, “An Investigation of Stiffness Modulation Limits in a Pneumatically Actuated Parallel Robot With Actuation Redundancy,” ASME/BATH 2015 Symposium on Fluid Power and Motion Control, Chicago, IL, Oct. 12, American Society of Mechanical Engineers, p. V001T01A063.
Yi, B.-J., and Freeman, R. A., 1992, “Synthesis of Actively Adjustable Springs by Antagonistic Redundant Actuation,” J. Dyn. Syst. Meas. Control, 114(3), pp. 454–461. [CrossRef]
Müller, A., 2010, “Consequences of Geometric Imperfections for the Control of Redundantly Actuated Parallel Manipulators,” IEEE Trans. Robot., 26(1), pp. 21–31. [CrossRef]
Vanderborght, B., Albu-Schaeffer, A., Bicchi, A., Burdet, E., Caldwell, D. G., Carloni, R., Catalano, M., Eiberger, O., Friedl, W., Ganesh, G., Garabini, M., Grebenstein, M., Grioli, G., Haddadin, S., Hoppner, H., Jafari, A., Laffranchi, M., Lefeber, D., Petit, F., Stramigioli, S., Tsagarakis, N., Van Damme, M., Van Ham, R., Visser, L. C., and Wolf, S., 2013, “Variable Impedance Actuators: A Review,” Robot. Autonom. Syst., 61(12), pp. 1601–1614. [CrossRef]
Van Ham, R., Vanderborght, B., Van Damme, M., Verrelst, B., and Lefeber, D., 2007, “MACCEPA, the Mechanically Adjustable Compliance and Controllable Equilibrium Position Actuator: Design and Implementation in a Biped Robot,” Robot. Autonom. Syst., 55(10), pp. 761–768. [CrossRef]
Jafari, A., Tsagarakis, N. G., and Caldwell, D. G., 2011, “Awas-II: A New Actuator With Adjustable Stiffness Based on the Novel Principle of Adaptable Pivot Point and Variable Lever Ratio,” 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, May 9, pp. 4638–4643, IEEE.
Hollander, K. W., Sugar, T. G., and Herring, D. E., 2005, “Adjustable Robotic Tendon Using a ‘Jack Spring’,” 9th International Conference on Rehabilitation Robotics, Chicago, IL, June 28, pp. 113–118, IEEE.
Lee, J., Ajoudani, A., Hoffman, E. M., Rocchi, A., Settimi, A., Ferrati, M., Bicchi, A., Tsagarakis, N. G., and Caldwell, D. G., 2014, “Upper-Body Impedance Control With Variable Stiffness for a Door Opening Task,” 2014 IEEE-RAS International Conference on Humanoid Robots, Madrid, Spain, Nov. 18, pp. 713–719.
Lim, W. B., Yeo, S. H., Yang, G., and Chen, I. M., 2013, “Design and Analysis of a Cable-Driven Manipulator With Variable Stiffness,” 2013 IEEE International Conference on Robotics and Automation, Karlsruhe, Germany, May 6, pp. 4519–4524.
Zhou, X., Jun, S.-K., and Krovi, V., 2015, “A Cable Based Active Variable Stiffness Module With Decoupled Tension,” ASME J. Mech. Robot., 7(1), p. 011005. [CrossRef]
Rice, J. J., and Schimmels, J. M., 2018, “Passive Compliance Control of Redundant Serial Manipulators,” ASME J. Mech. Robot., 10(4), p. 041009. [CrossRef]
Alamdari, A., Haghighi, R., and Krovi, V., 2018, “Stiffness Modulation in an Elastic Articulated-cable Leg-Orthosis Emulator: Theory and Experiment,” IEEE Trans. Robot., 34(5), pp. 1266–1279. [CrossRef]
Alizade, R. I., Tagiyev, N. R., and Duffy, J., 1994, “A Forward and Reverse Displacement Analysis of a 6-DOF In-Parallel Manipulator,” Mech. Mach. Theory, 29(1), pp. 115–124. [CrossRef]
Tsai, L.-W., and Tahmasebi, F., 1993, “Synthesis and Analysis of a New Class of Six-Degree-of-Freedom Parallel Minimanipulators,” J. Robot. Syst., 10(5), pp. 561–580. [CrossRef]
Behi, F., 1988, “Kinematic Analysis for a Six-Degree-of-Freedom 3-PRPS Parallel Mechanism,” IEEE J. Robot. Autom., 4(5), pp. 561–565. [CrossRef]
Ben-Horin, R., Shoham, M., and Djerassi, S., 1998, “Kinematics, Dynamics and Construction of a Planarly Actuated Parallel Robot,” Robot. Comput. Integr. Manuf., 14(2), pp. 163–172. [CrossRef]
Bonev, I. A., and Gosselin, C. M., 2002, “Geometric Algorithms for the Computation of the Constant-Orientation Workspace and Singularity Surfaces of a Special 6-RUS Parallel Manipulator,” 27th Biennial Mechanisms and Robotics Conference, Montreal, Quebec, Sept. 29–Oct. 2, pp. 505–514.
Wu, G., 2012, “Multiobjective Optimum Design of a 3-RRR Spherical Parallel Manipulator With Kinematic and Dynamic Dexterities,” J. Model. Identif. Control: Norwegian Res. Bull., 33(3), pp. 111–121. [CrossRef]
Neill, D. R., Sneed, R., Dawson, J., Sebag, J., and Gressler, W., 2014, “Baseline Design of the Lsst Hexapods and Rotator,” Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation, Vol. 9151, R. Navarro, C. R. Cunningham, and A. A. Barto, eds., International Society for Optics and Photonics, Bellingham, WA, p. 91512B.
Sawyer, B. A., 1968, “Magnetic Positioning Device,” U.S. Patent No. 3,376,578.
Mohamed, M., and Duffy, J., 1985, “A Direct Determination of the Instantaneous Kinematics of Fully Parallel Robot Manipulators,” J. Mech. Trans. Autom. Des., 107(2), pp. 226–229. [CrossRef]
Sima’an, N., Glozman, D., and Shoham, M., 1998, “Design Considerations of New Six Degrees-of-Freedom Parallel Robots,” Proceedings of the 1998 IEEE International Conference on Robotics and Automation, Vol. 2, pp. 1327–1333, Cat. No. 98CH36146.
Tsai, L.-W., 1999, Robot Analysis: The Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, New York.
Yi, B. J., Freeman, R. A., and Tesar, D., 1989, “Open-Loop Stiffness Control of Overconstrained Mechanisms/Robotic Linkage Systems,” Proceedings of the 1989 International Conference on Robotics and Automation, Vol. 3, Scottsdale, AZ, May 14, pp. 1340–1345.
Li, Y., and Kao, I., 2004, “Stiffness Control on Redundant Manipulators: A Unique and Kinematically Consistent Solution,” Proceedings of the IEEE International Conference on Robotics and Automation, 2004 (ICRA’04), New Orleans, LA, Apr. 26, IEEE, pp. 3956–3961.
Huang, S., and Schimmels, J. M., 1998, “The Bounds and Realization of Spatial Stiffnesses Achieved With Simple Springs Connected in Parallel,” IEEE Trans. Robot. Autom., 14(3), pp. 466–475. [CrossRef]
Liegeois, A., 1977, “Automatic Supervisory Control of the Configuration and Behavior of Multibody Mechanisms,” IEEE Trans. Syst. Man Cybern., 7(12), pp. 868–871. [CrossRef]
Walker, I. D., and Marcus, S. I., 1988, “Subtask Performance by Redundancy Resolution for Redundant Robot Manipulators,” IEEE J. Robot. Autom., 4(3), pp. 350–354. [CrossRef]
Li, L., Gruver, W. A., Zhang, Q., and Yang, Z., 2001, “Kinematic Control of Redundant Robots and the Motion Optimizability Measure,” IEEE Trans. Syst. Man Cybern. Part B (Cybern), 31(1), pp. 155–160. [CrossRef]
Liu, Y., Zhao, J., and Xie, B., 2010, “Obstacle Avoidance for Redundant Manipulators Based on a Novel Gradient Projection Method With a Functional Scalar,” 2010 IEEE International Conference on Robotics and Biomimetics, Tianjin, China, Dec. 14, pp. 1704–1709.
Euler, J. A., Dubey, R. V., and Babcock, S. M., 1989, “Self Motion Determination Based on Actuator Velocity Bounds for Redundant Manipulators,” J. Robot. Syst., 6(4), pp. 417–425. [CrossRef]
Khan, W. A., and Angeles, J., 2006, “The Kinetostatic Optimization of Robotic Manipulators: The Inverse and the Direct Problems,” ASME J. Mech. Des., 128(1), pp. 168–178. [CrossRef]
Zghal, H., Dubey, R. V., and Euler, J. A., 1990, “Efficient Gradient Projection Optimization for Manipulators With Multiple Degrees of Redundancy,” Proceedings of the 1990 IEEE International Conference on Robotics and Automation, Vol. 2, Cincinnati, OH, May 13, pp. 1006–1011.
Agarwal, S., Srivatsan, R. A., and Bandyopadhyay, S., 2016, “Analytical Determination of the Proximity of Two Right-Circular Cylinders in Space,” ASME J. Mech. Robot., 8(4), p. 041010. [CrossRef]
Ketchel, J. S., and Larochelle, P. M., 2008, “Self-Collision Detection in Spatial Closed Chains,” ASME J. Mech. Des., 130(9), p. 092305. [CrossRef]
Merlet, J. P., and Daney, D., 2006, “Legs Interference Checking of Parallel Robots Over a Given Workspace or Trajectory,” Proceedings of the IEEE International Conference on Robotics and Automation, 2006 (ICRA 2006), Orlando, FL, May 15, pp. 757–762.
Perreault, S., Cardou, P., Gosselin, C. M., and Otis, M. J.-D., 2010, “Geometric Determination of the Interference-Free Constant-Orientation Workspace of Parallel Cable-Driven Mechanisms,” ASME J. Mech. Robot., 2(3), p. 031016. [CrossRef]
Grioli, G., Wolf, S., Garabini, M., Catalano, M., Burdet, E., Caldwell, D., Carloni, R., Friedl, W., Grebenstein, M., Laffranchi, M., and Lefeber, D., 2015, “Variable Stiffness Actuators: The User’s Point of View,” Int. J. Robot. Res., 34(6), pp. 727–743. [CrossRef]
Hurst, J. W., Chestnutt, J. E., and Rizzi, A. A., 2010, “The Actuator With Mechanically Adjustable Series Compliance,” IEEE Trans. Robot., 26(4), pp. 597–606. [CrossRef]


Grahic Jump Location
Fig. 1

A Stewart–Gough type parallel manipulator with kinematic redundancy introduced through movable base anchor points

Grahic Jump Location
Fig. 2

Geometry of robot used in kinematic simulations

Grahic Jump Location
Fig. 3

Directional stiffness along x, y, and z directions while following the spiral in Eq. (36). (a) Fixed anchor points with constant joint stiffness, (b) fixed anchor points with variable joint stiffness, (c) using kinematic redundancy with constant joint stiffness, and (d) using both kinematic redundancy and variable joint stiffness.

Grahic Jump Location
Fig. 4

(a) Configuration in which the desired spatial stiffness was computed, (b) initial configuration, and (c) final configuration after optimizing the base anchor locations to satisfy the spatial stiffness. Note that the configuration in (c) matches well with its counterpart in (a).



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