Trajectory Planning of a One-Legged Robot Performing a Stable Hop

Abstract

In this paper, stable hopping of a one-legged, articulated robot with a flat foot is investigated. The robot has a special feature that before taking off, it goes through an underactuated phase in which the foot rotates about the unactuated toe on the ground. By having an underactuated phase, the robot can perform stable human-like hops with longer hopping distances. To devise a systematic trajectory design methodology for the robot, its dynamics was first derived, including the effects of ground-foot impact and hopping constraints. An optimization procedure is then proposed to plan the feasible actuated trajectories that not only meet performance requirements but also attain optimality with respect to the use of actuation energy. The hopping strategy and the planned trajectories are then verified by simulation and hardware implementation. Experiments indicate that the robot can perform stable hops with different hopping distances on level ground and can successfully hop up and down staircases.

Keywords

One-Legged Robot hopping non-linear optimization underactuation

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References

Ahmadi, M. and Buehler, M. ( 2006). Controlled Passive Dynamic Running Experiments with the ARL-Monopod 2. IEEE Transactions on Robotics, 22(5): 974-986.

Bessonnet, G. , Seguin, P. and Sardain, P. ( 2005). A Parametric Optimization Approach to Walking Pattern Synthesis. The International Journal of Robotics Research , 24(7): 523-536.

Chevallereau, C. and Aoustin, Y. ( 2001). Optimal reference trajectories for walking and running of a biped robot. Robotica, 19(5): 557-569.

Coleman, T.F. and Zhang, Y. ( 2007). Optimization Toolbox for use with MATLAB© (Version 2007b). The MathWorks, Inc.

Goswami, A. ( 1999). Postural stability of biped robots and the foot-rotation indicator (FRI) point. The International Journal of Robotics research, 18(6): 523-533.

Goswami, D. and Vadakkepat, P. ( 2009). Planar Bipedal Jumping Gaits With Stable Landing. IEEE Transactions on Robotics, 25(5): 1030-1046.

Gregorio, P. , Ahmadi, M. and Buehler, M. ( 1997). Design, Control, and Energetics of an Electrically Actuated Legged Robot. IEEE Transactions on Systems, Man, and Cybernetics , 27(4): 626-634.

Guo, Q. , Macnab, C.J. and Pieper, J.K. ( 2008). Hopping on Even Ground and Up Stairs with a Single Articulated Leg. Journal of Intelligent Robotic Systems,53(4): 331-358.

Hodgins, J. and Raibert, M. ( 1990). Biped gymnastics. The International Journal of Robotics Research (Special Issue on Legged Locomotion), 9(2): 115-132.

10.

Hurst, J.W. and Rizzi, A. ( 2008). Series Compliance for an Efficient Running Gait. IEEE Robotics and Automation Magazine, 15(3): 42-51.

11.

Kajita, S. , Nagasaki, T. , Kaneko, K. , Yokoi, K. and Tanie, K. ( 2004). A Hop towards Running Humanoid Biped. Proceedings of the IEEE International Conference on Robotics and Automation, pp. 629-635.

12.

Kajita, S. , Nagasaki, T. , Kaneko, K. and Hirukawa, H. ( 2007a). ZMP-Based Biped Running Control - The HRP-2LR Humanoid Biped Robot. IEEE Robotics and Automation Magazine, 14(2): 63-72.

13.

Kajita, S. , Kaneko, K. , Morisawa, M. , Nakaoka, S. and Hirukawa, H. ( 2007b). ZMP-based Biped Running Enhanced by Toe Springs. Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3963-3969.

14.

Koditschek, D.E. and Buehler, M. ( 1991). Analysis of A Simplified Hopping Robot. The International Journal of Robotics Research, 10(6): 587-605.

15.

Li, Z. and He, J. ( 1990). An Energy Perturbation Approach to Limit Cycle Analysis in Legged Locomotion Systems. Proceedings of the IEEE International Conference on Decision and Control, pp. 1989-1994.

16.

Lim, B. , Babic, J. and Park, F.C. ( 2008). Optimal jumps for biarticular legged robots. Proceedings of the IEEE International Conference on Robotics and Automation, pp. 226-231.

17.

M’Closkey, R. and Burdick, J. ( 1993). Periodic motions of a hopping robot with vertical and forward motion. The International Journal of Robotics Research , 12(3), 197-218.

18.

Playter, R.R. and Raibert, M.H. ( 1992). Control of a Biped Somersault in 3D. IEEE International Conference on Intelligent Robots and Systems, pp. 582-589.

19.

Poulakakis, I. and Grizzle, J.W. ( 2009). Modeling and Control of the Monopedal Running Robot Thumper . Proceedings of the IEEE International Conference on Robotics and Automation.

20.

Raibert, M. , Brown, H. and Chepponis, M. ( 1984). Experiments in Balance with a 3D One-Legged Hopping Machine . The International Journal of Robotics Research, 3(2): 75-92.

21.

Raibert, M. ( 1986). Legged Robots that Balance. Cambridge, MA, MIT Press.

22.

Roger, David F. ( 2001). An Introduction to NURBS with Historical Perspective . Morgan Kaufmann Publishers.

23.

Sato, Y. , Ohashi, E. and Ohnishi, K. ( 2005). Impact force reduction for hopping robot. Proceedings of the 31st Annual Conference of IEEE Industrial Electronics Society, pp. 6-10.

24.

Tajima, R. and Suga, K. ( 2006). Motion having a Flight Phase: Experiments Involving a One-legged Robot. Proceedings of the IEEE International Conference on Intelligent Robots and Systems, pp. 1726-1731.

25.

Ugurlu, B. and Kawamura, A. ( 2008). Real-time jumping trajectory generation for a one legged jumping robot. Proceedings of the 34th Annual Conference of IEEE on Industrial Electronics, pp. 1668-1673.

26.

Vermeulen, J. , Lefeber, D. and Verrelst, B. ( 2003). Control of Foot Placement, Forward Velocity and Body Orientation of a One-Legged Hopping Robot. Robotica, 21(1): 45-57.

27.

Vukobratovié, M. and Borovac, B. ( 2004). Zero-Moment Point - Thirty Five Years of Its Life. The International Journal of Humanoid Robotics,1(1): 157-173.

28.

Westervelt, E.R. , Buche, G. and Grizzle, J.W. ( 2004). Experimental Validation of a Framework for the Design of Controllers That Induce Stable Walking in Planar Bipeds. The International Journal of Robotics Research, 24(6): 55-68.

29.

Westervelt, E.R. , Grizzle, J.W. , Chevallereau, C. , Choi, J.H. and Morris, B. ( 2007). Feedback Control of Dynamic Bipedal Robot Locomotion . New York, Taylor and Francis, and Boca Raton, FL, CRC Press.

30.

Wu, T.-Y. and Yeh, T.-J. ( 2009). Optimal Design and Implementation of an Energy-Efficient Biped Walking in Semi-Active Manner. Robotica, 27(6): 841-852.

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