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Abstract

In this paper, we generalize our prior results in motion analysis to design gaits for a more general family of underactuated mechanical systems. In particular, we analyze and generate gaits for mixed mechanical systems which are systems whose motion is simultaneously governed by both a set of non-holonomic velocity constraints and a notion of a generalized momentum being instantaneously conserved along allowable directions of motion. Through proper recourse to geometric mechanics, we are able to show that the resulting motion from a gait has two portions: a geometric and a dynamic contribution. The main challenge in motion planning for a mixed system is understanding how to separate the geometric and dynamic contributions of motion due to a general gait, thus simplifying gait analysis. In this paper, we take the first step towards addressing this challenge in a generalized framework. Finally, we verify the generality of our approach by applying our techniques to novel mechanical systems which we introduce in this paper as well as by verifying that seemingly different prior motion planning results could actually be explained using the gait analysis presented in this paper.

Keywords

gait generation motion control underactuated robots non-holonomic motion planning mixed mechanical systems generalized momentum scaled momentum variable inertia snakeboard Stokes' theorem

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References

Bloch, A. , Baillieul, J. , Crouch, P.E. and Marsden, J.E. (2003). Nonholonomic Mechanics and Control. Berlin, Springer.

Brockett, R.W. (1981) Control theory and singular Riemanian geometry . New Directions in Applied Mathematics, Hilton, P. J. and Young, G. S. (eds). New York, Springer , pp. 11—27.

Brockett, R.W. and Dai, L. (1993). Nonholonomic kinematics and the role of elliptic functions in constructive controllability. Nonholonoimc Motion Planning, Li, Z. and Canny, J. F. (eds). New York, Kluwer, pp. 1—21.

Bullo, F. and Lewis, A.D. (2003). Kinematic controllability and motion planning for the snakeboard. IEEE Transactions on Robotics and Automation , 19(3): 494—498.

Bullo, F. and Lewis, A.D. (2004). Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems ( Texts in Applied Mathematics, vol. 49). New York, Springer.

Bullo, F. and Lynch, K.M. (2001a). Kinematic controllability and decoupled trajectory planning for underactuated mechanical systems. Proceedings of the IEEE International Conference on Robotics and Automation , Seoul, Korea, May, pp. 3300—3307.

Bullo, F. and Lynch, K.M. (2001b). Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems. IEEE Transactions on Robotics and Automation, 17(4): 402—412.

Bullo, F. , Lewis, A.D. and Lynch, K.M. (2002). Controllable kinematic reductions for mechanical systems: concepts, computational tools, and examples. Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, August.

Chitta, S. , Heger, F. and Kumar, V. (2004) Dynamics and gait control of a rollerblading robot . Proceedings of the IEEE International Conference on Robotics and Automation, vol. 4, May, pp. 3944—3949.

10.

Chitta, S. , Cheng, P. , Frazzoli, E. and Kumar, V. (2005). RoboTrikke: a novel undulatory locomotion system . Proceedings of the IEEE International Conference on Robotics and Automation, April, pp. 1597—1602.

11.

Choudhury, P. and Lynch, K.M. (2002). Trajectory planning for kinematically controllable underactuated mechanical systems. Proceedings of 2002 Workshop on the Algorithmic Foundations of Robotics, Nice, France.

12.

Iannitti, S. and Lynch, K.M. (2003). Exact minimum control switch motion planning for the snakeboard. Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 2, Las Vegas, NV, pp. 1437— 1443.

13.

Iannitti, S. and Lynch, K.M. (2004). Minimum control switch motions for the snakeboard: a case study in kinematically controllable underactuated systems. IEEE Transactions on Robotics, 20: 994—1006.

14.

Krishnaprasad, P.S. and Tsakiris, D.P. (2001). Oscillations, SE(2)-snakes and motion control: a study of the roller racer. Dynamical Systems, 16(4): 347—397.

15.

Krishnaprasad, P.S. and Tsakiris, D.P. (1994a). G-Snakes: nonholonomic kinematic chains on lie groups. Proceedings of the 33rd IEEE Conference on Decision and Control, vol. 3, Lake Buena Vista, Florida, December, pp. 2955— 2960.

16.

Krishnaprasad, P.S. and Tsakiris, D.P. (1994b). 2-module nonholonomic variable geometry truss assembly: motion control. Proceedings of the 4th IFAC Symposium on Robot Control (SYROCO'94), Capri, Italy, September.

17.

Krishnaprasad, P.S. and Tsakiris, D.P. (1995) Oscillations, SE(2)-snakes and motion control. Proceedings of the 34th IEEE Conference on Decision and Control , vol. 3, New Or-leans, LA, December, pp. 2806—2811.

18.

Lewis, A.D. (1999). When is a mechanical control system kinematic ? Proceedings of the 38th IEEE Conference on Decision and Control, vol. 2, December, pp. 1162—1167.

19.

Lugo, G. Differential geometry in physics. Technical Report, University of North Carolina.

20.

Marsden, J.E. and Ratiu, T.S. (1994). Introduction to Mechanics and Symmetry. Berlin, Springer.

21.

Marsden, J.E. , Montgomery, R. and Ratiu, T.S. (1990). Reduction, Symmetry and Phases in Mechanics (Memoirs of the American Mathematical Society, vol. 436). Providence, RI, American Mathematical Society.

22.

Mukherjee, R. and Anderson, D.P. (1993). Nonholonomic motion planning using Stoke's theorem . Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, May, pp. 802—809.

23.

Murray, R.M. and Sastry, S.S. (1993). Nonholonomic motion planning: steering using sinusoids . IEEE Transactions on Automatic Control, 38(5): 700—716.

24.

Nakamura, Y. and Mukherjee, R. (1989). Nonholonomic path planning of space robots. Proceedings of the IEEE International Conference on Robotics and Automation , vol. 2, May, pp. 1050—1055.

25.

Nakamura, Y. and Mukherjee, R. (1991). Nonholonomic path planning of space robots via a bidirectional approach. IEEE Transactions on Robotics and Autmation, 7: 500—514.

26.

Ostrowski, J. (1995). The mechanics of control of undulatory robotic locomotion. Ph.D. Thesis, California Institute of Technology .

27.

Ostrowski, J. and Burdick, J.W. (1998). The mechanics and control of undulatory locomotion . International Journal of Robotics Research, 17(7): 683—701.

28.

Ostrowski, J. , Lewis, A. , Murray, R. and Burdick, J.W. (1994). Nonholonomic mechanics and locomotion: the snakeboard example. Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, May, pp. 2391— 2397.

29.

Ostrowski, J. , Desai, J. and Kumar, V. (2000). Optimal gait selection for nonholonomic locomotion systems. International Journal of Robotics Research, 19(3): 225— 237.

30.

Shammas, E. , Choset, H. and Rizzi, A.A. (2007). Geometric motion planning analysis for two classes of underactuated mechanical systems. International Journal of Robotics Research, 26(10): 1043—1073.

31.

Tenenbaum, M. and Pollard, H. (1985). Ordinary Differential Equations. New York, Dover.

32.

Tsakiris, D.P. (1995). Motion control and planning for nonholonomic kinematic chains. Ph.D. Thesis, College Park, MD, University of Maryland, Department of Electrical Engineering .

33.

Walsh, G.C. and Sastry, S.S. (1991). On reorienting linked rigid bodies using internal motions. Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, December, pp. 1190—1195.

34.

Walsh, G.C. and Sastry, S.S. (1995). On reorienting linked rigid bodies using internal motions. IEEE Transactions on Robotics and Automation, 11(1): 139—146.

35.

Yamada, K. (1993). Arm path planning for a space robot. Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 3, July, pp. 2049—2055.

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