YAO Qijia, GE Xinsheng. Dynamics Modeling and Motion Planning for Snakeboard Systems Based on Differential Geometry[J]. Applied Mathematics and Mechanics, 2018, 39(1): 29-40. doi: 10.21656/1000-0887.380107
 Citation: YAO Qijia, GE Xinsheng. Dynamics Modeling and Motion Planning for Snakeboard Systems Based on Differential Geometry[J]. Applied Mathematics and Mechanics, 2018, 39(1): 29-40.

# Dynamics Modeling and Motion Planning for Snakeboard Systems Based on Differential Geometry

##### doi: 10.21656/1000-0887.380107
Funds:  The National Natural Science Foundation of China（11472058）
• Rev Recd Date: 2017-05-23
• Publish Date: 2018-01-15
• Dynamics modeling and motion planning for snakeboard systems were investigated, and a hybrid optimization strategy based on the genetic algorithm (GA) and the Gauss pseudospectral method (GPM) was presented. Firstly, the Euler-Lagrange equations for the snakeboard system were derived based on the Riemannian manifold and the affine connection theory in differential geometry. The configuration space of the snakeboard corresponds to the manifold space, the velocity space corresponds to the tangent space, the torque space corresponds to the cotangent space, and the inertia matrix provides a Riemannian measure on the manifold. The set of admissible velocities were represented by the appropriate bases to simplify the kinematics equations. Then the optimal motion planning problem was transformed into a nonlinear programming problem with the GPM. The optimal trajectory and the optimal control inputs were obtained with the sequential quadratic programming (SQP) algorithm. The GA was applied to generate the initial values of the GPM. Finally, through numerical simulation, the optimal trajectory agrees well with actual conditions, and the control inputs match the various constraints closely. The results demonstrate the effectiveness of the proposed method.
•  [1] OSTROWSKI J P, BURDICK J W. Controllability tests for mechanical systems with constraints and symmetries[J]. Journal of Applied Mathematics and Computer Science,1997,7(2): 101-127. [2] OSTROWSKI J P, DESAI J P, KUMAR V. Optimal gait selection for nonholonomic locomotion systems[J]. The International Journal of Robotics Research,1999,19(3): 225-237. [3] OSTROWSKI J P. Steering for a class of dynamic nonholonomic systems[J]. IEEE Transactions on Automatic Control,2000,45(8): 1492-1498. [4] LEWIS A D. Simple mechanical control systems with constraints[J]. IEEE Transactions on Automatic Control,2000,45(8): 1420-1436. [5] BULLO F, EFRAN M. On mechanical control systems with nonholonomic constraints and symmetries[J]. Systems & Control Letters,2002,45(2): 133-143. [6] BULLO F, LEWIS A D. Kinematic controllability and motion planning for the snakeboard[J]. IEEE Transactions on Robotics and Automation,2003,19(3): 494-498. [7] ASNAFI A R, MAHZOON M. Some flower-like gaits in the snakeboard’s locomotion[J]. Nonlinear Dynamics,2007,48(1/2): 77-89. [8] SHAMMAS E A, DE OLIVEIRA M. Motion planning for the snakeboard[J]. The International Journal of Robotics Research,2012,31(7): 872-885. [9] 刘延柱, 苗英恺. 活力板运动的动力学分析[J]. 力学与实践, 2008,30(3): 60-62.(LIU Yanzhu, MIAO Yingkai. Dynamic analysis of the motion of vigor board[J]. Mechanics in Engineering,2008,30(3): 60-62.(in Chinese)) [10] 丁洁玉, 潘振宽. 非完整约束多体系统时间离散变分积分法[J]. 动力学与控制学报, 2011,9(4): 289-292.(DING Jieyu, PAN Zhenkuan. Time-discrete variational integrator for multibody dynamic systems with nonholonomic constraints[J]. Journal of Dynamics and Control,2011,9(4): 289-292.(in Chinese)) [11] 郭宪, 马书根, 李斌, 等. 基于微分几何的蛇形机器人动力学与控制统一模型[J]. 中国科学: 信息科学, 2015,45(8): 1080-1094.(GUO Xian, MA Shugen, LI Bin, et al. Dynamics-control unified model of a snakelike robot based on differential geometry[J]. Scientia Sinica: Informationis,2015,45(8): 1080-1094.(in Chinese)) [12] 郭宪, 马书根, 李斌, 等. 基于动力学与控制统一模型的蛇形机器人速度跟踪控制方法研究[J]. 自动化学报, 2015,41(11): 1847-1856.(GUO Xian, MA Shugen, LI Bin, et al. Velocity tracking control of a snake-like robot with a dynamics and control unified model[J]. Acta Automatica Sinica,2015,41(11): 1847-1856.(in Chinese)) [13] 唐国金, 罗亚中, 雍恩米. 航天器轨迹优化理论、方法及应用[M]. 北京: 科学出版社, 2012.(TANG Guojin, LUO Yazhong, YONG Enmi. Spacecraft Trajectory Optimization Theory, Method and Application [M]. Beijing: Science Press, 2012.(in Chinese)) [14] ELNAGAR G, KAZEMI M A, RGZZAGHI M. The pseudospectral Legendre method for discretizing optimal control problems[J]. IEEE Transactions on Automatic Control,1995,40(10): 1793-1796. [15] BENSON D A. A Gauss pseudospectral transcription for optimal control[D]. PhD Thesis. Cambridge: Massachusetts Institute of Technology, 2005. [16] BENSON D A, HUNTINGTON G T, THORVALDSEN T P, et al. Direct trajectory optimization and costate estimation via an orthogonal collocation method[J]. Journal of Guidance, Control, and Dynamics,2006,29(6): 1435-1439. [17] HUNTINGTON G T, RAO A V. Optimal reconfiguration of spacecraft formations using the Gauss pseudospectral method[J]. Journal of Guidance, Control, and Dynamics,2008,31(3): 689-698.〖JP〗 [18] 廖一寰, 李道奎, 唐国金. 基于混合规划策略的空间机械臂运动规划研究[J]. 宇航学报, 2011,32(1): 98-103.(LIAO Yihuan, LI Daokui, TANG Guojin. Motion planning of space manipulator system based on a hybrid programming strategy[J]. Journal of Astronautics,2011,32(1): 98-103.(in Chinese)) [19] 孙勇, 张卯瑞, 梁晓玲. 求解含复杂约束非线性最优控制问题的改进Gauss伪谱法[J]. 自动化学报, 2013,39(5): 672-678.(SUN Yong, ZHANG Maorui, LIANG Xiaoling. Improved Gauss pseudospectral method for solving nonlinear optimal control problem with complex constraints[J]. Acta Automatica Sinca,2013,39(5): 672-678.(in Chinese)) [20] 董雪仰, 戈新生. 航天器太阳帆板展开过程最优控制的自适应Gauss伪谱法[J]. 应用数学和力学, 2016,37(6): 655-664.(DONG Xueyang, GE Xinsheng. The adaptive Gauss pseudospectral method for the optimal control of spacecraft solar array deployment[J].Applied Mathematics and Mechanics,2016,37(6): 655-664.(in Chinese))

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