基于微分几何的蛇板系统动力学建模与运动规划

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

研究了蛇板系统的动力学建模与运动规划问题,提出一种遗传算法与Gauss伪谱法相结合的混合优化策略.首先,基于微分几何中的Riemann(黎曼)流形与仿射映射理论,建立蛇板系统在其构型流形上的Euler-Lagrange(欧拉-拉格朗日)方程.蛇板的构型空间对应流形空间,速度空间对应流形切空间,力矩空间对应流形余切空间,惯量矩阵提供了流形空间上的一个Riemann度量.构造适当的基底描述蛇板系统的许可速度,可以使蛇板系统的运动方程得到简化.然后,利用Gauss伪谱法将蛇板系统运动规划问题离散为非线性规划问题,利用序列二次规划算法求解蛇板系统的运动轨迹与最优控制输入,其中,Gauss伪谱法的初值通过遗传算法得到.最后,通过数值仿真,蛇板系统的运动轨迹与实际情况吻合,最优控制输入也能很好地满足约束条件,验证了该混合优化策略的有效性.

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.