Nonholonomic motion planning for a free-falling cat using spline approximation

An optimal motion planning of a free-falling cat based on the spline approximation is investigated. Nonholonomicity arises in a free-falling cat subjected to nonintegrable velocity constraints or nonintegrable conservation laws. The equation of dynamics of a free-falling cat is obtained by using the model of two symmetric rigid bodies. The control of the system can be converted to the motion planning problem for a driftless system. A cost function is used to incorporate the final errors and control energy. The motion planning is to determine control inputs to minimize the cost function and is formulated as an infinite dimensional optimal control problem. By using the control parameterization, the infinite dimensional optimal control problem can be transformed to a finite dimensional one. The particle swarm optimization (PSO) algorithm with the cubic spline approximation is proposed to solve the finite dimension optimal control problem. The cubic spline approximation is introduced to realize the control parameterization. The resulting controls are smooth and the initial and terminal values of the control inputs are zeros, so they can be easily generated by experiment. Simulations are also performed for the nonholonomic motion planning of a free-falling cat. Simulated experimental results show that the proposed algorithm is more effective than the Newtoian algorithm.