This work presents the analysis of the properties of the shortest path control synthesis for the rigid body system. The systems we focus on in this work have only kinematic constraints. However, even for seemingly simple systems and constraints, the shortest paths for generic rigid body systems were only found recently, especially for 3D systems. Based on the Pontraygon's Maximum Principle (MPM) and Lagrange equations, we present the necessary conditions for optimal switches, which form the control synthesis boundaries. We formally show that the shortest path for nearby configurations will have similar adjoint functions and parameters, i.e., Lagrange multipliers. We further show that the gradients of the necessary condition equation can be used to verify whether a configuration is inside a control synthesis region or on the boundary. We present a procedure to find the shortest kinematic paths and control synthesis, using the gradients of the control constraints. Given the shortest path and the corresponding control sequences, the optimal control sequence for nearby configurations can be derived if and only if they belong to the same control synthesis region. The proposed procedure can work for both 2D and 3D rigid body systems. We use a 2D Dubins vehicle system to verify the correctness of the proposed approach. More verifications and experiments will be presented in the extensions of this work.

翻译：这项工作对僵硬机体系统最短路径控制合成的特性进行了分析。我们在此工作中关注的系统只有运动限制。但是,即使看似简单的系统和限制,普通硬体系统最短的路径直到最近才找到,特别是3D系统。根据Pontraygon的最大原理(MPM)和Lagrange方程式,我们提出最佳开关的必要条件,这些开关构成控制合成边界。我们正式表明,附近配置的最短路径将具有类似的连接功能和参数,即拉格兰特乘数。我们进一步表明,必要条件方程式的梯度可用于核查配置是否在控制合成区域或边界上。我们提出了一个程序,以便利用控制制约的梯度找到最短的运动路径和控制合成。根据最短路径和相应的控制序列,只有在附近配置属于同一控制合成区域的情况下,才能得出最优的控制序列。拟议的程序可以适用于2D和3D硬体机体系统。我们用一个程序来核查2D型飞行器的正确性试验。我们用一个程序来核查这一系统。我们用“更多的Dbin ” 的核查系统来核查。