This paper presents a method for optimal motion planning of continuum robots by employing Bernstein surfaces to approximate the system's dynamics and impose complex constraints, including collision avoidance. The main contribution is the approximation of infinite-dimensional continuous problems into their discrete counterparts, facilitating their solution using standard optimization solvers. This discretization leverages the unique properties of Bernstein surface, providing a framework that extends previous works which focused on ODEs approximated by Bernstein polynomials. Numerical validations are conducted through several numerical scenarios. The presented methodology offers a promising direction for solving complex optimal control problems in the realm of soft robotics.

翻译：本文提出了一种采用伯恩斯坦曲面逼近系统动力学并施加复杂约束（包括避障）的连续型机器人最优运动规划方法。主要贡献在于将无限维连续问题近似为离散问题，从而利用标准优化求解器进行求解。该离散化方法利用了伯恩斯坦曲面的独特性质，构建了一个扩展先前基于伯恩斯坦多项式逼近常微分方程研究的框架。通过多个数值场景进行了数值验证。所提出的方法为软体机器人领域中复杂最优控制问题的求解提供了有前景的方向。