This work presents a trajectory planning method based on composite Bernstein polynomials for autonomous systems navigating complex environments. The method is implemented in a symbolic optimization framework that enables continuous paths and precise control over trajectory shape. Trajectories are planned over a cost surface that encodes obstacles as continuous fields rather than discrete boundaries. Regions near obstacles are assigned higher costs, naturally encouraging the trajectory to maintain a safe distance while still allowing efficient routing through constrained spaces. The use of composite Bernstein polynomials preserves continuity while enabling fine control over local curvature to satisfy geodesic constraints. The symbolic representation supports exact derivatives, improving optimization efficiency. The method applies to both two- and three-dimensional environments and is suitable for ground, aerial, underwater, and space systems. In spacecraft trajectory planning, for example, it enables the generation of continuous, dynamically feasible trajectories with high numerical efficiency, making it well suited for orbital maneuvers, rendezvous and proximity operations, cluttered gravitational environments, and planetary exploration missions with limited onboard computational resources. Demonstrations show that the approach efficiently generates smooth, collision-free paths in scenarios with multiple obstacles, maintaining clearance without extensive sampling or post-processing. The optimization incorporates three constraint types: (1) a Gaussian surface inequality enforcing minimum obstacle clearance; (2) geodesic equations guiding the path along locally efficient directions on the cost surface; and (3) boundary constraints enforcing fixed start and end conditions. The method can serve as a standalone planner or as an initializer for more complex motion planning problems.

翻译：本文提出了一种基于复合伯恩斯坦多项式的轨迹规划方法，适用于在复杂环境中导航的自主系统。该方法在一个符号化优化框架中实现，能够生成连续路径并对轨迹形状进行精确控制。轨迹规划在一个将障碍物编码为连续场而非离散边界的代价面上进行。障碍物邻近区域被赋予更高的代价，这自然促使轨迹在保持安全距离的同时，仍能高效地穿越受限空间。复合伯恩斯坦多项式的使用在保持连续性的同时，实现了对局部曲率的精细控制，以满足测地线约束。其符号化表示支持精确导数计算，从而提升了优化效率。该方法适用于二维和三维环境，可应用于地面、空中、水下及空间系统。例如，在航天器轨迹规划中，它能够以高数值效率生成连续且动力学可行的轨迹，因此非常适用于轨道机动、交会对接与近距离操作、复杂引力环境以及星载计算资源有限的行星探测任务。实验表明，该方法能在多障碍物场景中高效生成平滑、无碰撞的路径，在无需大量采样或后处理的情况下保持安全间隙。优化过程包含三类约束：(1) 强制执行最小障碍物间隙的高斯曲面不等式；(2) 引导路径沿代价面上局部最优方向的测地线方程；(3) 强制固定起始与终止条件的边界约束。该方法既可作为一个独立的规划器，也可作为更复杂运动规划问题的初始化器。