We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time [Formula: see text] and shown to have norm [Formula: see text] which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manifold. So we compute the geodesic curvature and the Gaussian curvature of the Riemannian structure.

中文翻译：

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黎曼流形中使用最优控制的路径规划

我们考虑了黎曼流形中物体的运动规划，其中物体利用最优控制从初始点转向最终点。考虑到 Pontryagin 最小化原则，我们计算了将对象从初始点转向最终点所需的最佳控制。最优控制是关于时间 [公式：见文本] 并显示具有范数 [公式：见文本]，当作为 Pontryagin 原理的解的极值轨迹被弧长参数化时，应该是这种情况。极值轨迹应该是黎曼流形上的测地线。所以我们计算了黎曼结构的测地曲率和高斯曲率。