We consider the speed planning problem for a vehicle moving along an assigned trajectory, under maximum speed, tangential and lateral acceleration, and jerk constraints. The problem is a nonconvex one, where nonconvexity is due to jerk constraints. We propose a convex relaxation, and we present various theoretical properties. In particular, we show that the relaxation is exact under some assumptions. Also, we rewrite the relaxation as a Second Order Cone Programming (SOCP) problem. This has a relevant practical impact, since solvers for SOCP problems are quite efficient and allow solving large instances within tenths of a second. We performed many numerical tests, and in all of them the relaxation turned out to be exact. For this reason, we conjecture that the convex relaxation is exact, although we could not give a formal proof of this fact.

中文翻译：

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加加速度约束下的时间最优速度规划是凸问题吗？


我们考虑车辆在最大速度、切向和横向加速度以及加加速度约束下沿指定轨迹移动的速度规划问题。该问题是一个非凸问题，其中非凸性是由于加加速度约束造成的。我们提出了凸松弛，并提出了各种理论性质。特别是，我们证明了在某些假设下松弛是精确的。此外，我们将松弛重写为二阶锥规划（SOCP）问题。这具有相关的实际影响，因为 SOCP 问题的求解器非常高效，并且允许在十分之一秒内解决大型实例。我们进行了许多数值测试，所有这些测试结果证明松弛都是准确的。因此，我们推测凸松弛是精确的，尽管我们无法给出这一事实的正式证明。