It is often unnoticed that the predominant way to use collocation methods is fundamentally flawed when applied to optimal control in robotics. Such methods assume that the system dynamics is given by a first order ODE, whereas robots are often governed by a second or higher order ODE involving configuration variables and their time derivatives. To apply a collocation method, therefore, the usual practice is to resort to the well known procedure of casting an M th order ODE into M first order ones. This manipulation, which in the continuous domain is perfectly valid, leads to inconsistencies when the problem is discretized. Since the configuration variables and their time derivatives are approximated with polynomials of the same degree, their differential dependencies cannot be fulfilled, and the actual dynamics is not satisfied, not even at the collocation points. This paper draws attention to this problem, and develops improved versions of the trapezoidal and Hermite-Simpson collocation methods that do not present these inconsistencies. In many cases, the new methods reduce the dynamic transcription error in one order of magnitude, or even more, without noticeably increasing the cost of computing the solutions.

翻译：人们往往未曾注意到，当配点法用于机器人最优控制时，其主流应用方式存在根本性缺陷。这类方法假设系统动力学由一阶常微分方程描述，而机器人系统通常由包含构型变量及其时间导数的二阶或高阶常微分方程主导。因此，为了应用配点法，常规做法是采用将M阶常微分方程转化为M个一阶方程的经典技巧。这种在连续域中完全有效的数学操作，在问题离散化时会导致不一致性。由于构型变量及其时间导数采用同阶多项式近似，其微分依赖关系无法满足，实际动力学方程即使在配点处也未能得到满足。本文聚焦此问题，发展了改进型梯形配点法和埃尔米特-辛普森配点法，这些方法不存在上述不一致性。在许多案例中，新方法可将动力学转录误差降低一个数量级甚至更多，且不会显著增加求解计算成本。