  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.
