Publication
Collocation methods for second and higher order systems
Journal Article (2024)
Journal
Autonomous Robots
Pages
1-20
Volume
48
Number
2
Doc link
https://doi.org/10.1007/s10514-023-10155-z
File
Authors
Projects associated
Abstract
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 Mth 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 dynamics transcription error in one order of magnitude, or even more, without noticeably increasing the cost of computing the solutions.
Categories
nonlinear programming, optimal control, robot dynamics.
Author keywords
Collocation methods, trajectory optimization, optimal control, second and higher order systems
Scientific reference
S. Moreno, L. Ros and E. Celaya. Collocation methods for second and higher order systems. Autonomous Robots, 48(2): 1-20, 2024, to appear.
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