Technical Brief

DirCol5i: Trajectory Optimization for Problems With High-Order Derivatives

[+] Author and Article Information
Matthew P. Kelly

Biorobotics and Locomotion Lab,
Department of Mechanical Engineering,
Cornell University,
Ithaca, NY 14850
e-mail: mpk72@cornell.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received September 16, 2017; final manuscript received September 26, 2018; published online November 8, 2018. Assoc. Editor: Richard Bearee.

J. Dyn. Sys., Meas., Control 141(3), 034502 (Nov 08, 2018) (6 pages) Paper No: DS-17-1468; doi: 10.1115/1.4041610 History: Received September 16, 2017; Revised September 26, 2018

In this technical brief, we focus on solving trajectory optimization problems that have nonlinear system dynamics and that include high-order derivatives in the objective function. This type of problem comes up in robotics—for example, when computing minimum-snap reference trajectories for a quadrotor or computing minimum-jerk trajectories for a robot arm. DirCol5i is a transcription method that is specialized for solving this type of problem. It uses the fifth-order splines and analytic differentiation to compute higher-derivatives, rather than using a chain-integrator as would be required by traditional methods. We compare DirCol5i to traditional transcription methods. Although it is slower for some simple optimization problems, when solving problems with high-order derivatives DirCol5i is faster, more numerically robust, and does not require setting up a chain integrator.

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Grahic Jump Location
Fig. 2

Cart-pole swing-up: trace made by pendulum bob along the minimum-force and minimum-snap trajectories. Straight lines show the pendulum at uniformly spaced time intervals along the swing-up trajectory. Note that point p in the minimum-jerk and minimum-snap objective functions is the position of the tip of the pole.

Grahic Jump Location
Fig. 1

Cart-pole model: point mass cart and pendulum bob, connected by a rigid (massless) bar. A motor applies a horizontal force to the cart.

Grahic Jump Location
Fig. 3

Biped gait: optimal solution: periodic gait for the five-link biped



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