Optimal Gaits for Mechanical Rectifier Systems

• Published in: IEEE transactions on automatic control 2011-01, Vol.56 (1), p.59-71
• Main Authors: Blair, J, Iwasaki, T
• Format: Article
• Language: English
• Subjects: Aerospace engineering; Animals; Applied sciences; Biological control systems; Computer science; control theory; systems; Control theory. Systems; Cost function; Dynamics; Exact sciences and technology; Gait; Legged locomotion; Locomotion; Mathematical analysis; Mathematical models; motion-planning; Optimal control; Optimization; Optimization methods; Rectifiers; Robotics; Robots; Searching; Shape; Studies; Vehicle dynamics
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2010.2051074

The essential mechanism underlying animal locomotion can be viewed as mechanical rectification that converts periodic body movements to thrust force through interactions with the environment. This paper defines a general class of mechanical rectifiers as multi-body systems equipped with such thrust generation mechanisms. A simple model is developed from the Euler-Lagrange equation by assuming small body oscillations around a given nominal posture. The model reveals that the rectifying dynamics can be captured by a bilinear, but not linear, term of body shape variables. An optimal gait problem is formulated for the bilinear rectifier model as a minimization of a quadratic cost function over the set of periodic functions subject to a constraint on the average locomotion velocity. We prove that a globally optimal solution is given by a harmonic gait that can be found by generalized eigenvalue computation with a line search over cycle frequencies. We provide case studies of a chain of links for which snake-like undulations and jellyfish-like flapping gaits are found to be optimal.