Dynamical Model and Optimal Turning Gait for Mechanical Rectifier Systems

Dynamical Model and Optimal Turning Gait for Mechanical Rectifier Systems Saba Kohannim and Tetsuya Iwasaki Abstract—Animal locomotion can be viewed as me- chanical rectification due to the dynamics that convert periodic body movements to a positive average thrust, resulting in a steady locomotion velocity. This paper considers a general multi-body mechanical rectifier un- der continuous interactions with the environment, with full rotation and translation in three dimensional space. The equations of motion are developed with respect to body coordinates to allow for direct analysis of maneuvering dynamics. The paper then formulates and solves an optimal turning gait problem for a mechanical rectifier traveling along a curved path, with propul- sive forces generated by periodic body deformation (gait). In particular, the gait is optimized to minimize a quadratic cost function, subject to constraints on average locomotion velocity and average angular veloc- ity. The problem is proven to reduce to two separate, tractable minimization problems solvable for globally optimal solutions. The first problem solves for the opti- mal shape offset that results in turning, while the other solves for the optimal gait that results in locomotion along a straight path. A case study of a locomotor in a fluid environment is presented to demonstrate the utility of the method for robotic locomotor design. Index Terms—Robotics, optimal control, nonlinear systems, biological systems I. Introduction Robotic vehicle designs have been inspired by loco- motion of animals that can efficiently interact with the environment to produce a desired locomotion velocity, and can adapt to environmental changes through modifications in their body movements. Animal locomotion can be regarded as a type of mechanical rectification, in which sustained propulsive forces are produced through the inter- action of the environment with the animal’s periodic body motions (gaits) [1]–[3]. An essential problem in the design of robotic locomotors inspired from animal locomotion is determining a gait that optimizes an important perfor- mance or cost function while satisfying a desired trajectory constraint. This problem has been studied extensively in the literature for various mechanical rectifiers, but due to inherent difficulty, most existing results only provide solutions that are locally optimal. For locally optimal solutions, a standard approach is based on the nonlinear optimal control theory. In [4], Pontryagin maximum principle is used to characterize This material is based upon work supported by the National Sci- ence Foundation (NSF) Graduate Research Fellowship under Grant No. DGE-1144087 and an NSF grant no.1068997. The authors are with the Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, 420 Westwood Plaza, Los Angeles, CA 90095, {sabakohannim,tiwasaki}@ucla.edu. the optimal gait of a seven-link biped robot in terms of a two-point boundary value problem, which is solved by heuristic numerical methods. A similar method was used in discrete-time setting [5] to solve for snake-like link structures. Another popular approach is to reduce the problem to a finite dimensional parametric optimiza- tion by restricting the variables to the span of selected basis functions (e.g. Fourier series, polynomial, piecewise constant). In [6] and [7], this approach is taken to find optimal gaits for an underwater eel-like robot and for a nonholonomic snakeboard, respectively, by solving a stationarity condition using Newton iteration algorithms. The parametric approach is also used for biped robots with direct numerical optimizations through sequential quadratic programming [8], [9], steepest gradient descent method [10], and a commercial software package [11]. There are a few approaches for computing global solu- tions of certain optimal gait problems. One approach is in- spired by biology, where an optimization problem is formu- lated over a narrow set of possible gaits that are observed in animal locomotion. In particular, for serpent crawling and eel swimming, the animals exhibit undulatory body movements with waves traveling down the body, which can be parameterized by sinusoidally time-varying curvature with constant amplitudes and linearly decreasing phases over the slender body. Since the parameter space of various gaits is restricted, it is possible to find the globally optimal solution by simulations on gridded parameter points [12] or by analytical perturbation methods [13]. These meth- ods find reasonable gaits, but can miss better gaits that deviate from those observed in biology. Another approach to globally optimal gaits is to re- strict the class of underlying locomotion dynamics rather than the class of possible gaits. Reference [14] consid- ers a general class of mechanical rectifiers that are in continuous interactions with the environment, including swimming, flying, and slithering. A simplified bilinear model is developed under the assumption of small cur- vature deformation, capturing essential rectifier dynamics necessary for locomotion. An optimal gait problem is then formulated as a minimization of a quadratic cost function over all periodic body movements achievable with a given set of actuators, subject to a constraint on the average locomotion velocity. The globally optimal solution is obtained using generalized eigenvalue computation, and the method’s utility is validated by case studies of a link- chain rectifier swimming in water. The methods mentioned above are primarily used to find optimal gaits for locomotion along a straight line, but the