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Natural Oscillation for Undulatory Locomotion and Optimal Autonomous Pattern Formation


Pattern formation is everywhere in both biology and engineering. A few processes in which it

occurs are coupled oscillators, synchronization, multi-agent consensus, central pattern generators

(CPGs), and dynamic systems using repetitive movements such as animal-like gaits. How to determine the pattern most preferable for a given process is just as important as how to best achieved

a given pattern. ‘Preferred’ and ‘best’ being nothing more than vague adjectives with positive connotations, we explore these two ideas through research on two more precisely defined problems:

defining a ‘natural gait’ for undulatory locomotion and solving the optimal LTI output feedback

control problem for autonomous pattern formation.

Robotic vehicles inspired by animal locomotion are propelled by interactive forces from the

environment resulting from periodic body movements. The pattern of body oscillation (gait) can

be mimicked from animals, but understanding the principles underlying the gait generation would

improve our understanding of nature and allow for broader, more flexible engineering applications. We hypothesize that the traveling-wave oscillations observed in undulatory locomotion can

be characterized as a natural oscillation of the locomotion dynamics, and accordingly propose

a formal definition of a natural gait. We first identify the dynamics essential to undulatory locomotion, and define the mode shape of natural oscillation by the free response of an idealized

system. We then use body-environment resonance to define the amplitude and frequency of the

oscillations. Explicit formulas for the natural gait are derived, assuming uniformity of the body, to


provide insight into the mechanisms underlying undulatory locomotion. Examples of a swimming

leech and a fictitious swimmer with a non-uniform body illustrate how undulatory gaits similar to

those observed in nature can be produced as the natural gait, and how they can be modulated to

achieve a variety of swim speeds. Lastly, a linear feedback controller using damping compensation

is developed that makes the natural gait a limit cycle in the resulting nonlinear closed-loop system.

Next we solve the optimal, linear, output feedback problem in which the controller is autonomous and achieves pattern formation. Here the term ‘pattern’ denotes the behavior of the

individual plant states, e.g. periodic, constant bias, or stable. When applicable, it may also designates the relative magnitudes and phases of the multi-state target trajectory. The cost is defined as

the L2 norm of the transient portion of the impulse response. The solution is obtained by optimizing over freedom in the general solution to eigenstructure assignment theory, wherein the target

trajectory is determined by a ‘pattern generator’ embedded in the feedback loop. Freedom in the

pattern generator which corresponds to freedom in the target trajectory significantly increases the

complexity of the optimization problem with respect to the standard regulator problem. It also

allows for further cost reduction. We draw theoretically significant parallels and distinctions between this unique result and the classical optimal H2 norm result which combines a linear quadratic

regulator (LQR) and Kalman filter observer in accordance with the separation principle. A design

example illustrates the controllers ability to calculate and achieve the optimal target trajectory following a disturbance. We also demonstrate how adding nonlinearites to the controller results in a

limit cycle at a specified amplitude.

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