Pattern Formation
Objectives and Significance
A large number of dynamical components can be interconnected and interact with each other to form an integrated system with certain functionalities. Such complex systems are found in nature and have been created by human. A common feature of these systems is that a global pattern emerges as a result of local, distributed, dynamical interactions of components. It is desired to understand the mechanisms underlying this feature, analyze existing complex systems, and design and create innovative systems with new functionalities. The central pattern generator (CPG) is a representative example of such complex systems, in which a distributed network of neurons forms a nonlinear oscillator to create specific patterns for rhythmic body movements. This research aims to develop a dynamical systems theory that enables analysis and design of such network systems with a prescribed motion pattern through distributed computations embedded in the network interconnections. The theory will provide a foundation for designing controllers for robotic systems with multiple motion primitives that can be autonomously switched through sensory feedback by detecting changes in the environment. The theory could also be applied at a macroscopic level to coordinate multiple local subsystems (e.g. mobile robots) for global functionality.
Results
We consider a class of CPGs modeled by a set of interconnected identical neurons. Each neuron is modeled by a time lag followed by a threshold nonlinearity. Based on the idea of multivariable harmonic balance, we have found how the oscillation profile is related to the connectivity matrix that specifies the architecture and strengths of the interconnections. Specifically, the frequency, amplitudes, and phases are essentially encoded in terms of a pair of eigenvalue and eigenvector. This basic principle is used to estimate the oscillation profile of a given CPG model. Moreover, a systematic method is proposed for designing a CPG-based nonlinear oscillator that achieves a prescribed oscillation profile.
Example on left: A five-neuron CPG is designed with a constraint on the neural coupling structure. The model simulation verifies that the design achieves the target oscillation approximately.
The CPG design by the harmonic balance method is fairly effective for practical purposes but has several drawbacks. The temporal shape of the target oscillation has to be close to sinusoids, the method lacks theoretical guarantee for stability of the limit cycle oscillation, and the design relies on numerical optimization which could fail.
We have used an abstracted version of the neural dynamics to resolve all these issues. A simple oscillator with a scalar complex variable is developed from the Andronov-Hopf oscillator and is used as the basic neural unit in the network. Multiple limit cycles of prescribed profiles can be embedded in the network with stability guarantee, where the interconnection topology is arbitrarily specified. Moreover, the output signals can be shaped in an arbitrary manner.
Example on right: A four-segment, eight-neuron CPG is designed with a constraint on the neural coupling structure to achieve two stable limit cycles with arbitrary temporal shapes. The model simulation demonstrates stable transition from one limit cycle to the other.
A further generalization has been made to allow for heterogeneous subsystem dynamics, albeit in the linear setting. We formulate a general pattern formation problem as the design of a feedback controller such that selected outputs of a linear plant exponentially converge to a prescribed spatial/temporal pattern. We show that the problem reduces equivalently to an eigenstructure assignment problem, and provide a necessary and sufficient condition for existence of a feasible controller as well as a parameterization of all such controllers. This general theory is specialized to give a complete solution to a heterogeneous multi-agent synchronization problem. We reveal the importance of sensory feedback for adaptive pattern formation, and suggest an extension for achieving stable limit cycles by additional nonlinearities.
References
Linear Quadratic Regulator for Autonomous Oscillation
D.T. Ludeke and T. Iwasaki, American Control Conference, pp.4891-4896, 2019.
Design of Complex Oscillator Network with Multiple Limit Cycles
K. Ren and T. Iwasaki, IEEE Conference on Decision and Control, pp.115-120, 2018.
Pattern Formation Via Eigenstructure Assignment: General Theory and Multi-Agent Application
A. Wu and T. Iwasaki, IEEE Transactions on Automatic Control, vol.63, no.7, 1959-1972, 2018, accepted version
Design of Coupled Harmonic Oscillators for Synchronization and Coordination
X. Liu and T. Iwasaki, IEEE Transactions on Automatic Control, vol.62, no.8, 3877-3889, 2017, accepted version
Circulant Synthesis of Central Pattern Generators with Application to Control of Rectifier Systems
Z. Chen and T. Iwasaki, IEEE Transactions on Automatic Control, vol.53, no.1, pp.273-286, 2008, accepted version
Multivariable harmonic balance for central pattern generators
T. Iwasaki, Automatica, vol.44, pp.3061-3069, 2008.
Pattern Formation via Eigenstructure Assignment
Andy Wu, Ph.D Dissertation, UCLA, June 2016.
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