CPG model for autonomous decentralized multi-legged robot system—generation and transition of oscillation patterns and dynamics of oscillators

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Abstract

A central pattern generator (CPG) model is proposed for the gait-pattern generation mechanism of an autonomous decentralized multi-legged robot system. The topological structure of the CPG is represented as a graph on which two time evolution systems, the Hamilton system and a gradient system, are introduced. The CPG model can generate oscillation patterns depending only on the network topology and can bifurcate different oscillation patterns according to the network energy, which means that the robot can generate gait patterns by connecting legs and transit gait patterns according to such parameters as the desired speed.

Introduction

Legged animals can generate walk patterns (gaits) suitable to their walking speed. The gait-generating system is achieved by the central pattern generator (CPG), a group of neurons located in a central nervous system (Fig. 1) [1]. The CPG is modeled as a system of coupled nonlinear oscillators because of its periodical output (oscillation patterns) and is used to coordinate leg movements (Fig. 1) [2], [3], [4], [5], [6].

The autonomous decentralized multi-legged robot system has a leg as an autonomous partial system (subsystem) [7], [8]. Fig. 2 shows the autonomous decentralized multi-legged robot: NEXUS. Each subsystem has oscillators to control the leg movement. This robot system achieves the function of a CPG as a whole system by coupling with neighbor oscillators, i.e., by coupling with neighbor subsystems. Each subsystem does not need to discover the total number of subsystems or its position in the network. However, this robot system is expected to have properties such as a changeable number of legs, easy maintenance, failure tolerance, and environmental adaptability.

Many CPG models have been proposed in previous research. However, these models, which are divided into two types [5], cannot deal with changes in the number of oscillators and the number of legs. One type requires a phase relation of common gait patterns according to the number of oscillators [2], [3]. The other type is required to analyze how to change parameters and input in various cases [5], [6].

The purpose of our research is the construction of a CPG model with the following features in an autonomous decentralized multi-legged robot system:

  • 1.

    Gait patterns can be generated automatically according to the number of legs without embedded gait patterns.

  • 2.

    Transitions in gait patterns can be generated by changing a certain parameter, e.g., the network energy.

This paper is about gait control while the number of legs (the configure of the robot) changes. There are pioneering researches regarding to the locomotion of the self-reconfiguration robot [9]. The method proposed in this paper is different from the previously proposed methods in two aspects. First, the locomotion is achieved from the relationship of the neighbor legs. The other, each leg does not need to know the topology (where the leg is in the robot). In this paper, a method for constructing a CPG model is proposed from the viewpoint of the graph expression of the oscillator network and the oscillator dynamics. The oscillator and the interaction correspond to a vertex and an edge, respectively (Fig. 3). Thereby, the oscillation patterns result in the eigen value problem of the graph and the transition is achieved only by changing the oscillation energy (network energy). Furthermore, the locality of the subsystem is realized due to the locality of the oscillator dynamics.

The organization of this paper is as follows. In Section 2, a graph and a functional space are defined. The Hamilton system is introduced on the graph in Section 3, and the system is shown to obtain inherently multiple natural modes of oscillation. In Section 4, the method for constructing a CPG model is shown by using a simple example (hexapod). In Section 5, the Hamilton system is synthesized with a gradient system to select and transit the modes. Computer simulations of a hexapodal CPG model are demonstrated in Section 6.

Section snippets

Definition of a graph and function spaces

This section starts with some definitions of a complex function on a graph. Let a set of vertices be V and a set of edges be E. Then a graph G can be defined as a set of these two terms G=(V,E) (Fig. 3).

Generally, vertices in the graph correspond to the agents of an autonomous decentralized system. In the case of the CPG model, they correspond to the oscillators.

It is assumed that G is a finite and oriented graph with N vertices. That is, the number of edges is finite. Each edge connects the

Wave equation as the Hamilton system

A wave equation is introduced on the graph G, which is part of the Hamilton system. When a wave equation is used, the discrete natural frequency modes correspond to the gait patterns. Moreover, the discrete energy states correspond to the discontinuous walk speeds.

Schrödinger’s wave equation is introduced as a concrete wave equation. The reason is as follows:

  • 1.

    A complex variable is useful when we express the vibrating state.

  • 2.

    We can use the derivation in quantum mechanics as for the Hamiltonian

Natural oscillation modes that appear in a graph of six vertices

In this section, we consider a graph composed of six vertices as shown in Fig. 5. In this case, the incidence matrix Aa is as follows: Aa=−11000010−10000−101000010−10000−10100001−1.In the incidence matrix, the row corresponds to the vertex and the line corresponds to the edge. ‘1’ is allocated when the directed edge comes out of the vertex, and conversely ‘−1’ is allocated when getting in.

And then, the Hamiltonian operator of Eq. (2) becomes H=ΔA=2−1−1000−120−100−1020−100−1020−100−102−1000−1−12.

Synthesis of the Hamilton system and a gradient system

The Hamilton system merely produces a linear sum of the oscillating modes as we demonstrated in Section 3. The dynamic (a) and (b) is realized via the synthesis of the Hamilton system and a gradient system:

  • (a)

    One (or two) specific natural oscillation mode (or modes) appears alternatively out of multiple natural oscillation modes.

  • (b)

    Changes between gait patterns are performed properly by changing a certain parameter, such as the walk speed.

The procedures necessary to realize (a) and (b) are considered

Gaits generation and transition in computer simulations

We simulated generation and transition of hexapodal gaits with the connection as shown in Fig. 6. The parameters in Eq. (17) were α=1.0, β=0.1, γ=0.05, and the initial variables q(0), p(0) were generated, respectively, by a uniform random number of [−1,1].

The target value of the local Hamiltonian HT was HT=61−cos2πt400as shown in Fig. 8 (broken line). The results of this simulation (t=0–400) are shown in Fig. 8, Fig. 9.

Fig. 8 shows the changes of the total Hamiltonian value H (solid line) and

Discussion

In this paper, a method for constructing a mathematical CPG model was proposed. The dynamic of each oscillator is determined only by information about itself and neighbor oscillators. Nevertheless, the total system has natural oscillation modes as the total-order (gait pattern). The order depends on a scale of the total system, i.e., the number of oscillators. As a result, we can produce a CPG model adequate for an actual robot NEXUS (Section 1, points 1. and 2.).

The influence on the CPG model

Shinkichi Inagaki received the B.S. and M.S. degrees in Engineering from Nagoya University, Nagoya, Japan, in 1998 and 2000, respectively, and Ph.D. degree in Engineering from The University of Tokyo, Tokyo, Japan, in 2003. Since 2003, he has been the Research Associate at the Department of Electronic-Mechanical Engineering, Nagoya University. His specialties are autonomous decentralized systems and the application to robotics. He is a Member of SICE, JSME and JRS.

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Shinkichi Inagaki received the B.S. and M.S. degrees in Engineering from Nagoya University, Nagoya, Japan, in 1998 and 2000, respectively, and Ph.D. degree in Engineering from The University of Tokyo, Tokyo, Japan, in 2003. Since 2003, he has been the Research Associate at the Department of Electronic-Mechanical Engineering, Nagoya University. His specialties are autonomous decentralized systems and the application to robotics. He is a Member of SICE, JSME and JRS.

Hideo Yuasa received the B.S. and M.S. degrees in Engineering from Nagoya University in 1984 and 1986, respectively. He obtained Ph.D. from the university in 1992 and worked for the Department of Electronic-Mechanical Engineering, Nagoya University and Bio-Mimetic Control Research Center, RIKEN. Since April 1999, he has been the Associate Professor at the Department of Precision Engineering, The University of Tokyo.

His specialty was distributed autonomous systems, especially mathematical models and its application. He was famous in fundamental models of autonomous systems. He was a Program Co-chair of IAS-7. He died suddenly in September 2000.

Tamio Arai graduated from The University of Tokyo in 1970 and obtained Dr. of Engineering from the same university. He has been a Professor in Department of Precision Engineering since 1987 and now the Director of Research into Artifacts, Center for Engineering as a Director, in The University of Tokyo. His specialties are assembly and robotics, especially multiple mobile robots including legged robot league of RoboCup. He works in IMS programs in Holonic Manufacturing System. He is a Member of CIRP, IEEE/RAS, RSJ, JSPE, and Honorary President of JAAA.

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