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An eight-neuron network for quadruped locomotion with hip-knee joint control

Yide Liu, Xiyan Liu, Dongqi Wang, Wei Yang, shaoxing Qu

TL;DR

An eight-neuron network, composed of Stein neuronal models, capable of achieving five gaits and coordinated control of the hip-knee joints is designed, and the feasibility of this network is demonstrated by implementing motion control, gait transitions, and sensory feedback.

Abstract

The gait generator, which is capable of producing rhythmic signals for coordinating multiple joints, is an essential component in the quadruped robot locomotion control framework. The biological counterpart of the gait generator is the Central Pattern Generator (abbreviated as CPG), a small neural network consisting of interacting neurons. Inspired by this architecture, researchers have designed artificial neural networks composed of simulated neurons or oscillator equations. Despite the widespread application of these designed CPGs in various robot locomotion controls, some issues remain unaddressed, including: (1) Simplistic network designs often overlook the symmetry between signal and network structure, resulting in fewer gait patterns than those found in nature. (2) Due to minimal architectural consideration, quadruped control CPGs typically consist of only four neurons, which restricts the network's direct control to leg phases rather than joint coordination. (3) Gait changes are achieved by varying the neuron couplings or the assignment between neurons and legs, rather than through external stimulation. We apply symmetry theory to design an eight-neuron network, composed of Stein neuronal models, capable of achieving five gaits and coordinated control of the hip-knee joints. We validate the signal stability of this network as a gait generator through numerical simulations, which reveal various results and patterns encountered during gait transitions using neuronal stimulation. Based on these findings, we have developed several successful gait transition strategies through neuronal stimulations. Using a commercial quadruped robot model, we demonstrate the usability and feasibility of this network by implementing motion control and gait transitions.

An eight-neuron network for quadruped locomotion with hip-knee joint control

TL;DR

An eight-neuron network, composed of Stein neuronal models, capable of achieving five gaits and coordinated control of the hip-knee joints is designed, and the feasibility of this network is demonstrated by implementing motion control, gait transitions, and sensory feedback.

Abstract

The gait generator, which is capable of producing rhythmic signals for coordinating multiple joints, is an essential component in the quadruped robot locomotion control framework. The biological counterpart of the gait generator is the Central Pattern Generator (abbreviated as CPG), a small neural network consisting of interacting neurons. Inspired by this architecture, researchers have designed artificial neural networks composed of simulated neurons or oscillator equations. Despite the widespread application of these designed CPGs in various robot locomotion controls, some issues remain unaddressed, including: (1) Simplistic network designs often overlook the symmetry between signal and network structure, resulting in fewer gait patterns than those found in nature. (2) Due to minimal architectural consideration, quadruped control CPGs typically consist of only four neurons, which restricts the network's direct control to leg phases rather than joint coordination. (3) Gait changes are achieved by varying the neuron couplings or the assignment between neurons and legs, rather than through external stimulation. We apply symmetry theory to design an eight-neuron network, composed of Stein neuronal models, capable of achieving five gaits and coordinated control of the hip-knee joints. We validate the signal stability of this network as a gait generator through numerical simulations, which reveal various results and patterns encountered during gait transitions using neuronal stimulation. Based on these findings, we have developed several successful gait transition strategies through neuronal stimulations. Using a commercial quadruped robot model, we demonstrate the usability and feasibility of this network by implementing motion control and gait transitions.
Paper Structure (23 sections, 1 theorem, 17 equations, 14 figures, 8 tables)

This paper contains 23 sections, 1 theorem, 17 equations, 14 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Symmetry of networks
  3. Modeling CPG network
  4. Symmetries and $H/K$ theorem
  5. Four-neuron network with $\mathbf{D}_4$ symmetry
  6. Eight-neuron network and Stein neuronal model
  7. Expand four-neuron network to eight-neuron network
  8. Stein neuronal model and modifications
  9. Design couplings and model parameters
  10. Numerical simulations
  11. Five gaits
  12. Gait transition strategy
  13. Brief discussion
  14. Physical simulation
  15. Simulation framework
  16. ...and 8 more sections

Key Result

Theorem 1

($H/K$ Theorem) Let $\Gamma$ be the symmetry group of a coupled neuron network in which all neurons are coupled and the internal dynamics of each neuron is at least two-dimensional. Let $K\subset H \subset \Gamma$ be a pair of subgroups. Then there exist periodic solutions to some coupled neuron sys

Figures (14)

  • Figure 1: Phase relation for six types of quadruped locomotion gaits: walk, trot, pace, bound, pronk, and jump. In this work, the proposed eight-neuron network can achieve the above gaits except jump.
  • Figure 2: (a) The arrow represents the coupling effect from neuron A to neuron B. (b) The four-neuron network with one way-coupling. The graph-theoretic automorphism group is $\mathbf{Z}_4$. (c) The four-neuron network with two way-coupling. The graph-theoretic automorphism group is $\mathbf{D}_4$. Neurons 1, 2, 3, and 4 are assigned to control the left hind (LH), right hind (RH), right front (RF), and left front (LF) limbs of the quadruped, respectively.
  • Figure 3: (a) Consider each neuron in $\mathbf{D}_4$ four-neuron network as a small group. Each group is divided into two neurons with two-way couplings. The network is thus expanded into eight neurons. Both global and local symmetry are maintained. (b) The cube architecture of the eight-neuron network. The neurons in the top and bottom layers correspond to the hip and knee joints of a leg. The network has four types of couplings: $\alpha$, $\beta$, $\gamma$ and $\delta$.
  • Figure 4: (a)-(e) Signals of the eight-neuron network corresponding to the walk, trot, pace, bound, and pronk gaits, respectively. (f) Tests of five gaits against four types of perturbations. At 11s, a constant perturbation 0.1. At 12s, a random perturbation within $[-0.08,0.08]$. In 13-14s, a random noise within $[-0.008,0.008]$. In 14-15s, a random noise within $[-0.005,0.005]$.
  • Figure 5: (a) Unstable transition process in pronk-to-bound under Power Pair strategy with stimulated neurons 1 and 3. To solve this problem, stimulated neurons are selected as 1 and 2. (b) Invalid rhythm (234)(1) in walk-to-trot with Switch strategy. To solve this problem, Wait & Switch strategy is proposed. (c) Unstable transition process in bound-to-walk with Power Pair strategy. To solve this problem, Wait & Power Pair strategy is proposed. (d) Unstable transition process in bound-to-walk with Power Pair strategy. Wait & Power Pair strategy is also applied here to avoid instability, invalid rhythm, and wrong gait. (e)-(h) Some selected patterns of transitions with difficulties under Switch and Power Pair.
  • ...and 9 more figures

Theorems & Definitions (1)

  • Theorem