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

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 feasibility of this network by implementing motion control, gait transitions, and sensory feedback.

Keywords

central pattern generator gait transition quadruped robot

1. Introduction

Central pattern generator (CPG) is a small neural network composed of neurons with interactions (Ijspeert, 2008), and it has been widely demonstrated to exist in the central nervous system of vertebrates (Grillner, 1985; Grillner et al., 1998; Grillner and Wallén, 1985) and the ganglia of invertebrates (Orlovsky et al., 1999). CPGs can generate primary signals for rhythmic behaviors such as locomotion and respiration without sensory feedback, while sensory feedback can be involved to shape the signals (Yu et al., 2014).

In robotics, CPGs are widely modeled and implemented in programs or hardware for controlling the locomotion and behavior of various types of robots, such as salamander robot for investigating the neural mechanisms behind salamander swimming and walking (Ijspeert, 2020; Ijspeert et al., 2007; Thandiackal et al., 2021); integration with sensory feedback for bipedal (Righetti and Ijspeert, 2006b; Van der Noot et al., 2018), quadrupedal (Liu et al., 2011; Righetti and Ijspeert, 2008), and hexapod locomotion control (Manoonpong et al., 2007, 2008; Steingrube et al., 2010); control of flapping-wing robot (Bayiz et al., 2019; Chung and Dorothy, 2012) and fish robot (Li et al., 2015); integration with reinforcement learning for controlling quadruped robots (Bellegarda and Ijspeert, 2022; Shao et al., 2022) and snake-like robots (Liu et al., 2023); and so on.

Compared with model-based control (Di Carlo et al., 2018; Kim et al., 2019) and learning-based control (Lee et al., 2020; Tan et al., 2018), the most attractive superiority of CPG is its simplicity and computational efficiency. Previous works proved that the CPGs could be implemented on microcontroller (Li et al., 2015), oscillating circuit on chips (Dutta et al., 2019; Saito et al., 2017; Still et al., 2006), or even electronics-free pneumatic circuit (Drotman et al., 2021). When serving as a gait generator in locomotion control, CPG possesses intrinsic stability and adaptability. One only needs to compute a set of ordinary differential equations (ODEs) of the dynamical system to achieve coordinated rhythmic motion control of multiple joints of the robot. These characteristics ensure the continuity of the locomotion and gait transition.

While there are many projects involving quadruped robots using CPGs, the types of CPGs applicable to quadruped robots are limited. Most gait generators for quadruped robots can be generalized as generating four primary signals through a four-neuron CPG, each corresponding to one of the four legs, to control the phase relationship among legs. As for hip-knee joint control, each primary signal undergoes a mapping function to generate signals that are independently assigned to knee and hip joints. The type of CPG architecture based on a four-neuron network has many advantages, such as its simple structure, convenient tuning methods, ease of integration into a locomotion controller, and compatibility with combined approaches like sensor capability or learning-based control architecture. However, it has several limitations as described below:

First, the number of gait rhythms that can be generated is limited. It’s observed in quadruped animals that there are many types of gaits. For instance, six primary gaits observed in animals (Buono and Golubitsky, 2001; Golubitsky et al., 1998) are walk, trot, pace, bound, pronk, and jump (Figure 1), while most of the four-neuron networks can achieve no more than three gait types.

Figure 1.

Phase relation for six types of quadruped locomotion gaits: walk, trot, pace, bound, pronk, and jump (Buono and Golubitsky, 2001; Golubitsky et al., 1998). In this work, the proposed eight-neuron network can achieve the above gaits except jump.

Second, the number of joints/DoFs that can be controlled is limited. The common design for quadrupeds is to have three joints per leg: hip abduction-adduction, hip flexion-extension, and knee flexion-extension. However, in all four-neuron or even eight-neuron CPGs, only four signals can be utilized for phase relation control among four legs. Realizing the phase relationship between the knee and hip joints often requires a mapping method to transfer the signal of a single neuron into two or more joint position signals.

For engineering applications, despite these limitations, a four-neuron network remains the most straightforward and effective architecture within the CPGs. From a biological perspective, it is considered that control signals for multiple joints are likely produced by specific neurons rather than mapping functions (Golubitsky et al., 1998; Righetti, 2008), necessitating more complex network architectures. We believe that a CPG capable of generating various gaits and controlling multiple joints could offer new tools for gait generators in quadruped robots and stimulate research into new types of gaits and hip-knee coordinate control. These visions motivate us to overcome the above limitations and design novel CPG network architectures.

A feasible routine to design a CPG is to follow the symmetry principle. From a programming perspective, each neuron is a group of ODEs, which are also known as neuron models or oscillator models. The interrelations among neurons are named couplings, which can be represented by the coupling matrix. The symmetry of the CPG architecture is determined by the coupling matrix rather than the equation form of neurons. The whole CPG can be considered as a dynamic system, the rhythmic gaits correspond to the attractors of the system. The gait transition can be regarded as the system state switching from one attractor to the other. The relationship between the symmetry of the network and the periodic solutions (gaits) has been revealed by the H/K theorem, which will be further explained in the next section. Thus, the solution for achieving gait variety is to design the symmetry variety of the network architecture. Moreover, the phase relationship between hip and knee joints should be maintained for all gaits, which means another type of intrinsic symmetry should be considered.

In this article, we designed an eight-neuron network for quadruped locomotion control with hip-knee joint control. Here, the hip joint refers to the hip flexion-extension DoF without abduction-adduction. The proposed eight-neuron network has three key features:

(1) Gait diversity: Due to its more complex network architecture, the eight-neuron network enables a wider range of gaits. In this study, we utilized the eight-neuron network to achieve five gaits: walk, trot, pace, bound, and pronk. The regulation of gaits can be achieved by simply modifying the control variables within the mesencephalic locomotor region (MLR) module, its counterpart located in the midbrain of animals (Cabelguen et al., 2003).

(2) Control more DoFs: Most CPGs with four or even eight neurons are designed to control the phase relations among legs; each motor’s signal in a leg is obtained through post-processing by neurons. In engineering applications, this approach is more suitable for programmable foot trajectories. The eight-neuron network not only utilizes its first layer to accomplish similar tasks but also generates eight distinct control signals. Among these, every two neurons form a pair that can generate a group of signals directly assigned to both the knee and hip joints of a leg. This enables the achievement of phase relations among legs (i.e., gait) as well as phase relations between the hip and knee joints within each leg. These assignments are maintained during subsequent gait transitions.

(3) Joint position continuity: Since the neurons assigned to each joint remain unchanged, gait transition can be directly achieved by utilizing the MLR to regulate the eight-neuron network. We proposed four strategies by manipulating the variables in MLR to ensure the continuity and success of all 20 gait transitions.

The architecture of the network has two four-neuron layers, one for hip joints and the other for knee joints. Couplings of the network are designed based on the symmetry assumptions. The global symmetry refers to the phase relation among legs; the local symmetry refers to the phase relation between hip and knee joints in each leg. Stein neuronal model (Stein et al., 1974a, 1974b) is modified as the ODEs of the neurons. Control parameters for all gaits are verified by numerical simulation. Gait transitions are investigated exhaustively. For certain gait transitions that were neither encountered nor difficult to ensure success in previous studies, we uncover the patterns between the outcomes of gait transitions and the transition time. Based on these patterns, we propose several strategies to ensure successful transitions. We further demonstrate the stability and simplicity of the proposed network as a gait generator for quadruped locomotion control through physical simulations on a commercial quadruped robot model. The contributions of this work are:

(1) An eight-neuron network architecture for quadruped locomotion control.

(2) The realization of five gaits and transitions.

(3) The implementation of the network in the physical simulation.

The organization of this paper is as follows. In Section 2, we give an introduction to the symmetry theory of networks. We further propose that the four-neuron network with D₄ symmetry can generate five gaits. In Section 3, we demonstrate the process of expanding the D₄ four-neuron network into an eight-neuron network architecture. We also demonstrate the design of the ODEs of the neuron models. Section 4 presents the numerical simulation results of the eight-neuron network, including the gaits and transitions. The patterns and strategies of gait transitions are demonstrated in detail. Section 5 presents the physical simulation and the performance evaluation of the eight-neuron network by applying it to a commercial quadruped robot model. In Section 6, we demonstrate two sensor integration frameworks, which achieve direction control based on visual information and gait transition via reflex loop. Section 7 concludes this article.

2. Symmetry of networks

2.1. Modeling CPG network

From the perspective of biology, CPG is a small group of neurons that form a network. The connections among the neurons are called synapses. The behavior of the neuron can be modeled by the neuron models, which are generally a set of differential equations. The connections among the neurons can be modeled by the coupling effects, which can be further described by the coupling matrix. Modeling a CPG network can be separated into two steps:

(1) Design the network architecture.

(2) Design the neuron model.

The architecture of the network can be presented as a graph with nodes and edges. The nodes represent the state variables and the edges represent the interactions among these variables. CPGs are dynamic networks, the nodes represent the neurons and the edges represent the coupling effect of the neurons. Here, we give a brief example. As in Collins and Richmond (1994), the CPGs adopted the Hopf model, Van der Pol model, and Stein neuronal model, all these networks have the same architecture as shown in Figure 2(b). The coupling with arrows forms a unidirectional ring. An interesting phenomenon is that, despite the neurons of the CPGs being calculated by a variety of models, all of these CPGs are capable of generating walk, trot, and bound gaits. Additionally, plenty of robotic research has proved that the neurons of the CPGs can be modeled by a variety of forms of ODEs. Some of them are derived from the neurons’ behavior, and some of them are simply oscillator equations. These CPGs can perform similar gait types as long as they share the same network architecture.

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 Z₄. (c) The four-neuron network with two-way coupling. The graph-theoretic automorphism group is D₄. 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.

This can be explained as the symmetries of the network leading to the synchrony and phase relation of the neurons. The pattern of the network is determined by the architecture (or the symmetry) of the network, rather than ODEs. Thus, to design the CPG to achieve the desired gaits, the first step is to design the architecture of the network which determines the symmetries.

2.2. Symmetries and H/K theorem

The symmetry in CPG refers to the transformation that preserves the states of the system. In CPGs, two terms of symmetries are considered: the symmetry of the network and the symmetry of the gait rhythm. The symmetries of the network can be defined as a group of permutations that preserve network architecture. The gait rhythm corresponds to the periodic states of each joint in each leg (such as the angles of the joint). A brief introduction to the involved symmetry is given in Appendix B.

The H/K theorem (Buono and Golubitsky, 2001; Golubitsky and Stewart, 2006) is introduced here to demonstrate the relationship between the gait rhythms and the symmetry of the network. The theorem is stated as follows:

Theorem (H/K Theorem): Let Γ 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 ⊂ H ⊂ Γ be a pair of subgroups. Then there exist periodic solutions to some coupled neuron systems with spatiotemporal symmetries H and spatial symmetries K if and only if H/K is cyclic and K is an isotropy subgroup. Moreover, the system can be chosen so that the periodic solution is asymptotically stable.

Here, we make some brief notes for this theorem.

• CPG is considered as a network of neurons with couplings, which can generate gait rhythms.

• The symmetry of the CPG architecture is denoted as group Γ.

• The gait rhythms are time domain signals with two types of symmetry, the spatial and the temporal symmetries. The spatiotemporal symmetry of a specific gait is denoted as group H.

• Proving that a CPG architecture can generate a type of gait involves four steps: (1) Derive the symmetry group of the network Γ. (2) Derive the spatiotemporal symmetry group H of the gait, which should be a subgroup of Γ. (3) Derive the spatial symmetry group K of the gait, which should be a subgroup of H. K is an isotropy group, which corresponds to the spatial symmetry of the gait. (4) Calculate the quotient group H/K. H/K should be a cyclic group. H/K corresponds to the temporal symmetry of the gait.

For example, in classical CPG architecture Z₄ (Figure 2(b)), 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. The group generator is ω = (1324), there are four elements:

Z_{4} = {e, ω, ω^{2}, ω^{3}} .

(1)

For walk, trot, and bound gait in Collins and Richmond (1994), the H and K combinations are derived as:

• Walk: The H, K, and H/K are Z₄(ω), e, and Z₄(ω(1/4)), respectively. The 1/4 refers to the phase shift generated by the H/K cyclic group.

• Trot: The H, K, and H/K are Z₂(ω²), e, and Z₂(ω²(1/2)).

• Bound: The H, K, and H/K are Z₄(ω), Z₂(ω²), and Z₂(ω(1/2)).

2.3. Four-neuron network with D₄ symmetry

We aim to build a CPG for quadruped locomotion with multiple gaits: walk, trot, pace, bound, and pronk. Previous work (Golubitsky and Stewart, 2006) proved that in a network with Z₄ symmetry, the symmetry group of trot and pace was always conjugate. This conclusion implies that the trot and pace gaits are unable to coexist in a single Z₄ network, unless the coupling is modified (Song et al., 2023). Thus, we consider the symmetry of the network as D₄, which has more elements than Z₄. The corresponding network architecture is a four-neuron network with two-way coupling (Figure 2(c)). The generator of the D₄ group is ω = (1324) and k = (13)(24). The elements of the D₄ can be expressed as:

D_{4} = {e, ω, ω^{2}, ω^{3}, k, k ω, k ω^{2}, k ω^{3}} .

(2)

According to the H/K theorem, we need to find the appropriate cyclic quotient subgroup and isotropy subgroup of the D₄ that conforms to the gaits we need. For five desired gaits, the phase relation is illustrated in Figure 1, the derivation process is demonstrated below:

• Walk: The cyclic subgroup H/K can be derived from the phase relation among the legs, that is, Z₄(ω(1/4)).

• Trot: The spatial symmetry of the trot gait is diagonal equivalence, which implies the isotropic subgroup K of the trot gait is Z₂(k). The phase relation among the legs is 1/2; thus, the H/K is calculated as Z₂(ω²(1/2)).

• Pace: The spatial symmetry maintains the equivalence between neurons 1 and 4 and neurons 2 and 3. The isotropic subgroup of the pace is selected K = Z₂(kω²). The phase relation among the diagonal legs is 1/2, which corresponds to H/K = Z₂(k(1/2)).

• Bound: The spatial symmetry maintains the equivalence between neurons 1 and 2 and neurons 3 and 4. The isotropic subgroup of the bound is selected as K = Z₂(ω²). The phase relation among the diagonal legs is 1/2, which corresponds to H/K = Z₂(k(1/2)).

• Pronk: All neurons are in phase, and the isotropic subgroup of the spatial symmetry is K = D₄(k, ω). H/K = e.

The subgroup combinations of the above gaits are listed in Table 1. The symbol x_i represents the states of the neuron i, and the phase relations of other neurons compared with neuron 1 in different gaits are also listed. It is proved that the four-neuron network with symmetry D₄ meets the requirements of the spatiotemporal symmetries of all desired gaits. The symmetry of our network is set as D₄ for the following design process.

Table 1.

Spatiotemporal group H, subgroup K, and quotient subgroup H/K in D₄ for five gaits.

Gait type	H	K	H/K	x₂(t)	x₃(t)	x₄(t)
Walk	Z₄(ω)	1	Z₄(ω)	x₁(t + 1/2)	x₁(t + 1/4)	x₁(t + 3/4)
Trot	D₂(k, ω²)	Z₂(k)	Z₂(ω²)	x₁(t + 1/2)	x₁(t)	x₁(t + 1/2)
Pace	D₂(k, ω²)	Z₂(kω²)	Z₂(k)	x₁(t + 1/2)	x₁(t + 1/2)	x₁(t)
Bound	D₂(k, ω²)	Z₂(ω²)	Z₂(k)	x₁(t)	x₁(t + 1/2)	x₁(t + 1/2)
Pronk	D₄(k, ω)	D₄(k, ω)	1	x₁(t)	x₁(t)	x₁(t)

3. Eight-neuron network and Stein neuronal model

Based on the foundation that the four-neuron D₄ network with two-way coupling can generate five types of gaits, this section further introduces how to expand the network architecture from four-neurons to eight-neurons.

Most quadruped CPGs employ four-neuron networks, and the signals generated by neurons are assigned to the hip joints correspondingly. The signals of knee joints are generated by mapping functions based on the designed trajectories. The four-neuron network has its advantages; the simple architecture and computational efficiency allow it to be implemented into the locomotion controller easily.

Designing a network architecture more complex than the four-neuron architecture does not compromise these advantages. A network with additional neurons can have more symmetries, which may generate more gait types. Moreover, achieving hip-knee coordination through the inherent characteristics of the network rather than manually designed trajectories can offer more insights into gait transition mechanisms and performance, benefiting both robotics and biology.

We aim to build a CPG to achieve hip-knee control of the quadrupeds. The neurons of the network can generate signals for both the hip and knee joints of the robot. Each joint has an assigned neuron. Based on the gait existence proof of the four-neuron D₄ network, we can expand it to eight neurons. The two challenges that exist here are:

(1) Adding neurons while maintaining the D₄ symmetry.

(2) Designing symmetry to achieve the phase locking between the hip and knee neurons.

Thus far, we conclude that designing an eight-neuron network for hip-knee coordinated control requires consideration of two types of symmetry. One corresponds to the gait rhythm (among four legs), while the other corresponds to the phase relationship of the hip-knee joints. In subsequent sections, the former is referred to as global symmetry, while the latter is termed local symmetry.

3.1. Expanding four-neuron network to eight-neuron network

There are various approaches to expand a four-neuron network to the one composed of eight neurons. One straightforward method is to add four neurons and building couplings. However, this approach could potentially impact the global symmetry of the original network, and the local symmetry is also difficult to design this way.

In this paper, we increase the number of neurons in the network by splitting one neuron in the D₄ network into two neurons (Figure 3(a)). We take the foundation of the four-neuron D₄ network and split one existing neuron into a subgroup consisting of two neurons. Within this group, the two neurons are connected bidirectionally, establishing local symmetry Z₂, while preserving the original global symmetry. In an expanded eight-neuron network, the generator corresponding to local symmetry is λ = (15)(26)(37)(48), resulting in the group generator for the eight-neuron network being:

\begin{aligned} ω = (1324) (5768) \\ κ = λ \cdot (13) (24) (57) (68) = (17) (35) (28) (46) \end{aligned}

(3)

Figure 3.

(a) Consider each neuron in D₄ 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: α, β, γ, and δ.

The expanded eight-neuron network architecture can be further represented by a 3-D cube, which is shown in Figure 3(b). The original neuron assigned to a leg in D₄ four-neuron network is divided into two neurons with two-way couplings. The upper is for the hip joint and the bottom is for the knee joint. The neuron pairs 1 and 5 are assigned to control the hip and knee of the left hind leg of the quadruped, and the neuron pairs 2 and 6 for the right hind, 3 and 7 for the right front, and 4 and 8 for the left front, respectively.

The two-way couplings in D₄ four-neuron network are also separated into two layers, the top layer and the bottom layer both contain a one-way coupling ring but in different directions. New couplings are added between the top and bottom neurons to maintain the equivalence of the hip and knee neurons. The local symmetry formed by the couplings between neuron pairs can be derived from adding a self-coupling for each neuron in the D₄ four-neuron network (Golubitsky et al., 2005).

Considering the couplings between the top and bottom layer, there are four types of couplings in Figure 3(b):

• α: the coupling in the top layer

• β: the coupling in the bottom layer

• γ: the coupling from top to the bottom

• δ: the coupling from the bottom to the top

To maintain the D₄ global symmetry, couplings α and β should be equivalent, leading to the equivalence of the neurons in the same layer. Moreover, the neuron pair for a leg is also equivalent, leading to the equivalence of the couplings γ and δ. The local symmetry is formed by the γ and δ. So far, we still consider they are equal, further modification will be discussed in subsequent sections.

3.2. Stein neuronal model and modifications

In CPG design, the models of the neuron can be classified as human-designed oscillators and biological neuron models. The advantage of the oscillators is their relatively simple form, and the limit cycles are adjustable. The biological neuron models are typically proposed by neurobiologists. The equations of these models are relatively complex and can simulate certain behaviors of neurons. The parameters in the models often have clear biological and physical meanings.

In this work, we employ the Stein model, a biological neuron model first introduced in Stein et al. (1974a, 1974b). We choose this model because it effectively describes the pulse and step responses of neurons, and the parameters in the model have clear physical interpretations. The equations of this model can be represented as:

\begin{aligned} \dot{x_{i}} = a [- x_{i} + \frac{1}{1 + e x p (- f_{c i} - b y_{i} + b z_{i})}] \\ \dot{y_{i}} = x_{i} - p y_{i} \\ \dot{z_{i}} = x_{i} - q z_{i} \end{aligned}

(4)

where x_i is the membrane potential of the ith neuron, y_i, and z_i are terms that describe the chemical kinetics of the membrane potential change. a is a rate constant affecting the frequency of the neuron, in the eight-neuron network with two layers, there are parameters a^h and a^k, correspondingly. The superscripts h and k refer to hip and knee joints. f_ci is the driving signal for this neuron. b is the self-adaptation constant which determines the extent of the adaptation. p and q are the rate constants of the transitions referring to the sodium ionic accumulation (Hodgkin and Horowicz, 1960). In this work, the constants are set as b = −2000, p = 10, q = 30.

In the eight-neuron network, the coupling effects from the other neurons are included in the term of the driving signal f_ci. f_ci has three components: control signals for gait selection and transition from the MLR, coupling effects from the neurons in the same layer, and coupling effects from the neurons in the other layer. In this work, f_ci for hip and knee neurons are expressed as $f_{c i}^{h}$ and $f_{c i}^{k}$ as follows:

\begin{aligned} f_{c i}^{h} = f^{h} [1 + k_{1}^{h} \sin (k_{2}^{h} t) + α \sum_{j = 1}^{4} λ_{j i} x_{j} + δ \sum_{j = 5}^{8} λ_{j i} x_{j}] \\ For top layer: i = 1,2,3,4 \\ f_{c i}^{k} = f^{k} [1 + k_{1}^{k} \sin (k_{2}^{k} t) + β \sum_{j = 5}^{8} λ_{j i} x_{j} + γ \sum_{j = 1}^{4} λ_{j i} x_{j}] \\ For bottom layer: i = 5,6,7,8 \end{aligned}

(5)

where f(f^h, f^k) is an amplitude parameter,

k_{1} (k_{1}^{h}, k_{1}^{k})

and

k_{2} (k_{2}^{h}, k_{2}^{k})

determine the amplitude and frequency of the driving signal from the MLR.

α \sum_{j = 1}^{4} λ_{j i} x_{j}

and

β \sum_{j = 5}^{8} λ_{j i} x_{j}

refer to the coupling effects from the neurons in the same layer;

δ \sum_{j = 5}^{8} λ_{j i} x_{j}

and

γ \sum_{j = 1}^{4} λ_{j i} x_{j}

refer to the coupling effects from the neurons in the other layer. Parameters a, f, k₁, and k₂ control the gaits of the network.

3.3. Designing couplings and model parameters

The coupling parameters are set based on the network architecture and the neuron model. The coupling matrix λ_ij can be derived from the network architecture (Figure 3(b)) and is listed in Table 2.

Table 2.

Coupling matrix λ_ij of the eight-neuron network.

Neuron	1	2	3	4	5	6	7	8
1	0	0	1	0	1	0	0	0
2	0	0	0	1	0	1	0	0
3	0	1	0	0	0	0	1	0
4	1	0	0	0	0	0	0	1
5	1	0	0	0	0	0	0	1
6	0	1	0	0	0	0	1	0
7	0	0	1	0	1	0	0	0
8	0	0	0	1	0	1	0	0

Following the global symmetry, the coupling parameters α and β are both set as −0.15, representing inhibitory couplings among neurons in the same layer. As for local symmetry between neuron pairs, if γ = δ, it will lead to a Z₂ symmetry of the local neuron pair network. Even though a pair of hip and knee neurons are not strictly equivalent since they lie in different layers with opposite coupling rings, it still brings 1/2 phase-locking constraints. Thus in this work, γ and δ are set as −0.6 and −0.1, respectively. The biological assumption for these parameter selections is that the top layer of hip joints dominates the rhythm of the whole network, and the bottom layer “follows” the rhythm generated by the top layer. Set γ ≠ δ potentially breaks the global D₄ symmetry, since it leads to inequivalent in a pair of hip and knee neurons, but the following numerical simulation proves this inequivalent does not affect gait control.

So far the eight-neuron network has been constructed, and we consistently utilized this network in subsequent simulations. Table 3 lists the parameters of the network, along with their meanings and sources.

Table 3.

Parameters of the eight-neuron network.

		Parameter	Value or function	Meanings and sources
Neuron	Variable	a(a^h, a^k)	MLR parameters for gait control	Collins and Richmond (1994) and parameter search
		f(f^h, f^k)
		$k_{1} (k_{1}^{h}, k_{1}^{k})$
		$k_{2} (k_{2}^{h}, k_{2}^{k})$
	Constant	b	−2000	Neuron physiological parameters from Hodgkin and Horowicz (1960)
		p	10
		q	30
Network	Constant	α	−0.15	Symmetry consideration and parameter search
		β	−0.15
		γ	−0.6
		δ	−0.1

4. Numerical simulations

To demonstrate that the eight-neuron network model can effectively generate rhythmic gait signals and achieve gait transitions, we presented a series of numerical simulations. The numerical simulations are carried out in Python 3.12, and the ODEs of the network are calculated by the fourth Runge-Kutta method. The initial state of the eight-neuron network is listed in Table 6.

4.1. Five gaits

The eight-neuron network is capable of generating five gaits: walk, trot, pace, bound, and pronk. The spatiotemporal symmetries of these gaits are shown in Figure 1. The numerical simulation results of the gaits are shown in Figure 4. For each gait, the control parameters in simulation and the periods are listed in Table 7. The phase portraits of the hip-knee neuron pairs are also illustrated to prove that the eight-neuron network can generate stable phase-locking between the hip and knee neurons. This phase-locking feature is helpful in the quadruped locomotion control for it can replace the mapping function by directly sending the signals of the bottom layer to the corresponding knee joints. The video of five gaits is provided in video E1. The proposed network can also withstand a certain range of perturbation. Perturbation experimental details are in Appendix D.

Figure 4.

(a)–(e) Signals of the eight-neuron network corresponding to the walk, trot, pace, bound, and pronk gaits, respectively.

Compared with the Stein model four-neuron Z₄ network in Collins and Richmond (1994), it’s found that the value of a increased from the walk to the trot to the bound. However, pace gait is found to be not achievable in the four-neuron Z₄ network. In the numerical simulation, we observed the conjugate phenomenon of these two gaits, and we have studied in detail the rules involved in the transition between these two gaits. Therefore, although the parameters we set for the trot and pace are very close, we are still able to achieve a successful gait transition (see next subsection). However, we believe that in the eight-neuron network, there must be better combinations of control parameters for either trot and pace or other gaits.

Both trot and pace gaits have been observed in animals. Experiments also proved that some quadruped animals can learn pace gait by training (Błaszczyk and Dobrzecka, 1989), which implies that the CPG of the animals should support the rhythms of both trot and pace. From this aspect, the eight-neuron network which supports both trot and pace gaits is close to biology. Besides, assigned neurons to each joint with inner stability from the local symmetry would make the eight-neuron network in line with the biological characteristics of locomotion.

4.2. Gait transition strategy

Gait transition endows the animals to switch their speed, terrain adaptability, and energy consumption (Alexander, 2003; Xi et al., 2016). The most attractive feature of CPG is that the signals are continuous during gait transitions. From the aspects of dynamical systems, the gait transition of the CPG refers to the bifurcation of the system (Collins and Stewart, 1992, 1993; Golubitsky et al., 2012; Schöner and Kelso, 1988). The change of the control parameters of the ODEs (in this work, the control parameters are a, f, k₁, and k₂) leads to the system switch from one attractor to the other.

In the eight-neuron network with five gaits, there are a total of 20 types of gait transitions. It’s worth mentioning that the phase relation between a pair of knee and hip neurons is always maintained, even in situations of transition failure. Therefore, in the following discussion of transitions, our main focus will be on neurons 1–4 and not go into detail about neurons 5–8.

We applied four strategies to achieve all transitions, which are Switch, Power Pair, Wait & Switch, and Wait & Power Pair. These strategies are demonstrated below:

S.1 Switch: Directly change the control parameters (a, f, k₁, and k₂) to the target gait. This is the simplest strategy, the applicable transitions are walk-to-{bound, pronk}, {trot, pace}-to-{walk, bound, pronk}, and bound-to-pronk.

S.2 Power Pair: In addition to changing control parameters, the amplitude parameters of the driving signals for selected neurons are increased for brief periods and then return to their original values. This strategy has been previously proposed in Collins and Richmond (1994). In their four-neuron network, for transitions bound-to-walk and bound-to-trot, the selected neurons are a pair of diagonal neurons 1 and 3. In this work, the Power Pair is applicable for transitions pronk-to-{walk, bound}. For these two transitions, adopting the Power Pair at any moment can lead to success.

Increasing the driving signal can be considered as stimulating these neurons. In our strategy, the shape of the driving signal is defined by four parameters: gain ratio R_P, duration period T_P, and duty cycles of rising edge and falling edge, η_R and η_P, respectively. The method of constructing the stimulation is provided in Appendix C.

During the pronk-to-bound transition, stimulating neurons 1 and 3 as in the pronk-to-walk transition would cause signal instability (Figure 10(a)). At this condition, the amplitude of the neuron signals continues to fluctuate, and this state requires a long time (several seconds) to stabilize the walk gait. On the other hand, we found that if excitatory neurons 1 and 2 are selected, the gait transition can avoid such fluctuations. It can be observed from the waveform that, as neurons 1 and 2 are excited, their amplitudes increase and eventually enter the same phase, thus causing the rhythm to change to the bound gait, denoted (12)(34), which refers to the rhythm that neurons 1 and 2 in phase, 3 and 4 in phase, (12) and (34) out of phase. Therefore, we considered that when using the Power Pair strategy, the choice of stimulated neurons should be those in the same phase in the target gait (trot-(13)(24), pace-(12)(34)), and we maintained this approach in the strategies for {bound, pronk}-to-{trot, pace}.

Up to this point, we have found that the aforementioned strategies cannot be successful in a group transition: {all gaits}-to-{trot, pace} and bound-to-walk. These transitions can further be classified into Category #1: {walk, trot, pace}-to-{trot, pace}, Category #2: bound-to-walk, and Category #3: {bound, pronk}-to-{trot, pace}.

For Category #1, the difficulty arises from the conjugation of the trot and pace. It is observed that the gait after transitions has several failed situations. Taking the walk-to-trot transition as an example, we observed several types of failures after the switch, such as entering an unwanted target gait such as pace or entering an invalid rhythm, for instance, neurons 1, 2, 3 in phase, neuron 4 out of phase, which is denoted as (123)(4) (Figure 10(b)).

We have found that the system state after applying the Switch and the timing of strategy application are related, and these states exhibit periodicity. It is shown in Figure 10(e) that, as the transition time increases, the system continually switches between valid gaits and invalid rhythms. This pattern can be summarized as “trot → invalid rhythm → pace → invalid rhythm → trot → invalid rhythm → pace → invalid rhythm.” To overcome the potential gait transition failures caused by this conjugation and the invalid rhythms, we proposed the third strategy, Wait & Switch.

S.3 Wait & Switch: This is an improved strategy based on the Switch strategy. Upon receiving the transition command, the Switch is not executed immediately; instead, it is determined whether the current switch will result in the desired gait. If it would lead to an unwanted gait or invalid rhythm, the Switch is delayed until the next appropriate opportunity. The criteria for the applicability of Wait & Switch are provided in Appendix C. This strategy introduces a certain delay, which will not exceed a gait cycle, but it ensures the success of the gait transition.

For the transitions in Category #2 and #3, using either the Switch or the Wait & Switch strategy does not induce a change in rhythmicity. Direct use of the Power Pair can also lead to unwanted gaits or incorrect rhythms, thus we propose the last gait transition strategy, Wait & Power Pair.

S.4 Wait & Power Pair: This is an improved strategy based on the Power Pair. The method is similar to Wait & Switch. It does not execute immediately upon receiving the transition command but waits for the right moment to apply the Power Pair. The criteria for the applicability of the strategy are provided in Appendix C.

For Category #2: bound-to-walk, Collins and Richmond (1994) have previously reported that in the four-neuron network, using Power Pair to stimulate the ipsilateral neurons 3 and 4 for implementing bound-to-walk transitions carries a probability of failure. In our network, we have found that even though the selection are diagonal neurons 1 and 3, Power Pair still leads to several incorrect conditions.

A phenomenon that had not been observed in previous transitions is that the neurons would enter a chaotic state that lasts for an extended period, but eventually, it would return to a stable walking gait, which we refer to as an unstable transition process, which is shown in Figure 10(c). The signal after Power Pair can be categorized as: walk gait (stable and unstable) and bound gait and unstable transition process, and the pattern of the transition is shown in Figure 10(f).

Although the unstable signals display the rhythmicity of walk gait in the short term, the amplitude of the signal from each neuron is not stable, showing periodic variations. It is observed that, over a longer period, the disordered signals will gradually return to a stable walk gait. Considering this period may lead to the failure of the locomotion, we adopted the Wait & Power Pair to ensure the success of the transition.

For Category #3: {bound, pronk}-to-{trot, pace}, unstable transition process and invalid rhythms caused by conjunction are both observed. Taking bound-to-pace as an example, the unstable neuron signals are shown in Figure 10(d), and the patterns are shown in Figure 10(g). The patterns of pronk-to-trot are shown in Figure 10(h); it’s observed that the rhythms of the neuron signal after Power Pair are very complex. Many conditions may lead to the failure of the gait transition, which further proves the necessity of implementing gait transitions at appropriate moments by utilizing Wait & Power Pair.

The simulation results of the 20 gait transitions are demonstrated in Figure 9. The simulations are provided in video E2. The proposed four strategies ensure the success of transitions among all types of gaits and the continuity of neuronal signals.

4.3. Brief discussion

We can rank the five gaits according to the “strength” of their corresponding attractors and summarize the applicable situations for gait transition strategies (Figure 5). Low level gaits include walk, trot, and pace, high level gaits are bound and pronk. Based on this classification, we can summarize two regularities:

R.1 Switch and Wait & Switch are applicable to: {low level gaits}-to-{all gaits}, {each gait}-to-{gaits higher than itself}.

R.2 Power Pair and Wait & Pair Power are applicable to: {high level gaits}-to-{gaits lower than itself}.

Figure 5.

The strength levels of each gait and the corresponding gait transition strategies.

Another regularity can be summarized from the conjunction property of the trot and pace, which is:

R.3 Wait & Switch and Wait & Power Pair are applicable to: {all gaits}-to-{trot, pace}.

The only exception is the transition bound-to-walk. This transition also requires the use of the Wait & Power Pair strategy because unwanted bound gait and unstable transition process may occur. The following research will be focused on how to optimize the strategy parameters or the selection of stimulation to improve this phenomenon.

It is worth mentioning that, gait transitions can be achieved not only by changing neuron dynamic parameters from the MLR but also by altering the coupling parameters of the network. In this study, we do not employ the latter method because changing network coupling parameters is more suitable for the couplings with linearized form (Buono and Golubitsky, 2001; Rutishauser et al., 2008; Song et al., 2023). In the Stein model, we have integrated coupling into neuron dynamics, manifesting in a nonlinear form, as in equations (4) and (5). In this case, it becomes difficult to verify the eigenvalues of the coupling matrix, and using the coupling matrix to regulate bifurcation is also challenging.

In summary, we verified through numerical simulations that using the Stein neuronal model in an eight-neuron network can generate five types of gaits, and we verified the stability of the gaits by applying perturbations. For 20 types of gait transitions, we proposed four strategies of gait transition based on their types. We discussed in detail the design rationale, executing methods, and applicable conditions of these transition methods. All four gait transition methods involve changing parameters regulated by the MLR, thus ensuring that the signals generated by the neurons during all gait transitions are continuous.

5. Physical simulation

To demonstrate the feasibility of the eight-neuron network for gait control of a quadruped robot and to prove the stability of the network during gait transitions, we presented a series of physical simulations. We used PyBullet 3.2.6 (Coumans and Bai, 2024) as the simulator with Unitree Robot Go1 (Unitree, 2024a, 2024b) as the robot model. In the physical simulation, the hip-knee joint refers to the hip flexion-extension and knee flexion-extension DoF of the robot. The abduction/adduction DoF of the hip joint is locked.

5.1. Simulation framework

The simulation framework is shown in Figure 6(a). The parameters that control the gaits (Table 7) are considered as the MLR modules. After the gait parameters are set in MLR, the eight-neuron network generates the signals, and each neuron is assigned to a joint of the robot. The neuron signals are converted into joint position signals through the execution neural network. In this section, the direction control module is always set to move forward in open loop with the stride length command kept at 1 for both sides. In Section 6, the direction control module is replaced by a fuzzy controller. The stride length command can be adjusted to achieve different stride lengths between the left and right sides for direction adjustment.

Figure 6.

(a) Physical simulation framework. (b) The structure of the execution neural network. This network has been pre-trained with designed foot trajectories to convert neuron signals and their corresponding derivatives into joint position signals. (c) Trajectory designs and physical simulation results of five gaits with stride length command set to 1. (d) Taking the trot gait as an example, the stride length of the trajectory gradually increases as the command increases from 0.8 to 1.2.

Here, we provide a detailed description of the execution neural network and how it converts neuron signals into joint position signals. The execution neural network is a multilayer perceptron built with PyTorch. It consists of four layers: an 11-neuron input layer, two hidden layers with 24 neurons and 12 neurons, respectively, each using the ReLU function, and a single-neuron linear output layer. The 11-bit input vector consists of four parts (Figure 6(b)):

(1) A three-bit code that indicates whether the joint corresponding to the neuron is a hip or knee joint, fore or hind limb, and on the left or right side of the robot.

(2) A five-bit one-hot encoding representing the current gait (10000 for walk, 01000 for trot, and so on).

(3) A one-bit stride length command representing the desired foot trajectory stride length.

(4) A two-bit vector representing the current state of the neuron signal and its derivative. These two terms form a limit cycle (Figure 6(c)), which implies periodicity.

Since each neuron in the eight-neuron network is assigned to a corresponding joint, the execution network performs inference for each joint in each step of the simulation, conducting a total of eight inferences. The foot trajectories for each gait and stride length command are pre-designed. The joint signals for these trajectories are calculated through leg inverse kinematics and fed into the execution neural network for training (Figure 6(d)). Each gait is fed with five trajectories with different stride lengths, identified by stride length command ranges of [0.8, 1.2] (Figure 6(e)).

The simulation animations for five gaits are provided in video E3. The speeds of the gaits are listed in Table 4. Due to its impact-rich nature, the speed of the bound and pronk gait is not as stable as that of the others. One potential reason is that the quadruped robot model we are using does not include an elastic element. In previous studies, the elastic element has been considered an important factor in reducing impact during the touch-down process (Hyun et al., 2014; Remy et al., 2010). Except for the speed fluctuations in the bound and pronk gaits, all other gaits in the simulation exhibited stability. These results indicate that the eight-neuron network, combined with a simple simulation framework, can achieve stable motion control.

Table 4.

Physical simulation performance.

Gait type	Speed (m/s)	STD_speed (m/s)
Walk	0.41	0.06
Trot	0.80	0.08
Pace	0.58	0.09
Bound	0.68	0.17
Pronk	1.43	0.26

5.2. Physical simulation of gait transition

Based on the above simulation framework, we investigated the performance of the eight-neuron network during gait transitions. We first tested the success rate of 20 gait transitions. For each type of gait transition, we tested 500 transitions during the 10-11 second period at intervals of 2 × 10⁻³ seconds, covering at least four cycle lengths for each transition to ensure the generality of the results. The success rates are listed in Table 5. It can be observed that, except for a few transitions related to bound and pronk, the success rate of all other transitions achieving a 100% success rate. These results indicate that the eight-neuron network can achieve stable gait transition through a simple framework.

Table 5.

Success rate of 20 gait transitions in physical simulation.

		Gait before transition
		Walk	Trot	Pace	Bound	Pronk
Gait after transition	Walk	\	100%	100%	100%	99.4%
	Trot	100%	\	100%	100%	99.2%
	Pace	100%	100%	\	100%	100%
	Bound	89%	96%	96.2%	\	84.6%
	Pronk	99.4%	96.4%	79.6%	92.2%	\

We further investigated several representative processes of gait transitions in detail. The selected transition and the corresponding metrics such as speed, center of mass height, and posture are shown in Figure 7, and the animation of simulations is provided in video E4. The results imply that when a transition happens from a higher level gait to a lower level gait, the robot may experience excessive deceleration, but it eventually returns to a normal level. During the transitions, the height and posture will experience some disturbances, but they will eventually return to normal levels.

Figure 7.

The posture, speed, and center of mass height during the four gait transitions. Five trials are reported for each transition. All data is aligned according to the actual execution time of the transition. (a) walk-to-bound with Switch strategy. (b) pronk-to-walk with Power Pair strategy. (c) trot-to-pace with Wait & Switch strategy. (d) pronk-to-trot with Wait & Power Pair strategy.

6. Sensor integration

We proposed two frameworks that integrate sensory information. First, we focus on tasks involving visual information, directional control, and path tracking. Furthermore, we achieve gait transition by designing a reflex loop framework.

6.1. Path following using camera

In PyBullet, we installed a camera on the head of the robot and placed the robot on ground with an 8-shape path (video E5). By recognizing the path features through the camera, the robot can determine whether it is currently on the path. If the robot deviates from the path, a simple fuzzy controller adjusts the forward speed and direction by setting different stride length commands to the execution neural network (Figure 8(a)). For example, in the trot gait, if we want to drive the robot to turn left, we can set the stride length command to 0.8 for the left-side joints and to 1 for the right-side joints. The right legs will then perform a longer stride length than the left, resulting in a leftward turning motion. Under this framework, the robot can achieve path following task.

Figure 8.

Two CPG control frameworks with sensor integration. (a) The path information is obtained through a camera, and the stride lengths of the left and right legs are adjusted using a fuzzy controller to achieve the path following task. (b) Based on the measurement of joint torques, a four-layer reflex network is employed to achieve repeated gait transitions on flat and sloped ground.

6.2. Gait transition via reflex loop

The reflex loop has been demonstrated to integrate well with CPGs, enabling rapid responses to environmental stimuli through simple forms (Kimura et al., 2001, 2007; Liu et al., 2025; Owaki et al., 2013; Owaki and Ishiguro, 2017).We consider a scenario where the robot needs to perform uphill and downhill tasks, with the slope of the ground changing frequently. On flat ground, the robot uses a trot gait to increase its speed, while on slopes, it uses walk gait to enhance climbing stability. To achieve this, we designed a terrain-aware loop based on reflexes to regulate gait transitions (Figure 8(b)).The direction control in this framework is achieved using IMU; the method employs the same fuzzy controller as in the previous section. Here, we would like to emphasize that only the yaw axis information from the IMU is used. The terrain perception is entirely realized by the reflex loop, without utilizing the pitch axis information.

The reflex loop is designed as follows. We consider that there are torque sensors for all four hip joints, and the torque data can be obtained through current measurements in motors in real robot. First, the torque data from the four hip joints are sent to the corresponding sensor neurons in the R1 layer, which calculate the moving average torque data over 1 second. The data from the sensor neurons are then summed at the R2 layer with only one neuron, which is designed to measure the torque levels of all hip joints. Thus, it reflects the current ground slope. The signal neuron 13 R2 is further fed into two neurons in the R3 for activation. Neurons 14 and 15 are, respectively, used for torque detection during uphill and downhill movements. Neuron 16 is the sum of layer R3, and their representations are:

\begin{aligned} Neuron 14 : x_{14} = \frac{1}{1 + e^{- (x_{13} - θ_{u p})}} \\ Neuron 15 : x_{15} = \frac{1}{1 + e^{(x_{13} - θ_{down})}} \\ Neuron 16 : x_{16} = x_{14} + x_{15} \end{aligned}

(6)

Here, the activation thresholds are set as θ_up = 5 Nm and θ_down = −0.5 Nm through simulation measurement. Therefore, the activation of neurons 14 and 15 indicates that the robot is moving uphill or downhill, respectively. The activation of neuron 16 indicates that the ground has a slope, while its inactivation suggests a flat ground. On the other hand, we assume that the stimulation of the gaits in the MLR is generated by corresponding neurons. Here, we assume that only two neurons are involved: walk and trot. The activation of these two neurons is regulated by neuron 16. When neuron 16 is active, the trot neuron is inhibited; otherwise, the walk neuron is inhibited. Under the control of this framework, the behavior of the robot is manifested as follows: When the robot moves on flat ground, the torque levels are low. Therefore, neuron 16 is inactive. In the MLR, the walk neuron is inhibited, causing the robot to perform the trot gait. When the robot moves on sloped ground, the torque level is high, causing neuron 16 to become active and inhibiting the trot neuron, and the gait naturally transitions to a walk. When the robot returns to flat ground again, the gait transitions back to trot. Video E6 shows that the robot can switch between two gaits smoothly, and the direction is also maintained. This proves that the CPG, reflex loop, and IMU information are well integrated within this framework.

7. Conclusion

In this work, we proposed an eight-neuron network as CPG for quadruped locomotion control. We utilize the H/K quotient theory to design the symmetries of the network, enabling it to generate five gaits and control eight degrees of freedom of the knee and hip joints in a quadruped robot. We conducted detailed research on the gait transition of the network, explaining various conditions and patterns during gait transitions. Based on the discovered patterns, we designed several strategies to ensure the successful transition among all gaits. We investigated the potential applications of this network and demonstrated, through physical simulations, that the CPG can achieve smooth gait transitions and an extremely high success rate, and can be well integrated with sensor information and reflex loop to implement feedback control.

CPGs are believed to widely exist in the neural systems of animals. In the field of robotics, various CPGs have been designed by drawing inspiration from the neural principles of biology for gait generation. In this work, we refer to observed phenomena and principles in biology and strive to design the architecture and strategies of neural networks from multiple perspectives, including biology, robotics, and computational neuroscience. For instance:

• Based on the assumption of the knee joint following the hip joint, we designed the global and local symmetries of the neural network, as well as the foot trajectory modulation inspired by it.

• Based on the rhythmicity of gaits during transitions, we designed the stimulation selection method.

• Based on observed gaits in biology, we classified the conjugate characteristics and intensity levels of attractors corresponding to different gaits, and summarized the gait transition strategies employed.

We hope this research provides new tools for quadruped locomotion control and insights into biological CPG mechanisms. From the perspective of quadruped locomotion control, we use the top layer of the eight-neuron network for hip control and the bottom layer for knee control. However, there are other possibilities for the signal assignments of each neuron. For example, a pair of neurons could be used to control a pair of antagonistic muscles. Considering the symmetry of muscle movement, the network proposed in this paper can be further expanded, or larger networks can be constructed to fit the biological motion mechanism, by using the method of neuron groups described before (Figure 3(a)).

In conclusion, our proposed eight-neuron network serves as a new gait generator with multiple advantages, including high controllability, diverse gait types, stable signals, continuous signal transitions during gait transitions, and easy applicability. In this network, we reveal and overcome the potential challenges of the Stein neuronal model during gait transitions and provide complete strategies for smooth transitions between all gaits. This not only provides a new gait generator for quadruped robots but also offers an example of the design and modulation of novel CPGs based on symmetry. Meanwhile, the designed network may also assist in decoding nervous systems and understanding biological control mechanisms.

Future work will focus on the following three topics.

• Expanding the structure of the network, such as controlling additional degrees of freedom in the hip, ankle, and waist joints (Appendix I).

• Integrating with reinforcement learning for smooth walking tasks of humanoid robots.

• Applying the eight-neuron network to the motion control of the BioARS system (Liu et al., 2020).

BioARS is a project of constructing a small quadruped robot assembled from robotic insects (Liu et al., 2022, 2024). The simplicity and adaptability of CPG hold significant potential for the motion control of robots with onboard microcontrollers. By leveraging the low computational power and distributed nature of the proposed eight-neuron network, we aim to achieve joint control for this insect-scale assembled quadruped robot.

Supplemental Material

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

Supplemental Material for An eight-neuron network for quadruped locomotion with hip-knee joint control by Yide Liu; Xiyan Liu, Dongqi Wang, Wei Yang, and Shaoxing Qu in The International Journal of Robotics Research

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 12321002, 91748209, 52405026) and the Higher Education Discipline Innovation Project (B21034). The source code is published on Github at .

ORCID iDs

Yide Liu

Xiyan Liu

Dongqi Wang

Wei Yang

Shaoxing Qu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Nos. 12321002, 91748209, 52405026) and the Higher Education Discipline Innovation Project (B21034).

Declaration of conflicting interests

The authors declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: Part of the physical simulation framework described in this paper has been granted a patent in China (Patent Number: ZL202410320353.3). The authors declare no other potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental Material

Supplemental material for this article is available online

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Supplementary Material

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

Abstract

Keywords

1. Introduction

2. Symmetry of networks

2.1. Modeling CPG network

2.2. Symmetries and H/K theorem

2.3. Four-neuron network with D4 symmetry

3. Eight-neuron network and Stein neuronal model

3.1. Expanding four-neuron network to eight-neuron network

3.2. Stein neuronal model and modifications

3.3. Designing couplings and model parameters

4. Numerical simulations

4.1. Five gaits

4.2. Gait transition strategy

4.3. Brief discussion

5. Physical simulation

5.1. Simulation framework

5.2. Physical simulation of gait transition

6. Sensor integration

6.1. Path following using camera

6.2. Gait transition via reflex loop

7. Conclusion

Supplemental Material

Supplemental Material

Supplemental Material

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

Footnotes

Acknowledgments

ORCID iDs

Funding

Declaration of conflicting interests

Supplemental Material

Appendix

References

Supplementary Material

2.3. Four-neuron network with D₄ symmetry

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