Analysis of Parameter Transition Effects in CPG-Based Control for Multi-Joint Snake-like Robots  ^†

Abstract

Snake-like robots require body adaptation during locomotion when creeping through environments with obstacles. Central Pattern Generator (CPG) provides an effective way to generate rhythmic signals through parameter modulation. During body-shape adaptation, the body wave generated by the CPG can be modified by adjusting its parameters. In this paper, a CPG network based on Hopf oscillators is adopted, and the amplitude parameter is used for body-shape adaptation. However, the influence of amplitude variation during the transition process has not been fully understood. More specifically, when the amplitude parameter changes abruptly, the attractor shifts immediately, while the oscillator state cannot follow the new attractor instantaneously. This mismatch produces transient responses and waveform distortion during the transition process. To address this issue, a linear parameter transition method is introduced. The proposed method is subsequently extended to a coupled CPG network for controlling the multi-joint snake-like robots. Simulations are conducted under different parameter transition conditions. The results demonstrate that the parameter transition method strongly affects the transient torque response. Compared with abrupt parameter variation, the proposed linear transition method significantly reduces transient torque peaks. Additionally, the results further show that even a short transition interval is sufficient to achieve most of the torque reduction. Experiment results show that the proposed method can be applied to body-shape modulation and obstacle avoidance during snake-like robot locomotion.

1. Introduction

Snake-like robots have attracted considerable attention in recent years due to their unique locomotion capability in narrow and unstructured environments [1,2]. Their slender and flexible bodies allow them to traverse spaces that are difficult for wheeled or legged robots to access [3,4]. Therefore, snake-like robots have been applied to tasks such as exploration, inspection, and search operations in confined environments [5,6]. To move effectively in these environments, snake-like robots generate coordinated sinusoidal body waves that propagate along the body. The coordinated propagation of these body waves enables creeping locomotion and serves as the basis of snake-like robots’ movement [7,8,9]. In addition to locomotion control, kinematic modeling and error propagation in multi-link mechanical systems have also been investigated [10].

Central pattern generator (CPG) is a neural network capable of generating rhythmic motion without periodic external inputs [11,12,13]. In robotic applications, the output signals of central pattern generator models are often used as reference inputs for robot joints. Each joint follows these signals to produce coordinated oscillatory motion without requiring predefined trajectories [14]. Compared with predefined time-based input functions, CPG signals are generated through the interaction among oscillators rather than being explicitly specified as functions of time [15,16]. Therefore, CPG-based control approach has been widely applied to snake-like robots and other bio-inspired robots [17,18,19,20,21].

The central pattern generator can be implemented using different modeling approaches, such as neural network formulations [22], biologically inspired representations [23], and oscillator models [24]. Among these approaches, oscillator models are particularly noteworthy [25,26]. Different oscillator models have also been proposed, which include Van der Pol oscillators [27], Matsuoka oscillators [28], Kuramoto oscillators [29], and Hopf oscillators [30,31]. In the paper, a CPG network based on Hopf oscillators is adopted. Compared with many other oscillator models, the parameters of Hopf oscillators can be adjusted independently. Therefore, different locomotion patterns can be generated by modifying individual parameters. When multiple oscillators are coupled, phase differences naturally emerge among oscillators and generate traveling waves along the robot body. These traveling waves form the basis of creeping locomotion in snake-like robots [32]. As a result, Hopf oscillators provide a suitable platform for investigating the influence of parameter variation on locomotion generation.

In practical locomotion control, body-shape adaptation is often achieved through parameter modulation in CPG networks. For snake-like robots, parameters can be adjusted during locomotion to modify body shape when avoiding obstacles or moving through constrained environments. Most existing research focuses on waveform generation and phase coordination [33,34,35,36]. In contrast, relatively little attention has been paid to the transition process that occurs when CPG parameters are varied. In many applications, parameter variation is treated as a direct way to modify locomotion patterns. However, observations during parameter transition suggest that abrupt parameter variation can produce transient responses. These transient responses may cause waveform deformation and torque peaks during the transition process. Therefore, understanding the influence of parameter transition on locomotion generation remains an important issue.

Based on the above discussion, this paper investigates the influence of amplitude variation on creeping locomotion generation in a CPG network based on Hopf oscillators. A linear parameter transition method is proposed to reduce transient responses during parameter variation. The proposed method is subsequently applied to a coupled CPG network for controlling a multi-joint snake-like robot. Accordingly, simulations are conducted to analyze the influence of different parameter transition methods on waveform deformation and torque response. Additionally, experiments are performed to verify that the proposed method can be applied to a real snake-like robot for body-shape modulation and obstacle avoidance. The results demonstrate that the parameter transition method significantly affects the transient torque response during parameter variation and that the proposed linear transition method reduces torque peaks during body-shape transition.

The main contributions of this paper are summarized as follows:

• Abrupt amplitude variation can produce significant transient responses in coupled Hopf oscillators, which lead to torque peaks during parameter transition.
• The transient mismatch between the oscillator state and the shifted attractor is analyzed, which explains the waveform deformation and torque peaks observed during parameter transition.
• A linear parameter transition method is proposed to suppress transient responses during body-shape adaptation of snake-like robots.
• Simulation results show that transient torque peaks can be significantly reduced during parameter transition, and most of the reduction can be achieved with a transition duration as short as 0.1 s. Experimental results verify body-shape modulation during locomotion.

The remainder of this paper is organized as follows. Section 2 presents the mathematical model and analyzes the dynamic properties of the improved Hopf oscillator. The CPG-based control framework for snake-like robots is then described in Section 3. Section 4 demonstrates simulation results and corresponding analysis. Section 5 presents experimental validation. Section 6 provides discussion of the results. Conclusions are finally given in Section 7.

2. Mathematical Model of CPG Based on Improved Hopf Oscillator

The creeping motion of the snake-like robot is generated through the coordination between adjacent joints. Traveling waves are generated along the body through this coordination. A CPG network based on Hopf oscillators is adopted to generate these traveling waves. When control parameters are modified during movement, the attractor of the Hopf oscillator shifts immediately. However, the oscillator state cannot follow the updated attractor instantaneously. As a result, a transient mismatch appears between the oscillator state and the updated attractor. This phenomenon is not always obvious from the time-domain waveform alone. However, it is obvious from the phase portrait. To further explore the phenomenon, a mathematical model is established to analyze transient responses during parameter transition. Particular attention is given to attractor variation and the mismatch generated during parameter transition. A linear parameter transition method is then introduced to reduce transient responses during parameter transition.

2.1. Classical Hopf Oscillator and Limit Cycle Property

This section aims to analyze the impact of amplitude variation on the parameter transition of a Hopf oscillator. The focus is on attractor variations and the transient mismatches that occur during the transition process. To facilitate this analysis, the classical Hopf oscillator is described as:

$$\begin{array}{c}\hfill \left[\begin{array}{c}\dot{u}\\ \dot{v}\end{array}\right]\end{array}=\left[\begin{array}{cc}-\alpha ({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma )& -\omega \\ \omega & -\alpha ({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma )\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right],$$

(1)

where

$$r=\sqrt{u^2+v^2}.$$

(2)

Specifically, u and v are the oscillator states, r is the actual oscillation amplitude in the state space, $\rho$ is the amplitude-related parameter, $\omega$ represents the intrinsic frequency, $\alpha$ determines the convergence rate toward the limit cycle, and $\sigma$ acts as the bifurcation parameter. It should be emphasized that $\rho$ does not represent the instantaneous amplitude. Instead, it determines the radius of the attractor toward which the oscillator converges.

To reveal the amplitude evolution independently of the phase rotation, the radial dynamics is derived. Since $r^2=u^2+v^2$, differentiating both sides yields

$${\displaystyle \frac{d}{dt}}(r^2)=2u\dot{u}+2v\dot{v}.$$

(3)

Substituting Equation (1) into Equation (3) gives

$${\displaystyle \frac{d}{dt}}(r^2)=2u\left[-\alpha \left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)u-\omega v\right]+2v\left[\omega u-\alpha \left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)v\right].$$

(4)

The rotational terms cancel each other, i.e.,

$$-2\omega uv+2\omega vu=0,$$

(5)

and therefore

$${\displaystyle \frac{d}{dt}}(r^2)=-2\alpha \left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)(u^2+v^2)=-2\alpha \left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)r^2.$$

(6)

Using ${\displaystyle \frac{d}{dt}}(r^2)=2r\dot{r}$, the radial dynamics becomes

$$\dot{r}=-\alpha \left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)r.$$

(7)

This radial dynamics characterizes both the steady-state amplitude and the transient convergence behavior, and will be used to analyze how parameter variations affect the system dynamics.

The steady-state solutions are obtained from $\dot{r}=0$, which leads to

$$r\left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)=0,$$

(8)

which admits two equilibrium solutions:

$$r=0,r=\rho \sqrt{\sigma }(\sigma >0).$$

(9)

When $\sigma \le 0$, the only equilibrium is $r=0$. When $\sigma >0$, the origin $r=0$ becomes unstable, while a stable limit cycle emerges at

$$r^{*}=\rho \sqrt{\sigma },$$

(10)

where $r^{*}$ denotes the steady-state amplitude (i.e., the radius of the stable limit cycle). This steady-state amplitude defines the attractor of the system, which will shift when the parameter $\rho$ changes.

Linearizing Equation (7) around $r=r^{*}$ yields

$${\left.{\displaystyle \frac{d\dot{r}}{dr}}\right|}_{r=r^{*}}=-2\alpha \sigma <0(\sigma >0),$$

(11)

which indicates that the limit cycle is locally asymptotically stable. For simplicity, the bifurcation parameter $\sigma$ is normalized to $\pm 1$, as its magnitude can be absorbed into the amplitude parameter $\rho$. Based on these, Figure 1 illustrates the parameter-dependent attractor behavior of the Hopf oscillator. Thus, it provides a basis for understanding how parameter variations may induce changes in the system dynamics. When $\sigma =1$, the system converges to a stable limit cycle. In contrast, when $\sigma =-1$, the origin becomes the only stable equilibrium, and the oscillation amplitude decays to zero.

2.2. Parameter–Amplitude Relationship

From Equation (10), the steady-state amplitude satisfies

$$r^{*}(\rho )=\rho \sqrt{\sigma },$$

(12)

where $r^{*}(\rho )$ explicitly denotes the dependence of the steady-state amplitude on the parameter $\rho$. This relationship shows that the parameter $\rho$ directly determines the radius of the limit cycle, i.e., the position of the system attractor. This steady-state amplitude will be treated as the desired attractor when the parameter varies over time.

Differentiating with respect to $\rho$ gives

$${\displaystyle \frac{d{r}^{*}(\rho )}{d\rho }}=\sqrt{\sigma }.$$

(13)

According to Equation (13), under fixed $\sigma$, the steady-state amplitude varies linearly with the parameter $\rho$. Therefore, $\rho$ serves as a direct modulation variable for the oscillator amplitude. For our snake-like robot, the oscillation amplitude determines the joint-angle amplitude and thus directly affects body shape. As a result, changing $\rho$ provides a simple way to adjust body undulation during locomotion. It should be emphasized that $\rho$ does not represent the instantaneous amplitude of the oscillator. The actual amplitude is $r(t)=\sqrt{u^2+v^2}$, whereas $\rho$ determines the target radius of the stable attractor. When the parameter varies over time, the steady-state amplitude defines a time-varying attractor that the system attempts to track.

To illustrate the effect of parameter variation on the generated signals, Figure 2 compares the outputs of the CPG-based formulation with standard sinusoidal signals under abrupt parameter changes at different switching instants. It can be observed that the CPG output resembles a sinusoidal waveform under steady-state conditions. When parameters are changed during operation, the influence of parameter variation is not obvious from the time-domain waveform alone. This observation may lead to the intuitive assumption that parameter variation in CPG-based formulations does not introduce significant transient effects. To further investigate the influence of parameter variation, the attractor variation and the resulting mismatch will be analyzed in the Section 3.

2.3. Effect of Abrupt Parameter Change on System Dynamics

To analyze the transient behavior induced by parameter transitions, consider an abrupt change in the parameter $\rho$ at time $t=t_s$:

$$\rho (t)=\left\{\begin{array}{cc}{\rho }_0,\hfill & t<t_s,\hfill \\ {\rho }_1,\hfill & t\ge t_s.\hfill \end{array}\right.$$

(14)

Before the switching instant, the oscillator converges to the stable cycle with radius

$$r_0^{*}={\rho }_0\sqrt{\sigma }.$$

(15)

Immediately after the parameter change, the target attractor shifts to the new radius

$$r_1^{*}={\rho }_1\sqrt{\sigma }.$$

(16)

Since the system state $r(t)$ evolves continuously, it cannot instantaneously match the new attractor. This results in a mismatch between the current state and the shifted attractor. The post-switch tracking error can be defined as

$$e_s(t_s^{+})=({\rho }_0-{\rho }_1)\sqrt{\sigma }.$$

(17)

This mismatch induces transient responses during the transition process. Although the influence of parameter variation is not obvious from the time-domain waveform, a mismatch can still be observed in the phase-space trajectory. To illustrate this phenomenon, Figure 3 presents the phase-space trajectories of the Hopf oscillator under different parameter transition conditions. Specifically, as shown in Figure 3a, when no parameter change is applied, the oscillator converges to a stable limit cycle. In contrast, as illustrated in Figure 3b, an abrupt parameter change causes an instantaneous shift of the attractor. The oscillator state cannot follow the shifted attractor instantaneously. As a result, a mismatch appears in the phase-space trajectory. This mismatch is generated by the abrupt shift of the attractor during parameter transition.

2.4. Smooth Parameter Transition via Linear Scheduling

To address the transient mismatch identified in the Section 2.3, a smooth parameter transition strategy is introduced. Instead of directly switching the parameter $\rho$ from ${\rho }_0$ to $\rho 1$, a piecewise linear scheduling function is defined as

$$\rho (t)=\left\{\begin{array}{cc}{\rho }_0,\hfill & t<t_s,\hfill \\ {\rho }_0+{\displaystyle \frac{{\rho }_1-{\rho }_0}{\Delta T}}(t-t_s),\hfill & t_s\le t\le t_e,\hfill \\ {\rho }_1,\hfill & t>t_e,\hfill \end{array}\right.$$

(18)

where $\Delta T=t_e-t_s$ denotes the transition duration.

By construction, the parameter $\rho (t)$ is continuous at both $t=t_s$ and $t=t_e$. Therefore, the corresponding attractor radius evolves as

$$r_d(t)=\rho (t)\sqrt{\sigma }.$$

(19)

The tracking error can be defined as

$$e_r(t_s^{+})=r(t)-r_d(t).$$

(20)

From the definition of the desired attractor radius, substituting Equation (7) into the error definition yields

$${\dot{e}}_r(t_s^{+})=-\alpha \left({\displaystyle \frac{r^2}{{\rho }^2}}-\sigma \right)r-{\dot{r}}_d(t).$$

(21)

and within the transition interval $t_s\le t\le t_e$, it follows that

$${\dot{r}}_d(t)=\sqrt{\sigma }\dot{\rho }(t)=\sqrt{\sigma }{\displaystyle \frac{{\rho }_1-{\rho }_0}{\Delta T}}.$$

(22)

Compared with abrupt parameter switching, the key difference is that the attractor no longer undergoes an instantaneous jump. Conversely, the attractor position changes gradually during the transition process. Therefore, the mismatch between the oscillator state and the attractor is reduced.

Figure 4 demonstrates the phase space trajectory under linear parameter transition. Compared to the abrupt change in Figure 3b, this phase plane trajectory changes more smoothly and moves towards the new attractor without abrupt jumps. This result is related to the relative velocities of two processes: the convergence rate of the oscillator state and the moving rate of the attractor. Under linear parameter transition, the attractor position changes gradually rather than shifting instantaneously. Consequently, its rate of change remains within a finite range. The oscillator state can gradually follow the moving attractor. Therefore, transient mismatch is reduced during the transition.

3. CPG-Based Control Framework for Snake-like Robot

We introduce the improved Hopf oscillator method in Section 2. In this section, the method is further extended to a coupled CPG network. Phase differences are distributed along the body of the snake-like robots between adjacent oscillators. Thus, it assists in generating coordinated traveling waves during creeping locomotion. When the body parameter changes during locomotion, transient responses appear in the coupled oscillators. This behavior becomes more noticeable near the parameter transition point. Large transient torque peaks may reduce motion smoothness and increase the load on the actuators. The improved Hopf oscillators are subsequently coupled through a simple linear topology. Simulations are then conducted to analyze how different parameter transition conditions affect the CPG output signals. Particular attention is given to the influence of linear parameter transition.

3.1. Coupling Structure and Phase Coordination

To generate traveling waves along the robot body, a chain-structured CPG network is constructed, where each joint is driven by a Hopf oscillator. Adjacent oscillators are coupled to enforce a prescribed phase difference, which ensures coordinated spatial wave propagation. The coupled oscillator dynamics are defined as

$$\left\{\begin{array}{cc}\hfill \left[\begin{array}{c}\dot{u_i}\\ \dot{v_i}\end{array}\right]& =\left[\begin{array}{cc}-{\alpha }_i({\displaystyle \frac{r_i^2}{{\rho }_i^2}}-{\sigma }_i)& -{\omega }_i\\ {\omega }_i& -{\alpha }_i({\displaystyle \frac{r_i^2}{{\rho }_i^2}}-{\sigma }_i)\end{array}\right]\left[\begin{array}{c}{u}_i\\ v_i\end{array}\right]+T_{i,j}\hfill \\ \hfill T_{i,j}& =k_{i,j}{\displaystyle \sum _{i\ne j}^n}\left\{\left[\begin{array}{cc}cos({\theta }_{i,j})& -sin({\theta }_{i,j})\\ sin({\theta }_{i,j})& cos({\theta }_{i,j})\end{array}\right]\left[\begin{array}{c}{u}_j\\ v_j\end{array}\right]-\left[\begin{array}{c}{u}_i\\ v_i\end{array}\right]\right\},\hfill \end{array}\right.$$

(23)

where $k_{i,j}$ is the coupling weight between oscillators, ${\theta }_{i,j}$ is the prescribed phase difference, and n is the number of oscillators. The coupling structure imposes phase coordination across joints and enables the generation of traveling waves along the robot body. $T_{i,j}$ is considered as the input from the adjacent CPG model. This coupling structure enforces phase coordination between adjacent oscillators. To generate a serpentine traveling wave, a constant phase difference is imposed:

$${\theta }_{i,j}={\displaystyle \frac{2\pi K_n}{n}},$$

(24)

where n denotes the number of joints, $K_n$ represents the number of body waves. This formulation enables the generation of spatially distributed sinusoidal waves along the robot body.

The oscillator is directly mapped to the joint command of the i-th joint as

$$Joint\_angle\left[i\right]=u_i,$$

(25)

where $u_i$ is selected as the angular position reference. Through this mapping, the temporal oscillation generated by each Hopf oscillator is converted into creeping gait of the snake-like robot. The main parameters used in the CPG network are listed in

Table 1. Parameters of the coupled CPG model.

Parameters	Value
Number of Hopf oscillators	5
Convergence rate ${\alpha }_i$	3
Bifurcation parameter ${\sigma }_i$	1
Intrinsic frequency ${\omega }_i$	$\pi$
Coupling weight $k_{i,j}$	10
Phase difference ${\theta }_{i,j}$	$\pi /4$

. In this study, the number of oscillators is selected according to the robot configuration, and the phase difference is set to produce one body wave along the robot body.

Based on the coupling structure and phase difference settings, Figure 5a illustrates the chain-structured coupling topology, where each oscillator interacts with its adjacent oscillators to generate a traveling wave along the robot body. Figure 5b shows the relationship between the parameter $\rho$ and the steady-state amplitude of the oscillator. This result indicates that the parameter $\rho$ directly affects the body shape of the robot.

3.2. Effect of Abrupt Parameter Transition

To investigate the transient behavior induced by parameter variation, an abrupt change in the amplitude parameter $\rho$ is first considered. According to the analysis in Section 2, such a sudden change leads to an instantaneous shift of the system attractor. This discrepancy introduces a transient mismatch. In this work, $\rho$ is changed from $57.3^{\circ }$ (around 1 rad) to $28.6^{\circ }$ (around 0.5 rad) while maintaining a constant phase difference.

Under this direct transition, the oscillator attractor shifts instantaneously, whereas the system state cannot follow the new attractor immediately. As a result, the output signals exhibit pronounced transient distortion around the transition instant.

Figure 6a demonstrates the output signals generated under abrupt parameter transition. Waveform deformation can be observed around the transition instant. More specifically, Figure 6b provides an enlarged view of the output signals near the transition region. Although the mismatch is not immediately apparent from the overall waveform, it produces noticeable waveform deformation during the transition process. When such signals are directly applied as joint commands, the waveform deformation may influence the body-shape transition of the snake-like robot. Therefore, reducing the transient mismatch during parameter transition becomes important for body-shape modulation.

3.3. Linear Transition of the Amplitude Parameter $\rho$

To reduce the transient mismatch caused by abrupt parameter variation, we then introduce an improved Hopf oscillator model based on the linear parameter transition method. Based on the approach, the parameter $\rho$ is not switched instantaneously. Instead, it changes linearly within a predefined transition interval. As the parameter varies gradually, the system attractor evolves continuously, and the oscillator state can follow the moving attractor progressively.

Figure 7 demonstrates the output signals of the improved Hopf oscillator model under amplitude parameter variation. As shown in Figure 7a, the transition process becomes significantly smoother compared with the abrupt transition case in Figure 6a. Accordingly, no obvious waveform distortion is observed during the transition interval. Figure 7b further illustrates the behavior of each output signal during parameter variation.

$$\Delta t=t_e-t_s=1\mathrm{s},$$

(26)

where $t_s$ and $t_e$ denote the start and end times of the transition, respectively.

These results are consistent with the theoretical analysis. Under linear parameter scheduling, the attractor position changes gradually during the transition interval, which reduces the transient mismatch between the oscillator state and the desired attractor. Consequently, the amplitude parameter $\rho$ serves as an effective body-shape modulation variable, while the transition strategy affects the transient response during body-shape transition.

4. Simulation Analysis

4.1. Simulation Environment Setup

We conduct the simulations based on the Simscape Multibody toolbox in MATLAB R2022b to analyze the role of the proposed improved Hopf oscillator in body-shape adaptation of the snake-like robot. The simulations are solved using the variable-step ode45 (Dormand–Prince) solver with a maximum step size of $1\times {10}^{-3}$ s.

As shown in Figure 8, the simulation framework consists of the central pattern generator (CPG) controller, the snake-like robot, and the contact model between the robot and the ground. The framework CPG-based controller generates rhythmic signals based on the Hopf oscillator network and actuates the joints of the snake-like robot. Coordinated creeping gaits are then generated along the body of the snake-like robot. In the simulation framework of the snake-like robot, passive wheels are attached under each link. These wheels generate anisotropic friction between the robot and the ground. Forward creeping motion is generated based on the coordination of each joint of the robot. The main physical parameters of the snake-like robot are listed in

Table 2. Physical parameter of simulated snake-like robot.

Parameters	Value	Unit
Numbers of joints	5
Length of link	$L_{link}$ = 0.14	m
Hight of link	$H_{link}$ = 0.08	m
Width of link	$W_{link}$ = 0.05	m
Weight of link	$M_{link}$ = 0.1	kg
Width of wheel	$L_{wheel}$ = 0.012	m
Radius of wheel	$R_{wheel}$ = 0.06	m
Weight of wheel	$M_{wheel}$ = 0.01	kg
Friction coefficients	$\mu$ = 0.5

. The link parameters and friction coefficients were kept constant throughout the simulations. Meanwhile, the simulation duration is set to 10 s.

Within this framework, the amplitude parameter $\rho$ directly determines the steady-state oscillation amplitude of the system and affects body deformation during locomotion. Therefore, $\rho$ is treated as a key control parameter for body-shape adaptation. The following analysis compares abrupt parameter variation with smooth parameter transition.

4.2. Effect of Parameter Transition

Based on the previous theoretical analysis, abrupt modification of the amplitude parameter $\rho$ causes an instantaneous shift of the system attractor. This behavior may introduce noticeable transient oscillations in the joints of the snake-like robot during the creeping motion process. To verify this phenomenon, the parameter $\rho$ is directly switched at t = 5 s. The corresponding body shapes of the snake-like robot are presented in Figure 9. The results indicate that different $\rho$ values affect the body shapes of the snake-like robot directly.

The corresponding torque responses are shown in Figure 10. The torque responses are used to compare the transient behavior generated by different parameter transition methods under the same operating conditions rather than to evaluate the operating limits of a specific actuator. To observe the transient behavior of a single joint more clearly, the torque response of the fourth joint is presented in Figure 10a. A noticeable torque peak appears near the parameter switching point. Figure 10b further demonstrates that all joints exhibit large transient responses during the transition process. The torque responses are used to evaluate the transient behavior generated by the abrupt parameter transition. This behavior is consistent with our theoretical analysis. When the parameter changes abruptly, the system state cannot immediately follow the shifted attractor, which produces a large transient mismatch. Therefore, it increases the transient torque response during parameter variation and may affect the body-shape transition process.

To reduce this effect, the proposed linear parameter scheduling method is further investigated. Under the same parameter variation condition, the parameter $\rho$ changes gradually from 1 to $0.5$ within a transition interval of $\Delta t$ = 1 s. The corresponding torque responses are shown in Figure 11. Compared with the abrupt transition case in Figure 10, the torque peak around the parameter transition instant is reduced. More specifically, Figure 11a shows the torque response of the fourth joint, while Figure 11b presents the torque responses of all joints. The results indicate that the large torque peaks observed in Figure 10 become smaller when the linear parameter transition method is applied. Thus, it is consistent with the analysis in Section 2. Based on the linear parameter variation, the attractor position changes gradually during the transition interval. Consequently, the mismatch generated during parameter variation is reduced. As a result, the transient torque response can be significantly reduced during the body-shape transition.

4.3. Influence of Transition Time

The influence of the transition duration $\Delta t$ on system performance is further investigated. According to the analysis in Section 2, the effectiveness of smooth parameter scheduling depends on the rate of attractor variation. A shorter transition time corresponds to a faster attractor shift, which may violate the quasi-static tracking condition and induce transient mismatch. To evaluate this effect, different transition durations are applied, and the corresponding torque responses are analyzed. Figure 12a shows the total square-sum torque for a 5-joint (6-link) snake-like robot under abrupt transition and linear scheduling. When $\Delta t=0$, the transition is equivalent to a direct parameter change. Thus, it still results in a large torque peak. As $\Delta t$ increases, the torque response becomes significantly smoother, and the peak value decreases rapidly. Figure 12b further illustrates the relationship between $\Delta t$ and the total square-sum torque. It can be observed that even a small transition duration (e.g., $\Delta t=0.1$ s) is sufficient to significantly reduce the transient torque.

To examine the scalability of this behavior, the analysis is extended to a larger system with 8 joints (9 links), as shown in Figure 13. A similar trend is observed, where increasing $\Delta t$ effectively reduces the transient torque across all joints. These results are consistent with the theoretical condition derived in the Section 2, which indicates that smoother attractor evolution (i.e., smaller ${\dot{r}}_d(t)$) leads to reduced transient mismatch. However, increasing the transition duration also increases the time required for body-shape transition. Therefore, the transition duration should be selected according to the task requirements. For applications requiring rapid body-shape adjustment, a short transition duration (e.g., 0.1 s) already provides a significant reduction in transient torque. For applications where further torque reduction is preferred, a longer transition duration can be used, although the improvement gradually decreases as the transition duration increases.

4.4. Simulation Results

To evaluate the practical locomotion capability of the proposed control framework, obstacle-avoidance scenarios are simulated. Specifically, Figure 14 shows the body shape evolution of a 5-joint snake-like robot navigating around an obstacle. By adjusting the amplitude parameter $\rho$, the robot is able to smoothly reduce its body amplitude and adapt its shape to avoid collision. This behavior demonstrates that $\rho$ serves as an effective control variable for body-shape modulation.

Additionally, Figure 15 presents a more complex scenario involving an 8-joint (9-link) snake-like robot moving through narrow passages. The robot successfully adapts its body configuration by continuously modulating $\rho$, allowing it to pass through constrained spaces without collision.

These results further confirm that the amplitude parameter $\rho$ enables flexible body-shape adaptation, while the linear parameter scheduling strategy ensures smooth and stable transitions during motion. This is consistent with the theoretical and simulation analysis, where smooth parameter variation reduces transient mismatch and improves motion stability. Overall, the proposed framework provides a unified mechanism for adaptive locomotion. Therefore, the proposed method can both smooth motion generation and environment-aware body-shape adjustment.

5. Experimental Validation

5.1. Experimental Setup

Experiments are conducted on a snake-like robot prototype to demonstrate body-shape modulation during locomotion. As shown in Figure 16, the robot consists of multiple rigid links connected by actuated joints. Passive wheels are mounted under each link to introduce anisotropic friction, which enables forward propulsion. Each joint is driven by a servo motor, and the control signals are generated by the proposed Hopf oscillator-based CPG network.

The control architecture is implemented on a USB-to-RS485 communication interface, where the CPG signals are computed in real time and transmitted to the joint actuators. This setup ensures that the experimental system maintains the same control structure as the simulation framework, enabling a direct validation of the theoretical and simulation results. The amplitude parameter $\rho$ is used to regulate the oscillation amplitude and thus the body curvature, while the parameter transition strategy determines the smoothness of motion during adaptation.

5.2. Experimental Demonstration of Shape Modulation

A simple obstacle-avoidance experiment is conducted to demonstrate the feasibility of body-shape modulation using the proposed CPG-based control framework. Additionally, the link parameters and friction conditions are kept unchanged during the experiments. In this experiment, the amplitude parameter $\rho$ is adjusted online through linear parameter scheduling while the robot moves near a cylindrical obstacle. Figure 17 shows a sequence of snapshots during the experiment. As the robot approaches the obstacle, $\rho$ is gradually modified, which increases the oscillation amplitude and narrows the lateral body motion. This allows the robot to modify its body shape and pass around the obstacle.

This experiment demonstrates that the proposed parameter transition method can be applied to a real snake-like robot for body-shape modulation during locomotion. The objective of the experiment is to verify the feasibility of body-shape adaptation in a physical robot rather than to perform quantitative torque evaluation.

6. Discussion

We previously assumed that CPG-based methods could achieve smoother parameter transitions than conventional sinusoidal approaches. However, the theoretical analysis and simulation results in this paper demonstrate that abrupt parameter variation still strongly affects the transient behavior of coupled CPG systems. When the parameter changes abruptly, the attractor shifts immediately. The oscillator state, however, continues evolving around the previous trajectory for a short time. A transient mismatch then appears between the current state and the updated attractor. This mismatch becomes more obvious when oscillators are coupled together. Figure 2 shows that, compared with conventional sinusoidal signals, a single oscillator can still maintain continuous oscillation after parameter variation. However, phase coordinate between adjacent joints further amplifies the transient response during creeping locomotion. The simulation results of the snake-like robot also show that abrupt parameter transitions generate noticeable torque peaks and waveform distortion near the transition point. In contrast, the proposed linear parameter transition method reduces these effects. When the parameter $\rho$, which controls body deformation of the snake-like robot, changes gradually, the oscillator state remains closer to the moving attractor during the transition process. The resulting joint motion becomes smoother, and the torque variation is also reduced significantly. Similar behavior can be observed in both the phase portraits and the torque responses.

We also investigate the influence of different transition durations $\Delta t$. The results show that even a short transition interval can significantly reduce torque peaks. Longer transition durations further reduce transient torque. However, they also increase the time required for body-shape transition during obstacle avoidance. In this study, a transition duration of 0.1 s already achieves a substantial reduction in transient torque compared with direct parameter switching. Increasing the transition duration further provides additional reduction, although the improvement becomes less pronounced. The results indicate that the transient response during body-shape transition is affected not only by the control parameter itself, but also by the way the parameter changes over time. In the proposed framework, the amplitude parameter $\rho$ directly determines the body deformation, while the transition strategy influences the transient response during locomotion. In practical applications such as ducts and pipes, environmental perception may be required to obtain obstacle information and determine appropriate body-shape adjustments [37,38].

It should be noted that the simulation and experimental studies focus on different aspects of the proposed method. The simulation study quantitatively evaluates the influence of parameter transition on torque response, whereas the experimental study focuses on verifying body-shape modulation and obstacle avoidance on a physical robot platform. Our current work is based on predefined locomotion patterns and linear parameter transitions. More complex environments and feedback-based parameter adaptation are not considered. Future studies will investigate nonlinear transition strategies and online adaptation methods for dynamic environments.

7. Conclusions and Future Work

In the paper, we investigated parameter transition in an improved CPG control approach for multi-link snake-like robots, with a focus on body-shape adaptation during creeping locomotion. The main conclusions are summarized as follows:

(1)

The amplitude parameter $\rho$ was used to modify the body shape of the snake-like robot during locomotion. The relationship between the amplitude parameter and the oscillator output was analyzed based on the Hopf oscillator model.

(2)

The influence of parameter transition on the generated locomotion signals was investigated. The results showed that abrupt parameter variation produces transient mismatch during the transition process, which leads to waveform deformation and transient torque peaks.

(3)

A linear parameter scheduling method was introduced to reduce the transient response generated during parameter transition. The method was applied to a coupled CPG network for multi-link snake-like robots.

(4)

The proposed method was evaluated through simulations and experiments. The results demonstrated that transient torque peaks can be reduced during body-shape adaptation and that the method can be applied to body-shape modulation and obstacle avoidance during locomotion.

More specifically, simulation results quantitatively demonstrated the reduction of transient torque peaks during parameter transition. Experimental results further verified that the proposed method can be applied to body-shape modulation and obstacle avoidance on a physical snake-like robot. The results showed that even a short transition duration of 0.1 s significantly reduced transient torque peaks compared with direct parameter switching. Accordingly, most of the torque reduction can be achieved without introducing a long transition interval.

Future work will extend the proposed parameter transition method to snake-like robots with passive joints [39,40]. The influence of parameter transition on the interaction between active joints and passive joints will be investigated. In addition, this study focuses on the amplitude parameter $\rho$. Future research will consider transitions involving multiple CPG parameters, such as oscillation frequency and phase difference. Another direction is to adjust CPG parameters according to the surrounding environment, thereby allowing body-shape adaptation to be performed automatically during locomotion.