Non-holonomic constraints-based adaptive trajectory tracking controller for a multi-joint snake robot

Abstract

The multi-joint snake robot is poised to become a crucial asset for investigation, surveillance, and attack in national defense and military applications. This paper addresses the trajectory tracking challenge of a multi-joint snake robot characterized by high redundancy and multiple degrees of freedom in the plane. We propose an adaptive trajectory tracking controller that accounts for non-holonomic constraints. This controller estimates unknown environmental parameters, effectively mitigating the negative impacts of uncertain and time-varying conditions during robot movement, thereby ensuring stability. A suitable Lyapunov function is identified to confirm the controller’s stability. The trajectory tracking performance of the robot is thoroughly analyzed through simulations and experiments, demonstrating the effectiveness of the proposed adaptive controller.

Keywords

Multi-joint snake robot adaptive control trajectory tracking estimation

Introduction

A multi-joint snake robot is a bionic robot designed based on the structural characteristics and motion mechanism of biological snakes. This robot has the advantages of small volume, high flexibility, and various motion forms. It can move in narrow and unstructured terrain environments.^1,2 The slender multi-joint snake robot can hide in grass and rock cracks, which greatly improves the application of the multi-joint snake robot in the military and national defense fields.^3–5 Various complex environments will bring many restrictions and constraints to the task execution of the snake robot. Therefore, an adaptive control method suitable for the wheeled snake robot is considered to improve the robustness and motion ability of the robot.⁶

To study the movement of biological snakes on land, Yamada⁷ proposed the two-dimensional Serpenoid gait curve of the multi-joint snake robot and established the link model of series modular rotating joints. The link model in series will lead to the complex structure of the robot and the slow calculation velocity of the system. Moreover, the failure of any joint module may cause the paralysis of the whole snake. Rincon and Sotelo⁸ established the kinematic model and dynamic model based on the peristaltic gait of the four-link snake robot. However, this four-link model has a simple structure and is seriously limited by the geographical environment. By studying the motion mechanism of biological snakes, Marvi et al.⁹ found that the multi-joint snake robot could realize forward motion only when the tangential friction coefficient is greater than the normal friction coefficient, which greatly promoted the development of the motion control field of the robot. However, this cannot explain the principle of the robot lateral movement. Mohammadi et al.¹⁰ designed a constraint function of the multi-joint snake robot based on head rotation angle and forward velocity and realized the path tracking control of the robot. However, this method has not been verified in the prototype experiment. Considering the virtual holonomic constraint function, a linear feedback control law for the multi-joint snake robot is designed,¹¹ which realizes the purpose of path tracking. However, their study did not consider the impact of non-holonomic constraints. The Serpentine curve for controlling the motion of the snake robot is proposed in Ma¹² and Ma et al.¹³ Compared with the Serpenoid curve, the Serpentine curve has higher motion efficiency. However, they did not apply the Serpentine curve to robot trajectory tracking, which could not prove the trajectory tracking performance of the robot. Li et al.^6,14 solved the fluid-structure coupling obstacle avoidance problem of the snake robot in fluid from the simulation perspective by using the IB-LBM method. This method can realize the underwater obstacle avoidance trajectory tracking of the robot. Nevertheless, it does not verify the stability of the robot trajectory tracking from a theoretical point of view. Chirikjian and Burdick¹⁵ defines two helix direction angles from the geometric point of view, establishes the kinematics model of three-dimensional helical lateral movement of the multi-joint snake robot, and carries out simulation experiments on the model.

To realize the velocity tracking and head direction control of the snake robot, the parameters of gait function are designed by using the sliding mode control (SMC) method,¹⁶ which ensures the robustness and stability of the robot when the friction coefficient changes. Hatton and Choset¹⁷ analyzed the motion characteristics of the spiral side-shift of the multi-joint snake robot and improved the spiral curve motion function of the robot to the elliptic spiral curve motion function. Then, Yamada and Hirose¹⁸ and Branyan et al.¹⁹ put forward the three-dimensional conical spiral side-shift curve of the multi-joint snake robot. However, the robot needs control from the outside in the process of movement. Based on the Frenet Serret equation, Horigome et al.²⁰ established the three-dimensional link model of the arc-structure of the multi-joint snake robot and realized the smooth motion of the multi-joint snake robot in space. To consider the uncertainty and reduce the complexity of calculation, a time delay control (TDC) method for the snake robot based on virtual integrity constraints (VHCs) is proposed.²¹ The designed two-layer delay control is applied to the output system to realize the dynamic stability of virtual holonomic constraints. In addition to motion modeling and gait control, the trajectory tracking of the snake robot needs accurate direction guidance and position control. To determine the motion direction using limited position information, Børhaug²² proposed a line of sight (LOS) guidance method for the unmanned ship direction control. According to the mathematical and geometric relationship, this method converts the position error into the expected yaw angle of the unmanned ship and makes the motion position of the unmanned ship converge to the predetermined trajectory by adjusting the direction angle in real-time. After analyzing the model differences between the unmanned ship and snake robot, Yang et al.²³ uses the LOS method to guide the snake robot to move toward the virtual particle of the desired path, and realizes the trajectory tracking control of the robot prototype.

To reduce the restriction of unrestricted constraints on the multi-joint snake robot and improve the adaptive ability of the multi-joint snake robot to unknown environmental friction coefficient in the trajectory tracking process, an adaptive trajectory tracking controller of a multi-joint snake robot considering non-holonomic constraints is designed in this paper. The controller estimates the anisotropic friction coefficient in the system model through the motion state of the multi-joint snake robot, compensates the torque control input of the robot, reduces the negative impact of environmental friction on robot trajectory tracking, and enhances the robustness of the system. In addition, the robot can adjust the joint angle and joint angle velocity, reduce the overshoot phenomenon in the trajectory convergence process, improve the tracking efficiency of the robot, and ensure the asymptotic stability of the system.

The main contributions of this paper are as follows:

The proposed controller estimates the anisotropic friction coefficient of the system model through the robot motion state, uses the estimated value instead of the actual value to compensate the torque control input of the multi-joint snake robot, reduce the negative impact of the unknown ground environment on the robot, and enhance the robustness of the system.

The constant parameters in the Backstepping virtual control equation are selected by the adaptive control method, so that the controller can independently adjust the joint angle and joint angle velocity according to the convergence of the robot state, reduce the overshoot phenomenon in the trajectory convergence process, ensure the asymptotic stability of the system, and improve the trajectory tracking efficiency of the robot.

The rest of this paper is organized as follows: in Section 2, the mechanical model of a multi-joint snake robot is established. In Section 3, the control objectives for the joint angle, link angle, actuator torque, motion position and velocity of the robot are established. In Section 4, we design an adaptive trajectory tracking controller of the robot by the adaptive method and find a Lyapunov function, which proves the stability of the controller. In section 5, simulation and prototype experiments of the multi-joint snake robot are carried out to verify the effectiveness of the proposed controller. Section 6 draws a conclusion.

Modeling

Mechanical model

In this part, the mechanical model of the multi-joint snake robot with non-holonomic constraints is established.^24–26 The prototype structure of the snake robot consists of several link modules, which are connected by active revolute joints. These active joints enable precise control of the robot’s movements in the XY plane, allowing it to perform complex motion patterns. The joint drives the link to swing alternately so that the body of the robot presents a S-shape, to obtain the forward power. To establish the mathematical model of the snake robot, we consider that the robot is composed of N links with a length of 2 h. There is a joint between the links, the joint steering gear controls the drive of the link, and there are auxiliary wheels in the middle of each link. In the dynamic model of the multi-joint snake robot, the width of joints and links are not considered temporarily. A simplified model of the robot dynamics is shown in Figure 1.

Figure 1.

Dynamic model of the multi-joint snake robot.

The directional angle of the $j \in {1, \dots, N}$ th link in the multi-joint snake robot is $θ_{j}$ , which is the angle between the link and the horizontal direction.²⁷ The set of link angles is $θ = {[θ_{1} \dots θ_{N}]}^{T} \in R^{N}$ . The directional angle of the $j \in {1, \dots, N}$ th joint in the robot is $ϕ_{j} = θ_{j} - θ_{j + 1}$ , which is the angle of rotation between adjacent links.²⁷ The set of joint angles is $ϕ = {[ϕ_{1}, \dots, ϕ_{N - 1}]}^{T} \in R^{N - 1}$ . The joint centroid is $p = {[p_{x}, p_{y}]}^{T}$ . The link angles velocity of the robot is $v_{θ} = {[v_{θ_{1}}, \dots v_{θ_{N}}]}^{T} \in R^{N}$ , the joint angles velocity of the robot is $v_{ϕ} = {[v_{ϕ_{1}}, \dots v_{ϕ_{N - 1}}]}^{T} \in R^{N - 1}$ , the tangent vector velocity of the robot is $v_{t} \in R^{N}$ , the normal vector velocity of the robot is $v_{n} \in R^{N}$ , and the joint mass of the robot is m. The control input is $u = {[u_{1}, \dots u_{N - 1}]}^{T} \in R^{N - 1}$ , which is the rotational torque of the joints. The centroid angle is $θ = \frac{1}{N} \sum_{j = 1}^{N} θ_{j}$ . The viscous friction coefficients of the system are $μ_{1}, μ_{2} > 0$ , and the size of the viscous friction coefficients is affected by the geographical environment. The rotation coefficients of the joint are $λ_{1}, λ_{2} > 0$ .

A simplified model of the multi-joint snake robot with non-holonomic constraints is established as (1).^10,11

{\begin{matrix} \overset{\cdot}{ϕ} = v_{ϕ} \\ \overset{\cdot}{θ} = v_{θ} \\ {\overset{\cdot}{p}}_{x} = v_{t} \cos θ - v_{n} \sin θ \\ {\overset{\cdot}{p}}_{y} = v_{t} \sin θ + v_{n} \cos θ \\ {\overset{\cdot}{v}}_{ϕ} = - \frac{μ_{1}}{m} v_{ϕ} + \frac{μ_{2}}{m} v_{t} A_{e} D^{T} ϕ + \frac{1}{m} D D^{T} u \\ {\overset{\cdot}{v}}_{θ} = - λ_{1} v_{θ} + \frac{λ_{2}}{N - 1} v_{t} e^{T} ϕ \\ {\overset{\cdot}{v}}_{t} = - \frac{μ_{1}}{m} v_{t} + \frac{2 μ_{2}}{Nm} v_{n} e^{T} ϕ - \frac{μ_{2}}{Nm} ϕ^{T} A_{e} \bar{D} v_{ϕ} \\ {\overset{\cdot}{v}}_{n} = - \frac{μ_{1}}{m} v_{n} + \frac{2 μ_{2}}{Nm} v_{t} e^{T} ϕ \end{matrix}

(1)

where, $A_{e} = [\begin{matrix} 1 & 1 \\ ⋱ & ⋱ \\ 1 & 1 \end{matrix}] \in R^{(N - 1) \times N}$ , $D = [\begin{matrix} 1 & - 1 \\ ⋱ & ⋱ \\ 1 & - 1 \end{matrix}] \in R^{(N - 1) \times N}$ , $\bar{D} = D^{T} {(D D^{T})}^{- 1}$ , $e = {[1, \dots, 1]}^{T} \in R^{N - 1}$ .

Remark 1: The essence of the simplified model (1) is to transform the meandering movement of the snake robot into a normal linear displacement that ignores the rotation of the link. This simplified method is applicable to the meandering gait mode mentioned in Section 4. This simplified model considers various anisotropic friction for the nonlinear dynamic relationship between the model and the environment. However, on the premise that the friction model is effective, it is difficult to determine the accurate range of the limited link angle. Therefore, through the approximate method, the range of the link angle and joint angle of the snake robot is set in the hypothesis. Meanwhile, to reduce the impact of environmental friction on the simplified model, we will use the adaptive control method to solve this problem.

Coordinate transformation

In the snake robot dynamic model (1), $v_{θ}$ and $v_{n}$ will be affected by $ϕ$ , so the change of the robot body shape will affect the direction and normal velocity of the robot. These motion characteristics lead to the complexity of the design of the adaptive trajectory tracking controller. To simplify the design of the controller, the dynamic model of the multi-joint snake robot with non-holonomic constraints is optimized by coordinate transformation (2). In the process of a multi-joint snake robot rotating without lateral force,²⁸ the centroid position of the robot will move a certain distance along the tangent direction, and the new centroid position of the robot is $\hat{p} = {[{\hat{p}}_{x}, {\hat{p}}_{y}]}^{T} \in R^{2}$ . After coordinate transformation, the simplified model of the multi-joint snake robot with non-holonomic constraints can be written as (3). This model can make the motion path of the robot converge to the target trajectory and eliminate the lateral offset when the snake robot turns.

{\begin{matrix} {\hat{p}}_{x} = p_{x} + ε \cos θ \\ {\hat{p}}_{y} = p_{y} + ε \sin θ \\ {\hat{v}}_{n} = v_{n} + ε v_{θ} \end{matrix}

(2)

where $ε = - \frac{2 (N - 1) μ_{2}}{Nm λ_{2}}$ .

{\begin{matrix} \overset{\cdot}{ϕ} = v_{ϕ} \\ \overset{\cdot}{θ} = v_{θ} \\ {\hat{p}}_{x} = v_{t} \cos θ - {\hat{v}}_{n} \sin θ \\ {\hat{p}}_{y} = v_{t} \sin θ + {\hat{v}}_{n} \cos θ \\ {\overset{\cdot}{v}}_{ϕ} = - \frac{μ_{1}}{m} v_{ϕ} + \frac{μ_{2}}{m} v_{t} A_{e} D^{T} ϕ + \frac{1}{m} D D^{T} u \\ {\overset{\cdot}{v}}_{θ} = - λ_{1} v_{θ} + \frac{λ_{2}}{N - 1} v_{t} e^{T} ϕ \\ {\overset{\cdot}{v}}_{t} = - \frac{μ_{1}}{m} v_{t} + \frac{2 μ_{2}}{Nm} v_{n} e^{T} ϕ - \frac{μ_{2}}{Nm} ϕ^{T} A_{e} \bar{D} v_{ϕ} \\ {\hat{v}}_{n} = X v_{θ} - Y {\hat{v}}_{n} \end{matrix}

(3)

where $X = ε (\frac{μ_{1}}{m} - λ_{1})$ and $Y = \frac{μ_{1}}{m}$ are auxiliary simplified calculation constants.

Assumption 1: The link and joint angles in the robot (3) are bounded and satisfy $- \frac{π}{2} \leq θ \leq \frac{π}{2}$ , $- \frac{π}{2} \leq ϕ \leq \frac{π}{2}$ .

Assumption 2: The snake robot moves forward slowly with a winding gait, and always has a non-zero and positive definite forward velocity, which meets $v_{t} \in [v_{\min}, v_{\max}], \forall t \geq 0$ , $0 < v_{\min} \leq v_{\max}$ .

Lemma 1: If a scalar function $V$ exists, it satisfies: (1) $V$ has a lower bound, (2) $\overset{\cdot}{V}$ is semi-negative definite, (3) $\overset{\cdot}{V}$ are consistent and continuous for time. So, $\lim_{t \to \infty} \overset{\cdot}{V} = 0$ .²⁸

Design objectives

In this part, we will develop the control objectives of the adaptive trajectory tracking controller of the multi-joint snake robot with non-holonomic constraints.^10,11

Objective 1. Gait control: An input-output feedback controller is designed to make the gait tracking error of the multi-joint snake robot stable.

Objective 2. Joint angle control: The desired joint angle motion function $\tilde{ϕ}$ of the multi-joint snake robot is set up, and a virtual feedback controller is constructed to make the joint angle error $e_{ϕ} = ϕ - \tilde{ϕ} = 0$ of the robot stable.

Objective 3. Link angle control: The desired link angle motion function $\tilde{θ}$ of the multi-joint snake robot is set up to make the link angle error $e_{θ} = θ - \tilde{θ} = 0$ of the robot stable.

Objective 4. Actuator torque vector control: This is the key to the design of the controller. We set the desired values of the ideal actuator torque vector $v_{\tilde{ϕ}}$ and estimate values of the friction coefficient of the robot so that the actuator torque vector error $e_{u} = v_{ϕ} - v_{\tilde{ϕ}} = 0$ of the robot is stable.²⁶

Objective 5. Motion position and velocity control: The desired position ${\tilde{\hat{p}}}_{t}$ and tangent velocity ${\tilde{v}}_{t}$ of the robot are set up to make the motion position error $e_{p} = {\hat{p}}_{t} - {\tilde{\hat{p}}}_{t} = 0$ and velocity error $e_{v} = v_{t} - {\tilde{v}}_{t} = 0$ of the robot stable.

When the gait control and direction control of the robot are designed, we need to design objectives 1–5 successively. Simultaneously, an appropriate Lyapunov function needs to be found to prove the stability of the system. When objective 1 is realized, the multi-joint snake robot can achieve forward motion in a winding gait. When objective 2 is realized, the link angle of the robot converges to the desired value, making the robot move toward the specified path direction. When objective 3 is realized, the joint angle converges to the desired value, so that the swinging amplitude of the robot joints can be realized in the form of winding. When objective 4 is realized, the actuator torque converges to the desired value, so that the joints of the robot can maintain a stable rotation velocity. When objective 5 is realized, the robot will move forward along the desired path. Meanwhile, the motion velocity of the robot follows the desired value, that is, the position error and the motion velocity error converge to zero.

Adaptive controller design

In this part, we mainly design the adaptive trajectory tracking controller of the multi-joint snake robot to realize the gait control and direction control of the robot. The controller can estimate the system model’s anisotropic friction coefficient and replace the actual value with the estimated value to compensate for the robot’s torque control input, reducing the negative impact of the unknown ground environment. Now we will realize the five control objectives proposed in Section 3.

Gait control

The forward motion of the robot is realized by controlling the swing amplitude and swing frequency of each joint of the multi-joint snake robot.²⁰ The motion function of the $j \in {1, \dots, N - 1}$ th joint of the robot is (4).

{\tilde{ϕ}}_{j} = α \sin (ω t + (j - 1) δ) + ϕ_{0}

(4)

where $\tilde{ϕ} = {[{\tilde{ϕ}}_{1} \dots {\tilde{ϕ}}_{N - 1}]}^{T} \in R^{N - 1}$ . The swing amplitude of the sinusoidal motion of the robot joints is $α$ , and the swing frequency of the joints is $ω$ . The function of $α$ and $ω$ is to compensate for the movement velocity of the robot. The phase shift between the joints is $δ$ , and the joint offset is $ϕ_{0}$ . The function of $δ$ and $ϕ_{0}$ is to compensate for the movement direction of the robot. The motion direction and movement velocity of the multi-joint snake robot are directly affected by these parameters.^10,11

Joint and link angle controllers

To make the multi-joint snake robot move toward the specified path direction, we set up an ideal joint angle motion function $\tilde{ϕ}$ and construct a virtual feedback controller to realize the asymptotic stability of the joint angle error $e_{ϕ} = ϕ - \tilde{ϕ} = 0$ of the robot.

The virtual feedback controller is set as

v_{\tilde{ϕ}} = - {\hat{F}}_{ϕ} e_{ϕ} + \overset{\cdot}{\tilde{ϕ}}

(5)

where ${\hat{F}}_{ϕ}$ is the adaptive time-varying gain, and the updated law of ${\hat{F}}_{ϕ}$ is

{\overset{\cdot}{\hat{F}}}_{ϕ} = e_{ϕ}^{T} e_{ϕ}

(6)

The joint angle error of the robot is set to (7).

e_{ϕ} = ϕ - \tilde{ϕ}

(7)

where $e_{ϕ} = [ϕ_{1} - {\tilde{ϕ}}_{1}, \dots, ϕ_{N - 1} - {\tilde{ϕ}}_{N - 1}] \in R^{N - 1}$ .

After differentiating the link angle error of the robot, equations (8) and (9) are obtained.

{\overset{\cdot}{e}}_{ϕ} = \overset{\cdot}{ϕ} - \overset{\cdot}{\tilde{ϕ}} = v_{ϕ} - \overset{\cdot}{\tilde{ϕ}} = (v_{ϕ} - v_{\tilde{ϕ}}) - {\hat{F}}_{ϕ} e_{ϕ}

(8)

{\overset{\cdot\cdot}{e}}_{ϕ} = \overset{\cdot\cdot}{ϕ} - \overset{\cdot\cdot}{\tilde{ϕ}} \Rightarrow \overset{\cdot\cdot}{\tilde{ϕ}} = \overset{\cdot\cdot}{ϕ} - {\overset{\cdot\cdot}{e}}_{ϕ}

(9)

To make the multi-joint snake robot able to wind and swing, the desired link angle $\tilde{θ}$ of the robot to make the link angle error $e_{θ} = θ - \tilde{θ}$ stable. To guide the multi-joint snake robot to the ideal path, we introduce the LOS method, which converts target path tracking into target heading tracking, if the robot’s link angle can keep up with the target’s $\tilde{θ}$ . So the robot can gradually converge to the target path.^10,11 Referring to the desired link angle $\tilde{θ} = - \arctan \frac{{\hat{p}}_{t}}{Δ}$ of the robot, where the forward distance is $Δ$ . $Δ$ determines the velocity of the multi-joint snake robot to converge to the desired path.²⁶ Moreover, $Δ > 0$ represents the direction forward and $Δ < 0$ represents the direction backwards. Since $Δ$ directly affects the iteration velocity of the robot’s motion direction angle, the desired path of the robot can be achieved by adjusting the value of $Δ$ .

Δ > \frac{| X |}{| Y |} \cdot (1 + \frac{V_{\max}}{V_{\min}})

(10)

The link angle error of the multi-joint snake robot is differentiated to get

{\overset{\cdot}{e}}_{θ} = v_{θ} - \overset{\cdot}{\tilde{θ}}

(11)

The angle velocity error of the robot is

e_{ω} = v_{θ} - v_{\tilde{θ}}

(12)

{\overset{\cdot}{e}}_{ω} = {\overset{\cdot}{v}}_{θ} - {\overset{\cdot}{v}}_{\tilde{θ}}

(13)

Let

v_{\tilde{θ}} - \overset{\cdot}{\tilde{θ}} = - k_{θ} e_{θ}

(14)

where $k_{θ} > 0$ is positive constant. At this time, the link angle velocity error can be written as $e_{ω} = v_{θ} - \overset{\cdot}{\tilde{θ}} + k_{θ} e_{θ}$ .

The joint offset compensation of the motion function of the multi-joint snake robot is designed as

ϕ_{0} = \frac{1}{λ_{2} v_{t}} [\begin{matrix} λ_{1} v_{θ} - e_{θ} - k_{θ} e_{ω} \\ - \frac{λ_{2}}{N - 1} v_{t} \sum_{j = 1}^{N} α \sin (ω t + (j - 1) δ) + {\overset{\cdot}{v}}_{\tilde{θ}} \end{matrix}]

(15)

where

\begin{array}{l} \frac{λ_{2}}{N - 1} v_{t} e^{T} ϕ = \frac{λ_{2}}{N - 1} v_{t} e^{T} (ϕ - \tilde{ϕ}) + λ_{2} v_{t} ϕ_{0} \\ + \frac{λ_{2}}{N - 1} v_{t} \sum_{j = 1}^{N} α \sin (ω t + (j - 1) δ) \end{array}

(16)

To make the joints of the multi-joint snake robot maintain a stable rotation velocity, we set the ideal actuator torque vector $v_{\tilde{ϕ}}$ and the estimated values of the environmental friction coefficient to make the actuator torque vector error of the robot asymptotically stable.²⁶

The torque vector error of the actuator is differentiated to obtain

\begin{matrix} {\overset{\cdot}{e}}_{u} = {\overset{\cdot}{v}}_{ϕ} - {\overset{\cdot}{v}}_{\tilde{ϕ}} = - \frac{μ_{1}}{m} v_{ϕ} + \frac{μ_{2}}{m} v_{t} A_{e} D^{T} ϕ + \frac{1}{m} D D^{T} u - {\overset{\cdot}{v}}_{\tilde{ϕ}} \\ = - c_{1} {\overset{\cdot}{e}}_{ϕ} - \frac{v_{ϕ}}{m} (μ_{1} - Γ_{1}) + \frac{v_{t}}{m} A_{e} D^{T} ϕ (μ_{2} - Γ_{2}) - c_{2} e_{ϕ} \end{matrix}

(17)

To make the adaptive trajectory tracking controller of the robot adapt to the environment, we design the control input function (10)–(18).

u = m {(D D^{T})}^{- 1} (\bar{u} + \frac{Γ_{1}}{m} v_{ϕ} - \frac{Γ_{2}}{m} v_{t} A_{e} D^{T} ϕ)

(18)

where $\bar{u} = \overset{\cdot\cdot}{\tilde{ϕ}} - c_{1} {\overset{\cdot}{e}}_{ϕ} - c_{2} e_{ϕ}$ , $c_{1} = [\begin{matrix} c_{1, 1} \\ ⋱ \\ c_{1, N - 1} \end{matrix}], c_{1, j} > 0$ , $c_{2} = [\begin{matrix} c_{2, 1} \\ ⋱ \\ c_{2, N - 1} \end{matrix}], c_{2, j} > 0$ , and $j = 1, 2, \dots, (N - 1)$ , $\bar{u} = {[{\bar{u}}_{1}, \dots, {\bar{u}}_{N - 1}]}^{T}$ . $Γ_{1}$ and $Γ_{2}$ are the estimated value of $μ_{1}$ and $μ_{2}$ , its adaptive tuning law is (19).

{\begin{matrix} {\overset{\cdot}{Γ}}_{1} = - \frac{c_{3}}{m} v_{ϕ} e_{u}^{T} \\ {\overset{\cdot}{Γ}}_{2} = \frac{c_{4}}{m} v_{t} A_{e} D^{T} ϕ \end{matrix}

(19)

where $c_{3} > 0$ and $c_{4} > 0$ are positive constant gains. The estimator needs to adjust two parameters ( $c_{3}$ , $c_{4}$ ) to estimate the unknown friction coefficient. The estimated value can be adjusted to the system controller to compensate for the robot’s motion state and offset the negative impact of geographical environment on the robot.

Stability analysis

Proposition 1: The control input (18) of the joint torque of the snake robot, the offset compensation of the joint angle (15), and the adaptive tuning laws (6) and (19) can ensure that the angle system of the robot is asymptotically stable.

Proof

Lyapunov candidate function $L_{1}$ is

L_{1} = \frac{c_{2}}{2} e_{ϕ}^{T} e_{ϕ} + \frac{1}{2} {(\sqrt{c_{2}} {\hat{F}}_{ϕ} - f_{ϕ})}^{2}

(20)

where $c_{2} > 0$ and $f_{ϕ} > 0$ are positive gain. The differential of $L_{1}$ is

{\overset{\cdot}{L}}_{1} = - \sqrt{c_{2}} f_{ϕ} e_{ϕ}^{T} e_{ϕ} + c_{2} e_{ϕ}^{T} (v_{ϕ} - v_{\tilde{ϕ}})

(21)

Lyapunov candidate function $L_{2}$ is

L_{2} = \frac{1}{2} e_{θ}^{2} + \frac{1}{2} e_{ω}^{2}

(22)

{\dot{L}}_{2} = e_{θ} {\dot{e}}_{θ} + e_{ω} {\dot{e}}_{ω} = - k_{θ} e_{θ}^{2} - k_{θ} e_{ω}^{2} + \frac{λ_{2}}{N - 1} v_{t} e_{ω} e^{T} (ϕ - \tilde{ϕ})

(23)

Because,

| e^{T} (‖ ϕ - \tilde{ϕ} ‖) | \leq ‖ ϕ - \tilde{ϕ} ‖_{1} \leq \sqrt{N - 1} ‖ ϕ - \tilde{ϕ} ‖_{2}

(24)

\begin{matrix} {(\frac{λ_{2} v_{t}}{\sqrt{N - 1}} ‖ ϕ - \tilde{ϕ} ‖_{2} - \frac{1}{2} | e_{ω} |)}^{2} \\ = {(\frac{λ_{2} v_{t}}{\sqrt{N - 1}} ‖ ϕ - \tilde{ϕ} ‖_{2})}^{2} - \frac{λ_{2}}{\sqrt{N - 1}} v_{t} | e_{ω} | \cdot ‖ ϕ - \tilde{ϕ} ‖_{2} + \frac{1}{4} {| e_{ω} |}^{2} \geq 0 \end{matrix}

(25)

‖ ϕ - \tilde{ϕ} ‖_{2}^{2} = e_{ϕ}^{T} e_{ϕ}

(26)

So,

\begin{matrix} {\overset{\cdot}{L}}_{2} \leq - k_{θ} e_{θ}^{2} - (k_{θ} - \frac{1}{4}) e_{ω}^{2} + \frac{λ_{2}^{2} v_{t}^{2}}{N - 1} ‖ ϕ - {\tilde{ϕ} ‖}_{2}^{2} \\ = - k_{θ} e_{θ}^{2} - (k_{θ} - \frac{1}{4}) e_{ω}^{2} + \frac{λ_{2}^{2} v_{t}^{2}}{N - 1} e_{ϕ}^{T} e_{ϕ} \end{matrix}

(27)

Lyapunov candidate function $L_{3}$ is

L_{3} = \frac{1}{2} e_{u}^{T} e_{u} + \frac{1}{2 c_{3}} {(μ_{1} - Γ_{1})}^{2} + \frac{1}{2 c_{4}} {(μ_{2} - Γ_{2})}^{2}

(28)

\begin{matrix} {\overset{\cdot}{L}}_{3} = - c_{1} e_{u}^{T} e_{u} - c_{2} e_{u}^{T} e_{ϕ} + (μ_{1} - Γ_{1}) (- \frac{v_{ϕ} e_{u}^{T}}{m} - \frac{{\overset{\cdot}{Γ}}_{1}}{c_{3}}) \\ + (μ_{2} - Γ_{2}) (\frac{v_{t}}{m} A_{e} D^{T} ϕ - \frac{{\overset{\cdot}{Γ}}_{2}}{c_{4}}) \end{matrix}

(29)

Combined with (21), (27), and (29), the Lyapunov function $L$ is designed as

\overset{\cdot}{L} \leq - (\sqrt{c_{2}} f_{ϕ} - \frac{λ_{2}^{2} v_{t}^{2}}{N - 1}) e_{ϕ}^{T} e_{ϕ} - k_{θ} e_{θ}^{2} - (k_{θ} - \frac{1}{4}) e_{ω}^{2} - c_{1} e_{u}^{T} e_{u}

(30)

As long as $k_{θ} > \frac{1}{4}$ and $\frac{λ_{2}^{2} v_{t}^{2}}{N - 1} < \sqrt{c_{2}} f_{ϕ}$ are satisfied, then $\overset{\cdot}{L}$ is semi-negative definite. According to Lemma 1, the angle system of the snake robot is asymptotically stable.

Remark 3: $L$ contains $e_{ϕ}^{T} e_{ϕ}$ , $e_{θ}^{2}$ , $e_{ω}^{2}$ , $e_{u}^{T} e_{u}$ , ${(μ_{1} - Γ_{1})}^{2}$ and ${(μ_{2} - Γ_{2})}^{2}$ , while $\overset{\cdot}{L}$ does not include ${(μ_{1} - Γ_{1})}^{2}$ and ${(μ_{2} - Γ_{2})}^{2}$ . The system is a non holonomic constraint case. Errors $e_{ϕ}$ , $e_{θ}$ , $e_{ω}$ and $e_{u}$ converge, while errors $(μ_{1} - Γ_{1})$ and $(μ_{2} - Γ_{2})$ are stable.

Position and speed control

To control the motion position and velocity of the multi-joint snake robot on the desired path, we set up a desired position ${\tilde{\hat{p}}}_{t}$ and velocity ${\tilde{v}}_{t}$ of the multi-joint snake robot. By building a feedback controller, the motion position error $e_{p} = {\hat{p}}_{t} - {\tilde{\hat{p}}}_{t}$ and motion velocity error $e_{v} = v_{t} - {\tilde{v}}_{t}$ of the robot are stable.

The motion position and motion position error of the robot are differentiated to obtain (31) and (32).

{\overset{\cdot}{\hat{p}}}_{t} = v_{t}

(31)

{\overset{\cdot}{e}}_{p} = e_{v}

(32)

{\overset{\cdot}{e}}_{v} = - \frac{μ_{1}}{m} v_{t} + \frac{2 μ_{2}}{Nm} v_{n} e^{T} ϕ - \frac{μ_{2}}{Nm} ϕ^{T} A_{e} \bar{D} v_{ϕ} - {\overset{\cdot}{\tilde{v}}}_{t}

(33)

$e_{v} = - d_{1} e_{p}$ and $\hat{q} = e_{v} + d_{1} e_{p}$ are designed. We get $e_{v} = \hat{q} - d_{1} e_{p}$ , where $d_{1} > 0$ is a positive constant gain.

The Lyapunov candidate function $V_{1}$ . is designed.

V_{1} = \frac{1}{2} e_{p}^{2}

(34)

{\overset{\cdot}{V}}_{1} = e_{p} {\overset{\cdot}{e}}_{p} = e_{p} e_{v} = - d_{1} e_{p}^{2} + e_{p} \hat{q}

(35)

And,

v_{t} = e_{v} + {\tilde{v}}_{t} = \hat{q} - d_{1} e_{p} + {\tilde{v}}_{t}

(36)

The Lyapunov candidate function $V_{2}$ is designed.

V_{2} = \frac{1}{2} {\hat{q}}^{2}

(37)

where $\frac{2 μ_{2}}{Nm} v_{n} e^{T} ϕ - \frac{μ_{2}}{Nm} ϕ^{T} A_{e} \bar{D} v_{ϕ} - {\overset{\cdot}{\tilde{v}}}_{t} = {\overset{\cdot}{e}}_{v} + \frac{μ_{1}}{m} v_{t}$ and ${\overset{\cdot}{e}}_{p} = e_{v} = - d_{1} e_{p}$ .

\begin{array}{l} {\dot{V}}_{2} = - \frac{μ_{1}}{m} {\hat{q}}^{2} + d_{1} e_{p} \hat{q} (\frac{μ_{1}}{m} - d_{1}) + \hat{q} (- \frac{μ_{1}}{m} {\tilde{v}}_{t} + {\dot{e}}_{v} + \frac{μ_{1}}{m} v_{t}) \\ = - \frac{μ_{1}}{m} {\hat{q}}^{2} + d_{1} e_{p} \hat{q} (\frac{μ_{1}}{m} - d_{1}) + (e_{v} + d_{1} e_{p}) ({\dot{e}}_{v} + \frac{μ_{1}}{m} e_{v}) \\ = - \frac{μ_{1}}{m} {\hat{q}}^{2} + d_{1} e_{p} \hat{q} (\frac{μ_{1}}{m} - d_{1}) + e_{v} {\dot{e}}_{v} + \frac{μ_{1}}{m} e_{v}^{2} + d_{1} e_{p} {\dot{e}}_{v} + \frac{μ_{1}}{m} d_{1} e_{p} e_{v} \end{array}

(38)

Moreover ${\overset{\cdot}{e}}_{v} = - d_{1} {\overset{\cdot}{e}}_{p} = - d_{1} e_{v}$ , $e_{p} {\overset{\cdot}{e}}_{v} = - d_{1} e_{p} e_{v} = e_{v}^{2}$ , and $e_{p} e_{v} = - d_{1} e_{p}^{2}$ .

The Lyapunov candidate function $V_{3}$ is designed.

V_{3} = \frac{1}{2} {(μ_{1} e_{v} - \frac{1}{m} e_{p})}^{2}

(39)

{\overset{\cdot}{V}}_{3} = μ_{1}^{2} e_{v} {\overset{\cdot}{e}}_{v} - \frac{μ_{1}}{m} e_{v}^{2} - \frac{μ_{1}}{m} e_{p} {\overset{\cdot}{e}}_{v} + \frac{1}{m^{2}} e_{p} e_{v}

(40)

Combining the Lyapunov candidate function set above, $V$ can be obtained.

V = V_{1} + V_{2} + V_{3}

(41)

\begin{matrix} \overset{\cdot}{V} = - d_{1} e_{p}^{2} - \frac{μ_{1}}{m} {\hat{q}}^{2} + (- d_{1}^{2} + \frac{μ_{1}}{m} d_{1} + 1) e_{p} \hat{q} \\ - (μ_{1}^{2} d_{1} + \frac{μ_{1}}{m}) e_{v}^{2} - d_{1} (\frac{μ_{1}}{m} d_{1} + \frac{1}{m^{2}}) e_{p}^{2} \end{matrix}

(42)

When $- d_{1}^{2} + \frac{μ_{1}}{m} d_{1} + 1 = 0$ , that is ${md}_{1}^{2} - μ_{1} d_{1} + m = 0$ . We can get $d_{1} = \frac{μ_{1} + \sqrt{μ_{1}^{2} + 4 m^{2}}}{2 m} > 0$ .

\overset{\cdot}{V} = - \frac{μ_{1}}{m} {\hat{q}}^{2} - (μ_{1}^{2} d_{1} + \frac{μ_{1}}{m}) e_{v}^{2} - d_{1} (\frac{μ_{1}}{m} d_{1} + \frac{m^{2} + 1}{m^{2}} 1) e_{p}^{2}

(43)

So $\overset{\cdot}{V} < 0$ is negative definite. Therefore, the position and velocity system of the snake robot is asymptotically stable.

Remark 4: The velocity constraint in the multi-joint snake robot model can not be transformed into a position constraint through integration. This kind of non-holonomic constraint robot cannot move laterally without changing the attitude angle. Therefore, in the trajectory tracking control of the snake robot with non-holonomic constraints, we need to adjust the motion position of the robot by changing the motion direction angle. The complete structure of the adaptive trajectory tracking controller is shown in Figure 2.

Figure 2.

Trajectory tracking controller of the robot.

Simulations and experiments

In this section, we will carry out simulation experiments and prototype experiments to verify the effectiveness of the adaptive trajectory tracking controller of the multi-joint snake robot considering non-holonomic constraints.

Simulations

Following the shape of the multi-joint snake robot, the number of joints of the robot is set as $N = 6$ , the mass of each joint is $m = 0.2 kg$ , the joint length is $h = 0.2 m$ , and each joint is composed of a fixed cylinder and a universal joint that can be rotated. For the convenience of calculation, we take the ground viscous friction coefficients $μ_{1} = 0.2$ and $μ_{2} = 2.0$ as examples. The rotation coefficients are $λ_{1} = 0.5$ and $λ_{2} = 20$ . The initial tangential velocity of the robot is $v_{t} = 0 m / s$ , the initial normal velocity of the robot is $v_{n} = 0 m / s$ , the initial joint angle is $ϕ = 0 °$ , and the initial link angle is $θ = 90 °$ . The initial position is $(p_{x}, p_{y}) = (0, 0) m$ , the torque vector of the initial steering gear actuator is $v_{ϕ} = 0 ° / s$ , and the link angle velocity is $v_{θ} = 0 ° / s$ . The initial swing amplitude, initial swing frequency and initial phase of the multi-joint snake robot are $α = 0.05 m$ , $ω = 100 ° / s$ and $δ = 60 °$ , respectively. $F_{ϕ}$ is 1.6. To meet the requirements of $k_{θ} > \frac{1}{4}$ and $\frac{λ_{2}^{2} v_{t}^{2}}{N - 1} < \sqrt{c_{2}} f_{ϕ}$ , constant gains are set to $k_{θ} = 0.5$ and $f_{ϕ} = 18$ , $c_{1} = 5$ , and $c_{2} = 0.5$ , respectively.

In addition, to demonstrate the performance of the proposed method, we compared it with traditional trajectory tracking controllers (PID) and adaptive central pattern generator (CPG) controllers²⁶ in simulations. The link angle and the desired link angle curve of the multi-joint snake robot under the proposed controller with non-holonomic constraints are analyzed, as shown in Figure 3. There is a big deviation between the link angle $θ$ of the robot and the desired link angle $\tilde{θ}$ curve during the movement of the multi-joint snake robot before 10 s. When the robot moves to 15 s, the link angle of the robot can approach and track the desired link angle. When the robot moves from 5 to 10 s, under the proposed controller, the link angle error of the robot converges faster than under PID and CPG control. Under the control of the proposed method, $e_{θ}$ stabilizes after a 20 s fluctuation. However, due to interference and adjustable parameters, $e_{θ}$ eventually converges to around 0. The above results indicate that the proposed method can make the link angle error $e_{θ}$ of the robot more stable.

Figure 3.

$θ$ , $\tilde{θ}$ , and $e_{θ}$ curves of the robot.

The center of mass motion trajectory of the multi-joint snake like robot is shown in Figure 4. All three controllers can cause small-scale displacement of the robot’s center of mass motion trajectory within 10 s during the motion of a multi-joint snake robot, which is caused by small vibrations caused by frictional forces in the actual motion of the robot. In contrast, under the control of the proposed method, the trajectory of the robot’s center of mass gradually stabilizes after self-adjustment. The above results indicate that compared with other controllers, the proposed controller can more effectively reduce the offset of the robot centroid trajectory.

Figure 4.

Centroid motion trajectory.

The joint angle error curve of the multi-joint snake robot is shown in Figure 5. Under the control of the three controllers, the joint angle error of the robot gradually converges and tends to be stable during the movement of the robot after 10 s. The motion velocity of the multi-joint snake robot is directly affected by the friction in different terrain environments. The motion velocity error of the robot under the control of the different controllers gradually converges and tends to be stable during the movement of the robot after 10 s. The above results show that the proposed controller can reduce the velocity error of the robot at the initial moment.

Figure 5.

$e_{ϕ}$ and $e_{v}$ curves of the robot.

The torque control input of each joint of the multi-joint snake robot is shown in Figure 6. The control input changes obviously within 10 s, which can provide a faster starting velocity for the robot. After 10 s, the control input tends to be stable and presents a regular sinusoidal change, which is consistent with the winding motion characteristics of the snake robot.

Figure 6.

Control input $u$ .

The estimated values of viscous friction coefficients and control parameters of the multi-joint snake robot are shown in Figure 7. ${\hat{Γ}}_{1}$ converges to 2 at 10 s and remains stable. ${\hat{Γ}}_{2}$ converges to 1.6 at 20 s, and there is a small fluctuation in the stable curve, and the influence of this fluctuation on the control input can be ignored. ${\hat{F}}_{ϕ}$ converges to 0.2 at 5 s and stable. The converged estimated values can replace the real value to compensate the control input $u$ and ensure the adaptability of the robot to the changes of environmental friction.

Figure 7.

Estimates ${\hat{Γ}}_{1}$ , ${\hat{Γ}}_{2}$ , and ${\hat{F}}_{ϕ}$ .

The variance and root mean square (RMS) of the trajectory tracking performance under the action of different controllers are shown in Figure 8. Because the variance and RMS can reflect the degree of numerical dispersion and the offset between variables and expected values, the smaller the variance and mean square deviation of trajectory tracking performance, the better the control ability of the controller. The figure shows that compared with the other controllers, the proposed controller effectively estimates the friction coefficient, reduces the robot link angle, joint angle error, and motion speed error, and improves the system control accuracy.

Figure 8.

RMS error comparison of different controllers.

Experiments

In this part, we carry out a prototype experiment on the multi-joint snake robot with non-holonomic constraints to verify the stability of the adaptive trajectory tracking controller, as shown in Figures 9 –11. The planar motion capture system is used to measure the relative position coordinates of robots. The system is fixed above the experimental area and obtains the position information of the robot relative to the preset coordinate system on the ground through video images. The coordinate system is set on the plane of the experimental site and within the field of view of the camera system to ensure accurate capture of the robot’s motion trajectory. For the convenience of design, the multi-joint snake robot adopts $N = 6$ links, the length of which is $θ = \pm 60 °$ and the radius of which is $r = 65 mm$ . Six groups of auxiliary wheels are evenly distributed in each link, and the maximum rotation angle between joints is $θ = \pm 60 °$ .

Figure 9.

Joints and links of the multi-joint snake robot.

Figure 10.

Motion capture system.

Figure 11.

Prototype experiment of the multi-joint snake robot.

The robot is manufactured using 3D printing technology and moves forward by utilizing the friction between itself and the ground. A modular structure is adopted between the joints, with the main control module responsible for coordinating the robot’s movements. Each joint is equipped with angle sensors that record and provide feedback on joint rotation while calculating the robot’s motion direction angle in real time. The Bluetooth wireless transceiver module receives instructions from the upper computer and provides feedback on the robot’s status, while the joint modules execute the commands issued by the upper computer. Additionally, each link of the robot is equipped with an auxiliary wheel below it, which is designed to generate asymmetric friction in the normal and tangential directions, allowing the snake like robot to effectively move forward in a two-dimensional plane. They only provide auxiliary motion and do not directly provide power. Each joint has two actuators, with each motor controlling rotation in one direction. The control system of the multi-joint snake robot uses a bus to facilitate communication between the modules. To ensure fair comparisons, the same control parameters are used for components with identical structures, while original parameters are set according to relevant literature for different parts. Furthermore, to minimize the impact of randomness on the results, each algorithm is subjected to multiple experiments in the same hardware and software environment, including the operating system, computer processor, memory, hard disk capacity, and other factors.

To analyze the performance of the multi-joint snake robot in motion, we extract the performance indexes of the first joint and the sixth joint on the straight path of the robot. The robot will adjust the relationship between its actual position and the ideal position during the movement of the robot before 100 cm. When the robot moves to 100 cm, the actual position of the robot tracks the desired position almost perfectly, as shown in Figure 12. Meanwhile, the robot will adjust the relationship between the actual link angle and the desired link angle during the movement of the robot before 150 cm. When the robot moves to 150 cm, the actual link angle of the robot tracks the desired link angle almost perfectly. The multi-joint snake robot will have a small amplitude of oscillation in the initial period of movement, which is the self-adjustment process of the robot. Meanwhile, the motion position and link angle of the robot lag slightly compared with the ideal value, because the movement process of the robot will be affected by environmental factors. When the position and link angle of the robot are adjusted to reasonable values, the robot tracks the desired motion position and desired link angle under the control of the adaptive trajectory tracking controller.

Figure 12.

The position and link angle of the robot.

Conclusions

This paper presents an adaptive trajectory tracking controller for a multi-joint snake robot that effectively addresses non-holonomic constraints, achieving stable tracking in varying friction environments. The proposed controller demonstrates significant improvements in accuracy and robustness. However, its design is currently limited to two-dimensional environments and relies on accurate parameter estimation. Future work will focus on extending the controller’s application to three-dimensional scenarios and enhancing its adaptability under dynamic conditions.

Footnotes

Appendix

Handling Editor: Aarthy Esakkiappan

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors appreciate the support from National Natural Science Foundation of China with the Research Project on Human-Machine Collaborative Decision-Making (No.72192822) and Fujian Provincial Natural Science Foundation (2024J01278).

ORCID iD

Dongfang Li

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