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Abstract

Aiming at the problem of trajectory tracking between joints of the multi-joint snake-like robot in the flow fields, a trajectory tracking control law proposed based on the improved snake-like curve of a multi-joint snake-like robot to avoid obstacles in the flow fields is studied. Firstly, considering the external disturbance that the fluid environment may impose on the multi-joint snake-like robot system, from the point of view of probability, the fluid–solid coupling models of the obstacle channel and multi-joint snake-like robot are established in the flow field by using immersed boundary-lattice Boltzmann method algorithm, which solves the problem of nonlinear fluid motion that cannot be explained by solving the Navier-Stokes (N-S) equation. Then, a potential function is applied to the multi-joint snake-like robot so that the head of the robot can avoid obstacles in the fluid smoothly. By improving the snake-like motion equation, the snake-like curve trajectory tracking function of each joint of the multi-joint snake-like robot with time variation is obtained, which enables the tail joints of the snake-like robot to track the motion trajectory of the head joints. Finally, the effects of different flow field density, velocity, and Reynolds numbers on trajectory tracking of the multi-joint snake-like robot are studied by MATLAB simulations and experiments. The theoretical analysis and numerical simulation show that the designed trajectory tracking control law can make the multi-joint snake-like robot track the trajectory of the front joint when the robot encounters obstacles and make the robot stabilize the lateral distance, longitudinal distance, and direction angle, so as to effectively avoid obstacles. The simulation and experimental results verify the effectiveness of the trajectory tracking control law.

Keywords

Snake-like robot snake-like curve trajectory tracking fluid–structure interaction

Introduction

Snake-like robots is a kind of multi-joint, redundant, and limbless bionic robot which can imitate the movements of biological snakes.¹ Compared with traditional industrial robots, snake-like robots have the characteristics of stable motion and strong environmental adaptability.² In recent years, more and more attention has been paid to snake-like robots because of their wide application prospects in underwater exploration and target search.³ This article mainly solves the problem of trajectory tracking for multi-joint snake-like robot to avoid obstacles in fluid environment.⁴ After the head joint avoids obstacles, the tail joints can track the trajectory of the front joint to complete obstacle avoidance. As the multi-joint snake-like robot is subject to a variety of nonlinear fluid feedback forces in the flow field, it is of great importance to study the trajectory tracking of the multi-joint snake-like robot in the obstacle avoidance process.^5,6

In recent years, with the steady development of modern control methods, the trajectory tracking of mobile robots has also made many achievements in adaptive control, backstepping control, synovial structure control, and intelligent control.⁷ In the study of Guldner and Utkin,⁸ the Lyapunov function is used to design the controller for the dynamic model of the mobile robot, which makes the closed-loop system stable to the desired trajectory, but the algorithm has a large amount of computation and is not systematic enough. Zhongping and Nijmeijer⁹ combine the time-varying state feedback control law of robots with backstepping method and design a global tracking controller with a robot dynamic system. However, the method has to linearize the nonlinear system, which increases the computational difficulty. In the study by Kanayama and Fahroo,¹⁰ a “manipulation function” is proposed to linearize the tracking control system of mobile robots with the first and second derivatives of the path curvature, position errors, and position errors of the mobile robots as variables. However, higher order differential equations need to be solved in this method, which increases the complexity of the method. Zhongping and Nijmeijer¹¹ use cascaded nonholonomic systems in a more general sense and design a global tracking controller based on backstepping design method, but this method does not directly restrain interference information. In the study by Jongmin and Jonghwan,¹² the dynamic model of the mobile robot is established by using polar coordinate method, and a sliding mode control law is designed, which enables the closed-loop system of the mobile robot to track the desired trajectory, but when the state trajectory reaches the sliding mode surface, the system will travel back and forth on both sides of the sliding mode surface, resulting in jittering. In the study by Corradini and Orlando,¹³ a robust discrete sliding mode control law is proposed for the dynamic model. This method avoids jittering in principle, but it is only robust to matching disturbances. In the study by Fukao,¹⁴ a trajectory tracking controller based on inverse optimal control Lyapunov function is designed, which makes the closed-loop system robust to input uncertainties, but this method cannot guarantee the optimal performance of the control law.

In order to study the trajectory tracking problem of a multi-joint snake-like robot in the flow field, a trajectory tracking control law based on improved snake-like curve for multi-joint snake-like robot to avoid obstacles in the flow field is proposed. The proposed control law not only realizes the adaptive control of the robot in the flow field but also makes up for the deficiency of backstepping method which cannot solve the nonlinear control system directly, and it avoids the complexity of solving high-order differential by “manipulation function,” which can restrain both known and unknown disturbances. Firstly, in the stage of establishing an underwater 2-D flow field model, considering the external disturbance that may be imposed on the snake-like robot system in the fluid obstacle environment, the lattice Boltzmann method (LBM) is introduced to replace the Navier-Stokes (N-S) equation in traditional fluid calculation from the point of probability, and the LBM solves the problem of nonlinear fluid motion which cannot be explained by the N-S equation. Secondly, immersed boundary (IB) method is introduced to build a 2-D multi-joint snake-like robot model. The complex robot is modeled as an internal force in the flow field on a non-body-fitted Cartesian grid to realize the fluid–solid coupling control between the underwater nonlinear state feedback and the robot.¹⁵ Then, a potential function is added to the head of the multi-joint snake-like robot in order to make its head avoid obstacles smoothly, and the snake-like motion equation is further ameliorated to obtain the time-varying trajectory tracking function for the multi-joint snake-like robot, thus making the tail joints of the robot can follow the trajectory of its head joint.^16,17 In addition, a Lyapunov function is found to verify the trajectory tracking stability of tail joints of the snake-like robot. Finally, through the simulation and experiment of the multi-joint snake-like robot in fluid by MATLAB, the influences of disturbance parameters such as different flow field density, velocity, and Reynolds number on the trajectory tracking of the multi-joint snake-like robot are studied, and the steady-state characteristics of the lateral distance, longitudinal distance, and direction angles of different joints of the multi-joint robot are analyzed. Experiments show that the designed control law can not only overcome the external interference imposed on the multi-joint snake-like robot system but also track the movement trajectory of the front joint to avoid obstacles after the robot head joint avoids obstacles.^18,19

The rest of this article is organized as follows. The second section details the dynamic modeling of the obstacle channel in the flow field under the IB-LBM and the dynamic modeling of IB force source body of a multi-joint snake-like robot. The third section describes obstacle avoidance algorithm of the head of the multi-joint snake-like robot, and this algorithm makes each joint of the robot to track the trajectory of its front joint. Furthermore, the Lyapunov function is used to analyze the stability of the trajectory tracking between joints of the snake-like robot. The fourth section simulates obstacle avoidance trajectory tracking law of a multi-snake-like robot by MATLAB; studies the influence of different flow field density, velocity, and Reynolds numbers on a multi-joint snake-like robot trajectory tracking; and analyzes the errors of lateral distance, longitudinal distance, and direction angle of each joint of the snake-like robot. Thereafter, the fifth section verifies the proposed control method through experiments. Finally, the last section draws a conclusion.

Fluid–structure interaction model

Dynamic model

In the numerical model of the obstacle channel in the flow field, the dynamic model of fluid is modeled by the LBM method.^15,20 The discrete LBM equation of the single relaxation time model is

f_{i} (x + e_{i} Δ t, t + Δ t) - f_{i} (x, t) = - \frac{1}{τ} [f_{i} (x, ζ, t) - {f_{i}}^{eq} (x, ζ, t)] + Δ t \cdot G_{i}

where $f_{i} (x, t)$ is the particle distribution function of position x and time t, $Δ t$ is the time step, ${f_{i}}^{eq} (x, ζ, t)$ is the equilibrium distribution function, τ is the dimensionless relaxation time, and $Δ t \cdot G_{i}$ is the external forces term. ${f_{i}}^{eq} (x, t)$ and G_i are calculated as follows

{f_{i}}^{eq} (x, t) = ρ ω_{i} [1 + \frac{e_{i} \cdot u}{c_{s}^{2}} + \frac{{(e_{i} \cdot u)}^{2}}{2 c_{s}^{4}} - \frac{u^{2}}{2 c_{s}^{2}}] + O (u^{3})

G_{i} = (1 - \frac{1}{2 τ}) \cdot ω_{i} \cdot [\frac{e_{i} - u}{c_{s}^{2}} + \frac{(e_{i} \cdot u)}{c_{s}^{4}} \cdot e_{i}] \cdot f_{i}

where u is the velocity of the flow field, ρ is the density of the flow field, e_i is the discrete velocities, ω_i is the weight coefficient, and $c_{s} = Δ x / (\sqrt{3} \cdot Δ t)$ is the lattice coefficient.

Obstacle channel model

An obstacle channel model is established by the IB method. Among them, obstacles are replaced by small circles, and the obstacle channel is surrounded by small circles. The opening on the left side of the obstacle is large, which makes it easy for the multi-joint snake-like robot to enter the obstacle channel, as shown in Figure 1.

Figure 1.

Motion direction of a multi-joint snake robot in the obstacles channel.

Multi-joint snake-like robot model

The IB method is used to simulate the multi-joint snake-like robot model. Set a time step Δt and assign the velocity of the flow field to the IB point of the multi-joint snake-like robot through equation (4).¹⁵ After that, the position of the IB points of the multi-joint snake-like robot is updated at Δt time, as shown in equation (5)

U (x, t) = \int_{Ω} u (x, t) D (x_{f} - X (x, t)) d s

\frac{\partial X (x, t)}{\partial t} = U (x, t)

where Ω is the point immersed in the boundary body, $U (x, t)$ and $X (x, t)$ are the velocity and position of the snake-like robot immersed in the boundary body at moment t, and $u (x, t)$ is the velocity of the flow field. The force immersed in the boundary points is distributed to the nearby flow field points by Dirac-delta equation,^15,20 which makes the flow field update and change, as shown in Figure 2

f (x, t) = \int_{Ω} F (x, t) D (x_{f} - X (s, t)) d s

where the Dirac-delta approximation function $D (x_{f})$ is as follows

D (x_{f}) = δ (x - X_{j}) δ (y - Y_{j})

δ (r_{0}) = {\begin{matrix} \frac{1}{8} (3 - 2 | r_{0} | + \sqrt{1 + 4 | r_{0} | - 4 {r_{0}}^{2}}), & | r_{0} | \leq 1 \\ \begin{matrix} \frac{1}{8} (5 - 2 | r_{0} | + \sqrt{- 7 + 12 | r_{0} | - 4 {r_{0}}^{2}}), \\ 0, \end{matrix} & \begin{array}{l} 1 \leq | r_{0} | \leq 2 \\ other \end{array} \end{matrix}

where r₀ is the distance from the point of the immersed body to the grid point.

Figure 2.

Dynamic simulation steps of the snake-like robot.

In the calculation process, the tension of each IB point of the multi-joint snake-like robot is F_z, bending forces is F_w, shear forces is F_f, normal forces is F_j, and total forces is $F (s, t)$ ²⁰

F (s, t) = F_{z} + F_{w} + F_{f} + F_{j}

F_{z} = \frac{\partial}{\partial x} [K_{z} (| \frac{\partial X (x, t)}{\partial x} | - 1) \cdot \frac{\partial X (x, t)}{\partial x}]

F_{w} = - K_{w} \frac{\partial^{4} X (x, t)}{\partial x^{4}}

F_{f} = K_{f} \frac{S_{i} - S_{0}}{S_{0}}

F_{j} = {\begin{matrix} K_{j} \frac{X (x, t) - X_{w}}{{[min (| X (x, t) - X_{w} |)]}^{3}} & | X (x, t) - X_{w} | \leq γ \\ 0 & | X (x, t) - X_{w} | > γ \end{matrix}

where K_z is the tension forces coefficient, K_w is the bending forces coefficient, K_f is the normal forces coefficient, K_j is the shear forces coefficient, X_w is the position where the multi-joint snake-like robot immerses into the boundary body, S_i is the distance between the center point and the ith point of the head, S₀ is the distance between the center point and the head initial point, and γ is the effective cutoff distance of the multi-joint snake-like robot.

The snake-like robot is set to six joints, each of which is oval. In order to ensure that the joints of the multi-joint snake-like robot are not deformed in motion, the dynamic model of each joint of the snake-like robot is established by combining the elastic Hooke’s law and the Euler–Bernoulli constraint theory. Each IB point of each joint of the snake-like robot is connected with the center point of each joint by a spring. Here, the distance between the joint points is $d = 1$ , the length of the long axis is 14 cm, and the length of the short axis is 4 cm. In other words, the length and width of each joint of the robot are 14 and 4 cm, as shown in Figure 3.

Figure 3.

Multi-joint snake-like robot model.

Parameter setting

The obstacle channel of the multi-joint snake-like robot in the flow field is shown in Figure 4. Firstly, the flow field is set to 1000 × 160 cm channel, and the smaller obstacle channel is set in the middle. Then, the multi-joint snake-like robot passes through the obstacle channel from the left side of the flow field to the right side. Finally, the trajectory tracking control law of the multi-joint snake-like robot is studied by recording the motion trajectories of each joint and analyzing the angle changes of each joint. In the calculation process, the parameters are set, as shown in Table 1.

Figure 4.

Obstacle channel model of a multi-joint snake-like robot in the flow field.

Table 1.

Simulation parameters.

Name	Symbol	Parameter
Inlet velocity	$u_{0} (x, t)$	1.5 m/s
Normal forces coefficient	K_j	100
Shear forces coefficient	K_z	30
Tension forces coefficient	K_z	5
Bending forces coefficient	K_w	0.5
Amplitude	A	0.4
Frequency	$F_{r e}$	1000
Initial fluid density	ρ₀	1.0 g/cm³

Trajectory tracking control

Obstacle avoidance

Because a multi-joint snake-like robot needs to avoid obstacles in the channel, the IB-LBM method not only has limitations in obtaining the motion position and rotation angle of each joint of the snake-like robot but also has some difficulties in trajectory tracking.

In order to improve the trafficability of a multi-joint snake-like robot in the flow field of the obstacle channel, all IB points of the robot and the obstacle are optimized by adding potential function, which enlarges the prediction range of the obstacle avoidance of the multi-joint snake-like robot in the channel.¹⁶ Firstly, an influence distance is set for each obstacle point of the obstacle channel. Within this influence distance, the obstacle point repels the IB point of the multi-joint snake-like robot. The closer the distance is, the stronger the rejection is, while the farther the distance is, the less the rejection is. When the IB point of the multi-joint snake-like robot escapes from this influence distance, the obstacle point has no effect.

The negative gradient function of repulsive force is as follows

\begin{array}{l} F_{r} (X (x, t), n) = {\begin{matrix} K_{r} (\frac{1}{X (x, t) - X (n)} - \frac{1}{P}) \frac{1}{{(X (x, t) - X (n))}^{2}} \frac{\partial (X (x, t) - X (n))}{\partial X (x, t)}, & X (x, t) - X (n) \leq P \\ 0, & X (x, t) - X (n) > P \end{matrix} \end{array}

where K_r is the repulsive gain coefficient, $X (n)$ is the point position of the nth obstacle, and P is the local influence range of the obstacle point.

The multi-joint robot encounters the obstacle of the flow field, and it can perceive the position and distance of the obstacle in advance, which makes the multi-joint snake-like robot change its direction ahead of time and ensures that the robot has changed its direction of motion before reaching the obstacle point, thus avoiding the obstacle.

When the multi-joint snake-like robot is within the influence distance of the multi-obstacle points, the repulsive force is the resultant force $F^{'} (X (x, t), n)$ of the repulsive force of the multi-joint snake-like robot given by the effective obstacle points

F^{'} (X (x, t), n) = \sum_{i = 1}^{n} F (X (x, t), n)

The force acting on the multi-joint snake-like robot at $t + 1$ moment is the sum of inertia force $F (s, t)$ at t moment, direction force $f (s, t)$ given by velocity of the flow field and repulsion force $F^{'} (X (x, t), n)$ given by the obstacle point. Combination force is as follows

F (X (x, t), n, t + 1) = F (x, t) + f (x, t) + F^{'} (X (x, t), n)

The OB points of the multi-joint snake-like robot are optimized by adding potential function, which enlarges the prediction range of path planning of the multi-joint snake-like robot and improves the prediction ability of the obstacle avoidance in the channel.

Trajectory tracking control

According to the motion posture of a biological snake in water, and with reference to proposed by Professor Hirose of Tokyo Institute of Technology in Japan,²¹ the improved snake-like curve motion equation is obtained from the point of view of kinematics. Equation is as follows

M_{f} = A sin (F_{re} t + F_{i 1} + F_{i 0})

where A is the swing amplitude, $F_{i 0}$ is the initial phase of the ith joint at the tail, and $F_{i 1}$ is the differential phase of the ith joint of the tail.

The trajectory tracking control law requires that each tail joints to track the front trajectory of the previous joint, and the trajectory tracking error should be controlled within a certain ideal range.^22,23 The multi-joint snake-like robot is designed to consist of six joints. The trajectory tracking control law enables the ith joint to track the trajectory of the front joint, as shown in Figure 5. In this way, the trajectory tracking control law can not only overcome the shortcomings of traditional linear feedback and achieve real-time obstacle avoidance, but also overcome the shortcomings of traditional high-order derivation, and has great stability and robustness.

Figure 5.

Joint angle of a multi-joint snake-like robot, adapted from Kelasidi et al.²⁴

The improved snake-like curve gives the mathematical expression of the time-varying joint angles of a multi-joint snake-like robot in its meandering motion. In fact, the improved snake-like curve is not limited to the meandering motion of a multi-joint snake-like robot. Other motion modes of the improved snake-like curve can be evolved on the basis of the snake-like curve, including the turning, obstacle avoidance, and tracking control of the multi-joint snake-like robot.

In the inertial rectangular coordinate system, the component of position $X (x, t)$ and rotation angle ϕ is $(x_{i}, y_{i}, ϕ_{i})$ , and the component of velocity $U (x, t)$ and angle w is $(u_{i}, v_{i}, ψ_{i})$ . The dynamic model of the multi-joint snake-like robot is as follows²⁴

{\begin{cases} \dot{x} = u cos φ + v sin φ \\ \dot{y} = - u sin φ + v cos φ \\ \dot{ϕ} = ψ \end{cases}

The position error of trajectory tracking is as follows

{\begin{cases} x_{e} = x_{i} - x \\ y_{e} = y_{i} - y \\ ϕ_{e} = ϕ_{i} - ϕ \end{cases}

The speed error of trajectory tracking is as follows

{\begin{cases} u_{e} = u_{i} - u \\ v_{e} = v_{i} - v \\ ψ_{e} = ψ_{i} - ψ \end{cases}

The relationship between position error and velocity error of trajectory tracking is as follows

{\begin{cases} u_{i} = {\dot{x}}_{i} cos φ + {\dot{y}}_{i} sin φ \\ v_{i} = - {\dot{x}}_{i} sin φ + {\dot{y}}_{i} cos φ \end{cases}

The difference equation of position error is as follows

[\begin{array}{l} {\dot{x}}_{e} \\ {\dot{y}}_{e} \\ {\dot{ϕ}}_{e} \end{array}] = [\begin{array}{l} {\dot{x}}_{i} \\ {\dot{y}}_{i} \\ {\dot{ϕ}}_{i} \end{array}] - J [\begin{array}{l} u \\ v \\ ψ \end{array}]

J = [\begin{array}{l} cos φ & sin φ & 0 \\ - sin φ & cos φ & 0 \\ 0 & 0 & 1 \end{array}]

where J is a transformation matrix and satisfies $| J | = 1$ condition, when the tracking error of velocity vector is $u_{e} \to 0, v_{e} \to 0$ and the tracking error of position trajectory is $x_{e} \to 0, y_{e} \to 0$ .

For yaw trajectory tracking control of a multi-joint snake-like robot in the flow field, the longitudinal velocity of the multi-joint snake-like robot is controlled by tail thrust, and the velocity v is smaller than u, that is, $v \to 0$ . In this way, a simplified dynamic model is obtained

{\begin{cases} \dot{y} = - u sin φ \\ \dot{ϕ} = ψ \\ \dot{ψ} = - d_{1} ψ - d_{2} u sin φ - d_{3} φ_{e} \end{cases}

Where $d_{1} > 0, d_{2} > 0, d_{3} > 0.$

The selection of a Lyapunov function is as follows

L = L_{1} + L_{2} + L_{3} = \frac{1}{2} y_{e}^{2} + \frac{1}{2} b_{1} φ_{e}^{2} + \frac{1}{2} b_{2} ψ_{e}^{2}, b_{1} > 0, b_{2} > 0

where

φ_{e} = φ + c_{1} y_{e}, c_{1} > 0

ψ_{e} = ψ + c_{2} φ_{e}, c_{2} > 0

Because y_i is a step response, therefore, when the multi-joint snake-like robot is working, $t > 0$ , so ${\dot{y}}_{i} = 0$ and ${\ddot{y}}_{i} = 0$ .

The deformation $y_{e} = \frac{φ_{e} - φ}{c_{1}}$ of equation (26) is substituted into equation (28)

\begin{array}{l} {\dot{L}}_{1} = y_{e} {\dot{y}}_{e} = y_{e} ({\dot{y}}_{i} - \dot{y}) = - y_{e} \dot{y} = y_{e} u sin φ \cdot 1 \\ = (y_{e} u sin φ) (- c_{1} \frac{1}{φ} \frac{φ_{e} - φ}{c_{1}} + \frac{1}{φ} φ_{e}) \\ = (y_{e} u sin φ) (- c_{1} \frac{1}{φ} y_{e} + \frac{1}{φ} φ_{e}) \\ = - c_{1} u \frac{sin φ}{φ} y_{e}^{2} + u \frac{sin φ}{φ} y_{e} φ_{e} \end{array}

where $\forall φ \in (- π / 2, π / 2)$ , $u > 0$ . So $- c_{1} u \frac{sin φ}{φ} y_{e}^{2} < 0$ .

{\dot{φ}}_{e} = \dot{φ} + c_{1} {\dot{y}}_{e} = ψ - c_{1} u sin φ

\begin{array}{l} {\dot{L}}_{2} = b_{1} φ_{e} {\dot{φ}}_{e} = b_{1} φ_{e} (\dot{φ} + c_{1} {\dot{y}}_{e}) = b_{1} φ_{e} (ψ + c_{1} u sin φ) \\ = b_{1} φ_{e} [ψ + c_{1} u \frac{sin φ}{φ} (φ_{e} - c_{1} y_{e})] \end{array}

{\dot{L}}_{1} + {\dot{L}}_{2} = - c_{1} u \frac{sin φ}{φ} y_{e}^{2} + b_{1} φ_{e} [ψ + c_{1} u \frac{sin φ}{φ} φ_{e} + (\frac{1}{b_{1}} - c_{1}^{2}) u \frac{sin φ}{φ} y_{e}]

when $\frac{1}{b_{1}} = c_{1}^{2}$ .

{\dot{L}}_{1} + {\dot{L}}_{2} = - c_{1} u \frac{sin φ}{φ} y_{e}^{2} + b_{1} φ_{e} (ψ + c_{1} u \frac{sin φ}{φ} φ_{e})

Substituting equation (27) into equation (32)

\begin{array}{l} {\dot{L}}_{1} + {\dot{L}}_{2} = - c_{1} u \frac{sin φ}{φ} y_{e}^{2} - c_{2} b_{1} (1 - \frac{c_{1} u}{c_{2}} \frac{sin φ}{φ}) φ_{e}^{2} + b_{1} φ_{e} ψ_{e} \end{array}

So long as the condition $1 - \frac{c_{1} u}{c_{2}} \frac{sin φ}{φ} > 0$ is satisfied, $- c_{2} b_{1} (1 - \frac{c_{1} u}{c_{2}} \frac{sin φ}{φ}) φ_{e}^{2} < 0$ can be established.

\begin{array}{l} {\dot{L}}_{3} = b_{2} ψ_{e} {\dot{ψ}}_{e} = b_{2} ψ_{e} (\dot{ψ} + c_{2} {\dot{φ}}_{e}) \\ = b_{2} ψ_{e} [\dot{ψ} + c_{2} (\dot{φ} + c_{1} {\dot{y}}_{e})] \\ = b_{2} ψ_{e} [\dot{ψ} + c_{2} (ψ + c_{1} u sin φ)] \end{array}

\dot{L} = {\dot{L}}_{1} + {\dot{L}}_{2} + {\dot{L}}_{3} = - c_{1} u \frac{sin ϕ}{ϕ} y_{e}^{2} - b_{1} c_{2} (1 - \frac{c_{1} u}{c_{2}} \frac{sin ϕ}{ϕ}) ϕ_{e}^{2} + b_{2} ψ_{e} [\dot{ψ} + c_{2} (ψ + c_{1} u sin ϕ) + \frac{b_{1}}{b_{2}} ϕ_{e}]

when $d_{1} = c_{1} + c_{2}$ , $d_{2} = c_{1} c_{2}$ , $d_{3} = c_{2} c_{3} + \frac{b_{1}}{b_{2}}$ , $c_{3} > 0$ .

\begin{array}{l} \dot{ψ} = - d_{1} ψ - d_{2} u sin φ - d_{3} φ_{e} \\ = - (c_{1} + c_{2}) ψ - c_{1} c_{2} u sin φ - (c_{2} c_{3} + \frac{b_{1}}{b_{2}}) φ_{e} \\ = - c_{3} (ψ + c_{2} φ_{e}) - c_{1} c_{2} u sin φ - c_{2} ψ - \frac{b_{1}}{b_{2}} φ_{e} \\ = - c_{3} ψ_{e} - c_{1} c_{2} u sin φ - c_{2} ψ - \frac{b_{1}}{b_{2}} φ_{e} \end{array}

Substituting equation (24) into equation (36)

\begin{array}{l} \dot{L} = {\dot{L}}_{1} + {\dot{L}}_{2} + {\dot{L}}_{3} \\ = - c_{1} u \frac{sin φ}{φ} y_{e}^{2} - b_{1} c_{2} (1 - \frac{c_{1} u}{c_{2}} \frac{sin φ}{φ}) φ_{e}^{2} - b_{2} c_{3} ψ_{e}^{2} < 0 \end{array}

In conclusion, according to Lyapunov stability criterion, it is verified that the trajectory tracking control law can make the ith joint of a multi-joint snake-like robot track the motion trajectory of the front joint, which has certain stability and robustness.

Simulation

In order to verify the trajectory tracking characteristics of each joint of the snake-like robot based on the improved snake-like curve, the simulation experiments were carried out by using MATLAB to study the influence of disturbance parameters such as different flow field density, velocity, and Reynolds numbers on the trajectory tracking of the multi-joint snake-like robot. The lateral distance, longitudinal distance, and direction angle of each joint of the robot under different parameters were analyzed.^24,25 The trajectory tracking control block diagram of the multi-joint snake-like robot avoiding obstacles in the flow field is shown in Figure 6.

Figure 6.

Trajectory tracking control block diagram.

In this study, the water in the flow field was assumed to be an incompressible Newtonian fluid. The motion vector diagram of the multi-joint snake-like robot passing through the obstacle channel was shown in Figure 7. Before the multi-joint snake-like robot enters the channel, its tail joints were under great force and swing greatly, which provides continuous power for the snake-like robot. When the multi-joint snake-like robot enters the channel, the head swings to move forward, the head force increases, the force on the tail joint decreases, and the swing amplitude decreases. The tail joints follow the motion of the front joint, and each joint tracks the motion trajectory of the front joint. When the multi-joint snake-like robot moves out of the channel, the tail swing increases, and the snake-like curve motion mode was restored, which continues to provide power for the snake-like robot moving forward.

Figure 7.

Motion vector map of a multi-joint snake-like robot.

In this section, the trajectory tracking control law of each joint of a snake-like robot in the obstacle channel based on the improved snake-like curve was studied, which was mainly manifested in tracking path and tracking stability. Therefore, according to the motion trajectories of each joint of the snake-like robot, the difference between the front and back joint trajectories was calculated to verify the stability of the trajectory tracking control law of the multi-joint snake-like robot based on the improved snake-like curve.

Effect of flow density

The influence of different flow field densities on trajectory tracking and turning angle of a multi-joint snake-like robot based on the improved snake-like curve was studied, as shown in Figure 8. The trajectory tracking of the multi-joint snake-like robot was studied when kerosene $ρ = 0.8 g / c m^{3}$ , water $ρ = 1.0 g / c m^{3}$ , seawater $ρ = 1.027 g / c m^{3}$ , and sewage $ρ = 1.1 g / c m^{3}$ . It can be seen from Figure 8(a) that the motion of the multi-joint snake-like robot was free before entering the channel, and the joints swing based on the snake-like curve, which provides continuous power for the snake-like robot moving forward. when the snake-like robot enters the channel, each joint tracks the trajectory of the front joint because of the narrow position. When the snake-like robot swims out of the channel, the swing amplitude of the tail joints restores the swing motion of the snake-like curve. As can be seen from Figure 8(b), when the snake-like robot enters the channel, each joint tracks the direction angle of the front joint. The difference between the actual direction angle and the tracking direction angle was $φ_{e} < 0.2 rad$ , and the angle difference was within a very small range, which verifies the effectiveness of the trajectory tracking control law.

Figure 8.

Trajectory tracking errors in different densities of flow field. (a) Trajectory tracking position of each joint under different densities. (b) Trajectory tracking angles of joints at different densities.

This section studies the error stability of the snake-like robots’ trajectory tracking in the obstacle channel under different fluid densities, focusing on the trajectory tracking error of the snake-like robots when $ρ = 1.0 g / c m^{3}$ was analyzed, as shown in Figure 9. It can be seen that the trajectory tracking error of each joint decreases continuously after entering the channel, and finally remains within ±3 cm (black). The error was controlled within a reasonable range, which verifies the stability of the trajectory tracking control law.

Figure 9.

Trajectory tracking errors of each joint at $ρ = 1.0 {g/cm}^{3}$ .

The influence of flow field velocity

The effects of different flow field densities on the trajectory tracking and turning angles of a multi-joint snake-like robot based on improved snake-like curve are studied, as shown in Figure 10. The trajectory tracking of each joint of the multi-joint snake-like robot was studied at fluid velocities of $v = 0.75 m / s$ , $v = 1.0 m / s$ , $v = 1.5 m / s$ , and $v = 2.0 m / s$ . From Figure 10(a), it can be seen that the slower the fluid flow speed is, the more path coincidence paths are between the joints of the snake-like robot, that is, the smaller the deviation of the joint tracking trajectory is, the more stable the trajectory tracking control is. Conversely, the faster the velocity of the fluid is, the greater the force of the fluid on the snake body is. The greater the disturbance to the trajectory tracking of each joint of the snake-like robot, the greater the trajectory tracking deviation and the lower the stability. As can be seen from Figure 10(b), when the snake-like robot enters the channel, each joint tracks the direction angle of the front joint. The difference between the actual direction angle and the tracking direction angle was $φ_{e} < 0.2 rad$ , which verifies the effectiveness of the trajectory tracking control law.

Figure 10.

Trajectory tracking errors in different velocities of flow field. (a) Trajectory tracking position of each joint under different velocities. (b) Trajectory tracking angles of joints at different velocities.

This section studies the error stability of trajectory tracking of the snake-like robots in the obstacle channel under different flow velocities and focuses on the trajectory tracking error of the snake-like robots when $v = 1.5 m / s$ was applied, as shown in Figure 11. It can be seen that the trajectory tracking error of each joint decreases continuously after entering the channel and finally remains within ±3 cm (black). The error was controlled within a reasonable range, which verifies the stability of the trajectory tracking control law.

Figure 11.

Trajectory tracking errors of each joint at v = 1.5 m/s.

The influence of Reynolds numbers

The effects of the Reynolds number of different flow fields on the trajectory tracking and turning angles of a multi-joint snake-like robot based on improved snake-like curve are studied, as shown in Figure 12. The trajectory tracking of the multi-joint snake-like robots with the fluid Reynolds numbers $R = 500$ , $R = 1000$ , $R = 1500$ , and $R = 2500$ was studied. It can be seen from Figure 12(a) that when the multi-joint snake-like robot moves in a fluid environment with high viscosity and low Reynolds numbers, the high viscosity fluid will disturb the trajectory tracking of the multi-joint snake-like robot greatly and increase the errors of the trajectory tracking of the robot. When the Reynolds numbers increase to certain height, the trajectory tracking of each joint of the snake-like robot was relatively stable, and the trajectory tracking error can be controlled in a relatively low range. When the Reynolds numbers continue to increase and the fluid velocity increases, the force exerted by the fluid on the snake-like robot increases, that is, the external disturbance increases, which has adverse effects on the trajectory tracking of the robot, and makes errors of the joint trajectory tracking of the snake-like robot increase and the stability decreases. As can be seen from Figure 12(b), when the fluid Reynolds numbers were controlled within certain range, the difference between the actual direction angle and the tracking direction angle of each joint of the snake-like robot was $φ_{e} < 0.2 rad$ , which verifies the effectiveness of the trajectory tracking control law.

Figure 12.

Trajectory tracking errors in different Reynolds numbers of flow field. (a) Trajectory tracking position of each joint under different Reynolds numbers. (b) Trajectory tracking angles of joints at different Reynolds numbers.

This section studies the error stability of trajectory tracking of the snake-like robot in the obstacle channel under different Reynolds numbers of the flow field. It focuses on the trajectory tracking errors of the snake-like robot when $R = 1000$ was analyzed, as shown in Figure 13. It can be seen that the trajectory tracking errors of each joint decrease continuously after entering the channel and finally remain within ±3 cm (black). The errors are controlled within a reasonable range, which verifies the stability of the trajectory tracking control law.

Figure 13.

Trajectory tracking errors of each joint at $R = 1000$ .

Result analysis

In order to quantitatively analyze the influences of flow field density, velocity, and Reynolds numbers on obstacle avoidance performance of the snake-like robot, in this section, the trajectory tracking errors of the multi-joint snake-like robot are analyzed when the density is $ρ = 0.7 g / c m^{3}$ and $ρ = 0.9 g / c m^{3}$ , the speed is $v = 0.5 m / s$ and $v = 2.5 m / s$ , and the Reynolds numbers are $R = 750$ and $R = 3000$ . Because the tracking errors of the fifth joint of the multi-joint snake-like robot are larger than those of the other joints, the maximum error of the fifth joint of the multi-joint snake-like robot in tracking the fourth joint is compared and analyzed, as shown in Figures 14 to 16.

Figure 14.

Effect of density on tracking errors of the robot.

Figure 15.

Effect of flow field velocity on tracking errors of the robot.

Figure 16.

Effect of Reynolds numbers on tracking errors of the robot.

The influence of flow field density on obstacle avoidance performance of the snake-like robot is analyzed, as shown in Figure 14. It can be seen that the trajectory tracking errors of the robot increase with the increase of the density of the flow field after the robot entering the channel. However, when the flow field density is within the range of $0.7 g / c m^{3} < ρ < 1.1 g / c m^{3}$ , the tracking errors of each joint of the multi-joint snake-like robot are less than 3 cm.

The influence of flow field velocity on obstacle avoidance performance of the snake-like robot is further analyzed, as shown in Figure 15. It can be seen that the trajectory tracking errors of the robot increase with the increase velocity of the flow field after entering the channel. When the velocity of flow field is $v \leq 0.75 m / s$ , the trajectory tracking errors of the robot change little and are relatively stable. When the velocity of flow field is $v > 1.5 m / s$ , the trajectory tracking errors of the robot increase rapidly. When the velocity of flow field is $v = 2.5 m / s$ , the robot shows the phenomenon of trajectory tracking failure and collision with the wall.

The influence of Reynolds numbers on obstacle avoidance performance of snake-like robot is further analyzed, as shown in Figure 16. It can be seen that when the robot enters the channel, an inverted “V” shape distribution appears with the increase of Reynolds numbers. Too large or too small Reynolds numbers in the flow field will increase the trajectory tracking errors of each joint of the snake-like robot. When Reynolds numbers are $R \leq 750$ and $R \geq 3000$ , the trajectory tracking errors of the robot are large and the tracking instability will occur.

Experiment

In order to verify the trajectory tracking control law of a multi-joint snake-like robot based on improved snake-like curve in the flow field, experiments were carried out in a flume, as shown in Figure 17. The multi-joint snake-like robot was composed of six joints. The length of the joint was 14 cm. The robot was completely immersed in water and four obstacles were set. The trajectory tracking of each joint in the obstacle avoidance was studied. The kinematics parameters of the robot were detailed in the simulation parameter setting section.

Figure 17.

Trajectory tracking experiment of each joint.

In the experiment, the position and rotation angle information of each joint of the multi-joint snake-like robot were sent to MATLAB. The proposed trajectory tracking control law was combined with the direction controller to obtain the gait amplitude, and the joint angle information was converted into pulse width modulation (PWM) signal to drive the motor.

The obstacle avoidance trajectory of each joint of the snake-like robot based on the improved snake-like curve was shown in Figure 18. From Figure 18, we can see the trajectory tracking characteristic curve of each joint of the snake-like robot. In the experiment, the external wires will disturb the robot to a certain extent, and the experiment also proves that joints of the robot can also achieve trajectory tracking control in the presence of multiple disturbance. In conclusion, the trajectory tracking control law for the obstacle avoidance in the flow field was feasible for the multi-joint snake-like robot, and the trajectory tracking control law has good anti-disturbance ability.

Figure 18.

Trajectory tracking curve of each joint.

Conclusion

In this article, the fluid–structure coupling models of the flow field and a multi-joint snake-like robot were established by using IB-LBM control method. A potential function was added to the head joint of a multi-joint snake-like robot to make the head avoid obstacles smoothly. The trajectory tracking control law of a multi-joint snake-like robot based on the improved snake-like curve to avoid obstacles in the flow field was used to realize the pre-tracking of each joint of the snake-like robot. The stability and robustness of the trajectory tracking control law were proved by searching a Lyapunov function.

At the same time, the influence of disturbance parameters such as different flow field density, velocity, and Reynolds numbers on the trajectory tracking of a multi-joint snake-like robot was studied by MATLAB simulation and experiment, and the steady states of the transverse distance, longitudinal distance, and direction angle of different joints of the multi-joint robot were analyzed. The trajectory tracking control law based on the improved snake-like curve was validated. It can not only overcome the external interference imposed on the multi-joint snake-like robot system but also can track the trajectory of the front joint to complete obstacle avoidance.

Footnotes

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: This research was supported by The National Natural Science Foundation of China under grant number 5177041109.

ORCID iD

Dongfang Li

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Trajectory tracking control law of multi-joint snake-like robot based on improved snake-like curve in flow field

Abstract

Keywords

Introduction

Fluid–structure interaction model

Dynamic model

Obstacle channel model

Multi-joint snake-like robot model

Parameter setting

Trajectory tracking control

Obstacle avoidance

Trajectory tracking control

Simulation

Effect of flow density

The influence of flow field velocity

The influence of Reynolds numbers

Result analysis

Experiment

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References