In this work, I investigate the challenging problem of trajectory tracking for four-wheel steering intelligent mobile industrial robots under tire cornering stiffness perturbations. These industrial robots are widely deployed in smart manufacturing, logistics, warehouse management, agricultural production, disaster rescue, environmental monitoring, and military applications because of their high load capacity, environmental perception, localization, navigation, and actuator integration. Traditional mobile industrial robots often use a front-steering and rear-driving configuration, which suffers from large turning radii, poor stability, and limited flexibility. To overcome these limitations, four-wheel independent steering and four-wheel independent driving industrial robots have emerged. Each wheel of such an industrial robot is driven by an independent motor and participates in independent steering, yielding a smaller turning radius and higher maneuverability in complex and narrow spaces. However, when the front wheels steer, the industrial robot generates a rotation angle about its own axis, and the front and rear axle centers are no longer parallel. The extension lines of the two axle centers intersect at the steering center of the four-wheel steering intelligent mobile industrial robot. The self-rotation produces a yaw angle and centripetal force, and the rear wheels deviate from the normal path, causing sideslip. If the rear wheels steer in the same direction as the front wheels, the turning radius becomes larger, but the centripetal force of the rear wheels can be effectively counteracted, suppressing the self-rotation of the body and allowing the industrial robot to return to the correct driving trajectory. This configuration is suitable for improving stability at high speeds and reducing safety hazards caused by sideslip. Conversely, if the rear wheels steer in the opposite direction to the front wheels, the centripetal force of the body increases, but the steering center moves closer to the body, the turning radius becomes smaller, and the industrial robot can complete a turn within a relatively small range, which is suitable for improving flexibility at low speeds.
Tire cornering stiffness is a key factor affecting the motion stability and trajectory tracking of four-wheel steering intelligent mobile industrial robots. In traditional dynamic models, tire cornering stiffness is often treated as a deterministic value. However, under different operating conditions, the vertical load of each tire changes nonlinearly, and cornering stiffness varies with vertical load. This irregular mapping causes uncertain perturbations in cornering stiffness. In addition, tire pressure and road adhesion coefficient also induce perturbations in tire cornering stiffness, directly affecting the contact between the tire and the ground. For example, when a vehicle operates on icy and snowy roads, the road adhesion coefficient is low, cornering stiffness decreases, and slipping easily occurs, causing the motion trajectory to deviate. To address the problem of cornering stiffness perturbations, many researchers have conducted in-depth studies. Some have constructed three-degree-of-freedom nonlinear models combined with tire cornering stiffness parameters and proposed state feedback control strategies based on optimal control theory, improving vehicle handling stability and driving safety. Others have built corresponding relationships between front and rear tire slip angles and cornering stiffness, introduced dynamic cornering stiffness parameters into the dynamic model, and adopted model predictive control to optimally control the slip angles of front and rear wheels, overcoming instability caused by tire lateral force saturation and ensuring vehicle motion stability. For stability control problems caused by changes in ground adhesion, linear time-varying model predictive control strategies have been designed through locally linearized tire models, transforming changes in ground adhesion and wheel steering angle constraints into quadratic programming problems to obtain optimal control steering angles for front and rear wheels, effectively improving vehicle control stability.
In summary, to overcome the influence of tire cornering stiffness perturbations on the motion state of four-wheel steering intelligent mobile industrial robots, I establish a dynamic model considering cornering stiffness perturbations, design adaptive integral sliding mode control laws for the displacement loop and velocity loop respectively, and achieve high-precision trajectory tracking control. Finally, I verify the rationality and effectiveness of the method through simulations and tests.
Mathematical Model of Industrial Robots
I analyze the motion of a four-wheel steering intelligent mobile industrial robot. The lateral forces on the left front, right front, left rear, and right rear wheels are denoted by Flf, Frf, Flr, and Frr, respectively. The distances from the center of gravity to the front and rear axles are a and b. The front and rear wheel steering angles are δf and δr. The rotation center is O′. The longitudinal and lateral velocities are Vx and Vy. The velocity at the center of gravity is vc. The sideslip angle at the center of gravity is β. The yaw angle is φ. The yaw rate is r.
Lateral Motion Model
During driving, the lateral displacement of the four-wheel steering intelligent mobile industrial robot satisfies
$$ \dot{Y} = V_x \sin\varphi + V_y \cos\varphi $$
where Y is the lateral displacement of the industrial robot.
When the four-wheel steering intelligent mobile industrial robot turns or accelerates laterally, it generates a rotational speed about an axis perpendicular to the longitudinal axis of the body, namely the yaw rate. At the same time, under the action of lateral force, an angle with the forward direction is generated, namely the sideslip angle at the center of gravity, expressed as
$$ \beta = \arctan\left(\frac{V_y}{V_x}\right) \approx \frac{V_y}{V_x} $$
The tire cornering stiffness of the four-wheel steering intelligent mobile industrial robot directly affects the yaw rate and the sideslip angle at the center of gravity. If it is too large or too small, the industrial robot loses stability, thereby affecting its motion state and trajectory. The lateral forces of the four tires are proportional to the slip angles, satisfying
$$ \begin{cases} F_{lf} = k_{lf} \alpha_{lf} \\ F_{rf} = k_{rf} \alpha_{rf} \\ F_{lr} = k_{lr} \alpha_{lr} \\ F_{rr} = k_{rr} \alpha_{rr} \end{cases} $$
where klf, krf, klr, and krr are the cornering stiffnesses of the left front, right front, left rear, and right rear wheels, respectively. The slip angles αlf, αrf, αlr, and αrr satisfy
$$ \begin{cases} \alpha_{lf} = \alpha_{rf} = \beta + \frac{ar}{V_x} – \delta_f = \frac{V_y + ar}{V_x} – \delta_f \\ \alpha_{lr} = \alpha_{rr} = \beta – \frac{br}{V_x} – \delta_r = \frac{V_y – br}{V_x} – \delta_r \end{cases} $$
Considering the influence of Coriolis acceleration, the lateral velocity of the four-wheel steering intelligent mobile industrial robot satisfies
$$ \dot{V}_y = a_y – V_x r $$
where ay is the lateral acceleration.
Combining the above equations, the lateral motion equation of the four-wheel steering intelligent mobile industrial robot can be expressed as
$$ \begin{cases} \dot{Y} = V_x \sin\varphi + V_y \cos\varphi \\ \dot{V}_y = \frac{(k_{lf} + k_{rf})(V_y + ar)}{mV_x} + \frac{(k_{lr} + k_{rr})(V_y – br)}{mV_x} – \frac{k_{lf} + k_{rf}}{m}\delta_f – \frac{k_{lr} + k_{rr}}{m}\delta_r – V_x r \end{cases} $$
Yaw Motion Model
Combining the above equations, the yaw motion equation of the four-wheel steering intelligent mobile industrial robot can be expressed as
$$ \begin{cases} I_z \dot{r} = ak_{lf}\left(\frac{V_y + ar}{V_x} – \delta_f\right) + ak_{rf}\left(\frac{V_y + ar}{V_x} – \delta_f\right) – bk_{lr}\left(\frac{V_y – br}{V_x} – \delta_r\right) – bk_{rr}\left(\frac{V_y – br}{V_x} – \delta_r\right) \end{cases} $$
Simplifying the above equation yields the yaw motion equation of the four-wheel steering intelligent mobile industrial robot as
$$ \begin{cases} \dot{\varphi} = r \\ \dot{r} = \frac{(k_{lf} + k_{rl})(aV_y + a^2 r)}{I_z V_x} + \frac{(k_{lr} + k_{rr})(b^2 r – bV_y)}{I_z V_x} – \frac{ak_{lf} + ak_{rl}}{I_z}\delta_f + \frac{bk_{lr} + bk_{rr}}{I_z}\delta_r \end{cases} $$
Cornering Stiffness Perturbation Analysis
To achieve high-precision trajectory tracking control of the four-wheel steering intelligent mobile industrial robot, I must consider the influence of cornering stiffness perturbations. The tire cornering stiffness of the industrial robot is expressed as
$$ \begin{cases} k_{lf} = k_{lf0} + \Delta k_{lf} \\ k_{rf} = k_{rf0} + \Delta k_{rf} \\ k_{lr} = k_{lr0} + \Delta k_{lr} \\ k_{rr} = k_{rr0} + \Delta k_{rr} \end{cases} $$
where klf0, krf0, klr0, and krr0 are the nominal parts of the cornering stiffnesses of the left front, right front, left rear, and right rear wheels, respectively. Δklf, Δkrf, Δklr, and Δkrr are the uncertain perturbation parts of the cornering stiffnesses of the left front, right front, left rear, and right rear wheels, respectively.
According to the lateral motion equation, the lateral motion equation of the four-wheel steering intelligent mobile industrial robot considering cornering stiffness perturbations is
$$ \begin{cases} \dot{Y} = V_x \sin\varphi + V_y \cos\varphi \\ \dot{V}_y = \frac{(k_{lf0} + k_{rf0})(V_y + ar)}{mV_x} + \frac{(k_{lr0} + k_{rr0})(V_y – br)}{mV_x} – \frac{k_{lf0} + k_{rf0}}{m}\delta_f – \frac{k_{lr0} + k_{rr0}}{m}\delta_r – V_x r \\ + \frac{(\Delta k_{lf} + \Delta k_{rf})(V_y + ar)}{mV_x} + \frac{(\Delta k_{lr} + \Delta k_{rr})(V_y – br)}{mV_x} – \frac{\Delta k_{lf} + \Delta k_{rf}}{m}\delta_f – \frac{\Delta k_{lr} + \Delta k_{rr}}{m}\delta_r \end{cases} $$
According to the yaw motion equation, the yaw motion equation of the four-wheel steering intelligent mobile industrial robot considering cornering stiffness perturbations is
$$ \begin{cases} \dot{\varphi} = r \\ \dot{r} = \frac{(k_{lf0} + k_{rl0})(aV_y + a^2 r)}{I_z V_x} + \frac{(k_{lr0} + k_{rr0})(b^2 r – bV_y)}{I_z V_x} – \frac{ak_{lf0} + ak_{rl0}}{I_z}\delta_f + \frac{bk_{lr0} + bk_{rr0}}{I_z}\delta_r \\ + \frac{(\Delta k_{lf} + \Delta k_{rl})(aV_y + a^2 r)}{I_z V_x} + \frac{(\Delta k_{lr} + \Delta k_{rr})(b^2 r – bV_y)}{I_z V_x} – \frac{a\Delta k_{lf} + a\Delta k_{rl}}{I_z}\delta_f + \frac{b\Delta k_{lr} + b\Delta k_{rr}}{I_z}\delta_r \end{cases} $$
Let x1 = [Y, φ]T be the displacement state of trajectory tracking for the four-wheel steering intelligent mobile industrial robot, x2 = [Vy, r]T be the velocity state of trajectory tracking, and u = [δf, δr]T be the control command for trajectory tracking. Then the trajectory tracking equation of the four-wheel steering intelligent mobile industrial robot considering cornering stiffness perturbations can be described as
$$ \begin{cases} \dot{x}_1 = A x_2 + B \\ \dot{x}_2 = C + D u + d \end{cases} $$
where d is the cornering stiffness perturbation term in the trajectory tracking equation. The state matrices A, B, C, and D are expressed as
$$ A = \begin{bmatrix} \cos\varphi & 0 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} V_x \sin\varphi \\ 0 \end{bmatrix} $$
$$ C = \begin{bmatrix} \frac{(k_{lf0} + k_{rf0})(V_y + ar)}{mV_x} + \frac{(k_{lr0} + k_{rr0})(V_y – br)}{mV_x} – V_x r \\ \frac{(k_{lf0} + k_{rl0})(aV_y + a^2 r)}{I_z V_x} + \frac{(k_{lr0} + k_{rr0})(b^2 r – bV_y)}{I_z V_x} \end{bmatrix} $$
$$ D = \begin{bmatrix} -\frac{k_{lf0} + k_{rf0}}{m} & -\frac{k_{lr0} + k_{rr0}}{m} \\ -\frac{ak_{lf0} + ak_{rl0}}{I_z} & \frac{bk_{lr0} + bk_{rr0}}{I_z} \end{bmatrix} $$
$$ d = \begin{bmatrix} \frac{(\Delta k_{lf} + \Delta k_{rf})(V_y + ar)}{mV_x} + \frac{(\Delta k_{lr} + \Delta k_{rr})(V_y – br)}{mV_x} – \frac{\Delta k_{lf} + \Delta k_{rf}}{m}\delta_f – \frac{\Delta k_{lr} + \Delta k_{rr}}{m}\delta_r \\ \frac{(\Delta k_{lf} + \Delta k_{rl})(aV_y + a^2 r)}{I_z V_x} + \frac{(\Delta k_{lr} + \Delta k_{rr})(b^2 r – bV_y)}{I_z V_x} – \frac{a\Delta k_{lf} + a\Delta k_{rl}}{I_z}\delta_f + \frac{b\Delta k_{lr} + b\Delta k_{rr}}{I_z}\delta_r \end{bmatrix} $$
Adaptive Trajectory Tracking Control Law
I first design an integral sliding mode adaptive control law for the displacement loop of the four-wheel steering intelligent mobile industrial robot. Then I design an adaptive law to estimate the cornering stiffness perturbation term and design an integral sliding mode robust control law for the velocity loop to achieve high-precision trajectory tracking control of the industrial robot. The control system structure is shown in the previous figure.
Displacement Loop Control Law Design
The displacement tracking error of the four-wheel steering intelligent mobile industrial robot is expressed as
$$ \tilde{x}_1 = x_1 – x_{1d} $$
where x1d is the desired trajectory, x1d = [Yd, φd]T. Here, Yd is the desired lateral trajectory, and φd is the desired yaw angle, satisfying
$$ \varphi_d = \frac{V_x \ddot{Y}_d}{(1 + \dot{Y}_d^2)^{3/2}} $$
Because traditional sliding mode control has chattering phenomena, to improve the control effect, I design an integral sliding mode surface for the displacement loop of the four-wheel steering intelligent mobile industrial robot as
$$ s_1 = l_1 \tilde{x}_1 + l_2 \int \operatorname{sgn}(\tilde{x}_1) \tilde{x}_1^{q/p} dt $$
where l1 and l2 are sliding mode surface control coefficients, l1, l2 > 0. sgn(·) is the sign function. p and q are sliding mode surface adjustment parameters, p > q > 0.
I construct the Lyapunov function V1 as
$$ V_1 = \frac{1}{2} s_1^T s_1 $$
Taking its derivative and substituting the system equations, error definition, and sliding mode surface yields
$$ \dot{V}_1 = s_1^T \dot{s}_1 = s_1^T \left( l_1 A x_2 + l_1 B – \dot{x}_{1d} + l_2 \operatorname{sgn}(\tilde{x}_1) \tilde{x}_1^{q/p} \right) $$
For the displacement loop of the four-wheel steering intelligent mobile industrial robot, I design the virtual adaptive control law as
$$ x_{2d} = \frac{1}{l_1} A^{-1} \left( -l_1 B + \dot{x}_{1d} – l_2 \operatorname{sgn}(\tilde{x}_1) \tilde{x}_1^{q/p} – l_3 s_1 \right) $$
where l3 is a control law coefficient, l3 > 0.
Substituting the virtual control command into the derivative of the Lyapunov function yields
$$ \dot{V}_1 = -l_3 s_1^T s_1 \le 0 $$
According to the Lyapunov stability theorem, the designed virtual adaptive control law ensures that the displacement loop of the four-wheel steering intelligent mobile industrial robot stably tracks the desired trajectory x1d.
Velocity Loop Control Law Design
The velocity tracking error of the four-wheel steering intelligent mobile industrial robot is expressed as
$$ \tilde{x}_2 = x_2 – x_{2d} $$
where x2d is the velocity command of the four-wheel steering intelligent mobile industrial robot, namely the virtual adaptive control law of the displacement loop.
For the velocity loop of the four-wheel steering intelligent mobile industrial robot, I design the integral sliding mode surface as
$$ s_2 = l_4 \tilde{x}_1 + l_5 \int \operatorname{sgn}(\tilde{x}_2) \tilde{x}_2^{m/n} dt $$
where l4 and l5 are sliding mode surface control coefficients, l4, l5 > 0. n and m are sliding mode surface adjustment parameters, n > m > 0.
To overcome the uncertainty caused by cornering stiffness perturbations, I need to accurately estimate them. For the cornering stiffness perturbation term d, I design the adaptive law as
$$ \dot{\hat{d}} = l_4 s_2 $$
where d̂ is the estimated value of the cornering stiffness perturbation term d.
I construct the Lyapunov function V2 as
$$ V_2 = \frac{1}{2} s_2^T s_2 + \frac{1}{2} \tilde{d}^T \tilde{d} $$
where d̃ is the estimation error of d̂ with respect to the cornering stiffness perturbation term d, d̃ = d – d̂.
Taking its derivative and substituting the system equations, error definition, sliding mode surface, and adaptive law yields
$$ \dot{V}_2 = s_2^T \left( l_4 C + l_4 D u + l_4 d – l_4 \dot{x}_{2d} + l_5 \operatorname{sgn}(\tilde{x}_2) \tilde{x}_2^{m/n} \right) + \tilde{d}^T \left( \dot{d} – l_4 s_2 \right) $$
For the velocity loop of the four-wheel steering intelligent mobile industrial robot, I design the adaptive robust control law uc as
$$ u_c = \frac{1}{l_4} D^{-1} \left( -l_4 C + l_4 \dot{x}_{2d} – l_4 \hat{d} – l_5 \operatorname{sgn}(\tilde{x}_2) \tilde{x}_2^{m/n} – l_6 s_2 + l_7 \tilde{d} + l_8 \dot{\bar{d}} \right) $$
where l6 is the adaptive law control coefficient, l6 > 0. l7 and l8 are adaptive law adjustment parameters. ḋ̄ is the upper bound of the cornering stiffness perturbation term d, satisfying d ≤ ḋ̄.
Substituting the adaptive control law into the derivative of the Lyapunov function and simplifying yields
$$ \dot{V}_2 = -l_6 s_2^T s_2 + l_7 s_2^T \tilde{d} + l_8 s_2^T \dot{\bar{d}} + \tilde{d}^T \dot{d} $$
$$ \le -l_6 s_2^T s_2 + (l_7^2 + l_8^2) \|s_2\|^2 + (l_7^2 + 1) \|\tilde{d}\|^2 + l_8^2 \|\dot{\bar{d}}\|^2 + \frac{1}{2} \|\dot{d}\|^2 $$
When l7 < -1 and l8 < -1, I obtain V̇2 ≤ 0. According to the Lyapunov stability theorem, the designed adaptive robust control law effectively overcomes the influence of cornering stiffness perturbations and ensures that the velocity loop of the four-wheel steering intelligent mobile industrial robot stably tracks the velocity command x2d.
Stability Analysis
I construct the Lyapunov function V as
$$ V = V_1 + V_2 $$
Taking its derivative and substituting the virtual adaptive control law of the displacement loop, the adaptive law of the cornering stiffness perturbation term, and the adaptive robust control law of the velocity loop yields
$$ \dot{V} \le -l_3 s_1^T s_1 – l_6 s_2^T s_2 + (l_7^2 + l_8^2) \|s_2\|^2 + (l_7^2 + 1) \|\tilde{d}\|^2 + l_8^2 \|\dot{\bar{d}}\|^2 + \frac{1}{2} \|\dot{d}\|^2 $$
When l7 < -1 and l8 < -1, I obtain V̇ ≤ 0. According to the Lyapunov stability theorem, the designed virtual adaptive control law, the adaptive law of the cornering stiffness perturbation term, and the adaptive robust control law of the velocity loop ensure that the four-wheel steering intelligent mobile industrial robot stably tracks the desired trajectory x1d.
Simulation Results
To verify the control effect of the proposed adaptive trajectory tracking robust control on the four-wheel steering intelligent mobile industrial robot, I conducted comparative simulations in the MATLAB/Simulink environment.
Simulation Parameter Settings
In the simulation, I set the physical parameters of the four-wheel steering intelligent mobile industrial robot, as shown in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| m (kg) | 75 | Iz (kg·m2) | 112.8 |
| a (m) | 0.25 | b (m) | 0.35 |
| klf0 (N/°) | 58 | krf0 (N/°) | 58 |
| klr0 (N/°) | 60 | krr0 (N/°) | 60 |
In the simulation, I set the control law parameters, as shown in Table 2.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| p | 3 | q | 2 |
| n | 5 | m | 3 |
| l1 | 7 | l2 | 6 |
| l3 | 4 | l4 | 2 |
| l5 | 5 | l6 | 9 |
| l7 | -4 | l8 | -5 |
Because the tire cornering stiffness of the four-wheel steering intelligent mobile industrial robot is a comprehensive parameter determined jointly by the tire’s own physical characteristics, real-time motion state, and external environment, it undergoes significant perturbations under different loads, road adhesion coefficients, tire pressures, and temperatures. In the simulation, to verify whether different control methods can effectively overcome the influence of cornering stiffness perturbations, I set a nonlinear tire cornering stiffness perturbation term d to simulate different motion conditions of the four-wheel steering intelligent mobile industrial robot. The expression of d is
$$ d = \begin{bmatrix} (0.1 \sin(0.2t) + e^{-0.1t}) \, \text{m/s}^2 \\ (0.1 \cos(0.1t) + e^{-0.2t}) \, \text{rad/s}^2 \end{bmatrix} $$
Double Lane Change Trajectory Tracking Simulation
Because the four-wheel steering intelligent mobile industrial robot is more prone to lateral stability degradation during steering, I first conduct trajectory tracking simulation with a classic double lane change. I set the motion trajectory control command of the double lane change as
$$ \begin{cases} X_d = 10t \\ Y_d = 2\left(1 + \tanh\left[0.15(10t – 35)\right]\right) – 2\left(1 + \tanh\left[0.15(10t – 65)\right]\right) \end{cases} $$
According to the control command, the theoretical running trajectory of the double lane change is plotted. Because the cornering stiffness perturbation term is added during the simulation, the motion stability of the four-wheel steering intelligent mobile industrial robot inevitably decreases, and the running trajectory changes. I respectively adopt fuzzy PID control (FPIDC), model predictive control (MPC), linear time-varying model predictive control (LTV-MPC), and the proposed adaptive trajectory tracking robust control (ATTRC) to track and control the motion trajectory of the four-wheel steering intelligent mobile industrial robot. By observing the curves of yaw rate, sideslip angle at the center of gravity, and lateral displacement deviation, I evaluate the motion stability and trajectory tracking accuracy. The simulation results are shown in Table 3.
| Control Method | Max Yaw Rate Error (°/s) | Average Yaw Rate Error (°/s) | Max Sideslip Angle Error (°) | Average Sideslip Angle Error (°) | Max Lateral Displacement Error (m) | Average Lateral Displacement Error (m) |
|---|---|---|---|---|---|---|
| FPIDC | 9.745 | 2.171 | 0.128 | 0.032 | 0.686 | 0.175 |
| MPC | 7.779 | 1.655 | 0.108 | 0.026 | 0.545 | 0.145 |
| LTV-MPC | 3.806 | 0.764 | 0.052 | 0.012 | 0.247 | 0.073 |
| ATTRC | 1.113 | 0.204 | 0.015 | 0.003 | 0.063 | 0.022 |
From the simulation results and Table 3, I observe that in the time intervals (0–2) s and (8–10) s, the errors of yaw rate, sideslip angle at the center of gravity, and lateral displacement are almost zero, indicating that the four-wheel steering intelligent mobile industrial robot is less affected by cornering stiffness perturbations when moving in a straight line longitudinally. In the interval (2–8) s, the four-wheel steering intelligent mobile industrial robot enters a curve, generating lateral force, and the influence of cornering stiffness perturbations becomes larger, causing the errors of yaw rate, sideslip angle at the center of gravity, and lateral displacement to increase and deviate from the ideal values to varying degrees. Specifically, under FPIDC, the maximum errors of yaw rate and sideslip angle at the center of gravity of the four-wheel steering intelligent mobile industrial robot reach 9.745 (°/s) and 0.128°, respectively, and the average errors reach 2.171 (°/s) and 0.032°, respectively. The instability is the most severe, causing the maximum and average errors of lateral displacement to reach 0.686 m and 0.175 m, respectively, and the trajectory tracking accuracy is the worst.
Under MPC, the maximum errors of yaw rate and sideslip angle at the center of gravity of the four-wheel steering intelligent mobile industrial robot reach 7.779 (°/s) and 0.108°, respectively, and the average errors reach 1.655 (°/s) and 0.026°, respectively. This also causes the maximum and average errors of lateral displacement to reach 0.545 m and 0.145 m, respectively, and the trajectory tracking error is also relatively large. Under LTV-MPC, the maximum errors of yaw rate and sideslip angle at the center of gravity of the four-wheel steering intelligent mobile industrial robot are 3.806 (°/s) and 0.052°, respectively, and the average errors are 0.764 (°/s) and 0.012°, respectively. The motion stability is improved, but the maximum and average errors of lateral displacement are still relatively large, at 0.247 m and 0.073 m, respectively, and the trajectory tracking accuracy is still not ideal. Under ATTRC, the maximum errors of yaw rate and sideslip angle at the center of gravity of the four-wheel steering intelligent mobile industrial robot are 1.113 (°/s) and 0.015°, respectively, and the average errors are 0.204 (°/s) and 0.003°, respectively. The errors of the motion state are very small, showing high stability. The maximum and average errors of lateral displacement are 0.063 m and 0.022 m, respectively, demonstrating high trajectory tracking accuracy. This verifies the rationality of the proposed adaptive trajectory tracking robust control method.
Double Knot Line Trajectory Tracking Simulation
To further verify the tracking effect and superiority of the proposed control method in dealing with more complex running trajectories, I also conduct trajectory tracking simulation with a double knot line. I set the ideal motion trajectory control command of the double knot line as
$$ \begin{cases} X_d = 30 \sin(0.25t) \\ Y_d = 30 \sin(0.25t) \cos(0.25t) \end{cases} $$
Similarly, I respectively adopt FPIDC, MPC, LTV-MPC, and the proposed ATTRC to track and control the motion trajectory of the four-wheel steering intelligent mobile industrial robot. I evaluate the motion stability and trajectory tracking accuracy through yaw rate, sideslip angle at the center of gravity, and lateral displacement deviation. The simulation results are shown in Table 4.
| Control Method | Max Yaw Rate Error (°/s) | Average Yaw Rate Error (°/s) | Max Sideslip Angle Error (°) | Average Sideslip Angle Error (°) | Max Lateral Displacement Error (m) | Average Lateral Displacement Error (m) |
|---|---|---|---|---|---|---|
| FPIDC | 8.118 | 0.761 | 0.095 | 0.045 | 0.957 | 0.609 |
| MPC | 5.507 | 0.515 | 0.063 | 0.030 | 0.636 | 0.405 |
| LTV-MPC | 2.753 | 0.256 | 0.032 | 0.015 | 0.317 | 0.202 |
| ATTRC | 0.651 | 0.063 | 0.008 | 0.004 | 0.076 | 0.048 |
From the simulation results and Table 4, I observe that when entering the curves on the left and right sides of the double knot line, the running state of the four-wheel steering intelligent mobile industrial robot is still greatly affected by cornering stiffness perturbations. The yaw rate, sideslip angle at the center of gravity, and lateral displacement all show obvious errors, causing its running trajectory to deviate from the set track. Under FPIDC, the yaw rate and sideslip angle at the center of gravity of the four-wheel steering intelligent mobile industrial robot deviate the most from the ideal values, with maximum errors reaching 8.118 (°/s) and 0.095°, respectively, and average errors reaching 0.761 (°/s) and 0.045°, respectively. The instability causes the maximum and average errors of lateral displacement to reach 0.957 m and 0.609 m, respectively, which are significantly higher than the lateral displacement errors under the double lane change condition. This indicates that under more complex trajectory conditions, the same cornering stiffness perturbation intensifies the disturbance of the motion state and trajectory of the four-wheel steering intelligent mobile industrial robot, and FPIDC fails to effectively track and control it. Under MPC and LTV-MPC, although the motion stability of the four-wheel steering intelligent mobile industrial robot is improved, it is still affected by cornering stiffness perturbations, and the tracking accuracy of the running trajectory is still not ideal. Under the proposed ATTRC, the four-wheel steering intelligent mobile industrial robot effectively overcomes the influence of cornering stiffness perturbations. Even under more complex running trajectory conditions, it maintains high motion stability. The maximum errors of yaw rate and sideslip angle at the center of gravity are 0.651 (°/s) and 0.008°, respectively, and the average errors are 0.063 (°/s) and 0.004°, respectively. It can track the trajectory command more accurately, and the maximum and average errors of lateral displacement are only 0.076 m and 0.048 m, respectively, proving that this method has wide adaptability.
In summary, whether under the classic double lane change trajectory condition or the more complex double knot line trajectory condition, the proposed adaptive trajectory tracking robust control method can effectively overcome the influence of cornering stiffness perturbations on the motion state of the four-wheel steering intelligent mobile industrial robot and maintain high control accuracy, thus verifying the rationality and effectiveness of the proposed design method.
Test Example
To further verify the effect of the proposed adaptive trajectory tracking robust control method in engineering applications, I take the classic double lane change trajectory condition as a test example and use a four-wheel steering intelligent mobile industrial robot produced by Xianfa Technology (model: LZ-PRO) as the object for verification. To obtain the motion trajectory of the four-wheel steering intelligent mobile industrial robot in real time, I installed a high-precision all-constellation all-frequency GPS + BeiDou navigation positioning module produced by Wit Intelligent (model: WTRTK-980), whose horizontal positioning accuracy can reach up to 0.8 cm, and communicated with the host computer through a wireless module.
The specific test steps are as follows: (1) Compile the proposed control algorithm and import it into the controller of the four-wheel steering intelligent mobile industrial robot. (2) Place the four-wheel steering intelligent mobile industrial robot at the origin of the test site, with coordinates (0, 0) m. (3) Send the trajectory control command to the controller of the four-wheel steering intelligent mobile industrial robot through the host computer, so that the four-wheel steering intelligent mobile industrial robot moves according to the preset trajectory command under the action of the control algorithm. (4) Monitor the position coordinates returned by the four-wheel steering intelligent mobile industrial robot in real time and record them until it moves to the end point.
Similarly, I complete the trajectory tracking tests of FPIDC, MPC, and LTV-MPC in sequence according to the above method. Finally, I plot the trajectory tracking curves and trajectory tracking error curves obtained by these four different control methods in a two-dimensional spatial coordinate system. From the test results, I observe that under the action of the four control methods, the four-wheel steering intelligent mobile industrial robot can track the control command on the entire running trajectory, but under FPIDC, MPC, and LTV-MPC, the instability and sideslip phenomena are more severe, resulting in large trajectory tracking errors. The maximum trajectory tracking errors of FPIDC, MPC, LTV-MPC, and the proposed ATTRC are 0.704, 0.589, 0.325, and 0.077 m, respectively, and the average errors are 0.226, 0.174, 0.106, and 0.038 m, respectively. This indicates that the proposed control method exhibits stronger robustness, stronger stability, and higher control accuracy in the trajectory tracking process of the four-wheel steering intelligent mobile industrial robot, and has strong engineering practicability.
| Control Method | Maximum Trajectory Tracking Error (m) | Average Trajectory Tracking Error (m) |
|---|---|---|
| FPIDC | 0.704 | 0.226 |
| MPC | 0.589 | 0.174 |
| LTV-MPC | 0.325 | 0.106 |
| ATTRC | 0.077 | 0.038 |
Conclusion
To solve the problems of sideslip instability and large trajectory tracking errors that easily occur when four-wheel steering intelligent mobile industrial robots are in curves due to tire cornering stiffness perturbations, I analyze the motion state of the four-wheel steering intelligent mobile industrial robot and design integral sliding mode adaptive robust control laws on the displacement loop and velocity loop. This achieves stable control of the motion state and high-precision tracking of the motion trajectory of the four-wheel steering intelligent mobile industrial robot. Through relevant simulations and tests, I draw the following conclusions.
(1) In the classic double lane change trajectory tracking simulation, the proposed adaptive trajectory tracking robust control method has higher stability and control accuracy. The maximum errors of yaw rate, sideslip angle at the center of gravity, and lateral displacement are 1.113 (°/s), 0.015°, and 0.063 m, respectively. It can effectively overcome the influence of cornering stiffness perturbations. The average errors are 0.008 (°/s), 0.003°, and 0.022 m, respectively, demonstrating high trajectory tracking control accuracy that meets the design expectations.
(2) In the more complex double knot line trajectory tracking simulation, the maximum errors of yaw rate, sideslip angle at the center of gravity, and lateral displacement of the proposed adaptive trajectory tracking robust control method are 0.651 (°/s), 0.063°, and 0.076 m, respectively. It can still effectively avoid instability and sideslip phenomena. The average errors are 0.008 (°/s), 0.004°, and 0.048 m, respectively, still showing high trajectory tracking control accuracy, thus verifying that the proposed control method has stronger adaptability.
(3) The test example results show that the proposed adaptive trajectory tracking robust control method can ensure that the four-wheel steering intelligent mobile industrial robot stably tracks the control command, effectively avoiding sideslip and instability. The maximum and average errors of trajectory tracking are 0.077 m and 0.038 m, respectively, with high control accuracy. It can adapt to the needs of more application scenarios and provide technical support for promoting the development of the industrial robot industry.
In this work, I have shown that the integration of adaptive laws, integral sliding mode control, and robust compensation for cornering stiffness perturbations significantly improves the trajectory tracking performance of four-wheel steering intelligent mobile industrial robots. The proposed method is validated through both simulations and physical tests, and the results confirm its superior stability, accuracy, and engineering practicability. Future work will focus on extending the approach to dynamic environments, multiple industrial robots cooperation, and real-time implementation on embedded platforms.