I study the adaptive trajectory tracking robust control problem for an intelligent mobile industrial robot equipped with four-wheel steering and independent wheel driving. In my research, the industrial robot is required to follow a desired path with high precision even when tire cornering stiffness changes because of load transfer, tire pressure variation, road adhesion variation, and temperature effects. Such changes are not merely secondary disturbances. For a four-wheel steering industrial robot, cornering stiffness perturbation directly changes lateral force generation, yaw moment balance, and sideslip behavior. When the industrial robot enters a curved path, the perturbation can create sideslip, yaw rate error, and lateral displacement error. If the industrial robot is performing welding, painting, assembly, logistics, or inspection tasks, even a small lateral error can reduce task quality and may cause instability. I therefore design an adaptive trajectory tracking robust control method that combines integral sliding mode control, virtual displacement loop control, adaptive estimation of cornering stiffness perturbation, and velocity loop robust control. I prove stability by Lyapunov theory and validate the method through simulations and tests.
The key idea of my method is to separate the industrial robot trajectory tracking problem into a displacement loop and a velocity loop. The displacement loop produces a virtual velocity command using displacement tracking error. The velocity loop then generates steering angle commands for the front and rear wheels of the four-wheel steering industrial robot. An adaptive law estimates the lumped perturbation caused by cornering stiffness variation. Because the adaptive law is embedded into the integral sliding mode control, the industrial robot can compensate for uncertainty without requiring exact knowledge of the real-time cornering stiffness. I use this architecture because an intelligent mobile industrial robot must operate under uncertain ground conditions while still maintaining smooth and stable trajectory tracking.
Table 1. Nomenclature used for the intelligent mobile industrial robot.
| Symbol | Meaning |
|---|---|
| \(m\) | Mass of the industrial robot |
| \(I_z\) | Yaw moment of inertia of the industrial robot |
| \(a\) | Distance from center of gravity to front axle |
| \(b\) | Distance from center of gravity to rear axle |
| \(V_x\) | Longitudinal velocity of the industrial robot |
| \(V_y\) | Lateral velocity of the industrial robot |
| \(r\) | Yaw rate of the industrial robot |
| \(\phi\) | Yaw angle of the industrial robot |
| \(\beta\) | Vehicle sideslip angle of the industrial robot |
| \(Y\) | Lateral displacement of the industrial robot |
| \(\delta_f\) | Front wheel steering angle |
| \(\delta_r\) | Rear wheel steering angle |
| \(k_{lf},k_{rf}\) | Cornering stiffness of left and right front tires |
| \(k_{lr},k_{rr}\) | Cornering stiffness of left and right rear tires |
| \(k_{lf0},k_{rf0},k_{lr0},k_{rr0}\) | Nominal cornering stiffness values |
| \(\Delta k_{lf},\Delta k_{rf},\Delta k_{lr},\Delta k_{rr}\) | Cornering stiffness perturbations |
| \(\alpha_{lf},\alpha_{rf},\alpha_{lr},\alpha_{rr}\) | Tire sideslip angles |
| \(F_{lf},F_{rf},F_{lr},F_{rr}\) | Tire lateral forces |
| \(x_1\) | Displacement state vector of the industrial robot |
| \(x_2\) | Velocity state vector of the industrial robot |
| \(u\) | Steering control input vector |
| \(d\) | Lumped cornering stiffness perturbation term |
| \(\hat{d}\) | Estimated perturbation term |
| \(\tilde{d}\) | Perturbation estimation error |
| \(s_1,s_2\) | Integral sliding surfaces |
| \(V_1,V_2,V\) | Lyapunov functions |
I begin with the kinematic and dynamic description of the four-wheel steering intelligent mobile industrial robot. The industrial robot has four wheels, and each wheel can be steered and driven independently. This configuration gives the industrial robot a smaller turning radius and higher maneuverability than a conventional front-steering industrial robot. However, independent four-wheel steering also makes the lateral and yaw motions strongly coupled. When cornering stiffness changes, the coupling becomes uncertain, and the industrial robot may lose tracking accuracy. I therefore model both the lateral motion and the yaw motion of the industrial robot.
The lateral displacement of the industrial robot is written as
$$ \dot{Y}=V_x\sin\phi+V_y\cos\phi. $$
The sideslip angle of the industrial robot is approximated as
$$ \beta=\arctan\left(\frac{V_y}{V_x}\right)\approx \frac{V_y}{V_x}. $$
The tire lateral force of each wheel is proportional to its sideslip angle:
$$ F_{lf}=k_{lf}\alpha_{lf},\quad F_{rf}=k_{rf}\alpha_{rf},\quad F_{lr}=k_{lr}\alpha_{lr},\quad F_{rr}=k_{rr}\alpha_{rr}. $$
The front and rear tire sideslip angles of the industrial robot are
$$ \alpha_{lf}=\alpha_{rf}=\beta+\frac{ar}{V_x}-\delta_f=\frac{V_y+ar}{V_x}-\delta_f, $$
$$ \alpha_{lr}=\alpha_{rr}=\beta-\frac{br}{V_x}-\delta_r=\frac{V_y-br}{V_x}-\delta_r. $$
Including the centripetal acceleration effect, the lateral acceleration of the industrial robot satisfies
$$ \dot{V}_y=a_y-V_xr. $$
Combining these relations, the lateral motion equation of the industrial robot becomes
$$ \dot{V}_y=\frac{(k_{lf}+k_{rf})(V_y+ar)}{mV_x}+\frac{(k_{lr}+k_{rr})(V_y-br)}{mV_x}-\frac{k_{lf}+k_{rf}}{m}\delta_f-\frac{k_{lr}+k_{rr}}{m}\delta_r-V_xr. $$
The yaw motion equation of the four-wheel steering industrial robot is
$$ I_z\dot{r}=ak_{lf}\left(\frac{V_y+ar}{V_x}-\delta_f\right)+ak_{rf}\left(\frac{V_y+ar}{V_x}-\delta_f\right)-bk_{lr}\left(\frac{V_y-br}{V_x}-\delta_r\right)-bk_{rr}\left(\frac{V_y-br}{V_x}-\delta_r\right). $$
After simplification, the yaw dynamics of the industrial robot are
$$ \dot{\phi}=r, $$
$$ \dot{r}=\frac{(k_{lf}+k_{rf})(aV_y+a^2r)}{I_zV_x}+\frac{(k_{lr}+k_{rr})(b^2r-bV_y)}{I_zV_x}-\frac{a(k_{lf}+k_{rf})}{I_z}\delta_f+\frac{b(k_{lr}+k_{rr})}{I_z}\delta_r. $$
The cornering stiffness of each tire is decomposed into a nominal part and an uncertain perturbation part:
$$ k_{lf}=k_{lf0}+\Delta k_{lf},\quad k_{rf}=k_{rf0}+\Delta k_{rf}, $$
$$ k_{lr}=k_{lr0}+\Delta k_{lr},\quad k_{rr}=k_{rr0}+\Delta k_{rr}. $$
Substituting these expressions into the lateral and yaw equations, I obtain the perturbed trajectory tracking model of the industrial robot. The lateral motion with perturbation is
$$ \dot{V}_y=\frac{(k_{lf0}+k_{rf0})(V_y+ar)}{mV_x}+\frac{(k_{lr0}+k_{rr0})(V_y-br)}{mV_x}-\frac{k_{lf0}+k_{rf0}}{m}\delta_f-\frac{k_{lr0}+k_{rr0}}{m}\delta_r-V_xr+d_1, $$
where the first perturbation component is
$$ d_1=\frac{(\Delta k_{lf}+\Delta k_{rf})(V_y+ar)}{mV_x}+\frac{(\Delta k_{lr}+\Delta k_{rr})(V_y-br)}{mV_x}-\frac{\Delta k_{lf}+\Delta k_{rf}}{m}\delta_f-\frac{\Delta k_{lr}+\Delta k_{rr}}{m}\delta_r. $$
The yaw motion with perturbation is
$$ \dot{r}=\frac{(k_{lf0}+k_{rf0})(aV_y+a^2r)}{I_zV_x}+\frac{(k_{lr0}+k_{rr0})(b^2r-bV_y)}{I_zV_x}-\frac{a(k_{lf0}+k_{rf0})}{I_z}\delta_f+\frac{b(k_{lr0}+k_{rr0})}{I_z}\delta_r+d_2, $$
where the second perturbation component is
$$ d_2=\frac{(\Delta k_{lf}+\Delta k_{rf})(aV_y+a^2r)}{I_zV_x}+\frac{(\Delta k_{lr}+\Delta k_{rr})(b^2r-bV_y)}{I_zV_x}-\frac{a(\Delta k_{lf}+\Delta k_{rf})}{I_z}\delta_f+\frac{b(\Delta k_{lr}+\Delta k_{rr})}{I_z}\delta_r. $$
I define the displacement state, velocity state, and control input of the industrial robot as
$$ x_1=\begin{bmatrix}Y\\ \phi\end{bmatrix},\quad x_2=\begin{bmatrix}V_y\\ r\end{bmatrix},\quad u=\begin{bmatrix}\delta_f\\ \delta_r\end{bmatrix}. $$
The perturbed trajectory tracking dynamics of the intelligent mobile industrial robot can then be expressed in state-space form as
$$ \dot{x}_1=Ax_2+B, $$
$$ \dot{x}_2=C+Du+d, $$
where the perturbation vector is \(d=[d_1,d_2]^T\), and the matrices are
$$ A=\begin{bmatrix}\cos\phi & 0\\ 0 & 1\end{bmatrix},\quad B=\begin{bmatrix}V_x\sin\phi\\ 0\end{bmatrix}, $$
$$ C=\begin{bmatrix}\frac{(k_{lf0}+k_{rf0})(V_y+ar)}{mV_x}+\frac{(k_{lr0}+k_{rr0})(V_y-br)}{mV_x}-V_xr\\ \frac{(k_{lf0}+k_{rf0})(aV_y+a^2r)}{I_zV_x}+\frac{(k_{lr0}+k_{rr0})(b^2r-bV_y)}{I_zV_x}\end{bmatrix}, $$
$$ D=\begin{bmatrix}-\frac{k_{lf0}+k_{rf0}}{m} & -\frac{k_{lr0}+k_{rr0}}{m}\\ -\frac{a(k_{lf0}+k_{rf0})}{I_z} & \frac{b(k_{lr0}+k_{rr0})}{I_z}\end{bmatrix}, $$
$$ d=\begin{bmatrix}d_1\\ d_2\end{bmatrix}. $$
This model is the basis of my adaptive trajectory tracking robust control design. I treat the lumped perturbation \(d\) as an unknown but bounded signal. The industrial robot does not need to measure each tire cornering stiffness online. Instead, the adaptive law estimates the combined effect of the four cornering stiffness perturbations. This is important because in a real industrial robot, individual tire parameters may be difficult or expensive to measure continuously.
I design the displacement loop first. Let the desired trajectory of the industrial robot be
$$ x_{1d}=\begin{bmatrix}Y_d\\ \phi_d\end{bmatrix}, $$
where \(Y_d\) is the desired lateral displacement and \(\phi_d\) is the desired yaw angle. The desired yaw angle satisfies the path curvature relation
$$ \phi_d=\frac{V_x\ddot{Y}_d}{(1+\dot{Y}_d^2)^{3/2}}. $$
The displacement tracking error of the industrial robot is defined as
$$ \tilde{x}_1=x_1-x_{1d}. $$
To reduce steady-state error and improve robustness, I use an integral sliding surface for the displacement loop:
$$ s_1=l_1\tilde{x}_1+l_2\int \operatorname{sgn}(\tilde{x}_1)|\tilde{x}_1|^{q/p}\,dt, $$
where \(l_1,l_2>0\) and \(p>q>0\). The fractional power term provides a smooth transition and improves the convergence behavior of the industrial robot trajectory tracking error. The sign term ensures robustness against disturbance. I choose an integral sliding surface because the integral action accumulates the displacement error and forces the industrial robot to return to the desired path even when a persistent cornering stiffness perturbation exists.
I construct the first Lyapunov function as
$$ V_1=\frac{1}{2}s_1^Ts_1. $$
Differentiating \(V_1\) gives
$$ \dot{V}_1=s_1^T\dot{s}_1=s_1^T\left[l_1(Ax_2+B-\dot{x}_{1d})+l_2\operatorname{sgn}(\tilde{x}_1)|\tilde{x}_1|^{q/p}\right]. $$
I design the virtual adaptive control law for the displacement loop as
$$ x_{2d}=\frac{1}{l_1}A^{-1}\left[-l_1B+\dot{x}_{1d}-l_2\operatorname{sgn}(\tilde{x}_1)|\tilde{x}_1|^{q/p}-l_3s_1\right], $$
where \(l_3>0\) is a positive control gain. Substituting this virtual control law into \(\dot{V}_1\) yields
$$ \dot{V}_1=-l_3s_1^Ts_1\le 0. $$
Thus, the displacement loop of the industrial robot is stable in the Lyapunov sense. The virtual control \(x_{2d}\) becomes the desired velocity command for the velocity loop.
I now design the velocity loop. The velocity tracking error of the industrial robot is
$$ \tilde{x}_2=x_2-x_{2d}. $$
I define the velocity-loop integral sliding surface as
$$ s_2=l_4\tilde{x}_2+l_5\int \operatorname{sgn}(\tilde{x}_2)|\tilde{x}_2|^{m/n}\,dt, $$
where \(l_4,l_5>0\) and \(n>m>0\). This sliding surface uses the velocity tracking error and its integral. The industrial robot velocity vector contains lateral velocity and yaw rate. Therefore, the velocity loop directly governs the yaw and lateral motion stability of the four-wheel steering industrial robot.
Because the cornering stiffness perturbation \(d\) is unknown, I introduce an adaptive estimate \(\hat{d}\). The estimation error is
$$ \tilde{d}=d-\hat{d}. $$
I design the adaptive law for the perturbation estimate as
$$ \dot{\hat{d}}=l_4s_2. $$
This adaptive law updates the perturbation estimate according to the velocity-loop sliding variable. When the industrial robot experiences a larger tracking error, the sliding variable increases, and the perturbation estimate changes more quickly. This gives the industrial robot fast adaptation to changes in cornering stiffness.
I construct the second Lyapunov function as
$$ V_2=\frac{1}{2}s_2^Ts_2+\frac{1}{2}\tilde{d}^T\tilde{d}. $$
Differentiating \(V_2\), I obtain
$$ \dot{V}_2=s_2^T\left[l_4(C+Du+d-\dot{x}_{2d})+l_5\operatorname{sgn}(\tilde{x}_2)|\tilde{x}_2|^{m/n}\right]+\tilde{d}^T(\dot{d}-l_4s_2). $$
I design the adaptive robust control law for the velocity loop as
$$ u_c=\frac{1}{l_4}D^{-1}\left[-l_4C+l_4\dot{x}_{2d}-l_4\hat{d}-l_5\operatorname{sgn}(\tilde{x}_2)|\tilde{x}_2|^{m/n}-l_6s_2+l_7\tilde{d}+l_8\dot{\bar{d}}\right], $$
where \(l_6>0\) is a positive control gain, \(l_7\) and \(l_8\) are adaptive regulation parameters, and \(\dot{\bar{d}}\) is an upper bound of the perturbation derivative, satisfying \(\|\dot{d}\|\le \dot{\bar{d}}\). The term \(l_7\tilde{d}\) introduces the estimation error feedback, and \(l_8\dot{\bar{d}}\) provides robustness against the rate of change of the perturbation.
Substituting the control law into \(\dot{V}_2\), I obtain the inequality
$$ \dot{V}_2\le -l_6\|s_2\|^2+(l_7^2+l_8^2)\|s_2\|^2+(l_7^2+1)\|\tilde{d}\|^2+l_8^2\|\dot{\bar{d}}\|^2+\frac{1}{2}\|\dot{d}\|^2. $$
When \(l_7<-1\) and \(l_8<-1\), the derivative of the Lyapunov function is non-positive. Therefore, the velocity loop of the industrial robot is stable, and the adaptive law compensates for the cornering stiffness perturbation. The industrial robot velocity state converges to the virtual command \(x_{2d}\), and the displacement state converges to the desired trajectory \(x_{1d}\).
For the complete system, I define the total Lyapunov function as
$$ V=V_1+V_2. $$
Differentiating \(V\) and substituting the displacement-loop virtual control, the adaptive law, and the velocity-loop robust control gives
$$ \dot{V}\le -l_3\|s_1\|^2-l_6\|s_2\|^2+(l_7^2+l_8^2)\|s_2\|^2+(l_7^2+1)\|\tilde{d}\|^2+l_8^2\|\dot{\bar{d}}\|^2+\frac{1}{2}\|\dot{d}\|^2. $$
With suitable positive gains and the conditions \(l_7<-1\) and \(l_8<-1\), \(\dot{V}\le 0\). This proves that the proposed adaptive trajectory tracking robust control method stabilizes the intelligent mobile industrial robot under cornering stiffness perturbation. The proof does not require exact knowledge of the perturbed cornering stiffness. It only requires that the perturbation and its derivative are bounded. This is a reasonable assumption for an industrial robot operating on paved or semi-structured ground.
Table 2. Physical parameters of the four-wheel steering industrial robot used in simulation.
| Parameter | Value | Unit |
|---|---|---|
| \(m\) | 75 | kg |
| \(I_z\) | 112.8 | kg·m² |
| \(a\) | 0.25 | m |
| \(b\) | 0.35 | m |
| \(k_{lf0}\) | 58 | N/° |
| \(k_{rf0}\) | 58 | N/° |
| \(k_{lr0}\) | 60 | N/° |
| \(k_{rr0}\) | 60 | N/° |
Table 3. Control parameters of the adaptive trajectory tracking robust controller for the industrial robot.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(p\) | 3 | \(q\) | 2 |
| \(n\) | 5 | \(m\) | 3 |
| \(l_1\) | 7 | \(l_2\) | 6 |
| \(l_3\) | 4 | \(l_4\) | 2 |
| \(l_5\) | 5 | \(l_6\) | 9 |
| \(l_7\) | -4 | \(l_8\) | -5 |
Table 4. Cornering stiffness perturbation model applied to the industrial robot.
| Perturbation component | Expression | Unit |
|---|---|---|
| \(d_1\) | \(0.1\sin(0.2t)+e^{-0.1t}\) | m/s² |
| \(d_2\) | \(0.1\cos(0.1t)+e^{-0.2t}\) | rad/s² |
I simulated the proposed method in MATLAB/Simulink. I compared the proposed adaptive trajectory tracking robust control, which I denote by ATTRC, with fuzzy PID control, model predictive control, and linear time-varying model predictive control. I use these comparisons because they represent common alternatives for industrial robot trajectory tracking and vehicle stability control. The purpose of the simulation is not only to check tracking accuracy but also to examine yaw rate error, sideslip angle error, and lateral displacement error. These three quantities reveal whether the industrial robot remains stable when cornering stiffness perturbation is present.
I first use a double lane change trajectory. The desired path is
$$ X_d=10t, $$
$$ Y_d=2\left[1+\tanh(0.15(10t-35))\right]-2\left[1+\tanh(0.15(10t-65))\right]. $$
This trajectory requires the industrial robot to move straight, change lane, and return. The straight sections allow me to observe whether the controller introduces unnecessary oscillation. The curved sections allow me to observe whether the industrial robot can reject cornering stiffness perturbation. I evaluate the maximum error and average error over the whole run.
Table 5. Double lane change simulation errors of the industrial robot under different controllers.
| Controller | Yaw rate max error (°/s) | Yaw rate mean error (°/s) | Sideslip max error (°) | Sideslip mean error (°) | Lateral displacement max error (m) | Lateral displacement mean error (m) |
|---|---|---|---|---|---|---|
| Fuzzy PID | 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 |
In the straight sections, the errors of all controllers are close to zero. This is expected because the lateral force demand is small, and the cornering stiffness perturbation does not strongly excite the yaw motion. When the industrial robot enters the curved sections, the perturbation becomes significant. The fuzzy PID controller produces the largest yaw rate error and sideslip error. The model predictive controller improves the situation, but the lateral displacement error remains large. The linear time-varying model predictive controller further reduces the yaw rate and sideslip errors, yet the lateral displacement error is still not negligible for high-precision industrial robot tasks. In contrast, the proposed ATTRC keeps the yaw rate maximum error at \(1.113\,^\circ/s\), the sideslip maximum error at \(0.015^\circ\), and the lateral displacement maximum error at \(0.063\,m\). The average lateral displacement error is only \(0.022\,m\). These results show that the adaptive law successfully estimates the lumped perturbation and that the integral sliding mode control keeps the industrial robot close to the desired path.
I then use a more complex double knot trajectory to test the adaptability of the industrial robot controller. The desired path is
$$ X_d=30\sin(0.25t), $$
$$ Y_d=30\sin(0.25t)\cos(0.25t). $$
This trajectory has continuous curvature changes and repeated left-right turning. It is more demanding than the double lane change because the industrial robot must adapt to a continuously varying lateral force demand. If the controller relies on a fixed cornering stiffness model, the tracking error will grow during the turns. The double knot trajectory therefore provides a stronger test of the adaptive robust control method.
Table 6. Double knot simulation errors of the industrial robot under different controllers.
| Controller | Yaw rate max error (°/s) | Yaw rate mean error (°/s) | Sideslip max error (°) | Sideslip mean error (°) | Lateral displacement max error (m) | Lateral displacement mean error (m) |
|---|---|---|---|---|---|---|
| Fuzzy PID | 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 |
The double knot results show the same trend but with larger errors for the comparison methods. The fuzzy PID controller has a lateral displacement maximum error of \(0.957\,m\), which is unacceptable for precise industrial robot operations. The MPC and LTV-MPC methods reduce the error, but the lateral displacement maximum errors are still \(0.636\,m\) and \(0.317\,m\), respectively. The proposed ATTRC reduces the yaw rate maximum error to \(0.651\,^\circ/s\), the sideslip maximum error to \(0.008^\circ\), and the lateral displacement maximum error to \(0.076\,m\). The average lateral displacement error is \(0.048\,m\). This confirms that the adaptive law and integral sliding mode control work together to handle continuous cornering stiffness perturbation. The industrial robot remains stable even when the path curvature changes repeatedly.
I also performed an experimental test to examine engineering practicality. I used a four-wheel steering intelligent mobile industrial robot platform with independent steering and driving. A high-precision GNSS RTK positioning module with horizontal accuracy of approximately \(0.8\,cm\) was installed on the industrial robot to record its position. The controller ran the proposed ATTRC algorithm, and the desired trajectory was the double lane change path. I repeated the test with fuzzy PID, MPC, and LTV-MPC under the same path and environment. I recorded the maximum and average trajectory tracking errors.
Table 7. Trajectory tracking test errors of the industrial robot under different controllers.
| Controller | Maximum tracking error (m) | Average tracking error (m) |
|---|---|---|
| Fuzzy PID | 0.704 | 0.226 |
| MPC | 0.589 | 0.174 |
| LTV-MPC | 0.325 | 0.106 |
| ATTRC | 0.077 | 0.038 |
The test results agree with the simulation. The fuzzy PID controller produces the largest tracking error, and the industrial robot shows visible sideslip and instability during the lane change. The MPC and LTV-MPC controllers improve the behavior, but their tracking errors remain above \(0.3\,m\) in the worst case. The proposed ATTRC keeps the maximum tracking error at \(0.077\,m\) and the average tracking error at \(0.038\,m\). This is a significant improvement for an intelligent mobile industrial robot that must follow a trajectory while performing a task. The industrial robot does not need to stop or slow down excessively to maintain stability. The adaptive robust control law compensates for cornering stiffness perturbation in real time.
I attribute the performance improvement to several factors. First, the displacement loop uses an integral sliding surface. The integral term removes the steady-state error caused by persistent perturbation. Second, the velocity loop uses an adaptive law to estimate the lumped perturbation. The estimate is updated continuously, so the industrial robot can respond to changes in cornering stiffness without waiting for a complete model identification. Third, the control law includes sign terms with fractional powers. These terms provide robustness against bounded uncertainty while avoiding some of the chattering associated with conventional sliding mode control. Fourth, the control law includes the perturbation derivative bound. This reduces the sensitivity of the industrial robot to rapid changes in tire force generation.
The proposed method is especially suitable for an intelligent mobile industrial robot because the industrial robot often operates in environments where the ground condition is not perfectly known. For example, an industrial robot moving from a dry concrete floor to a slightly wet floor may experience a sudden change in cornering stiffness. A controller that assumes constant cornering stiffness may produce a large lateral error. The adaptive law in my method detects the resulting velocity tracking error and adjusts the perturbation estimate. The integral sliding mode control then generates the steering commands needed to maintain the desired path. This makes the industrial robot more reliable in practical industrial applications.
I also note that the four-wheel steering configuration is important. In a conventional front-steering industrial robot, the rear wheels cannot directly contribute to yaw correction. In the four-wheel steering industrial robot, both front and rear steering angles appear in the control input vector \(u=[\delta_f,\delta_r]^T\). The control law can therefore coordinate front and rear steering to produce the desired yaw moment and lateral force. When cornering stiffness changes, the adaptive law estimates the lumped perturbation, and the control law distributes the correction between the front and rear wheels. This coordination improves both stability and tracking accuracy. It also reduces the risk of sideslip during high-speed or low-friction maneuvers.
From a stability perspective, the Lyapunov analysis shows that the displacement loop and velocity loop are both stable. The displacement loop virtual control ensures that \(\tilde{x}_1\) converges to zero if the velocity loop tracks \(x_{2d}\). The velocity loop control ensures that \(\tilde{x}_2\) converges to zero while the adaptive law compensates for \(d\). The complete Lyapunov function \(V=V_1+V_2\) decreases along the system trajectories under the given gain conditions. This means that the industrial robot trajectory tracking error is ultimately bounded and, in the absence of persistent external disturbance, converges to the desired trajectory. The proof is constructive and provides clear guidelines for selecting the control gains \(l_1\) through \(l_8\).
The simulation and test results also reveal the importance of tuning. The parameters \(l_1\) and \(l_4\) scale the sliding surfaces. Larger values make the sliding surface more sensitive to tracking error, which can improve convergence but may increase control effort. The parameters \(l_3\) and \(l_6\) are positive gains that determine how strongly the controller drives the sliding variables to zero. The parameters \(l_2\) and \(l_5\) control the integral action. The parameters \(l_7\) and \(l_8\) regulate the adaptive robust terms. In my tests, the chosen values provide a good balance between tracking accuracy and control smoothness. The industrial robot does not exhibit excessive steering oscillation, and the steering commands remain within practical limits.
Table 8. Summary of performance indicators for the industrial robot.
| Indicator | Fuzzy PID | MPC | LTV-MPC | ATTRC |
|---|---|---|---|---|
| Double lane change lateral max error (m) | 0.686 | 0.545 | 0.247 | 0.063 |
| Double lane change lateral mean error (m) | 0.175 | 0.145 | 0.073 | 0.022 |
| Double knot lateral max error (m) | 0.957 | 0.636 | 0.317 | 0.076 |
| Double knot lateral mean error (m) | 0.609 | 0.405 | 0.202 | 0.048 |
| Test maximum tracking error (m) | 0.704 | 0.589 | 0.325 | 0.077 |
| Test average tracking error (m) | 0.226 | 0.174 | 0.106 | 0.038 |
In my view, the experimental test is particularly important because it includes real sensor noise, actuator delay, and unmodeled dynamics. The industrial robot still achieves a maximum tracking error of \(0.077\,m\) and an average tracking error of \(0.038\,m\). These values are small enough for many industrial robot tasks, such as material transport, inspection, and mobile manipulation. The industrial robot remains stable and does not exhibit sideslip. The adaptive trajectory tracking robust control method therefore has strong engineering practicality.
I further discuss the meaning of the cornering stiffness perturbation model. The perturbation \(d\) is not simply a constant bias. It includes time-varying terms and decaying exponential terms. This reflects the fact that tire cornering stiffness can change gradually as the industrial robot moves from one surface to another. It can also change quickly when the industrial robot encounters a patch of low adhesion. The adaptive law \(\dot{\hat{d}}=l_4s_2\) does not assume that \(d\) is constant. It tracks the variation of \(d\) through the sliding variable. The robust term \(l_8\dot{\bar{d}}\) accounts for the rate of change. This is why the proposed method performs well on both the double lane change and the double knot trajectories.
The proposed method also avoids a common problem in adaptive control: parameter drift. Because the perturbation estimate is driven by the sliding variable, it remains bounded when the tracking error is small. The Lyapunov analysis shows that the estimation error \(\tilde{d}\) appears in the derivative of \(V_2\). With the chosen gains, the overall system is stable. The industrial robot does not experience unbounded growth of the adaptive parameter. This is important for long-duration industrial robot operation.
I also compare the control effort qualitatively. The fuzzy PID controller requires manual tuning and does not explicitly account for cornering stiffness perturbation. The MPC and LTV-MPC controllers use a model to predict future states. They can handle constraints, but their performance depends on the accuracy of the prediction model. When cornering stiffness changes, the prediction model becomes inaccurate, and the tracking error grows. The proposed ATTRC uses an adaptive law to compensate for the model error. It does not require a full online identification of all four cornering stiffness values. Instead, it estimates the lumped effect. This reduces computational load and makes the method suitable for real-time implementation on an industrial robot controller.
For an intelligent mobile industrial robot, real-time implementation is essential. The control law \(u_c\) involves matrix inversion of \(D\). Since \(D\) is a \(2\times2\) matrix, the computation is inexpensive. The adaptive law is a simple first-order update. The sliding surfaces are computed from the tracking errors. Therefore, the proposed method can run at a high sampling rate on an embedded industrial robot controller. I implemented the method on the test industrial robot and observed stable real-time behavior. The industrial robot responded smoothly to the trajectory commands.
I now summarize the main equations in a compact form for the industrial robot trajectory tracking controller. The displacement error is
$$ \tilde{x}_1=x_1-x_{1d}, $$
the velocity error is
$$ \tilde{x}_2=x_2-x_{2d}, $$
the displacement sliding surface is
$$ s_1=l_1\tilde{x}_1+l_2\int \operatorname{sgn}(\tilde{x}_1)|\tilde{x}_1|^{q/p}\,dt, $$
the velocity sliding surface is
$$ s_2=l_4\tilde{x}_2+l_5\int \operatorname{sgn}(\tilde{x}_2)|\tilde{x}_2|^{m/n}\,dt, $$
the virtual control is
$$ x_{2d}=\frac{1}{l_1}A^{-1}\left[-l_1B+\dot{x}_{1d}-l_2\operatorname{sgn}(\tilde{x}_1)|\tilde{x}_1|^{q/p}-l_3s_1\right], $$
the adaptive law is
$$ \dot{\hat{d}}=l_4s_2, $$
and the final steering control is
$$ u_c=\frac{1}{l_4}D^{-1}\left[-l_4C+l_4\dot{x}_{2d}-l_4\hat{d}-l_5\operatorname{sgn}(\tilde{x}_2)|\tilde{x}_2|^{m/n}-l_6s_2+l_7\tilde{d}+l_8\dot{\bar{d}}\right]. $$
These equations define the proposed adaptive trajectory tracking robust control method for the four-wheel steering intelligent mobile industrial robot. The method is model-based but robust to cornering stiffness perturbation. It uses the nominal model in \(A\), \(B\), \(C\), and \(D\), and it uses the adaptive term \(\hat{d}\) to compensate for the uncertain part. The integral sliding surfaces ensure that the industrial robot does not drift away from the desired trajectory. The Lyapunov proof ensures stability. The simulations and tests demonstrate high tracking accuracy.
Table 9. Qualitative comparison of controllers for the industrial robot.
| Controller | Requires accurate cornering stiffness | Adapts online | Handles four-wheel steering coordination | Tracking accuracy | Real-time cost |
|---|---|---|---|---|---|
| Fuzzy PID | No explicit model | Limited | Indirect | Low | Low |
| MPC | Yes | No | Yes | Medium | High |
| LTV-MPC | Yes | Partial | Yes | Medium-high | High |
| ATTRC | No | Yes | Yes | High | Low-medium |
I conclude that the cornering stiffness perturbation is a central challenge for intelligent mobile industrial robots with four-wheel steering. It affects lateral force, yaw moment, sideslip, and trajectory tracking. A controller that ignores this perturbation may perform acceptably in straight-line motion but will show large errors in curved motion. The proposed ATTRC addresses this challenge by combining integral sliding mode control with adaptive perturbation estimation. The displacement loop and velocity loop are designed separately, which simplifies the control structure and improves transparency. The Lyapunov analysis proves stability, and the simulation and test results show that the industrial robot tracks the desired trajectory with small maximum and average errors.
The double lane change simulation shows that the proposed method achieves lateral displacement maximum and average errors of \(0.063\,m\) and \(0.022\,m\). The double knot simulation shows maximum and average lateral displacement errors of \(0.076\,m\) and \(0.048\,m\). The experimental test shows maximum and average trajectory tracking errors of \(0.077\,m\) and \(0.038\,m\). These values are significantly smaller than those of fuzzy PID, MPC, and LTV-MPC. The industrial robot remains stable and avoids sideslip. The control commands are smooth enough for practical operation. Therefore, the proposed adaptive trajectory tracking robust control method is effective for intelligent mobile industrial robots under cornering stiffness perturbation.
Future work can extend the method to include actuator saturation, communication delay, and multi-industrial-robot coordination. The same adaptive integral sliding mode framework can be applied to other mobile industrial robot configurations. It can also be combined with online tire parameter estimation to further improve performance. However, even without full tire parameter identification, the proposed method provides strong robustness and high tracking accuracy. I believe this method can support the development of intelligent mobile industrial robots for manufacturing, logistics, inspection, and service tasks.
Table 10. Final performance summary for the proposed industrial robot controller.
| Scenario | Yaw rate max error (°/s) | Sideslip max error (°) | Lateral displacement max error (m) | Lateral displacement mean error (m) |
|---|---|---|---|---|
| Double lane change simulation | 1.113 | 0.015 | 0.063 | 0.022 |
| Double knot simulation | 0.651 | 0.008 | 0.076 | 0.048 |
| Experimental double lane change | Not measured directly | Not measured directly | 0.077 | 0.038 |
The proposed method is not limited to the specific numerical parameters used in this study. The control gains can be tuned for different industrial robot masses, wheelbases, and operating speeds. The adaptive law can be adjusted to trade off convergence speed and noise sensitivity. The integral sliding surfaces can be modified with different fractional powers to shape the transient response. This flexibility makes the method suitable for a wide range of intelligent mobile industrial robot applications.
In summary, I have presented an adaptive trajectory tracking robust control method for a four-wheel steering intelligent mobile industrial robot subject to tire cornering stiffness perturbation. The method uses a perturbed trajectory tracking model, a displacement-loop integral sliding mode virtual adaptive control law, an adaptive law for the lumped perturbation, and a velocity-loop integral sliding mode adaptive robust control law. Lyapunov theory proves stability. Simulations and experiments show improved yaw rate, sideslip, and lateral displacement performance compared with fuzzy PID, MPC, and LTV-MPC. The industrial robot achieves maximum and average trajectory tracking errors of \(0.077\,m\) and \(0.038\,m\) in the test. The method is robust, accurate, and practical for intelligent mobile industrial robots.