Keywords

1 Introduction

Tracked vehicles are regarded as commonly ground unmanned mobile platforms in the domain of off-road strategies owing to their excellent traversability and maneuverability, to enhance navigation proficiency across diverse terrains. These scenarios encompass snow-covered terrains, unconsolidated sandy substrates, adhesive mud matrices, steep gradients, terrain laden with rubble, as well as intricate amalgamation thereof [1]. Nowadays, tracked vehicles are used in various industries, such as agriculture and military. Therefore, the study of automatic navigation and operation of tracked vehicles becomes imperative. As a pivotal technology in the realm of autonomous driving, trajectory tracking control also serves as a prerequisite for unmanned tracked vehicles to carry out missions effectively [2].

In previous studies, different strategies for vehicle trajectory tracking control have been proposed in the literature to achieve high-precision tracking, such as geometric and kinematic controllers, model based controllers, adaptive and intelligent controllers, and so on [3].

Initially, pure pursuit represents one of the foremost geometric controllers extensively employed by researchers [4]. It’s important to note that while geometric controllers offer these advantages, they may also have limitations, such as challenges in handling complex maneuvers or uncertainties in certain scenarios [4].

Model predictive control (MPC) serves as a technique capable of addressing system constraints and future projections. MPC aims to minimize tracking errors by utilizing a vehicle dynamics model to forecast the forthcoming vehicle nalysed within a defined prediction horizon [3]. Tang et al. [6] amalgamated model-based and data-driven control techniques, introducing a novel approach for trajectory tracking control in bidirectional independent electric-driven unmanned tracked vehicles. Model-based controllers also have limitations, including the need for accurate models, potential sensitivity to nalysed errors, and increased complexity in model development and maintenance.

Adaptive and intelligent controllers are typically employed in research or applications that demand a high degree of robustness against disturbances and variations. The fuzzy control rules of the agricultural machinery path tracking method utilizing the fuzzy adaptive pure pursuit model proposed by Li et al. [6] are established based on expert experiential knowledge, resulting in notable tracking errors that pose challenges for expeditious rectification. However, these controllers’ successful implementation requires proper design, training, validation, and consideration of potential limitations and challenges [8, 9].

In this paper, a trajectory tracking control method for unmanned tracked vehicles is proposed. The remainder of the paper is organized as follows. The kinematics analysis and nalysed of the tracked vehicle are carried out, and the expression of the controller is presented in Sect. 2. To demonstrate the performance of the proposed controller in path tracking, nalysed, and simulation experiments are carried out in Sect. 3, and the experimental results are nalysed. Finally, conclusions are given in Sect. 4.

2 Problem Description

In this paper, the trajectory tracking control of unmanned tracked vehicles is studied. For the subsequent construction of a path following controller and experimental verification, kinematic modeling is used in this paper, not only because it provides a simplified and accurate representation of vehicle motion, especially in cases where the primary concern is a path following and position accuracy rather than complex mechanical interactions, but also because it improves computational efficiency and allows for real-time implementation of control algorithms.

2.1 Modeling Premises and Assumptions

This section undertakes a kinematic analysis of tracked vehicles. To facilitate the analysis of tracked vehicle kinematics, specific assumptions are necessary regarding the research subject. Throughout the motion process, this study makes the following assumptions [10].

  1. (1)

    The contact surfaces of the tracks on both sides of the tracked vehicle are even, ensuring uniform contact with the ground, and frictional forces are uniform in magnitude across all points.

  2. (2)

    The resistance coefficient remains consistent between steering and straight-line travel.

  3. (3)

    The lengths of the tracks in contact with the ground on both sides are equal, and variations in track tension are not taken into consideration for their impact on ground pressure.

  4. (4)

    The contribution of aerodynamic resistance for the tracked vehicle can be disregarded.

2.2 Kinematic Analysis of Tracked Vehicles

It is imperative to establish a suitable coordinate system that accurately describes the motion state, pose information, and relative position details of the tracked vehicle to successfully carry out tasks such as path planning, obstacle avoidance, and tracking. This paper introduces a tracked vehicle kinematic model based on the instantaneous center of rotation, as depicted in Fig. 1.

Fig. 1
A graph of Y versus X. It presents the rotation around the origin in the Y X plane. Vector v forms angle alpha with the x vector. Omega is the rotational parameter. The gap between the two tracks is l. Point I C C is indicated.

Kinematic model of tracked vehicle

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Tracked vehicles commonly involve two types of coordinate systems: the absolute coordinate system \(F(XOY)\) and the body coordinate system \(f(xoy)\). In Fig. 1, \({\text{ICC}}\) represents the instantaneous rotation center of the tracked vehicle, \(R\) represents the instantaneous rotational motion radius, \(l\) is the center distance of both tracks, \(\omega \) represents the lateral angular velocity of the vehicle, \(\overrightarrow{v}\) is the instantaneous velocity vector, \(\theta \) is the heading angle and \(\alpha \) represents the angle between the velocity vector and the X-axis, the side-slip angle. \((X,Y,\theta )\) is the pose information of the tracked vehicle in the absolute coordinate system; \((\dot{X},\dot{Y},\dot{\theta })\) is the linear velocity and rotational angular velocity of the tracked vehicle in the \({\text{X}}\) and \({\text{Y}}\) axis directions in the absolute coordinate system.

Under conditions of low-speed motion, the slip motion and slip angle of tracked vehicles are disregarded. Based on Fig. 1, formula (1) can be derived [3].

$$\left[\begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{cc}cos\theta & 0\\ sin\theta & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}v\\ \omega \end{array}\right]$$
(1)

In tracked vehicles, the forward velocity of the vehicle's center point is equal to the average speeds of the left and right tracks:

$$v=\frac{{v}_{{\text{r}}}+{v}_{{\text{l}}}}{2}$$
(2)

When the time increment is very small and the change in the vehicle's rotation angle is minimal, the approximate formula is as follows:

$$\Delta \theta \approx sin\theta =\frac{({v}_{{\text{r}}}+{v}_{{\text{l}}})\Delta t}{l}$$
(3)

where, \({{\text{v}}}_{{\text{l}}}\) and \({{\text{v}}}_{{\text{r}}}\) represent the speed of the left and right track, respectively.

From Eq. (3), the angular velocity of the tracked vehicle's motion around the center of rotation \(\upomega \) can be obtained, which is also the rate of change in the heading angle.

$$\omega =\frac{\Delta \theta }{\Delta t}=\frac{{v}_{{\text{r}}}+{v}_{{\text{l}}}}{l}$$
(4)

The curved motion radius of the tracked vehicle can be derived from Eqs. (2) and (4).

$$R=\frac{v}{\omega }=\frac{l({v}_{{\text{r}}}+{v}_{{\text{l}}})}{{2(v}_{{\text{r}}}-{v}_{{\text{l}}})}$$
(5)

The relationship between the left and right wheel speeds of the tracked vehicle, the center point linear velocity, and the angular velocity can be obtained from Eqs. (2), (4), and (5) as follows.

$$\left[\begin{array}{c}v\\ \omega \end{array}\right]=\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ -\frac{1}{l}& \frac{1}{l}\end{array}\right]\left[\begin{array}{c}{v}_{{\text{l}}}\\ {v}_{{\text{r}}}\end{array}\right]$$
(6)

2.3 Pose Error Model for Tracked Vehicles

Tracking error is one of the most commonly used quantitative measures for assessing the effectiveness of a controller. It is typically characterized as the difference between the desired path and the actual vehicle position [3, 11]. The parameters for tracking error, as depicted in Fig. 2, are illustrated.

Fig. 2
A graph of Y versus X. It presents a curved line for a path. A tangent that forms angle theta ref with the x axis is drawn at (X ref, Y ref). It forms theta ref angle. The component with 2 tracks is tilted.

Position error diagram of the tracked vehicle

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Equation (7) is one of the commonly used formulas for determining tracking error, encompassing longitudinal, lateral, and heading errors.

$$\left[\begin{array}{c}{e}_{{\text{x}}}\\ {e}_{{\text{y}}}\\ {e}_{\uptheta }\end{array}\right]=\left[\begin{array}{ccc}cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{X}_{{\text{ref}}}-X\\ {Y}_{{\text{ref}}}-Y\\ {\theta }_{{\text{ref}}}-\theta \end{array}\right]$$
(7)

The differential equations for the pose error can be obtained from Eqs. (6) and (7).

$$\left\{\begin{array}{c}\dot{{e}_{{\text{x}}}}={e}_{{\text{y}}}-v+{v}_{{\text{ref}}}cos{e}_{\uptheta }\\ \dot{{e}_{{\text{y}}}}={e}_{{\text{x}}}\omega +{v}_{{\text{ref}}}sin{e}_{\uptheta }\\ \dot{{e}_{\uptheta }}=\frac{{v}_{{\text{ref}}}}{R}-\omega \end{array}\right.$$
(8)

2.4 Controller Design

In this section, the proposed LQR controller is introduced in detail. By employing Eq. (9) as the differential equation for pose error, with lateral error and heading error as state variables, and rotational angular velocity as input, a linearized equation can be derived:

$$\left\{\begin{array}{c}\left[\begin{array}{c}\dot{{e}_{{\text{y}}}}\\ \dot{{e}_{\uptheta }}\end{array}\right]=\left[\begin{array}{cc}0& {v}_{{\text{ref}}}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{e}_{{\text{y}}}\\ {e}_{\uptheta }\end{array}\right]+\left[\begin{array}{c}-L\\ -1\end{array}\right]\omega +\left[\begin{array}{c}0\\ {v}_{{\text{ref}}}\end{array}\right]\frac{1}{R}\\ y(t)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{e}_{{\text{y}}}\\ {e}_{\uptheta }\end{array}\right]\end{array}\right.$$
(9)

where, \(x(t)=\left[\begin{array}{c}{e}_{{\text{y}}}\\ {e}_{\uptheta }\end{array}\right]\), \(\dot{x(t)}=\left[\begin{array}{c}\dot{{e}_{{\text{y}}}}\\ \dot{{e}_{\uptheta }}\end{array}\right]\), \(u=\omega \), \(A=\left[\begin{array}{cc}0& {v}_{{\text{ref}}}\\ 0& 0\end{array}\right]\), \(B=\left[\begin{array}{c}-l\\ -1\end{array}\right]\)

3 Illustrative Examples

In this section, the proposed LQR controller is applied to a tracked vehicle simulation model to demonstrate its effectiveness and practicality as shown in Fig. 3. The simulation environment is MATLAB R2022b. The simulation results are shown in Figs. 4–7.

Fig. 3
A MATLAB simulation environment. circuit diagram of a simulation model. It takes inputs of theta, Y r and X r and has blocks labeled l q r e r r, p i d e r r, l q r, p i d, car system and predict.

Track vehicle driving simulation

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Fig. 4
A multiline graph of Y versus X presents 3 overlapped curves for reference, L Q R, P I D. The curves are horizontal at 0 till 20 meters and then ascend in an S-shaped manner. A zoomed in view of the transition is also presented. In the curved portion, the reference and L Q R curves overlap.

Contrast of steering trajectories

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Figure 4 presents a comparison of steering trajectories, illustrating the actual path followed by the tracked vehicle when different controllers are employed under the reference path. It can be observed that when the traditional PID controller is used, there is a certain amount of overshoot after the straight-line travel phase ends. Moreover, during sharp turns, the traditional PID controller struggles to accurately track the curvature, resulting in noticeable errors. In contrast, the LQR controller demonstrates improved steering performance, particularly when entering turns. Figure 5 presents the overall trajectory map used in the simulation experiments, illustrating the performance of different control algorithms throughout the complex route.

Fig. 5
A multiline graph of Y versus X presents 3 overlapping loops for reference, L Q R, P I D. The curves move horizontally at 0 meters till 50 meters, then increase, then again become parallel to the horizontal axis in the reverse direction, and join the origin point, forming a loop. Reference and L Q R overlap in a curved area.

Trajectory tracking simulation

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In Fig. 6, the actual vehicle motion under control signals is depicted, where “krz” represents the discrete input for the speed ratio. Compared to the output signal under PID control, the LQR control signal achieves superior control performance with a smaller overshoot and shorter settling time. Notably, the settling time is reduced by 5 s, and the overshoot is diminished to one-third. Furthermore, from the tracked vehicle’s Y-coordinate data, it is evident that LQR initially exhibits a slight deviation but subsequently achieves stability more rapidly through angle adjustments.

Fig. 6
Two line graphs. A. It plots stepped trends for I q r k r z and p i d k r z. The trend for I q r k r z becomes parallel to the horizontal axis after 10 units on the x axis. B. It plots 2 ascending trends for l q r X and p i d X and 2 horizontal trends at 0 for l q r Y and p i d Y.

Comparison of motion states

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Fig. 7
Two multiline graphs. A. It plots waveforms of descending amplitude for l q r and p i d. p i d trend has greater fluctuations. B. It plots 4 waveforms labeled I q r e r r e r r y, I q r e r r e r r theta, p i d e r r e r r y and p i d e r r e r r theta that saturate at 0.

Tracking performance comparison

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Figure 7 displays a comparison of tracking performance, demonstrating the control signals generated by the controllers under different error inputs. While the PID control system exhibits faster error reaching its peak value, it experiences a longer overall settling time and shorter oscillation periods. In contrast, the LQR control allows the error to reach zero within a shorter time frame of 15 s. Additionally, regarding the control signal for angular velocity output, LQR results in a smaller overshoot. This enables the tracked vehicle system to achieve the desired pose more rapidly when the tracking path undergoes sudden changes.

4 Conclusion

This paper establishes a tracked vehicle kinematic model based on the instantaneous center of rotation. A path tracking controller using a Linear Quadratic Regulator is designed, which simultaneously takes into account vehicle tracking accuracy and reduces the demand for model precision.

Through an analysis of the results obtained from MATLAB simulation experiments, it is observed that the approach presented in this paper achieves higher path tracking accuracy compared to traditional control methods. Under the same driving trajectory, notably improved steering performance is attained during corner entry. Overshoot is significantly reduced, the settling time is shortened, and steady-state error is eliminated.