**1. Introduction**

In articulated tracked vehicles, two double-tracked units are joined by an articulated mechanism. Unlike skid-steer vehicles, ATVs have articulated steering mechanisms driven by hydraulic actuators, which allow them to produce an articulation angle between the front and rear units of the ATVs and steer the front unit to the desired location [1]. ATVs have the advantage of a balanced driving force between both tracks, which results in minimal driving torque requirements in the steering maneuver compared to single and coupled tracked vehicles [2].

Research has been focused on off-road vehicles to enhance driving efficiency and ensure safety by using automated driving systems [3,4]. Several approaches have been applied to obtain a feasible kinematic trajectory for the off-ground vehicle [5]. As the articulated steering mechanism gives ATVs unique steering characteristics, the rear unit of ATVs contributes to the overall nonholonomic constraints. It is, therefore, impossible for traditional planning methods such as the RRT [6] or artificial potential field [7] to produce a feasible and smooth path for the ATVs. The Hybrid A-star algorithm could produce a smooth, kinematics feasible path for nonholonomic systems [8,9]. The Hybrid A-star is implemented in two stages, including the node search, to produce a kinematics-feasible trajectory. The second stage then locally improves the quality of the path using analytical expansion of the path.

For trajectory tracking, two types of modeling have been commonly used: kinematicsbased modeling and dynamics-based modeling. Because those ATVs operate at low speeds,

**Citation:** Hu, K.; Cheng, K. Trajectory Planning for an Articulated Tracked Vehicle and Tracking the Trajectory via an Adaptive Model Predictive Control. *Electronics* **2023**, *12*, 1988. https://doi.org/10.3390/ electronics12091988

Academic Editor: Shiho Kim

Received: 24 March 2023 Revised: 18 April 2023 Accepted: 19 April 2023 Published: 24 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

dynamic factors such as road-wheel load distribution and centrifugal force can be overlooked. The articulated steering vehicle (ASV) path-tracking deviation model has been extended from ordinary mobile robots in terms of speed deviation, lateral position deviation, and heading angle deviation. To simplify the automatic guidance of ASV, Nayl defined an improved path tracking model, considering the lateral displacement deviation, heading angle deviation, and curvature deviation [10]. A tracking error model, which includes both the position error and orientation error of both the front and rear units, has been applied in [11] to facilitate the control design for the rear unit of the ASV.

Based on the above path tracking error model, Ridley developed a full state feedback adaptive tracking method [12]. There are also numerous applications of complex controllers based on robust control [13], fuzzy control [14], and sliding mode control [15,16]. In addition, researchers proposed a linear switching control strategy that took advantage of the linearization of the ASV to overcome its nonlinear characteristics [17]. A simpler and more robust trajectory tracking controller, including the MPC controller, was suggested for the tracked vehicle [18]. A further advantage of the model predictive control (MPC) algorithm is that it takes constraints into account. Thus, the real-time status of the vehicle can be taken into account directly in the MPC controller. A switch controller consisting of multiple MPC controllers was proposed to account for the side angles of the ASV [19]. On the articulated vehicle platform, Kayacan implemented linear MPC, nonlinear MPC, and robust tube-based MPC algorithms for path tracking [20,21]. An articulated wheel loader achieved good tracking accuracy despite varying road curvature using the adaptive MPC method [22].

The steering characteristics of ATVs have been extensively studied in the recent research on ATVs in [23,24]. Furthermore, the design parameters regarding the steering characteristics of the ATVs were examined in [25,26]. A fuzzy-PID control system has been proposed to obtain the articulation angle of ATVs to track a predefined path [27]. After that, a closed-loop control of the steering torque of the ATV hydraulic-driven system was introduced to obtain the desired articulation angle [28]. Despite this, researchers have not studied motion control for ATVs during complex maneuvers in obstacle-filled environments. Using the trajectory provided by the path planner and optimization modules, this paper proposed a trajectorytracking control framework based on an adaptive MPC control framework for tracking the trajectory of ATVs. The following works have been completed:


This paper structure is presented as follows. In Section 2, we establish the kinematic model as well as the trajectory-tracking-error model for the ATVs. Section 3 presents the trajectory planner for ATVs based on the Hybrid A-star method and the trajectory optimized method based on the minimum snap method. Section 4 describes the two-layer trajectory-tracking controller consisting of a forward control method and an adaptive model predictive method. We discuss the simulation results of the proposed trajectory planner and the control framework in Section 5. The conclusion of this work is presented in Section 6.

#### **2. Autonomous Articulated Vehicle System**

The geometry of the articulated tracked vehicle is shown in Figure 1. The ATVs comprise two vehicles connected by articulating mechanisms and hydraulic steering actuators. Changing the articulation angle allows the ATVs to perform steering maneuvers. Additionally, the front and rear vehicle's tracks are controlled to maintain the longitudinal speed. The motion control of ATVs is intended to guide the front and rear units of the ATVs to the reference trajectory determined by the trajectory planner. The reference trajectory is defined as [*xr*(*t*), *yr*(*t*), *θr*, *ψr*], where [*xr*(*t*), *yr*(*t*), *θr*] are the center of the front vehicle's gravity, and *ψ<sup>r</sup>* denotes the orientation angle of the rear unit, respectively. In this work, the reference trajectory, namely, *<sup>q</sup><sup>r</sup>* = [*xr*(*t*), *yr*(*t*), *<sup>θ</sup>r*(*t*), *<sup>ψ</sup>r*(*t*)]*T*, and the derivatives of the reference trajectory are all continuous and bound. The longitudinal velocities of the front and rear vehicles are denoted by *υ*. ˙ *θ* and *ψ*˙ are the yaw rates of the two ATV units, respectively. Then, the state vector *q* = [*x*, *y*, *θ*, *ψ*] *<sup>T</sup>* denotes the ATVs' position and orientation.

**Figure 1.** The modeling of an articulated tracked vehicle system in this work is divided into theoretical mathematical modeling and virtual multi-body dynamic modeling based on the real vehicle system. The modeling depicts the steering of the ATVs driven by the hydraulic cylinders, which results in the change in articulation angle *γ*, the articulation angular rate *γ*˙ , and the yaw-rate response of the front unit and rear unit ˙ *θ* and *ψ*˙.

In this paper, we consider the path planning and tracking of ATVs in a structured environment, including the boundaries and obstacles described by the rectangle. The problem of achieving a viable path and accurate control can be divided into two stages, trajectory planning, and trajectory-tracking control. Several goals must be fulfilled in the trajectory planning process, such as continuous driving velocity profiles and establishing a feasible path to avoid obstacles. This planner provides a discrete and smooth reference path for the trajectory-tracking controller. Finally, the controller produces accurate velocity and steering angle for the ATVs to travel to the destination safely. Figure 2 shows the scheme of the whole work, including the path planning and the path tracking control. In the control system, the trajectory planning module provides the reference positions *Xr*,*Yr*, *θr*, *ψ<sup>r</sup>* and the kinematic reference states *υr*, ˙ *θ<sup>r</sup>* to the trajectory tracking controller as the external disturbance.

**Figure 2.** Overall scheme of the path planning and tracking modules for the articulated tracked vehicle system.

#### *2.1. Kinematic Vehicle Models*

In this section, the developed kinematic model is used to capture the main feature of the kinematics of the ATVs [15], which can be expressed as follows:

$$\begin{aligned} \dot{\chi} &= \bar{v}\cos\theta \\ \dot{y} &= \bar{v}\sin\theta \\ \dot{\theta} &= \frac{\bar{v}\sin\gamma + L\_f\dot{\gamma}}{L\_f\cos\gamma + L\_r} \\ \dot{\psi} &= \frac{\bar{v}\sin\gamma - L\_f\dot{\gamma}}{L\_f\cos\gamma + L\_r} \end{aligned} \tag{1}$$

where the variables *x* and *y* denote the coordinate of the geometry center of the front unit of the ATVs; *θ* and *ψ* denote the orientation angle of two parts of ATVs, respectively. *γ* and *γ*˙ denote the articulate angle and the articulate angle rate. The difference between the orientation angle of two units is defined as the articulation angle *γ*. To maintain the safety of the ATVs in the steering process, the steering control action, namely articulation angle and articulation angle rate, should be less than the maximum value.
