1. Introduction
Currently, the driverless technology of unmanned vehicles (UVs) has attracted more and more attention from industrial and academic circles because the efficient control schemes of UVs have great potential in guaranteeing vehicle safety performance and traffic efficiency [
1,
2]. Trajectory tracking control [
3,
4,
5] is a key link connecting environment perception and motion control for the implementation of UVs, and efficient trajectory tracking can improve the safety performances and ride comforts of UVs.
In the current research on the trajectory tracking problems, many intelligent optimization algorithms have been widely proposed for the trajectory tracking of UVs, such as the ant colony algorithm [
6], particle swarm optimization algorithm [
7], genetic algorithm [
8], graphic algorithm [
9,
10], etc. Although these algorithms can track the prescribed path, they mainly adopt the infinite iteration method that requires a large amount of computation to obtain the optimal solutions; therefore, the real-time performance and tracking accuracy of those methods may not be guaranteed. Herein, as a popular iterative optimized control approach, model predictive control (MPC) [
11,
12] can simultaneously incorporate system state constraints and control input constraints into the controller design process, i.e., at each sample time, the future system inputs and outputs can be obtained by updating the control plant and can be optimized via an appropriate optimization algorithm in the predictive horizon, wherein the system constraints are easily put in an explicit form.
As reported in the literature [
13,
14,
15,
16,
17,
18,
19,
20], the MPC method has been widely employed in the trajectory tracking controller development for different types of UVs. For instance, in the literature [
18], a control plant integrating the kinematic motions and the dynamic behaviors of an automated guided vehicle was established, and an MPC-based trajectory tracking controller was then designed. In [
19], an MPC method was presented to control the forward steering of autonomous vehicles by continuously linearizing nonlinear vehicle models. In addition, a direct data-driven MPC method has been proposed in [
20] to relax the laborious model identification procedure. A linear-parameter-varying MPC approach was proposed in [
21] for UVs, and the control scheme was validated through simulation and experiment investigations. Similarly, in order to estimate the driver intentions and the underlying behaviors, an MPC approach integrating with recurrent neural network and memory cell was proposed for an unmanned vehicle [
22], and the simulation results verified the improvements in trajectory tracking performances for this vehicle.
In short, most of the early literature [
18,
19,
20,
22] mainly focuses on the simulation investigations on trajectory tracking problems, and the real-platform or field test study is lacking. Moreover, due to the MPC execution load and extra modeling errors, it is hard to linearize the original (nonlinear) system around the current working point and then design an effective model-based predictive controller.
Fortunately, a nonlinear model predictive control (NMPC) approach has been extensively adopted in the practical path planning and trajectory tracking problem due to its inherent profits to deal with the nonlinear input constraints aiming at speeding up the online computation. In [
23], an NMPC approach was presented to address the safety issues regarding collision avoidance and lateral stability of unmanned ground vehicles in high-speed conditions. Further, a trajectory tracking NMPC strategy was proposed in [
24] to address the explicit state and input constraints for autonomous surface craft, and the real-time implementation of the NMPC was validated through the experimental results. In [
25], a distributed control scheme was provided to achieve the accurate tracking control of the autonomous underwater vehicle motion by using NMPC techniques. It can be concluded from these two studies that the NMPC algorithm is very appropriate for solving the nonlinear optimization problem with lower computational cost. Besides, it is inevitably encountered with the regular and irregular roads in the real world. With the NMPC method, the computational efficiency and control accuracy for a nonlinear control system can be guaranteed [
26].
Therefore, inspired by the literature mentioned above, this paper proposes a practical NMPC controller design for the trajectory tracking application of a UV considering the nonlinear road trajectory. The main contributions of this work are summarized as:
- (1)
A novel NMPC controller is proposed to achieve the accurate trajectory tracking of the UV under different prescribed roads, wherein the one-step Euler method is used to establish the nonlinear prediction model. As model predictive control is an iterative process, the Euler method has the advantages of a wide range of numerical solutions, a simple form that is easy to calculate, thus the tracking errors of forward velocity and yaw angle are minimized through a nonlinear optimization method;
- (2)
MATLAB simulations are carried out to verify the control performances of the proposed NMPC controller under two different driving scenarios, and the results show that the strategy can deal with the nonlinear road trajectory well, and can improve the tracking accuracy and the driving stability;
- (3)
A simple test platform consisting of a scaled-down real racing car, sensors, microcontroller, and host computer is built up to verify the effectiveness of the designed NMPC controller in UV trajectory tracking applications.
The rest of this article is arranged as follows. The UV’s kinematic model formulation is described in
Section 2. Next, the design procedure of this expected NMPC controller is provided in
Section 3. Then, in order to verify the control performance of the presented NMPC controller, both simulation and field test verifications are orderly conducted and discussed in
Section 4 and
Section 5. Finally, the conclusions and perspectives of this paper are presented in
Section 6.
2. The Unmanned Vehicle’s Kinematics Model
Here, a kinematic model of UV with two degree-of-freedoms (2-DOF) is adopted to perform the controller synthesis [
21]. To mimic a real unmanned vehicle’s kinematic behaviors, it is assumed that this UV is driven by a servo motor installed in a rear wheel, and is steered by a servo motor installed in a front wheel. By ignoring the side-slip angle when making the front-wheel steering operation, as well as considering the longitudinal speed to be a constant value, one can construct the 2-DOF kinematic model as shown in
Figure 1, in which point A
and B
stand for the center positions of the front axle and rear axle, and the other symbols used in this UV kinematic model are listed in
Table 1.
In terms of the kinematic relationships of this UV,
can be expressed by
The kinematic constraints between the rear- and front- axles of this UV are easily obtained
By integrating Equations (1) and (2), one obtains
By further transformation, we have the geometric relationship between points A and B satisfying
For simplicity, let
ω denote the derivative of
, and by substituting Equation (4) into the derivative of Equation (5), we have
By further integrating Equation (6) with Equation (3), we obtain
The expressions for
and
can be obtained as follows:
By combining Equations (4) and (7), we have
Thus, the state-space form of the kinematic equation for this UV can be formulated as
Define
as the system output vector and
as the control input vector, then Equation (10) is further rewritten as
It should be noted that the kinematics model can represent the relationship between the vehicle’s state of motion and the control input, thus the model predictive controller can achieve the purpose of predetermined control. Now, the kinematics model of the UV is completed, and the proposed controller design will be illustrated in the following.
5. Field Test Verification
In this section, an outdoor field test platform is constructed by ourselves in order to verify the effectiveness of those two MPC controllers. To facilitate the descriptions of this test platform,
Figure 12 displays the schematic diagram of the experimental setup, and the outdoor field test photograph of this UV on the different prescribed road trajectories are provided in
Figure 13. Different from the simulation verifications in
Section 4, the field tests are conducted under two different driving scenarios.
Herein, a scaled unmanned vehicle—a BT-4 racing car, is used as the control plant, and an Arduino board (MEGA2560R3) is adopted to develop our proposed NMPC controller. Moreover, there are two DC (direct-current) motors in this BT-4 vehicle, one is used for steering, and another one is used to drive this UV at the rear wheel. Both DC motors are rooted in the Arduino (MEGA2560R3) board, using 54 digital I/O pins and 6-bit pulse width modulation (PWM) drivers. Because of the limitations of this self-established outdoor field test setup, only the measured points of at the ground coordinates are collected to draw the tracking trajectories for this BT-4 vehicle, which are further utilized to compare and validate the tracking performances of the two different MPC controllers.
5.1. Test on Irregular Road
For the practical driving scenario, the irregular trajectory road (ITR) is often encountered. Thus, the outdoor field tests are conducted on a practical ITR, and the tracking response curves of this UV are shown in
Figure 14. In which, the solid green line denotes the reference trajectory road, the blue-dotted and the red-dashed lines denote the tracking response curves generated by the TMPC and NMPC controllers, respectively. Moreover,
Figure 15 reveals the tracking errors of
and
for the two MPC controllers regarding the reference trajectory road. It should be noticed that this UV runs starting from the point
at
= 1 m/s under the ITR.
It is obvious from
Figure 14 that the TMPC and proposed NMPC controller can track the ITR as possible as closely. Specifically, the blue-dashed line (denotes the tracking trajectory obtained by the NMPC controller) is much closer to the green-solid line (the reference trajectory) in comparison with the red-dashed line (denotes the tracking trajectory obtained by the TMPC controller). Furthermore, it is observed from
Figure 15 that the tracking errors of
and
generated by the two MPC controllers could gradually converge to zero states, and the tracking error curves by the NMPC controller present a flatter appearance compared to the corresponding tracking curves by the TMPC controller. Particularly, the tracking error ranged from −10 cm to 15 cm for the NMPC controller and −15 cm to 35 cm for the TMPC controller in the X-direction. Additionally, in the Y-direction, the tracking errors of
and
are varied from 0 to 25 cm for the NMPC controller, and −20 cm to 50 cm for the TMPC controller, respectively.
Similar to the analysis of simulation verifications in
Section 4, we herein use the RMSE comparisons of the tracking errors of
and
to evaluate the control performance of the designed controllers, which is presented in
Table 5. In addition to this, the histogram comparisons of the tracking errors of
and
are also provided in
Figure 16 to make a quantitative improvement of each tracking state for the TMPC and proposed NMPC controller under the ITR and the prescribed reference trajectory. Compared to the tracking performance of the TMPC controller, the UV using the NMPC controller can reduce its RMSE values of
and
by about 54.72% and 48.65%, respectively.
5.2. Test on Double-Circle Road
Except for the irregular road trajectory, it is very necessary to validate the control performances of the proposed NMPC controller under a general double-circle trajectory road (DCTR). An outdoor field test is performed under a prescribed DCTR with a radius of 2.5 m, which starts from
with
= 1 m/s.
Figure 17 displays the tracking response curves obtained by the TMPC and NMPC controllers, and
Figure 18 plots the tracking error curves of
and
for the two MPC controllers against the reference trajectory road.
From
Figure 17, it is easily seen that the two MPC controllers can nearly track the prescribed DCTR, and the tracking curves of the NMPC controller seem closer to the reference trajectory road. By a further observation from
Figure 18, the tracking errors of
and
generated by the two MPC controllers are flat on a whole, yet the tracking errors by the NMPC controller can yield a smoother tendency and can enter into a relatively stable state at a faster rate compared to the TMPC controller. Besides, the tracking error of
is varied from −10 to 15 for the NMPC controller, and from −30 cm to 40 cm for the TMPC controller in the X-direction. Moreover, the tracking error of
is ranged from −20 cm to 10 cm for the NMPC controller and from −40 cm to 30 cm for the TMPC controller in the Y-direction.
Similarly, the RMSE comparisons of the tracking errors of
and
under this DCTR are quantitatively compared and provided in
Table 6, and the histogram comparisons for the tracking errors of
and
are provided in
Figure 16 to assess the control performance of the designed controller. It is clear from
Table 6 and
Figure 19 that the RMSE values of the tracking errors
and
for the NMPC controller are reduced by about 56.63% and 48.73%, respectively, in comparison with those of the TMPC controller, which further illustrates that the NMPC controller has better control effect under the DCTR scenario.
6. Conclusions
In this paper, a practical NMPC design method is proposed to achieve the accurate trajectory tracking application of a scaled UV. This desirable controller is developed based on a 2-DOF kinematics model of the UV, and in the framework of a standard MPC, the one-step Euler method is utilized to construct the nonlinear prediction model, then a nonlinear optimization objective function is formulated to minimize the tracking errors of forward velocity and yaw angle from a time-varying preset reference road. Finally, both the comparative simulations and the outdoor field tests are carried out to confirm the superior performances of the designed NMPC controller against the TMPC controller for this UV. The comparison of results demonstrates that the improvements of the tracking indexes , , and for the UV with the presented NMPC controller are at least 45%, 1.35%, and 18.16%, respectively. Moreover, the field outdoor test results show that the improvements of tracking indexes x and y for the UV with the proposed NMPC controller are about 54% and 48%, respectively, compared with those with the TMPC controller. On a whole, it can be concluded that the NMPC controller outperforms better control performances regarding the TMPC controller.
Future study will focus on the trajectory tracking controller design of this UV considering the kinematics and dynamics properties simultaneously, and the experimental validation of the proposed controller in a real-time outdoor field test platform.