Comparison of Two-Wheel and Four-Wheel Steering Using Event-Triggered Predictive Motion Control and Scale Vehicles

With the rising popularity of electric vehicles, autonomous driving technology is also advancing rapidly [1,2]. In the field of autonomous vehicles, steering control plays a key role in ensuring precise navigation and safety [3–5]. While traditional two-wheel steering systems have been standard for decades, the advent of advanced technologies has brought four-wheel steering systems to the forefront as an alternative. Four-wheel steering aims to enhance handling, stability, and maneuverability compared to conventional two-wheel steering [6]. In four-wheel steering systems, both the front and rear wheels can be steered simultaneously. This provides an additional degree-of-freedom and control authority over the vehicle’s lateral dynamics, which enhances the vehicle’s maneuverability at slower speeds and increases stability at higher speeds [7].

A variety of control algorithms have been proposed for four-wheel steering vehicles, including linear quadratic control [8,9], decoupling control [10], and $H∞$ control [11]. Moreover, model predictive control (MPC) has been widely investigated in the context of autonomous driving, with the main advantage being its capability to solve complex optimization problems and constraints [12–15]. MPC can handle the highly nonlinear and complex dynamical processes during driving maneuvers. However, the optimization problem solved at each control step requires significant computation. Therefore, techniques to reduce the computation time are essential for real-time feasibility. To this regard, event-triggered MPC has been proposed to decrease the computation requirements of MPC [16,17]. Rather than performing optimization and updating the control at every step, the optimization is only solved when needed based on the system states satisfying a triggering condition. The stability and feasibility of the event-triggered MPC are extensively addressed in detail in papers [18,19]. These studies provide conditions under which the event-triggered MPC is feasible and stable. For example, the stability can be established by utilizing the Lyapunov stability theory with the cost function as Lyapunov functions. The efficacy of event-triggered MPC has been demonstrated in path-tracking problems using simulation [5] and vehicle experiments [20], demonstrating its benefits of reducing computation without major performance degradation.

However, these prior works have been focused on two-wheel steering systems only. In the authors’ previous work [21], MPC was implemented in a four-wheel steering system to test its advantage over two-wheel steering systems. However, Ref. [21] only considered tight steering maneuver, where the benefit of using four-wheel steering is rather obvious. This article builds upon the previous work and focuses on driving conditions that do not require excessive steering, examining whether four-wheel steering can still offer benefits. To enable experimental validation, a customized scaled vehicle platform was developed based on a 1/10th scale Tamiya TT-02 RC model car [22]. The major addition was an actuated rear-wheel steering mechanism to transform the vehicle from its original two-wheel steering system into one with a four-wheel steering system. For autonomous operation, the platform was outfitted with an NVIDIA Jetson Nano single board computer to run control algorithms and a Marvelmind ultrasonic-based indoor positioning system for real-time location measurements. Experimental results demonstrate that even in normal driving conditions, four-wheel steering systems can still offer control performance improvement and computation reduction in the event-triggered control framework. Moreover, efficiency improvement in four-wheel steering vehicles is also an interesting research topic. See, for example, Refs. [23,24], where significant energy improvement is reported by using optimal torque distribution methods, without compromising stability. In the future, this could be a potential area for testing on this platform.

The remainder of this article is organized as follows. Section 2 discusses the vehicle model for the MPC and the algorithm of event-triggered MPC. Section 3 introduces the setting up of the experimental platform. The numerical results are presented in Sects. 4, and Sec. 5 concludes this article.

In this article, we conducted tests on a scaled vehicle at low speeds. Given these conditions, the lateral tire forces have a minimal impact and can be reasonably omitted from our analysis. Please note that, for low speeds, kinematic models can be more effective for MPC, while dynamic models are generally better for high-speed maneuvers. This is due to the fact that at low speed, lateral tire forces are usually hard to accurately model, which can result in unrealistic prediction for MPC. It is also our observation through experiments that MPC with dynamic model would require a computational time that is too long for effective steering control, resulting in worse tracking performance. Therefore, in this article, we utilize a kinematic model for MPC.

In the case of path-tracking control, the primary objective is to minimize the offset between the vehicle’s actual and desired position. Therefore, in the cost function (2a), the first term penalizes deviation from the desired path. Meanwhile, to ensure stability, the second and third terms of cost function minimize the steering angles and the steering angle rate, respectively. The matrices $Qx$⁠, $Qu$⁠, and $Qd$ are the weights of path following error, steering efforts, and control activity respectively. In Eq. (2b), $x^t$ denotes the current state feedback, meaning at time $t$⁠, the initial states of the optimization problem are equal to the state feedback at time $t$⁠. Equations (2c), (2d), and (2e) are the constraints of the optimal control problem, where Eq. (2c) requires that at each step the state vector follows the kinematic model (1), Eq. (2d) limits the range of steering angle, and Eq. (2e) limits the steering angle rate. In Eq. (2d) when $k$ is equal to 0, $ut−1$ denotes the control action applied at the previous time-step. To address the optimal control problem mentioned earlier, the mpctools package along with casadi are used, both of which are open-source optimization tools as described in Refs. [25,26]. The mpctools package is divided into three key components: an estimator, a target calculator, and a regulator. Each of these elements provides access to the solvers within casadi.

Please note that, since only input constraints are consider in OCP (2), the feasibility of event-triggered MPC is automatically guaranteed if the original OCP is feasible. The stability of the event-triggered MPC has been established in the literature [18,19], by utilizing the Lyapunov stability theory with the cost function as Lyapunov functions. Please also note that though other control methods, such as proportional-integral-derivative (PID), linear quadratic regulator (LQR), or time-triggered MPC, can also be applied to demonstrate the difference between two-wheel and four-wheel steering control, event-triggered MPC is selected for investigation due to its capability to handle constraints compared to PID and LQR and its relatively low computational requirement compared to time-triggered MPC.

In this article, the scale vehicle platform based on the open-source jetracer [22] is used to validate four-wheel steering systems with event-triggered MPC, which is built from a 1/10th scale Tamiya TT-02 RC car (Fig. 2). Compared with the original Tamiya TT-02 RC car, the rear wheel steering system needs to be added. In particular, the added rear-wheel steering system is based on the front-wheel steering design with slight modification to fit the different architecture around the rear axles. Mounting points on the rear bumper originally used for cosmetic parts are repurposed to fasten replacement copies of the front-wheel bell cranks using a 3D-printed bracket with mounting pins. Because the bell cranks are positioned farther apart, the bell crank linkage design also has to be lengthened to match the separation of the mounting points and 3D printed. Replacement copies of both the steering links and drag links have to be shortened and reconnected with a 3D-printed bracket. The final lengths of the steering links must result in the wheels pointing straight forward when the steering linkage assembly is centered. After drilling holes in the rear bumper and fastening a second steering servo in place, the custom rear knuckles are attached to the suspension arms, and the aforementioned steering linkage components are assembled as shown in Fig. 3. The vehicle is driven by an electric motor powering all four wheels. Motor speed is controlled by a Tamiya electronic speed controller (ESC), and a servo multiplexer (switched from the RC transmitter) is used to select if the RC transmitter/receiver or the Jetson Nano/servo driver module supplies the control signals for the ESC and steering servos. More details about the setup of the platform can be found in Ref. [21].

To calculate vehicle state information for the MPC-based motion controls, a marvelmind indoor positioning system² is used, which consists of five beacons and a modem. One of the beacons is mounted on the scale vehicle, and the other four beacons are placed around the perimeter of the testing area, which are set as stationary beacons. Figure 4 shows the beacon positions during a typical test in the marvelmind software.

The system delay causes the actual position of the vehicle to be slightly ahead when control actions are applied. To address this issue, delay compensation is incorporated into the motion controls. The main causes of the delay are the time taken by the processor to update the control sequence and the latency of the marvelmind position data caused by filtering methods designed to reduce noise. Several measures can be taken to significantly reduce, but not entirely eliminate, latency in the marvelmind system. For example, including IMU data from the mobile beacon in the system results in a notable increase in latency. In this article, delay compensation is achieved by using the vehicle model to estimate the future vehicle state based on the measured vehicle state, vehicle velocity, and the current control set point. The MPC system uses this estimated vehicle state as the initial state $x0$⁠, expecting it to reflect the real vehicle state when the newly calculated control actions are applied more accurately.

The vehicle speed is controlled by a PID-based velocity control system, with all three (proportional, integral, and derivative) terms. However. as is typical with PID controller, the derivative term results in an amplification of noise from the error signal. To minimize this issue, a discrete low-pass filter is implemented using the form of the single-pole infinite impulse response filter in Ref. [28] for both the signal and for the calculation of the derivative term. In particular, let $V=bv+aV−1$⁠, where $V−1$ is the previous filtered velocity, $v$ is the raw velocity signal before filtering, and $a$ and $b$ are the filter coefficients. For this application, in addition to the standard PID terms, an additional control term, $S=Ksδf2$⁠, is added to increase the throttle value as steering angles increase to account for sources of resistance such as reduced efficiency of u-joints.

The scale vehicle testing included in this article uses an oval track (two half circles connected by straight sections) defined as an array of coordinates. To simplify the process of modifying the track, a python code was written to generate the track based off of several inputs. Because the marvelmind data are generated using one of the stationary beacons as the origin, it is useful to specify shifts in the $x$ and $y$ directions, as well as a rotation angle for the track (⁠$θ$⁠). In addition, the code allows the user to set the radius of the turns (⁠$R$⁠), the length of the straight sections (⁠$L$⁠), and the number of points included in each half circle section (⁠$N$⁠).

First, coordinates for the upper half circle are generated. For each of the points from $n=0$ to $n=N−1$⁠, the coordinates are calculated as $x=R×cos(nπ/(N−1))$ and $y=R×sin(nπ/(N−1))$⁠. The lower half circle coordinates are the same, but with negated $x$ and $y$ values. Then, the distance between points (⁠$d$⁠) is calculated as $d=(πr)/(N−1)$⁠. To ensure that the value of $d$ remains constant throughout the track, the requested length of the straight sections $l$ is modified to be the first multiple of $d$ that exceeds the requested length. The top/bottom half circles are shifted up/down by $L/2$⁠. The left and right straight section coordinates are generated to fill the space between the half circle sections while ensuring that the order of points results in a continuous track when the four sections are concatenated into a single array. After concatenation, the track can be rotated by multiplying each coordinate pair by a rotation matrix $R=[cosθ−sinθsinθcosθ]$⁠. Finally, all coordinates in the rotated track are shifted by the specified values to create the final array of track points used for motion controls testing.

The calibration process presented in this article focuses on the weights $Qx$⁠, $Qu$⁠, and $Qd$ for the cost function. In addition, only the relative magnitude of the weight affects the solution of Eq. (2a). Since $Qx$ is held as constant, this study only focuses on adjusting the values of $Qu$ and $Qd$⁠. In the two-wheel steering mode, both the control input $u=δf$ and the matrices $Qu$ and $Qd$ are scalars. Consequently, there are two calibration parameters $Qu$ and $Qd$ that need to be determined. In the scenario of four-wheel steering, the control input $u=[δf,δr]T$ spans a two-dimensional space. Therefore, in this case, both $Qu$ and $Qd$ are two-dimensional diagonal matrices and have four calibration parameters that need to be determined, which are $Quf$⁠, $Qur$⁠, $Qdf,$ and $Qdr$⁠. To generate the parameter matrix for testing, the design of experiment (DoE) method is taken into consideration. However, the factorial design method, which is the most common approach in DoE, is inadequate for problems involving four variables. Consequently, in this article, the Latin hypercube method [29], an alternative DoE approach better suited for multiple variables, is employed. The sets of parameters for the two-wheel and four-wheel steering systems are presented in Figs. 5 and 6, respectively. Both DoEs show a reasonable distribution over the design space. Note that more points are generated in the four-wheel steering system due to its large design space.

To collect data for these calibration sets, we implemented the two-wheel and four-wheel MPC controllers in separate scripts, with a target vehicle speed of $1.6$ m/s. The scripts automatically iterate through the DoEs, apply the calibration for a set time, and generate data files. Between each iteration, the first calibration (known to function reasonably well from hand calibration or previous DoE results) is reapplied to allow the system to recover if the previous DoE calibration performs poorly. In addition, data collection for a DoE calibration is ended early if the vehicle deviates from the desired path beyond a predefined threshold. The results for the two-wheel steering DoE shown in Fig. 5 are presented in Table 1, and the results for the four-wheel steering DoE shown in Fig. 6 are presented in Table 2. The cost index value $I$ is the normalization of $RMSE¯+MaxError¯$ with respect to the smallest value and is used to evaluate the performance of the sets of weights. The calibrations with the best performance according to the cost index are highlighted in bold, with the corresponding path-tracking performance shown in Fig. 7.

Table 3 compares the tracking performance of the two- and four-wheel steering systems under different values of $θ$ (the maximum tolerable error; see Sec. 2.3). From this table, it is evident that regardless of the threshold value applied, the four-wheel steering MPC consistently achieves a lower RMSE and Max tracking error. For instance, at $θ=0.0,$ the RMSE and Max error for the two-wheel steering system are 3.9 and 11.0, respectively, while for the four-wheel steering system, they are 3.7 and 8.6, respectively. This trend persists for higher values of $θ$⁠. At $θ=3.5$⁠, the RMSE and Max error for the two-wheel system rise to 12.3 and 26.8, whereas for the four-wheel system, they only increase to 9.6 and 20.3, respectively. In addition, the event-triggered mechanism in the four-wheel steering system reduces computational requirements more than in the two-wheel steering system. The results demonstrate that four-wheel steering MPC offers performance improvements over two-wheel steering MPC, and the event-trigger control mechanism provides the capability to lessen the computational load, even in normal driving conditions. This supplements the previous results [21] that focus on tight steering maneuvers only.

This article investigates the advantages of a four-wheel steering system under normal driving conditions, where the event-triggered MPC is applied to both two- and four-wheel steering systems. A modified 1/10th scale Tamiya TT-02 RC car with ultrasonic-based indoor positioning system is used for experiments. Contrary to the conventional belief that four-wheel steering systems are beneficial only in tight steering maneuvers, the results reported here show that, under identical settings of the event-triggered MPC, the four-wheel steering system offers better tracking performance while using less computational resources.

This work was supported in part by the Faculty Startup Fund at Oakland University and in part by National Science Foundation (Award No. 2237317).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.