Comparative Study of Trajectory Tracking Control for Automated Ground Vehicles via Model Predictive Control and Robust H-infinity State Feedback Control

: A comparative study of longitudinal and lateral control maneuverer in model predictive control (MPC) schemes and robust 𝐻 ∞ state feedback control (RSC) method for trajectory tracking of automated ground vehicles (AGVs) is presented in this paper. Both MPC-based and RSC-based tracking controller are designed on the same basis of longitudinal-lateral-yaw motions of a single-track vehicle model. The main objective is to compare the controllers’ performance of tracking accuracy of path and velocity under different test scenarios. The simulation is implemented on Carsim-Simulink joint platform using high-fidelity vehicle model and the mass uncertainties, sensor measurement noise and the performance in extreme driving conditions: 90 ° turn with big curvature are considered. The simulation results indicate that mass uncertainty and sensor measurement noise of lateral velocity have little effect on the RSC-based controller, while that have relatively great influence on MPC-based one. However, MPC-based controller shows a shorter response time and more accurate tracking performance than RSC-based scheme. Finally, for the test of 90 ° turn with curvature 0.02 𝑚 −1 , the maximum velocity that RSC-based controller can carry out has reached 22m/s, which is slightly better than MPC-based one: 21m/s.


Introduction
The increasing demands on mobility, efficiency and safety have extremely promoted the development of intelligent transportation system (ITS) in recent years [1]. As one of the most promising technologies of automotive industry, automated ground vehicles (AGVs) with the improved security and better road utilization have many potential Xiao-Lin Tang tangxl0923@cqu.eud.cn 1 State Key Laboratory of Mechanical Transmissions, College of Automotive Engineering, Chongqing University, Chongqing, China; 2 School of Mechanical Engineering, Beijing Institute of Technology, applications in many fields and have attracted many attentions from both industry and academic communities [2][3]. Generally, the typical architecture of an AGV system mainly incorporates two parts [4][5][6], namely, perception system and decision-making and each of them is generally divided into many subsystems that are shown in Figure 1. Trajectory tracking control is a fundamental issue for AGVs, which is devoted to track a predefined path and velocity profiles accurately and the errors (i.e. the lateral offset, heading and velocity errors) need to converge to zero during the manipulating process [7].
In recent years, much research work on trajectory tracking have been studied [8][9][10][11][12][13] and the main challenges of achieving accurate trajectory tracking lie in the following aspects: (1) non-holonomic property and multi-constrains of AGVs; (2) trade-off between vehicle model accuracy and computation efficiency; (3) uncertainties and time-varying parameters of vehicle dynamic model; (4) external disturbances of complex driving environment. Besides, the interaction between road and tires is also an important source of coupling. According to the vehicle dynamics, it's known that the maximal longitudinal and lateral tire force is determined based on the friction ellipse under certain road conditions [14]. Furthermore, for most existing literatures of trajectory tracking, the bicycle vehicle model which describes longitudinal, lateral and yaw motion is usually adopted for the controller design. But owing to the yaw motion caused by wheels steering, different dynamic and kinematic couplings can put huge impact on the controllers' performance [15]. It's well known that model predictive control and robust ∞ control theory are two effective techniques to address the problems mentioned above [16][17][18][19][20]. Figure 2 illustrates the research trend of trajectory tracking using MPC and robust ∞ control method of AGVs, which is derived by using keywords like 'trajectory tracking MPC' and 'trajectory tracking Robust ∞ control' to find the number of papers published in this fields from 2000 to 2019 in Web of Science.
On the one hand, MPC has become one of the most popular optimal control methods in these years due to its well performance in processing multi-constrained linear or non-linear system [21][22] and it is easy to be used at different levels of the process control structure and offers attractive solutions for tracking problems while guaranteeing stability [23][24]. Owing to its advantages, MPC has been widely implemented in autonomous industry including trajectory tracking issues. For example, the trajectory controller is usually designed by formulating the tracking task as a multi-constrained model predictive control (MMPC) [25][26][27][28][29]. In one word, MPC uses a mathematical dynamic process model to predict future system states and gets the optimal control values by formulating trajectory tracking issues as an optimal control problem and solving it with effective optimal algorithm. On the other hand, robust ∞ control has certain advantages in handling the presence of model uncertainties and external disturbances, which may result from variations of vehicles or environment parameters as well as the vehicle states [7]. Ref [30] manages to devise a robust ∞ path following control strategy for AGVs with considering signal transmission delay and data dropouts. In addition, the timevarying parameters, steering system backlash-type hysteresis, and uncertainties of nonlinear tire model are also considered in [31][32][33]. In short, the aim of robust ∞ control law is to design the feedback gain of the controller, which can make the system's output of external disturbance is as small as possible under the presence of model uncertainties.
Though MPC and robust ∞ control are two effective strategies in tackling trajectory tracking issues, they show many differences in some aspects. For instance, robust ∞ control does not consume too much on-board computation resources as its feedback gain is usually calculated offline, whereas MPC needs to solve complex optimization problems in order to get the optimal control values online. Therefore, the objective of this paper is to systematically compare the practicality of MPC method and robust ∞ state feedback control (RSC) approach when considering mass uncertainties, sensor measurement noise and the performance in extreme driving conditions. To the best of the present authors' knowledge, there has no comparative study of trajectory tracking control with MPC and RSC techniques. The main contribution of this paper is to evaluate MPC-based and RSC-based trajectory tracking controllers' performance and show their advantages and disadvantages with respect to robustness to parameters uncertainties and extreme driving conditions. In order to carry out a fair comparation between MPC and RSC, some details need to be explained in advance. Specifically, the most important performance index of controllers in this paper is tracking accuracy while satisfying system constrains. Besides, they both utilize the same tracking error model on account of combined longitudinal-lateral-yaw dynamic vehicle model to design control laws. Furthermore, the adjustable parameters of both controllers are obtained through trial and error method, trying to ensure that controllers can achieve optimal control performance (i.e. prediction horizon of MPC). The rest of the paper are organized as follows. In section 2, the trajectory tracking error model and combined longitudinal-lateral-yaw vehicle dynamics model are deduced. In section 3, the controllers based on MPC and RSC are designed to track path and velocity profiles, respectively. Then, the performance of MPC-based and RSC-based controllers is demonstrated by testing them in jointed Carsim-Simulink platform with the respect to parameter uncertainties and extreme driving condition in section 4 and finally, the conclusions are illustrated in section 5.  Figure 3 and the definition of parameters used in this paper is shown in Table   1. At any given time, we assume that the reference point is given by a tuple [ , , , ] and subscript represents the variables which are defined by reference profiles. As shown in Figure 3, the lateral offset can be calculated by: The heading error and velocity error are defined as Given the road curvature , the desired yaw rate is obtained: Besides, the derivatives of CG's positon can be obtained from the kinematic relationship:  Figure 3 Schematic diagram of trajectory tracking model and vehicle dynamic model

3-DOF Vehicle Dynamic Model
The schematic diagram of single-track vehicle model coupled 3-DOF (degree of freedom) vehicle dynamic model is also shown in Figure 3. According to Newton's laws, the vehicle's dynamics in the yaw plane can be described by the differential equations: where means the generalized longitudinal force of AGVs including wind drag and tire rolling resistance. And the front and rear lateral force and are the function of tire slip angles and can be calculated as follows where and denote the wheel slip angles, which can be obtained as Therefore, combined equation (6) with (7) and (8) , 1 = , 2 = .
In order to facilitate the controller design, rewrite the equation (9) to form the following linear state-space equation, which can be represented as: with ( ) =

Formulation of Augmented System
As analyzed above, the trajectory tracking task can be posed as predictive control problem with multi-constraints. Note that the model described in equation (10) is a linearized and continuous-time system. To facilitate MPC controller design, the continuous-time system needs to be transformed into discrete state-space mode with the fixed sampling period T.
Here, the zero-order hold (ZOH) method [11] is applied and the equation (10) can be depicted as discrete form: where ∈ 5×5 , ∈ 5×2 and ∈ 4×5 are the system coefficient matrices and means the time step.
Furthermore, to achieve a better control performance, the state vector ( ) and the increment input ∆ ( ) are usually coupled in an augmented vector, which can be represented as ̃( ) = [ ( ), ∆ ( )] . And the control input can be calculated by ( ) = ( − 1) + ∆ ( ) (12) Thus, the system equations (11) can be rewritten as the following equations: and ̃( ) = Therefore, the predicted output can be calculated by the following formulations: Here, we define the predicted outputs of the predictive statespace model at time step k as: Denote the performance outputs over the prediction horizon as a compact matrix form: with Γ = [̃̃,̃̃2, ⋯̃̃, ⋯ ,̃̃] ,

Formulation of Trajectory Tracking Problem Using MPC
The aim of MPC-based trajectory tracking controller is to make the predicted outputs as close as possible to the reference trajectory within the predictive horizon, and the reference trajectory ( ) is assumed to remain unchanged during an optimization window. The reference signals are described as: Thus, a typical tracking accuracy and control smoothness-oriented cost function over the predictive horizons is defined: 19) where diagonal matrices ̅ and ̅ are the positive definite weight matrices (i.e. ̅ > 0 and ̅ > 0 ) which can be regulated to achieve desired closed-loop performance. The first item in cost function reflects the tracking error between the predictive outputs and the reference trajectory. The second one refers to the penalty on the control inputs that is to make the control process smoother.
So far, the main task has converted to find an optimal control inputs ∆ ( ) that can minimize the cost function ℒ( , ) . However, AGVs have many inherent physical limitations needed to be taken into accounted. Specifically, these limitations on the capacity of control actuators or on the rate of control actuators result in the hard constrains of the trajectory tracking system. Besides, there are also some restrictions imposed on output variables due to environment conditions (i.e. road boundary, speed limit, etc.). Each type of constrains can make huge difference on the performance of the tracking system including the stabilization. According to the kinematics and dynamics of the vehicle model, the constrains for this trajectory tracking problem are specified as: where ∆ min , ∆ , min , ∈ 2 ×1 , , ∈ 4×1 and ( − 1) is the control inputs at time k-1.
The first and second inequality are adopted to prevent aggressive control strategy and avoid actuator saturation, respectively, and the third one is imposed to restrict output variables. Then, combined equation (19) with (20), the trajectory tracking problem is converted to solve the following optimal problem with multi-constrains: It's found that the optimization problem (21)  Then, at time step k, the above quadratic problem can be solved (i.e. quadprog function in Matlab) and will get a solution ∆ * . Once the solution is obtained, the first element of solution vector in vector, namely, ∆ * will be used and the optimal control inputs at this time step is computed as: ( ) = ( − 1) + ∆ * (23) Similarly, at next time step + 1 , new input measurements and updated system states produce new quadratic problem that needs to solve. In conclusion, by solving a quadratic problem at each time step, the MPC controller will get an optimal input for next time step. However, in most cases, there are some non-convex constrains need to process and the SCFS method in [34] can be used to convert the optimal problem with non-convex constrains to with convex constrains equivalently.

Preliminary Knowledge of RSC
We re-emphasize that the main objective of this paper is a comparative study of path-following control of an AGV employing different control strategies. Both MPC-based and RSC-based tracking controllers use standard design methods, and specifically, the RSC-based tracking controller design mainly adopts the theory applied in the literature [7] [20]. To carry out a fair comparison of tracking responses with those from MPC, RSC is also based on the linearized vehicle model (10) and the general objective function is as follows where the main task is to obtain a control law that can make the closed-loop system satisfy the asymptotically stability and fulfill the ∞ performance index that attenuate the effect of the external disturbance ( ).
Before the controller design, some essential lemmas and theorems also need to be introduced. Consider a polytopic LPV (linear parameter-varying) system which is described by state-space equations as follows: And it's equal to the following equations where the 0 , 0 and 0 are the values of 0 ( ) , 0 ( ) and 0 ( ) at the vertex of the parameters polytope. Theorem 1: Given a positive scalar 0 , the system described in equation (26) is asymptotically stable and meets the ∞ performance index (24), if and only if there exists symmetric definite matrices and a matrix 0 satisfying the following conditions The proof of this theorem can be found in [20]. The RSCbased controller design is on account of linear matrix inequality (LMI) method. So far, the preliminary knowledge has been prepared sufficiently for the RSC-based controller design.

Formulation of trajectory tracking problem
Similar to MPC-based controller, the constrains imposed on inputs should be taken into consideration [35]. Define as following Lyapunov equation and assume the following condition is satisfied where is the gain of the controller and is a given constant. The maximum inputs are denoted as , then, where is the largest eigenvalue of the matrix Based on Schur complement lemma [36] and equation (31), the following conditions can be obtained， For the system depicted by equation (10), it can be found that time-varying parameters , 1 , and are coupled in the system matrices. These parameters need to be processed before carrying out the design of RSC control law.
Here, the polytope model method in [7] where represents the vertices coordinates of the polytope and means the weighting factors, which can be denoted as: (37) where is the control gain, the key parameters need to be determined. Combined the equations (36) with (37), the closed-loop system can be obtained as According to theorem 1and equation (38), if the following condition is fulfilled, the controller will achieve the aim of ensuring system stability and ∞ performance index of the closed-loop system.
where the is a symmetric positive definite matrix and is a matrix with proper dimensions.
represents the performance index of system in attenuating disturbance.
So far, the key to solve the trajectory tracking issue has converted to calculate the controller gain ( ) by solving the inequality problems above. In this paper, we use the yalmip toolbox to solve this issue because it has simple syntax and is easy to use.

Platform Description and Test Setup
In this section, several simulation tests are carried out to compare the discrepancy of performance of the presented MPC and RSC control schemes, which are performed on the joint Carsim-Simulink platform with a high-fidelity and full-vehicle model. In this paper, we mainly compare their performance under the following test conditions respectively: 1) parameters uncertainties: mass, 2) sensors measurement error, 3) extreme driving condition: 90。Turn with big curvature. Besides, to quantify the tracking accuracy, the root mean square error (RMSE) is used in section 4.2 and section 4.3 and formula is given by where is the number of time periods, is the measured output and donates the reference output value. The vehicle and other related parameters used in simulation are listed in Table 2.

Parameters Uncertainties Case 1: Mass
Vehicle mass is a key parameter for the design of trajectory tracking controllers, which is varying when the number of passengers and cargo changes. In order to guarantee the safety of AGVs, the tracking controllers must show robustness to varying parameters in the vehicle model. In this section, we carry out the test with different mass (i.e. , + 30% , − 30% ). Note that in this section, the main objective is to illustrate their robustness to mass uncertainties and a straight line is used as the reference path. In addition, the initial lateral, angular and velocity errors are set as -1m, 0, -2m/s, respectively.
The simulation results are shown in Figure 4 and the quantifying results are given by the Table 3. From Figure 4 (a) and (b), it can be seen that MPC-based controller shows a faster response than RSC-based one and shown that mass is a key parameter which can influence the response time, namely, the heavier the quality, the slower the response. As shown in Figure 4 Table 3, it can be known that both controllers can track the desired trajectory accurately in a limited time, while in terms of tracking accuracy, MPC performs better.

Parameters Uncertainties Case 2: Sensor Measurement Error
In this paper, although we assume that the system state can be measured through sensors, the measurement is may be inaccurate with errors. For instance, the lateral velocity is difficult to obtained exactly without expensive equipment like dual antenna GPS. Therefore, in this section, the lateral velocity is set to have -30%, 0, +30% errors compared to accurate values, so the controllers' performance of robustness to state variable uncertainties can be tested. The initial lateral, angular and velocity error are also set as -1m, 0, -2m/s, respectively. Similarly, the simulation results are shown in Figure 5 and the quantifying results are given by the Table 4. Figure 5 (a), (b) and (c) display the response results of lateral error, longitudinal velocity error and angular error, respectively. It can be seen that RSC-based controller demonstrates well robustness and the sensor noise has little effect on tracking performance, which can be also illustrated in Table 4. By contrast, MPC-based one shows that its performance can be interfered by measurement noise significantly, but it can adjust the control outputs in time by online optimization and track the desired trajectory in the end. Apparently, the MPC-based controller shows a faster response than RSC-based one like the results in Figure 5. Besides, according to Table 4, MPC displays a better tracking performance than RSC in all the test situations of lateral velocity measurement error. Figure 5 (d) and (e) display the response results of lateral velocity and yaw rate, respectively. It can be known that for MPC-based controller, the value of lateral velocity and yaw rate is bigger than that of RSC-based one, which is result from the faster response of tracking desired trajectory. Figure 5 (f) and (g) demonstrate the control outputs of both controllers.     In this section, we increase the desired velocity continuously until the acceleration of vehicle is close to the , so as to obtain the critical velocity of the vehicle under MPC-based and RSC-based control algorithms. Figure 6  , it can be seen that RSC-based trajectory tracking controller can reach the maximum velocity 22m/s while guaranteeing the stability of the system. However, the maximum velocity tracking error has reached 1.5m/s because there is slight side slip. The maximum desired velocity that MPC-based controller can follow is about 21m/s. If the desired velocity is further increased, the vehicle will experience a significant side slip and controller loses its tracking ability in the end. Figure 7 (c) and (d) demonstrate the lateral velocity and yaw rate of both controllers, which are dominated in an accepted range [26] and the control outputs is shown in Figure 7   In this paper, a comparative study of trajectory tracking control using MPC and RSC techniques of an automated vehicle is presented. First of all, the tracking model with 3-DOF (i.e. longitudinal-lateral-yaw) vehicle dynamic model and linear tire model is deduced. Then, based on the model predictive control theory and robust ∞ control theory, the MPC-based and RSC-based trajectory tracking controller is proposed, respectively. This paper focuses on studying the tracking ability of MPC-based and RSC-based controllers and comparing the discrepancy between them under different scenarios. Therefore, the mass uncertainties, sensor measurement noise and the performance in extreme driving conditions: 90° turn with big curvature are considered. The simulation results show that mass uncertainty and sensor measurement noise of lateral velocity display little effect on RSC-based controller, while that put relatively greater influence on MPC-based one. But MPC-based controller shows a faster response speed and more accurate tracking performance than RSC-based controller. Finally, for the test of 90° turn with maximum curvature 0.02 −1 , the maximum velocity that RSC-based controller can carry out has reached 22m/s, which is slightly better than MPC-based one: 21m/s.