Shuping Chen， Huiyan Chen,? and Dan Negrut

（1. School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China；2. Department of Mechanical Engineering, University of Wisconsin-Madison, Madison,WI 53706, USA）

Abstract: In order to investigate how model fidelity in the formulation of model predictive control(MPC) algorithm affects the path tracking performance, a bicycle model and an 8 degrees of freedom(DOF) vehicle model, as well as a 14-DOF vehicle model were employed to implement the MPC-based path tracking controller considering the constraints of input limit and output admissibility by using a lower fidelity vehicle model to control a higher fidelity vehicle model. In the MPC controller, the nonlinear vehicle model was linearized and discretized for state prediction and vehicle heading angle, lateral position and longitudinal position were chosen as objectives in the cost function. The wheel step steering and sine wave steering responses between the developed vehicle models and the Carsim model were compared for validation before implementing the model predictive path tracking control. The simulation results of trajectory tracking considering an 8-shaped curved reference path were presented and compared when the prediction model and the plant were changed. The results show that the trajectory tracking errors are small and the tracking performances of the proposed controller considering different complexity vehicle models are good in the curved road environment. Additionally, the MPC-based controller formulated with a high-fidelity model performs better than that with a low-fidelity model in the trajectory tracking.

Key words: model predictive control；vehicle dynamics；path tracking；autonomous vehicle

Path tracking is an essential capacity for autonomous vehicles, which aims at controlling the vehicle to follow a desired path by calculating a proper steering angle, and generating optimal control commands to satisfy this path. An ideal control method is capable of maintaining the vehicle along the reference path by using its current measurements, and enforcing specific limits at the same time. In the literature concerning path tracking, a variety of controllers were proposed by using classical control theory, modern control theory and robust control theory, etc.,such as the PID controller[1], the optimal controller[2], the backstepping technique[3]and sliding mode controller[4]. However, the constraints of actuator saturation and physical limit were usually not considered in these controllers. Model predictive control method, which combines prediction models, a receding horizon optimization and a feedback correction, attracts a wide range of attentions from both academia and industry in recent years, due to its advantages of handling the problems considering input constraints and state admissible[5-6]. On the other hand, although previous work has introduced model predictive control algorithms for path following, more work is required to test the ability of controlling a vehicle simulated by different-fidelity models. To this end, the MPC algorithm is employed to tackle the path tracking problem and the robustness of the proposed MPC controller is evaluated through numerical simulations.

Model predictive control has its roots in optimal control. The basic concept of MPC is to use a dynamic model to forecast system behavior,and optimize the control move at the current time to produce the best performance in the future. Models are therefore central to every form of MPC[7]. The single-track bicycle model is usually used as the internal prediction model for the formulation of MPC algorithm, since it simplified the calculation of optimization and reduced the computational burden for real-time implementation[8-9]. However, a complex vehicle model is rarely considered in the controller, although it approximates the real plant system better. Furthermore, a realistic and precise model is of significant importance to develop vehicle control system and whether the controller designed based on a complex model is superior to that based on a simple bicycle model remains to be a question.With those motivations, the goal of this paper is to investigate how model fidelity of both the controller model and the plant model affects overall performance of the MPC-based path-tracking controller. We implemented the model predictive path tracking controller, a lower fidelity vehicle model, to control the plant, a higher-fidelity vehicle model, and considered the tracking path to be an 8-shaped curved trajectory in order to investigate the tracking performance of the proposed controller in the curved road environment.

The remainder of this paper is organized as follows: in Section 1, a linear tire model, a bicycle model, an 8-DOF vehicle model and a 14-DOF vehicle model used in this study as the controller or the plant are introduced. In Section 2,the formulation of MPC algorithm and the path tracking controller are specified with the basis on the developed models. In Section 3, the output responses between the developed vehicle models and Carsim model are compared for vehicle model validation before implementing the proposed controller with three different-fidelity models considering the reference path to be an 8-shaped curve trajectory. Then, the simulation results of the tests outlined for three cases are illustrated and the comparisons of path tracking performance when the internal controller vehicle model are provided and discussed. Section 4 concludes and presents potential future work directions.

1 Vehicle Dynamics Modeling

This section describes a linear tire model and different fidelity vehicle models used for simulations, including a bicycle model, an 8-DOF vehicle model and a 14-DOF vehicle model.

1.1 Tire model

Tire forces produce primary external influences and have highly nonlinear characteristics, it is essential to use a realistic tire model, especially when investigating large control inputs resulting into responses near the limits of the linear character scale of the tire. Except for aerodynamics forces and gravity, all the forces affecting vehicle behavior are provided by the tires. The tire lateral and longitudinal forces are assumed to depend on normal force, slip angle, surface friction, and slip ratio. However, when the slip ratio and slip angle are limited within small values,the tire model can be simplified as to generate linearized lateral force and longitudinal force[10-11].

Under this assumption, the tire lateral force can be given as Fc=Cα(μ,Fz)α, where Cαis tire cornering stiffness related to tire normal force Fzand tire-road friction coefficient μ, the tire slip angle is denoted by α; and tire longitudinal force can be given as Fl=Cx(μ,Fz)sf,r, where sf,ris the slip ratio of front tire or rear tire, Cxis the tire longitudinal stiffness which also related to the tire normal force and tire-road friction coefficient.

1.2 Bicycle model

The single-track vehicle model[5]or bicycle model[9-11]is often used in the controller design,the schematic of this model is depicted in Fig.1,which is based on the following set of simplifications and assumptions:

① The left and right tires at front and rear axles are lumped in a single wheel and the control input is simply the front tire steer angle (i.e.,δf=δ,δr=0) by neglecting the effect of steering system;

② The vehicle is assumed to move in the yaw plane and the vertical, the pitch and the roll motions are not considered by neglecting the effect of suspension;

③ The lateral acceleration coefficient (i.e.,the ratio between lateral acceleration and gravity acceleration) is limited within 0.4 to represent the vehicle lateral dynamics in the linear regime;

④ The left and right tire slip angles are assumed to be equal and the effects of aerodynamics and load transfer between the left and right tires are neglected.

Fig.1 Bicycle model[5, 9, 11]

With these assumptions, the bicycle model can be described as[10-11]

Assuming that the vehicle side slip angle and the tire slip angle are small. The approximations of linearized side slip angle and tire slip angle can be simplified as belows[5]

The position of the vehicle expressed in the global coordinates can be calculated as follows[10-11]

1.3 8-DOF vehicle model

An 8-DOF full vehicle model is often used as a simplified lower order model for studying vehicle handling in scenarios which do not involve significant longitudinal accelerations. In this section, a formulation for the 8-DOF model,which was adapted from various references and match the 14-DOF model reasonably and accurately, is presented[12].

The schematic of the 8-DOF full vehicle model is shown in Fig.2. The model has four degrees of freedom for the chassis and one degree of freedom at each of the four wheels representing the wheel spin dynamics. The DOF of chassis includes the longitudinal velocity, u, the lateral velocity, v, the roll angular velocity, ωx, and the yaw angular velocity, ωz. The pitch and heave motions are not modeled and the front and rear suspensions are represented simply by their respective equivalent roll stiffness (kφf/kφr) and roll damping coefficients (bφf/bφr)[12].

Fig.2 Schematic of 8-DOF full vehicle model[12]

The equations of chassis dynamics are presented as follows[12]

where

1.4 14-DOF vehicle model

In order to better represent the vehicle lateral dynamics and yaw dynamics as well as coupling of the yaw-roll motion due to the transient lateral load transfer during extreme maneuvers, a higher order model such as the 8-DOF model and 14-DOF model are also used in rollover studies.A 14-DOF vehicle model, which considers the suspension at each corner, has the same benefits of an 8-DOF vehicle model, with the additional capabilities of predicting vehicle pitch and heave motions. It also offers the flexibility of modeling nonlinear springs and dampers and can simulate the vehicle responses to normal force inputs in the case of an active suspension system.Moreover, the 14-DOF model, unlike the 8-DOF model, can predict vehicle behavior even after wheel lift-off and thus can be used in validating of rollover prediction/prevention strategies[12].

Fig.3 exhibits the schematic of 14-DOF vehicle model, which was used to investigate vehicle roll responses to steering and torque inputs. This schematic includes a 6 DOF model at the vehicle lumped mass center of mass and a 2-DOF one at each of the four wheels, including vertical suspension travel and wheel spin. The body is modeled as being rigid, with body-fixed coordinates attached at the center of mass and aligned in principal directions (coordinate frame 1). u,v,w indicate forward, lateral, and vertical velocities, respectively, of the sprung mass. The pitch angle, yaw angle and roll angle of the sprung mass are denoted by θ, ψ, φ. The roll angular velocity, pitch angular velocity and yaw angular velocity are denoted by ωx, ωyand ωz. The moments transmitted to the sprung mass along the ωx, ωyand ωzdirections are denoted by Mxij,Myijand Mzij, and the forces Fxsij, Fysijand Fzsijare the forces transmitted to the sprung mass along the longitudinal, lateral and vertical directions of coordinate frame 1, respectively[12].

The equation of motion for the 6 DOF of the sprung mass model can be derived from the direct application of Newton’s laws for the system as[12]

Fig.3 Schematic of 14-DOF model with one-dimensional suspension and coordinate frames[12]

where msis the sprung mass and the cardan angles θ, ψ, φ needed in the aforementioned equations are obtained by performing the integration of the following equations[12]

The rotational dynamics for each wheel can be given as, for instance, the right front wheel[13]

where Tdrfis the driving torque transmitted to the wheel, Tbrfis the wheel braking torque. In this paper, when implementing the MPC controller, the driving torque and braking torque are assumed to be zero.

2 MPC Controller Design

The essence of an MPC scheme is to apply a mathematical model of the system one desires to control to predict and optimize future system behavior. Such a prediction is accomplished by employing an internal model over a fixed finite time horizon, called the prediction horizon, from the current system state and inputs. At each sampling time, the controller generates an optimal control sequence, called control horizon, by solving an optimization problem and the first element of this sequence is applied to the plant. The repetition of this process over time by using the updated measurements creates a feedback loop which continually controls the system, pushing it towards an optimal path[14].

2.1 Linearization of the vehicle model

The general form around the operating point is

Using the Taylor series expansion at the operating point and ignoring higher order terms, we can obtain[15]

Subtracting Eq.(19) from Eq.(20) results into

Eq.(21) is called the linear error model[15].

In order to apply the model to the design of the MPC controller, the equation is described in the form of discretized state-space representation[15]:

where Ad=I +TA, Bd=TB and T is the sampling time.

Eq.(22) can also be described in the following form for the MPC controller design

The Jacobian matrices for 8-DOF vehicle model can be computed by a math function in MATLAB.

2.2 State prediction

The model used in the control system design is a state-space model. By using a state-space model, the current information required for predicting ahead is represented by the state variable at the current time.

The sequence of future incremental control inputs computed at time k is denoted by ΔUm,also represented as ΔUm. The control input varies for Nctime steps (i.e., the control horizon)and then is held constant up to the preview horizon. The predicted output for the predictive state-space model is denoted by ηm(k). The prediction model of performance outputs over the prediction horizon Npin a compact matrix form as[15-16]

where

2.3 Cost function definition

The cost function is capable of controlling the autonomous vehicle to track the desired path rapidly and smoothly. Thus, the system status deviation and the optimization of the control outputs should be combined into the controller.

Considering the soft constraints concept[17],the following objective function[15]can be obtained

where Q and R represent the weight matrices.The first term in Eq.(30) reflects the capability of tracking performance, while the second reflects the constraint on the change of the control output. After plugging Eq.(29) into it, the objective function can be given as

2.4 Constraint analysis

For any real world plant, control inputs are subject to physical limitations. Hence, to avoid large plant/model mismatch those limitations should be considered while computing control inputs. Considering the vehicle dynamic model, the constraints imposed on steering angle for the path-tracking problem are specified as follows[15]

① The constraints for control can be given as

② The constraints for control increments can be given as

③ The output constraints can be given as

In the cost function, the control increment in time domain is solved. Thus, the constraints should be presented in the form of control increment or the multiplication of control increment and transformation matrix. The relationship is defined as[15]

Assuming that

Combining Eqs.(36) – (38), we can obtain[15]

where Uminand Umaxare the lower and upper bounds of the control input, respectively.

After obtaining the solution to Eq.(32) in each control cycle, a series of control input increments in the control horizon can be calculated as[15]

The first element of the control sequences is taken as the actual control input increment of the controller[15].

3 Simulation Test Results

In this section, numerical simulations of the developed MPC path-tracking algorithm with different vehicle models as the prediction model in the controller and the plant are presented and analyzed.

3.1 Vehicle model parameters

The parameters of the vehicle model are listed as Tab.1.

Tab. 1 Vehicle modeling parameters

3.2 Vehicle dynamics model validation

Carsim is a software, in which the vehicle model is of high-fidelity and validated with experimental results on actual vehicles. To validate the developed models, we compared the simulation results of wheel steering responses between these vehicle models and the Carsim vehicle model. Assuming the speed is constant in the simulation, the bicycle model, 8-DOF model and 14-DOF model are implemented with input signals of step steering angle and sine wave steering angle. The vehicle responses are studied and compared to understand the effects of simplifications.

① Step steering simulation: the amplitude of step steering angle is 0.008 7 rad, i.e., 0.5° and the vehicle speed is set at a constant value 33.73 m/s. The vehicle output responses include yaw rate, lateral acceleration, lateral velocity and roll angle. The vehicle output response comparisons among bicycle model, 8-DOF model, 14-DOF model and Carsim model are given in Fig.4.

② Sine wave steering simulation: the amplitude of sinusoidal steering angle input is 0.008 7 rad, i.e. 0.5°, and the vehicle longitudinal velocity is fixed at 33.73 m/s. The output responses include lateral velocity, lateral acceleration, roll angle and yaw rate. The vehicle output response comparisons among Bicycle model, 8-DOF model, 14-DOF model and Carsim model are illustrated in Fig.5.

Based on the simulation results of roadwheel step steering and sine wave steering input,the output responses of developed vehicle models are approximate to the results of the Carsim vehicle model. During steady-state simulation,the value of yaw rate, lateral acceleration and lateral velocity increased a little from the bicycle model, 8-DOF model to 14-DOF model. Compared with other developed models, the 14-DOF model correlates better with the Carsim model.

Fig.4 Output responses between the developed vehicle models and Carsim model during step steer input at a speed of 33.73 m/s

Fig.5 Output responses between the developed vehicle models and Carsim model during sine wave steering at a speed of 33.73 m/s

3.3 Reference trajectory definition

The automated vehicle is assumed to drive at a constant longitudinal velocity during the path tracking process. Herein, an 8-shaped curved trajectory was utilized as the reference path in order to investigate the tracking performance of proposed controller in the curved road environment. The vehicle is controlled to follow a straight line first and then an 8-shpaed curves with the radius of 30 m at the speed of 10 m/s.In the MPC controller, the vehicle yaw angle(i.e., heading angle) ψ, the vehicle lateral position Y and the vehicle longitudinal position X are chosen as the tracking objectives in the cost function.

3.4 Path tracking simulation results

In order to investigate the performance and robustness of the proposed MPC-based path tracking controller considering different fidelity vehicle models, we implemented the path tracking controller in three cases that using a low-fidelity vehicle model to control a high-fidelity vehicle model.

According to the tire modeling, the relationship between the tire slip angle and cornering force is linear when the tire slip angle is small.With small angle assumption and the linear tire models, the front road wheel steering angle is limited in -8.6°≤α ≤8.6°, and the control increment is limited in -0.85°≤Δα ≤0.85°. As for the output state constraints, the lateral position boundary is –160 m ≤Y ≤ 1 m, the heading angle boundary is -3.2 rad ≤ψ ≤3.2 rad, and the longitudinal position is given as -1 m ≤X ≤90 m.

3.4.1 Case A: bicycle model as prediction model and 14-DOF model as the plant

Path tracking performance and steering control input signal for Case A are illustrated in Fig.6.

Fig.6 Trajectory comparison and control input for Case A

3.4.2 Case B: bicycle model as prediction model and 8-DOF model as the plant

Path tracking performance and steering control input signal for Case B are given in Fig.7.

3.4.3 Case C: 8-DOF model as prediction model and14-DOF model as the plant

Path tracking performance and steering control input signal for Case C are shown in Fig.8.

3.4.4 Trajectory tracking error comparison

The tracking errors of vehicle lateral position and longitudinal position for three cases are compared in Fig.9

Fig.7 Trajectory comparison and control input for Case B

Fig.8 Trajectory comparison and control input for Case C

Fig.9 Y, X and ψ tracking error comparisons for three cases

From the simulation results, the comparisons between the plant output trajectory and the desired path shown in Fig.6, Fig.7 and Fig.8 demonstrate that the automated vehicle tracks the 8-shaped curved path well and smoothly using the proposed MPC-based controller considering different fidelity models, subject to the constraints of input limits and output boundaries.The tracking errors of vehicle lateral position,longitudinal position and heading angle for three cases shown in Fig.9 are small for all three cases.Moreover, Case C, which used an 8-DOF model as the prediction model and 14-DOF model as the plant, outperformed Case A and Case B in the trajectory tracking.

4 Conclusions and Future Work

In this study, three MPC controllers are implemented in simulations for path tracking.Within this simulation, the design of model predictive path-tracking controller was employed in the scenario of tracking an 8-shaped curve trajectory. The bicycle model was applied as the MPC controller to control the 8-DOF model and 14-DOF model, respectively, and an 8-DOF model was also utilized as the MPC controller to control the 14-DOF vehicle model. The vehicle yaw angle, the lateral position and the longitudinal position were selected as the tracking objectives in the cost function. In order to validate the developed vehicle models, we also compared the output responses of these different fidelity models with those of the Carsim model using step steering input and sine wave steering input, respectively.

Based on the simulation results, the controller using the 8-DOF vehicle model performs marginally better than the one based on the bicycle model when the 14-DOF model is employed as the plant, and limited improvements on the tracking performance is observed when changing the 8-DOF model to 14-DOF model while the bicycle model is applied as the controller model. As a consequence, we conclude that, on the one hand, the bicycle controller can successfully navigate a vehicle along the given path as well and allows for faster calculation of optimal steering sequences, hence suitable for a possible physical implementation with real-time requirements. On the other hand, compared with the bicycle model,the 8-DOF vehicle model has a better trajectory tracking performance, which is further capable of considering the roll motion and lateral load transfer when the vehicle is cornering along the desired path. Furthermore, the implementation of 8-DOF vehicle model based on the MPC controller is more complicated than that of bicycle model based controller due to the higher order of the nonlinear dynamic model, the increased system state dimensions and more computational burden in optimization.

For the future research concerning path tracking and speed control, a time-variant speed reference will be applied and a vehicle that have both lateral and longitudinal controls will be studied.

Acknowledgments

The authors would like to thank Heran Shen of Columbia University for his comments on this paper.