Model Predictie Control for Autonomous Lateral Vehicle Guidance M. Sc. Jochen Prof. Dr.-Ing. Steffen Müller TU Berlin, Institute of Automotie Engineering Germany Statement: This paper will also be published during the 2017 AUTOREG conference. ABSTRACT Current actie drier assistance systems support the drier during lateral ehicle guidance in specific driing situations and across a limited speed range. On the way to automated driing it is necessary, to deelop suitable lateral controllers, which control the ehicle during different maneuer and speeds. This article will introduce a Model Predictie Lateral Controller which calculates an optimal manipulated ariable sequence for an inner low leel steering angle controller through a speed-dependent adaptation of the prediction model and the cost function weights to ensure a stable and precise path tracking performance. The real-time capability will be assessed and the performance of the proposed controller ealuated.
Introduction The lateral guidance of autonomous ehicles using Electric Power Steering (EPS) places high demands on the control accuracy of the lateral controller. This includes, in particular, a good reference tracking during different driing maneuers to aoid potential dangerous situations. To reduce the lateral displacement under consideration of the future desired heading angle and EPS limitations, an optimal manipulated ariable needs to be calculated. A possible control strategy in this context is the Model Predictie Control (MPC). The MPC allows to honor constraints of the steering angle δ and steering angle elocity dδ/dt during the calculation of the optimal manipulated ariable sequence. Due to the receding horizon principle, future course information are taken into account. [1] introduces a MPC with nonlinear prediction model for lateral ehicle guidance. The calculation of the nonlinear optimization problem causes a high computational burden, which is a drawback for practical implementations. [2] and [3] thus use linearizations of the nonlinear prediction model to calculate the manipulated ariable. The calculation of the lateral displacement references to the ehicles center of graity. The controller performance is tested with lane change and easie maneuers at 15 m/s. In experimental test ehicles the positon of the center of graity is load-dependent and therefore without prior measurements not precisely known. Thus a clearly defined geometric reference point on the ehicle is used such as the intersection between the central axis and the front axle axis. Within this work, the influence of the reference point on the controller design and controller performance will be examined. Furthermore, the operating area of the controller will be extended to different maneuers and elocities. The focus will be on low speeds (1 2 m/s) with small cure radius (< 10 m) as they occur during parking an maneuering on narrowest space, as well as highly dynamic maneuers with increased speed. reference heading angle = arctan (dy/dx) is calculated on the basis of the reference path information in xy-coordinates, which the car needs to follow. MPC dy δ ref EPS Figure 1. Path Control System. δ Vehicle The MPC receies the actual elocity, yaw rate yaw angle, lateral displacement dy and the steering angle at the front axle δ. The MPC manipulated ariable commands δ ref are propagated to a subordinated steering angle controller inside the EPS. Prediction Model An important element of the MPC is the prediction model, which is used to predict the future ehicle behaior. The key objectie is to predict the lateral displacement dy for future manipulated ariable commands δ ref. Figure 2 shows all releant ariables for the prediction of dy. The ariable represents the elocity ector of the front wheel which encloses the angle γ relatie to the inertial system. ref. path tangent Θ γ dy δ Figure 2. Single-track model and reference path. γ Path Control System Figure 1 shows the path control system consisting of Vehicle, EPS and the MPC lateral controller. The The discrete reference heading angle sequence i represents the discrete reference heading angle sequence for discrete time steps ahead of the ehicle. On the Basis of trigonometrical rela-
tionships the angle γ is calculated according to equation 1 and 2. (1) (2) The lateral displacement rate results from the ehicle elocity and the heading angle error ΔΘ = γ to: (3) The calculation of the states and will be performed through a dynamic and a kinematic singletrack model. The kinematic model describes the ehicle dynamics less accurate, but can simplify parameterization because it uses the wheel base as the only ehicle parameter and the model can also be used in case of zero elocity. Dynamic Single Track Model The linear single-track model according [4] describes the lateral ehicle dynamics with a statespace representation and the parameters cornering stiffness (c α ), mass moment of inertia around the ertical axis (J), ehicle mass (m) and the distances between CoG and front/rear axle ( h) as follows: (4) (5) (6) (7) (8) Kinematic Single Track Model Ignoring ehicle mass and inertia, and are geometrically defined using Ackermann steering angle according to equation 9 and 10. (9) (10) R corresponds to the cure radius and L = + h to the wheelbase. Steering Dynamics The reference alue δ ref calculated by the MPC needs to be controlled by the subordinated steering angle controller inside the EPS. The closed-loop steering angle controller dynamics can be modelled with a PT2 element (gain K, damping d and time constant T) in state-space representation according to equation 11 (11) Prediction Model in State-Space Representation Summarizing equations 3, 4 and 11 the whole prediction model for the MPC corresponds to: (12) (13) The elocity dependence of the matrices A() and E() results from the demand of a elocity adaptie lateral control. Through a range of = 1 30 m/s, a set of matrices were calculated. To analyze the transfer characteristics G(s) = dy(s)/δ(s), figure 3 shows the corresponding polezero plot for = 1 30 m/s. Due to the pole on the imaginary axis, the system is marginally stable. The impulse response corresponds for = 0 to a linearly rising function. With increasing speed, the poles shift to positions with lower damping ratio. The system still remains marginally stable. The task of the MPC is, to stabilize the system for 0 in compliance with manipulated ariable constraints.
dy MPC Im(s) in 1/s Ref. alues, soll dy 0 Ad ( ), Bd ( ) Model C, D d d y R ( k i) _ y(k) y P ( k i) Predictionmodel Kalman- Filter Measurements Optimizer xˆ ( k) u( k i) (k) u opt ref Figure 4. Structure of the MPC. Re(s) in 1/s Figure 3. Pole-zero plot of G(s) = dy(s)/δ(s). Substituting the dynamic single track model by the kinematic model, the reduced prediction model results to: Before eery prediction cycle the state ector of the predition model needs to be initialized with the measured plant outputs. Because the sidslip angle is not aailable as measured alue and all other measured outputs are oerlaid by interfering white noise, a Kalman-Filter is used to estimate all prediction model states. The Kalman-Filter and prediction model state space representations are adjusted, depending on the speed. The optimizer minimizes the cost function J according equation 18. (14) (15) For the controller design, the matrices B und E() are combined to the input matrix B() and afterwards discretized with the sampling rate T according to [5] with the equations 16 and 17. (18) (16) Model Predictie Controller (17) Figure 4 shows the inner structure of the MPC. The blocks Kalman-Filter and Prediction model include the discretized state-space representations 12 and 13 respectiely 14 and 15. The term J dy weights the lateral displacement, J Δδsoll the steering angle rate and J δsoll the absolute steering angle. Constant parameters are chosed as follows: MPC sampletime: T MPC = 20 ms Prediction horizon: n p = 50 Control horizon: n c = 20 Preiew distance: T pre = T MPC n p = 1 s MPC target alues: dy ref = 0 δ ref = 0
The sample time T MPC is equal to the sample time of the EPS-Steering and therefore garanties a smooth setpoint change without destabilizing the steering angle control loop. The choice of T pre is a compromise between the desired closed loop stability and the necessary processing performance that is required to compute a whole prediction cycle within T MPC in real-time. To assess the real-time capability, the MPC was run on a dspace MicroAutoBox II. With a turnaround time of approximately 5 ms, the MPC completed the prediction cycle within T MPC and therefore in realtime. The calculation of the optimal control sequenz (19) Figure 5. Lane change maneuer. Figure 6 shows the simulation results for dy max as also the optimized speed dependent weights. The required steering angle increases notably at low speeds. Therefore a reduction of r improes the tracking performance. In contrast, the weight q can be increased, because the closes loops system is unlikely to be destabilized at low speeds. is done by soling the differential equation (20) subject to (21) Equation 20 represents a quadratic optimization problem which can be soled efficiently with standard QP-solers [6,7]. Simulation Results To inestigate the control behaiour, a nonlinear steering- and single track model was used. First analysis hae shown, that stability an tracking performance are mainly influenced by the weights q and r. The weight r mainly influences the lateral acceleration rate. This represents a significant aspect of comfort [8]. This work howeer focuses on stability and tracking performance, therefore a constant weight of r = 1 is chosen. To examine the effects of the weights q and r on the controller performance, a lane change maneuer according to Fig. 5 is used. The maximum lateral displacement dy max is chosen as criteria for the controller performance. The maneuer was performed for speeds from 2 to 20 m/s. For eery speed the weights were aried to ealuate the influence on dy max. Figure 6. Weight optimization. At higher speeds the weight r stabilizes the ehicle by reducing the steering angle. Generally speaking, an increase of r and a reduction of q increases the stability of the closed loop system but effects the tracking performance negatiely at low speeds. Table 1 summerizes the relationship between ehicle speed, required steering angle and the resulting weights. Table 1. Qualitatie relationship between and weights Main challenge Required Weights δ ref r q low path tracking high low high high stability low high low
To analyse the control behaiour at different cure radii, the handling track from figure 7 is used as reference path. Figure 7. Reference path. At the drieway of the handling track, minimum cure radiuses up to 8 m occure. This area is passed through with 2 m/s and the optimized speed dependent weights r and q. The cure radiuses during the following handling track aries between 48 m and 101 m. The dotted line represents the results with the kinematic single track model. Because of the low lateral acceleration, both prediction models show a similar tracking performance. The relationship dy = ( + Θ soll ) can be used to predict the lateral displacement at the center of graity instead at the front axle. It can be seen, that the chosen reference point has a significant impact on the tracking performance. The low cure radius in the drieway results in a significant change of dy along the length. The predicted lateral displacement at the center of graity differs from the measured one at the front axle which results in a deteriorated tracking performance. Figure 9 shows the simulation results for cure radii until 101 m. To ealuate the tracking performance also at the limit of driing dynamics, a speed of 20 m/s for the handling track has been selected. Because of the high lateral accelerations, the kinematic single track model replicates the relationship between δ β and ψ only poorly. Figure 8 shows the simulation results for the handling track drieway with three different prediction models as a function of the arc length u. The solid line represents the results with the dynamic single track model and the calculation of dy at the front axle according to figure 2. Figure 9. Handling track with optimized weights. The resulting unprecise prediction of dy causes an increase of the lateral displacement. The dynamic single track model in comparison shows also for higher speeds a good tracking performance. CONCLUSIONS Figure 8. Handling track drieway with optimized weights. In this paper, a model predictie controller for lateral ehicle guidance has been presented, the influence of different prediction models analyzed and a speed dependent weight optimization implemented. The real-time capability has been tested using a dspace MicroAutoBox II. The controller shows at low speeds and tight cures as also during highly dynamic maneuer a stable and smooth reference tracking with low lateral displacement. Further studies will focus and analytical stability analysis of the closed loop system and the tradeoff between tracking performance and driing comfort.
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