ARX663667 1 13 Research Article Data driven automatic parking constrained control for four wheeled mobile vehicles Wenxu Yan1, Jing Deng1, and Dezhi Xu1 Abstract In this article, a novel data driven c[.]
Research Article Data-driven automatic parking constrained control for four-wheeled mobile vehicles International Journal of Advanced Robotic Systems November-December 2016: 1–13 ª The Author(s) 2016 DOI: 10.1177/1729881416663667 arx.sagepub.com Wenxu Yan1, Jing Deng1, and Dezhi Xu1 Abstract In this article, a novel data-driven constrained control scheme is proposed for automatic parking systems The design of the proposed scheme only depends on the steering angle and the orientation angle of the car, and it does not involve any model information of the car Therefore, the proposed scheme-based automatic parking system is applicable to different kinds of cars In order to further reduce the desired trajectory coordinate tracking errors, a coordinates compensation algorithm is also proposed In the design procedure of the controller, a novel dynamic anti-windup compensator is used to deal with the change magnitude and rate saturations of automatic parking control input It is theoretically proven that all the signals in the closed-loop system are uniformly ultimately bounded based on Lyapunov stability analysis method Finally, a simulation comparison among the proposed scheme with coordinates compensation and Proportion Integration Differentiation (PID) control algorithm is given It is shown that the proposed scheme with coordinates compensation has smaller tracking errors and more rapid responses than PID scheme Keywords Data-driven constrained control, automatic car parking, dynamic anti-windup, coordinates compensation Date received: 29 April 2016; accepted: 21 June 2016 Topic: Special Issue - Manipulators and Mobile Robots Topic Editor: Michal Kelemen Introduction Nowadays, with the fast development of science and technology and increase of people’s living standard, the number of cars is increasing very fast The space of car parking significantly reduced, and traffic accidents are particularly prone to happen when backing the car for these drivers who have less experience The university of Michigan study shows that manual parking is one of the important reasons why the driver is easy to cause the traffic accident.1 Thus, the automatic parking problem has become a hot research topic Automatic parking system can avoid parking accident It uses the sensor technology, computer technology, and automatic control technology to accurately perceive the parking environment and planning a optimal parking path and then lead the vehicle to the target by controlling the vehicle automatically complete the path tracking.2 Parking types include parallel parking, vertical parking, and oblique parking, as shown in Figure Parallel parking is most common in our daily life, so the parallel automatic parking system has become a hot spot of current research.3,4 Therefore, the work of this article is to study the parallel parking Parallel parking consists of three procedures5: (1) parking space detection, (2) path planning, and (3) path tracking Once drivers choose the automatic parking mode, the width and length of the parking space are detected by School of Internet of Things Engineering, Institute of Electrical Engineering and Intelligent Equipment, Jiangnan University, Wuxi, China Corresponding author: Dezhi Xu, School of Internet of Things Engineering, Institute of Electrical Engineering and Intelligent Equipment, Jiangnan University, Wuxi 214122, China Email: xudezhi@jiangnan.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 Figure Classification of parking: (a) parallel parking, (b) vertical parking, and oblique parking ultrasonic sensors mounted on the side of the car When the drivers stop a car at the beginning position, environmental geometry and target position are calculated automatically from the sensor information Then, a target parking trajectory is created by the automatic parking controller and the car is guided into the target parking space by controlling the electronic power steering.6 Four-wheeled vehicle is the most commonly used in our daily life Its dynamics model can be transformated into nonholonomic chained system,7 and the corresponding control algorithm can be designed based on the nonholonomic chained system But the automatic parking process has the characteristics of nonlinearity, time variation, multivariable, and so on It is difficult to establish an accurate model of the nonholonomic chained system Even if the model is established, the corresponding control algorithm is very complex and the computation is very large So it is difficult to popularize this method In the study by Chen et al.,8 the fuzzy-Proportion Integration Differentiation (PID) controller is designed for automatic car parking system, the front wheel angle is the output of the controller, and the speed error is the input of the controller In the study by Liang et al.,9 the automatic parking system is designed based on the fuzzy controller by adopting fuzzy control algorithm The vehicle coordinate position error and the orientation angle of the car error are the output of the controller, and the front wheel angle is the input of the controller Different fuzzy logic controllers are designed for different parking situations in the study by Li et al.,10 and the experimental verification is done on the car robot that on the basis of field programmable gate array In the study by Baturone et al.,11 the fuzzy controller is designed by adopting the position coordinate of the car, the direction of the vehicle body, vehicle speed, and trajectory curvature as inputs, and the marching direction, vehicle speed, and the target curvature size as outputs The experiment is done on a car robot In the study by Heinen et al.,12 the authors proposed a control method based on neural network by adopting the position coordinate of the car, the orientation angle of the car as the input of artificial neural network, and the vehicle speed and the front wheel angle as the output of International Journal of Advanced Robotic Systems artificial neural network The authors realized the virtual process of automatic parking in the simulation software But there are many types of cars on the market For each new type of the car, the parameters of PID controller need to be adjusted, the fuzzy control rules should be reformulated, and the neural network control need to be retrained Hence, the automatic parking system established by the above algorithm is poor in portability The data-driven control methods have been applied in several fields.13–16 Model-free adaptive control (MFAC), one of the data-driven control methods, was put forward by Hou In MFAC, combined with a innovative concept named pseudo partial derivative (PPD), an equivalent dynamic linearization data model is developed First, it takes the place of the general discrete time nonlinear system at each current operating point Next, the estimation of PPD is only conducted online through the input/output (I/O) data from the controlled system Eventually, the controller is designed At present, the MFAC method has been successfully applied in the fields of smart grid, chemical industry, welding process, wind power generation, urban expressway, artificial heart rate regulation, and so on.17–23 Theoretical analysis, simulation, and practical application show that the MFAC method is simple and practical, small computational burden, and strong robustness; it can deal with the control problem of unknown nonlinear time-varying systems.24 In this article, we propose a novel data-driven constrained control scheme with coordinates compensation for automatic parking system The proposed scheme consists of constrained control algorithm, parameter estimation algorithm, parameter reset algorithm, and coordinates compensation algorithm The design of the control scheme is only based on the I/O data of the automatic parking system, which does not include the vehicle model information Therefore, the proposed scheme is applicable to different vehicle types The proposed scheme gets a better control effect by adopting the coordinates compensation algorithm Because the control input is subject to change magnitude and rate constraints, a novel dynamic anti-windup compensator is used to deal with the saturation problem The simulation results have verified the superiority of the proposed constrained control scheme Problem formulation for automatic parking process Parking space detection The diagram of parking space detection is shown in Figure There are ultrasonic probes on the vehicle The ultrasonic probes in the front and rear of the vehicle are used to ensure parking safety The ultrasonic probes mounted on the side of the vehicle are used to detect the parking space Start the automatic parking system when the vehicle is ready to pull over, and the system detects the distance information by the ultrasonic probes mounted on Yan et al Figure Vehicle model P3 P which goes through the beginning location and is parallel to the x-axis Then make a circle O that is tangential to the straight lines P0 P1 and P3 P2 Denote the point of tangency of this circle O and line P3 P2 as P2 and the point of tangency of this circle O and line P0 P1 as P Then the path planning of parallel parking is illustrated as the curve in Figure From this diagram, the angle of line P0 P1 and the coordinates of P and P2 are got through some geometric calculations as Figure Diagram of parking space detection ẳ tan1 0:5Wc ỵ R2 cos R1 ð1 cos Þ Lp R sin R2 sin S P0 : R1 sin ; R1 ð1 cos Þ Figure Path planning of automatic parking the side of the vehicle If there is a vehicle parked at the roadside, the distance detected by the ultrasonic probe mounted on the right side of the vehicle is d1 , otherwise, the detection distance is d2 If there is a parking space, we can get the dashed-dotted line pulse as shown in Figure According to the vehicle speed and pulse duration, the length of parking spaces Lp can be obtained If Lp is greater than the minimal available length of parking space Lp , the parking space is identified as available Otherwise, the vehicle will continue to move forward to find other available parking spaces After finding the available parking spaces, the systems enter the stage of path planning R =cos R ỵ d ỵ 0:5W 1 c ; d ỵ 0:5Wc A P2 : @ tan The path planning of parallel parking consists of the following procedures The given parameters are the vehicle width Wc , the vehicle length Lc , and the length of parking space Lp (refer to Figure 3) Make a circle O at the center of (0, R ) and with a radius of R1 This circle is tangent to the x-axis at the origin O Make a circle O at the center of the corner O2 and with a radius of (3) (4) where d is the lateral distance between the two vehicles as shown in Figure Based on Figure 3, the minimal length of parking space Lp is given when Lp p ¼ Lc Lp ẳ R ỵ R2 ị sin þ Lc cos (5) where Lp0 p denotes the length of line P0 P1 , and for the , we consider the following equation with Lc sin ẳ R1 ỵ R2 ịcos ỵ 0:5Wc R1 Path planning (2) (6) Path tracking (1) Let us consider the four-wheeled car shown in Figure Obey the assumption that sideslip does not happen, the state space equation of four-wheeled car25,26 is obtained as x_ ¼ v cos > > > > < y_ ¼ v sin (7) > v tan > _ > > :¼ L where S is the safe distance between the two cars Draw the common tangent P0 P of both circles O1 and O ; the point of tangency with O1 is denoted as P0 and the point of tangency with O is denoted as P1 Draw a line where (x, y) is the coordinate of the center of the rear wheel axle, L denotes the wheel base, is the orientation angle of the car, and v and are the vehicle speed and the steering angle, respectively R2 ¼ 0:5Wc ỵ S International Journal of Advanced Robotic Systems In the practical application, equation (7) is usually transformed into the following discrete model xk ỵ 1ị ẳ xkị þ Tv cosððkÞÞ > > > > < yðk þ 1ị ẳ ykị ỵ Tv sinkịị (8) > Tv tankịị > > > : k ỵ 1ị ẳ kị ỵ L where T denotes sampling time In addition, the input ðkÞ of the vehicle cannot change too fast within a small time interval So the control input is subject to change magnitude and rate constraints as following ðkÞ max _ _ max _ ðkÞ (9) Based on the state space equation of four-wheeled car, the coordinate of the car ðx; yÞ is determined by the orientation angle of the car with little change in the parking speed Therefore, ensure the coincidence of the coordinate of the car and target trajectory by controlling the steering angle to control the orientation angle of the car is the control objective of automatic parking system kị ỵ Fe kị e k ỵ 1ị ẳ T kị~ where ~kị ẳ kị ^ðkÞ denotes the error of parameter estimation The adaptive update algorithm for the estimated parameters ^ðkÞ is chosen as ^k ỵ 1ị ẳ ^kị ỵ kịkị e k ỵ 1ị Fe kị (14) The gain kị is chosen as: 1 kị ẳ jjkịjj ỵ (15) where is a positive constant; hence, ðkÞ is positive definite for all the k Noticed that, with the assumption that jjðkÞjj g, ðkÞ can be lower bounded as jjkịjj ẳ >0 g2 þ In view of equations (13) and (14) and condition k ỵ 1ị kị, which denotes the slow time-varying system, the estimation dynamic error is obtained as ðkÞ ỵ Fe kị e k ỵ 1ị ẳ T kị~ ~k ỵ 1ị ẳ H ~kị Main results (13) (16) where H is given by In this section, we will put forward a novel data-driven constrained control scheme with coordinates compensation for automatic car parking system Major contributions in the following works are (1) an unknown PPD estimation algorithm, (2) proposed a data-driven constrained control algorithm via anti-windup scheme, and (3) the coordinates compensation algorithm of the target orientation angle of the car Theorem The equilibrium ½e ; ~T T ẳ ẵ0; 0T 21 T of the system (16) is globally uniformly stable Furthermore, the estimation error eo ðkÞ converges asymptotically to PPD parameter estimation algorithm Proof Consider the Lyapunov function First of all, the nonlinear system of automatic parking can be transformed into the following dynamic linearization data model k ỵ 1ị ẳ kịkị ỵ kịkị (10) where and are defined as the orientation angle of the car and the steering angle, respectively Parameters ’1 and ’ are called PPD k ỵ 1ị ẳ k ỵ 1ị kị and kị ẳ kị k 1ị The model (10) can be redescribed as k ỵ 1ị ẳ kị ỵ T kịkị (11) where kị ẳ ẵkị; kịT and kị ẳ ẵ kị; kịT The proposed parameter identification observer has the following structure: ^k ỵ 1ị ẳ ^kị ỵ T kị^ kị ỵ Ke kị (12) where e kị ẳ kị ^kị is the estimation error of out^ ðkÞT , and we choose the gain K as put, ^kị ẳ ẵ^ kị; making F ẳ K in the unit circle Therefore, by taking into account equations (11) and (12), the estimation dynamic error of output can be given by H ẳ I2 kịkịT kị and I denotes the ð2 2Þ identity matrix V1 kị ẳ Pe kị ỵ ~ T kị~ ðkÞ (17) where and P are positive constants and P is calculated by P F P ¼ Q with Q is a positive constant By taking into equation (16), we have V1 kị ẳ V1 k ỵ 1ị V1 ðkÞ ðkÞ~ T ðkÞðkÞ 2PFT ðkÞ~ ðkÞe kị ẳ PT kị~ ỵ PF eo2 kị þ ~T ðkÞð H T H Þ~ ðkÞþ Pe 20 kị ẳ Qe 20 kị T kịẵ T kịkị PðkÞ 2PFe ðkÞðkÞ Qje ðkÞj ẵ T kịkị Pjjkịjj ỵ 2PFjje kịjjkịjj c je ðkÞj c jjkịjj where kị ẳ T kị~ kị, c ¼ Q ð1= Þ, and c ¼ P P2 F Hence, V1 ðkÞ provided that , Q, and satisfy the following inequalities Q> ; P P2 F > Yan et al Noticed that V1 ðkÞ is negative definite in the variables eo ðkÞ and ðkÞ Since V ðkÞ in a decreasing and nonnegative function, it converges to a constant value V1 0, as k ! 1; hence, V ðkÞ ! This means that both eo ðkÞ and ~ðkÞ remain bounded for all the k, and limk!1 eo kị ẳ In order to make the parameter estimation, law (14) have a stronger capability in tracking time-varying parameter, a reset algorithm which13 should be considered is as follows ^ kịT ẳ ẵ^ ^ 1ịT ẵ^ ðkÞ; ’ ’1 ð1Þ; ’ " # ^ ðkÞ ^ kị if ẵ^ kị; 1ịị &; or sgn^ kịị 6ẳ sgn^ ^ ðkÞ ’ (18) ^ ð1Þ and ’ ^ ð1Þ where & is a small positive constant and ’ ^ ðkÞ and ’ ^ ðkÞ, respectively are the initial value of ’ A data-driven constrained control algorithm and stability analysis Observer tracking error is defined as ekị ẳ kị ^kị kị, under the dynamic constraints (9), and combining with (21), thus ek ỵ 1ị ẳ k ỵ 1ị ^k ỵ 1ị k ỵ 1ị ^ kịkị ẳ k ỵ 1ị ^ðkÞ ’ ^ ðkÞðkÞ Ke ðkÞ ðkÞ ’ ^ ðkÞð ðkÞ kịị ^ kịkị ẳ k þ 1Þ ^ðkÞ ðkÞ ’ ^ kị kị k ỵ 1ị Ke ðkÞ ’ (23) Substituting equation (22) into equation (23), we obtain k ỵ 1ị ^kị kị ek ỵ 1ị ẳ ^ kị ỵ ^ kịkị Ke ðkÞ ’ (24) On the basis of the observer (12), the data-driven unconstrained control algorithm is computed as kị ẳ k 1ị ^ kịkị ^ kị k ỵ 1ị ^kị Ke0 kị ỵ ^ 22 kị ỵ (19) where kị is the target orientation angle of the car is a given small finite positive number By taking into account the input constraints (9), the adaptive constrained controller is described as the following tị ẳ Sat k 1ị ỵ Satf ðkÞ ðk 1ÞÞ; (20) T _ ; T _ max ; ; max where T denotes the sampling time, and Satð Þ function is defined as > : c ac Because of the dynamic constraints in the close-loop control system, an anti-windup compensator is designed to adapt to the target orientation angle of the car ðkÞ The compensation signal ðkÞ is designed as follows The convergence and tracking performance analysis for data-driven constrained control law (20) and (22) are shown in theorem Theorem For given j ðkÞ ðk 1Þj , by using the data-driven constrained control law (20) and (22), the solution of close-loop observer error system (24) is uniformly ultimately bounded (UUB)27 for all the k with ultimate bound limk!1 jeðkÞj a2 =ð1 a1 Þ where is a given positive constant a1 ¼ ^ 22 kị ỵ a2 ẳ ^ kịkịj j ỵ ð1 Þ ðkÞ Ke ðkÞ ’ ^ 22 kị ỵ Proof Considering the absolute value of equation (24), it turns into jek ỵ 1ịj ẳ ^ 22 kị ỵ j k ỵ 1ị ^kị ðkÞ ^ ðkÞðkÞj Ke ðkÞ ẳ j k ỵ 1ị kị þ eðkÞ ^ ðkÞ þ ’ (21) ^ kịkịj ỵ ị kị Ke kị jekịj ỵ j ^ kị ỵ ^ kị ỵ where lies in unit circle Therefore, the controller (19) can be redescribed as ^ kịkịj ỵ ị kị Ke kị ẳ a1 jekịj ỵ a2 ^ kị kị kịị k ỵ 1ị ẳ kị ỵ ^ kị kị ẳ k 1ị ỵ 2 ^ kị ỵ ’ ^ ðkÞðkÞ ðk 1Þ ^ðkÞ ðkÞ Ke0 ðkÞ ’ (22) (25) Considering a Lyapunov function as V2 kị ẳ jekịj, from equation (25), one has V2 k ỵ 1ị ẳ jek ỵ 1ịj jekịj ẳ a1 ịV2 kị ỵ a2 International Journal of Advanced Robotic Systems Since a1 < and a2 is bounded, based on the lemma in,27 by using the constrained control law (20) and (22), the result of close-loop observer system (24) is UUB for all the k with ultimate bound limk!1 jeðkÞj a2 =ð1 a1 Þ Corollary Based on the controller (20) and (22), together with the observer (12) and adaptive laws (14), we can ensure that the system tracking error ekị ẳ kị kị is UUB with ultimate bound limk!1 j eðkÞj a2 =1 a1 ị Proof Since ekị ẳ ekị e ðkÞ (26) Considering the absolute value and limiting on both sides of eqaution (26), we obtain a2 lim j ekịj lim je kịj ỵ lim jekịj k!1 k!1 k!1 a1 (27) Hence the tracking error eðkÞ is UUB for all the k with ultimate bound limk!1 j eðkÞj a2 =ð1 a Þ The controller (22) can be divided into two parts: feedback control and feedforward control, which is described as kị ẳ 01 kị ỵ 02 kị where Feedback control: 01 kị ẳ k 1ị ỵ ^ kị ^ kị þ ’ ^ ðk þ 1Þ ðkÞ ðkÞ Ke ðkÞ Feedforward control: 02 kị ẳ ^ kị^ kịkị ^ kị ỵ Hence, we obtain the novel data-driven constrained control scheme of automatic parking system that consists of eqautions (14), (18), (20), and (22) The compensation algorithm of the target orientation angle of the car In the actual control process, there often exists tracking error between the actual orientation angle of the car and the target orientation angle of the car Hence, there will exist steady-state error between the actual parking trajectory and the target trajectory In order to eliminate the steady-state error, we will propose a coordinates compensation algorithm for automatic car parking systems The data-driven constrained controller corrects the steering angle according to the difference between the target orientation angle of the car and the current orientation angle of the car The compensation theory of the target orientation angle of the car is shown in Figure Backing a car from point E Figure The compensation theory of the target orientation angle of the car to point F The solid curve EF denotes the target trajectory and the dashed-dotted line EP(k) denotes the actual parking trajectory Point P(k) and point P k ỵ 1ị are defined as the current location of the car and the target location of next moment, respectively k ỵ 1ị and gkị are defined as the target orientation angle of the car of next moment and the orientation angle from point P k ỵ 1ị to point PðkÞ, respectively From Figure 5, we know that the actual position of car has separated from the target trajectory Hence, we propose the compensation algorithm of the target orientation angle of the car as the following ~ðk ỵ 1ị ẳ k ỵ 1ị ỵ gkị k ỵ 1ị (28) where is a positive adjustable parameter, used to adjust the intensity of compensation ~k ỵ 1Þ denotes the compensated target orientation angle of the car For the gðkÞ, we consider the following equation with ykị y k ỵ 1ị A (29) gkị ẳ arctan@ xkị x k ỵ 1ị where ðxðkÞ; yðkÞÞ denotesthe current coordinate of the car and x k ỵ 1ị; y k ỵ 1ị denotes the target coordinate of the car of next moment Hence, we obtain the novel data-driven constrained control algorithm with coordinates compensation as the following ^k ỵ 1ị ẳ ^kị þ ðkÞðkÞ e ðk þ 1Þ Fe kị (30) ^ kịT ẳ ẵ^ ^ 1ịT ẵ^ ’ ðkÞ; ’ ’ ð1Þ; ’ " #2 ^ ðkÞ ’ ^ ðkÞ &; or sgn^ kịị 6ẳ sgn^ if ẵ^ ðkÞ; ’ ’ ð1ÞÞ ^ ðkÞ ’ (31) ykị y k ỵ 1ị A gkị ẳ arctan@ xkị x k ỵ 1ị (32) Yan et al Figure Flowchart of the novel data-driven constrained control system design procedure for automatic parking gkị k ỵ 1ị Table Basic parameters of FAW-VW CC Parameter Representation Lc Wc L Lp max S Vehicle length Vehicle width Wheel base Length of parking space Maximal steering angle Safe distance ~k ỵ 1ị ẳ k ỵ 1ị ỵ Value 4.799 m 1.855 m 2.712 m 5.600 m 42.00 0.500 m kị ẳ k 1ị ^ kị ~k ỵ 1ị ^kị kị Keo kị ^ kịkị ỵ ^ 22 kị ỵ (34) Orientation angle of the car (°) 50 40 42.5 42 41.5 41 40.5 30 3.2 3.4 3.6 θ* PID algorithm proposed algorithm 43.5 43 42.5 42 41.5 16.2 20 16.4 16.6 16.5 12.5 10 16 12 2.7 2.8 2.9 11.5 11 18.8 19 19.2 −10 10 Time (s) 12 14 16 18 20 Figure The orientation angle of the car of FAW-VW CC automatic parking 20 proposed algorithm PID Steering angle β (°) 15 10 −5 (33) Figure The steering angle of FAW-VW CC automatic parking 10 Time (s) 12 14 16 18 20 International Journal of Advanced Robotic Systems 0.1 proposed algorithm PID 0.05 x−Axis error (m) −0.05 −0.1 −0.15 −0.2 −0.25 10 Time (s) 12 14 16 18 20 Figure Tracking error of x-axis 0.6 proposed algorithm PID 0.5 y−Axis error (m) 0.4 0.3 0.2 0.1 −0.1 10 Time (s) 12 14 16 18 20 Figure 10 Tracking error of y-axis proposed algorithm PID Error of θ (°) −1 −2 10 Time (s) 12 14 16 18 20 Figure 11 Tracking error of orientation angle of the car tị ẳ Sat k 1ị ỵ Satf kị k 1ÞÞ; _ (35) T ; T _ max g ; ; max In order to give a clear idea of the comprehensive proposed novel data-driven constrained control system design procedure for automatic parking, we provide a flowchart as shown in Figure Simulation results In order to verify the superiority of the novel data-driven constrained control scheme with coordinates compensation, a simulation comparison among the proposed constrained control scheme with coordinates compensation and PID control scheme is given for different two cars with constant parking speed PID controller adopts the position control algorithm as the following Yan et al x 10 −3 Compensation signal ρ(k) −2 −4 −6 −8 −10 −12 10 Time (s) 12 14 16 18 20 12 14 16 18 20 18 20 Figure 12 Compensation signal PPD1 1 1 10 Time (s) Figure 13 PPD1 parameter estimation PPD: pseudo partial derivative 0.05 0.05 PPD2 0.05 0.05 0.05 0.05 0.05 0.05 10 Time (s) 12 14 16 Figure 14 PPD2 parameter estimation PPD: pseudo partial derivative Table Basic parameters of Audi A6L Parameter Lc Wc L Lp max S Representation Vehicle length Vehicle width Wheel base Length of parking space Maximal steering angle Safe distance Value 5.015 m 1.874 m 3.012 m 5.600 m 42.00 0.500 m kị ẳ Kp ekị ỵ Ki k X ejị ỵ Kd ekị ek 1ịị (36) jẳ1 where ekị ẳ kị kị denotes the error of the orientation angle of the car Kp , Ki , and Kd denote the proportion parameter, integral parameter, and differential parameter, respectively 10 International Journal of Advanced Robotic Systems Orientation angle of the car θ (°) 50 * 40 θ PID algorithm proposed algorithm 44.5 41 40.5 40 16 30 15.5 2.7 2.8 44 43.5 43 3.2 2.9 3.4 3.6 16 16.2 16.4 16.6 20 13 12.8 12.6 12.4 10 18.9 19 19.1 −10 10 Time (s) 12 14 16 18 20 Figure 15 The orientation angle of the car of Audi A6L automatic parking 20 proposed algorithm PID Steering angle β (°) 15 10 −5 10 11 12 13 14 15 16 17 18 19 20 12 13 14 15 16 17 18 19 20 Time (s) Figure 16 The steering angle of Audi A6L automatic parking 0.1 proposed algorithm PID 0.05 x−Axis error (m) −0.05 −0.1 −0.15 −0.2 −0.25 −0.3 10 11 Time (s) Figure 17 Tracking error of x-axis Here, we choose the optimal PID parameters by taking into account the two car models, after repeated simulation test, and considering the principle of rapidity and small overshoot Parameter settings of the two algorithms are Kp ¼ 4:0, Ki ¼ 0:018, Kd ¼ 0:08, ^ 1ị ẳ 1, ^ 1ị = 0:05, & ¼ 104 , ¼ 0:003, ’ ¼ 0:01, ¼ 10, K ¼ 0.6, ¼ 0.96, ¼ 42 deg, max ¼ 42 deg, _ ¼ 20 deg= s, and _ max ¼ 20 deg= s The simulation consists of two parts: (1) automatic parking simulation of FAW Volkswagen CC (FAW-VW CC) and (2) automatic parking simulation of Audi A6L Yan et al 11 0.6 proposed algorithm PID y−Axis error (m) 0.5 0.4 0.3 0.2 0.1 −0.1 10 11 Time (s) 12 13 14 15 16 17 18 19 20 Figure 18 Tracking error of y-axis proposed algorithm PID Error of θ (°) −1 −2 10 11 Time (s) 12 13 14 15 16 17 18 19 20 Figure 19 Tracking error of orientation angle of the car Compensation signal ρ(k) × 10 −4 −0.2 −0.4 −0.6 −0.8 −1 −1.2 10 Time (s) 12 14 16 18 20 Figure 20 Compensation signal Automatic parking simulation results of FAW-VW CC The basic parameters of FAW-VW CC and parking space are shown in Table We should try to avoid the car steering wheel at the maximum rotation angle to protect the vehicle performance, therefore, the R1 is chosen as follows R1 ¼ L tanð max =1:1Þ (37) The automatic parking simulation results of FAW-VW CC with constant parking speed (v ¼ 1.5 km/h) are shown in Figures to 14 12 International Journal of Advanced Robotic Systems 1 PPD1 1 1 1 10 11 Time (s) 12 13 14 15 16 17 18 19 20 14 15 16 17 18 19 20 Figure 21 PPD1 parameter estimation PPD: pseudo partial derivative 0.05 0.05 0.05 PPD2 0.05 0.05 0.05 0.05 0.05 0.05 10 11 Time (s) 12 13 Figure 22 PPD2 parameter estimation PPD: pseudo partial derivative From Figure 7, we observe that the two schemes can both achieve the tracking of the target orientation angle of the car in the automatic parking process By the local amplification as shown in Figure 7, it can be seen that the tracking effect of the novel data-driven constrained control scheme with coordinates compensation is obviously better than PID control scheme From Figure 8, we observe that the steering angle of the two schemes is consistent with the change of the orientation angle of the car in the automatic parking process The front wheels of the two schemes are not swinging back and forth, which makes the parking process smooth The parking error comparisons of FAW-VW are shown in Figures to 11 From these figures, we can see clearly that the tracking error of the novel datadriven constrained control scheme with coordinates compensation is smaller than PID control scheme The parking error comparisons verified the validity of the coordinates compensation algorithm The anti-windup compensation signal ðkÞ is shown in Figure 12 In the whole tracking process, tuning PPD parameters are shown in Figures 13 and 14 Automatic parking simulation results of Audi A6L The basic parameters of Audi A6L and parking space are shown in Table R1 is same as equation (37) The automatic parking simulation results of Audi A6L with constant parking speed (v ¼ 1.5 km/h) are shown in Figures 15 to 22 From the simulation comparisons, we observe that the novel data-driven constrained control scheme with coordinates compensation has smaller tracking errors and faster response speed than PID scheme Conclusion A novel data-driven constrained control scheme is proposed for automatic car parking systems The design of the proposed scheme only depends on the steering angle and the orientation angle of the car, and it does not involve any model information of the car A coordinates compensation algorithm is also proposed, which reduces the tracking errors of proposed scheme A dynamic constraints unit with antiwindup compensator is designed to accommodate the reference trajectory ðkÞ The proposed constrained control algorithm has the real-time implementation advantages of Yan et al 13 not requiring any iterative calculations The simulation results have verified the superiority of the proposed scheme 11 Acknowledgements The authors sincerely thank the editor and all the anonymous reviewers for their valuable comments and suggestions 12 Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article 13 Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (61503156, 61403161, and 51405198) and the Fundamental Research Funds for the Central Universities (JUSRP11562 and NJ20150011) 14 15 16 References Green P Parking crashes and parking assistance system design: evidence from crash databases, the literature and insurance agent interviews In: 2006 SAE world congress, Detroit, MI, 3–6 April 2006 Sazawa M and Ohishi K Fast continuous path tracking control considering high precision and torque saturation In: IEEE Conference on Industrial Electronics, Porto, 3–5 November 2009, pp 3089–3094 Cheng K, Zhang Y and Chen H Planning and control for a fully-automatic parallel parking assist system in narrow parking spaces In: IEEE Conference on Intelligent Vehicles Symposium, Gold Coast, QLD, 23–26 June 2013, pp 1440–1445 Run RS, Yen JC and Chen C A study of automatic parallel parking system-from the viewpoints of user and manufacturer In: IEEE Conference on Circuits, Communications and System, Wuhan, China, 17–18 July 2011, pp 1–5 Jung HG, Lee YH, and Kim J Uniform user interface for semiautomatic parking slot marking recognition IEEE Trans Veh Technol 2010; 59(2): 616–626 Kim D, Chung W, and Park S Practical motion planning for car-parking control in narrow environment IET Control Theory Appl 2010; 4(1): 129–139 Liu K-Z and Horii M An experimental comparison of nonholonomic control methods: automatic parking benchmark In: IEEE Conference on Control Conference, Kos, 2–5 July 2007, pp 2039–2044 Chen CH, Hsu CW, and Yao CC A novel design for full automatic parking system Taipei, China, 2012 Liang Z, Zheng G, and Li J Application of fuzzy control strategy in automatic parking path planning Nanchang, China, 2012 10 Li THS, Chang S-J, and Chen Y-X Implementation of human-like driving skills by autonomous fuzzy behavior 17 18 19 20 21 22 23 24 25 26 27 control on an FPGA-based car-like mobile robot IEEE Trans Ind Electron 2003; 50(5): 867–880 Baturone I, Moreno-Velo FJ, Sanchez-Solano S, et al Automatic design of fuzzy controllers for car-like autonomous robots IEEE Trans Fuzzy Syst 2004; 12(4): 447–465 Heinen MR, Osorio FS, Heinen FJ, et al SEVA3D: using articial neural networks to autonomous vehicle parking control In: IEEE Conference on Neural Networks, Vancouver, Canada, 16–21 July 2006, pp 4704–4711 Hou Z and Jin S A novel data-driven control approach for a class of discrete-time nonlinear systems IEEE Trans Control Systems Technol 2011; 19(6): 1549–1558 Hou Z and Jin S Data driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems IEEE Trans Neural Netw 2011; 22(12): 2173–2188 Hou Z and Wang Z From model-based control to data-driven control: survey, classification and perspective Inf Sci 2013; 235: 3–35 Xu JX and Hou ZS Notes on data-driven system approaches, Acta Autom Sinica 2009; 35(6): 668–675 Zhao Y, Lu C, and Han YD MIMO model free adaptive wide area power system stabilizer design J Tsinghua Univ 2013; 53(4): 453–458,464 Xu D, Jiang B, and Shi P A novel model free adaptive control design for multivariable industrial processes IEEE Trans Ind Electron 2014; 61(11): 6391–6398 Xu D, Jiang B, and Shi P Adaptive observer based datadriven control for nonlinear discrete-time processes IEEE Trans Autom Sci Eng 2014; 11(4): 1037–1045 Lv FL, Chen HB, Fan CJ, et al A novel control algorithm for weld pool control Ind Robot 2010; 37(1); 89–96 Lu XP, Li W, and Lin YG Load control of wind turbine based on model-free adaptive controller Trans Chin Soc Agric Machinery 2011; 42(2): 109–114,129 Chi RH and Hou Z A model-free periodic adaptive control for freeway traffic density via ramp metering Acta Autom Sinica 2010; 36(7): 1029–1032 Gao B, Gu KY, Zeng Y, et al An anti-suction control for an intra-aorta pump using blood assistant index: a numerical simulation Artif Organs 2012; 36(3): 275–285 Hou Z and Xu JX On data-driven control theory: the state of the art and perspective Acta Autom Sinica 2009; 35(6); 650–667 Liu KZ, Dao MQ, and Inoue T An exponentially-convergent control algorithm for chained systems and its application to automatic parking systems IEEE Trans Control Syst Technol 2006; 14(6): 1113–1126 Muller B, Deutscher J and Grodde S Continuous curvature trajectory design and feedforward control for parking a car IEEE Trans Control Syst Technol 2007; 15(3): 541–553 Spooner J, Maggiore M, and Ordonez R Stable adaptive control and estimation for nonlinear systems New York: Wiley, 2002 ... 0.2 0 .1 −0 .1 10 11 Time (s) 12 13 14 15 16 17 18 19 20 Figure 18 Tracking error of y-axis proposed algorithm PID Error of θ (°) ? ?1 −2 10 11 Time (s) 12 13 14 15 16 17 18 19 20 Figure 19 Tracking... algorithm PID Steering angle β (°) 15 10 −5 10 11 12 13 14 15 16 17 18 19 20 12 13 14 15 16 17 18 19 20 Time (s) Figure 16 The steering angle of Audi A6L automatic parking 0 .1 proposed algorithm PID 0.05... 12 International Journal of Advanced Robotic Systems 1 PPD1 1 1 1 10 11 Time (s) 12 13 14 15 16 17 18 19 20 14 15 16 17 18 19 20 Figure 21 PPD1 parameter estimation PPD: pseudo partial derivative