Lateral Control for Autonomous Parking System with Fractional Order Controller

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1 JOURNAL OF SOFTWARE, VOL., NO., JUNE 75 Lateral Control for Autonomous Parking System with Fractional Order Controller Qingyang Chen College of Mechatronic Engineering and Automation, National Uniersity of Defense Technology Changsha, Hunan, China Tongtong Chen, Huiying Yu, Jinze Song, Daxue Liu College of Mechatronic Engineering and Automation, National Uniersity of Defense Technology Changsha, Hunan, China Abstract Lateral control in autonomous parking system, which is always at low speed, simultaneously requiring ery small following error, is a difficult problem to resole. To address this problem, this paper introduces the Fractional controller. What s more, to address the parameters tuning problem in design of the controller, an improed genetic algorithm is adopted. With the proposed method, simulation and real ehicle experiments are carried out. Results show that fractional controller can manage the ehicle to follow the preciously and that stably and robustly in autonomous parking process, which is at low speed. Index Terms Autonomous Parking; Fractional Order PI λ D ; Genetic Algorithm; Lateral Control I. INTRODUCTION With the deelopment of society, ehicle is becoming more and more popular in people s life and making people s liing coneniently. Howeer, as more and more people owning priate ehicle, lacking of parking space is a fatal problem. What s more, statistics in indicates that more than % of the ehicle accidents result from parking. To resole this problem, much research about the autonomous parking system has been done for a long decade. Paromtehic and Florencio present the continuous control in column parking system[,]. At the same time, many automobile manufacturers also spend much money and time on autonomous parking system, and there are many useful autonomous parking systems such as: Citroen s C3 City Park autonomous parking system, Valeo s ParkU autonomous parking system, Volkswagen s PAV (Park Assist Vision) autonomous parking system, Lexus LSL autonomous parking system and so on. Howeer, research on autonomous parking system is inchoate in China. Many researchers Supported by the National Natural Science Foundation of China under Grant Nos. 93, 777. Manuscript receied December, ; reised January, ; accepted January,. just focus on simulation of the autonomous parking process and most of them use fuzzy controller to control the ehicle[3,,5]. Concept of fractional order calculus has been proposed for 3 years. Recently, with the deelopment of fractional order calculus, more and more researchers hae considered describing the mechanical system by fractional order state equations. The idea of using fractional order controllers for dynamic systems comes from Oustaloup, who deeloped so-called CRONE controller[7]. Now, many other fractional order controllers hae been proposed, such as PI λ D controller[], TID controller[]. The fractional order controller is remarkably insensitie to change of the parameters and can behae robustly[3]. Een when parameters of the controller or the plant change in a special range, the fractional order controller can still get good performance. In this paper, the autonomous ehicle is a nonlinear system, and accurate dynamic model of the ehicle is complicated and not useful. Simple model is always adopted in real ehicle control, which results in some error in the dynamic model. With this case, traditional integer order controller can not satisfy the high-precision needs in the autonomous parking process. So this paper introduces the use of fractional controller for the lateral control of the autonomous parking system, and discusses the parameters tuning problem in design of the controller based on an improed genetic algorithm. This paper is organized as follows. In section, first we introduce the Fractional controller and how to calculate fractional order differential equation. Section 3 describes the design of the PI λ D controller in autonomous parking system, especially emphasizing the parameters tuning process based on an improed genetic algorithm. Some simulation results are gien, to erify the feasibility of the controller. In section, real ehicle experiments are carried out on an autonomous ehicle in kinds of situations, and results show the expected performance of the controller. Finally, a conclusion is gien in section 5. ACADEMY PUBLISHER doi:.3/jsw...75-

2 7 JOURNAL OF SOFTWARE, VOL., NO., JUNE II. FRACTIONAL ORDER PI λ D CONTROLLER AND FRACTIONAL ORDER CALCULUS First, consider the fractional order calculus gien as following[]: d re( ) > dt D () = re( ) a t = t a ( dτ) re( ) < Among aboe, is a complex number, a represents the lower limit andt represents the upper limit. Fractional order calculus is an expansion of integer order integration and differentiation. The most commonly used definition for the general fractional integrodifferential operator is the GL (Grumwald-Letniko) definition, which is gien by: [( t a)/ h] j adt = lim h ( ) f ( t jh) h j= j () Where [] i implies the integer part, implies a j quadratic polynomial. GL definition shows that fractional order differential operator proides a powerful description of the memory effect in eery substance: result at timet depends on the results from time a tot. So we hae to sae the result at eery time, it s the normal method named global- Memory. Obiously, as the time t increases, memory load and computation time will grow, which makes the computational efficiency to be ery low. Another point to notice is that as the time pointt further away fromt, less influence from time pointt will hae on the result of time point t. So when computing the result at time point t, results that of time far away from time point t can be ignored, and computation can be realized on a fixed time interal. This is so-called short-memory, which results the following expression from (): L [ ] h j Dt f() t ( ) f( t jh)/ h j= j (3) Where[ t L, t] is the fixed time interal, L is a time constant, h is the computation step. The short-memory method can well sole the computation efficiency problem, which makes the fractional order calculus to be useful in real situation. Fractional controller is an expansion of integer order PID controller, and both the rank of integrator and differentiator can be non-integer. The integer order controller is just an instance of fractional order controller. Because of the introducing of rank, λ, history data will be remembered and hae some effect on the temporal answer to compute. So the disturbance from the dynamical model of the plant or the noise from the sensors can be conquered, and to behae robustly. That is ery important to some situations which the accurate model can not be got or the accurate model is too complicated to use. In frequency domain, the transfer function of fractional controller definition is gien by: λ Cs () = Kp + Ki S + Kd S () Where, λ are the rank of the integrator and differentiator, which to be arbitrary nonnegatie real numbers, and Kp, Ki, K are the coefficients of the d normal PID control. III. DESIGN OF FRACTIONAL ORDER PI λ D In the autonomous parking process, the speed is ery low and always to be a constant. Path following mainly depends on the directional control of the autonomous ehicle. So it is reasonable to primary define the autonomous parking process as a lateral control problem of the autonomous ehicle[]. With the properties discussed aboe, we propose the use of fractional PI λ D controller for the lateral control of autonomous ehicle, and the block diagram is shown in Fig.. The fractional controller, model of the drier s delay and ehicle dynamical model is included in the diagram. To address the parameters tuning problem in the design process, an improed genetic algorithm is adopted and details of the algorithm will be discussed below. A. Lateral dynamical model of ehicle First we will discuss the dynamical model of the ehicle in the proposed control block diagram as shown in Fig.. Vehicle is a nonlinear system and the dynamical model is in fact ery complicated. To describe it appropriately, many simple lateral dynamical model hae been proposed, such as degree of freedom model (DOF)[], 3 DOF model, DOF model, 7 DOF model[9] and so on. As DOF model only has little parameters and mainly emphasizes the ehicle lateral control, it is widely used. This paper also adopts the DOF model to describe the motion of the ehicle, and representation of it as following []: m( γ ω β + γω ) + ( k+ k) β ( ak bk) = kδ (5) γ ω I zγ + ω ( ak bk) β + ( a k b k) akδ + = Thereinto, I represents the moment of inertia around z the z axis, m is mass of the ehicle, δ is the steering angle, a is the distance between front axle and the ehicle center of graity(cg) while b is the distance between rear axle and CG, k, k are the front and rear tires cornering stiffness respectiely, β is the sideslip angle of CG point, implies the ehicle speed, γ ω is the angular elocity of CG, and l implies the distance between front axle and rear axle. The transfer function between lateral acceleration and the δ is gien by: ACADEMY PUBLISHER

3 JOURNAL OF SOFTWARE, VOL., NO., JUNE 77 f () t + ks e Fractional order PI λ D controller δ Delay of the drier + Ts h y s Model of ehicle y() t y ς s + () s = G r δ Ts + Ts+ Gr = l ( + K ) lk lk K = ( ), C =, C = l C C mb ma = ( ), T = l + k CC + K T Figure. The proposed control block diagram of autonomous ehicle a+ ηb b+ ηa + C C η ( ) ς = C In this paper, we use stepper motor to drie the actuator, and its dynamical equation is gien by: δ =, (7) [ ] Where implies the turning angle of the actuator, which to be an arbitrary number between[ ]. B. Preiew-follower control Now we will discuss the error signal used for the fractional PI λ D controller, which is shown in Fig. as e. Obiously, the error signal should include the position error and the heading error between the ehicle and the. This paper will compute these errors by the preiew-follower control. Preiew-follower control[] is widely used in path following of autonomous ehicle, and the sketch map of the method is shown in Fig.. As shown in the figure, f(x) represents the planned trajectory and P represents the preiewed single-point on the. y represents the position error between the predicted ehicle trajectory and the preiewed single-point and θ represents the direction error. ( x, y) represents the ehicle position and Y () θ represents the heading of the ehicle at the computation interal. x represents the longitudinal elocity while y represents the lateral speed of the ehicle. T is the preiew time, which comes from the normal human driing model. y and θ are gien by: y = ( f ( x + xt) y yt) *cosθ () θ = arctg ( f ( x + xt )) θ The error signal for the lateral control is gien by the following expression: e= y+ k θ (9) Where k represents the weight between the position error and the heading error. C. Parameters tuning based on an improed genetic algorithm With the proposed control block diagram as shown in Fig., another important problem to resole is the design of the controller parameters. Now we will discuss the parameters tuning method based on an improed genetic algorithm. Genetic algorithm has been proposed by J.Holland and his colleagues in Michigan Uniersity for about forty years ago. As a simple and robust algorithm, genetic algorithm has been widely used in optimization, machine learning, planning and so on. This paper uses an improed genetic algorithm[] to tune the fractional order controller parameters as following: Coding: In this paper, we start soling the problem from constructing the initial population. As the parameters to be tuned include the normal P, I, D coefficients and the rank, λ, size of each indiidual is 5, and each indiidual is represented as follow: G G G G G G E = [ K, K, K, λ, ] () p i d f ( x) θ L P y O Figure. Sketch map of single point-preiew control. X ACADEMY PUBLISHER

4 7 JOURNAL OF SOFTWARE, VOL., NO., JUNE As the normal genetic algorithm, the initial population is generated by N indiiduals randomly: E kp ki kd λ E kp ki kd λ K = = () N N N N N N E kp ki kd λ Computation of the fitness function: Fitness alue of each indiidual represents the adapting capability to the enironment. In this paper, the fitness function is defined as follow: PO.. <. ( ω et ( ) dt+ ωtu ) () F = PO... ( ω et ( ) dt+ ωtu + *( PO...)) Whereω, ω implies the weights, t u is the setting time, PO.. is the oershoot of the system. In this paper, ω =, ω =. Reproduction: Compare the fitness of this generation and next generation. If the best fitness alue f in next generation is smaller than that in this generation, some copy process should be done. Assume that there are k sets in this generation, whose fitness alue is bigger than f next, we will copy the k sets from this generation to next generation, and randomly replace k sets or replace the k sets whose fitness alues are smallest in the next generation. Crossoer: This paper uses the method of arithmetical crossoer. With crossoer, the i th and the ( i + ) th indiiduals are gien by: TE(,:) i = * K( i +,:) + ( )* K(,:) i (3) TE( i +,:) = * K( i,:) + ( )* K( i +,:) Where represents the weight of the i th and the ( i + ) th indiiduals to the generated indiiduals. The crossoer probability p c is defined as follow: by: < ' pc = ( pc pc )( f fag) ' pc f f f fag Among aboe, pc =.9, pc =.. Mutation: The mutation rate p m = next ' pc f fag m ( p p )( f f) ag () pm in this paper is gien p f < f ag m m pm f fag f fag (5) Among aboe, pm =., pm =.. End: The algorithm will terminate when the best indiidual conerge in some threshold. When the ending condition is reached, the set whose fitness alue is the biggest is the best result. D. Simulations To erify the feasibility of the fractional controller with the improed genetic algorithm for the lateral control of the autonomous ehicle, simulation results will be gien in this section. This paper uses the single-point preiew control method and the improed genetic algorithm aboe to tune the fractional controller parameters. Table shows the correlatie parameters of the ehicle. To illustrate the predominance of the fractional controller, this paper uses fractional controller and integer order PID controller to simulate the block diagram in Fig. respectiely. Fig. 3 shows the conergence process of the genetic algorithm, represented by the best fitness alue and Fig. illustrates the simulation results of the unit-step response of these controllers, corresponding to the parameters computed from the genetic algorithm. From Fig. can see, oershoot of the system without controller is 33.5% and the setting time is.s, oershoot of the system with integer order PID controller is 9% and the setting time is.3s. Howeer, with fractional controller, oershoot of the system is.%, and the setting time is.s. Obiously, for the nonlinear plant with time delay, the performance of fractional order controller is better than that of the integer order controller. Another aspect, from Fig. 3 can see, een the best fitness alue of the initial population is far from the final result, it can get close to the answer quickly and finally stably at the best solution. So the improed genetic algorithm can well adjusted to the need of the parameters tuning problem. Reciprocal alue of Best Fitness TABLE I. PARAMETERS OF THE VEHICLE Parameter Value CC, /( ms / ) 3.7,55. /( km/ h).5 Th / s. ab, /( m ).,. η. k Time Figure 3. Conergence of the best fitness alue ACADEMY PUBLISHER

5 JOURNAL OF SOFTWARE, VOL., NO., JUNE 79. lateral control of the ehicle in kinds of situations at low speed, that to well satisfy the need of autonomous parking system.. cure of the unit-step response.... step response without controlloer step response of integer order controlloer step response of fractional order controlloer 3 5 time(s) Figure. Comparison of results of unit-step response IV. EXPERIMENT RESULTS Autonomous parking experiments are carried out on an autonomous ehicle as shown in Fig. 5. The ehicle is modified for autonomous task, and the subsystem of brake, fuel and steering is controlled by computer. Detection of the parking space is fulfilled by some ultrasonic sensors equipped on the ehicle. Width of the ehicle is.5m, and length of it is.m. Length of the parking space between two ehicles is 7.5m, which is normal for human drier. In the autonomous parking process, nominal speed of the ehicle keeps at.5km/h. Experiment results with fractional controller for the lateral control of the ehicle is shown in Fig., and the sampling period is.s. Because the process of autonomous parking lasts only seeral seconds, this paper uses the method of short- Memory to compute the fractional order calculus approximately and the mnemonic time lasts for s, so to meet the real time need of the control process. Fig. (a)-(c) show the experiment results in different floor situations, such as on the flat ground or uphill and downhill. From these results we can see that, the ehicle can follow the well in these normal situations. Another difficult case to mention is shown in Fig. (d), in which heading of the ehicle is not parallel to the parking space, but with a small angle. Howeer, the fractional controller can still mange the ehicle to follow the preciously and robustly. Other experiments such as in snow enironment or rainy day are carried out and results are shown in Fig. (e) and Fig. (f). In these situations, the dynamical model of the ehicle can be inaccurate and changeable, especially the force from the ground to the ehicle, making the lateral control of the ehicle more challengeable. Howeer, from these experiment results, we can see that the fractional controller for the lateral control of ehicle can still get good performance. Aboe all, the fractional controller can perform robustly and preciously for the Figure 5. The autonomous ehicle used in the parking experiments V. CONCLUSION We hae presented a noel fractional controller for the lateral control of the ehicle in the process of autonomous parking. To address the parameters tuning problem in design of the controller, an improed genetic algorithm is incorporated. Simulation and real ehicle experiment results show that the controller performs preciously and robustly. The ehicle can follow the well at low speed, een when the ehicle is at different state, such as the ehicle parallel to the parking space or not, the ehicle is at uphill or downhill, or the flat ground, so can meet the need of autonomous parking system. REFERENCES [] Paromtehic I. Steering and elocity commands for parking assistance[c].proceedings of the th IASTED International Conference on Robotics and Applications. : 7-3. [] Florencio M, Agostinho P,Sequeira J. S. Automatic parallel Parking of a car-like robots[c].37th International Symposium on Robotics.. [3] RUAN Jiu-hong,FU Meng-yin, LI Yi-bin,DENG Zhihong. Research of Intelligent Vehicle Lateral Control Algorithm Simulation Based on Fuzzy Logic and GA[J]. Journal of system simulation, Vol. No., Oct.. [] LIN Rui shen, WU Zhi jian, YAO Bi zheng. Multidimensional Fuzzy Controller of Truck Backer-upper Control System [J]. Journal Shanghai uniersity (Nature science edition), Vol. 5, No., Aug [5] Zhanjiang Li. Study on the Fuzzy Control of Automotie Automatic Parking [D]. Changchun: thesis for the degree of Master of Engineering, Jilin Uniersity, 7. [] I.Podlubny. Fractional-order systems and PI λ D - controller [J]. IEEE transactions on automatic control [7] A.Oustaloup, X.Moreau, M.Nouillant. The CRONE control of resonant plants: application to a flexible transmission. European Journal of Control.995. [] Konghui Guo. Vehicle Handling Dynamics[M]. Changchun: Jilin Science and Technology Press, 99. ACADEMY PUBLISHER

6 JOURNAL OF SOFTWARE, VOL., NO., JUNE [9] Yucheng Lei. Dynamics and Simulation of Automotie System[M]. Beijing: National Defence Industry Press, 997. [] Huang Yourui, Qu Liguo. Realization and tuning of PID controller[m]. Beijing: Science Press,. [] Song Jinze. Research on the key technology for autoparking system[d]. Changsha: thesis for the degree of Doctor of Engineering, National Uniersity of Defense technology. 9. [] D. Xue, Y.Q.Chen. A Comparatie Introduction of Four Fractional Order Controller[C], Proc. of the th IEEE World Congress on Intelligent Control and Automation(WCICA), Shanghai China, June -,. [3] Zeng Qingshan. Research on Fractional-order Control Systems and Its Application to Molten Carbonate Fuel Cell[D]. Shanghai, thesis for the degree of Doctor of Engineering, Shanghai Jiaotong Uniersity...5 x 5 9 x expected expected (a) (b).3 x 5. x expected expected steering angel (c) (d). x.75.5 x expected.3.5 expected (e) Figure. Real ehicle experiment results in different situations: (a) the ehicle moes on the flat ground and heading of the ehicle is parallel to the parking space. The left image shows the tracking performance of the and the right one shows the planned ehicle trajectory and the. (b) The ehicle moes at a downhill situation. (c) The ehicle moes at an uphill situation. (d) The ehicle moes on the flat ground but heading of the ehicle is not parallel to the parking space. (e) The ehicle moes in the snow enironment. (f) The ehicle moes on the wet ground in rainy day. (f) ACADEMY PUBLISHER

7 JOURNAL OF SOFTWARE, VOL., NO., JUNE Qingyang Chen was born in 9. He is a Ph.D. candidate in the College of Mechatronic Engineering and Automation at National Uniersity of Defense Technology. His research areas are intelligent ehicle, robot control and motion planning. Tongtong Chen was born in 97. He is a Master candidate in the College of Mechatronic Engineering and Automation at National Uniersity of Defense Technology. His research areas are enironment perception and robot control. Huiying Yu was born in 97. She is an instructor at National Uniersity of Defense Technology. Her research areas are information processing and communication engineering. Jinze Song was born in 979. He is a Dr at National Uniersity of Defense Technology. His research areas are intelligent ehicle, robot control and motion planning. Daxue Liu was born in 97. He is a Dr at National Uniersity of Defense Technology. His research areas are intelligent ehicle, robot control and enironment perception. ACADEMY PUBLISHER