Path Following Control for Experimental Used Car-like Robots†

Zhenyuan Yuan1

Abstract— This paper presents a method of path following fora car-like robot in an indoor setting using real-time feedbackfrom motion capture system. In particular, we implement amodified version of Proportional Control for path following.The major difficulty of the problem is due to the car’s non-holonomic motion, which makes path following very difficultdue to the differential constraint so that the car cannot gosideway. Path following is eventually realized by taking a linearcombination of the proportional gains over cross-track errorand heading error. The method has been experimentally testedon a laboratory-scale car-like robot in an indoor environment.The state feedback for the car is provided by a high-fidelity real-time motion capture system which can provide high-frequencyfeedback on the position, velocity as well orientation of the car.The finally tuned controlled is tested on several paths (e.g.,straight paths, paths composed of two parallel lines to simulatethe urban driving scenario of lane switching, etc.).

I. INTRODUCTION

Research on motion planning algorithm is experiencing atremendous growth recently in the field of autonomous cars.These motion planning algorithms, such as Rapidly exploringRandom Tree (RRT) in [1] and the multi-robot motionalgorithm in [2], are planning algorithm that determines theoutput according to the state of the robot. After the pathin generated, the vehicles need to follow the designated paththrough a close loop so that the whole autonomy is achieved.Path following for car-like robots becomes a complicatedissue because of the car’s non-holonomic motion, and thusit is usually implemented through rather complicated al-gorithm, in which the heavy computation requires, eventhough provides accurate regulation, slows down the speedof processing as well as requires high-fidelity on-boardhardware setup. In our lab, motion planning algorithm isalso the topic under research, and we need to verify thealgorithm through experiments on car-like robots by pathfollowing. In order to keep the experiments cost-effectivewhile keeping the dynamics of cars, we use 1/10 Radio-Controlled (RC) car as experimental-used car-like robot andaims to keeps the cost of other setup as low as possibleso that the platform can be replicated easily for future use.Thus in this paper, a computational-ease steering controlleris proposed to implement on car-like robots so that theautonomous platform is kept at low cost but retains goodperformance in following the path.

A lot of path following control researches are done intheory but rarely implemented in experiment. Breaking the

†This work is supported by Networked Robotic Systems Lab fromDepartment of Mechanical and Nuclear Engineering, University Park.

1Zhenyuan Yuan is with department of Electrical Engineering,The Pennsylvania State University, University Park, PA. 16802 USA.zqy5086@psu.edu

path needed to follow into points and computing outputsfor the car with respect to the closest point as in [3] and[4] theoretically fit our laboratory environment very well,but the theory had only been simulated in software, andthere are a lot of realistic conditions, such as transmissiondelay, computation capacity of the on board processor andthe steering accuracy, had not been taken into account.Existing path following algorithm like [5], even though isproved to be valid, is too computationally heavy, whichis unfeasible to be implemented on the microprocessor asArduino Uno on the car-like robots in our lab. Algorithmdeveloped in the experiment of car parking control using apath controller in [6] seems to be a solution to the problem;however, this algorithm is an adoption of control algorithmfor a two wheeled mobile robot. It is applicable on car-likerobots for motions on curvature such as parking, but whenit turns to urban driving scenarios such as lane followingswitching and right angle turning where the non-holonomicmotion constraint becomes effective, the controller goeshighly unstable as it will be shown in the experiment section.

PID (Proportional-Integral-Derivative) controller was firstintroduced in 1936, when Nicolas Minorsky was designinga control algorithm for the automated steering on the shipsto avoid the disturbance force from distracting the ship fromthe desired course [7]. As shown in Fig. 1, PID controllerimplement proportional, integral and derivative operationson an error between the output and the desired input sothat the output can be maintained at the desire level. PIDcontroller is now usually used in feedback loop and acts asa method of regulation, such as regulating temperature andpressure [8]. Similarly, PID controller is suitable for real-time speed control of electric motors as demonstrated in [9]and [10]. In [11], PID controller is used in steering controlfor car following taking the feedback as relative positionwith respect to the leading car. Similarly, PID controllercan also be used to address heading error between thecar and the reference path as well as the cross-track errorso that the computation is much relieved compared to thetransformations in [5].

In this paper, a path is given and a modification of PIDcontroller is presented as a method of path following forcar-like robots. The general function for PID controller is asin Fig. 1. But for this project,instead of taking one sourceof error, the new version of the controller, inspiring fromthe theory in [3], takes a linear combination of proportionalcontrol on two sources of error, heading error and cross-track error, while remaining economical in computation. Inthe rest of the paper, hardware setup and the function ofcontroller are explained in section II. The non-holonomic

Fig. 1.

Fig. 2. PID controller

system is analyzed in section III. In section IV, simulationsof different designs of the controller are displayed. In sectionV, experimental validation of the successful controller ispresented, and conclusion is drawn in section VI.

II. HARDWARE SETUP AND THE CONTROLLER

An overview of the project is shown in Fig. 3. The carused in the project is a 1/10 EXCEED RC 2.4GHz ElectricInfinitive RTR Off Road Truck. The parts original from thecar used in the project is the motor and the steering servo.The Electronic Speed Control (ESC) is replaced by an off-the-shelf motor driver. A platform is built upon the car inorder to hold the designed hardwares to satisfy the projectobjective. As shown in Fig. 4, the stacks of the boards makeup the center of control for the car. Beginning from the top ofthe layer is the circuit for steering control that is connectedto the steering servo. The steering servo is controlled by thePulse-Width-Modulation (PWM) signal sent from Arduino,which is the control signal of the controller so that the outputof the steering can keep the car on the path. The secondlayer is the motor driver. The third layer is equipped withBluetooth for transmission. The bottom one is the Arduino,which is the microcontroller of the system that actuate thecar.

The proposed controller has its block diagram shown inFig. 2. The designated path specifies the line location as wellas the direction the car should go. Taking these two setpointsas input, regulation is implemented on the cross-track errorand the heading error, which are the distance between the carand the path and the difference between the path directionand the car’s orientation, respectively. Then different gainsare implemented on the two errors to output an appropriatePWM signal to the steering servo for steering adjustment sothat the can can converge to the designated path.

III. NON-HOLONOMIC MOTION CONSTRAINT

Throughout the whole control problem, the inherent non-holonomic motion constraint of the car-like robot is provedto be the most challenging part. The configuration of car-like

robots can be modeled as in Fig. 5, where the rear wheels arefixed and always pointing to the moving direction. Since thecar-like robot in the experiment is usually run at low speedas 50 cm/s, the assumptions of non-slipping motion and non-sideway motion hold very well in the model but become themost difficult part to overcome in order to converge to thedesignated path. As explained in [1], using the same notationin Fig. 5 the motion of the car is described by the followingequations:

x = us cosθ (1a)y = us sinθ (1b)

θ =us

Ltanuφ (1c)

where x,y are the coordinates in the imaginary plane andθ is polar angle of the plane. These equations show that themotion of the car is related to the speed and the steeringangle as well as limited by the maximum curvature whenit is turning. Thus, the error of the car in path following isdescribed by two parameters, the distance from the desiredpath and orientation error. This makes it difficult whenapplying the traditional PID controller because it is usuallyimplement on one dimensional error, whereas the error inthis problem is two dimensional. It becomes even moredifficult when the real world disturbance factors such ascommunication delay, differential gearing and surface fictionare introduced into the system.

IV. SIMULATION

The purposed algorithm, taking the combination of cross-track error and the heading error as the input of thecontroller, is simulated in MATLAB before being carriedout in experiment. The main purposes of simulations aremimicking the behavior of the non-holonomic motion atdifferent situation, tuning and foreshadowing the gains inactual experiment. As shown in Fig. 6, the simulations arerun on a designated square in the same dimension of thetestbed, the car is represented by a point but retaining thecar features such as car length and maximum steering anglwhile its motion is simulated by the equations (1) and thedesignated path is assigned by the four corner points, denoted

Fig. 3. Project overview

Fig. 4. Layers of circuit boards on top of the car platform

Fig. 5. Simple Car Model (source: [1])

as (x1,y1),(x2,y2),(x3,y3),(x4,y4), and the equation of theline is followed by line equation

Ax+By+C = 0

in whichA = y j− y j+1

B = x j+1− x j

C = x jy j+1− x j+1y j

j is incremented when the car is getting close to the end pointof the line, in this case is 450 mm, to provide a smooth cornerturning. The calculation of the errors and the drawbacks ofusing only either one of them are explained in the followingsubsection.

A. CROSS-TRACK ERROR CALCULATION

Cross-track error was used as an input for PID controller,which is the distance between the car and the designatedpath, and the PID controller is adjusting output in order tominimize this error. The cross-track error e(t)d related totime is directly calculated as point-to-line distance as

e(t)d = (−1)sign(Ax+by+ c)|Ax+by+ c|√

A2 +B2

Fig. 6. Designated Trajectory in Simulation

Fig. 7. Simulation on cross-track error with optimal condition

in which, x and y are the corresponding coordinates of thecar, and the sign function is used to determined the side thesteering should turn. The steering servo functions within thePWM signal input range from 1000 to 2000: at 1500 PWMsignal input, the steering stays straight, and the steering turnsright when the input is higher than 1500 and turns left whenit is lower.

The condition is very limited in order for the car to behavewell by taking only the cross-track error as the feedback inputfor PID controller. This algorithm only works when the caris running at low speed under minimal communication delayand small initial offset. Successful simulation is shown inFig. 7, when the car speed is set at 500 mm/s, communicationdelay as low as 0.05 s and is started off very close to thedesignated path. When the car is run at higher or greatercommunication delay in the simulation, the system becomesas unstable as shown in Fig.8.

B. HEADING ERROR CALCULATION

The major drawback of using only cross-track error forregulation is that it does not take the orientation into account,which results in the misalignment between the car’s headingand the path’s direction even though the car is right on thetrack, and therefore the error is not reduced even thoughthe cross-track error is zero. The heading error is calculatedthrough the comparison between the heading direction andthe direction of the designated line. The PID controller takingthis error as input is trying to force the car to head in the samedirection as the designated path so the oscillatory motion can

Fig. 8. Simulation on cross-track error with nonideal condition

Fig. 9. Simulation on heading error

be avoided. The heading error e(t)h is calculated as

e(t)h = (−1)sign(θ − γ)(1− cos(θ − γ))

where θ is the heading angle of the car in the polarcoordinate system, and γ is the angle of the path. The signis used for indicating the side the steering should turn, thesame principle as in last subsection. The term 1−cos(θ−γ)is for limiting the difference within the range -1 to 1 so thatthe tunning process is easier.

As shown in Fig. 9, with only heading error for regulation,the car cannot be turned back into the line once it is off sincethe algorithm only makes the car goes in the same directionas the path.

C. COMBINATION OF CROSS-TRACK ERROR ANDHEADING ERROR

Obviously, regulation using cross-track error and regu-lation using heading error are complement to each other.When merging the two error calculation methods together,the drawbacks of both methods are complemented by theadvantages of the other. As it turns out, when taking thelinear combination of the two types of error, the car isregulated to follow the direction of the path as a result ofthe heading error, and it tends to converge back to the linewhen it is off as the cross-track error is in effect. Simulationresult can seen in Fig.10, and through the simulation, thenew algorithm is more tolerable to the range of speed andcommunication delay. The error of the combination e(t) iscalculated as

Fig. 10. Simulation taking the combinatio of both cross-track error andheading error

e(t) = Ae(t)d +Be(t)h

where A and B are the gains for cross-track error andheading error, respectively. The gain of the heading error isset as large as 1970 to keep the car move in a proper directionwhile the gain of the cross-track error is set as low as 0.028so that the car can converge back to the line when it is offbut the unstable behavior as shown in Fig. 8 is negligible.

V. EXPERIMENT

The proposed algorithm is tested in experiments equippedwith indoor motion capture system, Vicon, which providesposition information that can be translated to Cartesiancoordinate to MATLAB[12], as shown in Fig. 11. Throughthe same Cartesian coordinate system, designated path inlocated through code in MATLAB, which also communicateposition information with Vicon. As shown in the lane-switching scenario in Fig. 15, which is a good representationfor path following, the path designed for the car to completelane-switching motion is in Fig. 12 by MATLAB. The car,running at low speed, started from the bottom right cornerin Fig. 13 with offset by purpose. As captured by Vicon,the car trajectories are shown in Fig. 13 and Fig. 14. Thedots that are out of the continuous car path is due to thenoise of Vicon. Fig. 14 is the resulted plot when using onlycross-track error as an input for the controller; it is highlyunstable because of the non-holonomic motion constrain. Onthe other hand, the algorithm using the combination of cross-track error and heading error gives rise to stable control andsmooth performance in Fig. 13. Because of the transmissiondefect, the car cannot fully converge to the path even thoughthe command is instructing to on the left top corner inFig. 13, but compared the scale in the figure with the cardimension (39*30*14 cm3), the deviation is negligible. In theend, the gains tuned for the controller in actual experimentare 3050 for heading error and 0.225 for cross-track error.

VI. CONCLUSIONS

The main contribution of this paper is an steering controlalgorithm for autonomous car path following in urban sce-nario. The algorithm was validated in experiments, where thedesired performance is satisfying given the non-holonomic

Fig. 11. Vicon motion capture system

Fig. 12. Designated tracjectory in experiment

Fig. 13. Experiment in lane-swtiching scenario

Fig. 14. Car behavior using only cross-track error as input

Fig. 15. Lane-switching scenario

motion constrained. Experiments are held with the car start-ing off close to the designated path without opposite headingdue to the space limitation as well as the desire to have thecar converge quickly to the path, but these assumptions holdwell since motion planning algorithm would generate pathstarting from the car’s current position and heading. Thisalgorithm is proved to be valid for path following sectionunder motion planning , so in the coming research on-board computer (NVIDIA Jetson TK-1) will be embeddedon the car-like robots to execute on-board motion planningas well as minimize communication defect and integrate withsensors. In this way, the car, according to the motion planningalgorithm developed in the lab, will be able to generate andfollow a path on its own for a given task, achieving the fullautonomy.

ACKNOWLEDGMENT

This work is supported by Networked Robotic SystemsLab from Department of Mechanical and Nuclear Engineer-ing, supported by Dr. Asok Ray. The author would also thankfor the supervision from Dr. Minghui Zhu and the assistanceand guidance provided by graduate student, Nurali Viraniand Devesh Jha.

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