Vehicle Following with Minimal MemoryJohnson Carroll

Electrical and ElectronicEngineering Science

University of JohannesburgJohannesburg 2006, South Africa

Email: jcarroll@uj.ac.za

Abstract—A logical step on the path to full automation ofan automobile is following a lead vehicle. Most studies invehicle following use the chase vehicle’s estimated speed andsteering angle to calculate a series of approximate waypointsin an absolute coordinate frame to create a path for the chasevehicle to follow. However, state estimation requires increasedcomputational ability as well as a robust noise model. This papertakes the minimal approach, attempting to find effective pathsand steering strategies with minimal information storage throughuse of appropriate curve fitting parameters and measurements.The resulting control algorithm is tested through simulations onconstructed test tracks and real-world road data.

I. INTRODUCTION

There are many degrees of vehicle automation. At thelowest levels, common driver assistance technologies suchas automatic transmissions, ABS braking, and cruise controlautomate particular portions of the vehicle. At the highestlevel one finds fully automated, driver-less vehicles such asthe newly developed Google cars [1] and the competitors inthe DARPA Grand Challenge [2]. Between these extremes,there are a variety of intermediate levels that serve a varietyof purposes. One natural automation goal is vehicle following,in which the vehicle travels in the path of a lead vehicle.

Vehicle following has a variety of motivating applications,from driver-less platoons to automated highway systems [3].Modern production vehicles already employ active cruisecontrol systems that detect nearby vehicles and adjust speedappropriately [4]. Combining a robust active cruise controlsystem with an automated steering system produces a vehiclefollowing system.

This paper describes a new approach to simple vehiclefollowing control which does not require state estimationof the following vehicle, and greatly simplifies the controlalgorithm.

II. FOLLOWING PROBLEM

The vehicle following scenario involves two vehicles: thechase vehicle which is automatically controlled to safelyfollow in the path of the lead vehicle. ”Safely”, motivated byordinary road driving, is taken to mean that the chase vehiclemaintains a safe following distance and follows the lead car’spath with reasonable accuracy.

Fig. 1. Car following scenario with relevant measurements indicated:following distance r, relative angle θ, and relative heading φ.

A. Sensing

A variety of sensors are commonly used in vehicle automa-tion, including laser and radar range finders, single and stereocameras, GPS, and others. Additionally, internal sensors can beused to gather information about the chase vehicle itself, suchas speed, acceleration, temperature, etc. Together, the sensorsmust provide the control system with appropriate informationabout the position of the lead car and state of the chase car.

For the present study pitch and roll are neglected, and itis assumed that the lead and chase vehicles are on approxi-mately the same plane. As depicted in Figure 1, the followinginformation about the relative position of the lead car can beobtained:

• Relative distance r,• Relative angle θ,• Relative orientation/heading φ.

The only internal sensor considered here will be a speedome-ter. Additional sensors, such as inertial measurement units andGPS, can provide much more accurate readings and can becombined for more accurate velocity estimates. However, theminimalist approach considered here makes additional sensorsseem counterproductive.

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III. CONTROL ALGORITHM

The vehicle control problem is naturally decoupled intolongitudinal and lateral control problems. Control in the lon-gitudinal direction determines the torque output needed, whilethe lateral controller calculates the appropriate steering angle.During normal operation (i.e., when the car is not slidingor turning too sharply), the effects of the output torque onlateral motion are minimal, as are the effects of steeringon longitudinal motion. We therefore treat the two axes ascompletely separate control problems.

A. Longitudinal Control

Longitudinal control techniques are quite well established;as mentioned previously, many modern production cars utilisea active cruise control that serves the same purpose as thecurrent control study. Longitudinal control tries to maintaineither a constant distance between the lead and chase vehicles,or a constant headway time. The constant distance ensures thatthe lead car remains within the range of the chase car’s sensorsat any speed, but does not account for the safety of followingat a fixed distance at both low and high speeds. A constantheadway time, on the other hand, increases the followingdistance with speed. Since maneuverability also decreases withspeed, the constant headway time does not significantly impactthe ability of the chase car to accurately follow the lead car’spath. The headway time is approximated from the followingdistance and the chase car’s speed. In Section IV, the choiceof headway time is described in more detail.

Once the appropriate headway time is chosen, the outputtorque is determined by a PID controller with gain scheduling.Near the correct headway time the PID response is slow andsmooth, but as the headway time deviates from the nominalvalue, the controller responds more dramatically (see Figure2). The points at which the gains change must be chosen

Fig. 2. Graphical depiction of PID gain scheduling strategy.

based on the sensor accuracy, data rate, actuation delay, andcapabilities of the chase vehicle itself. If chosen appropriately,the chase vehicle must be able to consistently follow the leadvehicle through speed changes, but brake quickly enough toavoid collisions in an emergency stop.

B. Lateral Control

Lateral control is a much more complex problem. Thevariety of possible path configurations makes a simple PIDcontroller infeasible. The next two sections describe the tra-ditional lateral controller, which uses state estimation to finda series of waypoints approximating the path, and the newmemoryless controller.

1) State Estimation and Waypoints: The standard approachfor following a lead vehicle is to store a series of measure-ments and use internal sensors to estimate the change in chasevehicle position (see, for example, [5]). The measurementscan then be combined to form a series of waypoints thatapproximate the path taken by the lead vehicle. With theapproximate path of the lead vehicle in memory, the chase carcontroller can use one of a variety of path following algorithmsto find appropriate steering commands.

Fig. 3. A series of measurements is used to find waypoints which approximatethe lead vehicle path.

This approach requires accurate state estimation to pro-duce usable waypoint data. The combination of sensor dataincreases the sensitivity to sensor noise. More accurate stateestimates would require more complex sensors, such as aninertial measurement unit or GPS. Further, if the chase carbehaves unpredictably (e.g., under-steers more than expectedon a turn), a simple state estimate based on speed and currentsteering angle will produce incorrect path estimates.

Additionally, the waypoint method essentially requires in-formation about the position of the chase vehicle in an absolutereference frame. However, the vehicle following problem isintrinsically based in a relative reference frame: the positionand orientation of the lead vehicle is detected relative to the

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chase vehicle position and orientation, and the speedometerstipulates a moving reference frame.

2) Memoryless Path Estimation: In contrast, the proposedlateral controller does not require estimates of absolute posi-tion or orientation, nor does it store previous sensor measure-ments. Instead, it relies on the fact that the driving paths thatcan feasibly connect the chase car’s position and heading withthe lead car’s position and path are limited in variation.

At each measurement time, the controller uses a cubic splineto find a path between the chase car’s position (taken to be theorigin at every time) and the lead car’s position. The path isfurther required to coincide with the chase car heading (takento be directly along the longitudinal axis at every time) andthe lead car heading. Then the steering angle is chosen to beapproximately tangent to the spline curve.

The spline path is constructed as a function f : [0, 1]→ R2,where f(0) corresponds to the position of the chase car andf(1) corresponds to the position of the lead car. To derive theappropriate spline, we start with a generic cubic function:

f(γ) =

[f1f2

]=

[a1γ

3 + b1γ2 + c1γ + d1

a2γ3 + b2γ

2 + c2γ + d2

](1)

Recalling the definitions of the relative distance (r), therelative angle (θ), and the relative heading (φ), the geometricconstraints on the spline path are:

• f(0) =

[00

]; chase car position, always at origin,

• f(1) =

[r cos(θ)r sin(θ)

]; lead car position relative to chase car,

•df

dγ(0) =

[αc

0

]; matching chase car heading,

•df

dγ(1) =

[αl cos(φ)αl sin(φ)

]; matching lead car heading.

Here, αc and αl are parameters not constrained by the ge-ometry. Solving for the spline coefficients subject to theseconstraints yields:

a1 = αl cos(φ)− 2r cos(θ) + αc

b1 = −αl cos(φ) + 3r cos(θ)− 2αc

a2 = αl sin(φ)− 2r sin(θ)b2 = −αl sin(φ) + 3r sin(θ)c1 = αc

c2 = d1 = d2 = 0

(2)

As shown in Figure 4, the relative distance, orientation,and heading do not uniquely specify a spline path. Theundetermined parameters αc and αl must be chosen by thecontrol designer to best approximate feasible vehicle paths.Testing at various speeds indicates that the best performanceoccurs when αc and αl vary proportionally with speed:

αc = (chase car velocity)β1αl = (chase car velocity)β2

The selection of the parameters β1 and β2 is discussed inSection IV.

Fig. 4. Appropriate spline path with implied turning radius and infeasiblespline path with different parameters

This simple geometric algorithm has the distinct advantageof requiring only 35 mathematical operations at each mea-surement time to calculate the best steering angle from themeasurement data. By contrast, the waypoint strategy requiresa much more complex polynomial curve fit to find the bestpath for the estimated data points, then additional calculations(sometimes including optimisations) to find the best steeringangle to follow the path.

IV. MODELING AND SIMULATION

A. Vehicle Model for Simulation

In order to test the viability of the control algorithmdescribed above, the chase car was modeled in Simulink as arigid body with three degrees of freedom system. The controlsystem dictates desired steering angle and wheel torque, anddynamic and algebraic equations calculate the resulting vehiclemotion. All vehicle and sensor parameters were estimatedbased on the physical characteristics and performance ofthe University of Johannesburg’s BAR-1 hybrid vehicle [6].The sensor model was based on the SICK LMS291 laserscanner, which is configured to deliver 37.5 scans per secondwith ±1cm accuracy [7]. It is assumed that the position andorientation of the lead car is successfully calculated from everyscan, with additive random errors on the order of the laseraccuracy. Lead vehicle paths were pre-constructed and inputas time-indexed coordinates to the Simulink model. Distanceand angle measurements are then calculated based on the chasevehicle position at each time step.

B. Test Tracks

Two different test tracks were used to validate the simulationmodel. The first, shown in Figure 5, describes a figure-eight with different sized circles. Different radii and differentspeeds were tested in order to appropriately select the spline

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Fig. 5. A figure-eight test track with turn radii 30m and 50m

parameters and the headway time. Each test run consistedof three laps of the track, and the maximum and RMS pathdeviation was calculated for each run.

Then, as a test of real-world applicability, the second testtrack is a 175km stretch of road between Cape Town andBaardscheerders Bosch in southwestern South Africa. The datafor this track was obtained from time-stamped GPS data atOpenStreetMaps [8], so represents an actual driving scenariowith reasonable speed data.

C. Simulation Results

Figure 6 shows the RMS and maximum path error forthe vehicle at different headway times. The simulations usedthe figure-eight track with different turning radii at differentspeeds. The results clearly show that a headway time of 1second produces the best path results. Though this headwaytime is smaller than the usual recommended headway times forhuman drivers, 1 second is well within the braking capabilitiesof the automated vehicle.

Fig. 6. Plot of headway time vs RMS (lines) and maximum path error(squares) on figure-eight track with different speeds and different turn radii(indicated by different colors). Best results at 1s.

Figure 7 shows the RMS and maximum path error for thevehicle using different spline parameters. These tests wereperformed on a figure-eight track at 50 kph with turn radiiof 40m and 60m. The graph, in conjunction with similar testsat different speeds on different tracks, suggests that a goodchoice of parameters is β1 = 0.8 and β2 = 1.25. These valueswere used for all of the other tests cited here.

Fig. 7. Plot of spline parameters β1 (indicated by different colors) and β2(horizontal axis) vs RMS and maximum path error on figure-eight track withradii 40m and 60m at 50 kph.

On the real-world test track, the simulation resulted in anRMS path error of 0.127m. This error is only slightly largerthan the RMS errors found in [9], which used state estimationand complex path following algorithms on simple geometricpaths at low speed. As shown in Figure 8, most of the path

Fig. 8. Plot of road track with path deviation

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deviations were within 0.5m which, depending on vehiclegeometry and road size, is a reasonable result for such aconvoluted path. The maximum deviation is 1.966m, which israther large. However, closer examination in Figure 9 showsthat the lead car’s path takes a swerving path through whatmay well be a parking lot: not the ideal situation for automatedvehicle following.

Fig. 9. Detail of largest path deviation

V. CONCLUSION

This study demonstrates that autonomous vehicle followingwithout state estimation or memory is feasible. Future studieswill include implementation of the memory-less controller ona real-world vehicle, as well as more rigorous selection ofoptimal spline parameters. Additional studies will quantify thebenefits of state estimates, ideally finding the optimal numberof waypoints and thereby the preferred measurement rate forvehicle following applications.

REFERENCES

[1] J. Markoff, “Google cars drive themselves, in traffic,” The New YorkTimes, 9 October, 2010.

[2] website: DARPA Urban Challenge, accessed 14 April, 2011http://archive.darpa.mil/grandchallenge/index.asp.

[3] H. Raza and P. Ioannou, “Vehicle following control design for automatedhighway systems,” Control Systems, IEEE, vol. 16, no. 6, pp. 43 – 60,Dec 1996.

[4] W. Jones, “Keeping cars from crashing,” Spectrum, IEEE, vol. 38, no. 9,pp. 40 –45, Sep 2001.

[5] S. Gehrig and F. Stein, “Elastic bands to enhance vehicle following,” inIntelligent Transportation Systems, 2001. Proceedings. 2001 IEEE, 2001,pp. 597 –602.

[6] B. Andrews, A. Corregedor, M. Furrutter, S. Holte, C. Murcott, K. Smith,J. Carroll, F. Du Plessis, and J. Meyer, “Design and implementation ofan autonomous hybrid vehicle,” in IEEE Africon, Victoria Falls, Zambia,Sep 2011 - to be published.

[7] “Telegrams for configuring and operating the LMS2xx lasermeasurment systems v2.30,” [Online] accessed 13 March, 2011https://www.mysick.com/saqqara/pdf.aspx?id=im0015003.

[8] website: OpenStreetMap, accessed 14 April, 2011http://www.openstreetmap.org/.

[9] M. Spencer, D. Jones, M. Kraehling, and K. Stol, “Trajectory basedautonomous vehicle following using a robotic driver,” in Proc. AustralianConference on Robotics and Automation, Sydney, Australia, Dec 2009.

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