Automatic tractor guidance using carrier-phase differential GPS

Computers and Electronics in Agriculture

25 (2000) 53–66

Automatic tractor guidance using carrier-phasedifferential GPS

Thomas BellIntegriNautics, 130 Lester A6enue, Shillington, PA 19607, USA

Abstract

Previous research at Stanford University involving automatic aircraft landings using ahigh-precision form of differential GPS known as Carrier-Phase Differential GPS (CPDGPS)led to the development of a CPDGPS-based sensor system for automatic tractor control.With high accuracy and no drift in attitude (roll, pitch, and yaw) and/or position, CPDPGSoffers a cost-effective sensor option for automatic guidance systems. Stanford researchersfirst experimentally demonstrated automatic steering of a medium-sized Deere 7800 tractor inthe spring of 1996. Subsequent research included tractor control along spirals, arcs, andarbitrary curves, control on steeply sloped terrain, on-line identification of the steeringvalve’s ‘dead-zone’, and real-time identification/improvement of the tractor model. Typicalcontroller accuracies under full engine load with implement lowered were near 0 cm meanand 4–6 cm S.D. in the tracking error of the control point on the tractor from the desiredtrajectory (as measured by CPDGPS). Unbiased low-noise attitude measurements were vitalbecause the control point is not collocated with the GPS position antenna: an attitudemeasurement noise of only 1° (1 s) is shown to introduce additional position uncertainty ofup to 4 cm at the control point above the original GPS measurement (for a tractor of similarsize to the one used in this research). In other words, accurate position information alone isnot enough for a viable sensor system. These experimental results, enthusiastically receivedby farmers who have witnessed Stanford’s ‘GPS tractor’, show that CPDGPS could be thesole position and attitude sensor for a viable commercial tractor guidance system. © 2000Elsevier Science B.V. All rights reserved.

Keywords: Global positioning system; Carrier-Phase Differential GPS; Guidance system; Automatictractor control; System identification; Attitude measurement

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E-mail address: [email protected] (T. Bell)

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T. Bell / Computers and Electronics in Agriculture 25 (2000) 53–6654

1. Introduction

Tractor guidance research at Stanford University is an outgrowth of the NationalAeronautics and Space Administration’s (NASA) Relativity Gyroscope Experi-ment. The 28-year-old ‘Gravity Probe B’ experiment is designed to measure twophenomena predicted by Albert Einstein’s general theory of relativity — thegeodetic effect and the frame dragging effect — by launching four ultra-precisequartz gyroscopes into a polar orbit around the Earth and measuring theirprecession over time. The Global Positioning System (GPS) was proposed as apossible source of coarse position measurements of the probe while on-orbit. UsingGPS measurements required that satellite signals be ‘handed off’ from one GPSantenna on one side of the probe to another GPS antenna on another side as theprobe traveled in its orbit. Research into this problem led to the use of the GPSsatellite signals as a source for attitude (i.e. roll, pitch, and yaw) information. Theattitude research then led to an algorithm for using the GPS carrier signal forhigh-precision positioning. This type of ultra-precise GPS positioning is known asCarrier-Phase Differential GPS (CPDGPS). Stanford graduate students combinedthese attitude and positioning systems to replace the inertial guidance system in aUnited Airlines 737, which subsequently used the CPDGPS position and attitudeinformation to conduct 110 successful automatic landings (Cohen and Lawrence,1995; Pervan and Parkinson, 1997).

Several characteristics of CPDGPS-based attitude and positioning sensors madethem attractive to the aviation industry: low cost, high accuracy, and the absenceof drift or bias. These characteristics also made this sensor combination attractiveas a practical sensor system for automatic ground vehicle guidance. Professor B.Parkinson, principal investigator for the 737 flight trials, realized that agriculturewould be a realistic application of this sensor system. Tractor manufacturer Deereand Company also recognized the emerging potential of high-precision GPS forautomatic tractor guidance and agreed to fund a research effort at Stanford’sDepartment of Aeronautics and Astronautics to demonstrate automatic tractorguidance using only CPDGPS for both attitude and position. The project wasstarted in March 1995 by Professor Parkinson and Dr M. O’Connor and has hadan average ‘staff’ of two to three Ph.D. students. Although the project was notcreated to develop a pre-production prototype but rather to demonstrate a con-cept’s feasibility, the research effort has focused on designing a practical guidancesystem that could one day prove commercially successful.

2. GPS as a sensor for vehicle guidance

Many of the significant sources of position error such as the Department ofDefense’s deliberate dithering of the satellites’ time stamps and ephemerides (knownas ‘Selective Availability’, or SA; Parkinson and Spilker, 1996a) and ionosphericeffects cancel if the GPS measurement is taken relative to a reference antenna at aknown fixed location; the new position measurement is now relative to the reference

T. Bell / Computers and Electronics in Agriculture 25 (2000) 53–66 55

antenna. Implicit in the assumption that error sources will cancel is that the errorsare the same at both locations. This assumption’s validity deteriorates as the twoantennas are moved further apart or the time difference between measurementsincreases. In practice, the time difference between the two measurements can beminimized by immediately broadcasting reference station measurements or ‘correc-tions’ to the mobile vehicle via a radio link. Spatial degradation does increase as thevehicle moves further away from the reference station; 10 km is often quoted as theapproximate maximum effective range of reference station corrections. Althoughthe requirement to have a continuously available radio link between the referencestation and vehicle is a weakness of differential GPS systems, reliable and afford-able radio systems are commercially available.

The most commonly used form of differential GPS is code-differential GPS. Incode-differential GPS, the GPS receivers at the vehicle and reference station takephase measurements of the GPS pseudo-random noise (PRN) signal. The referenceand vehicle measurements are then differenced to provide a relative positionmeasurement. Code-differential systems offer 1- to 2-m accuracy and correctionsare valid for a large radius, usually on the order of 100 km (Parkinson and Spilker,1996b). Although code-differential positioning systems offer ease-of-use and arealready widely available, the meter-level accuracy is not good enough for manyhigh-precision agriculture applications.

CPDGPS is based on measuring the phase of the GPS carrier signal. Because theL-band GPS carrier signal has a wavelength of approximately 19 cm and GPSreceivers can track that carrier signal to within roughly 1 cm, CPDGPS offerscentimeter-level accuracy. The difficulty in CPDGPS, however, lies in the system’sinitialization. Both the reference and the vehicle receivers can measure the phase ofthe carrier signal, but neither knows the integer number of carrier waves betweentheir antennas for each satellite. Each satellite will have its own ‘integer ambiguity’,which must be resolved before the vehicle can begin to measure its relative positionprecisely. The integer ambiguity can be resolved through the observation of satellitemotion, vehicle movement past pseudo-satellite transmitters (‘pseudo-lites’) fixed atknown locations, or through a technique known as ‘wide-laning’. Wide-laning usesthe 347.82 MHz ‘beat frequency’ of the L1 and L2 signals to help resolve theinteger ambiguities. Interested readers are referred to Parkinson and Spilker(1996a,b) which provide a concise summary of the two techniques as well asdetailed information on the GPS system. Commercial carrier-phase GPS systemsare available from companies such as Trimble Navigation, Novatel, IntegriNautics,and Leica. In small quantities, these systems usually cost in excess of $30 000 forboth the reference station and ‘rover’ unit.

Measuring vehicle attitude using CPDGPS is a similar process. The phasedifference between antennas is used along with the satellites’ azimuth and elevationto measure the antennas’ angle relative to each satellite provided the antennas1 aremounted at known locations relative to each other. If at least four satellites areusable, the antenna array’s attitude can be resolved.

1 Although three GPS antennas is the minimum number required to resolve attitude, four antennas areoften used.


3. Vehicle guidance research

Automatic tractor guidance was defined as ‘accurate automatic control of thetractor or implement along a predefined trajectory’. The control system’s accuracywas judged by the mean and S.D. of the GPS-measured tracking error of a controlpoint from the desired trajectory. The control system’s performance was consideredaccurate enough when the mean tracking errors were less than 5 cm and the S.D.s

Fig. 1. Variable definitions for vehicle model.


Fig. 2. Automatic row guidance for four rows on level terrain: 0.4 cm average mean, 4.0 cm S.D. Dashedlines are 1 foot.

Fig. 3. Overhead view of automatic spiral guidance on level terrain.

were less than 10 cm. The desired trajectory was specified ahead of time relative tothe GPS reference antenna. This approach is different from other researchers’ whoare using vision-based guidance systems. These vision-based guidance systemsassume the computer has no prior knowledge of the desired trajectory, but mustinstead recognize the desired trajectory, usually by ‘watching’ the rows of the crop.The GPS positioning system used in this research measured the position of one ofthe tractor’s GPS antennas relati6e to a fixed reference station antenna. Theabsolute position of the tractor could be measured only if the absolute position ofthe reference station antenna was known.


4. Vehicle

O’Connor (1997) began by developing a highly simplified model of the tractorsuitable for controller synthesis (O’Connor et al., 1997). The model was based onthe following five states and control effort (Fig. 1):� Yaw error c. The yaw error from the desired trajectory.� Yaw error rate c: . The yaw error rate from the desired trajectory.� Steer angle d. The angle of the front wheels relative to the centerline of the

Fig. 4. Tracking error along experimental spiral trajectory. Mean error was −0.22 cm and S.D. (1 s)was 5.27 cm.

Fig. 5. Row guidance on sloped terrain, −0.8 cm mean, 6.4 cm S.D. (1 s).


Fig. 6. Height error for automatic control on a contour, 0.5 cm mean, 4.3 cm 1 s.

Fig. 7. Stanford’s John Deere 7800 experimental tractor.

vehicle. The model assumed the front wheels effectively acted as one (Wong,1993). The front wheels could not turn beyond roughly 35°.

� Steer angle rate d: . The front wheel angle rate.� Control signal u. The control signal drove a hydraulic pump, which actuated the

front wheels. Therefore, the control signal did not command a wheel angle butrather a wheel rate. To first order, the relationship between control effort andwheel rate can be described by d: c=u, where the ‘c’ subscript denotes ‘com-


manded’. Conceptually, u would correspond to how fast the driver was turningthe steering wheel.

� Tracking error d. The tracking error of the vehicle from the desired trajectory.

The rate of change of these states was described by:

Fig. 8. Equipment mounted inside the cab of the 7800 tractor.

Fig. 9. Electro-hydraulic steering valve and front wheel potentiometer.


c8 = 1tc

(c: c−c: )

d8 = 1tu

(d: c−d)

d: =V sin c+ l2 c: cos c

The subscript ‘c’ again denotes ‘commanded’. The lag state in yaw was created tocapture the effect of the tractor’s rotational inertia. If the front wheel anglechanged suddenly, the yaw rate would not change instantly, but rather behave as afirst order lag system: the front wheels would skid laterally, then slowly take holdand begin to yaw the tractor. The lag state in steering modeled the dynamicsof the hydraulic steering valve. Higher order valve dynamics were con-sidered negligible, and a first order valve model was considered an adequatedescription of actual valve response. The variable V was the forward speed of thetractor’s control point expressed in the tractor’s coordinate frame. The two timeconstants, tc and tu, were identified from experimental data. The relationshipbetween steer angle and yaw rate can be derived from vehicle geometry and ano-slip assumption:

c: c=1l1

V tan d

The variable l1 was the distance between the front axle and the tractor’s pivotpoint, which was assumed to be the center of the rear axle, and l2 was thelongitudinal distance between the tractor’s pivot point and the control point (Fig.1).

5. Tractor control along rows

In March of 1996, O’Connor used this linearized vehicle model to control aDeere Model 7800 tractor, detailed in Appendix A, along four 50-m rowsto a S.D. of approximately 2 cm and a mean error less than 1 cm. He published hisresearch results at the 1996 Precision Agriculture Conference (O’Connoret al., 1996). In June of that same year, he studied the effects of ahitched implement on the tractor by adding a three-shank subsoiler, or‘ripper’. Interestingly, the ripper did not degrade the controller performancesignificantly. Although the hard ground at the Stanford test site intro-duced additional tracking disturbances through the ripper, the ripper also helpedresist the tractor’s lateral movement from ground disturbances. Some relativelyrecent test results in Fig. 2 show typical experimental accuracy for tractorrow guidance along four straight rows with a large chisel plow and the engineunder full load. The large initial spikes come from the initial row acquisition. Theaverage tracking error for the four rows was 0.4 cm and the average S.D. was 4.0cm.


6. Tractor control along non-linear trajectories

The author began to explore the possibility of controlling the tractor alongnon-linear trajectories. Center-point irrigation techniques created a need forspiral tractor/implement trajectories, and locally linearized tractor models andcontrollers for both arcs and spirals were developed and later demonstratedexperimentally in September 1996. Fig. 3 shows an overhead view of an actualexperimental trajectory. Once again, the engine is fully loaded and a (towed) chiselplow was used. The tractor started from the outside and worked its wayinward. Fig. 4 shows the tracking error as a function of time. Near the center of thespiral, old beds created two large tracking errors as the tractor side-slipped intothem.

In the spring of 1997, the author began research into tractor control alongarbitrary trajectories, trajectories that might be encountered along the edge of afield that was bounded by a stream, for example. The premise was that a series ofclosely spaced points collected from a previous (human-controlled) pass would beused to define the curve, which was therefore assumed to be drivable. Parameterizedcubic splines were used to smooth through the data points to recreate a closeapproximation to the original trajectory provided the points were spaced closelyenough. A linear quadratic look-ahead control algorithm was developed andimplemented that looked at the next 8 s of the curve and fed that futureinformation back into the controller to guide the tractor accurately. Experimentalresults demonstrated that accurate control along curve paths without implement onlevel ground was not only possible at speeds up to 2.5 m/s, but that accuracy didnot degrade severely from control along straight lines. The results were published atthe International Association of the Institutes of Navigation’s 1997 conference (Bellet al., 1997).

7. Real-time model identification

In the summer of 1997, graduate student Andrew Rekow joined the project. Hisresearch led to a method that could identify the time-varying effectiveness of thetractor’s steering wheels in real-time (Rekow, 1998). The steering effectiveness wasa strong function of factors such as ground conditions, implement, and vehicleballasting. An effective on-line identification algorithm is important for the accu-racy of control over all conditions. Along with Dr V.K. Jones, he also developed aninnovative least-squares-based method to identify systems with an output dead-zone (Rekow et al., 1998). The type of system identified in the paper modeled thebehavior of the tractor’s electro-hydraulic valve quite well. Although the valve’sinput–output relationship could be calibrated and linearized through a look-uptable, the valve behavior could vary with hydraulic fluid temperature, pressure, andwear. An on-line identification algorithm could improve performance by accommo-dating these variations.


8. Control along sloped terrain

Recently, the author introduced a new form of bias estimation into the controlsoftware to enable more accurate control along sloped terrain. Fig. 5 shows a timehistory of the tractor’s tracking error along four straight rows over terrain slopedin excess of 14°. The tests were conducted with a large towed implement (chiselplow) at full engine throttle. The tracking error mean for all four rows, despite theslope, was only −0.8 cm. The average S.D. (1 s) was 6.4 cm. All of thesemeasurements were taken by GPS and therefore represent the control error. Therolling terrain, dust, and cost made independent measurement by other systemssuch as a laser tracker or high-quality inertial system nearly impossible, though avisual check confirmed straight rows.

In addition, the author also devised a way to farm the contours of hilly terrainwith no prior knowledge of the terrain. The GPS-measured height error for acontour 15 m above the reference station antenna is shown in Fig. 6. He success-fully demonstrated automatic guidance on sloped terrain and automatic contourfarming in May of 1998, and presented a paper at the 1998 Precision AgricultureConference in Minneapolis (Bell et al., 1998). The accuracy of the CPDGPS systemoffers farmers the opportunity to create high-precision three-dimensional terrainmaps of their farms by simply driving over the terrain. These maps could subse-quently be used to plan such things as more effective irrigation.

9. Error analysis of control point uncertainty

The point on the tractor to be controlled was the point on the ground directlybeneath the center of the rear differential. The position of the tractor was measuredby one of the four roof antennas (actually, the passenger-side antenna). The vectorfrom the roof antenna to the control point was transformed from its known valuein the tractor frame to the inertial frame using the GPS attitude measurement. Thistransformed vector, or ‘lever arm’, was added to the original roof antenna positionmeasurement so that the position measurement appeared to be taken at the controlpoint. In this manner, excessive tractor roll and pitch motion caused by roughterrain could be accounted for.

If the lever arm is denoted by r and the (non-linear) transformation matrix fromthe body frame b to the inertial frame i is denoted by Ti/b (Greenwood, 1988, p.357), then the position measurement r at the control point can be expressed as:

r icontrol=r i

antenna+Ti/b rb

=r iantenna+ri

Since the correlation between the uncertainties in the position measurement andthe attitude measurement is negligible, the uncertainty S at the control point is:

Sicontrol=Si

antenna+Silever arm


Assuming higher order moments are negligible, a first order approximation forthe lever arm uncertainty is:

Silever arm= [9ri ]T SU,r [9ri ]

where SU,r is the six-dimensional uncertainty matrix for both the attitude vector(U) and lever arm (r) measurements (Taylor, 1982). Since the two groups ofmeasurements are uncorrelated, SU,r is block-diagonal. The gradient operator 9 istaken with respect to not only attitude, but the lever arm as well.

The approximation for the lever arm uncertainty can be used with a particularlever arm correction to determine approximately just how much uncertainty theattitude measurement noise adds to the position measurement when the positionmeasurement is moved from the measurement point at the roof antenna to thecontrol point beneath the rear axle. For the following arbitrary conditions:� A lever arm correction of 0.5 m forward, 1.0 m right, and 2.0 m up (reasonable

measurements for tractors the size of the 7800).� No uncertainty in the lever arm. In other words, the lever arm is assumed to be

perfectly rigid.� Attitude measurements of 3.0° in roll, −2.0° in pitch, and 45.0° in yaw, 0.10°,

1 s noise in roll, pitch, and yaw measurements,the largest resulting uncertainty in any direction — the square root of the largestsingular value of the covariance matrix — is 4 mm. When the attitude measure-ment noise is increased to 1.0° 1 s, the resulting maximum uncertainty is 4 cm. Inother words, even with an attitude sensor that can measure roll, pitch, and yawwith measurement noises of only 1°, the lever arm-to-control point correction addsas much uncertainty as was in the original GPS position measurement. Further-more, as the quality of the attitude measurement decreases, the noisy attitudemeasurement becomes the dominant source of position measurement uncertainty at thecontrol point. An automatic centimeter-level tractor guidance system must eithertake the position measurement very close to the control point if a poor attitudesensor is to be used, or take the position measurement further away using anaccurate attitude sensor. Alternatively, the control point could be placed closer tothe position measurement antenna, but this may be unrealistic from a farmer’sperspective.

10. Future research

Future research includes implementing a kinematic model of a towed implementdeveloped previously so that the control point could be moved to the (towed)implement. Different types of tractors such as tracked or articulated tractors wouldrequire different models and may hold different control challenges. Previousmodeling work could safely neglect higher order dynamics because of the lowspeeds. Maintaining controller accuracy at higher speeds may require an expandedmodel and possibly inertial components.


11. Conclusion

Not only is high-precision tractor control possible using only CPDGPSsensors for position and attitude, but practical as well. Research at StanfordUniversity has demonstrated economical solutions under realistic control scenarios.Research has advanced control possibilities from straight lines on level ground toinclude spirals, arcs, and curves. Control on sloped terrain has been shownas possible. Advances in real-time identification algorithms have paved theway for real-time adaptive control. Based on the strong positive reception fromfarmers, the author believes that commercial automatic tractor control is only amatter of time.

Acknowledgements

The author wishes to thank Deere and Co. for their support of this research.Any opinions, findings, conclusions, or recommendations expressed in this publica-tion are those of the author and do not necessarily reflect the views of Deere andCo.

Appendix A. Experimental platform

Deere and Company loaned Stanford University a John Deere 7800 four-wheel-drive tractor shown in Fig. 7. Four GPS antennas are mounted on the roof of thetractor. The 108-kW tractor is considered a medium-sized tractor and has anenclosed cab. Inside the cab, students designed and built an aluminum equipmentrack that houses a Pentium 100 MHz PC running the Lynx real-time operatingsystem (Fig. 8). The computer, manufactured by Industrial Computer Source, is thelarge box in the lower right portion of the rack. Above the computer is an inverterto convert 12 V DC current to 110 V AC. The sunlight-readable display is mountedin the left side of the rack. IntegriNautics’ GPS receiver, mounted upside down, isbeneath the monitor. Underneath the nest of wiring lies Trimble’s TANS2 Vector(the GPS attitude system). A Pacific Crest 9600 baud radio modem is mountedbehind the IntegriNautics receiver. A Motorola MC68HC11 microprocessor (notvisible behind the monitor) converted the commanded steer rate from the computerinto a pulse-width modulated signal that was then amplified and sent to anOrthman electro-hydraulic valve that actuated the front wheels. A potentiometermeasured the front wheel angle, and that measurement was fed back into thecomputer also through the MC68HC11. The valve and potentiometer are bothshown in Fig. 9.

2 Trimble Advanced Navigation System.


References

Bell, T., O’Connor, M., Jones, V.K., Rekow, A., Elkaim, G., Parkinson, B.W., 1997. Realisticautofarming, closed-loop tractor control over irregular paths using kinematic GPS. In: Programmeand Papers of the 9th World Congress of the International Association of Institutes of Navigation,held November 18–21, 1997, Amsterdam, The Netherlands.

Bell, T., Bevly, D., Biddinger, E., Parkinson, B.W., Rekow, A., 1998. Automatic tractor row andcontour control on sloped terrain using Carrier-Phase Differential GPS. In: Proceedings of theFourth International Conference on Precision Agriculture.

Cohen, C.E., Lawrence, D.G., 1995. Automatic landing of a 737 using GNSS integrity beacons.Navigation 42 (3), 467–486.

Greenwood, D.T., 1988. Principles of Dynamics, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ.O’Connor, M., Bell, T., Elkaim, G., Parkinson, B.W., 1996. Automatic steering of farm vehicles using

GPS. In: Proceedings of the Third International Conference on Precision Agriculture, Minneapolis,MN, June 23–26, 1996, pp. 767–778.

O’Connor, M., 1997. Carrier-Phase Differential GPS for automatic control of land vehicles, Ph.D.thesis, Stanford University.

O’Connor, M., Bell, T., Elkaim, G., Parkinson, B.W., 1997. Real-time CDGPS initialization for landvehicles using a single pseudo-lite. In: Proceedings of the National Technical Meeting, The Instituteof Navigation, Santa Monica, CA, January 14–16, 1997, pp. 717–724.

Parkinson, B.W., Spilker, J.J. (Eds.), 1996a. Global Positioning System: Theory and Applications,American Institute of Aeronautics and Astronautics, Vol. I.

Parkinson, B.W., Spilker, J.J. (Eds.), 1996b. Global Positioning System: Theory and Applications,American Institute of Aeronautics and Astronautics, Vol. II.

Pervan, B.S., Parkinson, B.W., 1997. Cycle ambiguity estimation for aircraft precision landing using theGlobal Positioning System. J. Guidance Cont. Dyn. 20 (4), 681–698.

Rekow, A., 1998. CDGPS based identification of farm tractor steering effectiveness. In: IEEE 1998Position, Location, and Navigation Symposium, IEEE, pp. 570–574.

Rekow, A., Jones, V.K., Parkinson, B.W., 1998. LMS identification of systems with dynamics and anoutput deadzone. In: Proceedings of the American Control Conference, pp. 2770–2774.

Taylor, J.R., 1982. An Introduction to Error Analysis, University Science Books.Wong, J.Y., 1993. Theory of Ground Vehicles. Wiley, New York.

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Automatic tractor guidance using carrier-phase differential GPS