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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Error is always present in the GPS guidance of a tractor along a desired trajectory. One way to reduce GPS guidance error is by improving the tractor positioning. The most commonly used ways to do this are either by employing more precise GPS receivers and differential corrections or by employing GPS together with some other local positioning systems such as electronic compasses or Inertial Navigation Systems (INS). However, both are complex and expensive solutions. In contrast, this article presents a simple and low cost method to improve tractor positioning when only a GPS receiver is used as the positioning sensor. The method is based on placing the GPS receiver ahead of the tractor, and on applying kinematic laws of tractor movement, or a geometric approximation, to obtain the midpoint position and orientation of the tractor rear axle more precisely. This precision improvement is produced by the fusion of the GPS data with tractor kinematic control laws. Our results reveal that the proposed method effectively reduces the guidance GPS error along a straight trajectory.

Agricultural vehicle guidance has undergone impressive advances in recent years [

Control laws for autonomous tractor guidance can be developed by modeling the tractor with a simple kinematic model [

One way to improve tractor GPS guidance performance is to employ a more precise tractor positioning system, which can be obtained by means of: (i) a better receiver, (ii) more effective differential GPS corrections or (iii) the fusion of GPS with other local positioning systems such as electronic compasses or Inertial Navigation Systems (INS). These solutions generally result in a more expensive system. This article presents, in contrast, a simple and low-cost method to improve GPS tractor positioning using only a GPS receiver as the positioning sensor. The method is based on placing the GPS receiver in a forward position where changes of the tractor’s orientation influence the GPS trajectory more, to obtain a more precise tractor positioning taking into account the data received from the GPS receiver and the tractor kinematic laws. Although researchers such as Stombaugh

A tractor has front-wheel steering and the rear wheels are forward-driven without being steered. The inputs to this system are therefore the driving speed, u, and the front wheel steering angle, δ. The tractor behavior can be described with a vector in the space state q, defined by the expression:

Assuming no-slip conditions on the wheels, the kinematic model of the vehicle is given by:

The position of the rear axle midpoint and the orientation of the tractor have to be known so that guidance control may be performed more easily. In this article, this information is obtained from the positions supplied by the GPS receiver, which is placed in a forward position (

The tractor position (x, y, θ) at the midpoint rear wheel axle can be obtained from the positions obtained from the GPS receiver (x_{g}, y_{g}), either with the kinematic tractor model or from geometric relations.

The geometric relationship between (x, y, θ) and (x_{g}, y_{g}) is:

Differentiation with respect to time in

From

Then, the position and orientation of the tractor can be obtained in a recursive way using the position obtained from the GPS in a forward position and the previous tractor state, as follows:

Equations (_{g}[n], y_{p}[n]) can be geometrically computed, and any set that does not match the real position received by the GPS may be discarded.

The computational cost of the previous section can be reduced by an approximation, taking into account

_{g}[n], y_{g}[n]) in

Our experimental tests revealed that when (x[n], y[n]) is near to (x[n − 1], y[n − 1]) this approximation is accurate. This happens when the speed of the tractor is low and the GPS provides positions at 1 Hz, or when the GPS position rate is higher than 5 Hz at any usual tractor speed.

The control law employed by Noguchi

The following steps were taken to choose the control gains k_{1} and k_{2} for the experiments: (i) initially, k_{1} = 0 was fixed and a search was made for the maximum value of k_{2} that did not cause oscillations in the motion of the system. (ii) Subsequently, k_{2} was fixed as 70% of this maximum value. (iii) The selected value for k_{2} was used to find the maximum value of k_{1} before oscillation occurred in the motion of the tractor. Then, k_{1} was fixed as 80% of this maximum value. This tuning was done at a speed of 1 m/s. This experimental procedure is similar to the second Ziegler-Nichols experimental method for tuning PID controllers [

GPS receivers produce errors.

Relative errors are smaller than absolute errors. In agricultural tasks, farmers notice relative error, which has led companies such as John Deere or Trimble to provide information on relative errors in their GPS-guidance-systems specifications that they refer to as “pass-to-pass accuracy”. Relative errors in tractor GPS guidance systems are usually subject to some conditions, such as Dilution of Precision (DOP) and time between passes. The common time between passes in data sheet specifications is 15 minutes, because this is the habitual time between tractor passes in medium-sized plots.

The relative errors in these 900 points are used in the section Simulation Results with Experimental GPS Errors.

Taking the previous data into account, a low cost Haicom HI-204III GPS receiver can technically be used to guide a tractor when the time between passes is short and precision is not required. The distance between passes could differ by large amounts from the desired when a long time between passes occurs.

A 6400 John Deere tractor was equipped with devices to perform autonomous tractor guidance (

The methodology of this article comprises certain simulations and field tests to prove the performance of the proposed system. Some simulations of autonomous tractor guidance along a straight line and a step response, according to

In these simulations, the kinematic model of the tractor presented in _{1} = 0.08, k_{2} = 0.5) were fixed. Experimental errors of

Additionally, field tests were performed with the system described in the experimental system section. The guidance along a straight path, the step response impulse and the maximum stable speed were tested.

Simulations of autonomous tractor guidance along a straight line and a step response were performed on a rough plot. The initial conditions for the tractor were (x, y, ψ, δ) = (0, 0, 0, 0), and the desired trajectory in the step response was y = 2 m.

Simulations with different forward GPS guidance positions were performed in order to compute the mean and Root Mean Square (RMS) values of the error. Simulation time was 14 minutes (14 × 60 = 840 points) and the reference trajectory was a straight line.

Three kinds of tests were performed with the GPS placed ahead of the rear axle at three different distances: the first found the error in the continuous tracking of a straight trajectory; the second tested the step response behavior; the third measured the maximum speed that keeps the guidance control. The control law used for the tests is given by _{1} = 0.08 and k_{2} = 0.5, values experimentally obtained at 1 m/s speed, following the tuning procedure described in the methods section.

It is worth mentioning that it was necessary to shift some of the trajectories on the X and Y axes in

The standard deviation of the field test was also computed when the tractor followed the straight line trajectory and the results are shown in

The Haicom HI-204III guidance GPS receiver offers a rate of position data of 1 Hz. With this low rate, only low speed autonomous guidance is possible. In the third field test, the desired trajectory was a straight line. From an initial value of 1 m/s, the tractor speed was gradually increased. It was observed that the guidance error was increasing as the speeds increased. Moreover, the maximum speed at which the system remained stable was measured for each GPS position. These values are shown in

Results show that the proposed method improves the guidance performance. This improvement can be intuitively understood.

A more formal explanation of this precision improvement is that more information for the positioning is used in the proposed system; therefore, the tractor positioning is more precise. The data GPS receiver is only used to identify the position of the tractor when the GPS receiver is placed on the tractor rear axle. But with the GPS in a forward position, the rear position of the tractor is computed with the positions of the GPS receiver together with the kinematic model of the tractor laws. GPS data and kinematic model tractor laws are fused in the proposed method to position the tractor.

The proposed method employs only the GPS receiver to provide tractor positioning data. There was a steering angle encoder on the tractor. However, data from this sensor was only used by the steering system controller to help it achieve the desire steering angle.

The initial orientation of the tractor was provided to the system in our simulations and field tests, but an arbitrary value can be employed in the proposed method when the initial orientation of the tractor is not known. In this case, the proposed method would work poorly at the beginning of the trajectory, but after approximately one hundred iterations, it would work fine.

The Kalman filter can be used in GPS tractor guidance. This filter can be used to process (i) only the GPS receiver data or (ii) multiple sensors’ data. When it is used to process only the GPS receiver data, this filter produces an effect of smoothing [

Simulations and field test results of this article concur with the previous intuitive discussion. Concretely:

The guidance errors in both

Differences were neither observed in the step response trajectories, nor in the simulations, nor in the field tests. This makes good sense, because when the tractor reaches the step, the distance to the desired trajectory is a large value of around 10 meters. The control law will give the maximum value in the δ angle with values close to 10 meters. A more precise positioning will not therefore produce a different response when the step happens.

A cause that usually makes a real control system unstable is its noise level. From the data in

This article, then, proves that by placing the GPS ahead of the tractor, it is possible to obtain more precise tractor positioning, and subsequently improve the autonomous GPS guidance performance of a guidance system that uses only a GPS receiver as the positioning sensor. This more precise tractor positioning is achieved because the proposed method uses the received GPS data and the kinematic model of the tractor together for tractor positioning, whereas positioning is only obtained with the received GPS data when the receiver is placed on the rear axle. Our results are in agreement with Stombaugh

Some limitations should be considered. The proposed method has been tested in a guidance system that uses only a GPS receiver as the positioning sensor. Tractor guidance systems sometimes employ more positioning sensors, such as: (i) tilt and (ii) orientation sensors. When a tilt sensor is employed together with the GPS receiver, it is expected that the proposed method will also improve the positioning, but with a lower ratio. This is because in this situation the proposed method reduces only the GPS positioning error. It does not influence the positioning error caused by the oscillations that the tractor suffers when it goes through common rough farm terrain, because this error is eliminated by the tilt sensor. When a digital compass, a gyroscope, an INS or whatever other orientation sensor is employed together with the GPS receiver, it is expected that the proposed system will not improve the tractor GPS positioning. This is because the proposed system improves the precision of the orientation data obtained by the GPS positioning, but it is expected that these data were much more precisely provided by the orientation sensor.

By placing the GPS ahead of the tractor, as proposed in this article, it is possible to devise a simple and low cost method a more precise tractor positioning, and thus improve the autonomous GPS guidance performance. This more precise tractor positioning is achieved because the proposed method uses the received GPS data and the kinematic model of the tractor together for tractor positioning, whereas positioning is only obtained with the received GPS data when the receiver is placed on the rear axle. The proposed method improves tractor positioning when only a GPS receiver is used as the positioning sensor. The improvement in the positioning occurs in the tractor orientation and in the positioning with respect to the transverse axis of the tractor’s orientation. It is expected that the improvement ratio will decrease when a tilt sensor is employed and that no improvement will be achieved when the tractor positioning system includes an orientation sensor.

This work was partially supported by the regional 2010 Research Project Plan of the Junta de Castilla y León, (Spain), under project VA034A10-2. It was also partially supported by the 2009 ITACyL project entitled “Realidad aumentada, Bci y correcciones RTK en red para el guiado GPS de tractores (ReAuBiGPS)”.

Tractor schematic and variables description.

The position and orientation at the midpoint of the rear axle (x, y, θ) have to be computed from positions received from the GPS placed in a forward position (x_{G}, y_{G}).

Two near positions of the tractor when following a curved trajectory.

Different positions of the Haicom HI-204III navigation GPS in the field tests. The navigation sensor position is highlighted with a red circle.

Flow charts of the procedures to simulate

Mean and RMS errors obtained in the simulations with different forward GPS guidance positions.

| ||||||||
---|---|---|---|---|---|---|---|---|

0 | 2.6 | 0.46 | 0.45 | 0.43 | 0.41 | 0.39 | 0.29 | |

0 | 3.0 | 0.53 | 0.51 | 0.49 | 0.47 | 0.45 | 0.34 |

Standard deviation errors obtained in the field tests with different forward GPS guidance positions.

16 | 4.7 | 3.7 | |

32 | 5.7 | 4.8 |

For each advance from the rear axle, this table presents the maximum speed of the tractor that kept the system stable.

1.5 m/s | 2.4 m/s | 2.6 m/s |