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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/).

Low-cost GPS receivers provide geodetic positioning information using the NMEA protocol, usually with eight digits for latitude and nine digits for longitude. When these geodetic coordinates are converted into Cartesian coordinates, the positions fit in a quantization grid of some decimeters in size, the dimensions of which vary depending on the point of the terrestrial surface. The aim of this study is to reduce the quantization errors of some low-cost GPS receivers by using a Kalman filter. Kinematic tractor model equations were employed to particularize the filter, which was tuned by applying Monte Carlo techniques to eighteen straight trajectories, to select the covariance matrices that produced the lowest Root Mean Square Error in these trajectories. Filter performance was tested by using straight tractor paths, which were either simulated or real trajectories acquired by a GPS receiver. The results show that the filter can reduce the quantization error in distance by around 43%. Moreover, it reduces the standard deviation of the heading by 75%. Data suggest that the proposed filter can satisfactorily preprocess the low-cost GPS receiver data when used in an assistance guidance GPS system for tractors. It could also be useful to smooth tractor GPS trajectories that are sharpened when the tractor moves over rough terrain.

Global Positioning Systems (GPS) are nowadays used in many agricultural tasks [^{®} [^{®} [

The aim of this work is to smooth the tractor trajectories acquired by low-cost GPS receivers, by improving the precision of their position data by decreasing their quantization error. To do so, an implementation of the Kalman filter was applied to the trajectory data provided by a low-cost GPS receiver that was placed on a farm tractor.

The following points are introduced below for a better understanding of this implementation: (i) error considerations for GPS receivers; (ii) the quantization effects in low-cost GPS receivers; (iii) the kinematic model of a tractor; (iv) the Kalman filter; and (v) the Kalman filter tuning.

Two kinds of errors can be defined for GPS receivers and GPS guidance systems [

In guidance system applications, where the time between each pass is relatively short and the trajectories are not saved from year to year, precision can be considered the most important variable. In this way, low-cost GPS receivers with 10 m accuracy but sub-meter precision will be alternatives in such agricultural task.

The most common chipsets that low-cost GPS receivers and embedded mobile computing devices integrate are the

Geodetic coordinates are not appropriate for agricultural data processing and are usually converted to Cartesian coordinates. When positions in geodetic coordinates with a quantization of only 8 digits for latitude and 9 for longitude are converted to Cartesian coordinates such as Universal Transverse Mercator (UTM) [

On the basis of their professional experience with ^{®} [^{®} [

A classic tractor has two front wheels that steer as well as two rear wheels that are straight-driven. The behavior of this kind of tractor vehicle is typically modeled following the tricycle vehicle model [

The Kalman filter is an efficient, recursive, mathematical algorithm that processes, at each step, inaccurate observation input data and generates a statistically optimal estimate of the subjacent real system state, by employing a prediction model and an observation model [

The basic functioning of the filter is conceptualized into two stages. The first stage is called the prediction stage, as it produces an

_{k}_{k}_{k}_{k}_{k}

Kalman filter tuning consists of setting the relevant parameter values for the related noise covariance matrices _{k}_{k}_{1}_{2}…_{n}^{T}_{i,j}=cov(_{i}_{j}_{i}_{i}_{j}_{i}_{i}_{i}

There are two main approaches to address the filter tuning: static and dynamic. Static tuning approaches only tune the filter just before its use. Static procedures are based on techniques such as the Autocovariance Least Squares (ALS) method [

This section comprises the work carried out in this survey. Section 2.1 outlines the particularization of the Kalman filter along with the prediction model that is employed. Section 2.2 deals with the method employed to tune the filter, in order to achieve a suitable performance with artificial data. Section 2.3 explains the experimental system employed in real field tests, to check the behavior of the proposed system.

The system presented in this study uses a particularization of the Kalman filter applied to GPS receiver data, in order to achieve path smoothing and partial restoration of the lost resolution in positioning data. The prediction model of this system is based on the tricycle kinematic model, seen in

The system takes the array of GPS measurements _{k}_{GPS}_{GPS}_{GPS}_{GPS}^{T}_{GPS}_{GPS}_{GPS}_{GPS}_{k}_{k}_{k}_{k}_{k}_{k}^{T}

Based on the system state, _{k}

Labeling the a priori state estimate vector as
_{k}

It may be seen from _{k}_{k}

It is necessary to define the observation model matrix, _{k}

As the system state vector and measured magnitudes perfectly match, the particular observation model matrix _{k}_{4} denotes the identity matrix of size 4 × 4.

The Kalman filter was particularized for tractor guidance as detailed in Section 2.1, and the

The Kalman filter was tuned using Monte Carlo Sampling techniques [_{i}_{i}

The proposed Kalman filter performance was evaluated with artificial data, through the following steps (

Finally, the proposed Kalman filter performance was evaluated with real GPS data by following the next steps (^{®} programming environment.

The materials employed in the experimental tests of this article were: a low-cost GPS receiver, a precise GPS receiver, a laptop computer, and an agricultural tractor. The low-cost GPS receiver was a

Each one of the straight trajectories of the experimental tests was marked with three stakes and were joined by a cord. The stakes were driven into the center of the trajectory and at each extreme.

GPS receiver data were read and processed with an application running on the laptop. The application, developed using

The tuning results, following the method presented in Section 2.2, were obtained using: (i) the artificially generated paths (

These matrices,

Simulations and real tests were executed to evaluate the Kalman filter performance. As

The properties and improvements of this proposed method are shown with the distance errors histogram (

The RMSE and the 95

Another illustrative graph, showing the behavior of the filtering along straight lines, is the heading angle histogram. As

The standard deviation and the 95

Another heading angle histogram was also computed, this time with experimental data acquired from a low-cost

The standard deviation and the 95

The main finding of the present study is that implementation of the Kalman filter can reduce the quantization errors in the positioning of tractors equipped with some low-cost GPS receivers by 43%. Moreover, it reduces by 75% the standard deviation of the heading angle.

Several studies have shown that GPS accuracy can range from 1–2 cm to 100 m [^{®} [^{®} [

In a previous study, this research team has proposed a way of improving the precision of GPS tractor positioning [^{®} [^{®} [

Tractors usually move over rough surfaces, and then, although a tractor goes along a straight trajectory, the GPS receiver will acquire a sharp trajectory. This is due to the lateral vibrations experienced by the GPS receiver, which is placed over the tractor cab, two or three meters above-ground. At this position, the vertical displacement of the tractor wheels is converted into lateral vibrations. As a numerical example, a typical 10 cm vertical displacement of one rear wheel of the tractor can lead to a lateral displacement of the GPS receiver of up to 30 cm. Our implementation of the Kalman filter will be useful in these situations, because it will smooth the sharp trajectory due to vibrations and will provide more accurate heading information, close to the real data. Similar studies to remove noise from GPS data have been presented in the literature [

Overall, our data suggest that the proposed filter is adequate for data preprocessing of some low-cost GPS receivers, when used in GPS assisted-guidance systems for tractors. It also could be useful to smooth the GPS trajectories that are sharpened due to the tractor moving over rough terrain.

One limitation of this study is that the proposed Kalman filter reduces the quantization error in GPS receivers that provide geodetic latitude and longitude with eight and nine digits, but, in GPS receivers that provide more digits, the reduction in the quantization error does not exist or is negligible. Microelectronics technology progresses quickly. Hopefully, in a few years all GPS receivers in the market, high end and low cost, will be equipped with chipsets that provide positioning data with enough digits, so that the quantization effects are negligible. Nevertheless, today, our proposed Kalman filter is at present useful for processing the data of some low-cost GPS receivers. Moreover, in the future the filter will be useful for the preprocessing of GPS trajectories on tractors moving over rough surfaces. A second drawback of our proposed Kalman filter is that, besides smoothing errors, as a side effect real deviations are also smoothed. This effect could be negative in some systems, as for example the used in GPS assisted guidance of tractors.

Further studies could be conducted. A more detailed quantitative analysis about inherent delays of the proposed system, both along both straight and curve paths, could be addressed. Besides, because most low-cost GPS receivers provide positioning information at 1 Hz rate, simple modifications to the Kalman filter proposed in this paper could be employed to increase the positioning rate. Modifications could also be used to fuse the data from low-cost GPS receivers with local positioning systems as gyroscopes and compasses employing, for example, ^{®} or ^{®} boards. Since nowadays most modern smartphones also include gyroscopes and compasses, it will be possible to deploy this system using a smartphone that fuses the data from its own sensors.

In summary, certain low-cost GPS receivers, such as those equipped with

The authors declare no conflict of interest.

Graphic illustration of precision and accuracy concepts.

RMC Sentences acquired from two different GPS receivers. The yellow-highlighted numbers represent the latitude and longitude geodetic coordinates. It can be observed that the high end

Illustration of the quantization effect on the positions supplied by a GPS receiver, showing that quantified trajectories register (i) position errors; and (ii) speed errors, as shown by the variable distances between the blue rectangles; and (iii) heading error, which are higher in trajectories nearby, but different from, the direction of any coordinate axis.

Tractor schematic and description of variables.

Stage diagram of the Kalman filtering loop.

Black box diagram of the system implementation.

(

Illustration of the quantization and Kalman filter process conducted over an ideal sample trajectory. The RMSE is defined, according

(

Results of positioning improvement at a 5 Hz update rate, constant speed of 5 km/h (3.1 mph) and 60° heading angle along a straight path (

Histogram of distance errors, with 5 Hz update rate, using the simulations along the 18 straight lines shown in

Heading angle (

Heading angle (

Statistical parameters of the distribution of errors, before and after applying the proposed Kalman filter, along the 18 simulated straight paths (

Distances Root-Mean Square Error (RMSE) (cm) | 6.56 | 3.74 |

Distances 95 |
8.48 | 4.31 |

Statistical parameters of the heading angle distributions, before and after applying the proposed Kalman filter, along a straight path with a 60° heading angle, in simulations with artificial data.

Standard deviation (°) | 6.9362 | 1.8301 |

95 |
29.0661 | 7.2651 |

Statistical parameters of the heading angle distribution, before and after applying the proposed Kalman filter, along a straight path with a 60° heading angle, in real field tests.

Standard deviation (°) | 16.5594 | 4.1326 |

95 |
55.1185 | 14.7193 |