These oscillations are especially noticeable when they are acquired at low speeds and with headings close to the direction of a coordinate axis (Figure 3).Figure 3.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 …On the basis of their professional experience with Agroguia? [7] and Tractordrive? [8], the authors state that these oscillations negatively affect the use of low-cost GPS receivers in GPS assisted-guidance systems for tractors.1.3. Kinematic Model of a TractorA 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 [19]. In this model, the system inputs are the vehicle speed modulus, u, and the front-wheel steering angle, ��. The tractor behavior can be described with a vector state, q, defined by the expression:q=[x,y,��,u,��]T(1)and, with the equations of its kinematic model, assuming non-slip conditions on the wheels, given by:x�B=u?cos��x�B=u?sin�ȦȨB=u/L?tan��(2)where O �� (x,y) is the midpoint of the rear wheel axle, x and y represent the position in Cartesian coordinates of O, �� is the orientation of the vehicle with respect to the positive X-semiaxis, �� is the steering angle of the front wheels with reference to the vehicle’s forward direction, and L is the length from O to the center of the front axle, i.

e., the distance between both axles. Figure 4 shows a schematic of the system and the variables.Figure 4.Tractor schematic and description of variables.1.4. The Kalman FilterThe 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 Cilengitide state, by employing a prediction model and an observation model [20].The basic functioning of the filter is conceptualized into two stages. The first stage is called the prediction stage, as it produces an a priori system state estimate from the previous state, by using a system evolution prediction model.

The second stage, known as the update stage, takes into account measurements in the system to produce an a posteriori state estimate, by correcting the previous a priori estimate. This two-stage process starts with an initial estimated state, x^0?, and is repeated in a loop recursively until filtering ends (Figure 5).Figure 5.Stage diagram of the Kalman filtering loop.Figure 5 summarizes the steps in each stage of the Kalman filtering process and it presents the matrices that are involved and the steps followed to implement the Kalman filter [20,21].