ASAE Journal Article
Development and Validation of Globally Asymptotically Stable Control Laws for Automatic Tractor Guidance
J. Gomez-Gil, J.-C. Ryu, S. Alonso-Garcia, S. K. Agrawal
Published in Applied Engineering in Agriculture Vol. 27(6): 1099-1108 ( Copyright 2011 American Society of Agricultural and Biological Engineers ).
Submitted for review in September 2009 as manuscript number IET 8216; approved for publication by Information & Electrical Technologies Division of ASABE in June 2011.
The authors are Jaime Gomez-Gil, Lecturer, Department of Signal Theory, Communications and Telematic Engineering, University of Valladolid, Valladolid, Spain; Ji-Chul Ryu, Postdoctoral Fellow in the Department of Mechanical Engineering, Northwestern University, Evanston, Illinois, USA; Sergio Alonso-Garcia, Engineer, Department of Signal Theory, Communications and Telematic Engineering, University of Valladolid, Valladolid, Spain; and Sunil Kumar Agrawal, Professor, Mechanical Systems Laboratory, Department of Mechanical Engineering, University of Delaware, Newark, Delaware, USA. Corresponding author: Jaime Gomez-Gil, E.T.S. Ing. Telecomunicación, Camino del Cementerio s/n, 47011 Valladolid; phone: 0034-983423660; e-mail: email@example.com.
Abstract. Automatic guidance of agricultural vehicles requires the use of sensors and control techniques. While the Global Position System (GPS) is widely used as a position sensor, different control laws have been used in the scientific literature based on geometry, heuristics, and models. In this article, two new control laws for navigation control of agricultural tractors were developed with one for following straight lines and the other for tracking circular arcs. These two controllers have virtually the same structure based on the tractor's kinematic model. They have no singularity point in the state space. Also, global asymptotic stability is guaranteed with the control laws. Guidance with the proposed controllers in straight lines and circular arc trajectories was simulated and field-tested. The experiments were performed with a tractor equipped with an automatic GPS guidance system. In both simulation and experiment, the control laws led the tractor to converge to the desired trajectories of straight lines and circular arc paths under three different low speeds of 1, 2, and 3 m/s and four different initial conditions that are 90 ° apart. The simulation and experimental results show the practical validity of the proposed controllers.
Keywords. Control, Global Positioning System (GPS), Guidance, Agricultural vehicles.
Guidance and control of agricultural vehicles has experienced impressive advances in the last decade (Keicher and Seufert, 2000; Reid et al., 2000; Auernhammer, 2001; Zhang et al., 2002; Slaughter et al. 2008). These advances have been facilitated by the development of positioning systems using GPS (Lechner and Bumann, 2000; Borgelt et al. 1996), machine vision (Wilson, 2000; Subramanian et al., 2006), or both for improved precision and reliability (Kalman, 1960; Barshalom and Forman, 1987; Zhang et al., 1999).
Scientific literature provides information about different implementations employed in the guidance of agricultural tractors. Some used machine vision to look ahead and obtain the next via point along a path (Gerrish et al., 1997; Pilarski et al., 2002). This next via point can then be reached using a simple control law proportional to an angle difference between the vehicle orientation and the goal (Kise et al., 2005), by steering to reach the goal following a circular arc (Ku and Tsai, 1999; García-Pérez et al., 2008) or using the orientation information from the theory of screws (Wit et al., 2004).
Even though motion of a tractor is governed by non-linear models (Feng and He, 2005), PID controllers have yielded practical results (Benson et al., 2003; Subramanian et al., 2006). Other strategies are based on linearization of the desired path along with a P controller to reduce distance and orientation errors (Noguchi et al., 1997; Stoll and Kutzbach, 2000; Nagasaka et al., 2004). Other implementations use linearization, followed by a LQR controller (O'Connor et al., 1996; Thuilot et al., 2002). Fuzzy logic controllers and neural networks have also been used to control steering of agricultural vehicles (Cho and Ki, 1999; Cho and Lee, 2000; Ashraf et al., 2003).
Some of the control laws do not have the property of global asymptotic stability. In other words, the tractor may not converge to the desired trajectory when it starts far away from the desired trajectory. Some other controllers do have a mathematical proof of stability, but they have singularities in certain conditions where the controller will not work properly. To the authors' knowledge, no control laws have been reported to have these both properties in the agricultural scientific literature. Therefore, the goal of this study is to develop simple but effective controllers for agricultural tractors without these limitations. Furthermore, experimental validation of the proposed controllers is also an important objective of this study as the design of the controllers. These controllers are based on the nonlinear kinematic model of a tractor and were designed for following straight line trajectories and circular arcs, respectively, so it is expected that the tractor could follow any kinematically admissible trajectory. In addition, the two controllers have virtually the same structure. The performance of the controllers was assured using Lyapunov stability theory and was experimentally tested.
Materials and Methods
The vehicle has a steered front wheel and the rear wheels are forward-driven without being steered (fig. 1). Therefore, the inputs to this system are the driving speed, u, and the front wheel steering angle, d . The position of the midpoint O of the rear wheel axle is ( x, y ). ? is the orientation of the vehicle with respect to X-axis. d is the steering angle of the front wheel with respect to forward direction of the vehicle. L is the length from O to the center of the front wheel.
Figure 1 . The schematic of a farm tractor in Cartesian coordinates.
Assuming no-slip condition on the wheels, the kinematic model of the vehicle is given by:
( 1 )
This model is typically called the bicycle model. This kinematic model will be used for the controller design in the next section and the kinematic model-based controllers are suggested to be used at lower speeds, for example, under 4.5 m/s since they cannot represent dynamic effects such as slipping (Karkee and Steward, 2010).
In this section, two controllers are presented; one for a farming tractor to follow straight lines and the other for circular arcs. The two proposed controllers have virtually the same structure and their asymptotic global stability are proven using Lyapunov stability.
Controller for Following a Straight Line
When a farming vehicle follows a straight line, the coordinates can be reoriented, without loss of generality, so that x-axis is along the desired straight line and y-axis the deviation from this straight line. Therefore, the desired trajectory for this system is described by y = 0. Moreover, the driving speed u is considered to be a (positive) constant a since the driving speed is usually kept constant while operating in the field. With a positive driving speed, the problem of following a straight line can be considered a stabilization problem to the origin (0, 0) of y and ? .
On introducing an input transformation:
( 2 )
where ? is a new input, the kinematic variables that are of interest in the control are given by:
( 3 )
Theorem 1: For the system described in equation 3, the equilibrium point, y = 0 and ? = 0, is globally asymptotically stable if the control ? is chosen as:
( 4 )
where k 1 and k 2 are positive constant control gains.
Proof: Let us consider a Lyapunov function:
( 5 )
The time derivative of V is:
( 6 )
( 7 )
On substitution equation 4 into equation 7, can be expressed as:
( 8 )
( 9 )
Note that when ? = 0 as it expected from equation 7. Although is only negative semi-definite, it can be proven that the origin is a globally asymptotically stable equilibrium point by LaSalle's theorem since ( y , ? ) = (0, 0) is the only point in the invariant set of .
Note that the control input ? is a continuous function. In addition, it is independent of the velocity a . In fact, the driving speed u (t) does not have to be a constant because, in that case, and LaSalle's theorem still can be applied except when u = 0, which is a trivial case.
Finally, the original input d can be obtained from equation 2 as:
( 10 )
Controller for Following a Circular Arc
Using the polar coordinate system, the kinematic equation 1 described in the Cartesian coordinates can be converted into:
where ? and f are the polar coordinates as described in figure 2 and ? is the same in the orientation of the tractor as in Cartesian coordinates.
Figure 2. The farm tractor in the polar coordinates.
Then, let us introduce the error variables defined by:
where ? d denotes the desired radius of curvature. When a tractor follows an arc of a circle with radius, ? d , it satisfies:
Consequently, similar to the straight line following controller in the previous subsection, the problem of following a circular arc can now be considered a stabilization problem to the origin of ? e and ? e .
Equation 11 can be further rewritten in terms of ? e and ? e as:
By introducing an input transformation similar to the one in the design of the straight line following controller, given by:
the part of equation 15 that we are interested in is expressed as:
Note that these equations are in the same form as equation 3.
Theorem 2: For the system described in equation 17, the equilibrium point, ? e = 0 and ? e = 0, is globally asymptotically stable if the control ? is chosen as:
where k 1 and k 2 are positive constant control gains.
The proof of this theorem is the same as for Theorem 1 with substitution of ? e and ? e for y and ? , respectively.
The original input d can be obtained from equation 16 as:
( 11 )
Comparison with other Controllers
In this section, the proposed controller designed in the previous section is compared with two of the most commonly used controllers. Their control laws are expressed (with notations used in this paper) as
Controller I: . This control law applies a P control to correct for deviation distance and the orientation error (Noguchi et al., 1997; Stoll and Kutzbach, 2000; Nagasaka et al., 2004).
Controller II: . This control law is based on linearization along with a Linear Quadratic Regulator (O'Connor et al., 1996; Thuilot et al., 2002).
Controller I does not have a proof for global stability, hence, is limited in its performance. For example, the controller cannot make a tractor converge to the desired trajectory when it starts far away from the desired trajectory. Even though controller II has a mathematical proof for stability, it has singularities at cos ? = 0. In other words, if the orientation of the vehicle is oriented perpendicular to the desired trajectory, the steering angle input will be zero. In this situation, the vehicle goes straight and cannot follow the desired trajectory. In addition, even near the singularity, the steering angle input is close to zero. In this situation, even though the vehicle is able to eventually converge to the desired trajectory, the actual path of the vehicle may not be desirable. One example of such a case is shown in figure 3 that plots four trajectories with the same initial position ( x (0), y (0)) = (0, 5) m, but with different initial orientations of 0º, 45º, 80º, and 88º. This figure shows that Controller II does not work properly with initial orientations near 90º. Although different choices of control gains, vehicle speed and initial conditions may result in different trajectories, the phenomenon shown in the figure, i.e., large deviations from the desired path, will not disappear, since the steering angle input d becomes close to zero when cos ? is close to zero.
Figure 3. Trajectories of the proposed controller and controller II with different initial orientations.
Experiments were conducted on a two-wheel drive tractor (model 6400, Deere and Company, Moline, Ill.) with a maximum power of 73 kW (100 hp). The steering control system was implemented using a RE-30 Maxon DC motor with a GP 32 Maxon reducer (Sachseln, Switzerland). A pulley was used to join the DC motor with the steering wheel of the tractor as shown in figure 4b. The reduction rate was 14:1 for the reducer gear and 6:1 for the pulley.
(c) Figure 4. (a) Farm tractor used, (b) DC motor connected to the steering wheel, and (c) magnetic encoder.
An absolute magnetic encoder (US Digital MA3, Vancouver, Wash.) was used to measure the steering angle. A power stage (MD03) drives the DC motor with a maximum current of 20 A. A single GPS receiver Trimble R4 unit (Sunnyvale, Calif.) located at the middle of the tractor rear axle was employed. This unit received network RTK corrections from a Virtual Reference Station (VRS) by means of a GPRS connection. The GPS provided latitude, longitude location as well heading information with a rate of 5 Hz. A mean filter was applied to the last three heading data in order to smooth the orientation data. A portable laptop computer running Windows XP with an Intel T4200 processor was used to read positions from the GPS, convert the positions to the grid based Universal Transverse Mercator (UTM) coordinate system, calculate the steering angle d using the control law, and send the desired steering to the controller box by serial connection. Under the Windows XP operating system, National Instruments' Lab Windows CVI (Austin, Tex.) was used as the development environment for the implementation. The steering control was implemented in a Microchip PIC16F877A microcontroller inside the controller box. It reads the actual steering from the encoder, receives the desired steering from the laptop, and applies fuzzy logic to compute the desired speed and direction of the DC motor (Carrera-González et al., 2010). The detailed block diagram of the guidance implementation over the laptop and over the controller box is shown in figure 5. A maximum steering angle rate of 30 deg/s is achieved through this controller box. The detailed description of the controller box is beyond the scope of this article, so this article only analyzes the high level guidance control, that is, the outer loop of figure 5.
Figure 5. Block diagram of guidance implementation.
The experiments were done on a flat land with loose soil in Pozal de Gallinas, Spain. Due to the mechanical characteristics of the tractor steering, the maximum steering angle was limited to 30 ° . The tractor length was L = 2.3 m. The starting position ( x (0), y (0)) was (0, 5) m. The four initial conditions for the orientation ? (0) were 0, 90 ° , -180 ° , and -90 ° . The tests were performed at three different speeds of 1, 2, and 3 m/s for each initial condition.
Adjusting the Control Gains
In the (kinematics-based) simulations, the control gains can be determined by only considering the shape of the path the tractor would create while converging to the desired trajectory. However, the actual gains in practical implementation need to be selected more carefully since they can be affected by other factors such as a delay in executing the steering commands, the limited GPS update frequency and position errors, and other unmodeled dynamics. To choose the control gains k 1 , k 2 for the experiments, we took the following steps: (i) Initially, we fixed k 1 = 0 and searched for the maximum value of k 2 that did not cause an oscillation in the motion of the system at the forward speed of 1 m/s. (ii) Next, we took 80 % of this maximum value for k 2 in order to give some margin for tuning k 1 such that the system's motion stays stable when k 1 is increased. (iii) With this selected k 2 , we searched for the maximum value of k 1 that did not cause an oscillation in the motion of the tractor. This procedure of gain tuning worked satisfactorily throughout the experiments.
One can use different sets of control gains depending on the region where the tractor travels using a technique called gain scheduling because the control gains may need to be adjusted when the starting position is significantly changed. In our tests, although the gains should have been determined separately for each speed condition in order to have optimal performance, the same values of control gains were used for all three speed conditions of 1, 2, and 3 m/s in order to examine the influence of speed change to the system performance.
Controller for following a Straight Line
In the simulations the constant driving speed a was set at 1 m/s. The farm tractor's length L was selected to be 2.3 m which is the actual length of the tractor to be used in experiments. The initial conditions for ( x (0), y (0)) were given as (0, 5) m combined with four different initial conditions (0, 90 ° , -180 ° , -90 ° ) of ? (0) so that the initial errors in y and ? were present in order to check the control performance. Since the proposed controller is nonlinear, there is no straightforward way to choose appropriate control gains. By performing several simulations, we first found out the qualitative characteristic of each gain: (i) when k 2 was fixed, the increase of k 1 tended to shift the transient trajectory toward the negative X-axis before converging to the desired trajectory. In addition, it increased overshoot of the trajectory. (ii) The decrease of k 1 , when k 2 was fixed, caused slower convergence to the desired trajectory. (iii) when k 1 was fixed, the increase of k 2 caused slower convergence to the desired trajectory, which was a similar effect caused by the increase of k 1 . (iv) The decrease of k 2 , when k 1 was fixed, caused oscillation of the trajectory before the convergence. With these characteristics in mind, the control gains were chosen as ( k 1 , k 2 ) = (0.4, 1.1) by trial and error. The desired and actual trajectories under the proposed controller are shown in figure 6a.
Figure 6. Convergence to the desired trajectory of four trajectories with different starting orientations.
As expected, for all four different initial orientations the actual trajectories converge to the desired trajectory, y = 0, as show n in figure 6a. The computed control inputs d corresponding to each initial condition are presented in figure 6b.
The range of computed steering angles (fig. 6b) is not applicable for practical purposes because the largest steering angle that can typically be achieved mechanically is about 30 ° . Hence, the steering angle input d was limited to ±30 ° in the second simulation set presented in figure 7. That is, when the computed control input d exceeded ±30 ° , it was set to ±30 ° . In addition, the steering rate was constrained to 30 ° /s, which is also necessary for practical purposes.
Figure 7. Convergence to the desired trajectory of four trajectories with different starting orientations, a steering angle limit of 30 ° , a steering rate limit of 30 ° /s, and a speed of 1 m/s.
Controller for Following a Circular Arc
The simulations for this controller were performed under the same conditions: 1 m/s of the driving speed, L = 2.3 m, and the starting point of (0, 5) m with four different initial orientations ? of (0, 90 ° , -180 ° , -90 ° ). The control gains k 1 , k 2 can be chosen similarly by taking advantage of using the same qualitative nature due to the same structure of the control law. As a result, the same value of control gains, ( k 1 , k 2 ) = (0.4, 1) was used in this simulation . The desired radius of a circle ? d is taken ? d = 10 m. The desired and actual trajectories under the proposed controller are shown in figure 8a.
Figure 8. Convergence of four trajectories with different starting orientations to the desired trajectory.
The actual trajectories converge to the desired trajectory for all four different initial orientations as shown in figure 8a. The computed control inputs d corresponding to each initial condition are presented in figure 8b.
Since the range of computed steering angles (fig. 8b) is not applicable in practice, the steering angle limit of 30 ° as well as the 30 ° /s of the steering rate constraint was taken into account in another set of simulations presented in figure 9.
Figure 9. Convergence to the desired trajectory of four trajectories different starting orientations with a steering angle limit of 30 ° , a steering rate limit of 30 ° /s and a speed of 1m/s.
The adjustment of the control gains was done with the experimental system as described in the Materials and Methods section. The iterative process to obtain the gain parameters was performed at 1 m/s speed. ( k 1 , k 2 ) = (0.06, 0.25) was experimentally obtained for the controller to follow a straight line and ( k 1 , k 2 ) = (0.04, 0.3) was experimentally obtained for following a circular arc.
The trajectories under the proposed controller are shown in figure 10a. The computed control inputs d corresponding to each initial condition are presented in figure 10b. The simulation results with the same control gain used in the experiments are shown together with the experimental results. As in the simulations, for all four different initial orientations, the actual trajectories in the experiment converge to the desired trajectory of y = 0. The experiments were conducted five times for each initial condition, but only one of them is presented in the graph because the repeated experiments showed negligible difference.
The experiment result of the 0 deg initial orientation gives a step response performance with an input magnitude of 5 m. The settling distances to stay within ±5 % of the step input were 17, 22, and 26 m for 1, 2, and 3 m/s, respectively. In addition, the overshoots were 9%, 14%, and 48% and the peak distance (the distance at the maximum overshoot point) were 16.5, 12.5 and 12.9, respectively for 1, 2, and 3 m/s of the tractor speeds. The root mean square (RMS) error of the actual trajectory beyond the settling distance was 0.055, 0.058, and 0.067 m, respectively.
The difference between the experiments and simulations is mainly due to the delays in the guidance system. It is shown that in figure 10 that these delays become more significant at higher speeds.
Figure 10. Comparison between the actual and simulated trajectories of the tractor with four different starting orientations at at 1, 2, and 3 m/s speeds, with a steering rate constrains of 30 ° /s and a steering angle limit of ± 30 ° .
Figure 11. The computed steering angle input d used during the experiments and simulation for the 1 m/s trajectory in figure 10.
The trajectories under the proposed controller following circular arcs with a radius of curvature of 15 m are shown in figure 12. The test results in figure 12a show the convergence to the desired trajectories. The RMS error after the tractor reached within ±5 % of the initial distance offset, i.e., ±0.25 m, from the desired circle trajectory were 0.17, 0.33, and 0.40 m, respectively for 1, 2, and 3 m/s tractor speeds.
Figure 12. Comparison between the actual and simulated trajectories of the tractor with four different starting orientations at 1, 2, and 3 m/s speeds in the circular trajectory, with a steering rate limit of 30 ° /s and a steering angle limit of ± 30 ° .
Figure 13. The computed steering angle input d used during the experiments and simulation for the 1 m/s trajectory of figure 12.
Since the steering angle is typically limited to about 30 ° by its mechanism, to cope with this limitation the computed control input d was set to ±30 ° when it exceeded ±30 ° . The mathematical proof of the global asymptotic stability with such an input constraint is not provided in this article. However, interestingly, we observed that the system still converges to the desired trajectory in our experiments as well as in the simulations.
As it can be seen in figures 10 and 12, the trajectories of the tractor from the experiments show very similar qualitative behaviors to the simulation results. The trajectory differences depending on the tractor's speed are also well captured in the simulation results conducted with the constraints on the steering angle and its rate.
The improvement of the guidance accuracy could be made with the following suggestions: (i) the use of an inertial navigation system (INS) that provides relative positioning information to be fused with the GPS data, (ii) the use of a faster steering system that acts directly over the hydraulic system as opposed to typical steering systems which act mechanically over the steering wheel including the one used in this study, (iii) the use of a terrain compensator that eliminates the roll and pitch effects over the GPS antenna position, especially in non-flat terrains, and (iv) the use of multiple GPS receivers that allow more accurate calculation of orientation, roll and pitch data of the tractor.
In this article, two new control laws for agricultural tractor navigation control were developed; one for following straight lines and the other for circular arcs. Unlike other common controllers used for farming vehicles, the controllers proposed in this article have the following advantages: (i) global asymptotic stability is guaranteed via a mathematical proof and, (ii) it has no singularity. The performance of the controller was tested through simulations and experiments. From the experimental results along with the simulation, we can conclude that the proposed controllers which possess the two desirable properties mentioned above can be used for automatic guidance control of farming vehicles. Future work will include controller modifications by taking into account the following issues: (i) mathematical proof of convergence with the constraints on the steering angle and steering angle rate, (ii) use of full dynamic model, and (iii) robustness to wheel slip and parameter uncertainty.
This work was supported by VA064A08 and VA034A10-2 grants from Junta de Castilla y León, Spain. This research was partly supported by WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (No. R32-2008-000-10022-0).
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