CONTROLO th Portuguese Conference on Automatic Control

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1 CONTROLO th Portuguese Conference on Automatic Control University of Trás-os-Montes and Alto Douro, Vila Real, Portugal July 21-23, COMPENSATION OF SLIDING EFFECTS IN THE CONTROL OF TRACKED MOBILE ROBOTS R. González F. Rodríguez J.L. Guzmán M. Berenguel Departamento de Lenguajes y Computación, Universidad de Almería, Ctra. Sacramento s/n, E-04120, Almería (Spain) {rgonzalez,frrodrig,joguzman,beren}@ual.es Abstract: This paper deals with the trajectory tracking control problem of tracked mobile robots in presence of sliding conditions. The proposed control law consists in the modification of a well-known control algorithm based on the feedback linearization technique, where additional terms have been included in order to compensate the sliding effects. A kinematic model including sliding features has been validated by means of real tests using a tracked mobile robot available at the University of Almería (Spain). Furthermore, simulation results are presented comparing the modified control law with the original one. Keywords: Autonomous Mobile Robots, Robot Control, Sliding, Feedback linearization 1. INTRODUCTION Tracked Mobile Robots (TMR) are a kind of mobile robots designed specially to work in off-road situations. In most of these environments, TMR are affected by sliding situations, which are usually presented with rough and weak conditions of the terrain. Many research works have been developed for paved terrains where the sliding is low (less than 10%) and it can be partially neglected. However, when a mobile robot is designed to navigate in unpaved terrains, the sliding has a high influence in the guidance and the controllability of the robot (Wong, 2001). There exists numerous works where this control problem has been solved for Ackermann vehicles. For instance, in (Fang et al., 2006) the problem was addressed for an agricultural vehicle where adaptive control techniques were used to face the sliding lateral effects. Some solutions have also been proposed for wheeled mobile robots (WMR). (Klancar and Skrjanc, 2007) and (Shiller and Gwo, 1991) propose bounds on the velocity and acceleration to avoid the vehicle sliding. This consideration is not appropriate for TMR because the terrain where they operate is intrinsically loose and it produces a no controllable sliding. Considering that the TMR used in this paper works at low velocities, only longitudinal sliding effects need to be considered since lateral sliding only appears when the vehicle turns at high velocities (Kitano and Kuma, 1977), (Shiller et al., 1993). The proposed control algorithm consists in a linear feedback controller based on a modification of that described in (Canudas et al., 1997), where an additional factor is included in the feedback linearization procedure to compensate the longitudinal sliding effect. This is the main contribution of the paper. The motivation of this work comes from a research project at the University of Almería (Spain) which is devoted to the development of a tracked mobile robot (called Fitorobot) to perform different tasks in greenhouses, specially those related with spraying activities (González et al., 2007). Greenhouses are composed of sandy and loose soils, which have an important sliding component (10-15%) (Le, 1999). The paper is organized as follows: the kinematic model of a tracked mobile robot including the sliding effect is described in the section 2. The next section is devoted to describe the modified control law. Real tests for modeling validation and simulation results of the control algorithm are discussed in section 4. Finally, some conclusions are summarized.

2 KINEMATIC MODEL OF A TRACKED MOBILE ROBOT 2.1 Estimation of the sliding As was commented above, in this paper only longitudinal sliding effect has been considered because in a greenhouse the mobile robots usually works at relatively slow velocities ( [m/s]). Lateral sliding phenomenon is zero for straight-line movements and it only appears when the vehicle turns to high velocities (large centrifugal force) (Kitano and Kuma, 1977) and (Shiller et al., 1993). The longitudinal sliding can be determined using the relation between the theoretical angular velocity of the track and the current forward velocity of the vehicle (Wong, 2001) ẋ ẏ = cos θ 0 [ ] sin θ 0 vm cos θ 0 [ ] sin θ 0 σ (4) ω θ 0 1 m λ 0 1 where v m = r 2 [φ mr+φ ml ] is the linear or translational velocity and ω m = r 2 [φ mr φ ml ] is the angular or rotational velocity of the vehicle. σ and λ are the sliding terms defined as σ = rφ ri r + rφ l i l 2 λ = rφ ri r rφ l i l b (5) (6) From equations (4, 5,6) it can be observed how the term representing the sliding can be interpreted as a typical disturbance. i j =1 v m (1) rφ mj where j = r/l for right/left track respectively, i j is the sliding of associated track. In our real tests, the linear velocity (v m ) is measured using a radar sensor and, the angular velocity of the tracks (φ mj ) is measured using incremental encoders. 2.2 Kinematic Model with Sliding As well-known the motion of a mobile robot is expressed as the relative movement of the attached frame (R) to the mid-point of the vehicle (O ) with respect to a global or inertial base frame (G), such as Figure 1 shows. The classic kinematic model of a WMR can be recalled in (Canudas et al., 1997). Longitudinal sliding can be approximated as a factor reducing the rolling velocity of the tracks. That is, in presence of sliding the linear velocity of the tracks differs from the angular velocity by a factor of (1 i j ) in the following way v r = rφ mr (1 i r ) v l = rφ ml (1 i l ) (2) where v r and v l are the linear velocities for the right and left tracks respectively. Considering this effect, the classic kinematic model can be extended including the sliding effects as follows (Le et al., 1997) ẋ = r 2 [φ mr(1 i r )+φ m l(1 i l )] cos θ ẏ = r 2 [φ mr(1 i r )+φ m l(1 i l )] sin θ (3) θ = r b [φ mr(1 i r ) φ ml (1 i l )] After some basic manipulations the extended kinematic model can be represented as Fig. 1. Kinematic model of a TMR 3. CONTROL OF A TRACKED MOBILE ROBOT It is well known that a nonholonomic system, such as a TMR, cannot be stabilized by smooth static state feedback laws. This systems fails in the Brockett s Condition for the existence of a continuously differentiable control law (the dimension of the state space is three and the number of control signals are only two) (Brockett, 1983). In order to solve this fact, discontinuous feedback control laws (Bloch and McClamroch, 1989) and time-varying continuous feedback control laws (Samson and Ait-Abderrahim, 1991) have been commonly used. In (Canudas et al., 1997) a linear control law is presented based on the dynamic feedback linearization technique, which is used to linearize the kinematic model of a unicycle-type robot. This section describes a modification of the algorithm proposed by (Canudas et al., 1997) to consider and compensate sliding effects. The main idea consists in using the same design procedure proposed by (Canudas et al., 1997) but using the kinematic model described by (4) to consider sliding conditions. Figure 2 shows the diagram of the proposed control approach, being the main blocks described in the next section.

3 526 Fig. 2. Block diagram of the control approach 3.1 Trajectory Tracking The trajectory tracking problem can be seen as the problem where a mobile robot must follow a virtual mobile robot representing the desired positions and velocities (see Figure 3). Hence, the objective is to find a feedback control law (Canudas et al., 1997) such that u =(v ω)=f(ρ, ρ ref,v ref,ω ref ) (7) lim e(t) = lim [ρ(t) ρ ref(t)] = 0 (8) t t where ρ =[x, y, θ] T and ρ ref =[x ref,y ref,θ ref ] T. In (8), approximated tracking values are required instead of an exact error penalization because of this could lead to a very aggressive control signals. In this paper will be followed the notation of (Canudas et al., 1997). For this reason, the control signals are called (v, ω). For a real implementation of the controller in a mobile robot, these signals must be translated to angular velocities to each track, that will be controlled by two PIDs. Such as described in (Canudas et al., 1997) and considering a coordinate change for the global frame, the dynamic equation of the error can be expressed as e = e x cos θ sin θ 0 e y = sin θ cos θ 0 x ref x y ref y (9) e θ θ ref θ where e x is the longitudinal deviation or longitudinal error, e y is the lateral deviation or lateral error, and e θ is the orientation deviation or orientation error. These errors are graphically presented in Figure 3 where the reference virtual robot is represented in dotted lines and the real robot in solid lines. Substituting (4) in (9) and following the same steps proposed by (Canudas et al., 1997), if the resulting equation is derived with respect to the time, it is obtained that 0 (ω λ) 0 cos e θ ė = (λ ω) 0 0 e + sin e θ v ref Fig. 3. Longitudinal, lateral, and orientation errors v σ 0 0 (ω ref ω) λ (10) Considering the dynamic feedback linearization procedure, the control signal chosen for equation (10) is given by u 1 = v + v ref cos e θ + σ λe y (11) u 2 = ω ref ω + λ (12) where u 1 and u 2 are the called virtual control signals. Notice that in (Canudas et al., 1997) the virtual control signals include the terms depending on the linear velocity in the case of u 1 and angular velocity in the case of u 2. Now, this same change is performed but also including the sliding effects by means of σ and λ. Hence, using (11) and (12), the equation (10) can be represented as follows [ ė = 0 ω 0 (λ ω) ] [ 0 ] e + sin e θ 0 v ref + [ ] 1 0 [ ] u1 0 0 u (13) If (13) is linearized around the trajectory, that is with e =0, u =0, and e 3 being small enough, the following time-variant lineal system can be used (Canudas et al., 1997) [ ė = 0 ω ref (t) 0 (λ(t) ω ref (t)) 0 v ref (t) where u =[u 1 u 2 ] T. ] e Linear Control with Sliding Compensation [ ] u 0 1 (14) As commented above, this paper presents a modification of the well-known feedback control law addressed in (Canudas et al., 1997) in order to take into account sliding effects. This feedback control law is described by the following equations

4 527 u 1 = k 1 e x u 2 = k 2 sign(v ref )e y k 3 e θ (15) where k i are the time-dependent gains of the controller using the virtual control signals u 1 and u 2. Substituting (15) in (14) produces k 1 ω ref (t) 0 ė = (λ(t) ω ref (t)) 0 v ref (t) e 0 k 2 sign(v ref )e y k 3 e θ (16) Following the design procedure described in (Canudas et al., 1997), the values for the feedback gains k i,i= 1, 2, 3 can be calculated using the following equations k 1 =2δ(ωref 2 + βvref) (17) k 2 = β v ref + ω ref λ (18) k 3 =2δ(ωref 2 + βvref) (19) where β>0and δ>0allow to determine the desired closed-loop behavior of the system. Notice that, equations (17)-(19) differ with respect to those obtained in (Canudas et al., 1997) by means of the presence of λ parameter in equation (18). In this sense, k 2 is updated online based on the sliding disturbances and thus compensating its effect. Fig. 4. Slide on different grounds 4. RESULTS AND DISCUSSION 4.1 Model Validation Before addressing the control problem, equations (1) and (4) was validated with sliding estimation. In this way, different tests have been carried out using the TMR available at the University of Almería (Fitorobot). The model parameters for this vehicle are b =0.5[m] and r =0.1[m]. In order to validate the sliding estimation, two straightline tests were performed at a speed of 1[m/s] and for a distance of 40[m]. One of them was developed on paved ground. The second test was carried out in a greenhouse where the sliding values oscillate between 10% and 15%. Figure 4 shows the obtained results for these tests. The top plot presents the results for the paved ground where it can be observed how the sliding estimation is around zero, as expected. At the beginning of the test, sliding values around 4% are obtained because of the acceleration of the robot. The bottom plot shows the resulting estimation of the sliding for the unpaved ground. The sliding is around 10%-15% that is typical in this kind of terrain. The experiment to to verify the kinematic model was performed in the same unpaved ground that the previous test. The test consisted in an open-loop experiment where the robot was teleoperated. Then, the stored Fig. 5. Experiment to test both kinematic models data from the sensors were used to estimate the sliding and simulate the same TMR motions using the kinematic model with and without sliding. Figure 5 shows the reference trajectory (bold line). The trajectory determined using equations (4,??) fits appropriately the reference (dotted line). The trajectory using the classic kinematic model differs from the reference due to sliding (dashed line). 4.2 Simulations of the Controller An eight-shaped trajectory has been simulated to assess the performance of the proposed control law. For this test, a sliding of 15% has been supposed. The initial condition for the trajectory is (x, y, θ) =(0, 0, 0) and the controller parameters have been set to β =1 and δ =0.6 to reach a soft overdamped closed-loop behavior using the tuning rules described by (Canudas et al., 1997). Furthermore, the proposed controller has been simulated considering uncertainties in the sliding estimation. The supposed real sliding has been 15% and the estimated sliding 25%.

5 528 Fig. 6. (a)simulated Trajectory and (b) Values of λ parameter Fig. 7. Longitudinal, lateral and orientation errors Fig. 8. Control Signals (linear and angular velocities) Fig. 9. Virtual Control Signals

6 529 Figure 6a shows the reference trajectory of the vehicle (solid line), the tracked trajectory using the modified controller (dashed line), the controller with an error in the estimated sliding (dash-dotted line) and the original controller (dotted line). As can be seen, the original controller presents tracked errors and oscillatory behavior. Although, errors have been supposed in the estimated sliding, this case is still better than in the original case. The evolution of the λ parameter is depicted in figure 6b. As seen in (6) this parameter depends on the speed of the tracks and the sliding. It is expected that λ differs to zero at turns, when tracks rotate at different velocities. The value of λ in the two cases of the proposed controller is very similar because the sliding taken into account by the controller is the same. The errors can be better observed from Figure 7. The errors using the modified controller (solid lines) are smaller than those obtained with the original controller (dashed lines). The greater differences are noted in the longitudinal direction, as shown in the e x - plot. This is because the compensation of the sliding effect has been only taken into account in the longitudinal direction. Moreover, e y and e θ plots show an appreciable errors for the original controller. The controller with uncertainties (dash-dotted lines) is similar to the case where no error has been considered. The control signals (linear and angular velocities), are displayed in Figure 8. The effect of the sliding compensation can be seen from these plots. Longitudinal sliding decreases the linear velocity and the controllers must increase this component of the velocity to compensate this effect. The original controller (dashed line) oscillates when the linear velocity changes, while the proposed controller (dotted line and dash-dotted line) does not present large oscillations. Finally, Figure 9 shows the virtual control signals for the three tests. The modified controller presents less aggressive control signals against the negative effect of the sliding. At the instant times 8 and 40 seconds, the original controller presents great changes in the virtual control signals while these changes in the new modified controllers are much less noticeable. 5. CONCLUSIONS This paper presents the modification of the wellknown linear feedback controller described in (Canudas et al., 1997) with the aim of compensating sliding effects. A kinematic model for this kind of vehicles has been validated including the online sliding estimation. Afterwards, this model has been used in the proposed modified controller to compared it with the original control algorithm (Canudas et al., 1997) by means of a simulation. The obtained results have shown that the proposed modification improves the behavior of the original controller in presence of sliding. Real tests of both controllers will be attempted soon. 6. ACKNOWLEDGMENTS This work was supported by the Spanish CICYT and FEDER funds under grant DPI C The authors would like to thank Professor J. Sánchez- Hermosilla for his help and collaboration. 7. REFERENCES Bloch, A.M. and N.H. McClamroch (1989). Control of mechanical systems with classical nonholonomic constraints. Vol. 1. IEEE Conference on Decision and Control. IEEE. pp Tampa, USA. Brockett, R.W. (1983). Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory (R.W. Brockett, R.S. Millman and H.J. Sussmann, Eds.). pp Birkhauser. Boston. Canudas, C., B. Siciliano and G. Bastin (1997). 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