torque control with kriging estimation for the crab ... - Semantic Scholar

TORQUE CONTROL WITH KRIGING ESTIMATION FOR THE CRAB ROVER Bin Xu1 , Cedric Pradalier2 , and Roland Siegwart2 1 Department of Computer Science and Technology, Tsinghua University (THU), 100084,Beijing, China 2 Autonomous Systems Lab (ASL), Swiss Federal Institute of Technology Zurich (ETHZ), 8092 Zurich, Switzerland

ABSTRACT For the six-wheel CRAB rover, this paper presents the torque control with Kriging estimation to improve the terrainability by minimizing wheel slip. In previous works, we presented a torque control approach, effectively improving terrainability by minimizing the variance of the required friction coefficient based on the static model. Using a Kalman filter to estimate the velocity information, a correction torque can be computed by Kriging estimation. Kriging estimation was chosen because of the following important properties: no recursive computation and unbiased. Experiments are presented wearing tires with different friction coefficients. Key words: Torque control; Kriging estimation; Slip. Figure 1. The CRAB Space rover 1.

INTRODUCTION

This paper consider the terrainability [1] of planetary exploration rovers, and in particular the 6-wheel CRAB rover [3] (fig. 1), developed at the Autonomous Systems Lab – ETH Zrich as a concept study for the ExoMars platform [2]. In an unknown environment, the related terrainability [1] is influenced mostly by the kinematics of the rover suspension system and the control of the actuator. The main focus of the work is on the control design of the rover.

1.1.

• An Inertial Measurement Unit (IMU) is mounted on the body of the CRAB. It provides the orientation of the chassis. • The angular sensors measure the relative angle of a pivot joint. Three on each side are positioned on specific joints of the mechanical structure providing the angles. • The tactile wheels [7] provide the information about the wheel ground contact angles. This input to the model is crucial to compute the optimal torques in

CRAB Rover

The CRAB rover is characterized by a mechanical suspension system that is based mainly on parallel bogies as used by its two predecessors at ASL: the Shrimp [4] and the SOLERO [5]. The bogies are connected at the bottom next to the axis of the middle wheel and at the top through an articulated rocker. A differential mechanism between the left and right suspension levels the pitch angle of the chassis (Fig. 1). In order to provide the information for the static model which will be taken in the control scheme, the following sensors are required:

Figure 2. System overview

3.

uneven terrain.

1.2.

Slip Control

For wheeled rough-terrain robots, enhanced performances can be obtained in terms of obstacle climbing by maximizing the traction [6].The wheel velocities are synchronized in order to avoid them fighting each other. Although this approach is proven to be efficient, the limitation is that it doesn’t take the kinematics or the physical model of the rover into account. Thus the results are expected to be limited in very challenging terrain (3D). One case of slip phenomenon is that the applied torque is too high and the ground cannot sustain the created traction. Torque control tries to avoid this situation by assigning larger torques on wheels where the load is larger because more traction can be generated. The torque control based on PID correction is studied and implemented [8]. Based on this work we design the discrete Kriging control [9] to obtain the correction torque. In this case, the system uncertainty is estimated by the Kriging method [10, 11], complemented with a Kalman-filter based velocity estimator [12]. This paper is organized as follows: Section II presents briefly the static model of the CRAB system. The torque control with Kriging estimation is presented in Section III. The next section goes into the experiments and discussion. Finally, the conclusion and an outlook are included at the end of the paper.

2.

STATIC MODEL OF CRAB ROVER

The static model is based on the Newton-Euler formulation:

∑ Fi = 0 ∑ Ti = 0 i

(1)

i

The CRAB consists of 30 parts and can be characterized by 180 independent equations. As no interest in held into the internal variables, the final equation system is formulated as follows:

CONTROL SCHEME

The fundamental idea of the controller consists in minimizing slip by efficiently distributing the torques on the wheels, using a static model. For this reason, the controller is referred to as torque control. The control scheme is described in Fig. 3. The static model is employed for calculating the optimal torque Mo for a given state s. In order to track the designed velocity Vd , the adaptive controller based on Kriging estimation is added to the control loop where the calculation of correction torque Mc is generated.

3.1.

Torque Control

In the context of exploration in rough terrain, planetary exploration rovers can face very challenging terrain and in order to enhance their terrainability, all their wheels are motorized. This statement is true for the CRAB which, being equipped with six motorized wheels, is over-actuated. Considering the rover’s wheel, the forces in action are the tangential Ri and normal Ni force, the load Pi and the motor torque Ti . The tangential force is linked with the normal force as follow: Ri = µNi ,

(3)

with µ being the friction coefficient. This equation is true whether the wheel slips or not, but if µ is smaller or equal to the static friction coefficient µ0 , the wheel does not slip: µ ≤ µ0 . (4) In the torque control algorithm, the torques are subject to an optimization process based on the static model. The criterion for the optimization can be described as minimization of the variance of the required friction coefficient (Gi ). Gi is defined as the ratio between tangential (Ri ) and normal (Ni ) contact force. The equation for the ith wheel is: Ri Gi = (5) Ni The optimization is formulated as: !

~ 38×43 · ~N43×1 = ~b38×1 M

H = min

(2)

∑

Gi − G¯


2

(6)

i

where ~N corresponds to a vector containing remaining forces, including the wheel ground interaction forces and ~ and ~b correspond to the state of the wheel torques. M rover. This equation does not lead to a unique solution. Instead, one has to choose a criterion and find a set of forces ~N43×1 which minimise it. This choice being arbitrary, the static model is minimised with respect to the variance of the wheel torques.

where G¯ is the mean required friction coefficient. Eq. 6 determines the optimal set of torque minimizing the slippage risk.

3.2.

Kriging Estimators

In this paragraph, we recall some properties of Kriging estimators, as described in [9]. Given a domain

 V

Data Store





s

Kriging Estimation



Vd

Kalman Filter

e

Static Model & Optimization Mc

Adaptive Kriging Control

Mw

Correction Distribution





Mo



Rover

V

Figure 3. Control scheme D ∈ Rd , a random function Y is a collection of random variables{Y (x) |x ∈ D } that can be characterized by the set of all its k-variate cumulative distributions functions for any number k   n o F x1 ...xk ; y1 ...yk = Pr Y (x1 ) ≤ y1 ...Y (xk ) ≤ yk We denote by m(x) and σ (x, x0 ) the mean function and covariance function of Y


 σ (x, x0 ) = E (Y (x) − m(x)) Y (x0 ) − m(x0 ) Further, if it is a joint multivariate Gaussian distribution then, Y is a Gaussian random function. It is easy to see that a Gaussian random function can be specified by its mean function and covariance function as (7)

where N(., .) denotes the Gaussian distribution and Σ ∈ Rk×k is the covariance matrix, with Σi j = σ (xi , x j ). Suppose that the function y is a realization of Y , and the values of the variable y(x) are known at k points x1 ...xk . For a new point x0 , the value of y(x0 ) can be estimated by Simple Kriging Estimator as ySK x

0




=m x

0




+ ΣT0 Σ−1 (y − m)

(8)

where


T Σ0 = σ x1 , x0 ...σ xk , x0 ,


T y = y x1 ...y xk


T m = m x1 , ..., m xK To avoid the infinite data size, we take the recursive Kriging estimation [9] with the following assumption. Assumption 1 : Each unknown function di is the realization of a random function Di with covariance σi . The time correlation of Di , is local, that is σi (θi (k1 ), θi (k2 )) = 0, |k1 − k2 | > M where M is the positive integer.

Correction Torque

In this part we design the correction torque based on the adaptive Kriging control. The quite simple dynamics of CRAB is considered as follows: V (k) = V (k − 1) + Ta (k)   Mc (k) + Fd (k) = V (k − 1) + T mr

(9)

where Mc is correction torque, T is the time period, m is the mass of the CRAB rover, r is the radius of the wheel and Fd is the unknown disturbance acceleration. For CRAB, m is 44.1 kg and r is 0.098 m.

m(x) = E [Y (x)]

(Y (x1 ) · · ·Y (xk )) ∼ N((m(x1 ) · · · m(xk ))T , Σ)

3.3.

The velocity information is computed by a Kalman filter as follows: xˆk− = Axˆk−1 + Buk−1 Pk− = APk−1 AT + Q
xˆk = xˆk− + Kk zk − H xˆk−
−1 Kk = Pk− H T HPk− H T + R − Pk = (I − Kk H) Pk xk = Axk−1 + Buk−1 + ωk−1 zk = Hxk + υk The velocity error is e (k) = V (k) − Vd (k), where Vd (k) is the reference velocity. The error dynamics of the CRAB is given by e (k + 1) = = = =

V (k + 1) −Vd (k + 1) (10) V (k) + Ta (k) −Vd (k + 1) V (k) −Vd (k) + Ta (k) −Vd (k + 1) +Vd (k) e (k) + Ta (k) −Vd (k + 1) +Vd (k)

1 Selecting a (k) = [Vd (k + 1) −Vd (k) − KV e (k)], Eq. 10 T can be transformed as e (k + 1) = (1 − KV ) e (k)

(11)

If KV > 1, the error e (k) goes to zero exponentially. Then Mc (k) is derived as Mc (k) = mr (a (k) − Fd (k)) (12)   1 = mr (Vd (k + 1) −Vd (k) − KV e (k)) − Fd (k) T

Fd (k) is unknown in general but if we can estimate it accurately enough, then Eq. 11 can be approximately derived. In order to prevent a nonzero estimation error, we assume that   Fd (k) = Fd (k − 1) + Fe ~θ (k) (13) where Fe is the unknown function of ~θ (k), defined as follows: Figure 4. Description of the bump ~θ (k) = [V (k) , ∆V (k)]T ∆V (k) = V (k) −V (k − 1)

(14) (15)

And from Eq. 12,we have Mc (k) = mr (a (k) − Fd (k)) (16)    _ = mr a (k) − Fd (k − 1) − F e ~θ (k)

In  this paper, to  we adopt the Kriging method  estimate  _ _ ~ F e θ (k) . The estimation algorithm for F e ~θ (k) is the same one we used in [12]. Figure 5. Test I(rubber tire) 4. 4.1.

EXPERIMENT Metrics

4.2.

In this study, the following parameters for adaptive Kriging control are selected: M = 40, h = 1, τ0 = 0.01, KV = 3. The calculated Mc in Eq. 16 is distributed according to the normal force of Ni . Mw (i) =

Mc · Ni ∑ Ni

(17)

i

Environment Setup

The test terrain has a sinusoidal shape with different inclination angles (Fig. 4). Two bumps are fixed on the wooden floor. The bumps are scaled to reach a maximum height of 0.12 m which is slightly more than a wheel radius. The wheels are covered with ”tires”, as depicted in Fig. 8 (b) and (c). In practice, we use two kinds of tires made of rubber and paper. The length of the test terrain is approximately 2.3 m.

For the experiments, the controller performance is measured using the mean slip of the CRAB, determined by the measured traveled distance and the encoder value of the respective motor. Local slippage, such as sliding when the wheel is moving up and down the obstacles compensates itself in the final calculation. 6

6

∑ Li − ∑ di S=

i=1

i=1 6

(18)

6 ∑ di i=1

where S is the mean slip of the six wheel. For the ith wheel, di is the measured traveled distance and Li is distance based on the encoders value.

Figure 6. Test II(paper tire)

(a)

(b)

(c)

Figure 8. (a)Tactile wheel (b)Rubber tire (c)Paper tire 5. 5.1.

Figure 7. Test III with no tire

Results and Discussion

For Test I and II, CRAB is with equipped with rubber tires and paper tires respectively. This materials are chosen for their friction coefficient on our wooden obstacle. Rubber tires represent the case of a very high friction coefficient whereas paper tires correspond to a very slippy surface. The controller performs the best with 25.48% and 17.72% less slip than using a standard velocity control. Detailed results can be found in [12], but the key conclusion is that the advantage of our controller increases as the friction coefficient decreases. In Test III, the slippage of torque control is 5.3% while it is 5.7% for the velocity control. The reason is that during the movement, only the grousers of the wheels touch the carpet and provide the force to move on. Our next step will consist in testing on soft sand, where the wheels are likely to dig in. Overall, these results indicate that when the friction coefficient is small, torque control shows a significant advantage but when it is high, the performance is almost the same for the proposed torque control and the velocity control.

Conclusions

This paper present a torque controller design based on Kriging estimation for the CRAB planetary exploration rover. The focus is on the computation of the correction torque, assisted by a velocity estimator implemented using a Kalman filter. The controller has been evaluated and compared to a standard velocity controller. The overall performance evaluation shows that using this controller is advantageous in more slippery condition.

5.2. 4.3.

CONCLUSIONS AND FUTURE WORKS

Future Works

Although this approach has been validated in the controlled condition of laboratory experiments, more experiments will be required to complete the evaluation of the approach. In next steps, visual odometry will be integrated to provide more precise value of the vehicle speed. Furthermore, an automatic trajectory tracking will be designed and implemented on the CRAB rover. The control method could also be improved by running 6 independent controller, independently using Kriging to learn the control parameters for each wheel based on their respective velocity command. This would be particularly useful in cases where the rover is making tight turns and the required wheel velocity can vary widely.

6.

ACKNOWLEDGMENTS

The authors gratefully acknowledge Fabian Risch and Francis Colas for their support regarding the experimental work on the CRAB rover.

REFERENCES [1] D.S. Apostolopoulos, ”Analytical Configuration of Wheeled Robotic Locomotion”, CMU, 2001.

[2] T. Thueer, A. Krebs, P. Lamon and R. Siegwart, ”Performance Comparison of Rough-Terrain Robots - Simulation and Hardware”, Journal of Field Robotics, vol. 24:3, 2007, pp 251-271. [3] T. Thueer, P. Lamon, A. Krebs and R. Siegwart, ”CRAB - Exploration Rover with Advanced Obstacle Negotiation Capabilities”, 9th ESA Workshop on Advanced Space Technologies for Robotics and Automation (ASTRA), Noordwijk, The Netherlands, 2006. [4] T. Estier, Y. Crausaz, B. Merminod, M. Lauria, R. Piguet and R. Siegwart, ”An innovative space rover with extended climbing abilities”, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), 2000. [5] S. Michaud, A. Schneider, R. Bertrand, P. Lamon, R. Siegwart, M. Van Winnendael and A. Schiele, ”SOLERO: Solar powered exploration rover”, 7th ESA Workshop on Advanced Space Technologies for Robotics and Automation (ASTRA), Noordwijk, The Netherlands, 2002. [6] S.V. Sreenivasan and B.H. Wilcox, ”Stability and Traction Control of an Actively Actuated MicroRover”, Journal of Field Robotics, vol. 11:6, 1994. [7] A. Krebs, T. Thueer, E. Carrasco and R. Siegwart, ”Towards torque control of the CRAB rover”, International Symposium on Artificial Intelligence, Robotics and Automation in Space (i-SAIRAS), Los Angeles, USA, 2008. [8] A. Krebs, F. Risch,T. Thueer, C. Pradalier and R. Siegwart, ”Rover control based on an optimal torque distribution - Application to 6 motorized wheels passive rovers”, IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS), Taipei, Taiwan, 2010. [9] H. Wu, F. Sun, ”Adaptive Kriging control of discrete-time nonlinear systems”, IET Control Theory Appl., vol. 1:3, 2007, pp 646-656. [10] R.Murray-Smith, D. Sbarbaro, C. Rasmussen, and A. Girard, ”Adaptive, cautious, predictive control with Gaussian process priors,” 13th IFAC Symposium on System Identification, Rotterdam, Netherlands. Citeseer, 2003. [11] D. Sbarbaro and R. Murray-Smith, ”Self-tuning control of non-linear systems using Gaussian process prior models”, Switching and Learning in Feedback Systems, 2005, pp 140-157. [12] B. Xu, C. Pradalier, A. Krebs, R. Siegwart and F. Sun, ”Composite Control Based on Optimal Torque Control and Adaptive Kriging Control for the CRAB Rover ”, IEEE International Conference on Robotics and Automation (ICRA 2011, Shanghai, China, 2011.