Torque control of a poly-articulated mobilerobot during obstacle clearance

Pierre Jarrault, Christophe Grand and Phillipe Bidaud

ISIR - UPMC/CNRS (UMR 7222), Paris, France

Abstract This paper describes a control algorithm that optimisesthe distribution of joint torques of a polyarticulted robot while per-forming obstacle clearance of a large step. In this work, a specialclass of polyarticulted locomotion systems known as hybrid wheel-legged robots is considered. This type of system is usually redun-dantly actuated, involving internal forces that could be exploited toimprove the tipover stability and the traction forces needed to ad-dress more challenging obstacles. The proposed algorithm is basedon the forces distribution model including internal forces. Its objec-tive is to optimize a criterion representing the maximum allowabledisturbance while respecting the frictional contact conditions. Theperformance of this controller is evaluated in simulation.

1 Introduction

In this paper, we address the motion control of poly-articulated mobilerobots during obstacle clearance. We are focusing on a special class of mobilesystems, often called hybrid wheel-legged robots, which are designed in orderto increase both obstacle crossing and terrain adaptation capabilities.

Systems like HyLoS (Grand et al. (2002)) and the Workpartner (Halmeet al. (2003)) are examples of such hybrid locomotion systems. They arecomposed of 4 wheel-legs, each wheel-leg being a multi-dof serial chain endedby a driven and steerable wheel. Such robots have the ability to change theposition of their center of mass (CoM) and to modify the distribution of theircontact forces. Furthermore, they are often redundantly actuated systemsexhibiting internal forces that should be optimized.

The proposed motion controller is based on a torque control at jointlevel that addresses the combined optimization of the internal forces andthe CoM position, in order to maximize the contact stability (increasingtraction and avoiding tip-over).

V. Padois, P. Bidaud, O. Khatib (Eds.), Romansy 19 – Robot Design, Dynamics and Control, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1379-0_36, © CISM, Udine 2013

292 P. Jarrault, C. Grand and P. Bidaud

Figure 1. Hylos prototype

First, this paper introduces a stability criterion based on the contactforces slippage. Then, we present the formulation of the optimization offorces distribution. Lastly, simulation results are presented exhibiting theclearance capabilities over a step-like obstacle which height is greater thanwheel radius.

2 System modelling

2.1 Equations of forces distribution

We consider a system supported by n legs (Fig. 2). The ith leg is ina frictional contact with the ground at point Pi which coordinates pi areexpressed in the frame �p = (G,xp,yp, zp), attached to the chassis of therobot and located at CoM. The contact force of the ground on each leg is

denoted fi =[fxi fyi fzi

]twhere fxi

, fyiand fzi are the components

of the force along the contact frame’s axis �ci = (Pi,xi,yi, zi), such that ziis the contact normal and xi, yi are the tangential directions. The equationsdescribing the equilibrium of the system are given by:

G f = F (1)

where f =[f1

t ... fnt]t

is a [3n×1] vector containing all contact forcesand F is the set of external and inertial wrench applied on the platform. Gis a [6×3n] matrix giving the equivalent wrench to the contact forces at thecenter of mass in the frame �p (see Grand et al. (2010) for more details).

The contact forces must respect constraints related to actuator satura-tion and Coulomb friction law.

Torque Control of a Poly-articulated Mobile Robot… 293

Figure 2. Model of the systemFigure 3. Residual force on acontact

The actuators limits are defined as follow:{

JT f < τmax

−JT f < τmax(2)

where J = blockdiag(Ji), Ji being the Jacobian matrix of the ith leg andτmax is the torque limit vector.

The contact constraints defined by Coulomb’s friction law which areexpressed using a pyramidal form Kerr and Roth (1986) given by:

Aifi < 0 (3)

where

Ai =

0 1 µi√2

0 −1 µi√2

1 0 µi√2

−1 0 µi√2

2.2 Stability criterion

Previous works on legged robot stability mostly use geometrical stabilitycriteria based on the CoM or ZMP projection on the support polygon formedby the contact points. These methods do not consider the friction in contactsand thus are only suited for tip-over avoidance.

Extensive work has been done on the problem of forces distribution withfrictional contacts in grasping researches. They express the combinationof the constraints in the wrenches space associated with the CoM of thegrasped object. As a result, this gives all the wrenches that are resistible if

294 P. Jarrault, C. Grand and P. Bidaud

they are applied on the CoM. Different criteria derive from this representa-tion.

Kirkpatrick et al. (1992) uses the smallest wrench applicable on theCoM as a criterion to choose a grasp configuration. Whereas Yoshikawaand Nagai (1988) solves the forces distribution problem by maximizing theresidual ball radius in this space, making sure that the robot will be ableto add forces in order to support the largest possible disturbance.

However, as these methods do not consider individually each contactforce, the stability of some contacts may be reduced for the sake of globalstability, making modeling errors affecting the contact force control morelikely to cause slippage.

Thus we propose in this paper a stability criterion based on the smallestperturbation allowed at the contact level. This leads to to maximize ro-bustness with respect to modeling errors affecting the contact force control.Let us define vector d as:

d = A f (4)

where A is a matrix defined by A = blockdiag(Ai).The elements of d represent the tangential force that can be added to

the contact force of a leg in each tangential direction without breakingthe contact stability (Fig. 3). By maximizing the smallest element of d,we maximize the perturbations that can be sustained by the robot. Theoptimization criterion is then:

φ(f) = min(A f) (5)

where min(x) is the function returning the smallest element of the vectorx. If φ(f) is negative, at least one contact is sliding or broken.

3 Optimisation of forces distribution

Considering the elements given in the previous section, the forces distri-bution problem is formulated as a ”minimax” optimization procedure. In-deed, the objective is to maximize the smallest acceptable perturbation(φ = min(d)) subject to the constraints (1) and (2). This problem is trans-formed into its compact primal form (Cheng and Orin (1990)):

Find xf maximizing:

min(A[G+F+Nfxf

])

Subject to: [NT

f J −NTf J

]Txf < af

(6)

where:

Torque Control of a Poly-articulated Mobile Robot… 295

• G+F represents the particular solution of (1), G+ being the weightedpseudo-inverse of G,

• Nfxf represents the homogeneous solution of (1),Nf being the matrixcontaining the basis vectors of the null space of G,

• and aft =

[(τmax − JTG+F)t (τmax + JTG+F)t

]

We define xp = [X Z]T as the vector containing the longitudinal andvertical coordinates of the CoM position in the global frame (considering theglobal frame is chosen to have the Yaw of the robot equal to zero). In thiscase, the null space of G is entirely defined by the contact geometry. Thus,for fixed contact points, the null space of G is not modified by a change ofX or/and Z. Furthermore for a frontal crossing, i.e. the rotation betweenthe ground and the contact’s frame of the obstacle is around the lateral axisof the robot (yp), the particular solution can be stated as a combination ofthe CoM position (X, Z) and the contact angles (represented by f0) :

G+F = cx X + cz Z + f0 (7)

Thus, the CoM position can be added to the optimization variables:

Find x maximizing:

min(A[f0 +Nx

])

Subject toJTp x < a

‖pi‖ < Dmax

(8)

withN =

[Nf cx cz

]xt =

[xf

t xpt]

Jp =[NTJ −NTJ

]at =

[(τmax − JT f0)

t (τmax + JT f0)t]

4 Simulation results

The control algorithm is evaluated on the HyLoS2 robot, an actively ac-tuated polyarticulated system consisting of 4 legs supporting the chassis.Each leg is composed of 2 segments and a wheel with four actuated degreesof freedom (DoF): two for the leg movement, one for the steering and onefor the traction.

296 P. Jarrault, C. Grand and P. Bidaud

Figure 4. Simulation of the step crossing algorithm

The robot speed is sufficiently low to neglect the inertial effects on theCoM. The task of the optimisation problem (8) is only to compensate for

the gravity (F =[0 −Mg 0 0 0

]t) and the displacement of the

robot is created by an additional torque on the wheels. Considering this,the only CoM position’s parameter which is optimized is X. During thesimulation, the force optimization (6) is performed every 10 time steps ofthe simulation loop (1ms) while the position optimization (8) is performedevery 100 steps. The null space’s basis Nf is given by a singular valuedecomposition of the matrix G and the minimax optimization is performedby Dutta’s algorithm Dutta and Vidyasagar (1977). Given a new desiredconfiguration, the position of the CoM is controlled by adding a force to thetask vector F in the force distribution problem (6). This force is computedby a PD controller. The distance between two wheels on the same side isfixed and the static friction coefficient is set to 0.8.

The robot is asked to climb a step with a height representing 25% of thedistance between the wheels. The stability margin defined in (5) is plottedon Fig. 5 and the actual and desired horizontal position are given Fig. 6.

During phase 1, the robot is on a horizontal flat ground. The front wheelstouch the step and it enters in phase 2, the stability margin drops belowzero, indicating that it can not produce the necessary forces to start theclimbing motion. The desired position is calculated and the robot reachesthe optimal configuration. During this phase, the robot still uses the groundto sustain its weight. Once the stability margin has increased enough, therobot enter in phase 3 and starts to climb over the step. While climbing, theoptimal position of the CoM is updated, avoiding possible tip-over as therobot is rising. Once the front wheels have reached the top of the step, therobot is once again in a phase similar to phase 1 where all the wheels are onhorizontal planes. And the same phases follows: the rear wheels touch theobstacle, the robot reaches the optimal configuration and the wheels climbover the step.

Torque Control of a Poly-articulated Mobile Robot… 297

2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

0

5

10

15

20

25

30

simulation time (ms)

sta

bili

ty m

arg

in (

N)

1 2 3 1 2 3 1

Figure 5. Stability margin during a step crossing

2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

−0.1

−0.05

0

0.05

0.1

simulation time (ms)

Horizonta

l positio

n (

m)

current position

desired position

1 2 3 1 2 3 1

Figure 6. Desired and actual horizontal position of the CoM

5 Conclusion

The main objective of this work was to propose torque control of a poly-articulated mobile robot that exploit the redundant actuation to improvethe adhesion at the wheel-ground contacts while crossing frontal step-likeobstacle.

298 P. Jarrault, C. Grand and P. Bidaud

The control algorithm is based on the optimization of a criterion thatrepresents the robustness of the contacts stability in term of traction. Thiscriterion is based on the measure of the smallest perturbation sustainable atthe contacts level. A control algorithm that compute both the best forcesdistribution and the best geometric configuration of the robot relative tothis criterion have been described.

Simulations show good results of the control scheme for a step crossing,but an analysis of the posture trajectory shows some discontinuities thatshould be smoothed. Future works will address this problem by including apredictive controller and will focus on the practical implementation of thiscontrol algorithm on the actual platform HyLoS2.

Bibliography

Fan-Tien Cheng and David E. Orin. Efficient algorithm for optimal forcedistribution - the compact-dual lp method. In IEEE Transaction onRobotics and Automation, volume 6, April 1990.

S. R. K. Dutta and M. Vidyasagar. New algorithms for constrained minimaxoptimization. Mathematical Programming, 13:140–155, 1977.

C. Grand, F. Ben Amar, F. Plumet, and Ph. Bidaud. Stability control ofa wheel-legged mini-rover. In Proc. of CLAWAR’02 : 5th Int. Conf. onClimbing and Walking Robots, pages 323–330, Paris, France, 2002.

Christophe Grand, Faiz Ben Amar, and Frederic Plumet. Motion kinematicsanalysis of wheeled-legged robot over 3d surface with posture adaptation.Mechanism and Machine Theory, 45(3):477–495, March 2010.

Aarne Halme, Ilkka Leppnen, Jussi Suomela, Sami Ylnen, and Ilkka Ket-tunen. Workpartner : Interactive human-like service robot for outdoorapplications. The international journal of robotics Research, 22(7-8):627– 640, 2003.

J. Kerr and B. Roth. Analysis of multifingered hands. International Journalof Robotic Research, 4:3–17, 1986.

David Kirkpatrick, Bhubaneswar Mishra, and Chee-Keng Yap. Quantitativesteinitz’s theorems with applications to multifingered grasping. Discreteand Computational Geometry, 7:295–318, 1992.

Tsunoe Yoshikawa and Kiyoshi Nagai. Evaluation and determination ofgrasping forces for multi-fingered hands. In IEEE Conference on Robotic& Automation, volume 1, pages 245–248, 1988.