IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 5, OCTOBER 2013 1271 Wheel–Soil Interaction Model for Rover Simulation and Analysis Using Elastoplasticity Theory Ali Azimi, J ´ ozsef K¨ ovecses, and Jorge Angeles, Fellow, IEEE Abstract—A novel approach is proposed for the modeling of rigid-wheel and soft-soil interaction to efficiently compute normal and shear stress distributions in the contact area. The authors pro- pose a velocity field in the vicinity of the contact area based on the physical nature of the problem. Thereupon, the incremental changes to the stress field are computed by resorting to elastoplas- ticity theory and an appropriate already existing constitutive rela- tion for soil. The proposed approach leads to results that agree well with those obtained using well-established terramechanics models, while addressing some of their shortcomings. In addition, the pro- posed approach uses generalized velocities of the wheel as inputs, which makes it compatible with dynamic models of multibody sys- tems. The dynamic slip–sinkage behavior of the wheel and the semielliptical shape of the normal stress distribution under the wheel are natural outcomes of the proposed model. Experimental investigation under various ranges of wheel slippage shows good agreement with the data available in the literature. Index Terms—Contact modeling, dynamics, multibody systems, wheeled robots, wheel–soil interaction. I. INTRODUCTION M OBILE robotic systems represent key elements for fu- ture planetary exploration as well as earthly applications. Such robots have to operate on different types of unstructured terrain, among which soft deformable soil is of particular inter- est. In order to investigate the effect of deformable soil on the performance of rovers, appropriate models are required to rep- resent the interaction between wheel and terrain. In this context, soil reactions are required in response to the wheel movement. The Bekker model [1], [2] and the extension made by Wong and Reece [3] are two semiempirical terramechanics models that are widely used, as they have been experimentally vali- dated. The latter is referred to as the Wong–Reece (WR) model in this paper. These two models have a broad range of appli- cation in characterizing vehicles on soft terrain. Both models Manuscript received December 13, 2012; revised May 4, 2013; accepted June 8, 2013. Date of publication July 3, 2013; date of current version September 30, 2013. This paper was recommended for publication by Associate Editor M. Minor and Editor B. J. Nelson upon evaluation of the reviewers’ comments. This work was supported by the Natural Sciences and Engineering Research Council of Canada, CMLabs Simulations Inc., the Canadian Space Agency, and the McGill Engineering Doctoral Award Program. The authors are with the Department of Mechanical Engineering and the Cen- tre for Intelligent Machines, McGill University, Montreal, QC H3A 0G4, Canada (e-mail: ali.azimi@mail.mcgill.ca; jozsef.kovecses@mcgill.ca; angeles@cim. mcgill.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2013.2267972 have significant application in mobile robotics as well. For ex- ample, in the AESCO Soft Soil Tyre Model (AS 2 TM) [4], the Bekker model is used. Furthermore, Shibly et al. [5], Iagnemma and Dubowsky [6], Ishigami et al. [7], [8], and Hutangkabodee et al. [9] used the WR model in their rigid wheel–soil interac- tion studies. In addition, Wong and Asnani [10] compared the performance of several wheels of lunar vehicles by means of the NWVPM software package [11], in which normal (radial) stress distribution under the wheel is obtained using the Bekker model. A simplified version of the WR model was used by Iagnemma et al. [12] to identify cohesion and internal friction angle of soil for real-time applications to rovers operating on soft soil. Terrain parameter identification was also done by Ray [13], using the WR model. This WR model was also used by Ojeda et al. [14] for wheel slip detection and positioning error compensation. The motivation of this paper is the need of a model for wheel– soil interaction that is compatible with multibody dynamics models and simulation environments. In this context, we need to determine soil reactions—forces and moments—using the model and the state of the wheel. In addition, the computational cost should be modest. The Bekker and WR models have these features; however, as they were not developed for the aforemen- tioned conditions, they cannot represent some physical phe- nomena, which are explained below. Such models can still be used in multibody simulation by considering their limitations and using the necessary modifications. A possible implemen- tation of the Bekker and WR models in a multibody dynam- ics simulation environment was reported by Azimi et al. [15]. In addition, an extension of the model that includes operation on rough deformable terrain with compaction and hardening of soil (multipass effect) was developed by Azimi et al. [16]. Other examples of the application of these models in multibody simulation studies can be found in [17]–[21]. In addition to the aforementioned models, other models, based on continuum mechanics, can be employed. In this regard, soft soil is modeled as a continuum, in which wheel–soil contact can be analyzed by considering an appropriate constitutive re- lation for soil and using detailed finite-element discretization to calculate stress distribution and soil deformation in the contact area, as recently reported in [22]–[24]. In yet another class of methods, dry soil is modeled as cohesionless granular mate- rial, with wheel–soil contact analyzed with the discrete element method (DEM) [11]. Today, one of the issues with DEM in wheel–soil interaction modeling is the need to consider a large number of particles, which results in extremely high computing time, even with supercomputers [11]. For wheel–soil interac- tion, finite-element analysis (FEA) is computationally less de- manding than DEM. However, FEA is still inappropriate for 1552-3098 © 2013 IEEE