WHEEL MOTION RESISTANCE AND SOIL THRUST TRACTION OF MOBILE ROBOT

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1 8th International DAAAM Baltic Conference "INDUSTRIAL ENGINEERING April 202, Tallinn, Estonia WHEEL MOTION RESISTANCE AND SOIL THRUST TRACTION OF MOBILE ROBOT Petritsenko, A.; Sell, R. Department of Mechatronics, Tallinn University of Technology, Ehitajate tee 5, 9086 Tallinn, Estonia Abstract: The research presented in this paper is focusing on the wheel-terrain interaction of mobile robots. In particular four most common interaction cases are analysed and mathematical relationships in terms of forces are pointed out. The purpose of defining the wheel-terrain intaraction forces for different cases are to transfer them into the design library for helping designer in early design process. The design library is a part of early design framework and the mathematical relationships, in addition to other relations, are defined by novel System Eningeering description language - SysML. As a result of this study the library consists of mathematical models for most common wheel-terrain interactions of wheeled robots. Key words: mobile robot, motion, terrain, traction, modelling.. INTRODUCTION It is well known, that at beginning of design process of wheeled robot several parameters must be chosen and finally their compliance to the functional and performance requirements verified. Usually verification of chosen parameters, specified in requirements, is carried out by means of simulations of mathematical models with different complexity level or field tests. At the beginning and during the design process many parameters and mutual interaction must be evaluated, compared and finally selected for specific robotic applications. One of the crucial subsystems of mobile robot is a locomotion sub-system. The design and selection of conceptual locomotion system fix many further robot parameters, including key performance and terrainability parameters. In wheeled mobile robot, obviously one of the key components is wheel itself. When selecting wheel parameters (material, coverage, geometry, etc.) it is important to analyse the required terrain capabilities of the robot and performance criteria of passing the obstacles and holes. In addition, designer has to keep the mind open and do not relay only the conventional solution. For example in mobile robotics, many unconventional locomotion solutions are applied to improve the terrainability and flexibility in small scale locomotion end actuator. New solutions and wheel-terrain studies can be found in literature [], [2] including patented invention from Tallinn University of Technology, Department of Mechatronics [3]. In this paper we are focusing on the detail analyses of wheelterrain interaction and forces influencing on the different conditions in off-road terrain. The research results are mathematical models formulated into novel engineering description language SysML (System Modeling Language) [4]. This approach enables to reuse the common cases on later mobile robot design, including early stage simulations and candidate solution evaluation process.

2 2. WHEEL MOTION RESISTANCE 2. Motion resistance due to wheel-terrain interaction (R R ) When a robot moves on paved surfaces and roads it consumes energy to overcome the rolling resistance between the tires and the ground, as well as gravitational and inertial forces. Obstacles are dealt as a specific case and are simulated separately. Rolling resistance between the tire and the ground is attributed to tire slip, scrubbing in contact patch, deflection of the road surface and energy losses due to tire adhesion on the road and hysteresis. Rolling resistance varies with the type and material of tire tread, velocity of vehicle, and environmental parameters, such as temperature and humidity [5]. There exists four general wheel-terrain interaction cases [6] shown in Fig.. Fig.. Possible wheel-terrain interaction cases Notations for Fig. A) rigid wheel travelling over rigid terrain, B) deformable wheel travelling over rigid terrain, C) rigid wheel travelling over deformable terrain, D) deformable wheel travelling over deformable terrain. These four different wheel-terrain interaction cases involve different wheel resistance forces involved in mathematical models. Simulation models are generated according to mathematical model. This allows designer to select different cases according to simulation interest. Case A rigid wheel travelling over rigid terrain This model is simplified approach of wheel-terrain interaction. The model assumes that rolling resistance of the tire neglected and wheel-terrain interacts to each other at single contact point. This kind of model suits for metallic wheels or wheels with solid non-metallic tires. Case B deformable wheel travelling over rigid terrain In this wheel-terrain interaction case pneumatic tires are usually considered. Deformation of the terrain is generally neglected and it is considered that rolling resistance is caused primarily by the tire hysteresis, i.e. due to deflection of carcass of tire. When tire is rolling the carcass of tire is deflected in the area of ground contact patch and centre of normal pressure shifts in direction of rolling. The shift of centre of normal pressure causes increase of rolling resistance. Complex relationships between design and operational parameters of tire and its rolling resistance make it difficult to develop an analytical method for predicting of rolling resistance [7]. Determination of rolling resistance, therefore, relies almost on experiments. For modelling purposes, design library is composed consisting of different tire manufacturer s data, where average rolling resistances of pneumatic tires are given. Additionally, design library consists of data, where rolling resistance of tire at different inflation pressure is given as well. In the absence of tire manufacturer s data, the empirical formulas to calculate the rolling resistance of tire are used and given in [8]. Case C - rigid wheel travelling over deformable terrain In this model it is assumed that wheel travelling over off-road terrain is subject to sinkage. Wheel is classified as rigid if

3 deflection of under static loading is much lower than deformation of soil/terrain. This model fits to metallic wheels, solid nonmetallic tires and high-pressure pneumatic tires operating on weak soils. In case of pneumatic tires, the rolling resistance due to tire hysteresis is neglected. The main cause of the motion resistance is due the deformation of the soil. Motion resistance of tire consists then from two major contributions: the compaction resistance and the bulldozing resistance. Compaction resistance This form of wheel motion resistance can be analyzed considering the mechanics of a rigid wheel rolling into soft terrain. The motion resistance of rigid wheel is produced by vertical work done in making a rut of depth z as [7] ( 2n+2) 3W 2n+ D R c = (2n+2) (2n+) (2n+) kc ( 2n+) ) (3-n) (n+)b +kf () where b width of wheel contact area; k c, k f and n pressure-sinkage parameters of the specific terrains; W vertical force exerted by the tire to the terrain; D wheel diameter. The pressure-sinkage parameters for different soils are found from literature and included to the design library for modeling purposes. Bulldozing resistance Bulldozing resistance is developed when a substantial soil mass is displaced by a wheel. This type of resistance is very common when a wheel compresses the surface layers of soil and pushes compacted soil fore and aft of the tire [7]. Soil bulldozing phenomenon is apparent in the case of a wide wheel traversing very loose soils and has been estimated to cause a significant increase in total motion resistance for sinkage values greater than /6 of the wheel diameter. The bulldozing resistance on narrow tires is mitigated by the fact that a portion of the soil bulk is pushed to the sides of wheel. The bulldozing resistance can be calculated by implementing the theory of bearing capacity of soils subject to various criteria of failure. The equation to calculate the bulldozing resistance is given in [9] and not repeated here. Case D deformable wheel travelling over deformable terrain In this model it is assumed that tire is subject to sinkage and deflection of tire is in same order as deformation of off-road terrain. If the maximum contact pressure that terrain can support is greater than combined inflation pressure of tire and pressure due to the stiffness of the carcass, then the tire flattens at contact patch with the terrain. This model suits best for low to medium pressure pneumatic tires travelling over off-road terrains [7], [8]. The compaction resistance of deformable wheel travelling over deformable terrain is derived in [8] and is ( n+) bp n gr R c = (2) kc n ( n+ ) +kf where p gr average ground pressure which is usually provided by the tire manufacturer for a given inflation pressure and wheel loading. Eq. (2) shows that the compaction resistance of a flexible wheel is solely a function of width of contact patch (minimum width of tire in this case) and geophysical properties of soil. Additional resistance that has to be accounted is rolling resistance due to tire hysteresis as it was pointed out in case B. 2.2 Motion resistance due to slopes (R S ) Ground slopes add a component to the motion resistance of wheel which is proportional to the component of parallel to slope.

4 2.3 Aerodynamic motion resistance (R A ) Aerodynamic drag is dependent on aerodynamic factor and square of speeds. Aerodynamic drag of wheel can be neglected due to its low influence to wheel motion resistance. 2.4 Inertia resistance (R in ) The generation of inertia forces is closely related to traction forces that are available at wheel-terrain contact surface. Traction is limited by mechanical properties of terrain and loading at wheel/soil interface patch. Thus inertia properties of wheel and robot can be properly evaluated after determination of maximum tractive effort available at wheel-terrain contact patch. 3. SOIL THRUST AND TRACTION Vehicle motion relevant to the terrain is produced through traction (F K ). Caused by a physical process of adhesion and deformation, traction develops at the interface of a powered wheel with the ground. The maximum produced traction is limited by the adhesion between wheel and ground [7] and the torque-speed characteristics of vehicle s prime mover and drivetrain which basically determine that maximum torque and power transmitted to the wheel. F k F G z W N f r Direction of travel T F R x G - weight of the wheel, W- portion of the robotics weight applied to the wheel N- normal reaction of wheel, FR - sum of resistance forces due to wheel-terrain interaction, Fk - traction force, T - applied torque, R - radius of the wheel Fig. 2 Forces acting to the driven wheel For locomotion in soft soils traction is limited by mechanical properties of soil and loading at wheel/soil interface patch. Maximum force that can be sustained by the soil before excessive slippage occurs is known as soil thrust. The study of mechanics of traction generation provides equations for configuration of a robot s wheel and overall robot geometry. Forces acting on driven wheel generally are shown in Fig.2 Case A rigid wheel travelling over rigid terrain This model assumes that wheel-terrain interacts to each other at single contact point. Only force acting at wheel-terrain interaction point is frictional force. No slip of tire allowed in this model. Maximum traction developed is proportional to the wheel load and is limited due to Coulomb friction force F=μ K s(g+w) (3) Where µ s - sliding coefficient of friction, G and W weight of wheel and load to wheel respectively. This relationship is well known from classical mechanics of rigid bodies and is the simplest model to evaluate a maximum traction force. Case B deformable wheel travelling over rigid terrain When tire is rolling, carcass of tire is deflected in the area of ground contact patch. Deformation of rigid surface is neglected. Due to tire deflection the distance that tire travels when subject to driving torque will be less that in free rolling and longitudinal slip of tire occurs. The traction force developed is given then as simplified function of longitudinal slip of tire as [7]. μ p(g+w) F=μ K p(g+w) - (4) 4Cii where µ p - coefficient of adhesion; i slip of tire; C i longitudinal stiffness of tire. CASE C and D rigid and flexible wheel travelling over deformable terrain For rigid and flexible wheel travelling in soft soils traction is limited by mechanical

5 properties of soil and loading at wheel/soil interface patch. Maximum tractive force F is limited by the thrust produced by soil, which in turn is proportional to mechanical strength of soil. Data on the stress/strain relationships of disturbed soils, sand, snow, and saturated clays have verified the appropriateness of Janosi-Hanamoto relationship to describe shear stress-strain behavior of unprepared terrain [7],[0] as j - τ(θ)= ( c+p(θ)tanφ ) -e K (5) where c and φ are modulus of cohesion and angle of internal friction of soil, p(θ) normal pressure distribution in soil as function of contact angle between wheel and soil; j - the shear displacement and is function of slip velocity; K - shear deformation parameter, Integration of the Eq. (5) over contact patch area leads to total tractive effort as θ 0 K 0 ( ) F = τ θ cosθdθ (6) Of critical importance to the amount of forward thrust developed at tire-soil interface is geometric shape of the tire, which has to be determined before evaluation of the integral in Eq. (5). 3. Net traction Net traction of the driven wheel shown in Fig. 2 is the force equal to the difference between the tractive effort F K developed by the wheels and resisting forces as K n F = F R (7) i= Net traction is available force that can be used to generate acceleration of the wheels and robotics. 4. MODEL LIBRARY The framework supporting early design process is developed during the previous research []. One of the key factors of this framework is a design library which consists of unified mathematical models i par WheelTerrainInteraction [Resistance] R = ( G + W )sinα slope i i Slope:Resistance Do not have significant impact on low speed Aerodynamic:Resistance ( 2n+2) 3W 2n+ D R c= (2n+2) (2n+) (2n+) kc ( 2n+) ) (3-n) (n+)b +kf ( n+) bp n gr R c = Case C kc n ( n+ ) +kf Case D Bulldozing:Resistance Total:Resistance ΣR=FR Compaction:Resistance Givn in Ref. 9 Obstacle:Resistance Snow:Resistance Internal:Resistance Dependent on the Component selection Separate analysis Separate analysis Fig. 3 Simplified SysML parametric diagram of resistances in wheel-terrain interaction

6 transferred into the SysML graphical diagram format. The model design library offers pre-defined models for designer to start fast and efficient early design process on mobile robotic domain. In this paper we have described the wheel-terrain interaction cases which are the ground issue of the wheeled robots and have to have therefore well formulated and verified. The corresponding unified SysML parametric diagram is shown in Fig. 3. where the key factors are highlighted. This graphical representation can be used as a input for the simulations or transferred automatically into the analytical calculations. 5. CONCLUSIONS In this paper the major resistance forces and generation of traction forces in different basic wheel-terrain interaction cases are introduced. The mathematical description of them are generated and transferred into graphical diagram format of SysML. It allows performing efficient and optimal early design stage by offering the necessary simulation models at the beginning of the design stage, when many wheel parameters and wheel-terrain interaction cases must be evaluated, compared and finally selected for specific robotic application. 6. ACKNOWLEDGEMENT This research was supported by funding of the Estonian Science Foundation grant No REFERENCES. Zheng, L., et. Al. A Novel High Adaptability Out-door Mobile Robot with Diameter-variable Wheels, Proc. of the IEEE Int. Conference on Information and Automation Shenzhen, China, Chengbin Ma, Min Xu & Hao Wang, Dynamic emulation of road/tyre longitudinal interaction for developing electric vehicle control systems, Vehicle System Dynamics, 200, 49:3, Sell, R., Kaeeli, M., Wheel-Leg (Wheg), Patent no EE05283B, OMG SysML - System modeling Language specification, OMG document formal/ , Gillespie, T., D. Fundamentals of vehicle dynamics. Society of Automotive Engineers, Inc, Iagnemma, K., Dubowsky, S. Mobile Robots in Rough Terrain. Springer- Verlag Berlin Heideberg, Wong, J., Y. Theory of Ground Vehicles 4 th Ed. John Wiley & Sons, Inc; Hoboken, New-Jersey, Saarilahti M. Soil interaction models, In: Haarlaa R, Salo J, University of Helsinki, Pub. 3, Genta, C. Introduction to the Mechanics of Space Robots. Springer, Ray, L.R., Brande, D.B., Lever, J.H. Estimation of net traction for differential-steered wheeled robots. J. of Terramechanics, 2009, 46, Sell R., Coatanea E., Christophe F. Important aspects of early design in mechatronic, 6th International Conference of DAAAM Baltic Industrial Engineering, Tallinn, ADDITIONAL DATA ABOUT AUTHORS Raivo Sell, Ph.D, senior researcher, Tallinn University of Technology, Department of Mechatronics, Ehitajate tee 5, Tallinn, raivo.sell@ttu.ee Andres Petritsenko, Ph.D, researcher, Tallinn University of Technology, Department of Mechatronics, Ehitajate tee 5, Tallinn, andres.petritsenko@ttu.ee