Fast Algorithm for UGV Wheel Terrain Interaction Analysis

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  2008 10th Intl. Conf. on Control, Automation, Robotics and VisionHanoi, Vietnam, 17–20 December 2008 Fast Algorithm for UGV Wheel-Terrain Interaction Analysis T. H. Tran Dept. of Automation College of Engineering, Can Tho University Can Tho, Vietnam Q. P. Ha Faculty of Engineering University of Technology, Sydney Sydney, Australia  Abstract   —On-line prediction of unmanned ground vehicle (UGV) behaviour during interactions with terrain is essential for autonomous and safe operations of skid-steering UGVs. This paper presents a fast and accurate algorithm for a new interaction model to predict performance of a UGV running on a particular terrain. By approximating nonlinear relations involved in the interaction between the vehicle and terrain, a closed form of the interaction model is obtained to enable on-line computation. The minimum absolute error criteria are applied to secure accuracy for the proposed method. The development is compared with the other models in terms of both computation speed and accuracy.  Keywords—  UGV, terramechanics, wheel-terrain interaction. I.I  NTRODUCTION Unmanned ground vehicles (UGVs) have many potential applications, both in military and civil areas, such as reconnaissance, surveillance, target acquisition, and rescue. Most UGVs are currently teleoperated machines which require human intervention, thus, the range of applications is limited. Therefore, knowledge of the interaction between UGVs and terrain plays an important role in increasing the autonomy of UGVs and securing the safety for their locomotion. Many UGVs, especially in military applications, use skid-steering for all-terrain mobility, both in tracked and wheeled  platforms. Behaviour of a skid-sterring UGV operating on off-road areas depends strongly on interactions between the vehicle and terrain. For performance analysis, the theory of terramechanics has been applied to for several off-road vehicles [1, 2]. In general, experiments are used to derive normal stresses and shear stresses beneath wheels or tracks. The stresses are then integrated to yield reaction forces and moments of terrain acting on the vehicle. For track vehicles, a comprehensive theory for their skid-steering mechanics was developed by Wong and Chiang [3]. For skid-steering wheeled vehicles, a general approach is recently proposed to analyse their mobility performance [4]. By using the interaction model described therein, behaviour of vehicles operating on deformable terrain can be predicted. However, the prediction is time-consuming as the integated equations of stresses are not obtained in closed forms. Meanwhile, predicting on-line the vehicle performance is essential to avoid dangereous situations that may be encountered in autonomous navigation. Therefore, a fast approach to vehicle-terrain interaction analysis is desirable. With regard to this particular problem, there has been little research effort reported in the literature, see e.g., [5, 6]. Therein, based on the observation that the normal and shear stresses under a rigid driven wheel are nearly linear, a modified version of the basic wheel-terrain interaction equations is developed. By approximating the normal stress and shear stress  by straight lines going through their start and end points, as shown in Fig. 1a and Fig. 1c, solutions to the integral equations can be obtained and easily computed. Figure 1 indicates, however, that the distribution of normal and shear stresses is generally nonlinear and the method [5] can lead to large errors. In addition, this approach can be applied only if lateral forces on the wheels are ignored. In this paper, the idea proposed by Shibly et al.  [5] is refined to yield an efficient algorithm without losing accuracy. The stress is also approximated by a straight line going through its maximum stress point. But the other point of intersection is chosen so that effects of the approximation error on the vehicle  performance are minimised (Fig. 1b and Fig. 1d). The method is applied for all components of normal stress and shear stress in the longitudinal, lateral, and vertical directions and new equations for the reaction forces are obtained from that linearisation. The new vehicle-terrain interaction model, that does not involve integral equations, results in a fast algorithm for wheel-terrain interaction analysis. The development is verified by comparison with the srcinal model given in [4] and with the approximate model [5], in terms of both computation speed and accuracy. II.A  NALYTICAL B ACKGROUND Consider a driven rigid wheel running on a firm, deformable ground. Under the action of the vertical load and driving torque, the wheel compresses the soil to a sinkage  z   [1]. The normal stress, acting normal to the wheel-terrain contact  point, is related to sinkage  z   by: ( ) , nc  z k bk       += φ  θ σ   (1) ( ) ( ) ,coscos 1 θ θ θ  −=  r  z   (2) 674978-1-4244-2287-6/08/$25.00 c  2008 IEEE ICARCV 2008  where n  is the sinkage exponent, c k  , φ  k   are the pressure-sinkage moduli of the terrain, b  is the wheel width, r   is the wheel radius, θ   is the contact angle at a considered point, and θ  1  is the entry angle at which the considered point on the wheel rim first makes contact with the terrain. The maximum normal stress point, θ  m , separates the contact zone into front and rear regions ( θ  1 - θ  m  and θ  m - θ  2 ), where θ  2  is the wheel angular location at which the considered point loses contact with the terrain [7]. The normal stresses distributed on these regions are calculated by the following equations [8]: ( ) ( ) ,coscos 11 nnc r k bk  θ θ θ σ  φ  −      += (3) ( ) ( ) .coscos 112212 nmmnc r k bk       −      −      −−−      +=  θ θ θ θ θ θ θ θ θ σ  φ   (4) Experiments have shown that θ  2  is very small to be considered as zero and θ  m  is often assumed to be in the middle of the contact zone [7]. In that case, (4) becomes ( ) ( )( ) .coscos 112 nnc r k bk  θ θ θ θ σ  φ  −−      += (5) For loose sand, saturated clay, sandy loam, and most distributed soils, the shear stress, which is a tangential component of the stress at the wheel-terrain contact point, exhibits an exponential relationship with respect to the shear displacement: ( ) ( )( ) ( ) ,1tan /  K  j ec − −+=  φ θ σ θ τ   (6) where  K  is the shear deformation modulus,  j  is the shear displacement, c  and φ   are respectively cohesion and internal friction angle of the terrain. The above equations were extended to account for lateral motion and applied to all wheels of a skid-steering UGV [9], as shown in Fig. 2. Reaction forces acting on the i th  wheel ( i =1,2,…,8) along longitudinal, lateral and vertical directions are derived by integrating components of normal and shear stresses on these directions as ( ) , 1 02/2/    − += iiii bbii X  X  X   dyd r  F  θ  θ σ τ   (7) , 1 02/2/    − = iii bbiiY Y   dyd r  F  θ  θ τ   (8) ( ) , 1 02/2/    − += iiii bbii Z  Z  Z   dyd r  F  θ  θ σ τ   (9) 0 5 10 15010203040506070contact angle (deg)   n  o  r  m  a   l  s   t  r  e  s  s   (   k   P  a   ) normal stressApproximation a)  0 5 10 15010203040506070contact angle (deg)   n  o  r  m  a   l  s   t  r  e  s  s   (   k   P  a   ) normal stressApproximation b) 0 5 10 15012345contact angle (deg)   s   h  e  a  r  s   t  r  e  s  s   (   k   P  a   ) shear stressApproximation c) 0 5 10 15012345contact angle (deg)   s   h  e  a  r  s   t  r  e  s  s   (   k   P  a   ) shear stressApproximation d) Figure 1. Stress approximation used in the paper: a), c) method by Shibly et al  . [5]; b), d) our method θ  7   x 1   o 13 o 35 o 57  o 7   x 3  x 5  x 7   x 22   o 24 o 46  o 6 8 o 8  x 4  x 6   x 8 Y  X Oa 7 1 θ  V   X  V  Y  Ω  x 1 1 1 θ  θ  1 3 1 θ  θ  3 5 1 θ  θ  5  B Bd b X  Z  1 ω   L ω   L θ  7 7  r h (a)(b)  x 1 o 1135 o 3 o 5 o 7  P  1  P  1  P  3  P  5 V  P  7  χ O Center of mass Center of mass Geometric center Figure 2. Vehicle free-body diagram on deformable terrain Similarly, the vehicle’s turning moment is obtained as ( ) ( ) ( ) ,sin21231 8102/2/8102/2/1 11      =−=−+      +−          −−+++−−= ibbiiY i ibbii X  X i i Z  iiiii dyd r d a i E r dyd  y Br  M  θ θ  θ τ θ θ σ τ   (10) where  E  (  x ) gives the integer part of  x . Solving integrals in (7)-(10) is time-comsuming due to the nonlinearity of normal and shear stresses. Their closed forms are to be sought for speeding up the computation. 675  III.F AST ALGORITHM FOR VEHICLE - TERRAIN INTERACTION ANALYSIS In this section, normal stress and shear stress beneath a wheel are linearised so that their integrals can be obtained in a closed form. Criteria are first chosen to minimize the approximation error, and then methods for normal and shear stress approximation are developed. The approximate stress is then substituted into equations of the vehicle-terrain interaction model developed in [4] to derive a new version for predicting  behaviour of the vehicle on deformable terrain.  A. Normal stress approximation The distribution of the stress under the i th  wheel is dicomposed in front and rear regions as given in equations (4-7). As the stress is normal to the wheel rim, its components on the X and Z directions are ( )( ) . iii Z  iii X  ii θ θ σ σ  θ θ σ σ  cos,sin =−= (11) In most cases, the contact angle i θ   is small, and as a result, i  X  σ   is very small compared with i  Z  σ  . The largest influence of the normal stress is on the reaction force in the Z direction, as given in (9). If the normal stress is approximated so that its effect on the component i  Z   F  does not change, then the absolute error due to the approximation will be minimized. Applying this criterion, our proposed method is described in the following. A linear curve of the normal stresses in the front and rear regions can be generally expressed by ,ˆ,ˆ 222111 iiiiii ck ck  ii +=+= θ σ θ σ   (12) where iiii cck k  2121 and,,,are constants to be determined. As the stress and its approximate lines intersect at the maximum stress point, (12) becomes ( )( ) ( )( ) ,ˆ,ˆ 22222 22211111 111 iiiiiii iiiii iiiiiii iiiii ck k k  k ck k k  k  immi mmiimmi mmi +=+−= +−=+=+−= +−= θ θ σ θ θ  θ σ θ θ σ  θ θ σ θ θ  θ σ θ θ σ   (13) where ( ) ., 222111 iiiii iiiii mmmm k ck c θ σ θ θ σ θ  +−=+−= (14) The angular coefficients of the approximations are chosen so that ( ) ( )( ) ( ) ,coscosˆ,coscosˆ 20202111 11 iimiimiiiimiiimi  Ad d  Ad d  iiiiii iiiiii ====      θ θ θ θ θ θ  θ θ θ σ θ θ θ σ  θ θ θ σ θ θ θ σ   (15) where ii  A A 21 and are constants assigned for the expressions in the right hand side of (15). Substitution of (13) into (15) gives ( )  ( ) ( ) [ ] ( )  ( ) ( ) [ ] .coscosˆ,coscosˆ 22221111 2211 iimiiiii imiiiiimiiii iimi  Ad k d  Ad k d  iimmiiii iimmiiii =+−==+−=      θ θ θ θ θ θ θ θ  θ θ θ σ θ θ θ θ θ σ  θ θ θ σ θ θ θ θ θ σ   (16) The angular coefficients can then be derived from (16) as ( )( )( )( ) .coscossinsinsin,coscossinsinsin 22222221111111 iiiii iiiii iiiiii iiiii i mmmmmmmm  Ak  Ak  θ θ θ θ θ  θ θ θ σ  θ θ θ θ θ  θ θ θ σ  −+−−−=−+−−−= (17) Let i  x 1 θ   and i  x 2 θ   be the contact angles at which the normal stress intersects its approximations in the front and rear regions respectively. Simulations using the proposed method with experimental wheel data collected from a field test show that on a given terrain, the ratios ii  x 11 / θ θ   and ii  x 12 / θ θ   are almost constant with different values of i 1 θ  . This is a great advantage  because it is not necessary to recalculate angles i  x 1 θ   and i  x 2 θ  each time when the entry angle changes. Instead, the average values of these ratios can be used. As the intersections of the normal stress and its approximations can be derived from the angle ratios, the approximating lines can now be simply described as ( ) ( )( ) ,,where,ˆ 11111111111 iiiii iiiiii iiii mmm xm xi k ck ck  θ σ θ θ θ θ σ θ σ  θ σ  +−=−−=+= (18) 676    ( ) ( )( ) .,where,ˆand 22222222222 iiiii iiiiii iiii mmm xm xi k ck ck  θ σ θ θ θ θ σ θ σ  θ σ  +−=−−=+= (19) From the observation that the shapes of normal stress and its components in the X and Z directions are near-identical, the same intersection points in (18) and (19) can be used to approximate the stress components as follows ( ) ( )( ) ,,where,ˆ 11111111111 iiiii iiiiii iiii m X m X  X  m xm X  x X   X  X i X  X  k ck ck  θ σ θ θ θ θ σ θ σ  θ σ  +−=−−=+= (20) and similarly, ( ) ( )( ) ,,,ˆ 22222222222 iiiii iiiiii iiii m X m X  X  m xm X  x X   X  X i X  X  k ck ck  θ σ θ θ θ θ σ θ σ  θ σ  +−=−−=+= (21) ( ) ( )( ) ,,,ˆ 11111111111 iiiii iiiiii iiii m Z m Z  Z  m xm Z  x Z   Z  Z i Z  Z  k ck ck  θ σ θ θ θ θ σ θ σ  θ σ  +−=−−=+= (22) ( ) ( )( ) .,,ˆ 22222222222 iiiii iiiiii iiii m Z m Z  Z  m xm Z  x Z   Z  Z i Z  Z  k ck ck  θ σ θ θ θ θ σ θ σ  θ σ  +−=−−=+= (23)  B. Shear stress approximation Let us consider the shear stress under the i th  wheel. Components of the shear stress along X, Y, and Z direction are given as ,sincos,sin,coscos iii Z  iiY iii X  iii θ ϕ τ τ  ϕ τ τ θ ϕ τ τ  === (24) where ϕ  i  is the angle between the shear stress and its projection on the X-Z plane (Fig. 3). As noted above, the contact angle i θ   is small so that i  Z  τ  isvery small compared with i  X  τ  . When the vehicle runs in a straight line, the angle ϕ  i  is zero and so is i Y  τ  . In most cases, therefore, the largest effect of the shear stress is on its component along the X direction i  X  τ  . This component, in turn, is the main contribution to the longitudinal reaction force, or drawbar pull, expressed in (7). Again, our criterion is that if the shear stress is approximated so that its effect on i  X   F   does not change, then the absolute error of the approximation is the smallest. In general, the linear shear stresses can be written as ,ˆ,ˆ 442331 iiiiii ck ck  ii +=+= θ τ θ τ   (25) where iiii cck k  4343 and,,,are constants to be calculated. Because the stress and its approximate lines intersect at the maximum stress point, (25) becomes, ( )( ) ( )( ) ,ˆ,ˆ 44244 24233133 131 iiiiiii iiiii iiiiiii iiiii ck k k  k ck k k  k  immi mmiimmi mmi +=+−= +−=+=+−= +−= θ θ τ θ θ  θ τ θ θ τ  θ θ τ θ θ  θ τ θ θ τ   (26) where ( ) ., 244133 iiiii iiiii mmmm k ck c θ τ θ θ θ  +−=+−= (27) To satisfy the suggested criterion, the angular coefficients of the approximations are chosen so that ( ) ( )( ) ( ) .coscosˆ,coscosˆ 422311 2211 iimiiimiiiiimiiimi  Ad d  Ad d  iiiiii iiiiii ====      θ θ θ θ θ θ θ θ  θ θ θ τ θ θ θ τ  θ θ θ τ θ θ θ τ   (28) Substitution of (26) into (28) gives, ( )  ( ) ( ) [ ] ( )  ( ) ( ) [ ] .coscosˆ,coscosˆ 42423131 2211 iimiiiii imiiiiimiiii iimi  Ad k d  Ad k d  iimmiiii iimmiiii =+−==+−=      θ θ θ θ θ θ θ θ  θ θ θ τ θ θ θ θ θ τ  θ θ θ τ θ θ θ θ θ τ   (29) 677
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