The Bekker Model Analysis for Small Robotic Vehicles

Transcription

1 04TB-9 The Bekker Model Analysis for all Robotic Vehicles Copyright 004 AE International Grant R. Gerhart U Ary TACOM, Warren, MI 48397/ATTN: AMTA-TR-R/M 63 ABTRACT This paper uses the Bekker odel for land locootion analysis to copare ground vehicle vehicles with different running gear configurations. The Bekker odel is inherently phenoenological in nature and requires epirical data to both calibrate and validate the ethodology for realistic soil/terrain conditions. This foralis consists of two fundaental equations. The first uses the Coulob-Mohr law and a linear, one degree of freedo spring/ass/daper odel to predict terrain shear rates fro axiu vehicle tractive effort. The second epirically predicts soil sinkage as a function of ground pressure loading. The latter contains no phenoenological link to the continuu echanics of terrain aterials and conditions. INTRODUCTION Bekker s foralis was developed as a design tool to copare different types of ground vehicle obility perforance characteristics. No single vehicular locootion syste has optial obility perforance under all terrain conditions. Vehicle running gear design always involves design coproises or tradeoffs over a nuber of obility factors. Most future Ary robotic vehicle platfor concepts fall into two broad categories: wheeled and track systes. Wheeled vehicles are typically ore agile and aneuverable than tracked vehicles, but possess higher ground pressure levels and therefore are less obile over rough terrain. Tracked vehicles on the other hand have lower ground pressure, superior traction and are thus ore trafficable for off-road conditions. However, they are not as agile or echanically efficient as their wheeled counterparts due to (typically) larger ass and uch larger internal otion resistance. Both wheeled and tracked vehicles have been successful in negotiating roadways and oderately unstructured off-road terrain. Vehicles with a larger wheelbase, ground clearance and horsepower per weight ratios generally have uch better intrinsic obility perforance than saller systes. A coparison of vehicle types for equal size and weights indicates that wheeled systes are typically superior to track systes in agility, aneuverability, ride quality and terrain daage. Tracked vehicles have distinct advantages relative to stability, ground pressure, axiu vertical slope, and drawbar pull. election of running gear configuration usually becoes a choice between which obility characteristics are ost iportant for a vehicle s intended ission profile. Ride quality is not iportant to unanned or robotic vehicles unless payloads such as sensor systes or structural loading specifications are exceeded for rough terrain conditions. The vehicle need only have sufficient drawbar pull to transport itself and its payload. Low ground pressure is principally an advantage only in soft soil terrain conditions. Unanned systes generally weigh less and have a lower ground pressure than the larger anned cobat vehicles such as the ain battle tanks or infantry fighting vehicles. Agility and aneuverability are both advantageous for off-road conditions. Ground clearance, axiu slide slope angle and wheelbase are iportant for difficult obstacle negotiation challenges such as ditch crossings or large vertical steps. In general a coplete systes analysis is necessary to deterine the

2 Report Docuentation Page For Approved OMB No Public reporting burden for the collection of inforation is estiated to average hour per response, including the tie for reviewing instructions, searching existing data sources, gathering and aintaining the data needed, and copleting and reviewing the collection of inforation. end coents regarding this burden estiate or any other aspect of this collection of inforation, including suggestions for reducing this burden, to Washington Headquarters ervices, Directorate for Inforation Operations and Reports, 5 Jefferson Davis Highway, uite 04, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to coply with a collection of inforation if it does not display a currently valid OMB control nuber.. REPORT DATE 0 OCT 004. REPORT TYPE N/A 3. DATE COVERED - 4. TITLE AND UBTITLE The Bekker Model Analysis for all Robotic Vehicles 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR() Grant R. Gerhart 5d. PROJECT NUMBER 5e. TAK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME() AND ADDRE(E) UATACOM 650 E Mile Road Warren, MI PERFORMING ORGANIZATION REPORT NUMBER PONORING/MONITORING AGENCY NAME() AND ADDRE(E) 0. PONOR/MONITOR ACRONYM() TACOM TARDEC. DITRIBUTION/AVAILABILITY TATEMENT Approved for public release, distribution unliited 3. UPPLEMENTARY NOTE 4. ABTRACT 5. UBJECT TERM. PONOR/MONITOR REPORT NUMBER() 6. ECURITY CLAIFICATION OF: 7. LIMITATION OF ABTRACT AR a. REPORT unclassified b. ABTRACT unclassified c. THI PAGE unclassified 8. NUMBER OF PAGE 9 9a. NAME OF REPONIBLE PERON tandard For 98 (Rev. 8-98) Prescribed by ANI td Z39-8

3 optial set of obility characteristics for a particular ission profile. Unanned vehicles in particular need a new set of syste requireents and represent a separate set of design challenges as copared to their traditionally anned counterparts. to the soil surface, and the area of the track noral to the surface. This force is opposed by the soil resistance when the THE BEKKER MODEL The echanical behavior of soils varies considerably under a wide variety of environental conditions. For exaple coposition, oisture levels, porosity, teperature, etc., affect bulk soil echanical behavior relative to vehicle/terrain dynaics. It is also well known that for the sae aount of echanical loading, a tracked vehicle ay cross soft terrain without considerable slippage, whereas wheels ay slip significantly, or siply sink into the terrain. The aount of slip varies with soil type. The Bekker odel uses the relationship between certain physical soil characteristics and shearing strength to predict vehicle cross-country obility. Bekker considers wheels and tracks as siple loading surfaces having siilar fors, but different lengths and widths. He extrapolates the analogy between soil shear produced by laboratory crawlers to track vehicles as shown in Fig. a []. When the blocked track is oved relative to the soil ass in the laboratory shear box, the axiu shearing force is not developed instantaneously with the initiation of relative otion. Instead the soil ust be copacted to soe degree before reaching the final steady state echanical shearing stress. Thus the track grousers begin slipping before reaching the point of axiu vehicle traction. This transient condition is the basis for Bekker s siple -DOF odel for vehicle trafficability. The shear stress is the ratio between the vehicle traction force, which is parallel to the soil surface, and the area of the track noral to the surface. This tractive force is opposed by soil resistance as the grousers slip during the shearing process. The noral loading force of the vehicle copacts the soil, which affects the resistance it exerts against the track grousers as they push against the ground. In suary the track pushes against the soil, generates tractive forces that are deterined by echanical properties of the ediu. The shear stress is the ratio between the vehicle tractive force, which is parallel Figure oil hear Analogy vehicle weight copacts the soil and also affects the resistance it exerts against the grousers. In effect the track forces, which push against the soil, generate a soil resistance that is deterined by soil type and copaction. Vehicle weight generates ground pressure, which further copacts the soil and alters the soil resistance. Figure b shows a tracked vehicle in otion relative to the terrain. A grouser on the track first coes into contact with the ground at position. No shearing has occurred at the oent of initial contact. A shearing force develops in the lateral direction as the vehicle oves forward. The grouser position begins to ove - pushing against the soil and distorting it (). The soil distortion increases [] as the vehicle oves forward. Figure a shows three plots of soil shear stress as a function of soil shear for three different types of terrain conditions []. This epirical data is typical of different generic soil types. The curve labeled A is for a loose frictional or plastic soil such as wet clay. The shearing strength τ a of this

4 soil type is reached after the initial period of copaction, which takes place K ωt, and K b where K and K are coefficients of slippage. ubstitution of these paraeters gives the following result: ( K + ( K ) ) K ( K ( K ) ) K Ae + Ae τ. () We relate the coefficients A and A by setting 0 and τ 0, A + A 0. Also for 0, τ 0, and dτ/d K 3, d Ae K K K d τ ( K+ ( K )) K ( + ( )) ( K ( K )) K 3 + Ae ( K ( K )) K K. (3) A and A are given by the following expressions: K 3 A (4) K ( K ) A. Cohesion less oil B. Cohesive oil C. Mixture of A and B Figure Characteristics of soil deforation over a distance a. Beyond this point the stress reains nearly the sae irrespective of additional slip. oil type B consists of a dry coherent ass: dry clay or snow at very low teperatures. This type of soil quickly reaches its axiu shearing strength and then drops off rapidly. The curve C refers to a soil type that has interediate properties between A and B. Upon reaching a axiu value, it starts to lose its shearing strength but not as rapidly as B []. For odeling purposes, it is critical to develop a general equation for these curves. The graphs in Figure a correspond to the displaceent (x) and natural tie frequency (ωt) for an aperiodic vibration: x ( b+ ( b ) ) ω t ( b ( b ) )ωt Ae + Ae, () where b is the coefficient of daping. An equation describing soil stress assues the grousers slip during the shearing process. The shearing stress is (τ) and soil deforation (). We define τ x, K 3 A (5) K ( K ) ubstituting A and A into equation (): K 3 ( K + K ) ( K K ) K ( K τ e e ) (6) K ( K ) The axiu peak for the curve in Figure a is given in equation (7). ln( K ( K ) ) ln( K + ( K ) K ( K ) (7) The shear strength (τ) can be defined as the axiu value of the shear stress before the soil yields [3]. Figure 3 contains a Mohr - Coulob diagra plotting ground pressure as a function of shear stress for any orientation of the reference axis. The Mohr circle expands to a critical size before failure occurs. The line tangential to the failure point is the Mohr-Coulob failure liit. An equation for this line is y x + b where b is the coefficient of cohesion, is tan(φ), φ is the frictional angle, and x is noral stress or ground pressure. This line approxiates the axiu shearing

5 strength, τ, of a particular soil type, and it is used extensively strength in land locootion. τ c p tan(φ) (8) + is equal to the product of the slip speed and the tie duration in which it occurs. v t () However, the slip speed is equal to the difference between the tire or track and actual vehicle speeds: v v t v a () ( vt va ) t (3) and t d / vt, where d is the total distance along which the shearing has occurred in contact with the terrain. va d( ) iod (4) v The aount of soil distortion that takes place at a distance x fro the front of the ground contact area is equal to t ( x d) (5) i where,. (6) o x Figure 3 Mohr-Coulob Failure Line ince the portion contained in brackets (Eq. 6) is diensionless, the value of K 3/ K (K -) ust have the units of lb/in and the value of K 3 ay be expressed in the following anner. K 3 [ e K ( K + ( K )( c + p tanφ) ( K ) ) K ( K ( K ) ) K e Now Eq. (6) can be siplified: τ c + p tanφ) [ e y ] ax ( ( K + K ) K ( K K ) K ax e ] (9) (0) where y ax is the largest value for the expression within the brackets. The slip distortion and the aount of slip are related through the stress strain equations. The axiu shear distance ( ) Equation (6) gives a relationship between tractive force and slip. Figure 4 copares the shear force of a tracked vehicle for two different types of soil. The top graph is highly frictional, undisturbed, fir silt. For 0% slip, shear is produced along the entire ground contact surface; but the front portion of the track produces ost of the force []. As the vehicle begins to experience ore slip, ost of the shearing force is produced at the front portion of the track. The rear half of the track produces little shear and increases soil resistance by creating drag. The second type of soil has a high cohesive property such as wet clay. The track is producing shear forces in relatively equal aounts along its entire length for all percentages of slip. While in otion, a track or wheel develops a force produced by the shearing strength of the soil. This force H is called the gross tractive effort or soil thrust. The tractive effort is the integral of the shear forces along the slip distance. By substitution of Eq. (0),

6 H d 0 ( { e c + H p y ( K ax d 0 τ xdx tan φ ) ( e K ) K ix ( K + } dx K ) K ix (7) (8) cohesive properties, Therefore c0, and Eq. (9) is reduced to a single ter W tanφ. For this case as the vehicle weight is increased the aount of soil thrust increases proportionally. If the sae vehicle is operated in a plastic soil such as saturated wet clay, the frictional coponent of the soil is equal to zero (φ0). Equation (9) reduces to A c where A represents the contact surface area of the vehicle running gear. A higher value of thrust is only obtained by an increase in contact surface area. In suary, vehicles that traverse highly frictional soils benefit fro an increase in ground contact area. oil types with high oisture content or are very cohesive in nature, iprove vehicle obility with designs that increase ground contact surface area. An increase in weight for this type of soil is a liability [5] resulting in additional soil resistance and copaction. BDTM PREAD HEET We developed a spread sheet using the Bekker foralis to evaluate robotic vehicle obility perforance. It uses the linear one-degree of freedo (-DOF) Bekker odel that has been created in a spreadsheet forat. This odel assues that the soil is hoogenous and the loading effects on the soil are linear. Both track and wheeled vehicles can be siulated using this foralis. Iportant vehicle paraeters include tractive force, tractive effort, soil sinkage, drawbar pull, and tractive coefficients (DP/W). INPUT Figure 4 Tractive forces in different soil types Typically, heavier vehicles are able to generate larger tractive forces. Much experience by a nuber of investigators over the years gives credibility to this stateent, but is it valid for all soil types? In order to answer this question, consider Eq. (9). oil thrust is defined as the addition of two different soil strengths. The first originates fro its frictional properties while the second fro its cohesive properties, H A c + W tanφ. (9) If a soil type such as dry sand is chosen, a hoogenous saple would contain no The odel inputs are divided into three categories. The first set is general vehicle inforation. These paraeters include the width and length of the track or wheels which are in contact with the ground. These diensions correspond to the vehicle foot print in contact with the terrain. Other paraeters include the nuber of tracks or wheels, contact area, and vehicular weight. The second set of inputs describes the vehicle trafficability, or conversely, vehicle perforance in a given terrain. These paraeters define the strength, sinkage, and slippage that a vehicle would experience in a specific type of hoogenous soil. Most of these paraeters are obtained fro Bevaeter easureents, which is a device created by Bekker for this purpose []. These easureents

7 include the depth of the plate sinkage, the odulus of soil deforation in cohesional and frictional soil, the exponent of soil deforation, and the coefficients of slippage. A separate section in the spread sheet provides these paraeters for different types of soil. Other paraeters such as the coefficient of cohesion and the angle of friction are calculated fro the Mohr-Coulob failure line. OUTPUT The BDTM output paraeters are arranged into seven different categories. The first contains the axiu thrust force, which the terrain can support. This paraeter is the product of the Mohr-Coulob axiu ground pressure ultiplied by contact area. The Mohr-Coulob law in Eq. (9) contains the ters W tanφ for the frictional coposition of the soil and Ac due to cohesion. ince ost terrain is a ixture of these two properties, soil thrust is the su of these two ters. The next set of paraeters contains soil strength and ground pressure. The vehicle weight produces a noral force and ground pressure on the terrain. The Mohr-Coulob law deterines the axiu soil strength and ground pressure prior to plastic deforation of the terrain. Bekker derived fro the stress-stain curves that the soil behavior can be described in ters of the displaceent(x) and natural tie frequency (ωt) of an aperiodic vibration []. The equation for tractive force is derived fro this observation, and it is the product of the axiu shear force given by the Mohr-Coulob Law ultiplied by the percentage of slippage between the vehicle and terrain. Equation (8) expresses the tractive effort in ters of soil properties, contact area, load, and slip for a given type of soil defined by the constants K and K. Bekker derived an epirical forula fro his Bevaeter experiental data to evaluate sinkage in frictional and cohesive soils, z k c p / b k + φ / n, (0) where p is the ground pressure, b is the iniu diension of the track or tire, k c and k φ are frictional and cohesive odulus of soil deforation and n is the exponent of soil deforation. This equation explains why wider tracks or tires on vehicles with the sae ground pressure sink deeper into the terrain. Not all vehicle soil thrust results in useful work. Instead part of it dissipates into theral and frictional energy losses, which are caused by copaction, bulldozing, and dragging of the soil. An epirical expression for the losses overcoing copaction resistance ay be expressed by R c ( n + )( kc W + bk ) φ l n+ n, () where W is vehicle weight in pounds and l is the length of the tire or track in contact with the ground. It should be noted fro Eq. (), that longer contact areas produce saller copaction resistances. Bulldozing results fro the accuulation of a soil ass in front of the vehicle tires or tracks. In our analysis the bulldozing resistance is neglected and assued to be sall. iilarly, resistance fro soil trapping and dragging is also neglected in our analysis. The drawbar pull (DP) is the total thrust inus the soils resistances. It is custoary to view DP as a priary etric for vehicle locootion. If it is zero or negative, then the vehicle will have no net forward otion. In BDTM there are three different values for DP. The first considers the soil thrust developed using soil paraeters for siple geoetric track and tire geoetries. The second includes additional thrust that is generated by the action of grousers or treads. The Mohr- Coulob equation is odified for this result: ( ) H blc( + h/ b) + W tanφ ( h/ b) cot ( h/ b) () where b is the width, l is the length, h is the height of the grouser or tire tread, c is the coefficient of cohesion, and φ is the angle of friction. The last expression for DP in BDTM is the total tractive force evaluated for a specific percent slip at soe distance fro the front of the track or tire contact area. A coon etric for baseline evaluation noralizes these different expressions by dividing DP by total vehicle weight. These paraeters are referred to as traction coefficients, and they are one convenient ethod for coparing different vehicle types

8 The final set of outputs is cone index (CI) and obility index (MI) conversion factors. Waterways Experient tation (WE) developed a ethod to easure these soil paraeters. The cone index is an epirical paraeter, which is easured using a cone penetroeter device. The CI values are obtained by converting Bevaeter into CI paraeters using Eq. (3)[6]. The conversion was proposed by Janosi [7] and tested by WE in 964. highly frictional soil that produces nearly all the tractive effort for less than ten percent slippage. Type C on the other hand is a plastic or cohesive soil. Tractive effort is produced relatively uniforly regardless of the soil distortion or slippage experienced. CI.65 k c ( n + ) (( z +.5) z ) n n + + ( z +.5) z ( z +.5) z kφ + ( n + )( n+ ) n+ n + n n n (3) DATA ANALYI In BDTM there are four different graphing routines. The first plots traction coefficients as a function of k values. This process provides input paraeters in the calculation of traction coefficients for different soil strengths. Figure 5 shows both track and wheel versions of the sae vehicle in a priarily frictional soil. The tracked vehicle traverses this type of terrain with greater DP than the sae syste with wheels. Figure 6 Tractive Forces vs. Distance A. Cohesionless or loose frictional soil B. Mixture of A and C C. Cohesive or plastic soil Figure 7 Tractive Efforts vs. lip Figure 5 DP/W vs. K values The next two charts show tractive forces, which are produced for different configurations of track or wheel contact areas. The calculations in Fig. 6 are ade with 0, 0, 30, 40, and 00 percent slip values. Figure 6 shows a tracked vehicle in an undisturbed, settled, sandy loa. The peak tractive forces occur near the front portion of the track for these exaples. Figure 7 displays tractive effort per unit area as a function of percent slip for three different soil types. Type A is a EXAMPLE Two exaples in this paper are provided to deonstrate the capabilities of the odel. EXAMPLE The track vs. wheel tradeoff for ilitary vehicle obility has any different facets. Low-pressure pneuatic tires using adaptive tire pressure can draatically reduce ground pressure. We will use BDTM to copare this type of wheel running gear syste with a vehicle using tracks. In this section we look at different types of relatively sall robotic vehicles

9 traversing highly cohesive terrain such as wet clay. A sall, four-wheel robotic platfor with diaeter tires is shown in Fig. 8a. The rectangular foot print is 3 x4, weight is 000 lb. and the tires are separated by a distance of 36. Each tire has a total contact area of sq. in. and an overall surface contact area of 48 sq. in. The ground pressure of the vehicle is psi. Our odel predicts that the wheels will sink.3 below the terrain surface. At this depth, the soil resistance is larger than the axiu soil thrust generated by the vehicle. A negative value for DP indicates that the vehicle is incapable of oving forward at a nonzero speed. One possibility for iproving tractive effort ight be to increase the net payload or gross vehicle weight. When a 00 lb payload is added to a baseline vehicle weighing 000 lbs, the wheels sink deeper into the soil. The otion resistance increases and DP reains a negative nuber. This type of soil is soft and defors too easily to support the larger ground pressure. This technique does work, however, for ore rigid terrains with saller soil sinkage paraeters. When the diaeter of the tire is increased to a 3 x6 feet print as shown in Fig. 8b, the total ground contact area of the vehicle is increased to 7 sq. in. and the soil sinkage is reduced to.5. The soil resistance is reduced to a point where the vehicle is now able to ove with a positive DP. In suary an increase in tire diaeter, increases vehicle ground contact surface contact area, which ultiately increases platfor DP. Additional increases in wheel diaeter will ultiately increase the DP to the desired levels. At this point, however, there will be several tradeoffs aong various design paraeters including turning radius, engine torque and gearing, suspension design and roll over that will liit practical increases in wheel and tire diaeters. An alternative approach ight also be to add additional sets of wheels. Figure 8d displays the sae robotic platfor with a 6x6 wheel configuration. When diaeter wheels are used in the design, they produce a 3 x4 foot print. The aount of surface contact area is equivalent to the 4x4 wheel design with the larger tires. By increasing the size of the six tires to allow for a 3 x6 foot print, the platfor surface area is increased to 08 sq. in. The ground pressure decreases to 9. psi and the soil sinkage reduced to. The net vehicle tractive force increases while the DP doubles in value. The sae platfor with a track configuration is shown in Fig. 8c. Each track is 5 long and 3. The ground pressure decreases to 6 psi and soil sinkage to 0.6. The DP increases by a factor of 4 as copared to the original 4-wheeled vehicle, 4x greater than the platfor with enlarged tires, 4x larger than the six-wheeled vehicle with diaeter wheels, and x larger than the six-wheeled vehicle with larger diaeter wheels. When an additional 00-lb payload is placed on the tracked vehicle, the tractive effort reains unchanged while the DP decreases soewhat due to additional soil sinkage. Figure 8 Robotic Vehicles This exaple shows soe of the obility tradeoffs that can be ade in a plastic soil type such as wet clay. The vehicles ability to traverse in this type of terrain is dependent on the overall design of the running gear configuration and the aount of ground contact surface area. EXAMPLE This exaple looks at the tradeoffs between track and wheel running gear for highly frictional types of soil such as dry sand. The sae robotic vehicle platfors were also used in this analysis The tractive force and soil thrust fro a vehicle with tires that have a 3 x4 rectangular foot print are coparable to the sae vehicle outfitted with a pair of 3 x5 tracks. The track outperfors the tire by a ratio of.5/. If the 4x4 vehicle is outfitted with tires that have

10 a 3 x6 surface foot print, then this ratio decreases to./. The 6x6 vehicle with 3 x6 foot print tires has a ratio nearly equal to /. An interesting observation fro these two exaples is that wheel vehicles, which experience ore slip relative to the terrain, have perforance siilar to tracked systes with coparable size and weight. This phenoenon occurs near slip values of 33% slip for the 3 x4 foot print and 4% for the case of oversized tire. These exaples point out another interesting feature of the wheels vs. tracks tradeoff. When vehicle payload increases for tracked platfors; the tractive force, soils thrust and DP becoe larger under soe conditions because the ground contact area is sufficiently large to prevent excessive soil sinkage and iniize soil resistance. When payload increases for the 4x4 vehicle, the tractive force and soil thrust both increase less rapidly than the soil resistance. This condition leads to a net reduction in DP. Reducing vehicle payload fraction drastically iproves the perforance of the wheel relative to the track platfors. These exaples show quite clearly that track running gear echaniss perfor uch better than their wheel counterparts for heavy platfors. The larger ground contact areas lead to saller ground pressures that reduce soil resistance and produce larger tractive forces and net DP. The track systes, however, generate uch larger internal frictional losses and are generally less reliability for off road conditions. Light weight wheeled vehicles, however, can perfor coparable to track vehicles under any off road conditions assuing the ground pressure is low enough to prevent excessive soil resistance to terrain sinkage. Under these conditions wheel vehicles have any distinct advantages including energy efficiency, aneuverability and agility. CONCLUION This paper explores the Bekker foralis as a ethodology to exaine obility tradeoffs between wheel and track platfors. We developed a odel, BDTM, as a design tool to copare different types of running gear echaniss for sall robotic vehicle platfors. It is a siple, linear one-degree of freedo (- DOF) odel that has been created in a spreadsheet forat. We use BDRM priarily as an analytical tool for off-road vehicle obility perforance evaluation. This ethodology suppleents the NATO Reference Mobility Model (NRMM), which is the Ary s priary obility perforance evaluation tool for off road platfors. The NRMM odel is inherently epirical in nature, and its foundation rests upon a huge historical data base that has been acquired over any years. The Bekker odel, although siplistic in its forulation, is easy to use and provides a phenoenological understanding of any essential features in vehicle terrain echanics. Our particular interest in the Bekker Model involves perforance evaluations of sall, unanned ground vehicle systes that weigh under 500 lobs. Most epirical data bases have very little data for this size and weight class of vehicles. The Bekker odel has been very useful in aking rudientary design tradeoffs for very sall platfors weighing 00 pounds or less where very little epirical data is available. One can calculate tractive force, tractive effort, drawbar-pull, soil sinkage, safe weight pressures, ground pressures, and percent slippage for these systes. REFERENCE. Bekker, M.G. 960 Off-the-Road Locootion University of Michigan Press, UA. Bekker, M.G. June 955 Wheels or Tracks Autoobile Engineer 3. Whitlow R 990 Basic oil Mechanics ( nd Ed) Longan Group, UK p07 4. Wong, J.Y. 989 Elsevier cience Publisher B.V., The Netherlands 5. Bekker, M.G. Land Locootion Research Report 3, 957 Terrain Evaluation in Autootive Off-the- Road Operations OTAC, Detroit 6. Bekker, M.G. 969 Introduction to Terrain-Vehicle ystes University of Michigan Press, UA 7. Janosi, Z and Hanaoto, B. 96, Proc. Of the st International Conference on the Mechanics of oil- Vehicle ystes, Edizioni Minerva Tecnica, Torino, Italy.