Dynamic Stability of Off-Road Vehicles: Quasi-3D Analysis

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1 28 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, Ma 9-23, 28 Dnamic Stabilit of Off-Road Vehicles: Quasi-3D Analsis Moshe Mann and Zvi Shiller Abstract This paper presents a method to determine the stabilit of off-road vehicles moving on rough terrain. The measures of stabilit are defined as the maimum speed and acceleration under which the vehicle does not slide or tip over. To compute these stabilit measures, we propose a quasi 3D analsis b decomposing the vehicle dnamics into three separate planes: the aw, pitch, and roll planes. In each plane, we compute the set of admissible speeds and acceleration for the planar vehicle model, contact model, and ground force constraints. The intersection of the admissible sets provides the total range of feasible speeds and accelerations along the vehicle s path, from which we obtain the stabilit margins. Numerical results for a vehicle traversing a simulated terrain demonstrate the effectiveness of the approach. I. INTRODUCTION Man current missions require an autonomous robot to traverse rough terrain in minimum time, such as the DARPA s Grand Challenge []. The risk of instabilit becomes a crucial factor in its motion planning when both speed and terrain roughness are significantl high. The robot s susceptibilit to slide, tip over, or lose contact with the ground increase with its speed and unevenness of the ground. Analing and ensuring the stabilit of the robot or vehicle at nonero speeds is the goal of this work. The success of a motion planner demonstrabl depends on keeping the velocities and accelerations of the vehicle within their admissible range. Our goal is to compute the range of feasible speeds and accelerations for off road vehicles and present it as a unified measure of dnamic stabilit. Although various stabilit margins have been proposed over the ears, [9] [3], none of them consider vehicle dnamics, friction constraints, and terrain surface characteristics, and hence are not suitable to evaluate vehicle stabilit during high speed motion on rough terrain. A few motion planners take into account onl the geometr of the terrain [5], and of the vehicle [3][4] [4]. These planners, however, assume quasistatic motion, where the vehicle s speed is considered negligible. The first dnamic motion planner for off road vehicles was developed in []. It calculates the global optimal trajector for a point mass vehicle on general three dimensional terrain. This paper etends [] and [8] to a 3D vehicle model. The approach consists of projecting the vehicle s motion onto three planes: the aw, pitch, and roll planes, and then analing each plane separatel. The velocit and acceleration The authors are with the Department of Mechanical Engineering and Mechatronics, The Paslin Robotics Research Laborator, Ariel Universit Center, Ariel 47, Israel. This work was partl supported b grant of the Israeli Space Agenc through the Ministr of Science Culture and Sports of Israel. shiller@ariel.ac.il limits are computed for each plane using a vector-algebraic procedure, and are then combined to ield the total dnamic stabilit margins. II. THREE DIMENSIONAL TERRAIN AND VEHICLE MODEL A. Coordinate sstems Figure shows the global and vehicle fied coordinate frames. The vehicle travels along a path parameteried b its arc length s, such that its center of mass along the path is a parametric function of s. The vehicle s orientation along the path relative to the global frame is described b the rotation matri R B. The vehicle s angular velocit is epressed b rotations around the vehicle fied frame, as shown in Figure. Z Fig.. Y X World and bod fied coordinate frames B. Angular velocities The vehicle s angular velocit ω is computed b differentiating the rotation matri of its bod fied frame relative to the inertial frame ([5]): n S(ω)=R BT Ṙ B () where S(ω) is a skew smmetric matri, consisting of the elements of ω, and Ṙ B is approimated b the finite differences of two successive rotation matrices along the path. C. Path coordinates To facilitate the analsis of the vehicle s stabilit, the acceleration of its center of mass, ẍ, is best described in terms of its speed ṡ and acceleration s along the path: ẍ = st +(ṡ 2 /ρ)n (2) where t is a unit vector tangent to the path, n is a unit vector normal to the path and pointing towards the center of path curvature, as shown in Figure, and ρ is the radius /8/$ IEEE. 23

2 of curvature. The vehicle s angular speed and acceleration are epressed as: ω = Θ s ṡ (3) ω = Θ s s + Θ ss ṡ 2 (4) where Θ s is obtained b differentiating R B with respect to s: S(Θ s )=R BT (R B ) s (5) and Θ ss is its partial derivative with respect to the arc length s. D. Inverse kinematics Given the terrain map and vehicle s location and orientation in the the inertial coordinate sstem, we wish to calculate the vehicle s orientation and the direction of the ground normals. To this end, we model the vehicle as a triccle b lumping the two front wheels into one center wheel, located at f. We leave the radius of the wheels out of the inverse kinematics problem since the equations are highl nonlinear when the normals at the three contact points are not parallel to each other. Once the contact points are calculated, we add the average height of the wheels to the vehicle to obtain an approimation of its actual position. This is a reasonable approimation for relativel smooth terrain. Thus, given c and c of the vehicle s center and its aw angle ψ, we solve for seven unknowns ( f, f, l, l, r, r, c ), using three equations for the location of the center of mass in world coordinates: c c = f f + R B d, (6) c f h three kinematic chains f l = 4d 2 + l 2 (7) f r = 4d 2 + l 2 (8) l r = 2l, (9) and the aw angle relation [2] sin(ψ)r B(2,2) + cos(ψ)r B(3,) =, () where d, l, and h are the vehicle half length, width, and height, respectivel, and f, r, l are the locations of the front, right, and left wheel ales, respectivel. The steering angle is approimated b projecting the vehicle at each point along the path onto the previous vehicle s plane, then using the approach presented in [2]. III. PLANAR PROJECTIONS For the quasi 3D approach, we split the vehicle s motion into three separate planes: the aw (ψ), pitch (θ), and roll (φ) planes, as shown in Figure 2, where β is the steering angle. The unit tangent vector t, curvature vector κ, and gravit vector g, are projected onto all three planes using the respective projection matrices: [ ] [ ] [ ] e T R ψ = e T e T, R θ = e T e T, R φ = e T () where e,e,e are the unit vectors comprising the vehicle s bod fied frame. The projections of the tangent, curvature, Yaw Plane Pitch Plane Roll Plane Fig. 2. β g ψ g θ g φ Vehicle as seen in the aw, pitch, and roll planes and gravit vectors in each plane are computed b premultipling each vector b the respective projection matri. IV. CONTACT MODEL To maintain contact with the ground, without slip, the contact force between the wheel and the ground must be confined to the elliptic friction cone shown in Figure 3: f 2 s µ s 2 + f2 r µ 2 r f 2 n (2) where µ s and µ r are the longitudinal and lateral coefficients of friction, and f n is the normal force applied b the ground. Fig. 3. f s f r The ground forces on the wheels are bounded b a conic ellipsoid. The elliptic cone projects onto the aw plane as an ellipsoid, and onto the pitch and roll planes as a friction cone. The cone is linearl approimated as a pramid, where the friction constraints are reduced to: n f s µ s f n (3) f s µ s f n (4) f r µ r f n (5) f r µ r f n (6) This approimation, though not conservative, allows us to easil solve for the feasible speeds and accelerations. V. EQUATIONS OF MOTION Both the vehicle dimensions and the ground forces are projected onto the pitch, roll, and aw planes, with the respective plane indicated b the subscript. 232

3 A. Pitch Plane The forces acting on the lateral plane of the vehicle undergoing longitudinal motion are shown in Figure 4. The equations of motion are: f θ + f 2θ + mg θ = mt θ s + mκ θ ṡ 2 (7) r θ f θ + r 2θ f 2θ = I θ (θ s s + θ ss ṡ 2 ) (8) where I θ is the moment of inertia around the pitch plane. f θ r θ r 2θ Fig. 4. mg θ f 2θ The vehicle in the pitch plane. The friction cones shown represent (5)-(6). B. Roll Plane Similarl, for the roll plane shown in Figure 5, we have: f φ + f 2φ + mg φ = mt φ s + mκ φ ṡ 2 (9) r φ f φ + r 2φ f 2φ = I φ (φ s s + φ ss ṡ 2 ) (2) Note that t φ is rather small, since the path direction is generall almost perpendicular to the pitch plane. The friction cones shown represent constraints (3)-(4). Fig. 5. f φ rφ r2φ mg φ f 2φ The vehicle in the roll plane. C. Yaw Plane The vehicle is modeled in the aw plane as a biccle, as shown in Figure 6. We make a simplifing assumption that the normal force in the aw plane on each wheel is half of the gravitational force mg k normal to the plane, where: g k = g e (2) Constraints (3)-(6) then become: µ r 2 mg k f r µ r 2 mg k (22) µ s 2 mg k f s µ s 2 mg k (23) These constraints are linear approimations of the projected friction ellipses shown in Figure 6. The resulting equations f 2ψ β Fig. 6. r 2ψ r ψ f ψ The vehicle in the aw plane. of motion are: f ψ + R(β)f 2ψ + mg ψ = mt ψ s + mκ ψ ṡ 2 (24) r ψ f ψ + r 2ψ (R(β)f 2ψ )=I ψ (ψ s s + ψ ss ṡ 2 ) (25) where [ ] cos(β) sin(β) R(β)= (26) sin(β) cos(β) VI. SOLUTION METHOD To determine the constraints on the vehicle s speed and acceleration, we map the contact force constraints onto constraints on s and ṡ. Because this problem is indeterminate, since in each plane we have three equations of motion and four unknowns, we substitute the proper combination of ground force constraints into the equations of motion to result in linear constraints in the s ṡ 2 plane. The solution method in the pitch and roll plane is equivalent to [7], and we review it here to etend it in a similar manner to the aw plane. The graphical approach [6] is another convenient wa to solve the problem. A. Pitch A planar friction cone C ma be epressed as an intersection of two half planes. An ground force f θ C can be epressed as a linear combination of the unit vectors e parallel and n normal to the boundar of the left and right half planes, as illustrated in Figure 7: f θ = k l e l + c l n l (27) f θ = k r e r + c r n r (28) For f θ to be in C, the coefficients in (27, 28) must simultaneousl satisf the two inequalit constraints: c l (29) c r. (3) Since (29, 3) appl to each contact point, there are a total of four such constraints for the (planar) vehicle. We now substitute the ground forces, each represented b one half plane of its friction cone, (27) or (28), into the equations of motion (7)-(8) to result in three equations in four unknowns (k,c,k 2,c 2 ). Eliminating two unknowns (k,k 2 ) produces one equation in two unknowns (c,c 2 ) of the form: a s + a 2 ṡ 2 +(a 3 c + a 4 c 2 + a 5 )= (3) where: [a a 2 ] = m[r θ e r 2θ e 2 ] I θ [θ s θ ss ][e e 2 ] [t θ κ θ ] (32) 233

4 f C For f ψ to be in R, it is necessar that the coefficients in (35-38) simultaneousl satisf the inequalit constraints: e l n l er n r c r (39) c t (4) c l (4) c b (42) Fig. 7. Each friction cones is the intersection of two half planes. Substituting one of equations (35-38) for each wheel into the equations of motion (24,25) and eliminating two unknowns ields a scalar equation in s,ṡ 2 similar to (3), with the coefficients: [a 3 a 4 ]=[r θ n r 2θ n 2 ] [r θ e r 2θ e 2 ][e e 2 ] [n n 2 ] (33) a 5 = [r θ e r 2θ e 2 ][e e 2 ] mg. (34) Equation (3) represents a straight line in the s ṡ 2 plane for given c,c 2.Ifa 3 a 4 >, then varing c,c 2 from to would span a half plane in the s ṡ 2 space. Thus, (3) maps the selected force constraints (the two half planes) on f θ and f 2θ to constraints on s ṡ 2. Repeating (3) for all possible combinations of half planes of the rear and front friction cones would produce four half planes in the s ṡ 2 space. The intersection of these half planes represents the set of admissible speeds and accelerations that satisf all the ground force constraints. This will be further discussed in Section VII. B. Roll The set of admissible speeds and accelerations due to constraints on the ground forces in the roll plane are obtained similarl to the procedure outlined for the pitch plane, ecept that we now use the friction cones and equations of motion of the roll plane (7-8). C. Yaw In a similar manner, the rectangular approimation of the friction ellipsoid in the aw plane can be visualied as the intersection of four half planes, whose edges form the right, top, left, and bottom edges of the friction rectangle R, as shown in Figure 8, where e r and e s are the unit vectors in the respective rolling and lateral directions, and n r and n s (not shown) are the respective normals. An force f ψ R must belong to the following four half planes: f ψ = k r e r +( 2 µ smg k c r )n r (35) f ψ = k t e s +( 2 µ rmg k c t )n s (36) f ψ = k l e r +( 2 µ smg k + c l )n r (37) f ψ = k b e s +( 2 µ rmg k + c b )n s (38) [a a 2 ] = [r e r 2 e 2 ][e e 2 ] (43) I ψ [ψ s ψ ss ] [a 3 a 4 ] = [r n r 2 n 2 ] [r ψ e r 2ψ e 2 ][e e 2 ] [n n 2 ](44) a 5 = ([r ψ n r 2ψ n 2 ] (45) [ ] [r ψ e r 2ψ e 2 ][e e 2 ] µ ) [r ψ e r 2ψ e 2 ][e e 2 ] mg ψ where e,e 2,n,n 2, µ, µ 2 are selected for ever combination 2µ s mg k 2µ r mg k -e s R -e r e r e s Fig. 8. The friction rectangle in the aw plane is the intersection of four half planes. of half planes that contain the ground forces on each wheel. VII. STABILITY MARGINS Intersecting the half planes in the s ṡ 2 space, produced b all combinations of the ground force constraints in each of the three planes, results in the set of Feasible Speeds and Acceleration (FSA), shown schematicall in Figure 9 []. An given pair of speed and acceleration that lies outside of the FSA region is unattainable, i.e. violates at least one contact constraint. We define the dnamic stabilit margin (DSM) as the maimum feasible speed, and the static stabilit margin (SSM), as the minimum feasible acceleration range at ero speed along ever point of the path, s, asshown in Figure 9, where the stabilit margins are indicated b the boundaries of the FSA: { ma(ṡ),ṡ DSM(s)= 2 FSA if ma(ṡ) (46) otherwise µ 2 234

5 { min( smin, s SSM(s)= ma )ṡ= if s ma >, s min < = otherwise (47) point the static stabilit margin in the aw plane is the most restrictive, as its FSA is a narrow polgon. The roll plane, on the other hand, is the least restrictive. The orientations of the s DSM FSA s 2 Fig.. The FSA s for each plane Fig. 9. The Feasible range of Speed and Acceleration (FSA) with the Static Stabilit Margin(SSM) and the Dnamic Stabilit Margin (DSM). The SSM and DSM are important indicators of the stabilit of a vehicle traversing a preset trajector. For eample, a vehicle on a steep incline will have a low SSM, indicating that slippage is likel to occur if the vehicle accelerates too rapidl. A vehicle on a conve surface will have a low DSM, which indicates that a sufficientl high velocit will result in the vehicle losing contact with the surface. VIII. EXAMPLES To demonstrate the utilit of the FSA and stabilit margins, we simulate a three dimensional vehicle model traversing a serpentine path on a random crater and hill, as shown in Figure. The vehicle is illustrated as a triccle in concurrence with its kinematic model. The three separate Fig.. A vehicle traversing a crater and a hill FSA s for each pitch, roll, and aw plane, are shown in Figure for the point at which the vehicle is at the top of the hill in Figure. One sees from Figure that at this vehicle in each plane are shown in Figure 2. Matching the configuration in the pitch plane with the corresponding FSA indicates that the result is in good agreement with previous two dimensional stabilit analsis [8], as a vehicle on conve ground is epected to have a finite velocit limit. The SSM Fig. 2. Roll Plane Pitch Plane Yaw Plane The vehicle s orientation in the pitch, aw, and roll planes and DSM for the entire trajector are shown in Figure 3. As epected, the aw motion is the most constraining along most of the path, and the aw stabilit margins are at their lowest when the vehicle makes a turn in the crater. For the case shown, the SSM drops to ero along the downward slope towards the end of the path, indicating a slope too steep to sustain the vehicle in a static state. The pitch DSM is lowest at the top of the hill, and the roll SSM is monotonousl high along the entire path. Figure 4 shows the vehicle over a road bump. The DSM for this path is shown in Figure 5. The DSM drops as each wheel passes over the bump. It reaches around m/s as the front wheel moves over the bump. Moving at an higher speeds would cause the font wheel to loose contact with the ground. The numericall computed velocit limit (DSM) was verified with a dnamic simulation of a vehicle with the same mass properties, moving over the speed bump at various speeds. Starting at m/s, the vehicle moved over the bump without loosing contact with the ground, as epected. The degree of contact was determined b the minimum normal force developed between the front wheel and ground. Increasing the speed resulted in a lower normal force, until reaching ero, or separation between the front wheel and ground. Figure 6 shows the minimum normal force, normalied b the stead state normal force (roughl half the vehicle weight), as a function of the vehicle speed. 235

6 5 Static Stabilit Margin Pitch FSA Roll FSA Yaw FSA m/s Dnamic Stabilit Margin Dnamic Stabilit Margin 5 Pitch FSA Roll FSA Yaw FSA m Fig. 5. The DSM for the road bump Normal force on front wheel Fig The respective SSM s and DSM s for each vehicle model Normalied normal force Speed m/s Fig. 6. Normal force on front wheel as a function of speed Fig. 4. A vehicle traversing a road bump The normal force drops to ero at.75m/s. This is slightl higher than the m/s velocit limit predicted b our stabilit analsis. This difference is epected since the vehicle used in the dnamic simulation uses had suspension and inflated tires, whereas our vehicle model has no suspension and assumes rigid wheels. Repeating the simulation for a smoother bump (not shown), resulted in a closer agreement between the simulated velocit at which separation begins and the computed velocit limit. Although these results are preliminar, the do validate the effectiveness of our approach to predicting the dnamic stabilit of road and off-road vehicles. Further studies are required to better understand the effect of suspension on the velocit limit. IX. CONCLUSIONS This paper presented a unified method for computing the range of feasible speeds and accelerations for an all terrain vehicle undergoing motion in three dimensions using the quasi-3d approach. The vehicle s motion is projected onto the aw, pitch, and roll planes. The stabilit of the vehicle is analed in each separate plane b substituting the ground force constraints into the equations of motion for each plane. This produces the set of admissible speeds and acceleration (FSA) in each plane, which when overlaed results in the overall FSA. This in turn provides the static and dnamic stabilit margins along the path. An eample of a vehicle moving along a curved path over mountainous terrain demonstrates the approach. [2] J. Craig. Appendi b. In Introduction to Robotics: Mechanics and Control. Prentice Hall, 2nd edition, 989. [3] E. Garcia, J. Estremera, and P.G. de Santos. Stabilit margins for walking machines. Robotica, 3:595 66, 22. [4] H. Hacot. The kinematic analsis and motion control of a planetar rover. Master s thesis, MIT, Ma 998. [5] A. Liegeois and Ch. Moignard. Geometric reasoning for perception and action. Springer Verlag, 993. [6] M. Mann and Z. Shiller. Dnamic stabilit of off road vehicles: A geometric approach. In Proceedings of the IEEE International Conference on Robotics and Automation, pages , Ma 26. [7] M. Mann and Z. Shiller. On the dnamic stabilit of off road vehicles. In ROMANSY, pages , Ma 26. [8] M.P. Mann. Dnamic stabilit of off road vehicles. Master s thesis, Technion, Israel Institute of Technolog, Haifa, Israel, August 27. [9] R.B. McGhee and A.A. Frank. On the stabilit properties of quadruped creeping gaits. Journal of Mathematical Biosciences, 3(2):33 35, 968. [] Z. Shiller and Y.R. Gwo. Dnamic motion planning of autonomous vehicles. In IEEE Transactions on Robotics and Automation, volume7, pages , April 99. [] Z. Shiller and M. Mann. Dnamic stabilit of off road vehicles. In Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Sstems (IROS), pages , October 24. [2] Z. Shiller and S. Sundar. Emergenc maneuvers of ahs vehicles. Paper 95893, SAE Transactions, Journal of Passenger Cars, 4: , 995. [3] T. Simeon. Motion planning for a non-holonomic mobile robot on 3-dimensional terrain. In Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Sstems, pages , November 99. [4] T. Simeon and B. Dacre-Wright. A practical motion planner for all terrain mobile robot. In Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Sstems, Jul 993. [5] M.W. Spong, S. Hutchinson, and M. Vidasagar. Robot Modeling and Control, chapter 2. John Wile and Sons, 25. REFERENCES [] Darpa s grand challenge. as of March