Autonomous terrain parameter estimation for wheeled vehicles

Transcription

1 Autonomous terrain parameter estimation for wheeled vehicles Laura E. Ray a a Thayer School of Engineering, Dartmouth College, 8 Cummings Hall Hanover NH 3755 ABSTRACT This paper reports a methodology for inferring terrain parameters from estimated terrain forces in order to allow wheeled autonomous vehicles to assess mobility in real-time. Terrain force estimation can be used to infer the ability to accelerate, climb, or tow a load independent of the underlying terrain model. When a terrain model is available, physical soil properties and stress distribution parameters that relate to mobility are inferred from vehicle-terrain forces using multiple-model estimation. The approach uses Bayesian statistics to select the most likely terrain parameters from a set of hypotheses, given estimated terrain forces. The hypotheses are based on the extensive literature of soil properties for soils with cohesions from 7 kpa. Terrain parameter estimation is subject to mathematical uniqueness of the net forces resulting from vehicle-terrain interaction for a given set of terrain parameters; uniqueness properties are characterized in the paper motivating the approach. Terrain force and parameter estimation requires proprioceptive sensors accelerometers, rate gyros, wheel speeds, motor currents, and ground speed. Simulation results demonstrate efficacy of the method on three terrains low cohesion sand, sandy loam, and high cohesion clay, with parameter convergence times as low as.2 sec. The method exhibits an ability to interpolate between hypotheses when no single hypothesis adequately characterizes the terrain. Keywords: Terrain parameter estimation, autonomous terrain characterization, proprioceptive sensors. INTRODUCTION As the Army moves towards a future in which corps of robots are ubiquitous in military campaigns, there is a practical need for technologies by which a few soldiers can reliably control tens of robots in the field to enhance situational awareness and combat effectiveness. Such technologies will demand that robots have self-awareness of the terrain that they travel on, their mobility on the terrain, and limitations to mobility, in order to avoid immobilization and allow for autonomous or supervisory operation. The envelope of operation maximum achievable translational velocities, accelerations, and maneuverability on a given terrain depends on a robot s multi-body dynamics and its interaction with the terrain, from which external forces and moments on the robot are generated. The multi-body dynamics are, in general, reasonably well-known. Vehicle-terrain interaction models and their parameters are generally not as well known, particularly for lightweight robots. In this paper, we develop a terrain characterization approach that uses proprioceptive sensors to estimate physical and empirical terrain parameters of a given terrain model. First, we estimate net traction force vs. slip and resistive torque vs. slip generated at the wheel-terrain contact. From these forces and moments, we estimate resistance and thrust due to the terrain at each wheel of a four-wheel drive robot under mild assumptions regarding the dynamics of the robot and the normal and shear stress distribution along the wheel-terrain contact. Then, given a terrain model, the estimated forces and moments are used along with that model to estimate physical and empirical terrain parameters. From these parameters, the stress distributions under the wheels can be estimated, and the trafficability of the terrain may be inferred. The key feature of the methodology is that net traction (or drawbar pull), resistive torques, and wheel slips are first estimated through proprioceptive sensors without needing to assume an underlying semi-empirical or physics-based vehicle-terrain model. The vehicle-terrain forces and moments vs. slip behavior are valuable in and of themselves to infer the robot s capacity to accelerate, climb, or tow a load. The terrain parameters are determined using a Bayesian approach which compares the estimated force-slip characteristics to those resulting from one or more hypothesized terrain models and associated parameter sets. The methodology is summarized in Figure. The method is demonstrated in this paper through computer simulation using the Bekker rigid-wheel terrain model [-3]. This model incorporates up to eight parameters and involves semi-empirical relations between normal stress and sinkage, and a Mohr-Coulomb criterion that relates shear stress and normal stress. Bekker model parameters for soils

2 Proprioceptive sensor inputs Ground & wheel speeds, accelerations, motor currents Force vs. slip estimation Bayesian estimation Physical terrain properties Terrain model(s) Terrain parameter sets (hypotheses) Fig. Two-step terrain parameter estimation approach: Terrain force vs. slip estimation followed by Bayesian hyothesis selection, given competing terrain model(s) and parameter sets. ranging from cohesiveless sand to firm clay can vary by several orders of magnitude [2]. Moreover, there is no apparent relationship between two important parameters soil cohesion and friction angle within the Mohr-Coulomb equation, thus making it difficult to develop a nonlinear optimization approach to terrain parameter estimation. The methodology for terrain parameter estimation presented here considers mathematical uniqueness issues, namely, whether there is a sufficiently unique mapping between a set of terrain parameters and the net forces and moments generated on the vehicle. We show weak uniqueness of this mapping, and based on this, we propose Bayesian multiple-model estimation (MME) for estimating terrain parameters. The method is demonstrated by computer simulation of a four-wheel drive, differential-steered robot, with Bekker theory governing the tire-terrain interaction, and sets of terrain parameters reported in the literature forming terrain hypotheses. A number of papers present work on terrain characterization using proprioceptive sensors. An online parameter estimation method is developed in [4] to determine soil cohesion and internal friction angle for a rigid-wheel planetary rover. This method uses a simplified model of the shear and normal stress distribution under the wheel to pose a leastsquares estimator whose inputs are normal load, wheel torque, sinkage, wheel rotational speed and wheel linear speed. The simplified model assumes symmetric stress distributions along the wheel-terrain contact patch. The method presented in this paper does not use sinkage sensors, simplifying instrumentation required for terrain parameter estimation. [5] uses a Newton Raphson technique to identify internal friction angle, shear deformation modulus, and pressure-sinkage coefficients for a wheeled vehicle traversing unknown terrain. [5] takes a similar approach to [4], but uses off-line identification. Real-time or near-real time parameter estimation is required to prevent immobilization or delays in the robot mission. [6] addresses terrain trafficability characterization through an empirical metric between motor current and yaw rate, evaluating the metric with a small, differential-steered commercial robot. They show a relationship between motor current vs. rate-of-turn that differs on various terrain, e.g., gravel, sand, dirt, and grass. Data are elicited by performing a controlled, quasi-steady turning maneuver. [6] also develops a neural terrain classification approach using rate gyros, accelerometers, motor current, and voltage. These papers present common themes of direct identification of terrain parameters governing a given semi-empirical terrain model and exploiting the net effect of the terrain on observed motion to determine terrain characteristics. This paper reviews terrain force estimation on deformable terrain from [7] and then presents a methodology for extracting compaction resistance and gross thrust from estimated forces. Section 3 discusses mathematical uniqueness issues in terrain parameter estimation and presents the multiple-model estimation approach. Section 4 provides simulation results of terrain parameter estimation during longitudinal driving for three terrains. 2. TERRAIN FORCE ESTIMATION 2. Estimation of Net traction and Resistance Torques Vehicle-terrain forces are estimated using an extended Kalman-Bucy filter (EKF) following the procedure presented in [7]. Estimated forces include resistive torques on each wheel, per-side drawbar pull, and per-axle lateral forces. The procedure is reviewed here for the simpler case of a vehicle undergoing longitudinal motion and thus neglects lateral force. A four-wheel drive, differentially steered robot is modeled with longitudinal rigid-body dynamics in body-fixed coordinates given by

3 ( F + F + F F ) v x = v yr + xfl xfr xrl + xrr () m tw [ ( Fxfr + Fxrr ) ( Fxfl + Fxrl )] M res r = r (2) I zz 2 ( T T b ω ) ω fl = fl rfl w fl (3) I w ( T T b ω ) ω fr = fr rfr w fr (4) I w ωrl = ( Trl Trrl bwωrl ) (5) I w ωrr = ( Trr Trrr bwωrr ). (6) I w x = [ v x r ω fl ω fr ωrl ωrr ] is the system state, comprised of longitudinal velocity, yaw rate, and four wheel velocities; F xfl, F xfr, F xrl, and Fxrr are the net longitudinal tire forces (gross traction minus resistance) at each wheel; and T rfl, T rfr, T rrl, and T rrr are the resistive torques about the y axis through each wheel due to the wheel-terrain interaction. A restoring moment about the z-axis through each wheel, e.g., stiffness-based realignment due to the vehicle-terrain response, is modeled by a single aggregate restoring moment M res r in eq. 3. M res > is a yaw damping term arising from the restoring moment provided by the vehicle-terrain interaction, and bwω ( ) are wheel damping terms, e.g., due mechanical damping in the drivetrain. T fl, T fr, T rl, and T rr are applied wheel torques after consideration of gearbox losses. t w is the track width, I zz is the yaw moment of inertia, and I w is the wheel moment of inertia. Eq. -6 together with net traction and resistive torque on each wheel from a terrain model form a fully specified system or truth model used to simulate performance in section 4. While only longitudinal motion is considered, retention of the yaw dynamic equation imposes constraints on the per-side longitudinal forces and thus this equation is retained. The set of unknown forces and moments in eq. -6 includes T rfl, T rfr, T rrl, and T rrr, F xfl, F xfr, F xrl, and F xrr. An extended Kalman-Bucy filter (EKF) is constructed by augmenting the vehicle dynamics with second-order random walk models of the form Fˆ = Fˆ F ˆ + w F ˆ for each of the four resistive torques and for the per-side net traction forces, where w is a white process noise vector. Assuming that per-wheel net traction is proportional to normal loads at each wheel, F xfl, F xfr, F xrl, and Fxrr are estimated from per-side net traction and normal load. Normal load can be estimated from the static weight transfer and measured accelerations as in [8,9]. The measurement set z = [ ax, ω fl, ω fr, ωrl, ωrr, v, r] comprised of longitudinal acceleration at the center-of-mass, wheel angular velocities, ground speed of the center of mass, and yaw rate renders the augmented state, comprised of x, four resistive torques, and per-side net traction observable. Motor currents provide the applied wheel torques through the torque constant and gearbox ratio, minus losses due to gearbox efficiency. Details regarding implementation of the EKF are reported in [7] for deformable terrain and [8,9] for Ackerman steered vehicles on rigid terrain and thus these details are omitted here. (7)

4 2.2 Estimation of resistance and gross traction from net traction Figure 2 shows the applied torque T and normal load W on a driven, rigid wheel that give rise to normal stress distribution σ (θ ) and shear stress distribution τ (θ ). From these stress distributions, net forces develop at the vehicleterrain interface, commonly referred to as drawbar pull F x (thrust minus resistance) acting at the wheel axle and resistive torque T r. The effective forces, shown in Fig. 3a, are translated to a point along contact patch as in Fig. 3b. These forces are the resultant integrals of σ (θ ) and τ (θ ) over the contact patch and are related to estimated net traction and resistive torque from the EKBF by an unknown angle θ f. The effective radial and tangential forces resulting from the normal and tangential stress distributions are assumed to act at a common angle θ f. Justification of this assumption is taken from the Mohr-Coulomb criterion that relates the maximum shear stress in the material to the normal stress [2]: τ = c +σ tan φ) (8) max ( max Maximum shear stress is related to normal stress through two soil properties, soil cohesion c and angle of internal shearing resistance or internal friction angle φ [], and thus neglecting effects of shear displacement, maximum shear and normal stress should be approximately coincident. Since the normal components of T r and net traction where F r, F r and F t must balance W, F x. The tangential force is given by Rw is the wheel radius. The normal load and net force are related to F r and F t, and θ f F x w θ f is derived from the EKBF estimated resistive torque Tr F t = (9) R F t as = F cos( θ ) F sin( θ ) () t f r f W = F t sin( θ f ) + Fr cos( θ f ) () are given by the solution to eq. 9-. From these, terrain resistance, which acts opposite to the velocity W ω θ 2 θ θ T τ(θ) V z σ m(θ m) Fig.2 Applied forces and moment and resulting stresses on a driven, rigid wheel in deformable terrain. σ(θ) W T r T F x ω V z W θ f T ω F t V z Fig. 3 Net result of vehicle-terrain interaction represented as (a) net traction force and resistive torque tangential forces F r and F t acting at θ f. F r Fx and T r and (b) radial and

5 vector, is given by Rc = F r cos( θ f ). (2) F r, F t, and θ f provide scalar variables representing the net effect of σ (θ ) and τ (θ ) on the vehicle; however, since the stress distributions can be asymmetric, it is not generally that θ f = θ m, the angle at which the maximum shear and normal stress occurs. 2.3 Terrain Model for Terrain Parameter Estimation Estimation of net forces and moments in section 2. assumes no a priori knowledge of a vehicle-terrain model of the shear and normal stress distributions. However, given a model, the force and moment estimates can be used to identify parameters of that model, assuming that a sufficiently unique mapping between the terrain parameters and resulting forces and moments exists. This issue of uniqueness of the mapping between terrain parameters and net forces is addressed in Section 3 to inform the approach selected for terrain parameter estimation. Bekker theory [-3] is used to demonstrate the terrain parameter estimation methodology. For a rigid wheel moving longitudinally at constant speed on horizontal, deformable terrain as in Fig. 2, the shear stress-shear displacement relationship governs shear stress and is given by [2] τ ( θ ) = ( c + σ ( θ ) tanφ)( e j / K ) (3) j( θ ) = Rw [ θ θ ( i)(sinθ + sinθ )]. (4) where j is the shear displacement, K is the shear deformation modulus, i is the wheel slip, and remaining parameters are defined in Fig. 2. [] relates the normal stress to sinkage through empirically-determined pressure-sinkage parameters k c, k φ, and n: n z σ ( z) = ( kc + kφb) (5) b b is the wheel width, and z is sinkage. The maximum normal stress achieved along the contact patch is given by the empirically-determined relationship [3] θm = ( c + c2i) θ. (6) c and c2 introduce two additional empirical parameters. The normal stress is transformed to a function of θ to integrate over the contact patch. For the region in front of and behind σ m, respectively, the normal stress distribution is [3] n R ( k + ) w (cosθ cosθ) n c kφb θ2 θ < θm b σ ( θ ) = n R θ θ ( k + ) w cos 2 c kφb θ ( θ θm ) cosθ θm θ < θ b θm θ2 At constant velocity, the following static equilibrium conditions hold [3]: (7) θ θ W = R w b σ ( θ )cosθdθ + τ ( θ )sinθdθ θ2 θ2 (8) θ θ D = Rw b τ ( θ )cosθdθ σ ( θ )sinθdθ θ 2 θ 2 (9)

6 = 2 θ Tr Rwb τ ( θ ) dθ (2) θ 2 W is the normal load on the wheel, D is the drawbar pull or net force (gross traction minus resistance) available to the vehicle to tow a load, accelerate or climb hills, and Tr is the resistive torque. Eqs. 3-2 provide the terrain inputs for deformable terrain simulation results presented in Section 4. Note that the integrals in eq. 8-2 have no closed form solution. 3. Terrain Model Behavior and Uniqueness Issues 3. TERRAIN PARAMETER ESTIMATION In order to relate terrain force estimates to terrain parameters, we first investigate the behavior of the terrain model as a function of terrain properties and normal load on the wheel. To do so, we choose three terrain types that bound a range of soil cohesions reported in [2] from the order of to 7 kpa. Table reports terrain parameters for these three soils and sources for these parameters. We evaluate the shear and normal stress distributions arising from these terrain parameters for vehicles of mass kg and kg at a slip ratio of.95. The mass is assumed to be distributed evenly over four wheels (W = 245 N and 245 N, respectively) with rigid wheels of diameter.58 m (2 in) and width.5 m (6 in). By holding the wheel size constant, we investigate the geometry of the stress distributions as a function of normal pressure. Figure 4 shows shear and normal stress distributions for i =.95 for each of the three terrains in Table and each normal pressure along with drawbar pull as a function of slip on each terrain. Figure 4 shows that () the maximum shear and normal stress occur at approximately the same angle θ m on low cohesion soils; and (2) normal and shear stress distributions are approximately linear with θ over the two regions of increasing and decreasing stress for low cohesion soils. These observations are not valid for lean clay. While these observations appear to be insensitive to normal load, the net force or drawbar pull depends significantly on normal load, as expected. In Fig. 4, the heavy vehicle would not develop positive drawbar pull on the dry sand and thus would be immobile on this soil. Its mobility would be borderline on sandy loam. The light vehicle experiences positive net traction on all three soils at sufficiently high slip; however, the drawbar pull vs. slip characteristics vary from a linear relationship with slip to a saturating relationship. These observations suggest that the force-slip response mapped from terrain characteristics could be useful in estimating underlying terrain parameters. [4] approximates shear and normal stress distributions on all soil types as linear in order to develop approximate closedform solutions to eq These solutions are then used to determine soil cohesion and friction angle using a least squares approach. [4] also assumes stress distributions are symmetric, i.e., θ m occurs at the midpoint between θ and θ 2, which is not in Figure 4 or in [3]. Terrain parameters c and c 2 capture the asymmetry in the stress TABLE Terrain Parameters Parameter Dry Sand Sandy Loam Lean Clay c (kpa) φ (deg) k c ( kpa/m^n-) k φ (kpa/m-n) n...2 c c K (m) Parameters for dry sand, sandy loam, and lean clay from [2] except c and c 2, which are not reported in [2] and are taken from [3]. c and c 2 were not found in any source for sandy loam or clay and are taken as similar to c and c 2 reported for compact sand in [3].

7 Dry sand stress (kpa) Dry sand stress (kpa) (a) m = kg (b) m = kg 5 Sandy loam stress (kpa) Sandy loam stress (kpa) 2 distribution. The assumption of a symmetric distribution provides a unique relationship between stress distributions σ (θ ) and τ (θ ) and resultant forces of eq. 8-2 for the underlying model of eq. 3-7, i.e., given parameters c and φ, stress distributions provide unique resultant Tr and Fx in Fig. 3a. When the distribution is asymmetric, mild uniqueness is preserved, under the condition that c > ; for a completely cohesionless soil, shear stress asymmetry results in identical values of T r in eq. 2 when the shear stress distribution is approximately linear and is reflected vertically around the midpoint of θ and θ 2, i.e., there are two possible shear stress distributions that give rise to the same resistive torque. Although a completely cohesionless soil may not be of interest, mobile robots do operate on soils of very low cohesion, and Tr is only weakly unique for such soils; measurement uncertainty in sensors from which soil parameters are estimated therefore present a challenge for terrain parameter estimation in low cohesion soils. Uniqueness issues are also evident in intersecting curves for drawbar pull vs. slip ratio in Figure 4, i.e., for certain slip ratios, drawbar pull is identical on different terrains. Dry sand and lean clay present similar drawbar pull at ~4% slip for the lighter vehicle, and sandy loam and lean clay present similar drawbar pull at ~3% slip for the heavy vehicle. Thus, operating around these slip ratios could present a challenge for estimating terrain parameters. 3.2 Terrain Parameter Estimation Even when a linear representation of stress distributions to provide closed-form solutions to eq. 8-2 is valid, the relationship between terrain parameters and net forces remains nonlinear and the parameter space is large. Terrain parameter estimation could be formulated as a nonlinear optimization problem to determine a terrain parameter set that minimizes a cost function of the error between the estimated terrain forces and terrain forces resulting from a given terrain model. To reduce the parameter space, the linear approximation of normal stress could be retained for low cohesion soils and the maximum normal stress (and its location) estimated as a parameter eliminating the empirical parameters k c, k φ, n, c and c 2. Terrain parameters could then be estimated from EKF net force estimates by minimizing a cost function F ( p) subject to pl p pu, e.g., F normal shear Lean clay stress (kpa) Lean clay stress (kpa) ( Wˆ W ( p) ) + α ( Tˆ T ( p) ) + ( Fˆ F ( ) 2 ( p) = 2 r r α3 x x p) α (2) W ˆ, Tˆ r, and Fˆ x are the measured normal load and EKF estimates of Tˆ r and Fˆ x, αi are weighting parameters, and W ( p), Tr ( p), and F x ( p) are the results of eq. 8-2 with linear normal and shear stress distributions for a terrain parameter vector p. pl and pu are upper and lower bounds on p. Bounds on parameters could be selected based on the range of parameters reported in [2]. Bounds on θ m can be chosen dynamically so that θ > θ m and bounds on σ m are dependent on the average normal stress W / R w θ. θ2 is assumed to be zero, i.e., rut recovery is assumed to be negligible Drawbar pull (kn) slip ratio dry sand sandy loam lean clay slip ratio Fig. 4 Shear and normal stress distributions on three terrain types for two normal loads. Drawbar pull (kn)

8 This nonlinear optimization approach was implemented, but the weak uniqueness of the mapping between terrain parameters and terrain forces affects the estimation of cohesion, friction angle, and maximum normal stress; high estimates of friction angle in eq. 8 are offset by low estimates of σ max, thus while the product σ max tanφ may be estimated, it is difficult to estimate the individual parameters of the product through nonlinear optimization. Moreover, the linear approximation of stress distributions breaks down for high-cohesion soils. We therefore develop an alternate approach that does not require approximation of the stress distributions. While it is possible that a nonlinear optimization approach would benefit from data for more than one normal force, a requirement for such data does not readily permit an on-line solution. [2] reports sets of Bekker terrain parameters { c, φ, kc, kφ, n } for 2 terrains. Using the Bekker rigid-wheel terrain model as a forward model and these terrain parameters sets as hypotheses, we propose a multiple-model estimation (MME) approach to terrain identification as a computationally-efficient alternative to nonlinear optimization that is more robust to weak uniqueness described above. For each hypothesis, a mapping of the terrain parameter set to terrain forces is determined a priori as a function of slip and normal load. This mapping could be stored in computer memory. Then, the most likely set of terrain parameters from among the hypotheses is determined by recursive implementation of Bayes rule as follows. Let p j, j = to N comprise N hypothesized terrain parameter vectors with probability mass function initialized as Pr( p j ) = / N. The conditional probability mass function for parameter set p j subject to a vector Fˆ k of terrain force estimates from the EKBF at time k evolves according to Bayes rule as where pr( Fˆ k p j ) Pr( p j Fˆ k ) Pr( p j ) k = Pr( p j Fˆ k ) = (22) N pr( Fˆ k pi ) Pr( pi Fˆ k ) i= T ˆ r ( j ) k ( j ) ( ) k p S r p pr Fk p j = e m / 2 (2π ) S (23) S is the covariance matrix of the residual rk ( p j ) = Fk ( p j ) Fˆ k and Fk ( p j ) is the vector of terrain forces mapped for parameter set p j and estimated slip î. The most likely parameter set is given by p k = N i= Pr( p Fˆ ) p (24) Eq are implemented recursively, i.e., at each time step k new estimated forces and wheel slips are used to update the probability mass function for each hypothesis. Note that the success of this method depends heavily on the quality of the hypotheses and thus the method should be evaluated for cases where no hypothesis precisely represents the underlying parameters, but the hypotheses bound the underlying parameter set. i k i 4. EVALUATION OF TERRAIN PARAMETER ESTIMATION To evaluate terrain parameter estimation, we simulate longitudinal acceleration of a vehicle of mass m = kg with wheel diameter.58 m (2 in) and width.5 m (6 in). The applied torque to each wheel at t = is large, and the torque decreases linearly with time over two seconds. This input produces wheel slips of % at t = decreasing as torque is decreased and soliciting the characteristic net force vs. slip and resistive torque vs. slip at each wheel. Zero mean, Gaussian process noise is injected into each dynamic equation -6. Zero mean, Gaussian, measurement noise is injected to simulate measured acceleration, wheel speeds, yaw rate, and ground speed. Measurement noise variance is chosen according to laboratory measured values for transducers used in experiments to validate terrain force estimation [7]. Measurement and process noise covariance are assumed to be known.

9 4 Dry Sand 5 Sandy Loam 8 Lean Clay F xfl (N) 2 2 estimate 4.5 front left slip F xfl (N) 5 estimate 5.5 front left slip F xfl (N) estimate 2.5 front left slip T rfl (N m) 3 2 estimate.5 front left slip T rfl (N m) estimate.5 front left slip estimate 5.5 front left slip Fig. 5 Actual and estimated net traction (top) and resistive torque (bottom) vs. slip on sandy loam for an applied torque input decreasing linearly over time. The EKF is simulated at a sample frequency of Hz. Representative force vs. slip estimation results for each terrain in Table are given in Fig. 5 as and estimated net traction force vs. slip and and estimated resistive torque vs. slip at the front-left wheel. Figure 5 shows that the EKBF is able to track the external force and moments due to terrain within limits imposed by measurement noise and filter transients. Evaluation of the terrain force estimation, including physical testing is given in [7] and thus is omitted here. Thrust and resistance are estimated from net traction using eq. 9-. Using estimated thrust, resistance, and resistive torque vs. slip, Bayesian terrain parameter estimation is evaluated for the three terrains reported in Table. The parameter set includes all eight terrain model parameters p = [ c φ kc kφ n c c2 K]. The first five parameters are reported for 2 terrains in [2], and these 2 terrain parameter sets form the hypothesis set for the Bayesian MME approach. Parameters c, c 2,and K are not reported in [2] for these terrains. K is reported in [2] as varying from cm (firm sandy terrain) to 2.5 cm (sandy terrain),.6 cm for clay at maximum compaction, and cm for fresh snow. Thus, where no data exist, we take K =.25 m for sandy soils, K =. m for sandy loams, K =.6 m for clayey soils, and K =.5 m for snow within the 2 terrain parameter hypotheses. c and c2 are summarized in [3] for sandy terrain from three sources and range from c =.43 and c 2 =.32 for compact sand to c =.38 and c 2 =.4 for dry sand. In the absence of data for the 2 terrains in [2], we take c =.43 and c 2 =.32 for sandy loams and clayey soils and c =.38 and c 2 =.4 for sandy soils and snow. Given N = 2 hypothesized terrain parameter vectors, among which are the three terrains in Table, the Bayesian multiple-model estimator is evaluated for these three terrains. EKF terrain force estimates from a single wheel are used in eq , with two candidates for the terrain force vector. Candidate is Fˆ [ ˆ ( ) ˆ k = F fl k Trfl ( k)], i.e., the estimated net traction or drawbar pull (thrust minus resistance) and resistive torque at the front left wheel, and candidate 2 is Fˆ [ ˆ ( )cos( ( )) ( ) ˆ k = Ftfl k θ f k Rcfl k Trfl ( k)], i.e., gross traction, resistance, and the resistive torque at the front left fl wheel We also consider the case where the three sets of terrain parameters in Table representing the underlying terrain characteristics are not in the hypothesis set. In place of exact hypotheses, terrain parameters for these three terrains are modified, with random white noise with standard deviation of. of the nominal terrain value added to each parameter. Thus, in this case, no hypothesis precisely matches the underlying terrain, but at least one hypothesis should approximate the underlying terrain. In implementation of terrain force estimation, we delay multiple-model estimation for 2 samples (.2 sec) to allow the EKBF transients to decay and we measure convergence from the start of multiple-model estimation. T rfl (N m) 2 5 5

10 Results of simulation evaluation are summarized as follows. When exact terrain parameter sets in Table are included in the set of 2 hypotheses, the Bayesian estimator converges to the correct hypothesis in at most two iterations, or.2 sec for all three terrains. When inexact terrain parameter sets in Table are represented among the hypotheses by adding Gaussian white noise so that no hypothesis matches the underlying set, terrain parameter estimates converge to a single hypothesis for dry sand and sandy loam, namely the hypothesis produced by modifying underlying terrain parameters with Gaussian white noise. Convergence occurs within two iterations (.2 sec) for both candidate terrain force vectors. For lean clay, terrain parameters converge to a single hypothesis in fewer than five iterations for the candidate terrain force vector. However, that hypothesis is not the one generated by injecting Gaussian noise. The terrain for the hypothesis that multiple-model estimation converges to is described in [2] as a clayey soil with parameters n =.3, k c = 2.7 ( kpa/m^n-), k φ = 556 (kpa/m-n), c = kpa, and φ =.34 deg, i.e., a soil with parameters similar to lean clay. Results for the high cohesion soil are attributed to weak uniqueness of the net forces and resistive torque resulting for various clayey soils represented in the hypothesis set. For lean clay and candidate 2 terrain force vector, which incorporates both thrust and resistance components instead of net longitudinal force, the converged parameters are represented by a linear combination of two or three hypotheses according to eq. 24. An example of parameter convergence and probability mass function convergence for lean clay is given in Figure 6, which shows convergence to within two hypotheses in two iterations (.2 sec). These two hypotheses are those for Heavy Clay [2] reported above and the hypothesis generated from lean clay parameters with Gaussian noise added; thus, classification of terrain is achieved for all three terrain types within two iterations. Probability mass functions at t = 2. sec for these hypotheses are.35 and.64, respectively. As an additional measure of performance, Figure 6 shows the normal and shear stress distributions at.95 slip for the actual underlying terrain assumed and for the parameters to which the multiple-model estimator converges, based on the candidate 2 force vector, for the case when the hypothesis set does not include the actual terrain. For all three terrains, estimated stress distribution features (maximum stress and location of maximum) are within 2% of actual. Results in Figure 7 are representative of best case (lean clay) and worst-case (dry sand) estimation of stress distributions. 5. CONCLUSION This paper develops a Bayesian multiple-model estimation methodology for identifying terrain parameters from estimated terrain force vs. slip characteristics. Simulation evaluation of the methodology for a Bekker rigid-wheel terrain model shows that it can identify the hypothesis or hypotheses best representing terrain characteristics from a set of hypotheses, even if the exact terrain parameters are not represented within the hypothesis set. Good convergence is exhibited in simulation. The methodology requires no approximations of the underlying shear and normal stress distributions. It poses a low computational burden when mappings from the hypothesized terrain parameter sets to terrain forces characteristics are pre-computed as functions of wheel slip and normal load. Computation scales as O(N), where N is the number of hypotheses, thus additional hypothesis could be drawn from the literature or created empirically without imposing undue additional computation. In this paper, the methodology is demonstrated for an underlying Bekker terrain model for driven rigid wheels; however, the method is not constrained so that the model structure posed for each hypothesis needs to be consistent; hypotheses can take the form of alternative model structures and parameters. Moreover, the terrain identification example indicates that gross traction, resistance force, and resistance torque from a single wheel provides sufficient information for terrain identification, thus terrain parameters could be identified independently for each wheel. Finally, the methodology can also be used as a rapid classifier, as it is able to distinguish characteristics of terrain, e.g., clayey soil vs. sandy loam vs. sand with excellent convergence. ACKNOWLEDGEMENT This research is supported by the National Institute of Standards and Technology under Grant No. 6NANB4D44 and Grant No. 6NANB6D63 awarded through the Institute for Security Technology Studies, and by the Army Research Office under contract No W9NF

11 2 2 8 kc (kpa/m n ) 5 5 kphi (kpa/m n) 5 5 soil cohesion c (kpa) time (sec) 4 Estimated actual.5 time (sec).25.5 time (sec) heavy clay lean clay phi (deg) time (sec) n time (sec) Conditional probability time (sec) Fig. 6 Terrain parameter convergence for simulation of vehicle on lean clay, with candidate 2 force vector used in multiple-model estimation. Left to right, from top: Estimates of five soil parameters compared with actual values, convergence of the dominant conditional probabilities for hypotheses representing heavy clay (from Wong 2) and lean clay. 25 Dry sand 25 Sandy Loam 3 Lean Clay Normal stress (kpa) Normal stress (kpa) Normal stress (kpa) Shear stress (kpa) estimated Shear stress (kpa) Shear stress (kpa) Fig. 7 Comparison of shear and normal stress distributions as described by actual and estimated terrain parameters. REFERENCES [] Bekker, M.G., Theory of Land Locomotion. University of Michigan Press, Ann Arbor, MI. (956). [2] Wong, J.Y., Theory of Ground Vehicles (3 nd ed). Wiley-Interscience, Wiley & Sons, New York (2). [3] Wong, J.Y. and Reece, A.R., Prediction of Rigid Wheel Performance based on the Analysis of Soil-Wheel Stresses Part I. Performance of Driven Rigid Wheels, J. Terramechanics, 4(), 8-98, (967). [4] Iagnemma, K., Kang, S., Shibly, H., Dubowsky, S., Online Terrain Parameter Estimation for Wheeled Mobile Robots with Application to Planetary Rovers, IEEE Trans. on Robotics, 2(5), , 24.

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