ADAPTIVE META-LEARNING FOR IDENTIFICATION OFROVER-TERRAIN DYNAMICS

Somrita Banerjee1, James Harrison1, P. Michael Furlong2, Marco Pavone1

1Stanford University, 450 Jane Stanford Way, Stanford, CA 94305, USA1E-mails: somrita@stanford.edu, jharrison@stanford.edu, pavone@stanford.edu

2NASA Ames Research Center, Moffett Field, MS 269-3, CA 94043, USA, E-mail: padraig.m.furlong@nasa.gov

ABSTRACTRovers require knowledge of terrain to plan trajectories thatmaximize safety and efficiency. Terrain type classificationrelies on input from human operators or machine learning-based image classification algorithms. However, high levelterrain classification is typically not sufficient to preventincidents such as rovers becoming unexpectedly stuck ina sand trap; in these situations, online rover-terrain inter-action data can be leveraged to accurately predict futuredynamics and prevent further damage to the rover. This pa-per presents a meta-learning-based approach to adapt prob-abilistic predictions of rover dynamics by augmenting anominal model affine in parameters with a Bayesian re-gression algorithm (P-ALPaCA). A regularization schemeis introduced to encourage orthogonality of nominal andlearned features, leading to interpretable probabilistic esti-mates of terrain parameters in varying terrain conditions.

1 INTRODUCTIONExtraterrestrial rovers must traverse terrain that is oftenrugged and uncertain in order to achieve mission goals. Toensure a high degree of safety and maximize efficiency, itis crucial for the rover to be equipped with as much knowl-edge as possible about the terrain it will traverse. In modernrover systems, sources of information include terrain typeclassification, hazard detection, and excessive slip detec-tion. Terrain type classification relies on input from humanoperators or machine learning-based image classificationalgorithms [1], [2], as well as proposed approaches to clas-sify terrain using vibration measurements [3]. However,terrain classifications are not sufficient to guarantee safetyfor path selection. Discrete categories of terrain mask thecomplexity of vehicle-terrain interactions that are unique toindividual vehicles and different environmental conditions.To truly avoid incidents like NASA’s Spirit rover becomingstuck in a sand trap [4], and Curiosity’s wheels being dam-aged by sharp rocks [5], wheel-terrain dynamics must beconsidered. Using proprioceptive or visual measurements,unexpected wheel-terrain interaction effects such as exces-sive slip or sinkage can sometimes be detected as they oc-cur [6, 7, 8, 9] but in order to predict and prevent such un-desirable wheel-terrain interactions, it is crucial to main-tain an adaptive model of the terrain parameters that gov-ern wheel-terrain interaction. Having an estimate of terrainparameters allows the rover to adapt its control and plan-ning strategy to a given terrain [10], accurately predict thetraversability of neighboring terrain regions [11], and pre-vent further damage to the rover.

Related Work. Analytical models governing wheel-terrain interaction were first introduced by Bekker, Wong,and Reece [12, 13, 14] and have since been expanded toincorporate multiple physical effects [15]. However, thesemodels are complex, so efforts to estimate terrain parame-ters with the limited hardware on-board a rover have moti-vated approaches using linear-least squares regression [16]or a Newton-Raphson method [17] on simplified terrame-chanics models. These approaches require access to sen-sor information about the rover’s forward velocity froman IMU or visual odometry, wheel angular velocity fromtachometers, sinkage from camera measurements, and amethod for measuring or computing the vertical load andtorques on the wheels [16]. These methods provide goodonline estimates of terrain parameters on a variety of soils,but cannot model non-linear interactions, and moreover,they do not provide any measure of uncertainty for theirestimates.

Since wheel-terrain interaction models have multiplesources of uncertainty, efforts to validate these modelsagainst experimental data have relied on fitting stochas-tic semi-empirical models [18], [19] which benefit froma measurement of uncertainty for terrain parameter esti-mates. Such measures of uncertainty can be obtained bymodeling the wheel-terrain interactions as a Gaussian Pro-cess (GP) [19] but GPs are limited in their ability to incor-porate physical priors, and have high computational cost.Instead, Bayesian neural network models such as ALPaCA[20] have been shown to be successful in modeling un-certain dynamics for safety critical systems while remain-ing computationally efficient [21, 22]. ALPaCA performsBayesian linear regression on neural network features thatare learned via meta-learning (or “learning-to-learn”). Bytraining on a variety of physically plausible simulations,the ALPaCA model is capable of learning expressive fea-tures capturing behavior that may be difficult to representanalytically. In this work, we develop a nominal model thatrelates the dynamics of the rover to the terrain parameters.This model is based on work by Iagnemma et al. [16] and islinear in terrain parameters. Next, we augment the nominalmodel with an ALPaCA model [20], to produce a modelthat yields accurate predictions as well as Bayesian poste-riors reflecting prediction uncertainty. A similar approachis employed in [23, 24, 25] of relating the pose error to slipestimates, with the differences being that our model esti-mates terrain parameters instead of slip directly, providingan understanding of the terrain, and our model yields notonly predictions but also probabilistic estimates to charac-

5054.pdfi-SAIRAS2020-Papers (2020)

mailto:somrita@stanford.edumailto:jharrison@stanford.edumailto:pavone@stanford.edumailto:padraig.m.furlong@nasa.gov

terize uncertainty in our predictions.

Problem Formulation. We are focused on developing amodel to learn, adapt, and predict a rover’s dynamics onunknown terrain and provide online estimates of the phys-ical parameters that govern wheel-terrain interactions. Thetwo key parameters of interest are cohesion c and internalfriction angle φ that determine the maximum terrain shearstrength [16] and therefore, a rover’s ability to safely tra-verse the terrain. Since rovers will be encountering noveland unmodeled terrain, it is crucial for safety that ourmodel provide probabilistic predictions and adapt online.

We assume that the sensor suite on-board the rover, includ-ing wheel encoders, IMU, and vision system, gives us in-formation about the rover’s current position, velocity, slip,and sinkage, though we hope to relax the requirement onknowledge of slip and sinkage in future work.

Contributions. The key contributions of this paper areas follows. We develop a framework for wheel-terrain dy-namics modeling that is capable of rapid adaptation, byfirst developing a nominal model that is linear in parame-ters. In order to capture the non-linear interactions betweenthe rover dynamics and terrain parameters, we augment thelinear model with new more complex meta-learned neu-ral network features using a Bayesian regression algorithm(ALPaCA). We present an alternative form of the ALPaCAmodel, which we name P-ALPaCA, which enables joint in-ference of the parameters of the ALPaCA model and of thenominal model. We introduce a new regularization schemeto promote orthogonality between the nominal model fea-tures and the meta-learned features, providing interpretableterrain parameter estimates. Lastly, we demonstrate ourframework on simulations of rocker-bogie dynamics onvarying terrains.

2 BACKGROUNDThe dynamics of a rover are different on different terrains.Therefore, predicting rover dynamics is, in its most gen-eral form, a system identification problem, where Bayesianregression can be leveraged to produce probabilistic predic-tions that adapt online. Approaches using Gaussian Processregression have been used for model identification [19] butare computationally inefficient for large numbers of sam-ples and cannot easily incorporate prior knowledge fromthe existing models of rover dynamics and wheel-terraininteraction that we discuss in sections 2.2 and 2.3. Boththese problems are addressed by using a Bayesian meta-learning algorithm, ALPaCA [20], that we discuss in sec-tion 2.1 which allows for efficient regression while incor-porating prior knowledge.

2.1 System Identification via Meta-LearnedNeural Network

Our system identification approach builds upon ALPaCA[20], a Bayesian meta-learning algorithm. This approachseparates the online adaptation process—a convex opti-mization problem that can be solved analytically—from the

offline non-convex feature learning process.

The basis of this method is meta-learning, in which it isassumed there exists a distribution over “tasks”. For exam-ple, in system identification, these tasks may correspond todifferent system dynamics. Meta-learning aims to use datacollected from several tasks to improve the efficiency andaccuracy of learning in a new task sampled from the samedistribution.

We will write the system dynamics as

x+ = f (x,u;θ) + � (1)

where θ ∼ p(θ) corresponds to the task and may be seen asparameters of the environment, and � is zero-mean Gaus-sian noise, uncorrelated in time, with covariance Σ� . Wewrite x and u to denote state and action respectively. Weassume episodic interaction with the system, where θ issampled at the beginning of the episode, and held fixedthroughout. While this is a somewhat unrealistic assump-tion in the context of online terramechanics modeling, wewill discuss relaxations of this assumption in Section 5.Moreover, several works have addressed extensions beyondthis assumption that may be applied on top of the methodsdeveloped herein [26, 27]. We assume θ is not directlyobserved, and we do not know f (·, ·; ·). Our goal in themeta-learning problem setting is to learn some approxima-tion of (1) that is capable of being adapted online. As tran-sitions are observed in an episode, we wish to use experi-ence within an episode to infer θ and improve subsequentpredictions.

The approach of ALPaCA is to perform Bayesian linear re-gression on learned neural network features. The predictivemodel takes the form

x̂+ = Kφ(x,u;w) + � (2)

where φ(·, ·; ·) is a neural network with weights w, andK ∈ Rnx×nφ is a matrix. We write x̂+ to denote the pre-dicted next state. The matrix K may be seen as the lastlinear layer of the neural network. Critically, the ALPaCAmodel maintains a distribution over K reflecting the epis-temic uncertainty—uncertainty that can be reduced withmore data collection. Thus, the ALPaCA model returnsa predictive distribution for x̂+ as opposed to a point pre-diction. The central idea of ALPaCA models lies in up-dating the last layer of the network online. This updatemay be performed by recursive least squares, similar to theKalman filter. This form of updating is not novel in sys-tem identification; indeed, recursive least squares updat-ing on neural network basis functions has been a commonapproach in nonlinear identification since the 1980s [28].Where ALPaCA varies from these classical approaches isto train the neural network by backpropagating the train-ing loss through the model update, to ensure the featuresare broadly useful across multiple tasks and across time.As such, ALPaCA combines ideas from classical adaptivecontrol with ideas from recurrent neural networks. Fun-damentally, in the ALPaCA model, the matrix K serves tosummarize all information within one episode and account

5054.pdfi-SAIRAS2020-Papers (2020)

Figure 1 Forces acting on planar rocker-bogie

for variations between tasks, whereas the neural networkfeatures are learned to be useful across all tasks and are notupdated online.

We will now make concrete the above (informal) discus-sion. We assume access to operational data in a variety ofenvironments, obtained via previous experience or simula-tion. ALPaCA fixes a matrix normal prior on K, which wewrite

p(K) =MN(K̄0,Λ−10 ,Σ�) (3)where K̄0 denotes the prior mean and Λ0 is a prior preci-sion (inverse of covariance) matrix. Note that the matrixnormal distribution over the matrix K is a normal distribu-tion over the vectorized matrix, with a particular covariancestructure. We refer the reader to [20] for more details. LetDt = {(xτ,uτ,xτ+1)}t−1τ=0 denote the state/action data ob-served within an episode at time t. Given this data, theposterior over K is

p(K | Dt) =MN(K̄t,Λ−1t ,Σ�). (4)where K̄t,Λ−1t may be computed via recursive least squaresthrough update rules

Λ−1t = Λ−1t−1 −

11 + φTt Λ

−1t−1φt

(Λ−1t−1φt)(Λ−1t−1φt)

T (5)

Qt = xtφTt + Qt−1 (6)

whereK̄t = Λ−1t Qt (7)

and where φt = φ(xt−1,ut−1). Given this posterior, theposterior predictive distribution may be computed, yielding

xt+1 ∼ N(K̄tφt+1, (φTt+1Λ−1t φt+1 + 1)Σ�). (8)Given this posterior predictive distribution, we can evaluatethe likelihood of the remaining data in the episode (that hasnot been conditioned upon) to update the neural networkweights and the prior. The prior for the model relies onexisting physics models of the rover dynamics and wheel-terrain interactions, which we discuss next.

2.2 Rover Dynamics and Kinematics

In order to develop a model for rover dynamics on differ-ent terrains, we require knowledge of the kinematics of therover. Though our approach would work on any configu-ration of the rover, in this work we consider laterally sym-metric rocker-bogie motion, i.e. right and left wheels haveidentical motion and driving torques. This is a justifiedassumption for low-acceleration motion in a straight-line

Figure 2 Rigid wheel on deformable terrain.or very large radius turns. With knowledge of the roverkinematics and terrain geometry, the forward net force Fx(see Fig. 1) can be related to the normal Ni and tractive Tiground reaction forces through

Fx = ST[N1 N2 N3 T1 T2 T3

]TS =

[−s1 −s2 −s3 c1 c2 c3

]T (9)where ci = cosαi and si = sinαi [29].

2.3 Wheel-Terrain InteractionsIn this section, we describe how the ground reaction forcescan be related to the terrain parameters. Fig. 2 shows a freebody diagram of a rigid wheel travelling on deformable ter-rain. For simplicity, the wheel is assumed to be rigid andmoving quasi-statically with low acceleration. The interac-tion model for a rigid wheel traversing deformable terrainis obtained by considering the force and moment balance[16]

W = rb

θ2∫

θ1

σ(θ) cos (θ) dθ +

θ2∫θ1

τ(θ) sin(θ) dθ

DP = rb

θ2∫

θ1

τ(θ) cos(θ) dθ −θ2∫

θ1

σ(θ) sin(θ) dθ

M = r2b

θ2∫θ1

τ(θ) dθ

(10)

where W is vertical load applied to the wheel, DP is draw-bar pull applied to the wheel from a suspension system, Mis wheel drive torque produced by an actuator, θ is the ar-bitrary angular location of wheel-terrain contact measuredfrom a vertical axis, θ1 is the angle at which the wheel firstmakes contact with the terrain, θ2 is the angle at which thewheel loses contact with the terrain, σ is radial stress nor-mal to the wheel-terrain contact, τ is shear stress tangent tothe wheel-terrain contact, r is wheel radius, and b is wheelwidth.

In quasi-static equilibrium, the normal force N equals theweight W and the traction force T counteracts the drawbarpull DP, henceforth N and T will be used in place of W andDP. By considering linear approximations of the shear and

5054.pdfi-SAIRAS2020-Papers (2020)

normal stress, Iagnemma et al. [16] find analytical expres-sions of these forces

N = f0[σm f1 − τm f2 − c f3

]T = f0

[σm f2 + τm f1 − c f4

]M =

r2b2

(τmθ1 + cθm)

(11)

where σm, τm are the maximum radial and shear stresses,f0−4 are all functions of θ1 and θm which is the angle atwhich maximum stress occurs, which are given by

f0 =rb

θm(θ1 − θm)f1 = −θm cos θ1 + θ1 cos θm − θ1 + θmf2 = θm sin θ1 − θ1 sin θmf3 = θ1 sin θm − θm sin θm − θmθ1 + θ2mf4 = θ1 cos θm − θm cos θm + θm − θ1.

(12)

The radial and shear stresses themselves are functions ofthe wheel radius r, wheel width b, slip i, sinkage z, the ter-rain parameters of interest - cohesion c and angle of internalfriction φ - as well as other terrain parameters (kc, kφ, n, k)via

σm =

(kcb

+ kφ

)(r(cos θm − cos θ1))n

τm = (c + σm tan φ)A

A = 1 − exp(− r

k[θ1 − θm − (1 − i)(sin θ1 − sin θm)]

).

(13)The model exhibits low sensitivity to the other terrain pa-rameters (kc, kφ, n, k) and these are replaced with represen-tative values [14]. The sinkage

z = r(1 − cos θ1) (14)is determined from the angle of initial contact between thewheel and the terrain θ1 through wheel geometry. The an-gle of maximum stress

θm = (c1 + ic2)θ1 (15)

is related to the angle of initial contact θ1 using terrain pa-rameters c1, c2. Leveraging these relationships, the forcesfrom Eqs. 11 are expressed as functions of wheel geom-etry, slip i, sinkage z, the terrain parameters of interest -cohesion c and angle of internal friction φ - as well as otherterrain parameters. Therefore, with knowledge of the ver-tical force N, torque M, sinkage z, and slip i, the terrainparameters for cohesion c and angle of internal friction φare estimated via a least squares regression[

ctan φ

]= K†2K1 (16)

where K1,K2 are functions of N,M, i, z [16]. We buildupon this regression model to relate the rover dynamics di-rectly to the terrain parameters for cohesion c and angle ofinternal friction φ in the following section.

3 TECHNICAL APPROACHIn this section, we build a nominal linear model relat-ing the rover dynamics and terrain parameters. We then

incorporate this model in to an ALPaCA-based learningframework. To allow for adaptation of the terrain param-eters in the nominal model, we propose an extension toALPaCA in section 3.2 that incorporates models with un-known parameters and performs joint inference on theseparameters. In section 3.3, a regularization scheme is in-troduced to reduce correlation between nominal model fea-tures and meta-learned features, thereby maintaining the in-terpretability of the terrain parameters.

3.1 Linear Model Relating Terrain Parametersand Dynamics

We continue to assume that noisy measurements of slip iand sinkage z are available to the rover, but we relax therequirement in [16] of measuring the vertical force N andtorque M through a force sensor or of computing them on-board. Since force sensors may not be available, and thecomputation adds overhead, we instead relate the terrainparameter estimates directly to the dynamics of the rover.Assuming the same linear stress distribution as in Section2.3, we can find the ground normal forces Ni and tractionforces Ti (for i = {1, 2, 3}) as linear in parameters c andtan φ

Ni = Qi

ctan φ1 where

Qi =[− f0 · ( f3 + A f2) − f0σmA f2 f0σm f1

]Ti = Ri

ctan φ1 where

Ri =[f0 · (− f4 + A f1) f0σmA f1 f0σm f2

](17)

whereQ andR are 3x3 matrices with non-linear functionsof slip i and sinkage z. These can then be used to find theforward force using (9),

Fx = ST[QR

] ctan φ1 (18)

This enables us to use rover dynamics to estimate the ter-rain parameters or, conversely, to use terrain parameters topredict rover dynamics via[

ẋv̇

]=

[v

Fx/m

]=

0 0 v1mST [QR]

ctan φ1 (19)

Euler numerical integration is used to yield discrete timedynamics with a chosen time step size of 0.1 seconds thatis sufficiently accurate for the low velocities of the rover.This linear model relating dynamics and terrain parametersis used as a prior for the ALPaCA model described next.

3.2 P-ALPaCA: Extending ALPaCA with aParameterized Prior Model

We now extend the previously proposed ALPaCA model tohandle linearly parameterized nominal models. We use theterm nominal to refer to a model obtained outside of the

5054.pdfi-SAIRAS2020-Papers (2020)

meta-learning framework. In this paper, we will considermodels derived via simplified dynamics equations, as dis-cussed in the previous subsection. Previous work [22, 30]has incorporated prior knowledge into ALPaCA models inthe form of a fixed (non-parameterized) nominal model. Inparticular, previous work has considered system dynamics

x+ = f (x,u;θ) + � = g(x,u) + h(x,u;θ) + �, (20)

where g(·, ·) is the nominal model, capturing prior knowl-edge of system dynamics, and h(·, ·;θ) is the unknowncomponent of the dynamics that we wish to learn, whichvaries across episodes/tasks. Previous work has taken g(·, ·)as fixed, and parameterized h(·, ·; ·) with an ALPaCA model

h(x,u; K) = Kφ(x,u) (21)

where K ∈ Rnx×nφ .In this work, we investigate integrating nominal modelswith unknown parameters into the previously developedALPaCA modeling approach. We investigate models of theform

x+ = g(x,u;θnom) + h(x,u;θnet) + �, (22)where θnom denotes parameters of the nominal model andθnet denotes (online adapted) parameters of the neural net-work. This formulation will enable joint inference of un-known parameters from a physical model (e.g. masses) andparameters of the neural network model, which can capturefeatures not appearing in the nominal, simplified physicalmodel. Importantly, joint inference of these parameters isnon-trivial, and represents the core machine learning con-tribution of this paper.

We will work with a discrete-time nominal model that isaffine in the parameters [31], of the form

g(x,u;knom) = Φnom(x,u)knom + φnom(x,u) (23)

where Φnom(x,u) is a nx × nnom matrix of basis functions,φnom(x,u) is a constant (in the parameters) vector of lengthnx, and knom ∈ Rnnom is a set of nominal parameters. To fitwith this model, we introduce an alternate version of theALPaCA model of the form

h(x,u;knet) = Φnet(x,u)knet (24)

with knet ∈ Rnnet . This yields the dynamics model

x+ =[Φnom(x,u) Φnet(x,u)

] [knomknet

]+ φnom(x,u) + �

(25)for which we write kT = [kTnom,k

Tnet]

T and Φ(x,u) =[Φnom(x,u),Φnet(x,u)]. This approach, critically, resultsin parameters that are shared for each output dimension.Note that k ∈ Rnnom+nnet ; we write nφ = nnom + nnet. In thenominal ALPaCA model, each output dimension is deter-mined by a shared set of basis functions and a unique (foreach output dimension) set of parameters. In this model,the parameters are shared for all output dimensions, andeach dimension has unique neural network features. Dueto the reduced use of adaptive parameters, we refer to thismodel as Parsimonious ALPaCA, or P-ALPaCA.

Bayesian inference in this model closely follows from stan-

dard results for linear-Gaussian systems (see, e.g., the dis-cussion of Bayesian linear regression in [32]). We fix aprior k0 ∼ N(k̄0,Λ−10 ). Then, the mean and precision (in-verse of the covariance) are updated as

Λt = ΦTt Σ−1� Φt + Λt−1 (26)

k̄t = Λ−1t (Λt−1k̄t−1 + Φ

Tt Σ−1� (xt − φt)) (27)

where Φt = Φ(xt−1,ut−1) and φt = φnom(xt−1,ut−1). Inthe standard ALPaCA formulation the Woodbury identityis applied to yield an efficient update rule. Applying thesame approach here does not yield computational efficiencyimprovements, as we do not avoid a matrix inverse in theupdate. Thus, we maintain the natural parameters of themultivariate Gaussian during updating via

qt = qt−1 + ΦTt Σ−1� (xt − φt) (28)

and the precision update (26). To perform prediction, wemay compute k̄t = Λ−1t qt. Then, the posterior predictivedistribution is

xt+1 = N(Φt+1k̄t + φt+1,Φt+1Λ−1t ΦTt+1 + Σ�) (29)which may be derived based on the standard equationsfor mean and variance of a random variable under lineartransformations. Given this posterior predictive, the likeli-hood may be computed as in the standard ALPaCA meta-learning algorithm, and used to train the neural networkfeatures as well as the prior over k. Due to space con-straints we do not outline the full training algorithm in thispaper; we refer to [20] for more details.

3.3 Interpretability via Feature Orthogonality

We have so far presented an approach to combine a “grey-box” nominal system model, derived via simplified physicsof the rover system, with an entirely black-box neural net-work model without physical interpretation. If our only ob-jective is accurate state prediction, then our approach so faris sufficient. More likely, however, the physical parametersof the grey-box model provide information that is of useto system operators, scientists, or elsewhere in the plan-ning and control architecture. For example, inferred ter-ramechanics properties may be critical to identify regionsas safe or unsafe for traversal. Thus, it is necessary thatour added black-box model not result in poor estimates ofthe physical parameters in the grey-box model. Such anoutcome is not guaranteed; collinearity between features inlinear regression results in weights that are highly unsta-ble under minor perturbations [33]. This collinearity mayresult in the parameters of the neural network model be-ing highly correlated with the grey-box model, resulting inmisleading estimates of the physical parameters. To reducethis collinearity, we add a regularization term to the lossfunction to promote feature orthogonality.

Let φTi,t denote the i’th row of Φt1. Our orthogonality regu-

1Note the notation conflict with the previous subsection; we will ig-nore the constant (with respect to parameters) term in the nominal modelin this section for ease of presentation.

5054.pdfi-SAIRAS2020-Papers (2020)

Figure 3 Estimate of terrain parameters on loose and compact sand

larization takes the form∑tasks

nx∑i=1

∥∥∥∥∥∥∥I − 1TT∑

t=0

φi,tφTi,t

∥∥∥∥∥∥∥2

F

(30)

where ‖ · ‖F denotes the Frobenius norm and T is the lengthof the episode. There are several things to note about theabove. The norm of the innermost sum over time is a MonteCarlo approximation of the constraint

Ex∼µ,u∼π[φi(x,u)φi(x,u)T

]= I (31)

where π is the controller used for data collection and µis the occupancy measure (i.e. the probability distributionover states) induced by the system dynamics and the con-troller. Thus, the addition of this regularizer may be in-terpreted as a Lagrangian relaxation of the neural networktraining loss with a constraint requiring the basis functionsto be orthogonal, as

Ex∼µ,u∼π[φi j(x,u)φik(x,u)T

]= 0 for j , k (32)

where φi j denotes the j’th entry of φi. As orthogonal-ity of features implies no multicollinearity, this regulariza-tion term will minimize the effect and thus improve inter-pretability of the model.

4 EXPERIMENTSSimulations were conducted of a rover in rocker-bogie con-figuration traveling over two types of terrain - compact sandand loose sand - using equations for the wheel-terrain inter-action and terrain parameters from Wong et al. [14]. Fora given rover body position, the rover dynamics were sim-ulated by computing the ground reaction forces for each

wheel from applied torques and matching against the inte-grals from Wong’s model which were solved using Simp-son’s rule. The ground reaction forces were then used tocalculate the slip and sinkage for each wheel, which, alongwith rover position and velocity, were the inputs to themodels. Rover kinematics was used to compute the netforward force and resultant change in rover velocity. Thepurpose of the simulations was to compare the performanceof the nominal linear model against the model augmentedwith P-ALPaCA features and analyze the effect of enforc-ing feature orthogonality.

The simulation2 was written in Python and the machinelearning models were implemented using the PyTorch li-brary [34]. Each of the models consisted of two hiddenlayers of size 128 and were trained for 1000 iterations inbatches of 20 sampled dynamics transitions each to mini-mize the negative log likelihood of the posterior predictivedistributions. The first model was P-ALPaCA without anyadded features, the second model was P-ALPaCA with twoadded nominal features corresponding to the linear modelfeatures for velocity from Eq. 19, and the third model wasP-ALPaCA with two added nominal features as well as anadditional regularization term to promote orthogonality be-tween features as discussed in Section 3.3. The P-ALPaCAmodels with nominal features had an additional loss termto encourage the parameters associated with cohesion andinternal friction angles to be positive.

The state consisted of velocities between 0 to 1 m/s and the

2The code for all of our experiments is available at https://github.com/StanfordASL/rover-meta-learning.

5054.pdfi-SAIRAS2020-Papers (2020)

https://github.com/StanfordASL/rover-meta-learninghttps://github.com/StanfordASL/rover-meta-learning

Figure 4 Prediction error for velocity

control actions were applied wheel torques between 0 and5 Nm, wheel slips between 0 and 1, as well as sinkage be-tween 0 and 0.1 m. The rover mass was 10 kg, the wheelradius was 0.4 m, and wheel width was 0.1 m. During train-ing, the parameter for cohesion was randomly selected foreach batch between 0.5 and 5 kPa, except the band between0.7 and 1.3 kPa which are the typical values of cohesion forloose and compact sand [14] and were reserved for testing.The internal friction angle was randomly selected from therange 0 to 60 degrees, excluding the range from 20 to 45degrees for testing. The other terrain parameters were heldfixed at the values for loose sand (n = 1.1, kc = 0.9 kPa,kφ = 1523.4 kPa, k = 0.025, c1 = 0.18, c2 = 0.32) andcompact sand (n = 0.47, kc = 0.9 kPa, kφ = 1523.4 kPa,k = 0.038, c1 = 0.43, c2 = 0.32) [14] during training andperturbed by 5% during testing to ensure robustness.

First, we compare the ability of the models to predict thedynamics of the rover given an estimate of the terrain pa-rameters. As shown in Fig. 4, the linear model as well asall the P-ALPaCA models have low prediction error for ve-locity of the rover, with the largest being the error for thelinear model on loose sand, which is less than 5%.

Second, we compare the terrain parameter estimates pro-vided by the models using the dynamics of the rover, aswell as the estimates provided from Iagnemma et al.’smodel [16]. The ground truth of vertical load and wheeltorques are passed from the simulator to Iagnemma’smodel, while the other models utilize the position and ve-locity of the rover directly to make predictions. As seenin Fig. 3, both the linear model and Iagnemma’s modelprovide good estimates for cohesion and angle of internalfriction. The higher error for Iagnemma’s model on loosesand is due to their assumption that the angle of maximumshear stress is located halfway from the angle of initial con-tact, which becomes a looser assumption for higher cohe-sion terrains. While both P-ALPaCA models augmentedwith the linear model provide uncertainty estimates, the

Figure 5 Effect of regularization strength

terrain parameter estimate shows better convergence whenorthogonality of the features is enforced explicitly throughthe regularization term discussed in Section 3.3.

The addition of the regularization scheme results in inter-pretability of the meta-learned features. P-ALPaCA withorthogonal regularization not only provides dynamics pre-dictions but also an estimate of the terrain parameters usedin the model, aiding in interpretability of the model. Mod-els with orthogonal regularization produce lower parameterestimation error than models without the regularization andlower dynamics prediction error than the nominal linearmodel. Increasing the orthogonal regularization strength,i.e. increasing the weight on the orthogonal loss, results ina trade-off between lower parameter estimation error andhigher dynamics prediction error, as seen in Fig. 5.

5 DISCUSSION AND CONCLUSIONS

Our approach of augmenting a nominal model affine in pa-rameters with meta-learned neural network features fromALPaCA leverages Bayesian regression to adapt to roverdynamics. We maintain interpretability of the predic-tions by encouraging orthogonality of the nominal modelfeatures and the meta-learned features, resulting in goodterrain parameter estimates. Our experimental resultsdemonstrate that the combination of the grey-box nomi-nal model with a properly regularized black-box neural net-work model results in achieving the best of predictive ac-curacy and accuracy in parameter inference. Thus, our pro-posed approach is a highly flexible and performant systemidentification framework that is capable of application tomany systems.

Developing terrain parameter estimation models thatquickly adapt to novel terrains is a step towards greater au-tonomous operations of rovers, while increasing safety andefficiency. Directions for future work include incorporatingmore sources of information such as vision systems, detect-ing both sharp and gradual changes in terrain types, test-ing on physical prototypes, incorporating bulldozing, highslope, and multipass effects, and developing combined pa-rameter estimation and adaptive slip prediction models.

5054.pdfi-SAIRAS2020-Papers (2020)

AcknowledgementSomrita Banerjee and James Harrison were supported bythe Stanford Graduate Fellowship. The authors were par-tially supported by an Early Stage Innovations grant fromNASA’s Space Technology Research Grants Program. Theauthors wish to thank Abhishek Cauligi, Thomas Lew, andApoorva Sharma for their helpful feedback.

References[1] Higa S, Iwashita Y, Otsu K, Ono M, Lamarre O, Didier A

and Hoffman M (2019) Vision-Based Estimation of DrivingEnergy for Planetary Rovers Using Deep Learning and Ter-ramechanics. In: IEEE Robotics and Automation Letters.

[2] Rothrock B, Kennedy R, Cunningham C, Papon J, HeverlyM and Ono M (2016) SPOC: Deep Learning-based TerrainClassification for Mars Rover Missions. In: AIAA SPACEConferences & Exposition.

[3] Brooks CA and Iagnemma K (2005) Vibration-Based Ter-rain Classification for Planetary Exploration Rovers. In:IEEE Transactions on Robotics.

[4] Lorenz RD and Zimbelman JR (2014) Moving on sand. In:Dune Worlds. Springer-Verlag, first edition.

[5] Arvidson RE, Iagnemma KD, Maimone M, Fraeman AA,Zhou F, Heverly MC, Bellutta P, Rubin D, Stein NT,Grotzinger JP and Vasavada AR (2017) Mars Science Labo-ratory Curiosity Rover Megaripple Crossings up to Sol 710in Gale Crater. In: Journal of Field Robotics.

[6] Biesiadecki JJ, Leger PC and Maimone MW (2007) Trade-offs Between Directed and Autonomous Driving on theMars Exploration Rovers. In: Int. Journal of Robotics Re-search.

[7] Angelova A, Matthies L, Helmick D, Sibley G and Perona P(2006) Learning to predict slip for ground robots. In: Proc.IEEE Conf. on Robotics and Automation.

[8] Helmick D, Angelova A and Matthies L (2009) TerrainAdaptive Navigation for planetary rovers. In: Journal ofField Robotics.

[9] Gonzalez R, Apostolopoulos D and Iagnemma K (2018)Slippage and immobilization detection for planetary explo-ration rovers via machine learning and proprioceptive sens-ing. In: Journal of Field Robotics.

[10] Iagnemma K and Dubowsky S (2000) Mobile Robot Rough-Terrain Control (RTC) For Planetary Exploration. In: Proc.ASME Biennial Mechanisms and Robotics Conference.

[11] Kang S (2003) Terrain Parameter Estimation andTraversability Assessment for Mobile Robots. Master’sthesis, Massachusetts Inst. of Technology.

[12] Bekker MG (1969) Introduction to Terrain-vehicle Systems.Univ. of Michigan Press, first edition.

[13] Wong JY (2008) Theory of ground vehicles. John Wiley &Sons, fourth edition.

[14] Wong JY and Reece AR (1967) Prediction of rigid wheelperformance based on the analysis of soil-wheel stressespart I. Performance of driven rigid wheels. In: Journal ofTerramechanics.

[15] Ding L, Deng Z, Gao H, Tao J, Iagnemma KD and LiuG (2015) Interaction Mechanics Model for Rigid DrivingWheels of Planetary Rovers Moving on Sandy Terrain withConsideration of Multiple Physical Effects. In: Journal ofField Robotics.

[16] Iagnemma K, Kang S, Shibly H and Dubowsky S (2004)Online terrain parameter estimation for wheeled mobilerobots with application to planetary rovers. In: IEEE Trans-actions on Robotics.

[17] Hutangkabodee S, Zweiri YH, Seneviratne LD and Althoe-fer K (2006) Soil parameter identification for wheel-terraininteraction dynamics and traversability prediction. In: Int.Journal of Automation and Computing.

[18] Bauer R, Leung W and Barfoot T (2005) Experimental andsimulation results of wheel-soil interaction for planetaryrovers. In: IEEE/RSJ Int. Conf. on Intelligent Robots & Sys-tems.

[19] Lee JH (2015) Statistical modeling and comparison with ex-perimental data of tire–soil interaction for combined longi-tudinal and lateral slip. In: Journal of Terramechanics, 58.

[20] Harrison J, Sharma A and Pavone M (2018) Meta-LearningPriors for Efficient Online Bayesian Regression. In: Work-shop on Algorithmic Foundations of Robotics.

[21] Fan DD, Nguyen J, Thakker R, Alatur N, Agha-mohammadiA and Theodorou EA (2019) Bayesian Learning-BasedAdaptive Control for Safety Critical Systems. In:arXiv:1910.02325.

[22] Lew T, Sharma A, Harrison J and Pavone M (2020)Safe Model-Based Meta-Reinforcement Learning: ASequential Exploration-Exploitation Framework. In:arXiv:2008.11700.

[23] Rogers-Marcovitz F, George M, Seegmiller N and Kelly A(2012) Aiding off-road inertial navigation with high perfor-mance models of wheel slip. In: IEEE/RSJ Int. Conf. onIntelligent Robots & Systems.

[24] Rogers-Marcovitz F, Seegmiller N and Kelly A (2012) Con-tinuous vehicle slip model identification on changing ter-rains. In: R:SS Workshop on Long-term Operation of Au-tonomous Robotic Systems in Changing Environments.

[25] Seegmiller N, Rogers-Marcovitz F, Miller G and Kelly A(2013) Vehicle model identification by integrated predictionerror minimization. In: Int. Journal of Robotics Research.

[26] Harrison J, Sharma A, Finn C and Pavone M (2020) Contin-uous meta-learning without tasks. In: arXiv:1912.08866.

[27] Nagabandi A, Clavera I, Liu S, Fearing RS, Abbeel P,Levine S and Finn C (2019) Learning to Adapt in Dynamic,Real-World Environments Through Meta-ReinforcementLearning. In: International Conference on Learning Rep-resentations (ICLR).

[28] Ljung L and Söderström T (1983) Theory and practice ofrecursive identification. MIT press.

[29] Hacot H (1998) Analysis and traction control of a rocker-bogie planetary rover. Master’s thesis, Massachusetts Inst.of Technology.

[30] Harrison J, Sharma A, Calandra R and Pavone M (2018)Control Adaptation via Meta-Learning Dynamics. In:NeurIPS Workshop on Meta-Learning.

[31] Atkeson CG, An CH and Hollerbach JM (1986) Estimationof inertial parameters of manipulator loads and links. In:International Journal of Robotics Research.

[32] Deisenroth MP, Faisal AA and Ong CS (2020) Mathematicsfor machine learning. Cambridge University Press.

[33] Greene WH (2003) Econometric analysis. Pearson.[34] Paszke A, Gross S, Chintala S, Chanan G, Yang E, DeVito

Z, Lin Z, Desmaison A, Antiga L and Lerer A (2017) Auto-matic differentiation in PyTorch. In: Conf. on Neural Infor-mation Processing Systems - Autodiff Workshop.

5054.pdfi-SAIRAS2020-Papers (2020)

IntroductionBackgroundSystem Identification via Meta-Learned Neural NetworkRover Dynamics and KinematicsWheel-Terrain Interactions

Technical ApproachLinear Model Relating Terrain Parameters and DynamicsP-ALPaCA: Extending ALPaCA with a Parameterized Prior ModelInterpretability via Feature Orthogonality

ExperimentsDiscussion and Conclusions