TERRAIN AIDED NAVIGATION FOR PLANETARY EXPLORATION MISSIONS

Jakob Schwendner and Javier Hidalgo

Robotics Innovation Center, DFKI, Bremen, Germany

ABSTRACT

A central part of current space activities is to learn moreabout our solar system, its origins, its resources and itsconditions for harbouring life. A key technology for per-forming surface exploration of celestial bodies are mo-bile robotic systems, as they are able to withstand theharsh conditions with reasonable effort. One importantrequirement for performing navigation of a mobile robotis the ability to localise the system within a known ref-erence. Visual methods, which have proven useful inthis context, can add constraints on processing powerand environmental conditions. In this paper an alterna-tive approach is presented which only uses inertial sen-sors and encoders in order localise within a known map.The method is evaluated in a simulated lunar environmentbased on LRO digital elevation maps. The results showan average localisation error of 11 m for a travelled dis-tance of 2.3 km, assuming low-precision MEMS gyros.The proposed method can be applied to perform resourceefficient localisation in situations where visual methodsfail or are too costly to perform.

Key words: Terrain Aided Navigation; Exploration; Lo-calisation; LRO DEM.

1. INTRODUCTION

Mobile robotic systems will without a doubt become evenmore relevant for space exploration missions than theyalready are [11]. High cost for manned programs and re-cent success of robotic mission (e.g. MER [Mai06]) arelikely to form a shift towards robotic missions. One ofthe key aspects of exploration systems is mobility. Apartfrom the physical capabilities to negotiate complex anddifficult terrain, the aspect of navigation is of great impor-tance in this context. Navigation requires the three keycomponents of localisation, mapping and control. Thesubject of this paper is the problem of robot localisation.A mobile robot requires the knowledge of its positionwith respect to its goal, as well as a model of its surround-ing to generate steering commands which will lead it tothe goal. This is true regardless of whether the systemis tele-operated or performs its navigation autonomously.Earthbound activities can make use of the Global Posi-

tioning System (GPS), which provides a means of local-isation with respect to the earth fixed frame. Althoughmissions for mobile surface exploration systems oftenhave an orbiter component, they can not provide suffi-cient accuracy in localising the robot. Most potential mis-sion targets however have a map generated from orbiters.Most probably these maps are not provided with an accu-racy detailed enough for control of the system. They dohowever provide a global reference map, which can putmission targets in a spatial context and aid global pathplanning [Mur10].

Most previous successful approaches for mobile robotnavigation in space are vision based [MMJ+07]. A greatnumber of positive sides like environmental awarenessand exploitation of the images for science are on the up-side of this. There are however some design constraintsin order to make the use of vision based navigation fea-sible. Image processing does require a certain amount ofprocessing resources and a favorable positioning of thecameras [HK97].

One possible alternative could be the use of Terrain AidedNavigation [GGB+02, Lon82, Hos78] methods, whichuse a-priori map knowledge to compensate drift errorsfrom dead-reckoning navigation. These approaches havebeen applied to the problem of aerial localisation with theaid of radar sensors and terrain height maps. Matchingof 3D optical sensor information to known terrain maps[Car10] has been shown to provide absolute referencing.A proof of concept for non-visual methods was given by[CVGL07] on a four legged robot. In [SJ11] it was shownthat it is possible to localise using only inertial sensor andencoder data on a 50 m by 30 m sand field to within an ac-curacy of 1.5 m. The map resolution was 5 cm/pixel, whichis not realistic for orbiter based maps.

In this paper a method based on [SJ11] is presented,which is able to perform non-visual global referencing ofa ground based rover using maps from an orbiter. Inertialsensor readings and encoder values are the only sensordata used to perform the localisation in a Digital Eleva-tion Map (DEM) from the LRO [CBF+07] mission. Thefeasibility of the approach is demonstrated in a simulatedenvironment which takes the lunar DEM and a simulatedmodel of the Asguard v4 [JSK+11] system. The simu-lation takes gyro noise into account to generate realisticheading errors for the odometry model. The method isapplied to the simulated data, and compared to the con-

tact point based odometry model also presented in thispaper.

The contributions of this work are (i) a method to extractterrain slope from an odometry model, (ii) a measurementmodel for evaluating terrain slope against a DEM modeland (iii) evaluation of these methods in a Bayesian filterframework for estimating the position in a km scale sim-ulated scenario.

While the proposed method should not be seen as a com-plete replacement of visual navigation, it can augmentsuch systems. It can also provide a localization solutionwith a bounded error for missions where visual process-ing is not feasible due to resource or engineering con-straints.

2. METHOD

2.1. Overview

The method which is proposed in this paper uses a parti-cle distribution in order to represent the pose uncertaintyof a planetary exploration rover. The rover provides infor-mation from the wheel encoders and wheel position infor-mation in case of a non-rigid chassis. Further, the roveruses inertial sensors and sensor fusion in order to esti-mate the direction of the gravity vector. The fused inertialsensors also provide an estimate of the heading, which isprone to drift unbounded because of accumulated gyro er-rors. This information is used in a probabilistic odometrymodel of the system, in order to forward project the statedistribution. A terrain slope measurement is generatedbased on accumulated environment contact informationover time. This terrain slope information is comparedagainst a digital elevation model of the environment inorder to correct the pose distribution.

2.2. Odometry Model

The odometry model provides a way to make a predictionof the pose delta of the robot, so the change in positionand orientation, between two time steps. Odometry mod-els have seen extensive research (e.g. [GZZD05]) andusually depend on the class and configuration of the sys-tem. The work presented in this paper uses an environ-ment contact based approach. The underlying assump-tion is that a set of contact pointsC, which have a defined,time variate position expressed in the body frame of therobot B stay fixed in the environment. What is requirednow is the full transform

CBk

Bk+1=

[RBk

Bk+1TBk

Bk+1

0 1

](1)

from the body frame B at time step k, to the body frameB at time step k + 1. T and R are the rotational andtranslational parts of the transform.

The rotational part of the transform is already knownthrough the estimate of the inertial measurement unit.The IMU provides a rotation RW

IMU from the IMU fixedframe to a local geo fixed frame W , and the transforma-tion CB

IMU from the IMU frame to the body frame B isknown. Then the rotational part of the delta can be givenas

RBk

Bk+1= (RW

IMUkRIMU

B )−1RWIMUk+1

RIMUB (2)

Finding the translation is based on the assumption that thecontact points are fixed in the environment. By rotatingthe contact points from two time steps into the same refer-ence frame, the frame translation is given by the positiondifference of the same contact point at two different timesteps. For a single point, the assumptions that the contactpoint stays fixed in the environment always holds. How-ever, for multiple points the problem is over-specified,and the translation is required such that the squared sumof the individual displacements is minimal. It can be eas-ily verified that this is just the average of all the displace-ment, so that finally the translational delta can be givenas

TBk

Bk+1=

1

|C|∑c∈C

ck −RBk

Bk+1ck+1 (3)

This delta transformation is assumed to be the mean µO

of the odometry transform. To model for different typesof errors, a simple Gaussian distribution with covarianceΣO is parametrized based on certain dynamic and staticaspects.

ΣO = diag

A

cos−1(ez ·RWBkez)

|TBk

Bk+1|

Rz(RBk

Bk+1)

1

(4)

In (4) ez is the unity vector in the direction of the z-axis.A is a configuration matrix, which relates the tilt of therobot, the delta translation and the change in heading tothe pose errors. A robot which is situated on an incline ismore likely to slip than a robot on a flat surface. Equally,higher translational velocities, which are expressed in alarger translation over a fixed time period, are more likelyto generate errors due to system dynamics. The actualvalues for the coefficients of A depends on the systemand the environment.

Based on the mean and covariance values, the final odom-etry distribution can be given as

p(xk|uk, xk−1) = N (µO,ΣO) (5)

where uk is the odometry input consisting of the contactpoint values and the orientation reading from the IMU.

2.3. Measurement Model

In Section 2.2 a method was developed to estimate thetransformation between the body frame in two consec-utive time steps. This is an estimate and prone to er-rors from different sources. These errors accumulate overtime and lead to drift in the pose estimate of the robot.Depending on the quality of the sensor data, the valueswhich are generated are locally stable. The principle ideabehind the measurement model used in this approach isto extract information on the contact points with the en-vironment and place them in a locally consistent correla-tion. Further, it is assumed that the measurements fromthe IMU can provide information on the pitch and roll ofthe robot, since they are able to compensate for drift er-rors by measuring the gravity vector. This is not possiblefor the heading error. The contact points can be related tothe body frame and the gravity vector. By fitting a planethrough the contact points, an estimate of the slope of theterrain the robot has traversed can be performed. Given amap with slope information, and a position and heading,the measurement from the contact points can be related tothat map, by comparing the normals of the respective sur-faces. This information can then be used in the particlefilter to estimate the likelihood of a particular hypothesis.

Let COBk

be the transform from the body frame at timek to an arbitrary odometry frame O, such that CO

Bk+1=

COBkCBk

Bk+1. The set S of contact points in the odome-

try frame can be constructed by transforming the contactpoints for the last n time steps.

S =⋃

i=1..n

⋃c∈Ck−i

COBk−i

c (6)

In this approach n is chosen so that the total translation|TBk−n

Bk| < tmeas is smaller than a threshold value.

The next step is to fit a plane through the set S of con-tact points. There are multiple methods on how this canbe done. For simplicity the following system of linearequations is chosen

∑p∈S

p2x pxpy pxpypx p2y pypx py 1/|S|

x =∑p∈S

[pxpzpypzpz

](7)

which when solved for x results in a scaled normal of thesurface n = (−xx,−xy, 1) which can be normalized ton = n/|n|. The surface normal is expressed in the arbi-trary odometry frame O. For the slope to be comparableto the slop information from the map, the normal is ro-tated into the body frame. Rather than using the full RO

Brotation, only the component around the z-axis is appliedto preserve the direction of the gravity vector. This resultsin the measured normal in body coordinates as

zk = Rz(−R−1z (ROB))n. (8)

The measurement model is used to estimate the likeli-hood of a particular state hypothesis xk given the mea-surement and the map p(xk|zk,m). The measurement in

this case is the surface normal in body coordinates, andthe map is a digital elevation map of the environment.In order to compare the slope measured by the odome-try model to the map, the map needs to provide slopeinformation as well. If this information is not availabledirectly, it can be estimated based on the neighbouringmap cells. Let mpq be the elevation value of the map cellat index position pq which correspond to the x and y co-ordinates of the state variable xk. The slope values in xand y direction can then be directly derived given the cellsize c, such that the surface normal at the position of xkis

nm =

(− (mp+1q −mp−1q)

2c,− (mpq+1 −mpq−1)

2c, 1

).

(9)Which again is normalized such that nm = nm/|nm|.Similar to the measured normal from the odometry, themap normal needs to be rotated into body coordinates, inorder to be comparable. The comparison is performed bycalculating the angle between the two normals. The er-ror due to measurement uncertainties and other effects isassumed to be normally distributed so that for the likeli-hood of the measurement we can finally give

p(xk|zk,m) = φ(1

ζcos−1(zk ·Rz(−R−1z (xk))nm)),

(10)where ζ is the measurement noise factor and φ the stan-dard normal distribution.

2.4. Particle Filter

The pose of the robot is modelled as an uncertainty dis-tribution over x, y and θ, which represent the position onthe plane and the heading of the robot. The z-position isnot modelled explicitly, since it can be derived from theelevation model, given the assumption that the robot isin contact with the ground all the time. Due to the na-ture of the problem, the probability distribution can notbe modelled using Gaussians, and a sampling representa-tion is used instead. A particle filter is then used to up-date the representation according to the odometry and themeasurement model. The odometry performs a forwardpropagation of the pose samples in the distribution. Theodometry model provides a mean and an error distribu-tion over the pose deltas, so the change of pose betweentwo time steps. In the projection step of the filter, eachof the pose particles gets updated with a sample from theodometry model distribution. In that way the prior errordistribution is increased by the error that is added fromthe odometry model. The update step is called repeatedlyin a fixed time interval. Whenever the robot has travelleda distance larger than tmeas, the measurement model isused. According to (8), the measurement of the slope ofthe environment in body coordinates is calculated as zk,and updated according the update model p(xk|zk,m) de-fined in (10). This measurement model provides a likeli-hood value for the state xk given the measurement and themap. Each particle has a relative weight value, which gets

updated by the measurement function. In this way parti-cles that represent a pose with a high likelihood receive alarger weight than those with a lesser likelihood. After awhile, certain particles will have a very low weight, anddo not contribute significantly to representing the proba-bility distribution. This can be measured by the effectivenumber of particlesNeff. Whenever this value falls belowa certain threshold, the particles get resampled accordingto their relative particle weight.

Algorithm 1 Particle Filterχk = χk = ∅D = Ifor all m = 1 . . .M do

∆T ∼ p(xk|uk, xk−1)

x[m]k = x

[m]k + ∆T

D = ∆TDif Dtrans > tmeas thenw

[m]k = w

[m]k−1p(zk|x

[m]k )

D = Iend ifχk = χk+ < x

[m]k , w

[m]k >

end forNeff = 1∑M

i=1

(w

[i]k

)2

if Neff < Nthr thendraw i with probability ∝ w[i]

k

add x[i]k to χk

elseχk = χk

end if

The Particle Filter used is a Sequential Importance Re-sampling (SIR) [GSS93] filter, where an initial set ofparticles is generated using a random distribution overX ∼ N (µ0, Σ0). A more in depth discussion of thefilter in the context of this application can be found in[GGB+02].

3. EXPERIMENTS

In order to evaluate the feasibility of the approach forlocalisation in an lunar exploration scenario a represen-tative set up was generated in a simulation environment.The Lunar Reconnaissance Orbiter (LRO) [CBF+07] car-ries instruments for geological inspection of the lunarsurface. The LOLA instrument generates globally con-sistent elevation data. Further, the narrow angle cam-era (NAC) produces high-resolution images in the orderof 0.5/[m/pixel]. This image data can be used to extracthigh resolution digital elevation maps (DEM) from thelunar surface [MBB+10]. One of these maps from theAitken Basin near the lunar south pole was selected (seeFig. 1). The map was generated by the United StatesGeological Survey (USGS) and made publicly availablethrough the LMMP website 1. The identifier for the map

1http://lmmp.nasa.gov

is [LRO NAC DEM 60S200E 150cmp.tif]. From thismap, a 1500 m by 1500 m segment was extracted for fur-ther processing. The coordinates of the map corners are:

(160◦14′11.28”W, 59◦39′50.92”S)

(160◦14′11.28”W, 59◦42′49.00”S)

(160◦8′15.12”W, 59◦39′50.92”S)

(160◦8′15.12”W, 59◦42′49.00”S)

The rendered image shown in Fig. 2 provides an impres-sion of the local shape of the environment.

During the experiments a gyroscope model is used in or-der to be precise. The model is balanced between accu-racy and complexity covering the commonly encounterederrors in inertial sensors but does not include g-sensitivedrift, scale factor asymmetry or other errors which de-pend on the design of a particular sensor and technol-ogy. The angle random walk and the rate angle randomwalk are considered the dominant stochastic errors forgyros. The first error is affected by white noise whilethe second one affects the bias instability directly alter-ing the gyros drift. It is a variation of Farrenkopf’s gyromodel [Far78] which has been widely accepted and usedin the aerospace sector for many years. The model isgiven as:

ω(t) = (I3×3 +M(SF,MA))ω(t) + βω(t) + nrw(t)(11a)

βω(t) = nrrw(t) (11b)

where ω(t) is the sensor readout, ω(t) is the ideal value,βω(t) is the bias and nrω(t) and nrrω(t) are indepen-dent zero-mean Gaussian white noise of the characterizedstochastic processes defined with

E{nrω(t)nTrω(τ)} = I3×3N

2δ(t− τ) (12a)

E{nrrω(t)nTrrω(τ)} = I3×3K

2δ(t− τ) (12b)

where E denotes expectation, δ(t − τ) is the Dirac deltafunction, N and K are the angle random walk and raterandom walk coefficients in rad/

√s and rad/s/

√s re-

spectively. M is the deterministic errors matrix cor-responding to the misalignment (MA) and scale factor(SF ). Assuming to be temperature independent theseerrors can be calibrated and therefore the matrix M hasbeen set to zero in this case.

The resulting DEM was used in the MARS [RKK09]simulation environment. The rover which was used inthe simulation is modelled after the Asguard v3 system[JSK+11]. The simulation provides true position andorientation values as well as information about contactpoints with the environment. The position values fromthe simulation are only used as reference. The orienta-tion values have been perturbed by additional noise froma simulated gyro. The parameters for the gyro modelhave been chosen to be K = 1.5 · 10−4rad/s/

√s and

Figure 1. The experiments were performed on dig-ital elevation maps (DEM) from the lunar southpole. A 1500 m by 1500 m patch with a resolu-tion of 1.5 m/pixel was selected at [160◦14’11.28” W,59◦39’50.92” S]. The patch was extracted froma DEM (LRO NAC DEM 60S200E 150cmp.tif), whichwas made publicly available by the United States Geo-logical Survey (USGS) through the LMMP website.

N = 5.7 · 10−4rad/√s. These parameters have been ex-

tracted experimentally using the Allan variance [EHN08]technique 2 from the commercially available XSens-MTI.The gyro rate error is evaluated for a single gyro only,integrated to produce the heading error and finally ap-plied to the orientation values from the simulation. Thenoise added orientation data as well as the contact infor-mation is sufficient information for the odometry modeldescribed in Section 2.2. The odometry result is the basereference against which the filter performance is com-pared. The filter described in Section 2.4, receives thesame inputs as the odometry model, but also holds a ref-erence to the DEM model. The initial condition of theparticle filter is a sampling around the start position ofthe trajectory, which is considered known. The orienta-tion with respect to the map is considered unknown andneeds to be recovered by the filter. The tmeas variable wasset to the cell size of the DEM.

There are a number of ways to represent the particle posedistribution as a single position value. For simplicity rea-son, the particle centroid has been chosen. This is theweighted average of the particle position and orientationvalues.

2http://cran.r-project.org/web/packages/allanvar/

Figure 2. A rendered image of the DEM which was usedfor the experiments. The height values are not to scale.The difference between the lowest and the highest pointfor this 1500 m by 1500 m patch is 118 m

-100 0 100 200 300 400 500 600-100 0

100 200

300 400

500 600

-10-5 0 5

10 15 20 25

z (m)

true position

x (m)

y (m)

z (m)

Figure 3. A rigid body simulator [RKK09] was used torun a model of the Asguard system [JSK+11] on the lu-nar DEM. The graph shows the trajectory of the robotin the simulated environment, starting and ending at theorigin. The trajectory has a total length of 2300 m. Thedifference between the lowest and the highest point on thetrajectory is 32 m.

4. RESULTS

The MARS simulator was able to generate position andorientation values as well as environment contact pointsfor the Asguard v3 model in real-time. The total sim-ulation time for the provided trajectory is 70 min. Thetotal distance the rover has travelled is 2300 m, which re-sults in an average velocity of 0.54 m/s. The loop the roverfollows, starts and ends at the same position. The over-all height difference between the highest and the lowestpoint of the trajectory is 32 m. A plot of the trajectory canbe seen in Fig. 3.

The orientation values are perturbed by gyro noise. Theresults of the gyro noise modelling can be seen in Fig. 4.The integration of the rate noise results in a final headingerror of approximately 6 rad. The gyro noise parameterswere chosen to reflect characteristics of low-precisionMEMS gyroscopes, and thus represent a very basic sen-sor configuration. The trajectory the odometry followscan be seen in Fig. 6. The error in heading has a signifi-cant impact on the trajectory of the odometry. The global

-0.02

-0.01

0

0.01

0.02

0 500 1000 1500 2000 2500 3000 3500 4000 4500-6

-4

-2

0

2

4

6ang

ula

r ra

te e

rror

(rad

/s)

ang

ula

r err

or

(rad

)

time (s)

gyro rate errorangular error

Figure 4. In order to simulate realistic sensor condition,the true orientation provided by the simulation was per-turbed by simulated gyro noise. The graph shows the an-gular rate noise over the course of 70 min of runtime. Theplot also shows the angular error, which is the integrationof the rate error.

Figure 5. The simulated sensor data was processed by thelocalisation filter proposed in the methods section. Theillustration shows the true pose (black) and the centroidof the particle distribution (blue). The particles repre-sent the uncertainty distribution of the current pose of therobot.

position estimate has a significant deviation compared tothe reference trajectory, and is only meaningful locally.

The particle filter based approach receives the DEM ofthe environment as additional information to performglobal referencing. The data provided by the simulationwas processed according to the description given in themethods section. A total number of 500 particles wasselected and the odometry parameters adapted to the sim-ulation and gyro noise environment. Fig. 5 shows a visu-alisation of the particle filters internal state. The currentpose estimate is represented as a set of particles. The dis-tribution of the particles represents the uncertainty overthe pose. Because the gyro error is modelled in the errormodel of the odometry, the particles spread in the typi-cal half-moon shape for orientation uncertainty. Particleswhich correlate the most with the reference map are givenhigher weights and are reinforced against low weight par-

-100

0

100

200

300

400

500

600

700

-100 0 100 200 300 400 500 600 700

y p

osi

tion (

m)

x position (m)

odometryparticle filtertrue position

Figure 6. The result of the localisation experiment areshown in this graph. The true position in x and y-direction is plotted in blue. The odometry (red) ex-hibits significant deviation from the true position, mainlycaused by the gyro noise which results in heading errors.The estimate of the particle filter (green) is able to trackthe position and exhibits a bounded error behavior.

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000 3500 4000 4500

posi

tion e

rror

(m)

time (s)

odometry errorparticle filter error

Figure 7. Comparison of odometry error to the filter er-ror. While the odometry error grows up to 500 m, the filtererror stays within a 40 m boundary.

ticles. In this way, the particle distribution is able to trackthe true position of the rover much closer, which can beseen in Fig. 6.

Fig. 7 shows a comparison of the odometry error with theerror from the filter. The error profile for the odometry ismuch smoother, but grows up to an error of 500 m, whilethe filter error stays within a 40 m boundary. The error iscalculated as the length of the distance vector between theposition on the true pose vs the position on the estimateat any particular time step. The min and max values forthe filter error are 0.9 m and 38.8 m, with an average of11.7 m. The experiments were performed on an Core-i7 processor with 2.8 GHz clock frequency. The runtimefor the filter was around 10 min for the 70 min simulationtime log. The code was not optimized for this particulartask and could be made significantly faster.

5. CONCLUSION

Mobile robotic systems will remain to have a signifi-cant influence on planetary and lunar exploration activ-ities. The ability to navigate reliably is a crucial factorin this. While visual methods are a valid option in manycases, they may fail to perform under certain conditionsor prove to be too resource intensive. In this paper, amethod was proposed, that allows to perform global lo-calisation using digital elevation models of the surface tobe explored. These models are often available throughorbiters or other sources like DEMs from a lander de-scent. The experiments performed show that it is feasibleto track the global position for large travel distances of upto several kilometres. For a distance of 2.3 km, the filterapproach was able to track the position with a maximumerror of 40 m, and an average error of 11 m. This accuracyis sufficient for example to return to a lander module au-tonomously after several kilometers of travel. Also, thismethod does not make use of visual data other than theDEM models. No visual data needs to be processed onthe rover, making it very resource efficient. The methodwill also work in complete darkness, as well as other con-ditions like atmospheric clouds or dust storms.

The results provided give a proof of concept in simula-tion. Although the principle approach has been verified indifferent scenarios, more experimental work is requiredto detail the effectiveness under realistic conditions, aswell as the resource requirements. The current method ofusing the centroid of the particles for estimating best posecan be improved by taking particle history and clusteringinto account.

ACKNOWLEDGMENTS

The work presented here is part of the Project ”Intelli-gent Mobility”, which is funded by the Federal Ministryfor Economics and Technology (BMWI) through the Ger-man Space Agency (DLR) grant number 50 RA 0907.

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