KINEMATICS MODELING OF A HYBRID WHEELED-LEGPLANETARY ROVER

Javier Hidalgo and Florian Cordes

DFKI - Robotics Innovation CenterRobert-Hooke-Str. 5, 28359 Bremen, Germany

ABSTRACT

It is the goal of this manuscript to describe and analyze acomplete kinematic model for hybrid wheeled-leg roversand its applicability to Sherpa, a flexible rover with acomplex actuation system. Differential kinematic equa-tions of the hybrid legs are combined to form the com-posite equation for the rover motion. The model capturesthe 6 DoF pose (position and orientation) while travers-ing uneven terrains for hybrid systems and independentlyactuated joints. The kinematics model is analyzed in or-der to correctly propagate rover pose as input for a poseestimator in localization. Initial results from simulationare discussed for Sherpa navigation kinematics towardsefficient pose estimation and dead reckoning.

Key words: robot motion, wheel odometry, roverkinematics, dead reckoning and planetary rovers.

1. INTRODUCTION

Up to now planetary rovers deployed on extraterrestrialsurfaces made use of passive suspension systems (mostlythe ”Rocker-Bogie” suspension). Wheeled mobile robotsare more energy efficient than legged robots on hard andsmooth surfaces while active articulated robots with so-phisticated mobility systems enable traversal over un-even terrain. Development of hybrid wheeled-leg solu-tions for the next generation of more capable systemswith active suspension are currently ongoing. The ATH-LETE [WLB+07] family of rovers with completely ac-tive suspension or hybrid solutions with actively and pas-sive suspension as in Scarab [BWW08] or Chariot1 areonly some of the developments currently in progress.

Rover kinematics plays fundamental roles in design, dy-namic modeling, dead reckoning, control and wheel slipdetection. As the control of an actuated suspension sys-tem is more complex, a motion model of the chassis isrequired to correctly command the desired trajectory andpropagate the pose of the wheels as well as to reduce slipby proper maneuvering [TM05] [SSVO09]. Most efforts

1http://www.nasa.gov/mission_pages/constellation/main/lunar_truck.html

on kinematics have concentrated either on planar two-dimensional space, i.e., translation in the x-y plane andheading [Vol, GFASH09] or in more sophisticated sys-tems with passive articulate joints [TM05]. Typically,the kinematics modeling can be solved by two methods,the geometric or the transformation approach. The ge-ometric approach is more intuitive but it is restrictive tothe particular platform lacking of generalization [CW90].The transformation approach is general and consistent byapplying a series of transformations and Jacobian matri-ces to relate motion in the joint space to the Cartesianspace, which main contribution is by Muir and New-man [MN86].

Extensive research with different reasoning is availablein literature. Alexander et al [AM89] present a pla-nar rigid body model considering a variable number ofwheels. Campion et al [CBDN96] classified ordinarymobile robots into five types taking into considerationgeneric parts of the model equations. Other research intowheel-ground contact angle and pose estimation of robotsmoving on uneven surfaces can also be found [LS07].

As of today, the challenges and problems of hybrid struc-tures have not been extensively described. The approachpresented in this paper extends [TM05, MN86] in or-der to solve the complete kinematics of hybrid wheeled-leg rovers. The methodology is applied to the particu-

Figure 1: The fully integrated Sherpa robot in a lunartesting environment.

Figure 2: Coordinate frames for the Sherpa rover. W isthe world reference frame, B is the body frame, Si is theswing leg-base frame, Ai is the wheel axle frame and Cithe wheel contact frame

lar case of Sherpa which has a complex mobility system(see Figure 1). A six degrees-of-freedom (DoF) solu-tion is pursued to propagate rover pose and velocity ina three dimensional space depending on the articulatedjoint configuration. The paper starts by describing a gen-eral kinematics model in Section 2. Afterwards, Section 3analyzes its applicability to Sherpa. Different kinematicsforms as the navigation kinematics (rover motion model)and the slip kinematics (wheel slip vector detection) areexplained in Section 4. Uncertainty modeling and deadreckoning are presented in Section 5 to finally concludewith some results.

2. KINEMATIC MODELING

Wheeled-leg mobile robots consist of a number of legsconnected to a main body and having a wheel at the endfor rolling capabilities. Four main coordinate frames aredefined, a robot body frame (B) attached to the rover cen-ter of mass, a swing leg-base frame (S) at the base of theleg, a wheel axle (A) frame attached to the wheel axleand a wheel contact frame (C) defined as a single pointof contact between the wheel and the ground (see Fig-ure 2). The B frame is related to a fixed navigation-frame(W) by the pose vector U = uW,B = (x y z φ θ ψ) Eachwheel frame Ai = 1, 2, 3, ...,m is related to the B frameby the transformation matrix TB,Ai

(q) which depends onthe particular leg kinematics in joints space representedby the vector q = [q1 q2 ... qn−1] where n is the numberof degrees of freedom of each leg.

The wheel is considered to be a rigid disc. A key dif-ference between planar and articulated rovers for uneventerrain is the wheel contact angle δi at each C-frame de-fined as the angle between the normal vector to the ter-rain plane and the wheel z axis in A-frame (see Figure 3).Whereas the contact angle is always zero for planar mo-bile robots moving on flat terrains, it is a main distinctionfor articulated rovers and it is more important for hybridwheeled-leg rovers which their suspension system allowsactively control the angle.

The C-frame is related to the A-frame by the transforma-

Figure 3: General coordinate frames for wheel axle andcontact point on inclined terrain. Ai is depicted here asx-axis pointing forward.

tion matrix TAi,Ci(δi) defined as a rotation of the contact

angle δi along the y axis and a translation in z-axis de-fined by the wheel radius r. The corresponding transfor-mation matrix is given in (where c denotes cosine and ssine):

TAi,Ci(δi) =

cδi 0 sδi −rsδi0 1 0 0−sδi 0 cδi −rcδi

0 0 0 1

(1)

The leg forward kinematics solution relating thewheel contact point with respect to the body is de-fined by the matrix multiplication TB,Ci

(q, δi) =TB,Ai

(q)TAi,Ci(δi). Hybrid rovers also provide rolling

capability which entails a change of the Ci frame overtime increment ∆t. The wheel motion is composite ofa translation of the contact point along the x axis byrqni + ξi, where qni is the wheel rotation angle corre-sponding to the nth DoF of the leg i and ξi is the slip inthe x direction. The slip vector can be modeled in threedimensions including lateral slip ηi along the y axis androtational slip ζi along the z axis. The resulting transfor-mation matrix is (where Ci is Ci(k−1) and Ci is Ci(k)):

TCi,Ci=

cζi −sζi 0 rqni + ξi

sζi cζi 0 ηi0 0 1 00 0 0 1

(2)

The transformation of the body frame B with respect tothe wheel contact point frame including the wheel rota-tion is:

TCi,B = TCi,Ci(q

ni, εi)TCi,Ai(δi)TAi,B(q) (3)

where εi = [ξi ηi ζi] is the slip vector and the matrixTCi,Ai = (TAi,Ci)

−1 and TAi,B = (TB,Ai)−1

Mobile robots are commonly commanded by desired ve-locities. The mapping between the rover Cartesian spacerate vector u = uB,B =

[x y z φ θ ψ

]and the joint

space rate vector q, the wheel rotation rate qni, wheel slip

rate vector εi and the wheel-ground contact angle rate δiis solved by the Jacobian matrix. The velocity kinematicsis deduced by applying the transformation matrix and thederivation of the point position relative to a coordinatesystem. Defining the transformation of the rover body attime step k − 1 (B) to rover body at time step k (B) asTB,B = TB,Ci

TCi,CiTCi,B , the derivative is

TB,B = TB,CiTCi,Ci

TCi,B +

TB,CiTCi,Ci

TCi,B +

TB,CiTCi,Ci

TCi,B (4)

Considering that for a general body in motion the posi-tion and orientation rates of TB,B are defined by the skewmatrix:

TB,B =

0 −ψ θ x

ψ 0 −φ y

−θ φ 0 z0 0 0 0

(5)

Equating matrices in Equation 4 and 5 the velocitykinematics is obtained. The equation can be reorderedin order to have the leg Jacobian matrix Ji of the form

[x y z φ θ ψ

]T= Ji

[q q

ni εi δi

]T(6)

It defines the contribution of each leg to the body rovermotion allowing the analysis of each joint to the resultingfinal velocity in u. The Ji matrix size is 6×(n+4) wheren corresponds to the DoF of the leg. Finally, the compos-ite rover equations are obtained combining the Jacobianmatrices for all legs into a sparse matrix equation as

I6I6...I6

xyz

φ

θ

ψ

= J

qqnεδ

≡ Eu = Jp (7)

where E is a 6m × 6 matrix that is obtained by stackingm 6 × 6 identity matrices, q is the m(n − 1) × 1 vectorof rover joint angle rates, q

nis the m× 1 vector of wheel

rotation velocities, ε is the 3m×1 slip vector and δ is them × 1 vector of wheel-ground contact angle rates. Therover Jacobian matrix J is a 6m × (mn + 4m) matrixobtained from the individual leg/wheel Jacobian matricesJi, i = 1, 2, ...,m and p is a (mn + 4m) × 1 vector ofcomposite angular rates for m leg/wheels of the rover.

Figure 4: Detail of a Sherpa leg with the number of DoF

3. SHERPA ROVER

Sherpa [CDK11] is a hybrid wheeled-leg rover that ispart of a multi-robot exploration system [CBLK10]. Therover is equipped with four legs with 4 DoF each, on theseso called swing units are mounted the rover’s wheelswith 2 further DoF, one for steering the wheel and onefor driving the wheel, respectively. More detailed infor-mation on Sherpa and the RIMRES project is providedin [CDK11, CRK12].

Sherpa is a highly capable planetary rover that is intendedto serve as the central mobile unit in a multi-robot explo-ration scenario. It is able to manipulate payload-itemsthat can be used to build up scientific or infrastructuralcomponents as well as for enhancing the capabilities ofthe mobile systems. The manipulator on top of Sherpa isused for handling the payload-items and for locomotionsupport [DRC11].

The wheeled-legs that constitute Sherpa’s suspensionsystem provide a total of 6 DoF each, summing up to24 active DoF in the locomotion system. The first DoF(q1) seen from the central body is called Thorax joint.The movement range is ±90 around the vertical axis.The Basal joint (q2) is designed as a parallel structure,that is responsible for lifting the wheel. These two DoFare the main DoF for (re-)positioning the wheel with re-spect to the body. The two auxiliary DoF WheelTilt (q3)and WheelFlip (q4) are used to orient the Wheel to theground. By positioning a designated foot plate on topof the WheelSteering (q5), the WheelFlip can be used toflip over the whole wheel and use a secure foot point2.WheelSteering (q5) and WheelDrive (q6) are used forsteering and driving the wheel respectively. Figure 4 andTable 1 details the DoFs.

3.1. Sherpa Kinematic Model

Figure 5 illustrates the coordinate frames of a single legof Sherpa according to Denavit-Hartenberg (D-H) nota-tion. The D-H parameters θ, d. a and α are presentedin Table 2 with units radians and millimeters. The pres-ence of a parallel structure in the leg requires the defi-nition of a virtual axes (Ba2) which rotation angle (q2)

2This feature is currently not implemented mechanically.

Table 1: Sherpas degrees of freedom of the suspension system and their specification

DOF Name AngleLimit ()

Maximum Angu-lar Velocity (/s)

Repeatable PeakTorque/Force

Estimated std. de-viation ()

1 Thorax (Th) ±90 12.0 250Nm 0.082 Basal (Ba) ±60 13.3 600N 0.083 WheelTilt (WT ) ±30 6.7 168Nm 0.24 WheelFlip (WF ) ±180 23.4 76Nm 0.45 WheelSteering (WS) ±90 75.0 34Nm 0.056 WheelDrive (WD) cont. rot. 165.0 50Nm 0.02

is the negative value of the Basal joint (Ba). Wheel Tiltframe is rotated Φ radians along the z-axis in order toavoid the elbow between the virtual Basal joint and theWheel Tilt. Wheel Flip frame appears to be outside ofthe mechanical structure between the steering and its pre-decessor frame in the chain (WT ) with a displacement of−HWF units along its x-axis. This choice of the coor-dinate frame ensures the minimum number of frames torepresent the transformation between the wheel leg swingbase (S) and the wheel contact point (C)

The resulting transformation from Sherpa body to thewheel contact point can be written as

TB,Ci(q,δi) = TB,SiTSi,Ai

(q)TAi,Ci(δi) i = 1, 2, 3, 4

(8)

The transformation from the body frame (B) to the leg

Table 2: D-H parameters for Sherpa Leg

Frame θ d a α

Th q1 0 LTh Π/2

Ba q2 0 L 0

Ba2 −q2 − Φ 0 D 0WT q3 + Ω 0 HWF Π/2

WF q4 + Π/2 LWT + LWF 0 Π/2

WS q5 + Π/2 −HWD LWD 0

LTh = 90mm offset between the Th and the Ba frameL = 520mm forward distance between Ba and Ba2LBa = 88mm forward offset between Ba2 and WTLWT = 50mm horizontal offset between WT and WFLWF = 67mm horizontal distance between WF and WSLWD = 161mm offset between WS and AHWF = 29mm vertical displacement between WT and WFHWD = 100mm vertical offset between WS and AΦ = arctan (HWT

LBa) offset angle along the z-axis between

Ba2 and WF frameD =

√Lba2 + HWT 2 displacement between the rotated

Ba2 frame by Φ and the WF frame

swing base (S) (denoted in Equation 8 by TB,Si) is a fixed

transformation depending on the leg position. Therefore,forward-right leg has a rotation of 45, forward-left of−45, rear-left of 135 and rear-right of−135 along thez-axis and a corresponding translation along x and y-axis.In order to be consistent to the right-handle frame nota-tion and keep the x-axis in the Ci-frame pointing towardsthe forward motion of the wheel rotation, the transforma-tion betweenAi andCi frames given by Equation 1 variesdepending on the leg side. It results on a positive (right-side legs) or negative (left-side legs) 90 angle rotationalong z-axis, which gives the following transformationmatrices

TAi,Ci(δi) =

0 cδi sδi −rsδi−1 0 0 00 −sδi cδi −rcδi0 0 0 1

i = 1, 3 (9)

TAi,Ci(δi) =

0 −cδi sδi −rsδi1 0 0 00 sδi cδi −rcδi0 0 0 1

i = 2, 4 (10)

It is noted that wheel-contact transformation depends onthe articulated joints values q and the wheel-contact an-gle δi. The Jacobian matrices are calculated as explainedin Section 2 using Equation 8. The computation of thevelocity kinematics requires the derivative of transforma-tion matrices, which for Sherpa rover there are a consider-able amount of DoFs (10 DoF per leg coming from joints,wheel-ground contact angle and slip vector). The cas-cade velocity derivative corollary [MN86] is applied inorder to simplify the derivation. Since TB,Si

(and there-fore TSi,B) is independent of time, the time derivative ofTB,B is

TB,B = TB,SiTSi,Ci

TCi,B +

TB,CiTCi,Ci

TCi,B +

TB,CiTCi,SiTSi,B (11)

where the first row represents the rate change of the wheelcontact point at time k−1 with respect to the body frame.

Figure 5: Coordinate frames for rover’s leg at zero position. Note that the orientation of wheel contact point frame (C)depends on the leg side.

It involves leg lifting and is only applicable when walkingmode is used by Sherpa (i.e, overcoming big obstacles),since walking machines do not provide continuous con-tact to the ground. The second row is the rate change ofthe wheel contact point due to wheel rotation. The thirdrow is the Sherpa body frame rate change related to theground contact point when adapting to the terrain.

This kinematic model defines different wheel Jacobianmatrices depending on the locomotion mode. When therover is walking over its wheels, no rotation is performedand only the slip vector rate defines the derivation ofTCi,Ci

. The derivative transformation TSi,Cidefines the

first part of the walking cycle and TCi,Simodels the

second part. In the majority of cases, when the robotis rolling over its wheels, the wheel contact point onlychanges due to wheel rotation and the first row of Equa-tion 11 is set to zero while the third row defines the jointsrate change due to actively terrain adaptation of the chas-sis.

The resulted TB,B given by Equation 11 is equaled to theEquation 5 and the leg Jacobian of the form in Equation 6is obtained for the Sherpa rover. Further, the compositerover equations are formed by combining the four legsJacobian matrices in the form of Equation 7.

4. SHERPA KINEMATIC EQUATIONS

Different forms of rover kinematics can be obtained de-pending on the specific state of interest. This article fo-cuses on two forms since the purpose is to properly modelSherpa rover kinematics towards an efficient pose estima-tion during path following. This will be done by describ-ing useful forms of Equation 7 for Sherpa, referred asnavigation kinematics and slip kinematics. Finally, un-certainty modeling and step-integration is performed toestimate rover pose over time using dead reckoning meth-ods. Other forms of interest as leg-inverse kinematics andactuation kinematics to command desired rover velocitiesare also valuable to analyze [TM05], but they are beyond

the scope of this paper.

4.1. Navigation Kinematics

Navigation kinematics relates rover pose rate to the jointsand sensed rate quantities. The navigation kinematics isoften referred as motion model and is the basics for deadreckoning systems. The objective of the method is to es-timate the rover pose. Moreover, it is useful for the un-derstanding of the role of different quantities contributingto the final rover pose.

In order to simplify the model, the navigation kinematicsfocuses on the navigation form of Equation11 whenSherpa is rolling over its wheels and adapting its legs tothe uneven terrain. Therefore, the first row of Equation11is set to be zero since any walking movements are an-alyzed for the purpose of this work. Joint angle mea-surements are available in Sherpa rover since absoluteencoders are installed in each joint and relative encodersare available in all wheels for measuring wheel rotationrates. Sherpa rover is also equipped with an IMU thatcan provide information on the pitch and roll angle, sincethey are able to compensate for drift error by estimationof the gravity vector using accelerometers. This is nottotally possible by the heading error. Sensor availabilitydefines sensed and not-sensed quantities and Equation 7separates as

[Es En]

[usun

]= [Js Jn]

[pspn

](12)

Rearranging into not-sensed (right-side) and sensed (left-side) quantities, the resulting equation is obtained

[En −Jn]

[unpn

]= [−Es Js]

[usps

]≡ Aχ = Bγ

(13)

where A and B are matrices dimensions depends on thesensing capabilities of the rover system which will directinfluence on the existence of a solution. The study therank of A refers the sensing analysis of the navigationequations. There is no solution if the matrix A has notfull-rank and therefore the system is undetermined. How-ever, if rank[A|B] = rank[A] the system is determinedand a unique solution exits. When the system is overde-termined, rank[A|B] > rank[A], there is more than onesolution and least-square method is applied to solve theequations minimizing the error. If the matrix A has fully-rank and rank[A|B] > rank[A], there is ample sensingbecause it provides extra sensing capabilities. This ex-tra information is very useful for slip detection and erroranalysis where the equations are inconsistent [TM05].

All the joints angles and wheel rolling rates are sensedquantities, as well as the pitch φ and roll θ angles rates.The slip vector ε is not a sensed quantity and the wheel-ground contact angles δ are defined here as unknownvalues even though some techniques can be used to es-timate these angles [LS07] or by the installation of forcesensors in wheel axles. Not-sensed quantities of the vec-tor u are x, y, z and ψ. Here, the slip vector ε is modeledas only rotation along its z-axis ζi since it is assumedno wheel slip with nonholonomic constraints. Then rowscorresponding to x and y-axis elements of the slip vectorcan then be removed for the navigation kinematics. Theresulting matrices En, Es, Jn and Js have dimensions24 × 4, 24 × 2, 24 × 8 and 24 × 24 respectively. Thematrix A and B have dimensions 24 × 12 and 24 × 26respectively and χ is a 12 × 1 vector corresponding tothe not-sensed quantities and γ is a 26 × 1 vector corre-sponding to the sensed quantities (i.e., rover joints, wheelrolling and pitch and roll angles). The solution for theEquation 13 is obtained using least-square method wherethe error vector is given as

e = Bγ −Aχ (14)

the solution is based on minimizing the error vector E =eTCe, where C is the constant matrix which could be theweighting matrix block diagonal, but for simplificationhere C ≡ I . The solution to Equation 13 is given by

χ = (ATCA)−1ATCBγ (15)

The desired quantities of Sherpa pose x y z are extractedby taking the first element of the solution vector χ. Theleast-square solution provide an optimal solution by min-imizing the error e in velocity. This solution is applica-ble to dead reckoning methods. A large error representslarger navigation uncertainty, while a small error impliesa more accurate solution.

4.2. Slip Kinematics

The detection of the slip vector ε is important to iden-tify the terrain, correct odometry errors and reduce un-desirable motions as well as detect the called ’slip-sinkage effect’. Similar to the navigation kinematics, theslip kinematics equation can be obtained per each roverleg/wheel. No-sensed values are worked out together atthe right-side of the equation and sensed values are in theleft-side

[In −Jin]

[εiδi

]= [−Is Jis]

[uqqni

]Aiχi = Biγi , i = 1, 2, ...,m (16)

The analyses if the existence of a solution is similar tothe navigation equations. The study the rank of Ai refersthe sensing analysis of the slip equations. There is nosolution if the matrix Ai has not full-rank and thereforethe system is undetermined. The wheel slip rates could befully detected if rank[Ai|Biγ] = rank[Ai] or equivalentto express the residual error equal to zero as

Ai(ATi Ai)

−1ATi − IBiγi = P (Ai)− IBiγi = 0 (17)

where P (Ai) is the projection matrix to the column spaceof the matrix Ai. When P is coincident with the identitythe error is zero and a unique solution is found for the slipvector. For Sherpa rover, it is assumed that rover pose uis know using sensor capabilities as visual odometry orinertial sensors. The wheel contact angle is unknown andthe joint rate angles and wheel rolling rate are known.With this configuration the resulting matrices Is, Jis andJin have dimensions 6× 6, 6× 6 and 6× 4 respectively.The matrix Ai and Bi have dimensions 6× 4 and 6× 12respectively, with not-sensed vector χi of dimension 4×1and sensed vector γi of dimension 12× 1.

5. EXPERIMENTS

Experiments with the real hardware are costly to performand a proof-of-concept is needed as a first approach be-fore testing with the real system. In order to evaluatethe feasibility of the approach, a simulated terrain andthe Sherpa rover in MARS (Machina Arte Robotum Sim-ulans) [RKK09], a simulation and visualization tool fordeveloping control algorithms and designing robots, isused. The simulator consists of a core framework con-taining all main simulation components, a GUI, OpenGLbased visualization and a physics core that is currentlybased on ODE simulation environment. The simulationprovides true position and orientation values, as well asinformation about wheel contact points with the ground.It is not the purpose of this work to accurately model sur-face conditions, the wheel-terrain interaction is based on

simple models. The output of the simulated rover is thesensed quantities as joint-angle vector q, wheel rollingangles q

ni, contact angles δi and ground truth for the posevector U .

5.1. Position Estimate Uncertainty and Results

Determining the uncertainty associated with robot’s in-ternal sensors is especially important since sensed valuesare usually corrupted by measurement errors. The uncer-tainty of the wheel-contact point is estimated with respectto the body frame. The information should be statisticallyrelevant in order to be useful. A common assumption isto work with the first and second order centered momentsof a calculated pdf ρν , which is the expectation value νand the covariance matrix

∑νν (std. deviation is detailed

in Table 1). Each DoF in the leg is modeled as a randomvalue in the direction of the joint rotation which affectsto the final pose uncertainty as a composition of transfor-mations.

The angle uncertainty is then treated as a couple(ν,∑νν). A classical first order approximation of Ja-

cobians described by Pennec et al in [PT97] is used tomodel the resulted noise of a chain of transformationsalong the leg. Therefore, rigid motions and the handlingof uncertainty are done by composition of noise covari-ance matrix instead of additivity. The detailed derivationof the method is not reported here for brevity and can befound in [PT97].

The same noise propagation is used in the dead reckon-ing process. The least-squared solution of the navigationkinematics is the exact solution for the rover velocitiesunder wheel no-slip assumption. The dead reckoning up-date calculation is described as

U(k) = U(k− 1) +∆t

2RW,B(u(k− 1) + u(k)) (18)

where RW,B is a rotation matrix from W -frame to B-frame. It is assumed that the Sherpa motion is ade-quately modeled by constant accelerations since the robotis being actuated by constant force/torque generators ineach sampling period ∆t (the same sampling period asthe dead reckoning process).The dead reckoning integra-tion is erroneous when wheel slip occurs and visual tech-niques could be used to estimate the wheel slip vectordescribed in Equation 17.

Two tests are presented here: a squared path and a ser-pentine path performed on inclined flat terrain by 15.Figure 6 depicts the pose estimator for the navigationkinematics and the dead reckoning method when thesquared path is performed. An uncertainty of 0.12 m in xaxis after 19 meters path is estimated considering internalerrors coming from the encoders. Figure 7 show the pitchand roll angles propagation when initial pose is given andthe angular rates φ θ ψ are not-sensed for a serpentine

Figure 6: Pose estimation of Sherpa while performing asquared path on inclined terrain. The ellipse represent therovers belief at different times.

Figure 7: Estimated pitch and roll angles when perform-ing a serpentine path on inclined terrain.

Figure 8: Wheel track and slip vector estimation for theFront Right wheel on one of the turns of the serpentinepath. Black arrows represent the direction of the esti-mated slip vector.

path. The navigation kinematics is used to estimate thechange in attitude. Estimation of the wheel slip vectorusing Equation 17 is shown in Figure 8 demonstratingthe applicability of the slip kinematics.

6. CONCLUSION

A methodology for the kinematic modeling and the poseestimation problem of a hybrid wheeled-leg robot is pre-sented. Additionally, the wheel slip vector detection bythe slip kinematics is also analyzed. The insight intokinematic modeling takes more importance for localiza-tion in space where methods can not make use of theGlobal Positioning System (GPS). Therefore, a six DoFsolution in a three dimensional space is desirable for posi-tioning during complex maneuvers and long term naviga-tion by means of dead reckoning methods. The followingobjectives have been achieved. (1) Calculation of theleg/wheel forward kinematics and Jacobian matrices forthe Sherpa rover. (2) Equations for the navigation andthe slip kinematics for the Sherpa rover (3) Uncertaintymodeling and propagation along the kinematic model dueto measurement errors and (4) six DoF dead reckoningpropagation. By using this method, all degrees of free-dom are captured. Further analysis on the influence ofeach rover leg joint on the resulted robot pose is desired.Experiments with the real system should be performed toallow a more in-depth analysis in real environments.

ACKNOWLEDGMENTSThe project RIMRES is funded by the German Space Agency(DLR, Grant number: 50RA0904) with federal funds of theFederal Ministry of Economics and Technology (BMWi) in ac-cordance with the parliamentary resolution of the German Par-liament. The research described in this paper was performedwithin the ESA Networking Partnering Initiative (NPI), grantno. 4000103229/11/NL/P.

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