Contact Dynamics of Internally-Actuated Platforms for theExploration of Small Solar System Bodies

Robert G. Reid⇤, Loris Roveda⇤⇤, Issa A. D. Nesnas⇤, Marco Pavone⇤⇤⇤

⇤Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USAe-mail: {rgreid, nesnas}@jpl.nasa.gov

⇤⇤Institute of Industrial Technologies and Automation (ITIA - CNR), Milan, Italye-mail: loris.roveda@itia.cnr.it

⇤⇤⇤Dept. of Aeronautics and Astronautics, Stanford University, Stanford, CA, USAe-mail: pavone@stanford.edu

Abstract

Directed in-situ science and exploration on the sur-face of small Solar System bodies requires controlled mo-bility. In the microgravity environment of small bodiessuch as asteroids, comets or small moons, the low gravi-tational and frictional forces at the surface make typicalwheeled rovers ine↵ective. Through a joint collabora-tion, the Jet Propulsion Laboratory together with StanfordUniversity have been studying microgravity mobility ap-proaches using hopping/tumbling platforms. They havedeveloped an internally-actuated spacecraft/rover hybridplatform, known as “Hedgehog,” that uses flywheels andbrakes to impart mobility. This paper presents a modelof the platform’s mobility, analyzing its three main statesof motion (pivoting, slipping and hopping) and the con-tact dynamics between the platform’s spikes and variousregolith simulants. To experimentally validate the model,an Atwood machine (pulley and counterbalance) was usedto emulate microgravity. Experiments were performedwith a range of torques on both rigid and granular surfaceswhile a high-speed camera tracked the platform’s motion.Using parameters measured during the experiments, theplatform was simulated numerically and its motion com-pared. Within the limits of the experimental setup, themodel is consistent with observations; it indicates the abil-ity to perform controlled forward motions in microgravityon a range of rigid and granular regolith simulants.

1 Introduction

There is increasing interest in exploring small bodiessuch as asteroids, comets, small moons and near-Earth ob-jects for both science and human missions [1, 2]. One ofthe biggest challenges associated with this type of targetis the microgravity environment that makes surface mo-bility very challenging. For the smallest potential targets,the gravitational forces can be as low as 10�6 g [3].

Small forces normal to the surface and loose or low-friction regolith may make it di�cult for wheeled rovers toproduce enough traction, thus restricting them to very lowvelocities [4]. Moreover, the potential exists for wheelsto bind on rocky surfaces, whereby a momentary energybuild-up could cause the rover to leave the surface in anuncontrolled manner (or exceed escape velocity). Severalresearch groups have considered legged mobility as an al-ternative to wheels, with anchoring devices at the tip of itsappendages [5], however such highly articulated systemsare complex, making thermal management challenging.

Other researchers have adopted approaches that usemicrogravity to their advantage, including hopping ortumbling techniques for surface mobility. In 2005, JAXA

Figure 1. : NASA/JPL Hedgehog platform prototypewithout avionics, covers or solar panels. The array of pro-truding spikes provide the contact interface with the sur-face and protect the platform.

attempted to deploy the MINERVA rover onto the surfaceof Asteroid (25143) Itokawa from the Hayabusa space-craft. While the 0.6 kg MINERVA lander didn’t reach thesurface, JAXA had planned to demonstrate controlled mo-bility with a single flywheel mounted on a turntable [6].In late 2014 JAXA is planning to launch Hayabusa 2 toAsteroid (162173) 1999 JU3 carrying both MINERVA 2and DLR’s MASCOT lander [7]. MASCOT is a 13.5 kgbox-shaped lander with an actuated o↵-center mass. Af-ter landing it will use the actuated mass to right itself, andoptionally later for uncontrolled mobility.

Our team has been developing a platform to allowcontrolled mobility in microgravity [8, 9, 10]. The plat-form uses internal actuation (three mutually-orthogonalflywheels) to generate reaction torques that create mobil-ity. The experimental results in this paper were obtainedusing the Jet Propulsion Laboratory’s prototype, shown inFigure 1. Its external array of spikes protect the platformduring motion while providing the primary contact inter-face with the surface.

In previous work using an earlier prototype [8], Allenet al. showed that such platforms could produce tumblingmotions in emulated microgravity (0.024 g). Using an ex-perimental setup similar to the one described in this pa-per they demonstrated planar 3 degree-of-freedom (DOF)motions that they compared to simulations. Validationswere limited to a comparison of the torque levels requiredto initiate a tumbling/hopping motion. They simulatedtheir prototype by applying the torque profiles measuredin their experiments while scaling the profiles to best cor-relate the results (mean scale factor was 106%). In our ex-periments we collect significantly more data, including theplatform’s 6 DOF pose at 120 Hz, and directly comparedsimulated and experimental trajectories. Our comparisonsalso include an analysis of the sensitivities to surface fric-tion coe�cients.

Small bodies may have complex and varying mor-phologies: from rocky or icy surfaces to fine granularregolith. Surface observations of the Asteroid Itokawaare currently the highest quality observations of complexrocky and granular morphologies [11]. The ability of aplatform to move over such surfaces depends on the mag-nitude of the reaction forces between a platform’s contactsand the surface. Further, the ability to direct the motiondepends on having su�cient friction, hence reaction forceto accelerate across the surface.

Koenig et al. analyzed both linear and angular actua-tion of platforms using a Coulomb friction model for sur-face contact [10]. They provide an analysis of forces andthe resulting constraints on motion for a platform that ei-ther pivots over its contact points, or slips, and describe aset of viable hopping angles. In this work, we use numer-ical simulations and high-speed video of experiments toshow the dynamic nature of transitions between pivoting

and slipping motions, including the possibility of control-ling the hopping angle within the viable range describedby Koenig et al. We increase the fidelity of the contactmodel, and approximate compliance in both the surfaceand the platform, by adding a spring-damper componentthat acts normal to the surface [12]. We note that smallbodies are unlikely to be smooth and homogeneous, how-ever the model may be su�cient to analyze mobility onrigid parts of their surface.

For complex or non-rigid surfaces including granularregolith, pebbles or ice, realistic contact models are harderto derive and validate. In microgravity environments, theforces interacting with the regolith are likely to be muchstronger than the local surface gravity field [13]. The mainfactors a↵ecting contact dynamics with granular regolithare i) the compressive strength of the regolith, ii) the dy-namic drag term resulting from the transfer of momentumfrom the platform to the ground (proportional to the granu-lar media density and the square of velocity of the contact)and iii) the sliding friction at the contact interface. DLRdeveloped a “soil” contact model for MASCOT, based onplanetary rover wheel-soil interactions [7]. They used ter-ramechanics theory to evaluate the contact dynamics be-tween MASCOT’s rectangular faces and regolith in mi-crogravity. Contact models for granular regolith remainan open research question.

High-level mission architectures to small Solar Sys-tem bodies have been explored. One such architecture in-cludes a “mother” spacecraft with one or more deployableplatforms [9, 14]. Aside from mobility, there are variousother challenges deploying platforms on small bodies in-cluding soft-landing on the surface. Tardivel and Scheeressimulate trajectories for approaching small bodies [15]and, using high-level stochastic models, they consider apassive “pod” bouncing around on the surface for manyhours until coming to rest.

The closest terrestrial equivalent to the Hedgehog pro-totype is “Cubli”, a 15⇥15⇥15 cm cube with 3 orthogonalflywheels [16]. To operate in the Earth’s gravity, Cublirequires orders of magnitude more torque, which it cre-ates using a braking mechanism. Gajamohan et al. havedemonstrated Cubli hopping onto one corner and balanc-ing in an inverted pendulum configuration. Using precisetorque control they have demonstrated a sequence of con-trolled tumbles and maneuvers while balancing.

In the next section, we give an overview of our plat-form design principles including our current Hedgehogprototype. Section 3 presents our model for the platform’sdynamics including its three main states of motion (pivot-ing, slipping and hopping) and the regolith contact modelduring these states. The experimental setup used to val-idate our model is presented in Section 4, with variousresults and analysis in Section 5 and Section 6.

2 Platform Description

The spacecraft/rover hybrid we describe here is a mo-bility platform that uses internal flywheels to producetumbling or hopping motions. It is supported and pro-tected from the surface by an array of external spikes. Weare considering designs that are about ⇡ 0.5 m in diameterand ⇡ 5 kg in mass. The design is scalable depending onpayload accommodations [9]. Our current Hedgehog pro-totype, shown in Figure 1, has 24 external spikes forminga rhombicuboctahedron and providing an octagonal cross-section of 0.4 m in diameter. The platform is normallysupported by four spikes forming a square base and oc-casionally supported by three spikes forming a less stabletriangular support. We expect the final mass with coversand solar panels to be around 10 kg.

The platform uses no external propulsion. Instead,it is actuated by three mutually-orthogonal flywheels en-closed in the body. Benefits of this design include sim-pler environmental sealing and thermal management [9].Using a linear combination of torques, the three flywheelmotors can produce reaction torques at the surface in anyorientation without additional moving parts. Regardingmobility, the design is expected to be safer and more capa-ble than wheeled rovers. The internal torques can producemomentarily large reaction forces at the surface resultingin increased acceleration across the surface. By dynam-ically controlling the flywheel torques, the platform canperform motions such as small tumbles across the surfaceor large ballistic hops.

3 Dynamic Analysis

This section derives a dynamic model for the plat-form with equations of motion provided for its three mainstates: pivoting, slipping and hopping. We define a con-tact model for the pivoting and slipping states when theplatform’s spikes contact the regolith surface. Figure 3provides a schematic of the dynamic model, while Table1 lists the parameters used throughout this paper.

To simplify the analysis, the platform’s mass distri-bution is assumed to be uniform with its center of masscoinciding with its geometric center. By operating onlya single flywheel, the platform’s motion is restricted to a

Figure 2. : Three main states of motion.

Figure 3. : Platform and contact schematic.

Parameter Definitionq = (

x,y,✓) Platform posem

hh

Platform massJ

hh

Platform inertia matrixl Spike length from the center to the pivot⌧ Torque applied by flywheel

µs

, µd

Static and dynamic friction coe�cientsK Surface sti↵nessC Surface dampingg Gravity at surface (m/s2)

Table 1. : Platform and environment parameters.

plane. It’s initial pose, parameterized by q = (x,y,✓), is⇣

0, lcos ⇡8 ,0⌘

at rest. The regolith surface is assumed to beflat and homogeneous, while contact interfaces with thespikes are modeled as uniform points. The contact equa-tions comprise of a spring-damper component [12] that isnormal to the surface, and a Coulomb friction componentthat acts tangential to the surface.

Figure 2 illustrates a typical hopping motion. Thetorque ⌧, applied to the platform at rest, causes it to pivotover its two leading spikes until the horizontal compo-nent of the contact force becomes greater than the staticfriction force . A period of slipping motion typically fol-lows where the platform rotates forward and the same twospikes are dragged backwards in contact with the surface.Given su�cient energy, the platform will then leave thesurface, hopping forwards in a ballistic trajectory. Theprofile of the torque ⌧ (

t

) determines whether a tumble ora hop is executed. In this analysis, a constant torque isapplied for the duration of any contact with the surface.

Transitions between states are determined by the pres-ence of interactions (contacts) between spikes and the sur-face. If the tip of a spike intersects the surface, the reactionforce on the spike at the contact consists of a normal com-

ponent F

n

and a tangential component F

t

. The Coulombmodel for static friction provides the upper bound on F

t

to maintain a pivoting motion: F

t

< µs

F

n

. Once the plat-form transitions to a slipping motion, the dynamic frictioncoe�cient places a lower bound on F

t

to keep slipping:F

t

> µd

F

n

. If no spikes are intersecting the surface, theplatform is in a ballistic (hopping) trajectory.

We model the platform with a Lagrangian approachwhere q = (

x,y,✓) is a vector describing the platform’spose:

d

dt

@T

@q� @T@q+@D

@q+@V

@q=@�L

@�q(1)

T is the kinetic energy, D is the dissipated energy, V

is the potential energy, L is the virtual work of externalforces and @�L@�q is the Lagrangian component of the exter-nal forces. We derive each of these terms in the followingsections.

3.1 Contact DynamicsContact points are defined during motions where one

or more spikes penetrate the surface. Our Hedgehog plat-form has an octagonal cross-section, where eight spikes oflength l are separated by 45�. The spikes are rigidly con-nected to the platform, and therefore rotate with ✓. Thecontact angle from vertical, �, for each spike, is:

� (✓,n) = ✓+⇡

8+⇡

4n, n = 1, 2 . . .8 (2)

The inset in Figure 3 shows a spike contact point,(x

c

,yc

). The penetration depth for the spike, y

c

, is a func-tion of the platform’s height, y, and spike rotation, �:

y

c

= y� lcos� (3)

Similarly the horizontal position of the contact, x

c

, is:

x

c

= x+ lsin� (4)

For a spike penetrating y

c

into the elastic surface K,the stored potential energy is defined:

V

c

=12

Ky

2c

8 y

c

< 0 (5)

The dissipative energy is defined by a damping term,proportional to the square of the spike’s penetration veloc-ity:

D

c

=12

Cy

c

2 8 y

c

< 0 (6)

Here the damping term is only applied when y

c

< 0, asthe spike is actively penetrating into the surface and dissi-pating energy. During slipping motions each contact pro-duces the friction force F

f

= �sign

(x

c

)µd

F

n

. This createsa virtual work term:

L

c

= F

f

x

c

(7)

3.2 Slipping and Hopping DynamicsHopping is the simplest motion state since no surface

contact exists (y

c

> 0). During hopping, the platform un-dergoes unrestrained 3 DOF ballistic motion, q = (

x,y,✓).During slipping, the platform’s pose is constrained suchthat contact points slide along the surface, however it stillundergoes 3 DOF motion. Thus the same dynamic equa-tions are used for both slipping and hopping. For hoppingmotions the contact-related terms V

c

, D

c

and L

c

are zero.The kinetic energy of the platform is defined:

T =12

m

hh

⇣x

2+ y

2⌘+

12

J

hh

✓2 (8)

The potential energy in the system is defined by agravitational term and the term due to contact V

c

(if any):

V = m

hh

gy+V

c

(9)

If a contact exists, the dissipative energy D = D

c

isgiven in equation 6. The virtual work L that results fromapplying a torque ⌧ on the flywheels is given by ⌧✓, andwhile slipping, the work done against friction, L

c

, fromequation 7:

L = ⌧✓+L

c

(10)

Using equations 2, 3 and 4, the contact-related termsV

c

, D

c

and L

c

can be rearranged in terms of x, y and ✓.Applying the Lagrangian approach, the contact dynamicsare:

m

hh

x = F

f

(11)m

hh

y+m

hh

g+K

⇥y� lcos(✓)

⇤

+C

hy+ lsin(✓)✓

i= 0 (12)

J

hh

✓+ lsin(✓) [K (y� lcos(✓))

+C

⇣y+ lsin(✓)✓

⌘i= ⌧�F

f

cos(✓) (13)

3.3 Pivoting DynamicsDuring a pivoting motion, the platform’s pose q =

(x,y,✓) is constrained to rotate on an arc with a radius l

centered on the contact point. To simplify calculations,the constrained pose is parameterized using 2 DOF: theplatform rotation, ✓, and the vertical penetration of thespike at the contact point, y

c

0, given in equation 3.The kinetic, T , and potential, V , terms are given in

equation 8 and 9. The dissipative energy in the contactD = D

c

is given in equation 6. Di↵erentiating and substi-tuting equations 2 and 3 into T , V and D results in func-tions of y

c

, y

c

, ✓, and ✓. The virtual work L is given bythe torque, ⌧, that the flywheel applies during the pivotingmotion:

L = ⌧✓ (14)

By applying the Lagrangian approach, the 2DOF piv-oting contact dynamics are:

m

hh

hy

c

+ lsin(✓)✓+ lcos(✓)✓2i

+Cy

c

+Ky

c

+m

hh

g = 0 (15)⇣m

hh

l

2+ J

hh

⌘✓+m

hh

lsin(✓)yc

+m

hh

glsin(✓) = ⌧ (16)

4 Experimental Setup and Procedure

Experiments were performed to validate the analyti-cal model presented in Section 3. The setup is shown inFigure 4. An Atwood machine was configured as a grav-ity o✏oading test bed. The platform was secured throughits center of mass by a shaft and a pair of bearings sothat it could rotate freely. Two high-tensile steel wireswere connected to the bearings, looped over two pulleysand connected to a counterbalance suspended on the otherside. The large diameter light-weight pulleys were free torotate around low-friction bearings. With the counterbal-ance o↵setting most of the platform’s mass, experimentswere performed on a range of regolith simulants.

Using the test bed, we were able to simulate gravitylevels on the order of 10�2 g. However, this type of plat-form (described in Section 2) targets surface gravity lev-els several orders smaller (from ⇡ 10�4 g on Mars’ moonPhobos down to 10�6 g on small asteroids). The flywheelmotors in the prototype produce a maximum of 0.22 Nmtorque, which can only produce tumbles in the test bed.To apply larger torques and analyze hopping, the proto-type includes electromechanical brakes that deceleratedthe flywheels to produce ⇡ 6.0 Nm torque.

The total mass of the prototype was 5.3 kg, and withthe matching counterbalance mass, the additional inertialimited mobility further. The additional mass also createdartificial stresses in the chassis impacting braking perfor-mance. To reduce the mass and corresponding chassisstress, only one of the three flywheel and brake assemblieswere installed for the experiments. The motor controllerswere installed in the prototype, however the avionics wereo↵-board to minimize weight. A light-weight 3-wire um-bilical was used to communicate with the platform, sup-ported to prevent disruption to the platform’s dynamicsduring motion. Power was supplied from on-board lithiumpolymer batteries.

The configuration allowed for two translational andone rotational degrees of freedom, providing about 40 mmof free motion in simulated microgravity before tangentialforces from the pendulum distorted the trajectory. Thisprovided enough range of motion to allow the initial tra-jectory to be characterized. Data recorded during experi-ments included the velocity of the flywheel at 600 Hz andthe platform’s rotation rates and accelerations with an in-ertial measuring unit (IMU) at 100 Hz. The platform’s

Figure 4. : Gravity o✏oading test bed: an Atwood ma-chine o↵sets the weight of the Hedgehog platform allow-ing di↵erent gravity levels to be simulated.

pose was tracked in 6 DOF at 120 Hz using an externalhigh-speed video camera and visual fiducial markers [17].The tracking quality was monitored throughout the exper-iments and millimeter pose accuracy was maintained. Thevideo was also used to identify motion state transitions.

Experiments were performed at two simulated grav-ity levels (0.015 g and 0.0085 g). For each gravity, levelthree torques (0.125 Nm, 0.22 Nm and 6.0 Nm) were eval-uated on four di↵erent surfaces. To simulate rocky and icyregolith, a limestone brick and steel plate were used. Tosimulate granular regolith, beds of fine sand and JSC-1ALunar simulant [18] were used.

The experimental procedure included preparationssuch as aligning the pulleys to minimize friction, and thenallowing the platform to come to rest on the surface, sup-ported by four spikes. For the granular regolith exper-iments the media was “flu↵ed” and leveled before eachtest. Granular media properties were measured with asmall conical penetrometer.

4.1 Gravity O✏oading Setup DynamicsDuring dynamic motions the gravity o✏oading setup

causes the counterbalance to move in the vertical (y) di-rection, while the platform’s center of mass moves in thex, y plane. The rigid coupling between the masses, and

the platform’s pendulum-like motion creates a potentiallycomplex dynamic system. While the Lagrangian compo-nents are tractable, by only considering the first 40 mm ofthe platform’s motion, the dynamics are simplified. Withthe o✏oading pulleys 2.2 m above the platform, the arcangle 40

2200 ⇡ 1.0� is small enough to use small-angle ap-proximations.

To compare simulated and experimental results wemodel the counterbalance mass, m

b

, by adding two termsto the Lagrangian equations. The velocity of the counter-balance, y

b

, is approximated by:

y

b

� �y (17)

The kinetic and potential energy terms for the counterbal-ance are hence:

T

b

=12

m

b

y

2b

V

b

= m

b

y

b

g

(18)

4.2 Numerical SimulationsWe performed numerical simulations using the plat-

form’s analytical model, derived in Section 3, and the of-floading setup model from Subsection 4.1. The discrete-time simulation was programmed in Matlab with a fixed1 ms time-step and parameters that were measured exper-imentally. Simulated trajectories appeared most sensitiveto the static and dynamic friction coe�cients, however we

0 5 100

5

10

150.0085g, Steel Surface

y [m

m]

x [mm]

Simulation µ

s = 0.45

Simulation µs = 0.5

Simulation µs = 0.55

0.125 Nm Experiments 1!5

0.22 Nm Experiments 1!5

6.0 Nm Experiments 1!5

Figure 5. : Initial platform trajectories. Both experi-mental and simulated trajectories are shown in simulated0.0085 g. As the torque increases (0.125 Nm, 0.22 Nmand 6.0 Nm) the trajectory angle increases also. Sensitiv-ity to the static friction coe�cients (µ

s

= 0.25) is indicatedby the bounding ±10% simulations.

note these were the hardest to measure accurately. In eachof the simulations we evaluated this sensitivity by sim-ulating ±10% of the friction coe�cients. The dynamicfriction coe�cients were fixed to half of the static value:µ

d

= µs

/2 .

5 ResultsFigure 5 shows the platform’s initial trajectories while

completing motions on the steel surface (µs

= 0.25) witha simulated gravity level of 0.0085 g. Experimental andsimulated trajectories match well. Di↵erences begin toappear in some trajectories where the prototype appearsto push away from the surface faster. This may be causedby vibrations through the platform as the flywheels ac-celerate/decelerate. Spike vibrations at the contact pointcause it to “skip” over the surface and greatly reduce thedynamic friction coe�cient. Pendulum-like e↵ects, due tothe o✏oading setup, are observed over larger trajectories(> 50 mm). As the applied torque increases the trajectoryangle increases also, to a maximum of about 60 degrees.

To further validate our model the state transitions be-tween pivoting, slipping and hopping are plotted in Figure6. Even for a short tumble in 0.0085 g the simulation pre-dicts the platform will break contact with the ground andperform a brief hop. Analyzing the high-speed video con-firms this in 5 of 6 cases. Similarly the majority of thetransition angles in the experimental occur within 5 de-grees of the simulated angles. Figure 6 includes markerswhere the pendulum e↵ect begins to a↵ect motion in theexperiments. State transitions after these markers are notexpected to match well. The 6.0 Nm torque simulationsshow a harmless artifact in the transition to hopping thatis not observed in the experiments.

To demonstrate that the platform can produce for-ward motions on a variety of surfaces, the initial trajec-tory angles, maximum linear and angular velocities wereplotted in Figure 8. Most of the low-torque experiments(0.125 Nm and 0.22 Nm) produced slow, tumbling mo-tions that can be identified by the ⇡ 22.5 degree initial tra-jectory angle. Five experiments exceed this angle, indicat-ing hopping trajectories. Four of these experiments usedthe higher torque from the brakes (6.0 Nm), for whichFigure 6 confirms that the platform begins to slip almostimmediately, which is consistent with the predictions in[10] for high-energy hops. Figure 7 extrapolates the ini-tial hopping trajectory from these experiments, suggest-ing what the unrestrained hopping trajectories could havelooked like. Simulated trajectories agree well with the ex-trapolated experimental data. Four experiments in Fig-ure 8 have negative trajectory angles, corresponding to thebrake being used on granular simulants. In each of theseexperiments the platform made good forward progress,however sank into the simulant slightly due to initial con-ditions.

0 5 10 15 20 25 30 35

Pivoting

Slipping

Hopping

Pivoting

Slipping

Hopping

0.0085g, Steel Surface, 0.125 Nm

! [°]

Experiments

Simulations

0 5 10 15 20 25 30 35

Pivoting

Slipping

Hopping

Pivoting

Slipping

Hopping

0.0085g, Brick Surface, 0.125 Nm

! [°]

Simulations

Experiments

(a) (b)

0 5 10 15 20 25 30 35

Pivoting

Slipping

Hopping

Pivoting

Slipping

Hopping

0.0085g, Steel Surface, 0.22 Nm

! [°]

Experiments

Simulations

0 5 10 15 20 25 30 35

Pivoting

Slipping

Hopping

Pivoting

Slipping

Hopping

0.0085g, Brick Surface, 0.22 Nm

! [°]

Simulations

Experiments

(c) (d)

0 5 10 15 20 25 30 35

Pivoting

Slipping

Hopping

Pivoting

Slipping

Hopping

0.0085g, Steel Surface, 6.0 Nm

! [°]

Simulation µs = 0.225

Simulation µs = 0.25

Simulation µs = 0.275

Experiments 1!5

Pendulum Effect

Experiments

Simulations

0 5 10 15 20 25 30 35

Pivoting

Slipping

Hopping

Pivoting

Slipping

Hopping

0.0085g, Brick Surface, 6.0 Nm

! [°]

Simulation µs = 0.45

Simulation µs = 0.5

Simulation µs = 0.55

Experiments

Pendulum Effect

Simulations

Experiments

(e) (f)

Figure 6. : Motion state transitions during a tumble (a, b, c and d) and hop (e and f) with gravity level 0.0085 g. Left (a,c and e) the steel surface with µ

s

= 0.25 and right (b, d and f) the brick surface with µs

= 0.50. The torques applied toproduce the hops/tumbles are (a and b) 0.125 Nm, (c and d) 0.22 Nm, (e and f) 6.0 Nm. Sensitivity to the static frictioncoe�cients are indicated by the bounding ±10% simulations.

6 ConclusionsWe have presented an analytical model for the Hedge-

hog platform, considering its three main states of motion(pivoting, slipping and hopping) and the contact dynam-ics between the platform’s spikes and rigid surfaces. Ex-periments were performed at two simulated microgravitylevels (0.0085 g and 0.015 g), and with a range of torques(0.125 Nm, 0.22 Nm and 6.0 Nm). Numerical simulationsusing parameters measured in the experiments were com-pared and show close correlation with the experimentalobservations, within the limits of the experimental setup.Experiments on four surfaces (brick, steel, sand and JSC-1A) indicate that the platform can perform forward mo-tions in simulated microgravity. The correlation observedbetween torque and trajectory angles suggests that dynam-ically controlling the torque may enable hopping angles tobe controlled.

One source of discrepancies between the analyticalmodel and experiments results from vibrations inducedby the flywheels. These vibrations disrupt the Coulombfriction at the spike’s contact, e↵ectively decreasing thestatic friction coe�cients. Modeling the spike’s vibra-tion modes may resolve part of this discrepancy. Small ir-regularities in the surfaces, particularly the brick surface,

100 200 300 400 500 600

100

200

300

x [mm]

y [m

m]

0.015g and 0.0085g, Steel Surface, 6.0 Nm

0.015g Experiments 1!50.0085g Experiments 1!5

Simulation µs = 0.45

Simulation µs = 0.5

Simulation µs = 0.55

(a)

100 200 300 400 500 600

100

200

300

x [mm]

y [m

m]

0.015g and 0.0085g, Brick Surface, 6.0 Nm

0.015g Experiments 1!50.0085g Experiments 1!5

Simulation µs = 0.45

Simulation µs = 0.5

Simulation µs = 0.55

(b)

Figure 7. : Extrapolated hopping trajectories on (a) thesteel surface and (b) the brick surface at both gravity levels(0.015 g and 0.0085 g). The parabolic hops have been ex-trapolated from the trajectory angles and velocities mea-sured in the experiments. These hops were not possibledue to the experimental setup.

further amplify this e↵ect, however they are di�cult tomodel. Errors in parameter measurements, slightly o↵-center platform mass (non-uniform mass distribution) andthe assumption of a point contact (l varies with angle � fora spherical spike) may contribute to small discrepanciesthat were observed. Experiments with the granular sim-ulants indicate the platform can make forward progresson lightly-packed regolith, however, multiple tumbles areneeded to provide more realistic initial conditions for suc-cessive tumbles and to confirm that the platform will not

0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm

!40

!20

0

20

40

60

80

[Degre

es]

Trajectory Angle

0.0085g mean value0.0085g tests0.015g0.015g tests

JSC!1ASandSteelBrick

(a)

0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm0

0.05

0.1

0.15

0.2

0.25

[m/s

]

Maximum Magnitude of Absolute Velocity

0.0085g mean value0.0085g tests0.015g0.015g tests

JSC!1ASandSteelBrick

(b)

0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm 0.125 Nm 0.22 Nm 6.0 Nm0

1

2

3

4

5

6

[rad/s

]

Maximum Angular Velocity

0.0085g mean value0.0085g tests0.015g0.015g tests

JSC!1ASandSteelBrick

(c)

Figure 8. : (a) The initial platform trajectory angle,(b) maximum linear velocity magnitude and (c) angu-lar velocity. This figure captures all experiments per-formed: two gravity levels (0.015 g and 0.0085 g) withthree torques (0.125 Nm, 0.22 Nm and 6.0 Nm) evaluatedon four di↵erent surfaces (brick, steel, sand and JSC-1A).

continue to sink. This is not possible with the currentgravity o✏oading setup.

Future work includes addressing deficiencies in thegravity o✏oading setup. Our collaborators at Stanford arecurrently developing an actuated 6 DOF gantry, where thevertical axis uses a voice-coil to increase the fidelity ofgravity o✏oading. Actively “flu�ng” granular simulantswith air fluidization may allow contact dynamics to be ex-plored in microgravity, however designing comprehensivemicrogravity mobility experiments remains a challenge.Another avenue for future work is to simulate both theplatform and granular media with multi-body discrete el-ement methods, however computational requirements andmodel validation remains an open research problem. Inthe near-term our team is planning to further validate theHedgehog platform’s mobility in a campaign of parabolicflights on a zero-g aircraft.

AcknowledgmentsWe would like to thank Kun ho Kim for his inital

analysis of the motion states and their transitions. Thisresearch was carried out at the Jet Propulsion Laboratory,California Institute of Technology, under a Contract withthe National Aeronautics and Space Administration.

References

[1] T. Yoshimitsu, T. Kubota, S. Aakabane, I. Nakatani,T. Adachi, H. Saito, and Y. Kunii, “Autonomous nav-igation and observation on asteroid surface by hop-ping rover MINERVA,” in i-SAIRAS, 2001.

[2] A. F. Cheng, “Near Earth asteroid rendezvous: mis-sion summary,” Asteroids III, pp. 351–366, 2002.

[3] D. Scheeres, S. Broschart, S. Ostro, and L. Ben-ner, “The dynamical environment about Asteroid25143 Itokawa: target of the Hayabusa Mission,” inAIAA/AAS Astrodynamics Specialist Conference and

Exhibit, 2004.

[4] B. H. Wilcox and R. M. Jones, “The MUSES-CNNanorover mission and related technology,” in IEEE

Aerospace Conference Proceedings, vol. 7, 2000.

[5] B. H. Wilcox et al., “ATHLETE: A cargo handlingand manipulation robot for the moon,” Journal of

Field Robotics, vol. 24, no. 5, pp. 421–434, 2007.

[6] T. Yoshimitsu, T. Kubota, and I. Nakatani, “MIN-ERVA rover which became a small artificial solarsatellite,” AIAA/USU Conf. on Small Satellites, 2006.

[7] C. Dietze, F. Herrmann, S. Kuß, C. Lange, M. Schar-ringhausen, L. Witte, T. van Zoest, and H. Yano,“Landing and mobility concept for the small aster-oid lander MASCOT on asteroid 1999 JU3,” in 61st

International Astronautical Congress, 2010.

[8] R. Allen, M. Pavone, C. McQuin, I. A. Nes-nas, J. Castillo, T.-N. Nguyen, and J. Ho↵man,“Internally-actuated rovers for all-access surfacemobility: Theory and experimentation,” in Proc.

IEEE Conf. on Robotics and Automation, 2012.

[9] M. Pavone, J. C. Castillo-Rogez, J. A. Ho↵man,I. A. D. Nesnas, and N. J. Strange, “Space-craft/Rover hybrids for the exploration of small so-lar system bodies,” in Aerospace Conference, 2013

IEEE, 2013.

[10] A. W. Koenig, M. Pavone, J. C. Castillo-Rogez,and I. A. Nesnas, “A Dynamical Characterizationof Internally-Actuated Microgravity Mobility Sys-tems,” in IEEE Conf. on Robotics and Automation,2014.

[11] H. Yano, T. Kubota, H. Miyamoto, T. Okada,D. Scheeres, Y. Takagi, K. Yoshida, M. Abe,S. Abe, O. Barnouin-Jha, et al., “Touchdown of theHayabusa spacecraft at the Muses Sea on Itokawa,”Science, vol. 312, no. 5778, 2006.

[12] G. Gilardi and I. Sharf, “Literature survey of con-tact dynamics modelling,” Mechanism and machine

theory, vol. 37, no. 10, 2002.

[13] B. Quadrelli, H. Mazhar, and D. Negrut, “Modelingand Simulation of Anchoring Processes for SmallBody Exploration,” in AIAA SPACE 2012 Confer-

ence & Exposition, 2012.

[14] J. Castillo-Rogez, M. Pavone, I. Nesnas, and J. Ho↵-man, “Expected science return of spatially-extendedin-situ exploration at small solar system bodies,” in2012 IEEE Aerospace Conference, 2012.

[15] S. Tardivel and D. J. Scheeres, “Ballistic deploymentof science packages on binary asteroids,” Journal of

Guidance, Control, and Dynamics, vol. 36, 2013.

[16] M. Gajamohan, M. Muehlebach, T. Widmer, andR. D’Andrea, “The Cubli: A reaction wheel based3D inverted pendulum,” in European Control Con-

ference (ECC), vol. 2, 2013.

[17] B. Atcheson, F. Heide, and W. Heidrich, “CALTag:High Precision Fiducial Markers for Camera Cali-bration,” in 15th

International Workshop on Vision,

Modeling and Visualization, 2010.

[18] X. Zeng, C. He, H. Oravec, A. Wilkinson, J. Agui,and V. Asnani, “Geotechnical properties of JSC-1Alunar soil simulant,” Journal of Aerospace Engineer-

ing, vol. 23, no. 2, 2010.