modeling and simulation of asteroid capture using …modeling and simulation of asteroid capture...

Proceedings of the ASME 2015 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2015August 2-5, 2015, Boston, USA

DETC2015-47840

MODELING AND SIMULATION OF ASTEROID CAPTURE USING A DEFORMABLEMEMBRANE CAPTURE DEVICE

Havard Fjær Grip, Miguel San Martin, Abhinandan Jain,Bob Balaram, Jonathan Cameron, and Steven Myint

NASA Jet Propulsion Laboratory, California Institute of Technology4800 Oak Grove Dr., Pasadena, California 91109

ABSTRACTThe National Aeronautics and Space Administration have

recently been investigating a mission concept known as the As-teroid Redirect Mission, aimed at collecting a large amount ofasteroid material and transporting it into lunar orbit for inspec-tion by human astronauts. Of the two mission options that havebeen considered, one involves the capture of an entire near-Earthasteroid in the 10-m class by a robotic spacecraft. The space-craft would first make contact with the asteroid through a de-formable membrane, before securing it inside a large flexiblebagging mechanism. In this paper we describe the developmentand implementation of a model designed for simulation of thecapture process, which includes a low-complexity representationof the interaction dynamics.

INTRODUCTIONThe National Aeronautics and Space Administration (NASA)

has recently been investigating a mission concept known as theAsteroid Redirect Mission (ARM), aimed at collecting a largeamount of asteroid material and transporting it into lunar orbitfor inspection by human astronauts. Two options for gatheringasteroid material have been under evaluation: in Option A, anentire near-Earth asteroid in the 10-m class would be captured,whereas in Option B, a boulder would be collected from the sur-face of a larger parent asteroid.1

We focus here on the task of capturing an entire asteroid

1Since initial submission of this paper, Option B has been officially selectedby NASA.

under Option A. This task is challenging due to a number offactors, most of all the significant uncertainty that exists withrespect to the mass, shape, composition, and rotational state ofthe eventual target asteroid. This uncertainty drives the searchfor a robust capture solution in terms of mechanical design andclose-proximity guidance, navigation, and control (GN&C).

A particular difficulty is the complex rotational dynamics ofthe asteroid. Due to the large relaxation time constant of smallspinning bodies [1], the asteroid is expected to tumble rather thanspin about a fixed axis, inevitably resulting in non-trivial dynami-cal interactions as the spacecraft makes contact with the asteroid.The capture system must be designed to handle these interactionswithout giving rise to excessive loads on the spacecraft; of par-ticular concern in this context are the solar array drive assemblies(SADAs), which tolerate only limited bending moments.

Capture SystemAn early study by the Keck Institute for Space Studies at

Caltech [2] concluded that the capture of an asteroid could bestbe achieved by using a flexible bagging mechanism capable offully enveloping the asteroid and securing it to the spacecraft.Such a system would be suitable both for a solid rock and a rub-ble pile, ensuring robustness against uncertainty in asteroid com-position.

Building on this idea, a design team at NASA’s Jet Propul-sion Laboratory has developed a more complete concept aimed atallowing the spacecraft to establish and maintain contact with theasteroid during the bag closure, while keeping the transient dy-namics and the resulting loads on the spacecraft at an acceptable

1 Copyright c© 2015 by ASME

Folded&bag&membrane&Trampoline&membrane&

Sensor&access&&opening&

FIGURE 1. Illustration of the capture mechanism concept. The mech-anism is stowed for cruise and deployed prior to capture by unfoldingsix “petals” supporting the hexagonal trampoline and the capture bag.The mechanism is shown here in a half-deployed state, with the capturebag still folded.

level. In this concept, the spacecraft first makes contact with theasteroid through a well-damped “trampoline,” situated within thecapture bag and realized as a deformable membrane suspendedfrom a finite number of actuated winches, as illustrated in Fig-ure 1. Once initial contact has been made, the spacecraft appliesthrust against the asteroid in order to maintain contact while thetransient dynamics settle out and the capture bag closes. Duringthis process, the trampoline acts as a shock-absorbing cushionbetween the spacecraft and the asteroid.

Topic of This PaperIn this paper we focus on modeling and simulation of the

dynamics of the asteroid capture event, particularly during thefirst few minutes after contact is made through the trampoline.The goal of this effort is to provide a tool for GN&C-centric in-vestigations of the capture problem, including the design of theapproach strategy, gains, and thrust levels, as well as large-scaleexplorations of parameters and initial conditions through Monte-Carlo simulations.

Methods for simulating elastic membranes include finite-element models and spring meshes (see, e.g., [3] and referencestherein). These types of approaches result in high-order mod-els whose computational requirements can be prohibitive for thepurposes outlined above. We therefore choose a simplified ap-proach that ignores the dynamics of the trampoline. At any giventime, the shape of the trampoline is taken as a function only ofthe instantaneous relative pose of the asteroid and the trampolinesuspension points, the implicit assumption being that any inter-nal dynamics in the trampoline evolve and settle on a time scale

much faster than the remaining dynamics of the system.

The shape of the trampoline, as a function of the relativepose of the asteroid and the trampoline suspension points, isestimated based on a convex hull. To compute the forces andtorques arising from the interaction, we associate with a givendeflection of the trampoline a particular potential energy, andwith a given rate of deflection a loss of energy due to viscoelasticdamping, similar in principle to the modeling of a linear spring-damper. Based on the resulting energy balance and the conserva-tion of momentum, we calculate restoring and damping forces,and augment these with friction forces based on a Coulomb fric-tion model distributed across points on the contact surface.

Clearly, the modeling procedure described above involvesa level of abstraction away from the actual mechanical imple-mentation of the trampoline. In particular, we make certain as-sumptions regarding how the trampoline responds to deflectionsin terms of the storage of potential energy and loss of energydue to damping. We note, however, that the mechanization ofthe trampoline, using actively controlled winches, provides con-siderable flexibility in shaping the response of the trampoline tomeet requirements that may be derived through simulation. Con-versely, it is possible to modify the properties of the simulatedtrampoline by applying different profiles for potential energy andviscoelastic damping.

For additional context on the asteroid capture problem, werefer the reader to an earlier paper [4], which discusses prior gen-erations of the capture system concept as well as simulation stud-ies aimed at determining feasibility of the capture problem.

NOMENCLATUREIn our derivations we shall use multiple frames of reference

interchangeably. We use superscripts to indicate the frame ofreference for a particular two- or three-dimensional vector; forexample, xa refers to the vector x decomposed in frame a. Therotation matrix from frame a to frame b is denoted by Rb

a, so thatRb

axa = xb. We denote by ωcab the angular velocity of frame b

with respect to frame a, decomposed in frame c.

For a vector x ∈ R3, we denote by S(x) the skew-symmetricmatrix such that for any y∈R3, S(x)y = x×y. For a scalar x∈R,we define S(x) =

[0 −xx 0

].

TRAMPOLINE MODELIn this section we describe the derivation of the trampoline

model that will be used as part of the asteroid capture simulation.For clarity of presentation, we describe the derivation of a 2Dmodel before extending the same modeling principles to the 3Dcase.


2D ModelWe consider here a two-dimensional trampoline suspended

at two fixed points, together with a rigid body that can comein contact with the trampoline. Our goal is to model the forcesand torques acting on the body as a result of this contact, thusallowing us to simulate the evolution of the state of the bodyover time. We assume that the body shape is described by a two-dimensional polygon, represented by an ordered set of vertices.The body is free to move in two-dimensional space, as describedby the following dynamical system:

pi = Ribvb,

mvb =−mS(ωib)vb + f b,

Rib = Ri

bS(ωib),

Jωib = τ,

where pi ∈R2 is the position of the body center of mass (CM), de-composed in the inertial reference frame i; vb ∈R2 is the velocityof the CM, decomposed in the body-fixed frame b; Ri

b ∈ SO(2)is the rotation matrix from the body-fixed frame to the inertialframe; ωib ∈ R is the angular rate of the body-fixed frame rel-ative to the inertial reference frame; f b ∈ R2 and τ ∈ R are theforces and torques acting on the body; and m ∈ R and J ∈ Rare the mass and inertia of the body. The modeling task at handconsists of determining f b and τ as a function of the state of thesystem. For the purpose of the remaining discussion, we assumethat f b and τ are the result only of interactions with the trampo-line; other forces (such as gravity) can be added as desired.

Trampoline Shape As discussed in the introduction, wedo not model the internal dynamics of the trampoline; instead,we assume that the shape of the trampoline is a function of thepose of the body relative to the trampoline suspension points. Inparticular, to estimate the shape, we take the convex hull of thesuspension points of the trampoline and the vertices of the bodythat are located below the straight line connecting the trampolinesuspension points.2 From the boundary of the convex hull, de-scribed as a finite set of line segments, we remove the segmentconnecting the two suspension points to arrive at our estimate ofthe trampoline shape. Figure 2 illustrates the result of this pro-cess.

Let `=∑nj=1 ` j denote the length of the trampoline, where ` j

denotes the length of a single segment, in order from left to right(see Figure 2). Note that the number of segments n varies withtime. The rate of change in ` can be computed as ˙= ∑

nj=1

˙j =

2If no vertices are located below the line connecting the suspension points,then the body is not in contact with the trampoline and no forces act on the body;the remainder of this derivation assumes contact between the body and the tram-poline.

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FIGURE 2. The figure shows the trampoline shape obtained by takingthe convex hull of the trampoline suspension points and the vertices ofthe rigid body located below the suspension points.

˙1 + ˙n, where we have used the fact that `2, . . . , `n−1 representsegments connecting vertices of the rigid body, whose lengthsdo not change with time.

We are interested in writing ˙ as a function of the state vari-ables of the system. To this end, let pi

sL and pisR denote the

positions of the left and right suspension points, respectively.Furthermore, let rb

L denote the constant vector from the CM ofthe rigid body to the leftmost vertex in contact with the tram-poline. Similarly, let rb

R denote the vector from the CM to therightmost vertex in contact with the trampoline. We can nowwrite `1 = ‖pi +Ri

brbL− pi

sL‖. Hence,

˙1 =(pi +Ri

brbL− pi

sL)ᵀ(pi + Ri

brbL)

`1= ei

Lᵀ(Ri

bvb +RibS(ωib)rb

L)

= eiLᵀ(vi +S(ωib)ri

L),

where eiL is the unit vector pointing from pi

sL to pi +Ribrb

L, andwhere we have used the identity Ri

bS(x) = S(x)Rib for an arbitrary

scalar x. Similarly, we can write ˙n = eiRᵀ(vi +S(ωib)ri

R), whereei

R is the unit vector pointing from pisR to pi +Ri

brbR. We can now

write ˙= eiLᵀ(vi +S(ωib)ri

L)+ eiRᵀ(vi +S(ωib)ri

R).

Restoring and Damping Forces We decompose theforces and torques in the body reference frame as f b = f b

N + f bf

and τ = τN +τ f . The force f bN and the torque τN represent normal

forces and corresponding torques, which arise due to the trampo-line’s resistance to deflection and viscoelastic damping due tochanges in deflection. The force f f and the torque τ f representfriction forces and corresponding torques.

We start by modeling f bN and τN , temporarily letting f b

f = 0and τ f = 0. We assume that the trampoline, when deformed,


responds in a manner similar to a linear spring-damper; in par-ticular,

1. the trampoline has a potential energy equal to 12 k(`− `0)

2,where k > 0 is a constant, and `0 = ‖pi

sR− pisL‖ is the nom-

inal length of the trampoline; and2. the system experiences a loss of energy from viscoelastic

damping equal to −c ˙2, where c > 0 is a damping constant.

Consider now the total kinetic and potential energy of the system:

E =12

k(`− `0)2 +

12

mvbᵀvb +12

Jω2ib.

The rate of change in energy is

E = k(`− `0) ˙+ vbᵀ f bN +ωibτN

= k(`− `0)(

viᵀ(eiL + ei

L)

−riLᵀS(ωib)ei

L− riRᵀS(ωib)ei

R

)+ viᵀ f i

N +ωibτN .

Since the forces on the rigid body are due to reaction at the tram-poline suspension points and act through the trampoline fabric,they can be decomposed as two forces of magnitude FL and FRalong the unit vectors ei

L and eiR: f i

N = f iL + f i

R = FLeiL +FRei

R. Itis easy to confirm that the torque associated with these forces isgiven by

τN = τL + τR =−riLᵀS(1) f i

L− riRᵀS(1) f i

R

=−FLriLᵀS(1)ei

L−FRriRᵀS(1)ei

R.

Expanding the desired expression E =−c ˙2 for the rate of energyloss, we now obtain

k(`− `0)(

viᵀ(eiL + ei

L)− riLᵀS(ωib)ei


R

)+ viᵀ(FLei

L +FReiL)−FLri

LᵀS(ωib)ei

L−FRriRᵀS(ωib)ei

R

=−c ˙(

viᵀ(eiL + ei

L)− riLᵀS(ωib)ei


R

).

It is easy to verify that this equation is solved by FL = FR =−k(`− `0)− c ˙. Hence, the normal force and correspondingtorque on the rigid body are given by

f bN =−

(k(`− `0)+ c ˙)(eb

L + ebN),

τN = (k(`− `0)+ c ˙)(rbLᵀS(1)eb

L + rbRᵀS(1)eb

R).

Friction Force Modeling In addition to the restoringand viscoelastic damping forces, the body will be subject to fric-tion forces as it slides against the trampoline surface. Frictionphenomena are highly complex and difficult to model in general,and more so when a deformable trampoline surface is involved.We base ourselves on the simple but commonly applied Coulombfriction model, in which the friction between two sliding surfacesis modeled as f f = −µFNes, where µ is the coefficient of fric-tion, FN is the normal force between the surfaces, and es is theunit vector pointing in the direction of sliding.

In many cases the system being modeled involves bodiesthat are in contact at several places or over a surface area witha non-uniform friction coefficient or sliding direction. In thesecases, it is common to distribute the total normal force over adistributed set of reference points and to apply the Coulomb fric-tion model separately at each point. We choose a similar strategyhere.

For each vertex of the rigid body that is in contact with thetrampoline (i.e., that is included in the convex hull), we definetwo unit vectors normal to the two line segments joined at thatvertex. The result is a set of unit vectors nb

1, . . . ,nbm, which we

use to distribute the total normal force across the correspondingvertices. To this end, we need to find FN1, . . . ,FNm, such that thefollowing holds:

f bN =

m

∑j=1

FN jnbj , τ =−

m

∑j=1

FN jrbjᵀS(1)nb

j ,

where rbj represents the vector from the CM of the body to the

vertex corresponding to vector nbj .

The above expression gives rise to the generally underdeter-mined set of equations

[nb

1 · · · nbm

−rb1ᵀS(1)nb

1 · · · −rbmᵀS(1)nm

]F1...

Fm

=

[f bN

τN

].

In order to solve the above for the normal forces that are non-negative, a complementarity problem can be posed [5–7] andsubsequently solved to obtain the solution. We instead choosea less robust but faster approach based on the Moore-Penrosepseudoinverse. This approach has the drawback of potentiallygenerating illegal solutions that contain negative normal forces;in our implementation, a check was therefore added to verify thatthe solutions were legal.

Based on the computed normal forces, the Coulomb fric-tion model is now applied twice at each vertex in contact withthe trampoline, once for each of the normal vectors originatingthere. The sliding velocity is in each case computed by projecting


FIGURE 3. The figure shows the trampoline shape obtained by takingthe convex hull of the trampoline suspension points and the vertices ofthe rigid body located below the suspension points.

the velocity of the vertex onto the line segment from which thenormal vector was derived. The torques arising from the frictionforces are computed in the obvious way.

3D ModelWe now extend the modeling technique to the 3D case. We

assume that the body shape is described as a polygon mesh, in theform of a finite set of vertices joined by triangular surfaces. Thedynamics of the body is now described by the following Newton-Euler equations of motion:

pi = Ribvb,

mvb =−mS(ωbib)v

b + f b,

Rib = Ri

bS(ωbib),

Jωbib =−S(ωb

ib)Jωbib + τ

b,

where pi ∈R3 is the position of the body’s CM in the inertial ref-erence frame; vb ∈ R3 is the velocity at the CM with respect toa body-fixed reference frame; Ri

b ∈ SO(3) is the rotation matrixfrom the body-fixed frame to the inertial frame; ωb

ib ∈ R3 is theangular velocity of the body-fixed frame with respect to the in-ertial frame; f b ∈ R3 and τb ∈ R3 are the forces and the torquesacting on the body; m∈R is the mass; and J ∈R3×3 is the inertia.

Trampoline Shape Similar to the 2D case, we estimatethe shape of the trampoline by the convex hull of the trampolinesuspension points and the vertices of the body located below thetrampoline surface. From the boundary of the convex hull, nowdescribed by a set of triangles, we remove those triangles that areformed only from trampoline suspension points, to arrive at ourestimate of the trampoline shape. Figure 3 illustrates the resultof this process.

Let A = ∑nj=1 A j denote the area of the trampoline, where A j

denotes the area of a single triangle forming part of the trampo-line. The area of a single triangle is given by A j =

12‖ρ

ij1×ρ i

j2‖,

where ρ ij1 and ρ i

j2 are vectors corresponding to two of the trian-gle’s three sides.

Observing the fact that triangles joining only vertices of therigid body are of constant area, the rate of change in A can becomputed as A = ∑ j∈I A j, where I ⊂ 1, . . . ,n enumerates thosetriangles for which at least one endpoint corresponds to a tram-poline suspension point.

To calculate A j, j ∈ I , as a function of the state vari-ables of the system, let the vectors ρ i

j1 and ρ ij2 be chosen such

that both originate at trampoline suspension points, located atpi

s j1 and pis j2, respectively (see Figure 4). Furthermore, let rb

j1

and rbj2 denote the respective vectors from the CM to the as-

teroid vertices that form the endpoints of ρ ij1 and ρ i

j2. Thenρ i

j1 = pi +Ribrb

j1− pis j1 and ρ i

j2 = pi +Ribrb

j2− pis j2; moreover,

ρ j1 = Ribvb + Ri

bS(ωbib)r

bj1 = vi + S(ω i

ib)rij1 and ρ j2 = Ri

bvb +

RibS(ωb

ib)rbj2 = vi + S(ω i

ib)rij2, where we have used the identity

RibS(xb) = S(xi)Ri

b for x ∈ R3. Based on this, it is straightfor-ward to calculate

A j =(ρ i

j1×ρ ij2)

ᵀ

4A j

(S(ρ i

j1−ρij2)v

i

+(S(ρ i

j2)S(rij1)−S(ρ i

j1)S(rij2))

ωiib).

Restoring and Damping Forces As in the 2D casewe decompose the forces and torques as f b = f b

N + f bf and

τb = τbN + τb

f , and start by modeling f bN and τb

N . We associatea potential energy with a deflection of the trampoline from thenominal, and an energy loss with a change in deflection; how-ever, the deflection is now described in terms of the trampolinearea. In particular, we assume that

1. the trampoline has a potential energy equal to 12 k(A−A0)

2,where k > 0 is a constant, and A0 is the nominal area of theundeflected trampoline; and

2. the system experiences a loss of energy from viscoelasticdamping equal to −cA2, where c > 0 is a damping constant.

Consider now the total kinetic and potential energy of the system:

E =12

k(A−A0)2 +

12

mvbᵀvb +12

ωbibᵀJω

bib.


Trampoline+suspension+point+

Body+vertex+

Trampoline+suspension+points+

ȡM�ȡM�ȡM� ȡM�

SVM�/SVM� SVM�

SVM�

FIGURE 4. Triangles connecting trampoline suspension points tobody vertices either contain a single suspension point and two body ver-tices (left) or two suspension points and a single body vertex (right).

The rate of change in energy is

E = k(A−A0)A+ vbᵀ f bN +ω

bibᵀτ

bN

= k(A−A0)

(−viᵀ

∑j∈I

14A j

S(ρ ij1−ρ

ij2)(ρ

ij1×ρ

ij2)

+ωiibᵀ

∑j∈I

14A j

(S(ri

j1)S(ρij2)−S(ri

j2)S(ρij1))(ρ i

j1×ρij2)

)+ viᵀ f i

N +ωiibᵀτ

iN .

Since the forces on the rigid body are due to reaction at the tram-poline suspension points and are transmitted through the trampo-line fabric, they can be decomposed as n forces, f i

N = ∑ j∈I f iN j,

one for each triangle connecting the suspension points to therigid body. Each force f i

N j is in turn decomposed as a sum offorces acting along the vectors ρ i

j1 and ρ ij2: f i

N j = f iN j1 + f i

N j2 =

Fj1eij1 + Fj2ei

j2, where eij1 = ρ i

j1/‖ρ ij1‖ and ei

j2 = ρ ij2/‖ρ i

j2‖.The torque arising from these forces is similarly decomposedas τ i

N = ∑ j∈I τ iN j, where τ i

N j = τ iN j1 + τ i

N j2 = Fj1S(rij1)e

ij1 +

Fj2S(rij2)e

ij2.

Expanding the desired expression E = −cA2 for the rate ofenergy loss, we now obtain

k(A−A0)

(−viᵀ

∑j∈I

14A j

S(ρ ij1−ρ

ij2)(ρ

ij1×ρ

ij2)

+ωiibᵀ∑∈I

14A j

(S(ri

j1)S(ρij2)−S(ri

j2)S(ρij1))(ρ i

j1×ρij2)

)+ viᵀ

∑j∈I

(Fj1eij1 +Fj2ei

j2)+ωiibᵀ

∑j∈I

(Fj1S(rij1)e

ij1 +Fj2S(ri

j2)eij2)

=−cA

(−viᵀ

∑j∈I

14A j

S(ρ ij1−ρ

ij2)(ρ

ij1×ρ

ij2)

+ωiibᵀ

∑j∈I

14A j

(S(ri

j1)S(ρij2)−S(ri

j2)S(ρij1))(ρ i

j1×ρij2)

).

Clearly this equation can be solved if we can solve the followingindividual equation for each j ∈ 1, . . . ,n:

viᵀ(Fj1eij1 +Fj2ei

j2)+ωiibᵀ(Fj1S(ri

j1)eij1 +Fj2S(ri

j2)eij2)

=a

4A j

(viᵀS(ρ i

j1−ρij2)(ρ

ij1×ρ

ij2)

−ωiibᵀ (

S(rij1)S(ρ

ij2)−S(ri

j2)S(ρij1))(ρ i

j1×ρij2)),

where a := k(A−A0) + cA. This equation is in turn solved ifwe can find a common solution to the following individual equa-tions:

Fj1eij1 +Fj2ei

j2 =a

4A jS(ρ i

j1−ρij2)(ρ

ij1×ρ

ij2) (1)

and

Fj1S(rij1)e

ij1 +Fj2S(ri

j2)eij2

=− a4A j

(S(ri

j1)S(ρij2) −S(ri

j2)S(ρij1))(ρ i

j1×ρij2). (2)

We will start by solving (1); we will then show that the solutionis also a solution of (2).

Equation (1) can be rewritten as

[ei

j1 eij2][Fj1

Fj2

]=−a

2S(ei

j)(ρij1−ρ

ij2), (3)

where eij = ei

j1×eij2. Pre-multiplying by the non-singular matrix

[eij1,e

ij2,e

ij]ᵀ and noting that ei

jᵀei

j1 = 0, e jᵀei

j2 = 0, eijᵀS(ei

j)= 0,


eij1ᵀei

j1 = 1, and eij2ᵀei

j2 = 1, we obtain the equivalent equation

1 bb 10 0

[Fj1Fj2

]=−a

2

eij1ᵀ

eij2ᵀ

0

S(eij)(ρ

ij1−ρ

ij2),

where b := eij1ᵀei

j2. Eliminating and inverting the resulting left-most matrix yields

[Fj1Fj2

]=− a

2(1−b2)

[1 −b−b 1

][ei

j1ᵀ

eij2ᵀ

]S(ei

j)(ρij1−ρ

ij2)

=− a2(1−b2)

[ei

j1ᵀ(I− ei

j2eij2ᵀ)

eij2ᵀ(I− ei

j1eij1ᵀ)

]S(ei

j)(ρij1−ρ

ij2).

Expanding the expression, this yields the forces

f iN j1 =−

k(A−A0)+ cA2(1− (ei

j1ᵀei

j2)2)

eij1ei

j1ᵀ(I− ei

j2eij2ᵀ)S(ei

j)(ρij1−ρ

ij2),

f iN j2 =−

k(A−A0)+ cA2(1− (ei

j1ᵀei

j2)2)

eij2ei

j2ᵀ(I− ei

j1eij1ᵀ)S(ei

j)(ρij1−ρ

ij2).

Next, we show that this solution also satisfies (2). To thisend, we first note that (2) can be rewritten as

[S(ri

j1)eij1 S(ri

j2)eij2][Fj1

Fj2

]=−a

2(S(ri

j1)S(ρij2)−S(ri

j2)S(ρij1))

eij. (4)

We now consider two distinct possibilities: (i) triangle j includestwo trampoline suspension points; and (ii) triangle j includes twobody vertices (see Figure 4). In the first case, ri

j1 = rij2, and hence

we can write (4) as

S(rij1)[ei

j1 eij2][Fj1

Fj2

]=−S(ri

j1)a2

S(ρ ij2−ρ

ij1)e

ij

=−S(rij1)

a2

S(eij)(ρ

ij1−ρ

ij2).

It is easy to see that this equation is satisfied since (3) is satisfied.In the second case, we can write ri

j1 = sij +ρ i

j1 and rij2 = si

j +ρ ij2,

where sij is the trampoline suspension point included in triangle

j. Noting that S(ρ ij1)e

ij1 = 0 and S(ρ i

j2)eij2 = 0 we can rewrite

(4) as

S(sij)[ei

j1 eij2][Fj1

Fj2

]=−a

2(S(si

j +ρij1)S(ρ

ij2)−S(si

j +ρij2)S(ρ

ij1))

eij

=−S(sij)

a2

S(eij)(ρ

ij1−ρ

ij2)

− a2(S(ρ i

j1)S(ρij2)−S(ρ i

j2)S(ρij1))e

ij.

Considering the factor (S(ρ ij1)S(ρ

ij2)− S(ρ i

j2)S(ρij1))e

ij in the

last term, we can write

(S(ρ ij1)S(ρ

ij2)−S(ρ i

j2)S(ρij1))e

ij

=1

2A j(S(ρ i

j1)S(ρij2)−S(ρ i

j2)S(ρij1))(ρ

ij1×ρ

ij2)

=1

2A j(S(ρ i

j1)S(ρij2)S(ρ

ij1)ρ

ij2−S(ρ i

j2)S(ρij1)

2ρ

ij2)

=1

2A j(−S(ρ i

j1)S(ρij2)

2ρ

ij1−S(ρ i

j2)S(ρij1)

2ρ

ij2) = 0,

where we have used the identity S(x)S(y)2x = −S(y)S(x)2y. Itnow follows that

S(sij)[ei

j1 eij2][Fj1

Fj2

]=−S(si

j)a2

S(eij)(ρ

ij1−ρ

ij2),

which is satisfied since (3) is satisfied.

Friction Force Modeling The modeling of frictionforces is carried out in the same way as for the 2D case. Foreach vertex of the body that is in contact with the trampoline, wedefine a normal vector based on each adjacent triangular surface,and we use the collection of all the normal vectors to distributethe normal force calculated in the previous section. For each ofthese normal forces, Coulomb’s friction model is then applied atthe corresponding vertex, with the sliding direction calculated byprojecting the velocity of the vertex onto the surface from whichthe normal vector was calculated.

Simulation Figure 5 shows a sequence of images froma Matlab simulation in which a 100-kg object with a maxi-mum diameter of approximately 4.5 m falls in Earth gravity andbounces off the trampoline. The hexagonal trampoline has anominal area of approximately 146 m2 and is parameterized withk = 30 N/m3, c = 1.5 Ns/m3, and µ = 0.2. The object has a tri-inertial mass geometry and is initialized in a tumbling rotationalstate.


FIGURE 5. Excerpt from Matlab simulation of an object bouncing off a trampoline in Earth gravity. The object was dropped from a height of 10 mabove the trampoline with an angular velocity of ωb

ib = [0.5,1.0,−0.2]ᵀ rad/s. The images show the period from t = 1.2 s to t = 2.52 s in 0.12 sincrements, in order from top left to top right, continuing from bottom left to bottom right.

ASTEROID CAPTUREThe ARM Option A mission concept, as outlined in the intro-

duction, consists of two elements: an unmanned robotic missionto capture an asteroid and bring it into lunar orbit, and a mannedmission to visit the asteroid in lunar orbit. The spacecraft thatwould be to accomplish the robotic mission, referred to as theAsteroid Retrieval Vehicle (ARV), consists of a hexagonal centralbus approximately 3 m in diameter and 6 m in height and witha mass of approximately 15 t, with the asteroid capture devicemounted at one end (see Figure 6). Two large solar panels areused to power the ARV’s solar-electric propulsion system. Eachsolar panel is carried on a boom mounted on a gimbal mechanismknown as a solar array drive assembly (SADA). Mechanically, theSADAs represent a weak point on the spacecraft, as they can tol-erate only limited bending moments during the asteroid captureevent.

Potential asteroid targets, limited by the dimensions of thecapture bag and the propulsive capacity of the spacecraft, maymeasure up to 13 m in diameter and have a mass of up to 1000 t.While the asteroid is expected to tumble, the instantaneous spinrate of any potential target will be within known limits. In [4]it was shown that a properly designed capture mechanism maybe capable of handling a tumbling asteroid rotating at 2 RPMwithout exceeding an assumed SADA bending moment limit of1765 Nm. However, the margins in this scenario were small forcertain types of rotational patterns. The requirement on the targetspin rate was since reduced to 0.5 RPM.

Approach Strategy As discussed in [4], the transientdynamics of the combined asteroid-spacecraft system after con-tact is to a large extent determined by the strategy chosen forapproaching the asteroid. Due to the possibly complicated rota-tional state and a priori unknown asteroid shape, choosing theoptimal approach strategy is nontrivial. Key considerations in-clude minimizing the relative motion between the spacecraft andthe asteroid at the time of contact; ensuring sufficient thrusterleverage for subsequent despin; keeping the asteroid relativelycentered in the bag during approach; and minimizing transients

2D+Modeling+

3+

Rigid+structure+

Irregularly+shaped+asteroid+

Trampoline+with+damping+and+fric?on+

Constant+thrust+aAer+contact+

FIGURE 6. Illustration of spacecraft with trampoline capture device.Upon initial contact with the asteroid, the spacecraft applies constantthrust to maintain contact until the bag has been closed around the as-teroid.

as the spacecraft adjusts itself to accommodate the asteroid shapeafter contact.

A natural strategy would be to perfectly match the motion ofthe asteroid and make contact at some pre-determined locationwith zero relative motion. However, this strategy is complicatedin terms of GN&C and would consume a substantial amount offuel. Two alternative approach strategies are discussed in [4]; oneinvolves approaching along the angular momentum vector whilespinning at a rate that approximately minimizes the relative mo-tion, whereas the other involves matching the instantaneous an-gular velocity of the asteroid at the moment when contact is made(requiring precise timing and prediction of the asteroid motion).

One of the main goals of the trampoline capture design is tomake the capture process more robust by minimizing sensitivityto the parameters of the approach. It is thought that the flexi-


bility of the trampoline fabric can absorb a significant amountof relative motion without imparting large forces on the space-craft. Meanwhile, the constant thrust applied toward the asteroidis akin to an “artificial gravity,” and should allow the spacecraftto naturally accommodate the shape of the asteroid and settleagainst it in a stable configuration.

Asteroid Capture Simulation We use the trampolinemodel developed in the previous sections as part of an overallmodel for simulation of the dynamics of the asteroid capture pro-cess. In this model, the asteroid is represented as a rigid bodywith configurable inertial properties, and with a geometric shapedescribed by an arbitrary triangle mesh. The spacecraft bus ismodeled as a solid cylinder. The solar panels are modeled assolid cylinders on mass-less booms, connected to the bus via balljoints endowed with suitably parameterized spring-dampers. Thetrampoline is mounted from six suspension points, spaced 60◦

apart, attached to a rigid structure protruding from the spacecraft.Straightforward modifications to the trampoline model are madeto account for the fact that the suspension points are no longerstationary, but instead move with the spacecraft. The spacecraftis subject to forces and torques equal and opposite to the ones im-parted on the asteroid according to the trampoline model derivedabove.

The simulations are based on the Darts/Dshell architecturefor spacecraft dynamics simulation, developed at JPL (see, e.g.,[8]). The simulation runs at speeds faster than realtime on a desk-top computer, and produces concurrent high-quality 3D visual-izations of the capture.

SIMULATION RESULTSFigure 7 shows a sequence of images from an example as-

teroid capture simulation in Darts/Dshell. The target is a 1000-tpear-shaped asteroid with a tri-inertial mass geometry, which isinitialized with a spin rate of 0.5 RPM. The hexagonal trampo-line has a nominal area of approximately 146 m2 and is param-eterized with k = 1.5 N/m3, c = 15 Ns/m3, and µ = 1.0. Thespacecraft approaches at 5 cm/s along the angular momentumvector, while flying in a corkscrew pattern to keep the asteroidgeometrically centered within the bag. Upon contact, the space-craft thrusts against the asteroid with a force of 475 N. The simu-lation is performed using a fourth-order Runge-Kutta integrationmethod with a step size of 0.1 s. The SADA bending momentsremain well within the acceptable limits in the course of the sim-ulation.

Simulations were run with a number of different asteroidsof different mass, shape, spin state, as well as different frictioncoefficients between the asteroid and the trampoline. The resultsshow that the spacecraft tends to settle against the asteroid andcatch up with its rotational motion. As expected, the constant

thrust results in the spacecraft orienting itself toward a stableconfiguration, typically resting on a flatter area of the asteroid.

For high-friction cases (µ close to 1.0), very little slippagebetween the asteroid and the trampoline is observed, and thespacecraft quickly catches up to the motion of the asteroid. Forlow-friction cases (µ close to 0.1), significant amount of slippageis observed, especially for asteroids with a relatively smooth sur-face, and the spacecraft takes longer to catch up to the motion ofthe asteroid.

As contact is maintained over several minutes, an interest-ing phenomenon is observed; namely, that the spacecraft tendsto converge toward a “flat spin,” where the symmetry axis isroughly perpendicular to the axis of rotation of the overall sys-tem. This is unproblematic as long as the thrust level is cho-sen large enough to supply the necessary centripetal accelera-tion, and in fact it is desirable with respect to minimizing fuelconsumption during the subsequent de-spin phase.

The loads on the spacecraft are generally small, and with anappropriately chosen thrust level after initial contact, the SADAbending moments remain well below critical levels.

CONCLUSIONSIn this paper we have presented a modeling technique that

allows for fast simulation of asteroid capture using a capture de-vice similar to a trampoline. As mentioned in the introduction,the technique involves a level of abstraction away from the actualmechanical implementation, in particular by assuming a certainpotential energy and viscoelastic damping profile. Nevertheless,considerable freedom exists in bringing the model and the phys-ical mechanism into alignment, both by modifying the energyprofiles assumed in this paper and by modifying the physical mo-tor profiles at the trampoline suspension points.

Visually, the model produces realistic results; however, val-idation against physical or higher-fidelity models would be nec-essary to confirm that the model captures the dynamics of thespacecraft-asteroid system with sufficient accuracy. Moreover,extensive Monte-Carlo simulations would be needed to buildconfidence in the overall technical solution. We refer readers toa separate paper [9] for a description of one-fifth scale hardware-in-the-loop asteroid capture test bed built at JPL as part of theOption A study.

ACKNOWLEDGMENTThe research described in this paper was carried out at the

Jet Propulsion Laboratory, California Institute of Technology,under a contract with the National Aeronautics and Space Ad-ministration. We would like to thank the members of the capturemechanism design and GN&C teams, with special thanks to BrianWilcox (capture mechanism design lead).


FIGURE 7. Excerpt from an example Darts/Dshell simulation of an asteroid capture. The spacecraft approaches a pear-shaped asteroid at 5 cm/swhile flying a corkscrew pattern to keep the asteroid geometrically centered in the bag. The images show the period from t = 0 s to t = 170 s in 10 sincrements, in order from top left to top right, continuing from middle left to middle right and from bottom left to bottom right.

REFERENCES[1] Burns, J. A., and Safronov, V. S., 1973. “Asteroid nutation

angles”. Monthly Notices of the Royal Astronomical Society,165, pp. 403–411.

[2] J. Brophy, F. Culick, F. Friedman, et al., 2012. Asteroidretrieval feasibility study. Tech. rep., Keck Institute for SpaceStudies, California Institute of Technology, 04.

[3] Van Gelder, A., 1998. “Approximate simulation of elas-tic membranes by triangulated spring meshes”. J. GraphicsTools, 3(2), pp. 21–42.

[4] Grip, H. F., Ono, M., Balaram, J., Cameron, J., Jain, A.,Kuo, C., Myint, S., and Quadrelli, M., 2014. “Modelingand simulation of asteroid retrieval using a flexible capturemechanism”. In Proc. IEEE Aerospace Conf.

[5] Anitescu, M., and Potra, F. A., 1997. “Formulating dynamicmulti-rigid-body contact problems with friction as solvablelinear complementarity problems”. Nonlinear Dynamics,14(3), pp. 231–247.

[6] Trinkle, J. C., 2003. “Formulation of Multibody Dynamicsas Complementarity Problems”. In ASME International De-sign Engineering Technical Conference.

[7] Mylapilli, H., and Jain, A., 2014. “Evaluation of Comple-mentarity Techniques for Minimal Coordinate Contact Dy-namics”. In Proceedings of the ASME 2014 International

Design Engineering Technical Conferences & Computersand Information in Engineering Conference.

[8] Lim, C., and Jain, A., 2009. “Dshell++: A componentbased, reusable space system simulation framework”. InProc. Third IEEE International Conference on Space Mis-sion Challenges for Information Technology.

[9] Wilcox, B., Litwin, T., Carlson, J., Shekels, M., Grip, H. F.,Jain, A., Lim, C., Myint, S., Dunkle, J., Sirota, A., Fuller, C.,and Howe, A. S., 2015. “Testbed for studying the captureof a small, free-flying asteroid in space”. In Proc. AIAASPACE Forum 2015. Submitted.


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