HIGH-FIDELITY SMALL BODY LANDER SIMULATIONS Stefaan Van Wal and Daniel J. Scheeres∗ University of Colorado Boulder Celestial A

HIGH-FIDELITY SMALL BODY LANDER SIMULATIONS

Stefaan Van wal and Daniel J. Scheeres∗ Simon Tardivel

University of Colorado Boulder California Institute of Technology Celestial and Spaceﬂight Mechanics Laboratory Jet Propulsion Laboratory 431 UCB, CO 80309, USA CA 91109, USA

ABSTRACT ders [1]. In the coming decade, the Hayabusa-2, OSIRIS- This paper provides an overview of the elements required for REx, and AIDA missions will continue to expand our knowl- high-fidelity simulations of the motion of landers deployed to edge of small bodies and carry out remote sensing, sample the surfaces of small bodies. Previous work on the motion return, and orbit deflection operations at their respective tar- of spherical landers is expanded to the general case of arbi- gets. Hayabusa-2 also carries the MASCOT and MINERVA trary shapes, which is challenging due to the high restitution lander/rover, which will carry out surface mobility operations of the necessary gravity, shape, and surface rock models. We using momentum exchange mechanisms. This allows them implement established techniques that reduce the numerical to return scientific measurements from multiple sites on their burden of handling shape models by distinguishing between target, asteroid Ryugu [2, 3]. Similar cubesat landers are pro- models used for gravity field evaluations and surface interac- posed to be included on the AIDA mission [4]. tions. Significant attention is given to the implementation of a robust collision model for the non-eccentric collisions of ar- One important aspect in the design of lander/rover opera- bitrary lander shapes. A C++ implementation of this software tions is the release and deployment strategy used to deliver the enables quick verification of deployment and surface mobility craft to the surface of its target. This directly affects the lan- strategies in the presence of uncertainties in the lander release der hardware (e.g. battery) and the inherent risk to the moth- conditions, lander properties, and surface properties. ership. Similarly, simulations of surface mobility operations will provide insight into the dynamics and controllability of Index Terms— Small body, lander, rover, simulation, de- a rover following deployment to the surface. Previous work ployment, collision has showed successful, low-risk deployment to a target along its unstable manifolds [5]. This strategy has been extensively 1. INTRODUCTION verified with numerical simulations for spherical landers, using a statistical model of the surface rock distribution [6, 7]. The small bodies of our Solar System have matured into com- While this provides a tractable approximation to the motion of mon targets for large space missions. Specifically, the mis- landers with arbitrary shapes and establishes important sim- sions to these asteroids, comets, and small moons aim to ad- ulation techniques, we must expand these simulations to be dress three distinct goals: first, the pristine condition of most able to handle such shapes and generate accurate rock distri- small bodies provides insight into the early conditions of the butions. Solar System, shedding light on its formation. Secondly, missions to small bodies provide a way of validating planetary In this work, we present an overview of the elements re- defense strategies to deflect hazardous near-Earth objects. Fi- quired to carry out high-fidelity simulations of these small nally, the analysis of small body (sub-)surfaces provides in- body landers with arbitrary shapes. In Sec. 2, we briefly re- sight into the feasibility of in-situ resource utilization tech- view the equations that express the lander motion. This mo- niques. tion is governed by gravitational forces, of which the evalua- Over the past two decades, the NEAR-Shoemaker, Haya- tion is detailed in Sec. 3, along with techniques to make this busa, and Rosetta spacecraft have established a core under- evaluation computationally feasible. Similarly, we present standing of the origins, characteristics, and dynamics of small techniques to convert the high-resolution surface shape and bodies. These mission mainly performed remote sensing op- rock models into tractable subsets that allow for collision de- erations about their respective targets; Rosetta’s Philae lan- tection in Sec. 4. Sec. 5 then reviews a robust strategy for der further demonstrated the feasibility of small body lan- computing the impulsive forces that act in these collisions. The applications of these high-fidelity simulations are dis- ∗This research was supported by NASAs SSERVI program (Institute for the Science of Exploration Targets) through institute grant number cussed in Sec. 6, and Sec. 7 finally provides some conclusions NNA14AB03A. and future work directions for this research.   2. FRAMEWORK Q0 −Q1 −Q2 −Q3 dQ 1 Q1 Q0 −Q3 Q2  0 =   (5) Simulations of the motion of a lander in the neighborhood of dt 2 Q2 Q3 Q0 −Q1 Ω a targt small body first require set-up of the proper reference Q3 −Q2 Q1 Q0 frames and equations of motions. For simplicity, this is done dΩ here for unitary bodies, though the expressions given are eas- = 0 (6) ily expanded to binary systems. dt where [I] is the inertia matrix of the lander, and where the 2.1. Reference Frames target body is assumed to be in uniform rotation, as implied by Eq. 5. In Eq. 2, g is the gravitational attraction of the target In order to set up the state variables and equations of mo- body, and aE is the external acceleration, which may include tion of the lander-target system, we first define three reference sources such as solar radiation pressure and accelerations dur- frames: ing continued contact with the surface. Here, we set aE = 0, i.e. we ignore contact motion of a lander on the surface of its N 1. The inertial frame , with origin at the target center of target, and only consider impulsive forces that occur during mass and axes fixed inertially. collisions. 2. The rotating target frame T , with origin at the target These equations of motion are propagated numerically us- center of mass and axes fixed to the target. ing a fifth-order Runge-Kutta method, with a fourth-order lo- 3. The rotating lander frame L, with origin at the lander cal truncation error estimate, i.e. the RK5(4) method. The center of mass and axes aligned with the lander princi- integrator is further equipped with event detection capability pal inertia axes. in order to detect collisions between the lander and its target body. We implement this integrator in C++ to enable fast We may now define the state variables that describe the simulations of a large number of trajectories. dynamical state of the spacecraft-target system. As surface interactions between the two bodies are computed relative to the target surface, the spacecraft motion is expressed primar- 3. GRAVITY ily in the T -frame. The system state consists of: As the time required for a lander to come to rest on the small 1. The spacecraft position r, expressed in the T -frame. body surface following release from a mothership is on the 2. The spacecraft linear velocity r˙, expressed in the T - order of hours, the gravitational attraction of the target body frame. is the dominant force on the lander. We therefore ignoring 3. The spacecraft attitude quaternion q, expressed in the perturbing forces such as solar radiation pressure. In this sec- T -frame. tion, we review our modeling of the small body gravitational 4. The spacecraft angular velocity ω, expressed in the field, and present techniques to mitigate numerical challenges L-frame. inherent in these models. These techniques were developed 5. The target attitude quaternion Q, expressed in the N - with the specific purpose of lander/rover simulations by [6]. frame. 6. The target angular velocity Ω, expressed in the T - 3.1. Polyhedron Model frame. The complex and irregular gravitational field of the target small body is implemented using the constant-density polyhe- 2.2. Equations of Motion dron model [8]. This model represents the small body shape The equations of motion of the state elements, in their respec- through a collection of vertices Pi, edges Eij, and facets Fijk. tive reference frames, are equal to: In order to evaluate the gravitational field of such a polyhe- ˆ dron, we first compute the outward-pointing normals NFijk dr ˆ = r˙ (1) of each facet and NEij of each edge with a ray-tracing tech- dt nique. Using these normals, we compute the dyad of every dr˙ edge and facet, as: = g − Ω × Ω × r − 2Ω × r˙ + a (2) dt E   E = Nˆ Nˆ T + Nˆ Nˆ T (7) q0 −q1 −q2 −q3 Eij Fijk Eij Fijl Eji dq 1 q q −q q 0 =  1 0 3 2  ˆ ˆ T   (3) FFijk = NFijk NF (8) dt 2 q2 q3 q0 −q1 ω ijk q3 −q2 q1 q0 As the normals and dyads are invariant, they are pre- dω computed prior to simulations in order to reduce the compu- = −[I]−1 (ω × [I]ω) (4) dt tational overhead. When evaluating the gravitational field at Fig. 1. Illustration of (left) a high-resolution shape model and (right) a low-resolution gravity model of asteroid Itokawa.

some position r, we first compute the per-edge factor LEij 3.2. Linearization and per-facet factor ωFijk of all edges and facets of the target polyhedron: The first technique applies a linearization of the gravity field. When the lander velocity or integration step size are small, ri + rj + eij the lander moves only a small distance between successive LEij = ln (9) ri + rj − eij integration steps. As a result, we may approximate the gravitational field at some r using a previous evaluation at some ωFijk = 2 arctan ($) (10) r0, provided that the distance ∆r = kr − r0k is small. When with: this is true, the gravitational potential and attraction at r are r · (r × r ) $ = i j k (11) approximately equal to: rirjrk + ri (rj · rk) + rj (rk · ri) + rk (ri · rj) U(r) ' U(r0) + (r − r0) · g(r0) where ri is the vector from vertex Pi to the position r, (15) i.e. ri = r − Pi, and ri = krik. Using this, we may finally + (r − r0) · Γ(r0) · (r − r0) evaluate the gravitational potential U, gravitational attraction g, and gravity gradient matrix Γ at the position r as [8]: g(r) ' g(r0) + (r − r0) · Γ(r0) (16)

1 X Through the selection of some ∆rmax, we control the al- U(r) = Gρ rEij · EEij · rEij · LEij lowable error of this gravity field linearization. In verification 2 Eij tests, we find that setting ∆r to 1/100th of the mean vol- (12) max 1 X umetric radius of a target body yields a relative error in the − Gρ r · F · r · ω 2 Fijk Fijk Fijk Fijk gravitational acceleration on the order of 10−5. Fijk X g(r) = −Gρ E · r · L Eij Eij Eij 3.3. Gravity Model Resolution Eij (13) X The second technique involves the reduction of the shape +Gρ FF · rF · ωF ijk ijk ijk model used for gravity field evaluations. Indeed, the gravity Fijk X X field of a high-resolution model can be approximated by a Γ(r) = Gρ EEij · LEij − Gρ FFijk · ωFijk (14) reduced-resolution model of the same shape. In order to con- Eij Fijk struct these models, we first triangulate the largest ellipsoid

where rEij and rFijk are the vectors from the position r that fits entirely within the original shape model. The vertices to respectively any vertex of edge Eij and facet Fijk. High- of this ellipsoid are then projected outwards until they inter- resolution small body shape models consist of several hun- sect the original shape model; these intersection points define dred thousand facets; evaluations of Eqs. 12 through 14 for the vertices of the reduced model. This model is then trans- a these models are therefore computationally expensive and lated such that its center of mass matches that of the original significantly increase the runtime of simulations. Therefore, model. Finally, as the volume of the new model differs from we implement two techniques that were developed by [6] and that of the original model, we adjust its density such that its shown to be highly effective at reducing this computational total mass matches that of the original model. In Fig. 1, we load. For a more detailed discussion, the reader is referred to illustrate a high-resolution shape model and its corresponding this work. reduced gravity model. As we can see, the reduced model maintains the general shape of the body, but has lost small local variations. By controlling the number of vertices in the initial ellipsoid triangulation, we control the resolution of the final shape model and the corresponding gravity field error. In verification tests, we find that 162-vertex models yield a surface gravity error of approximately 1% for relatively smooth small body shapes. For more irregular models, a 642-vertex model yields the same error. Using these two techniques, the computational cost of gravity field evaluations is significatly reduced, while maintaining control over the associated error. Fig. 2. Illustration of the grids that comprise (left) the Z-atlas 4. SURFACE and (right) the X-atlas [7].

Lander/rover spacecraft operating on the surface of a small body dissipate energy due to interactions with the surface, pri- correct surface features are active when the lander closely ap- marily through collisions. As the shape and orientation of the proaches the target surface, and enables collision detection local surface strongly affects these interactions, it is necessary with a high-resolution surface at low computational costs, as to consider high-resolution shape models in computations of we only ever perform distance computations to a small subset the surface interaction; simplified models may not accurately of the total number of surface features. capture local topography that may serve as basins of attraction or rejection for objects moving on/over the surface. Addition- 4.2. Rock Distribution ally, it has been shown that rocks and boulders on the small body surface play an important role in the dissipation process, This division of the small body surface into local worlds can both from an energetic and a topographic point of view [6, 7]. be further exploited when generating rocks on the small body A model of the surface rock distribution must therefore be surface. The inclusion of such rock distributions is not a novel included in addition to a high-resolution shape model. technique and has been shown in previous work to be an im- Once again, this introduces issues with regard to compu- portant factor in lander dissipation. In this work, the presence tational burden, which are most significant when performing of rocks is accounted for through a stochastic model that im- collision detection between a lander and the small body sur- poses a degree of randomness of collisions between a lander face, as this detection requires distance computations between and the small body surface [6]. While this strategy is nu- all features (vertices, edge, and facets) of both the lander and merically very effective and performs well at modeling the the small body. Fortunately, the burden can be significantly first few impacts of a lander trajectory, it has a number of reduced using three techniques: the division of the ‘global’ drawbacks. Specifically, a stochastic model limits the surface small body surface into a number of smaller, ‘local worlds’, slopes at which a lander can come to rest, and cannot produce the procedural generation of rocks on these local worlds, and repeatable collisions of a lander at the same surface location. the use of bounding spheres. These limitations are most prominent when the range of motion of the lander, i.e. its velocity, is small. In order to re- move these limitations, we must include a ‘full’ rock model 4.1. Atlas in which rocks are shaped and placed on the surface, and con- Prior to performing simulations, we follow the technique of sidered as surface features in collision detection. Depending [6] and construct the atlas of the target small body. This is on the minimum considered radius of a rock, the total num- done by creating a latitude-longitude grid, where individual ber of rocks on a small body may well exceed the millions, cells contain those target surface features that fall within the making it infeasible to both generate and store these rocks in cells’ latitude-longitude span. In order to establish a consis- memory. tent grid size across the surface, we create one atlas relative Instead, a single local world will contain only a manage- to the rotation axis of the target (‘Z-atlas’), and a second atlas able few hundred rocks; the exact number follows directly relative to an axis in its equatorial plane (‘X-atlas’), as illus- from the surface area of the local world and the statistical dis- trated in Fig. 2. Each of the two atlases spans ±45◦ of lati- tribution of the rocks. Similarly, we can invert the rock size tude, thus covering the entire small body surface with similar distribution and apply a random number generator to shape cell sizes. and place the rocks on the local world. By controlling the When propagating the lander motion, the simulation con- seed of the random number generator, we ensure that this pro- tinuously updates the ‘active’ local world based on the cur- cedural generation is consistent, i.e. that the same rocks are rent lander latitude and longitude. This technique ensures the always created on the same facet. This holds for any arbitrary Fig. 3. Illustration of a small body surface with an active local world marked in red, with procedurally generated rocks. rock model, as long as the rocks on a given facet are generated the minimum distance between the lander and the active lo- independently of the geometry surrounding that facet. Fig. 3 cal world, and examines whether the surface penetration con- illustrates an arbitrary active world with generated rocks. The straint has been violated. If it has, the integrator converges on visualization software used to generate this and subsequent the instant just before the collision, and then handles the col- figures was developed in [6] and expanded in [7]. In this par- lision event before continuing with a successive integration ticular model shown, rocks are created as regular icosahedra arc. with a superimposed random variation on their vertices, that Here, we review the methodology applied to handle these are then partially sunk into the surface. This method of creat- collisions. Although simple algebraic collision laws can ro- ing and shaping rocks is consistent with those used to generate bustly capture the impulsive forces present in collisions of the a priori stochastic model in [6], but has not been applied a spherical body, these laws cannot be applied in the non- for procedural generation before. eccentric collisions of a body with an abritrary shape where the contact point, body center of mass, and normal force, are 4.3. Bounding Spheres not aligned. When this is the case, algebraic laws often result in energy increases or spin reversals in such collisions The efficiency of collision detection can be increased even [9]. Instead, we follow the approach of [10], where a numer- further through the use of bounding spheres. Specifically, we ical integration is performed over the span of a collision to define bounding spheres around each local world, as well as correctly compute the impulsive forces. around the lander, each sphere encompassing its entire con- tents. Following this definition, collisions between the lander and the active local world are possible only when their bound- 5.1. Geometry ing spheres intersect. Therefore, it is sufficient to compute the Using the converged, pre-collision lander state, we define the distance between the centers of the bounding spheres in most collision frame C, with the origin at the contact point H be- cases; the ‘full’ distance computation between all features of tween the lander and the small body surface, and orthonormal the lander and the local world is only carried out when the axes {ˆt1, ˆt2, nˆ}. The nˆ axis is normal to the tangent plane of bounding spheres are found to intersect or contain one an- collision; the two tangential axes ˆt1 and ˆt2 (arbitrarily) span other. This process is illustrated in Fig. 4. In a similar process, this plane. We further define the collision vector c = H − r. we create bounding spheres for the surface rocks to enable ef- In order to handle the collision, the quantities r˙, ω, and c are ficient collision detection between an arbitrarily-shaped lan- transformed into the C-frame. Assuming that the rigid body der and several hundred rocks. and surface are mutually impenetrable at the point of contact, the lander will be subject to a contact impulse P, which af- 5. COLLISIONS fects its linear and angular velocity as:

As discussed before, we propagate the motion of a lander in Mdr˙ = dP (17) the small body environment using an integrator with event capability. Indeed, at every time step, the integrator computes [I]dω = c × dP (18) Fig. 4. Illustration of the lander-target minimum-distance computation when the bounding spheres (left) do not intersect and (right) do intersect.

where M is the mass of the lander and [I] is its inertia 5.2. Normal Force matrix, expressed in the C-frame. The contact point velocity v is determined by both the linear and angular velocity of the During the collision, the lander is subject to a normal impulse lander: that enforces the non-penetration constraint. Assuming that the contact region is infinitesimally small and nondeformable, it can be shown that the normal impulse must always be com- v = r˙ + ω × c (19) pressive and monotonously increases during the (infinitesimally short) contact period. As such, we may resolve the which is again expressed in the C-frame, such that the collision in terms of the normal impulse p, by using it as a third component of v represents the normal contact point ve- time-like variable for integration. It is noted that, due to the locity, v which must always be negative at the start of a col- 3 definition of the collision frame, the normal impulse p = P . lision. We will make use of the lander inverse inertia matrix, 3 A collision consists of two phases: compression and resti- m∗, a symmetric matrix whose elements are defined as: tution. During compression, the normal impulse reduces the normal contact point velocity v3 to zero. In this phase, some 3 3 3 3 of the lander’s energy is removed by the work Wc done by the ∗ −1 X X X X −1 normal force, and stored in the lander. The resitution phase mij = M δij + εikmεjlnIkl cmcn (20) k=1 l=1 m=1 n=1 begins when the contact point reaches v3 = 0, during which the normal force continues to act and returns part of the stored where δij is the Kronecker delta, εijk is the permutation energy through the work Wr. The ‘duration’ of a collision in −1 tensor, and Ikl are the elements of the inverse of the inertia terms of the total normal impulse provided is determined by matrix1 [I]. Using this definition, we find for the elements of the energetic coefficient of restitution, 0 ≤ e ≤ 1, which is m∗: defined as:

2 Wr  ∗ −1 2 −1 2 −1 −1 e = (22) m11 = M + c3I + c2I − 2c2c3I W  22 33 23 c  ∗ −1 2 −1 −1 −1 m12 = c2c3I13 − c3I12 + c1c3I23 − c1c2I33  ∗ −1 2 −1 −1 −1 We note that this definition is different from the more m13 = c2c3I12 − c2I13 − c1c3I22 + c1c2I23 ∗ −1 2 −1 2 −1 −1 (21) commonly used kinetic coefficient of restitution, which is de- m22 = M + c3I11 + c1I33 − 2c1c3I13  fined in terms of the contact point normal velocity. m∗ = c c I−1 − c c I−1 − c2I−1 + c c I−1  23 1 3 12 2 3 11 1 23 1 2 13 When resolving a collision, we thus first propagate the  ∗ −1 2 −1 2 −1 −1 m33 = M + c2I11 + c1I22 − 2c1c2I12 respective equations of motion until v3 = 0 and compute the work done by the normal force during compression, as the The remaining elements are found by symmetry, i.e. area under the v3(p) curve. Using the energetic coefficient ∗ ∗ ∗ ∗ ∗ ∗ m21 = m12, m31 = m13, and m32 = m23. of restitution, we can then determine the work done by the normal force during restitution, which determines when the 1Not to be confused with the inverse of the elements of the inertia matrix. collision will terminate. 5.3. Presence of Sliding sufficiently large, slip may halt during either compression or restitution, i.e. s → 0. In addition to the normal force, a lander will also be subject When this happens, the contact point will either stick for to a Coulomb friction force that attempts to drive the tangen- the remainder of the collision, or immediately resume slip in tial contact point velocity to zero. In order to determine the some direction φˆ. To determine which of the two cases oc- magnitude of this friction force and related impulse, we make curs, we compute the critical coefficient of friction, µ¯, which use of the slip velocity, s: provides the minimum value of µ required for permanent q stick: 2 2 s = v1 + v2 (23) pα2 + β2 where v1 and v2 are the tangential contact point velocity µ¯ = (28) components in respectively the directions of ˆt1 and ˆt2. When γ s 6= 0, we say that the contact point is in slip; when s = 0, we where: say that it is in stick. When slip occurs, the direction of slip  ∗ ∗ ∗ ∗ α = m11m23 − m12m13 can be defined by the angle φ measured in the tangent plane,  β = m∗ m∗ − m∗ m∗ (29) ˆt 22 13 12 23 from 1, as:  ∗ ∗ ∗ ∗ γ = m11m22 − m12m12 v φ = arctan 2 (24) v1 5.4.2. Continued stick where, numerically, we make use of the atan2 function If µ ≥ µ¯, the contact point will remain in stick under the to determine the proper quadrant of φ. Respectively, the two influence of a friction impulse µ¯ ·dp in a direction φ¯−π, such tangential velocities can be expressed in terms of the slip ve- that the contact point velocity equations of motion become: locity and angle as:

v1 = s cos φ  dv (25) 1 = 0  dp v2 = s sin φ dv2 dv : dp = 0 (30)  dv3 = −µm∗ cos φ¯ − µm∗ sin φ¯ + m∗ 5.4. Collision Equations of Motion dp 31 32 33 The state of the lander during collision is described by the where the angle φ¯ can be computed as: contact impulse P, the contact point velocity v, and the lander velocities r˙ and ω. The latter two can be integrated following α φ¯ = arctan (31) Eqs. 17 and 18; the equations of motion that govern P and β v must be selected depending on the slip/stick state of the which is only a function of the collision geometry and the contact point, and are described here. mass properties of the rigid body. Correspondingly, the impulse equations become: 5.4.1. Slip  dP1 =µ ¯ cos (φ¯ − π) When the contact point is in slip, the contact impulse can be  dp dP2 ¯ propagated with the following equations of motion: dP : dp =µ ¯ sin (φ − π) (32)  dP3 = 1  dP1 = −µ cos φ = −√µv1 dp  dp v2+v2  1 2 dP : dP2 = −µ sin φ = −√µv2 (26) dp v2+v2 5.4.3. Slip reversal  1 2  dP3 dp = 1 If instead µ > µ¯, slip will resume immediately after reaching where 0 ≤ µ ≤ 1 is the coefficient of friction. Corre- the s = 0 point. In this second phase of slip, the direction ˆ spondingly, the contact point velocity varies as: of slip φ will be constant, and can be found as the root of h(µ, φ) = 0 for which g(µ, φ) > 0. The two functions are defined as:  dv1 ∗ ∗ ∗ dp = −µm11 cos φ − µm12 sin φ + m13  ∗ ∗ ∗ 2 dv2 ∗ ∗ ∗ g(µ, φ) = m13cφ + m23sφ − µm11c φ dv : dp = −µm21 cos φ − µm22 sin φ + m23 (27) ∗ ∗ 2 (33)  dv3 ∗ ∗ ∗ −2µm sφcφ − µm s φ dp = −µm31 cos φ − µm32 sin φ + m33 12 22 ∗ ∗ Eqs. 26 and 27 must be integrated simultaneously to cor- h(µ, φ) = −m13sφ + m23cφ ∗ ∗ ∗ 2 2 (34) rectly resolve the collision. When the coefficient of friction is +µ (m11 − m22) sφcφ + µm12 s φ − c φ where sφ = sin φ and cφ = cos φ. If it has been de- results at low computational effort, when this methodology is termined that slip reversal occurs, the collision can be prop- implemented numerically in C++. agated using the equations of motion for slip, as given by While the use of τ as a time-like variable is highly effec- Eqs. 26 and 27, by simply substituting φ = φˆ. tive in increasing the accuracy with which the point of stick is reached, it can no longer be used once the contact point is in 5.5. Numerical Issues stick. This is easily understood by inspecting Eq. 35, which becomes undefined when s = 0, as in that case dp/dτ = 0 Inspection of Eq. 26 reveals a singularity that occurs when and dτ/dp. As such, we must continue integration of the col- s → 0, that is, when the contact point approaches stick. In- lision with p as time-like variable, once stick has indeed been deed, when s → 0, also v1 → 0 and v2 → 0, such that the achieved. This requires changing back from polar to Carte- quantities φ, dP1/dp, and dP2/dp become ill-defined. A nu- sian coordinates. merical integrator will therefore be unable to properly reach Using this model, we can robustly handle collisions be- the point of stick where s = 0. tween a lander with an arbitrary shape and the surface of a To resolve this issue, we use the method by [11] and intro- small body. While collisions make up the bulk of surface in- duce the ’stretching’ variable τ, which is related to the normal teractions between a lander and a small body, it is also possi- impulse through: ble for the lander to be in continuous contact with the surface, which occurs when the normal contact point velocity is zero. µ µ dp s This regime of motion has not yet been implemented, in other dτ = dp = dp ⇒ = (35) p 2 2 s τ µ words, simulations are terminated when the normal contact v1 + vx d velocity drops below some v3,min following a collision. It is This blow-up transformation effectively rescales the nor- noted that for arbitrary lander shapes, this regime only occurs mal impulse such that the singularity at s → 0 is shifted to for a very brief time at the very end of a deployment sequency, τ → ∞, allowing for a stable integration. To avoid further and therefore does not notably affect the final settling position numerical issues that occur with the computation of s from and time of a lander. v1 and v2, we also perform a change of variables from the Cartesian (v1, v2) to the polar (s, φ) coordinates. The equa- 6. RESULTS tions of motion for the contact point velocity thus become: By combining all of the elements described in the previous  sections, we obtain the capability of performing simulations  ds = s −m∗ c2φ − (m∗ + m∗ )cφsφ  dτ 11 12 21 of lander/rover spacecraft in the small body environment and  m∗ m∗  −m∗ s2φ + 13 cφ − 23 sφ account for all its intricacies. This simulation capability has  12 µ µ dφ ∗ 2 ∗ 2 ∗ ∗ (36) a wide range of applications, the most important of which is dτ = −m21c φ + m12s φ + (m11 − m22)cφsφ  m∗ m∗ the verification of a release strategy.  + 23 cφ − 13 sφ  µ µ Due to the challenges in spacecraft navigation around  m∗  dv3 dv3 dp ∗ ∗ 33 small bodies, as well as variations in any release mechanisms,  dτ = dp · dτ = s −m31cφ − m32sφ + µ the position and velocity of a lander/rover at the moment of The corresponding impulse equations, using τ as time- release from its mothership is inherently uncertain. As our like variable, are found with the chain rule as: simulation software is capable of generating large numbers of simulations at moderate computational cost, it is possible to  dp dp dp 1 = 1 · = −s cos φ carry our sets of Monte Carlo simulations in which the release  dτ dp dτ dp2 = dp2 · dp = −s sin φ (37) conditions are randomly distributed in agreement with the ex- dτ dp dτ pected uncertainties. Analysis of the resulting deployment  dp3 = dp3 · dp = s dτ dp dτ µ trajectories will then reveal whether the selected strategy (e.g. While this technique was introduced by [11] in order to nominal release altitude and velocity) results in successful study the behavior of colliding objects near the point of stick, deployments. Additionally, we can obtain statistics about tra- we find that it is highly efficient at dealing with the singular- jectory characteristics such as the duration of deployment, the ity at s → 0. As it is impossible to integrate until τ → ∞, reachable landing zone, and the magnitude of the impulsive we must select some limiting value at which to terminate the forces acting on the lander; elements that may in turn affect integration. This is most easily done by selecting some small hardware selection of the lander/rover. , such that the onset of stick occurs when s ≤ . The choice In addition to uncertainties in the release conditions, these of a particular value for will affect the computational bur- simulations also allow for uncertainties in the surface and lan- den of carrying out the collision, but also the accuracy of the der properties. For example, it is possible to study the effect impulse P that is required to drive the slip velocity to zero. In of variations in the coefficients of restitution and friction on practice, we find that a value of ∼ 10−10 produces accurate the resulting surface spread and deployment time. Addition- Fig. 5. Sample deployment of a cube-shaped lander to asteroid Itokawa.

Fig. 6. Sample deployment of various lander shapes. ally, we can study the effect of variations in the applied rock contact motion; the brief period of motion at the very end of a model. All these variations can be applied to the small body deployment in which the lander dissipates its final velocities in both a global sense and a local sense. For example, the co- and comes to a full stop on the small body surface. efficient of resitution could be assumed to constant across the entire small body surface, or certain latitudes may be given a 8. REFERENCES higher coefficient than others. These techniques enable mission designers to take into account all available information [1] J. Biele, S. Ulamec, M. Maibaum, R Roll, L Witte, E Ju- on a target body when designing a deployment strategy, and rado, P Munoz,˜ W. Arnold, H.-U. Auster, and C. 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