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JOURNAL OF GUIDANCE , CONTROL , AND DYNAMICSVol. 28, No. 2, MarchApril 2005

Control of Hovering Spacecraft Near Small Bodies:Application to Asteroid 25143 Itokawa

Stephen B. Broschart and Daniel J. Scheeres

University of Michigan, Ann Arbor, Michigan 48109-2140

The small gravitational forces associated with a minor celestial body, such as an asteroid or comet nucleus, allowvisiting spacecraft to implement active control strategies to improve maneuverability about the body. One form of active control is hovering, where the nominal accelerations on the spacecraft are canceled by a nearly continuouscontrol thrust. We investigate the stability of realistic hovering control laws in the body-xed and inertial referenceframes. Two implementations of the body-xed hovering thrust solution are numerically tested and comparedwith an analytical stability result presented in the previous literature. The perturbation equations for inertialframe hovering are presented and numerically simulated to identify regions of stability. We nd that body-xedhovering can be made stable inside a region roughly approximated by the bodys resonance radius and inertialframe hovering is stable in all regionsoutside theresonance radius. A casestudy of hoveringabove Asteroid(25143)Itokawa, target of the Hayabusa mission, is also presented.

Introduction

A T the Japanese, European, and U.S. space agencies, missionscurrentlyexist to send spacecraft to explore minor celestialob- jects such as asteroids, comets, and small planetary satelli tes. It issuspected that theelemental composition of these objects, relativelyuncorrupted by time, may offer scientists insight into the early his-tory of our solar system. Thus, successful missions to such objectsare of critical importance to the ongoing goal of understanding theformation and evolution of our solar system.

These objects are generally quite small compared to the planetsand moons that have been the targets of most space missions todate. Because of their small size and correspondingly weak grav-itational pull, they are also usually quite irregular in shape. Theseattributes present great challenges to the traditional orbiter-typecontrol strategiesusedto date.Orbital trajectoriesabout small,irreg-ularly shaped bodies are generally quite complex and nonperiodic. 1Further, stability for close orbits above many small bodies is guar-anteed only for a limited range of latitudes, 2 possibly preventingstudy of relevant regions of the bodys surface. Also, because thegravitational forces are small, solar radiation pressure becomes animportant perturbing force, driving orbits to instability for objectssmaller than a particular size. 3

These problems with orbital control strategies for missions tosmall bodies necessitate a paradigm shift in the way we controlspacecraft. One solution that has been proposed is hovering. 4,5 Thisactive control strategy uses thrusters in a nearly continuous mannerto null the nominal accelerations on thespacecraft, creating an equi-librium point at the desired position. This approach is feasible foroperating near a small body because of the relatively weak gravita-tional forces involved. Unlike orbital control approaches, hoveringcontrol can be used to explore any size of body at any latitude, lim-ited only by fuel restrictions.The perturbing forcesof solar radiationpressure and solar gravity are also less of a concern in hovering, be-cause these forces can be actively nulled by the hovering controller.

Received 15 July 2003; revision received 7 January 2004; accepted forpublication 12 January 2004. Copyright c 2004 by Stephen B. Broschart.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on conditionthat thecopierpay the$10.00per-copy feeto theCopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0731-5090/05 $10.00 in correspondence with the CCC.

Graduate Student, 2016 FXB, Department of Aerospace Engineering;sbroscha@umich.edu.Associate Professor, 3048 FXB, Department of Aerospace Engineering;

scheeres@umich.edu. Associate Fellow AIAA.

In this paper, we investigate two types of hovering: hovering inthe small-body-xed frame and in the inertial frame. Hovering inthe body-xed frame would be ideal for a spacecraft operating veryclose to the bodys surface. Because body-xed hovering xes thespacecrafts position relative to the body, this control could be ad-vantageous for obtaining high-resolution measurements of a partic-ular area on the bodys surface. Body-xed hovering also simpliesdescent and ascent maneuvers that may be necessary for a samplereturnmission. A spacecraft visiting a small body may also be inter-ested in hovering in the inertial frame, where the small body rotatesbeneath the spacecraft. Inertial-frame hovering may be ideal for aspacecraft mapping a bodys surface or for use as a holding orbitfrom which to stage other maneuvers. For hovering to be a viablemission option, it must be shown to be stable and robust over a range

of situations and implementable under realistic control constraints.This paper uses analytic and numerical analysis to identify regionsof stable operation for a spacecraft subject to hovering control.

We begin by formulating the solutions of the two-body problemcorresponding to body-xed and inertial-frame hovering. Next, wepresent criteria for stability of body-xed hovering developed in aprevious paper. 5 This is followed by a discussion of the numericalsimulation tool wehavedevelopedto aidour study. We then compareresults of numerical simulation for two different implementationsof body-xed hovering with the predictions of the analytical stabil-ity criteria. Next, we present analytical and numerical analysis of hovering in the inertial frame. Finally, we perform a case study of both hovering approaches for Asteroid (25143) Itokawa (formerly1998 SF36), the target of the Japan Aerospace Exploration Agency(JAXA) Hayabusa (formerly MUSES-C) mission. 6, 7

Problem FormulationIn this analysis, we model the spacecraft dynamics near a small

body using a two-body gravitational model in which the spacecrafthas negligible mass. The equations of motion for this system intheuniformlyrotating, small-body-xed Cartesiancoordinateframewith origin at the bodys center of mass are

x 2 y = V x +T x +

2 x (1)

y +2 x = V y +T y +

2 y (2)

z = V z +T z (3)

343

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344 BROSCHART AND SCHEERES

where represents the rotation rate of the small body, V is its grav-itational potential, and T x , T y, and T z represent the component of the spacecraft thrust in each respective direction. It is assumed thatthe small body has constant density and is rotating uniformly abouta xed axis ( z) corresponding to its maximum moment of inertia.We further orient our coordinate frame by aligning x and y withthe principal axes associated with the bodys minimum and inter-mediate moments, respectively. A position in this space, [ x , y, z]T ,is denoted by the vector r . The only forces present in this system

are gravitational forces, inertial forces, and forces from the space-crafts propulsion system. In this paper, we ignore third-body forcesof smaller magnitude, such as solar radiation pressure and solargravity. These additional perturbations will be considered in futurework.

We investigate two solutions to these equations. The rst solu-tion, which we call body-xed hovering, creates a xed equilibriumpoint in the body-xed frame at the desired position, or hoveringpoint, by enabling thrust. Under this control, the spacecrafts posi-tion remains xed relative to the rotating small-body when placedat the nominal hovering point, r 0 =[ x 0 , y0 , z0]T . Enabling thrust isdened as

T = V x

0

2 x 0 , V y

0

2 y0 , V z

0

T

(4)

The second solution to Eqs. (13) that we study is a retrograde,constant-latitude, circular orbit whose orbital period is equal to thesmall bodys rotation period. This solution corresponds to a space-craft hovering in the inertial reference frame, where the spacecraftposition remains xed in inertial space while the small body rotatesbeneath it. In the body-xed frame, this solution is specied as

r0 I (t ) = r0 I (0) [cos 0 cos( t + 0), cos 0 sin( t + 0), sin 0]T (5)

where 0 and 0 arethe initiallatitude andlongitude (measuredfromthe positive x axis), respectively. For inertial hovering, the necessarythrust vector is dened as that necessary to make this circular orbita solution to the equations of motion:

T (t ) = V x r 0 I (t )

, V y r 0 I (t )

, V z r 0 I (t )

T

(6)

Previous WorkSawai et al. 5 developed a set of criteria analytically that charac-

terize the stability of a hovering trajectory in the body-xed frame.Through their analysis of the problem, they determined that theHessian matrix of the gravitational potential at r 0 determines thestability of hovering in the body-xed frame subject to their ideal-ized controller:

2 V r2 0 =

2V x 2

2 V x y

2 V x z

2V x y

2 V y2

2 V y z

2V x z

2 V y z

2 V z2 0

(7)

The conditions for stability they developed are

32v23 z +2 1 +2 2 +2 0 (8)1 +2 2 +2 2v21 z 2 +2 2v22 z 1 +2 0

(9)

1 2 2 2 +34v23 z v23 z +2

82v23 z 1 +2 + 2 +2 > 0 (10)

where 1 , 2 , 3 are the three eigenvalues of the Hessian matrixat r 0 and v1 , v2 , v3 are the corresponding eigenvectors (subscripts x , y, and z refer to the components of these eigenvectors). Theeigenvectors andeigenvalues are arranged so that the third set refersto the eigenvector nearly aligned with the gravitational attractiondirection. Of the remaining two, the eigenvector/eigenvalue pairwith the larger eigenvalue is the rst set.

These stability criteria were developed by linearizing the equa-tions of motion about r 0 and assuming one-dimensional, innitely

tight control of motion along v3 , such that r(t )T

v3 0. These threecriteria dene a theoretical region of stability for body-xed hov-ering above any arbitrary-shaped body. This region roughly cor-responds to the locus of initial hovering points inside the bodysresonance radius. The resonance radius r r is dened as the distancefrom the bodys rotation axis in the equatorial plane at which, fora spherical body with the same gravitational parameter , the cen-trifugal force is equal and opposite to the gravitational attraction:

r r = (/ 2)13 (11)

Illustrations of the region dened by these sufciency criteria forvarious small-bodies can be seen in Ref. 5.

The assumption of innitely tight control is not realistic, as it isimpossible for an actual spacecrafts control system to implement.However, similar control canbe implemented by useof a dead-bandthrust control on altitude, in which the spacecrafts movement isconned to a nite region around a target altitude. Allowing motionacross the region dened by such a dead-band may have signi-cant effects on system dynamics. Thus, an analysis using the fullnonlinear equations of motion is necessary to validate these results.We further the previous work by numerically simulating the non-linear equations of motion to determine the stability of hoveringunder realistic control constraints. We compare and contrast theseresults with the previous ndings to determine in which regions theanalytical stability criteria produce valid results.

Numerical Simulation ToolOur numerical simulation package was developed to aid us in

the analysis of spacecraft trajectories near arbitrary-shaped rotatingbodies. The basic function of the code is to integrate the body-xedequations of motion [Eqs. (13)] for a spacecraft in an arbitrarypotential eld subject to a chosen control law. The code for thissimulation was written using Matlab and Simulink software. Theequations of motion are integrated using Matlabs ode45 function(DormandPrincemethod) witha relative toleranceof 10 6 (m) andan absolute tolerance of 10 8 (m).

Thesimulation wasdeveloped to work withtwo differentasteroidshape and gravity models: the triaxial ellipsoid and the polyhedron.Triaxialellipsoidsare oftengoodapproximationsof real small-bodyshapes. Simulations using this shape can help us to understand fun-damental phenomena associated with this system that would be dif-cult to decouple from other effects in the truly arbitrary-shapedcase. More complex geometries can be specied to the simulationas n -faced polyhedra. Polyhedra can model a much wider range of shapes than ellipsoids, allowing depressions, ridges, cliffs, caverns,andholes on thebody. Polyhedron modelscurrently exist fora num-ber of real asteroids. This shape model can be very accurate, withresolution increasing with the number of faces used. Assuming thesmall body has constant density, previous works exist that exactlydene the potential around ellipsoidal 8 and polyhedral 9 shapes. Themethods described in these papers are implemented in our simula-tion to calculate gravitational potential, attraction, the Hessian, andthe Laplacian.

To ensurethat thesimulation works as intended,we have runa se-ries of tests with known results and compared the simulation outputto the expected answer. First, we conrmed that the spacecrafts an-gular momentum andenergy abovea sphericalbody were conservedthroughout thesimulation if no thrustwasapplied. Ournext test wasto verify that the simulation was computing the correct body forceson the spacecraft. This was done by analytically solving for theforces on the spacecraft at a given point and then adding open-loop

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BROSCHART AND SCHEERES 345

thrust of equal magnitude and opposite direction to the spacecraft.If the simulation is computing the forces on thespacecraft correctly,this open-loop thrust should cancel those forces and the spacecraftshould not move. The simulation was found to correctly producethis result. Our nal verication test was done to ensure that thecontrollers built into the simulator were working as intended. Thiscan be determined by examining the various simulation outputs.Thrust should be enabled in the proper direction whenever the cri-teria for activation of control are met. We performed this test for

many different initial conditions and found the controllers to work as expected.

Stability of Body-Fixed HoveringWe have investigated the stability of body-xed hovering sub-

ject to two different implementations of the nominal thrust solution[Eq.(4)]. We compare thenumerical regionof stabilityobtainedwiththe region dened by the stability criteria [Eqs. (810)] derived inthe previous work. 5

Because we are working with numerical data, determiningstability characteristics of a trajectory is inherently difcult. Withnite-time simulation data, it is impossible to verify true stabil-ity or instability. Here, we will be using a modied form of Lya-punov stability suitable to evaluate our trajectories. Beginning witha perturbation from equilibrium smaller than a given magnitude,we quantify stability by the size of the region centered at the initialhovering point that contains the entire trajectory for a given simu-lation time. We will dene instability not as unbounded motion butas motion that extends beyond a specied region during the simula-tion time. Because of potentially long time constants of instability,this analysis seeks only to verify stability for a xed duration in theregions predicted by the analytical stability criteria and not to verifyinstability in the complementary space.

Numerical Simulations of Hovering with Gravitational DirectionThrusting and Sensing with Open Loop Control

Therst body-xed hovering controller we evaluated is a slightlymodied version of the realistic hovering controller suggested inRef. 5. This controller, which we call the gravitational direction

thrusting and sensing with open loop (GDTS w/OL) hovering con-troller, is a combination of open-loop control thrust to eliminate thespacecrafts nominal acceleration and dead-bandcontrol on altitudemeasured along v3. The dead-band controller is dened by a tar-get altitude h 0 and a tolerance factor . If the spacecrafts altitudeis outside the band dened by h 0 , thrust is enabled in orderto return the spacecraft into the dead-band. A spacecraft with aninternal model of the small body can compute the Hessian matrixof the potential and determine v3 . In Ref. 5, it is suggested thatthe open-loop control need only cancel the centrifugal force on thespacecraft. However, we have chosen to use the open-loop controlto null the full nominal acceleration on the spacecraft, as proposedin Ref. 4, because we have found this type of control to conne thespacecrafts range of motion more tightly. The thrust vector T forthe GTDS w/OL controller is

T =T o +T db (12)where

T o = V x 0

2 x 0, V y 0

2 y0 , V z 0

T

(13)

T db =T m v3 , if (h 0 h) > T m v3 , if (h 0 h) < 0, otherwise (14)

where T m is the constant magnitude of the dead-band thruster andh is the spacecraft altitude. The vector v3 should be chosen to pointaway from the body. This controller mimics the ideal controllerused to derive the conditions for stability of body-xed hovering[Eqs. (810)] in the previous work. In that ideal case, the dead-bandtolerance factor for this controller is zero. Therefore, a comparison

of the performance of this controller with the analytical stabilityconditions is a direct test of the validity of the idealized result for aspacecraft subject to realistic control.

Using an ellipsoidal small-body shape model, we ran simulationsusing the GDTS w/OL controller for random initial positions in thebodys three symmetric planes ( XY , X Z , and Y Z ). The ellipsoidused in these simulationswas15 7 6 km,a rough approximationof thesize of Asteroid (433) Eros, which wasvisited duringNASAsNear Earth Asteroid Rendezvous (NEAR) mission. A bulk density

of 3 g/cm3

and a 10-h rotation period were used, which are notconsistent those of with (433) Eros but were chosen to move theresonance radius further from the body for clarity. (With Erosstrue rotation period of 5.27 h and bulk density of 2.4 g/cm 3 , theresonance radius is roughly 15.7 km, barely beyond the long endof the body.) The dead-band tolerance factor was 10 m. For eachinitial spacecraft position, the simulation was run 10 times withdifferent velocity error each time. Each Cartesian component of velocity error was chosen randomly from a uniform distributionbetween 1 and 1 cm/s.Data were collected from independent simulations of two differ-ent durations, 20,000 and 50,000 s (roughly 5.5 and 13.9 h, respec-tively). These relatively short simulation times (on the order of onerotation period) are justied by noting that a spacecraft operatingin the body-xed frame very near the surface would have little rea-son to remain in one position for long periods of time. This typeof maneuver would likely be used during a descent or a translationacross the surface, whereas longer term hovering station-keepingmaneuvers would more likely be carried out in the inertial frame.

Oursimulations show that in the X Z and Y Z planes the analyticalcriteria for stability were valid. There were no regions of numeri-cally unstable motion that encroached upon the predicted region of stability. We foundthat numerical stability extendedto someregionsoutside the area dened by the stability criteria in these planes butthese trajectories are not necessarily stable in the long term and,therefore, do not suggest error in the stability conditions.

Figures 1 and2 present results from simulations in thebodys XY (equatorial) planefor durations of 20,000 and 50,000 s, respectively.In these gures, each data point represents the average size over 10trials of the smallest solid angle such that the entire trajectory iscontained within it. For this analysis, we will consider averages of less than 0.4 deg to be stable. The region inside the bold line is thelocus of initial positions satisfying the analytical stability criteria.Note that for an ellipsoidal shape model, the equations of motionexhibit a longitudinal symmetry; that is, results are the same for anytwo points 180 longitude apart.

We can see in these gures that the stability criteria do not holdin the XY plane subject to the GDTS w/OL body-xed hoveringcontroller. Here, instabilities arise above the bodys leading edgesin regions satisfying the stability conditions. The leading edges aredened as the two quadrants of longitude on the bodys surface thatextend from the tip of the largest semimajor axis of the ellipsoidto the intermediate semimajor axis in the direction of z x. As thesimulation duration increases from 20,000 to 50,000 s, we can seethat the area of instability encroaches further upon the region satis-fying thestability criteria. On theother hand, the regionof predictedstability above the bodys trailing edge is unaffected and remainsstable. We also nd the region of stability off the trailing edge to besignicantly expanded for shorter duration (20,000 s) hovering.

Analysis of GDTS w/OL Controller ResultsWe attribute the discrepancy between the ideal region of stability

and our numerical ndings to the Coriolis accelerations introducedby relaxing the innitely tight control in the gravitational direc-tion assumed in the analytical work. An interaction between thedead-band orientation, the control direction, and the Coriolis ac-celerations causes the degradation of stability above the ellipsoidsleading edges and the improvement in stability above the trailingedge that are seen in our simulations.

The orientation of the dead-band region for hovering in the XY plane is shown in Fig. 3. We dene a coordinate system for the dead-band dynamics consisting of the normal vector to the dead-band

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346 BROSCHART AND SCHEERES

Fig. 1 Stability results for GDTS w/OL body-xed hovering controller. Angular deviation 20,000 s ( XY plane).

Fig. 2 Stability results for GDTS w/OL body-xed hovering controller. Angular deviation 50,000 s ( XY plane).

boundary, ndb ,theunitvectoralongtherotationaxis, z,andthevectortransverse to thedead band, tdb = z ndb . Thesurface normalwherethe altitude-sensing-direction vector vs intersects the bodys surfacedenes n db , the local orientation of the dead band. This direction,in general, is not aligned with the control direction vc . Therefore,when control thrust is applied, it will have some component along tdb . This transverse thrust component will be equal and opposite atthe twodead-band boundaries. If thespacecraft hits both boundariesan equal number of times, this transverse acceleration will have nonet effect on the spacecrafts motion along the dead band. However,

because of Coriolis forces, which effectively rotate the spacecraftsvelocity vector, this maynot be the case. This rotation, coupled withthe control thrust component along the dead band, can either causethe spacecraft to repeatedly hit one boundary of the dead band orencourage it to bounce back and forth between the boundaries.

Figure 4a illustrates the dynamics above the leading edge. Abovean ellipsoids leading edge subject to theGDTS w/OL hovering con-troller, the relative orientation of n db and vc is such that at the mini-mumaltitudeboundary of thedead-band,a componentof thecontrolthrust is applied in the negative tdb direction and at the maximum

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BROSCHART AND SCHEERES 347

Fig. 3 Dead-band orientation diagram.

altitude boundary, thrust is applied in the positive tdb direction. TheCoriolis force actingon thespacecraft, F c =2( z r) , effectivelyrotates the spacecrafts velocity vector clockwise in the plane de-ned by n db and tdb . When the spacecraft reaches a boundary, thecomponent of control thrust along tdb will effectively rotate the re-turn velocity vector clockwise from the direction it would have hadif there had been no thrust component along tdb . As the spacecraftmoves toward its next boundary crossing, Coriolis forces will ro-tate the velocity in the same clockwise direction. Once transientsdue to initial velocity errors wear off, the combination of these twoeffects will cause the spacecraft to hit the same boundary on suc-cessive occasions until the nonlinear effects of gravity and changesin dead-band orientation eventually turn it around. This successivebouncing motion is linearly unstable and leads to large oscilla-tions from the nominal hovering point. Examination of numericalsimulations shows that this bouncing movement always takes overin thesteady state forhovering above theleadingedge of anellipsoid

subject to the GDTS w/OL controller.Figure 4b illustrates the dynamics above the trailing edge. Abovethe bodys trailing edge, the relative orientation of n db and vc is op-posite, such that at the minimum altitude boundary, a component of the control thrust is applied in the positive tdb direction and at themaximum altitude boundary, thrust is applied in the negative tdb di-rection. As opposed to above the leading edge, the thrust componentalong tdb in this congurationcausesthe velocityvector reected off the dead-band boundary to be rotated counterclockwise from whereit would have been if there were no transverse thrust component. Inthis case, the clockwise rotation of the velocity vector caused by theCoriolis force will counter the effect of the control thrust, encour-aging the spacecraft to move toward the other dead-band boundary.Again, in the steady state, this motion always takes over above anellipsoids trailing edge, causing a chattering effect, where thespacecraft never hits the same boundary on successive occasions.This chattering hasa stabilizingeffect that keepsthe spacecraft closeto the initial hovering point.

The analysis here is done in the equatorial plane for the sake of simplicity, but this effect occurswith weakening strength all thewayto the bodys X Z and Y Z planes. In these planes, thrust does notdirectly induce motion that causes disruptive Coriolis accelerations.This is why our numerical results in the X Z and Y Z planes agreewell with the analytical predictions, in whose derivation Coriolisforces essentially do not exist. The strength of this Coriolis effectabove either the leading or trailing edge is determined by , , andthe angle between n db and vc . If we dene this angle as , the lessnegative is (measured from n db and positive for orientations suchthat vT c tdb > 0), the weaker the destabilizing effect of control thrustwill be above the leading edge, whereas the more positive is, thestronger the focusing effect willbe above the trailing edge. If we usea symmetric, universal method of choosing control direction, suchas gravitational direction or initial acceleration, we can postulate

that some optimal control law exists that will adequately focus theleading edge dynamics, while not weakening the focusing effect off the trailing edge too much. This optimal control direction would bea function of small-body parameters as well as the hovering dura-tion. Ideally, a spacecraft would use a method of choosing controldirection above the leading and trailing edges such that the angle is always positive. We have been able to show through numeri-cal simulation that shifting the control direction above the leadingedge so that is positive does indeed produce the desired stabiliz-

ing effect. It should be noted that should never approach 90 degbecause there would then be no thrust in the direction normal to thedead-band to maintain the spacecrafts altitude.

Numerical Simulations of Hovering with Initial AccelerationThrusting and Normal Sensing Control

We have also performed numerical simulations using another im-plementation of the body-xed hovering solution that we call initialacceleration thrusting and normal sensing (IATNS) hovering con-trol. We still usea dead-bandcontroller based on altimeter readings,but we eliminate the open-loop thrust that cancels the spacecraftsnominal accelerations. For the IATNS controller, we also changethe direction of the dead-band control thrust to be aligned with thenominal accelerationvector to more directly counter thenatural mo-tion of the spacecraft. The direction of altitude measurement is alsochanged from along v3 to measure altitude in the direction normalto the surface at the hovering point. This is the most robust directionin which to measure altitude because the component of altitude thatchanges with thesurface topology is minimized for small deviationsin spacecraft position.

The IATNS controller offers advantages over the GDTS w/OLcontroller. The elimination of the open-loop thrust will result in fuelsavings because with an open-loop thrust, any inaccuracies in its ap-plication will result in a constant push toward one side of the deadband that must be negated with thrust from the closed-loop con-troller, effectively causing the control to work against itself. Also,removing the open loop causes the spacecrafts motion to proceedvery closely along the initial acceleration direction from the initialhovering point. This lessens the deviation from the initial hovering

point dueto transienteffectscaused by initial velocityerrors becausethe spacecraft will more quickly move into steady-state motion inthedead band. Sensing altitude in thedirectionnormal to thesurfaceat the initial hovering point keeps the dead-band orientation vector ndb closer to the control direction vc , which our analysis suggestsshould improve performance above the bodys leading edge. Thiscontroller may also offer advantages in a surface-descent scenario,as there will exist some angle between the thruster plume and thesurface normal. This angle could help avoid contamination of thesurface regolith by the thrusters in a sample collecting mission andallow descent imaging to be less obscured by the thruster plume.The thrust vector for the IATNS controller is

T

=T m a0 , if h 0 h > T

m a

0, if h

0 h