zem/zev feedback guidance applicationto fuel-eﬃcient ...the gravitational potential of a constant...

ZEM/ZEV Feedback Guidance Application to

Fuel-Efficient Orbital Maneuvers Around an

Irregular-Shaped Asteroid

Matt Hawkins,∗ Yanning Guo,† and Bong Wie‡

Iowa State University, Ames, Iowa, 50011, USA

The new vision for advanced missions to asteroids, including soft landing, presents manychallenges that have essential differences from previous experiences with planetary land-ing. This paper focuses on two subjects pertaining to asteroid proximity operations: high-accuracy modeling of the gravitational environment and fuel-efficient guidance and controlalgorithm design. Both a spherical harmonic expansion method and a polyhedron shapemodel are used for modeling the gravitational environment of an irregular-shaped asteroid.The effects of Coriolis and centripetal accelerations are also examined. The ZEM/ZEV(Zero-Effort-Miss/Zero-Effort-Velocity) feedback guidance algorithm is in general not anoptimal control scheme, however it is conceptually simple and easy to implement, and inmany cases it approaches optimality. Two mission phases, orbital transfer between observa-tional orbits and soft landing, are numerically simulated using different implementations ofthe ZEM/ZEV algorithm. These simulations show that the ZEM/ZEV algorithm is suitablefor asteroid proximity operations, and important considerations for using the algorithm arediscussed.

Nomenclature

C̄nm Zonal spherical harmonic coefficients (normalized)P̄ Normalized associated Legendre functionS̄nm Tesseral spherical harmonic coefficients (normalized)ω Asteroid angular velocity vectorλ Longitude angleA Control acceleration vector expressed in an inertial framea Control acceleration vector expressed in a body frameD Disturbance acceleration vector expressed in an inertial framed Disturbance acceleration vector expressed in a body framef Generalized acceleration vector expressed in a body frameG Gravitational acceleration vector expressed in an inertial frameg Gravitational acceleration vector expressed in a body framen Face-normal vectorR Position vector expressed in an inertial framer Position vector expressed in a body frameV Velocity vector expressed in an inertial framev Velocity vector expressed in a body frameZEM Zero-effort miss distanceZEV Zero-effort velocity error

∗Ph.D. Candidate, Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall. AIAA Student Member.†Visiting Student (2010-2012). Ph.D. Candidate, Department of Control Science and Engineering, Harbin Institute of

Technology, Harbin, Peoples Republic of China 150001‡Vance Coffman Endowed Chair Professor, Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall.

AIAA Associate Fellow.

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American Institute of Aeronautics and Astronautics

techtechText BoxAIAA 2012-5045

e Unit vector along a coordinate directionEe Edge matrixFf Face matrixp Co-state vectorre field point-to-edge position vectorrf field point-to-face position vectorμ Gravitational parameter∇ Gradient operatorωf Solid angle subtended by polyhedron face∂ Partial derivative operatorφ Latitude angleψ Asteroid attitude angleρ Asteroid densityτ Dummy variable for integrationAx Inertial X acceleration componentAy Inertial Y acceleration componentd Signed distanceG Universal gravitational constantH Hamiltonian functionJ Performance indexLe Potential of a 1-D wireM Asteroid massr Radial coordinateR0 Nominal asteroid radiuss Path lengtht Timetgo Time-to-goU Gravitational potentialu Radial velocity componentv Transverse velocity componentVx Inertial X velocity componentVy Inertial Y velocity componentX Inertial X coordinatex Body-fixed x coordinateY Inertial Y coordinatey Body-fixed y coordinate

I. Introduction

Proximity operations in the vicinity of near-Earth objects (NEOs) represent an emerging area of research.Unmanned robotic probes can be used to explore smaller bodies such as asteroids and comets. Scientificinterest in these objects mainly focuses on what they can tell us about the evolution and diversity of oursolar system. Such missions also are invaluable in the area of asteroid deflection. Given sufficient warning,an observation probe could be sent to a potentially hazardous object. By flying close to the body, detailedimages of the surface can be taken as well as readings on the gravitational forces present. Such data can tellus more about the physical makeup of the body, which would help determine the most efficient deflectionmethod.

In order to evaluate the overall performance of the guidance and control system designed for the asteroid’sproximity operations, one must establish a high-fidelity dynamic model to simulate the asteroid gravitationalenvironment, where the gravitational acceleration and the centrifugal acceleration due to asteroid’s rotationmust be taken into account. A common method to describe the gravitational potential is the sphericalharmonic expansion method, from which the gravitational acceleration can be determined. Previous studieshave shown that it is the best model to simulate the asteroid’s gravity field outside a circumscribing sphereabout the asteroid. When the spacecraft completes proximity operations like soft landing or close flyby, thepolyhedron shape model is more desirable to provide correct estimates of the asteroid’s gravity field.1,2

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Bryson and Ho discussed optimal control laws for a simple rendezvous problem, considering both freeterminal velocity and constrained terminal velocity.3 They also discussed the relationship between optimalcontrol and proportional navigation guidance. Battin also discussed an optimal terminal state vector controlfor the orbit control problem, directly compensating for the known disturbing gravitational acceleration.4

D’Souza further examined an optimal control algorithm in a uniform gravitational field, and developed acomputational method to determine the optimal time-to-go.5 Ebrahimi et al. proposed a robust optimalsliding mode guidance law for an exoatmospheric interceptor, using fixed-interval propulsive maneuvers.6 Inthis paper, gravity was considered to be an explicit function of time. One major contribution of Ebrahimi etal. was the new concept of the zero-effort-velocity (ZEV) error, analogous to the well-known zero-effort-miss(ZEM) distance. The ZEV is the velocity error at the end of the mission if no further control accelerationsare imparted. Furfaro et al. later employed the ZEM/ZEV concept to construct two classes of non-linearguidance algorithms for a lunar precision landing mission.7 Guo et al.8 showed that in a uniform gravitationalfield, the ZEM/ZEV logic is basically a generalized form of various well-known optimal feedback guidancesolutions such as soft landing on an asteroid,9 intercept or rendezvous,3 terminal guidance,4 and planetarylanding.5 The performance of the ZEM/ZEV logic for an asteroid intercept mission with precision targetingrequirements was evaluated by Hawkins et al.,10 and compared with the performances of classical missileguidance methods like proportional navigation guidance (PNG) and augmented proportional navigationguidance (APNG).

Two proximity operations are considered in this paper. Optimal orbital transfer from a high-altitudeorbit to a low-altitude orbit is considered first. A soft landing on the surface of the asteroid starting fromthe low-altitude orbit is then considered. A fuel-efficient way to maintain the low-altitude orbit is describedby Winkler et al.11

For the asteroid soft landing mission, the gravitational acceleration is neither constant nor an explicitfunction of time, but is instead a function of position, velocity and the rotation of the asteroid. TheZEM/ZEV algorithm is not an optimal solution when the gravitational acceleration is a function of position.However, the ZEM and ZEV terms can be obtained by numerically integrating the dynamic equations,and the ZEM/ZEV algorithm can accomplish the control mission in a near-optimal manner. Therefore thegeneralized ZEM/ZEV algorithm is adopted to fulfill this complex control mission.

In this paper the equations of motion, which take the rotational motion of the asteroid into consideration,are presented first, followed by the asteroid’s gravitational modeling. After briefly reviewing the general-ized optimal control problem and the ZEM/ZEV algorithm, two different ZEM/ZEV algorithms for orbitaltransfer are presented, followed by four different types of ZEM/ZEV optimal feedback control algorithms forthe asteroid soft landing mission. These control laws are used to simulate proximity operations about theasteroid 433 Eros.

II. Problem Description

A. Coordinate Systems

Two coordinate frames are considered in this study, the asteroid’s body-fixed frame (x, y, z), and theinertial frame (X, Y , Z). Typically the body-fixed frame is used for landing, and the inertial frame is usedfor orbital transfers. The body-fixed frame has its origin at the center of mass of the asteroid, with the x, y,and z axes along the axes of minimum, intermediate, and maximum inertia, respectively. When using thespherical harmonic expansions for gravitational modeling, it is convenient to use the radius, r, latitude, φ,and longitude, λ. It is assumed that the asteroid spins about the z-axis with spin rate ω. For this paper,it is also assumed that the asteroid spins at a constant rate about a constant axis, which is reasonable forthe asteroid considered over the time span considered. An illustrative figure using the body-fixed coordinatesystem is given in Figure 1.

The inertial reference frame shares its origin with the body-fixed frame. The inertial axes are denotedas X, Y , and Z. The two frames are co-aligned at time t = 0. Unless otherwise specified, the asteroid isassumed to have an inertial orientation angle ψ = 0 at t = 0. When using the the inertial reference frame,the asteroid will be shown at its reference attitude. It is important to note that in general the asteroid willhave rotated during the time shown. Due to the asteroid’s rotation, the spacecraft’s angular position inthe inertial frame is ψ + λ. Figure 2 illustrates the inertial and body-fixed axes for the asteroid 433 Eros,considered in this study.

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Spacecraft

Figure 1. Coordinate system definition.

X

Y

x

y

Figure 2. Inertial and body-fixed frames.

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B. Dynamic Equation

The equations of motion of the spacecraft are expressed in the asteroid’s body-fixed frame as

ṙ = v

v̇ = a+ g (r)− 2ω × v − ω × (ω × r) + d (1)

where r is the spacecraft position, v is the spacecraft velocity, a is the applied control acceleration onthe spacecraft, g (r) is the gravitational acceleration due to the asteroid, d includes disturbances, such asmodeling uncertainties and perturbations from solar effects, and ω is the angular velocity vector of theasteroid. It is assumed that the asteroid has a uniform angular velocity about its axis of maximum inertia.

After neglecting disturbances, the gravitational acceleration can be combined with the Coriolis accel-eration term, 2ω × v, and the centripetal acceleration, ω × (ω × r), into a generalized acceleration term,f (r,v,ω). Equation 1 then becomes

ṙ = v

v̇ = a+ f (r,v,ω) (2)

where r = [x, y, z]T.

The equations of motion of the spacecraft in the inertial frame can also be expressed as

Ṙ = V

V̇ = A+G (R, ψ) +D (3)

where R = [X, Y, Z]T. Uppercase is used to denote the vectors expressed in an inertial coordinate system.

In this case, the gravitational acceleration depends on both the asteroid’s attitude angle ψ, which definesthe gravitational field within the inertial frame, and the spacecraft’s position within the inertial frame, R.For an asteroid with constant angular velocity about the Z-axis, we have

ψ = ωt (4)

The two frames are related by

r = CR where C =

⎡⎢⎣ cos (ωt) sin (ωt) 0− sin (ωt) cos (ωt) 0

0 0 1

⎤⎥⎦ (5)

C. Asteroid Shape and Gravity Model

1. Spherical Harmonic Expansion Method

The gravitational potential of an asteroid can be modeled with a spherical harmonic expansion as

U =GM

r

∞∑n=0

n∑m=0

(R0r

)nP̄mn (sinφ)

[C̄nm cos (mλ) + S̄nm sin (mλ)

](6)

where n and m are the degree and order, respectively, of the polynomials, G is the universal gravitationalconstant, M is the asteroid mass, R0 is the nominal radius, and r, φ, and λ are the radius, latitude,and longitude, respectively. P̄mn (·) is the normalized associated Legendre function, and C̄nm and S̄nm arenormalized spherical gravitational coefficients.

The gravitational acceleration, needed for simulation of asteroid proximity operations, can be obtainedby computing the gradient of the gravitational potential

g (r, φ, λ) = ∇U (r, φ, λ) = ∂U∂r

er +1

r

∂U

∂φeφ +

1

r cosφ

∂U

∂λeλ (7)

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Substituting the expressions for the unit vectors er, eφ and eλ into Equation 7 gives

g (r, φ, λ) =

⎡⎢⎣cosφ cosλ − sinφ cosλ − sinλcosφ sinλ − sinφ sinλ cosλ

sinφ cosφ 0

⎤⎥⎦⎡⎢⎣

∂U∂r

1r∂U∂φ

1r cosφ

∂U∂λ

⎤⎥⎦ (8)

where

∂U

∂r= −GM

r2

∞∑n=0

n∑m=0

{(n+ 1)

(R0r

)nP̄mn (sinφ)

[C̄nm cos (mλ) + S̄nm sin (mλ)

]}

∂U

∂φ=

GM

r

∞∑n=0

n∑m=0

{(R0r

)n [P̄m+1n (sinφ)−m tanφP̄mn (sinφ)

] [C̄nm cos (mλ) + S̄nm sin (mλ)

]}

∂U

∂λ=

GM

r

∞∑n=0

n∑m=0

{(R0r

)nmP̄mn (sinφ)

[S̄nm cos (mλ) + C̄nm sin (mλ)

]}

2. Polyhedron Shape Model

Werner and Scheeres12 give a comprehensive treatment of the polyhedron shape model, which will now bebriefly described. The gravitational potential of a constant density polyhedron at a field point r can bedescribed as

U =1

2Gρ

∑e∈edges

LerTe Eere −

1

2Gρ

∑f∈faces

ωfrTf Ffrf (9)

where ρ is the asteroid’s density, re and rf are column vectors denoting the position from the field point toany point on the edge and face, respectively, and Ee and Ff are defined as

Ee = n̂A(n̂A12

)T+ n̂B

(n̂B21

)TFf = n̂f n̂

Tf

where the face-normal vector nf and the edge-normal vector on face f , nfij are defined as

nf = (r2 − r1)× (r3 − r2) = r1 × r2 + r2 × r3 + r3 × r1

nfij = (rj − ri)× nfFigure 3 shows the face-normal and edge-normal vectors for a section of the polyhedron.Let ri be the vector from the field point to vertex Pi, with length given by |ri| = ri, and eij the length

of the edge between vertices Pi and Pj . The potential of a 1-D straight wire is

Le =

∫e

1

rds ln

ri + rj + eijri + rj − eij (i, j = 1, 2, . . . , N) (10)

Let ωf be the signed solid angle subtended by each polyhedron face, viewed from the field point. It isfound as

ωf =det ([ri rj rk])

rirjrk + rirTj rk + rjrTk ri + rkr

Ti rj

(i, j, k = 1, 2, . . . , N) (11)

The gravitational acceleration at the field point r can be found as

g (r) = ∇U (r) = −Gρ∑

e∈edgesLeEere +Gρ

∑f∈faces

ωfFfrf (12)

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Point 2

Point 1

P

1

Facet A

Facet B

Figure 3. Illustration of face- and edge-normal vectors.

The above equation can be expressed in terms of only faces of the polyhedron

g (r) = Gρ∑

f∈faces(ωfdf − Lijdeij − Ljkdejk − Lkideki) n̂f (13)

where df is the signed distance between the field point and the face, Lij , Ljk, and Lki are the edge potentials,and deij , dejk, and deki represent the signed distances from the projection of the field point in each faceplane to each of the three edges. Each d parameter is positive if it lies in the outward-pointing side of thecorresponding edge, otherwise it is negative. The d parameters are computed as

df = n̂Tf rf deij = n̂

Tijri dejk = n̂

Tjkrj deki = n̂

Tkirk (14)

Also of interest when computing the gravitational attraction is the Laplacian. It can be shown thatEquation 6 can be differentiated to give

∇2U = −Gρ∑

f∈facesωf (15)

The sum −∑ωf is zero when the field point is inside the polyhedron, and equals −4π inside the polyhe-dron. This term can be computed for almost no cost, since ωf is already found at every step, and providesa convenient check to see if the spacecraft is ever inside the asteroid.

3. Gravity Model Comparison

The asteroid Eros, considered in this study, was visited by the NEAR Shoemaker mission. Its shape andphysical characteristics have been extensively studied,13 and high-fidelity models exist for both the sphericalharmonic expansion and the polyhedron.14 These models allow comparisons between the two approaches.

There are advantages and drawbacks to using either the spherical harmonic expansion model or thepolyhedron model, which are now discussed. The harmonic model is much less computationally intensive,as it requires summing over the degree and order of the model. The double-summation to 16th degree andorder involves 153 terms. The main drawback of the harmonic model is that it is only valid outside thesmallest bounding sphere about the asteroid, and is inaccurate near this sphere. The harmonic model, then,is suitable for orbital operations.

The polyhedron model is much more computationally intensive, as it involves summing over the faces ofthe model. The lowest-resolution model for asteroid Eros has 1708 faces. The polyhedron model also assumesconstant density, which is a reasonable approximation for Eros, but may not be for other asteroids. Adjustingfor this would require introducing one or more additional polyhedrons, while it requires only modifying the

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coefficients of the harmonic expansion. The polyhedron model is more accurate near bounding sphere, andis the only valid model inside this sphere, so it is suited to landing and other close proximity operations.Additionally, the polyhedron model provides a simple check to ensure that the spacecraft is outside of theasteroid.

Both models are useable in either the body-fixed or the inertial frame. The gravitational accelerationcan be found by use of simple coordinate transformations, without need to rotate the asteroid model itself.

III. The ZEM/ZEV Guidance Algorithm

A. Generalized ZEM/ZEV Feedback Guidance Algorithm

In general, the gravitational acceleration is a function of the position and velocity of the spacecraft, as well asthe attitude of the target asteroid. This will not lead to a tractable solution of the optimal control problem.If instead the gravitational acceleration is assumed to be a function of time, optimal feedback algorithmscan be found. Lowercase variables will be used in this discussion, as it is independent of the reference frame.

For a mission from time t0 to tf , the optimal control acceleration is determined by minimizing the classicalperformance index of the form

J =1

2

∫ tft0

aTa dt (16)

subject to Equations 2 or 3 and the following given boundary conditions:

r (t0) = r0 r (tf ) = rf

v (t0) = v0 v (tf ) = vf (17)

The Hamiltonian function for this problem is

H =1

2aTa+ pTr v + p

Tv (g (t) + a) (18)

where pr and pv are the co-state vectors associated with the position and velocity vectors, respectively. Theco-state equations say that the optimal control solution can be expressed as a linear combination of theterminal values of the co-state vectors. Defining the time-to-go, tgo, as

tgo = tf − t (19)the optimal acceleration command at any time t is

a = −tgopr (tf )− pv (tf ) (20)By substituting the above expression into the dynamic equations to solve for pr (tf ) and pv (tf ), the

optimal control solution with the specified rf , vf , and tgo is finally obtained as

a =6 [rf − (r+ tgov)]

t2go− 2 (vf − v)

tgo+

6∫ tft

(τ − t)g (τ) dτt2go

− 4∫ tft

g (τ) dτ

tgo(21)

The zero-effort-miss (ZEM) distance and zero-effort-velocity (ZEV) error denote, respectively, the dif-ferences between the desired final position and velocity and the predicted final position and velocity if notadditional control is commanded after the current time. For the assumed gravitational acceleration g (t),the ZEM and ZEV have the following expressions

ZEM = rf −[r+ tgov +

∫ tgot

(tf − τ)g (τ) dτ]

(22)

ZEV = vf −[v +

∫ tft

g (τ) dτ

](23)

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The optimal control law, Equation 21, can be expressed as

a =6

t2goZEM− 2

tgoZEV (24)

For certain missions where the terminal velocity is not specified, the optimal control law, in terms ofZEM only, can be obtained as

a =3

t2goZEM (25)

The optimal control law to regulate only the terminal velocity, which will be used for orbital transfer, interms of ZEV only, can also be obtained as

a =1

tgoZEV (26)

Since the gravitational acceleration cannot be simply modeled as a function of time, the ZEM andZEV must be found by some means. Hawkins et al.15 describes how the ZEM and ZEV can be found bynumerically integrating the dynamic equations, or can be approximated with an error state transition matrix(STM). The ZEM and ZEV are updated in real-time, accomplishing the control mission at a near-optimallevel while maintaining acceptable computational complexity.

For highly nonlinear systems, predicting the future states is prone to errors. Another alternative formof the ZEM/ZEV algorithm can be adopted for this situation. Rather than predicting the effect of thenonlinear terms, the effects of these terms are directly compensated for at all times. The algorithm thusapproaches feedback linearization behavior. The control algorithm, Equation 21, then simply becomes thefollowing form suggested by Battin4

a =6 [rf − (r+ tgov)]

t2go− 2 (vf − v)

tgo− g (r) (27)

B. ZEM/ZEV guidance for orbital transfer

There are two implementations of ZEM/ZEV guidance to achieve orbital transfer. For proximity operationsnear an asteroid, it is assumed that the angular position at orbit insertion is unimportant. The longitudeof the spacecraft would be relevant for spacecraft in a resonant orbit, but for this paper resonant orbitsare avoided. The first orbital transfer formulation uses polar coordinates, and does not specify the angularposition at orbit insertion. The second formulation uses Cartesian coordinates, and although no particularangular position is needed, one must be chosen to for the guidance law to work. For the short mission timeof the orbital transfer, the asteroid can be modeled as a point mass, and differences in the gravitational forcedue to the point-mass model become disturbances for the control system to overcome. The two guidanceschemes are described next.

1. ZEM/ZEV guidance in polar coordinates

The objective of the orbital transfer problem here is to transfer a spacecraft from one circular orbit to anothercircular orbit. The terminal constraints are that the spacecraft should be placed at a specified distance fromthe asteroid with corresponding circular orbital velocity. The final radial velocity is zero, and the angularposition is free. Due to the nature of the constraints, polar coordinates are used. The standard dynamicalmodels for this type of orbit raising problem are described by

ṙ = u

u̇ =v2

r− μ

r2+ ar (28)

v̇ = −uvr

+ at

where r, u, and v represent the distance of the spacecraft from the center of mass of the asteroid, the radialvelocity, and the transverse velocity, respectively, and ar and at are the control accelerations in the radial

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and transverse directions, respectively. The gravitational parameter for the asteroid is given as μ. Recallingthe harmonic expansion model, we see that μ = GM . The required terminal states, as described above, are

r (tf ) = rf u (tf ) = 0 v (tf ) =

√μ

rf(29)

The orbit raising problem is somewhat unusual in that the control requirements are different alongthe radial and tangential axes. In the radial direction, there are position and velocity requirements asusual. In the tangential direction, we have the rare case where only the velocity is specified. For asteroidproximity operations, the gravitational environment is highly nonlinear, so the ZEM/ZEV algorithm withdirect compensation of gravitational terms, Equation 27, can be used as:

ar =6

t2go(rf − (r + tgou))− 2

tgo(uf − u)−

(v2

r− μ

r2

)

at =1

tgo(vf − v)−

(−uv

r

)(30)

It is seen that the law is a combination of the ZEM/ZEV law, Equation 24, in the radial direction, andthe ZEV law, Equation 26, in the tangential direction. For the orbital transfer problem between circularorbits, recall also that uf = 0.

2. ZEM/ZEV guidance in Cartesian coordinates

Despite being posed in such a way as to ignore the final angular position, the compensating ZEM/ZEVlaw suffers from the nonlinear effects of the gravitational field. Guo et al.16 shows this for a Mars orbitaltransfer example, and suggests ways to overcome this. The method suggested there is to use an offline optimalsolution to generate a series of waypoints to track. This is less practical for the asteroid orbit problem, as theoptimal trajectory depends on both the longitude of the spacecraft, and the total angle change commanded.For the asteroid orbital transfer mission, a second method of simply finding a point on the target orbit andusing ZEM/ZEV guidance in Cartesian coordinates, is suggested.

For the ZEM/ZEV orbital transfer problem in Cartesian coordinates, consider the following spacecraftdynamic equations

Ẋ = Vx

Ẏ = Vy (31)

V̇x = G (X,Y, ψ) +Ax ≈ −μ X(X2 + Y 2)

32

+Ax

V̇y = G (X,Y, ψ) +Ay ≈ −μ Y(X2 + Y 2)

32

+Ay (32)

where (X, Y ) and (Vx, Vy) denote the position and velocity components in the inertial frame, and (Ax, Ay)are control accelerations along the (X, Y ) axes. Although the true gravitational field is represented bythe harmonic expansion signified by G (X,Y, ψ), the nonlinearities are such that it is better to use theapproximate values on the right hand side of Equation 32 for numerical propagation of the orbit.

The terminal conditions for the asteroid orbital transfer problem are as follows:

X (tf ) = Xc Y (tf ) = Yc

Vx (tf ) = Vxc Vy (tf ) = Vyc (33)

The equations of motion are strongly coupled, and an analytic optimal control algorithm does not exist.The ZEM/ZEV algorithm, Equation 24, can control the terminal position and velocity at a specified finaltime. These encompass all of the requirements of the orbital transfer problem, making it a good candidate

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for solution with ZEM/ZEV guidance. Expressed in the X- and Y -coordinates, the proposed ZEM/ZEV lawbecomes [

AxAy

]=

6

t2go

[ZEMxZEMy

]− 2

tgo

[ZEVxZEVy

](34)

where the ZEM and ZEV are obtained by subtracting the predicted terminal states (with no further controlaccelerations) from the required terminal states, as follows:

[ZEMx

ZEMy

]=

[Xc − X̃fYc − Ỹf

](35)

[ZEVx

ZEVy

]=

[Vxc − ṼxfVyc − Ṽyf

](36)

C. ZEM/ZEV guidance for soft landing on an asteroid

For an asteroid rendezvous or soft landing mission, the terminal velocity is by definition zero. For anypractical asteroid proximity operation, the actual gravitational acceleration of the target asteroid will notbe exactly known a priori, and its magnitude is often small in comparison to other accelerations and distur-bances. There are several options for dealing with nonlinearities and disturbances using ZEM/ZEV logic.

The simplest way to formulate a guidance law is to outright ignore the gravitational and apparentaccelerations, treating them as disturbances for the ZEM/ZEV feedback law to overcome. The ZEM/ZEVlaw with zero gravitational acceleration is called ZEM/ZEV-z.

The next option is to directly compensate for the Coriolis and centripetal accelerations. This law makesuse of the spin rate of the asteroid without getting into the details of the gravitational field. The ZEM/ZEVlaw that compensates for the Coriolis and centripetal accelerations is called ZEM/ZEV-a.

Even though it is not worth invoking the modeled gravitational field, as it will inevitably differ fromthe true field, it is still possible to account for gravity in the ZEM/ZEV law. The generalized gravitationalacceleration is the acceleration due to a point mass. This generalized gravitational acceleration can be directlycompensated for. The ZEM/ZEV law that also accounts for generalized gravity is called ZEM/ZEV-g.

Finally, the usual predictive ZEM/ZEV law can be used. As with ZEM/ZEV-g, it is not necessary toconsider the gravitational model. The predicted final states are found by propagating the equations of motionwith point-mass gravity. The predictive ZEM/ZEV law is called ZEM/ZEV-p.

We now have the four ZEM/ZEV laws as follows:

1. ZEM/ZEV-z:

a =6

t2go(rf − r) + 4

tgov (37)

2. ZEM/ZEV-a:

a =6

t2go(rf − r) + 4

tgov + 2ω × v + ω × (ω × r) (38)

3. ZEM/ZEV-g:

a =6

t2go(rf − r) + 4

tgov +

GM

|r|3 r+ 2ω × v + ω × (ω × r) (39)

4. ZEM/ZEV-p:

a =6

t2go(rf − r̃f ) + 4

tgo(v − ṽf ) (40)

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IV. Simulations and Results

A. Low-Thrust Orbital transfer

A low-thrust (non-impulsive) orbital transfer mission from a 100-km orbit to a 35-km orbit around asteroidEros was simulated using both the polar and the Cartesian forms of the ZEM/ZEV algorithm. The sphericalharmonic expansion gravitational model was used. The orbits are prograde in the XY -plane. In general,prograde planar orbits are the least stable,17 so guidance that works for such orbits will be able to workwith any other inclination. Once the spacecraft reaches the 35-km orbit, further proximity operations canbegin. A detailed description of an algorithm to maintain a 35-km orbit via feedback control can be foundin Winkler et al.11

The orbital transfer mission starts with the spacecraft on the positive X-axis, and the asteroid at anattitude angle of zero. The mission time is 41,234 seconds, corresponding to one half of the period ofa Hohmann transfer between the two circular orbits. The exact mission time is not critical, as the trueoptimal solution would depend on angular positions in addition to transfer time. For such a short missionphase, a range of transfer times gives reasonable performance. The polar coordinate ZEM/ZEV algorithmdoes not require a final anomaly angle. For the Cartesian ZEM/ZEV algorithm, angular change is chosen as135◦. The Cartesian version is limited to angular changes of less than 180◦.

Figure 4 shows the transfer orbit trajectories for the two different algorithms. Also shown is the asteroidat its initial attitude. The normalized acceleration vector is shown every 1/10th of the total time. Bothalgorithms command mostly radial acceleration at the beginning. Later on the Cartesian algorithm is able tofind smaller commands than the polar algorithm. Figure 5 shows these acceleration histories. Finally, Fig-ure 6 shows the performance index histories. It is seen that the Cartesian algorithm has a lower performanceindex.

For a given mission time, the Cartesian algorithm is usually to be preferred. This is somewhat unexpected,as it imposes more constraints. Although not shown here, the polar algorithm is able to make transfers ofgreater than 180◦ and reduce the performance index. There is a practical limit to this, as for longer missionsthe polar algorithm will command the spacecraft to travel past the lower orbit, risking collision.

20

40

60

80

100

30

210

60

240

90

270

120

300

150

330

180 0

100-km Orbit35-km OrbitTransfer Orbit

(a) Polar ZEM/ZEV.

20

40

60

80

100

30

210

60

240

90

270

120

300

150

330

180 0

100-km Orbit35-km OrbitTransfer Orbit

(b) Cartesian ZEM/ZEV.

Figure 4. Transfer orbit trajectories.

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0 1 2 3 4 5

x 104

-2

-1

0

1

2

x 10-4

Time (s)

cont

rol a

ccel

erat

ion,

m/s2

Radial accelerationTangential acceleration

(a) Polar ZEM/ZEV.

0 1 2 3 4 5

x 104

-2

-1

0

1

2

x 10-4

Time (s)

cont

rol a

ccel

erat

ion,

m/s2

Radial accelerationTangential acceleration

(b) Cartesian ZEM/ZEV.

Figure 5. Transfer orbit acceleration histories.

0 1 2 3 4 5

x 104

0

1

2

3

4

5

6x 10

-4

Time (s)

J (m

2 /s3

)

PolarCartesian

Figure 6. Performance index comparison.

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B. Soft landing on an asteroid

1. Ideal spherical asteroid

In order to demonstrate the effectiveness and optimality of the ZEM/ZEV guidance algorithm, a landing onan ideal spherical asteroid is first analyzed. The asteroid is assumed to have a diameter of 400 meters, anda total mass of 1 × 1011 kilograms. The rotation period of the asteroid is 10 hours. Since the asteroid is auniform sphere, the gravitational field is exactly known, as it is simply the gravitational field due to a pointmass. The optimal solution is found using GPOPS, an open-source optimization software package. For moreinformation on GPOPS, the reader is referred to Reference 18.

An asteroid landing problem using the ZEM/ZEV-p law, Equation 40, was numerically simulated. The

lander is initially on an equatorial orbit, with initial velocity of v0 = [0, 0.1492, 0]Tm/s, and an initial posi-

tion of r0 = [300, 0, 0]Tm. The final conditions for the soft landing are: tf = 2000 s, rf = [0, 141.3, 141.4]

T

m, vf = [0, 0, 0]Tm/s. Figure 7 shows the trajectories using the ZEM/ZEV approach and the open-loop

optimal approach generated using the GPOPS software package. Figure 8 compares the control historiesand performance index for the ZEM/ZEV law and the open-loop optimal solution. It is seen that the ac-celeration histories are nearly identical along the x- and z-axes, with small deviations along the y-axis. Theperformance index comparison verifies that the ZEM/ZEV approach is indeed nearly optimal.

-300-200

-1000

100200

300

-300-200

-1000

100200

300

-300

-200

-100

0

100

200

300

Trajectories of vehicle

y (m)

z (m

)

Figure 7. Soft-landing trajectories on a spherical asteroid using ZEM/ZEV and GPOPS.

2. Gravity field analysis for asteroid Eros

The asteroid 433 Eros considered in this study has an irregular shape. The longest dimension is more thantwice as long as the shortest dimension. There are also significant concavities. The harmonic expansion isonly valid outside the smallest bounding sphere, so the polyhedron model must be used for operations near thesurface. The contributions from various sources to the overall gravitational acceleration are now compared.NASA provides polyhedron shape models for various numbers of faces, ranging from 1708 to 20700.14 Herewe choose 7790 faces to generate results with acceptable accuracy and computational efficiency. Eros’sdensity is considered to be a uniform value of 2760 kg/m3. The rotation period is 5.27 hours, giving anangular velocity of 3.31× 10−4 rad/s.

Three acceleration terms will be considered. The generalized gravitational acceleration is the gravitationalforce from a point mass on the surface of Eros. This is the gravitational acceleration used by the controllaws. Using Equation 13, the acceleration from the polyhedron shape model can be found. Finally, giventhe angular velocity, the contribution from centripetal acceleration can be found. The Coriolis accelerationterm does not allow for a similar analysis as it is a function of the spacecraft’s velocity. In any case, for soft

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0 500 1000 1500 2000-4

-2

0

2

4

6x 10

-4

x (m)

a x (m

/s2 )

OptimalZEM/ZEV

(a) ax.

0 500 1000 1500 2000-1

-0.5

0

0.5

1x 10

-4

x (m)

a y (m

/s2 )

OptimalZEM/ZEV

(b) ay .

0 500 1000 1500 2000-1

0

1

2

3

4x 10

-4

x (m)

a z (m

/s2 )

OptimalZEM/ZEV

(c) az .

0 500 1000 1500 20000

1

2

3

4

5

6x 10

-5

Time (s)

J (m

2 /s3

)

Peformance Index

Optimal, J=5.2e-5ZEM/ZEV, J=5.2e-5

(d) J .

Figure 8. Control history and performance index comparison.

landing, the velocity approaches zero as the spacecraft nears the surface, so the contribution from Coriolisacceleration at the surface cannot be compared with the other terms.

Figure 9 shows the generalized gravitational magnitude at the surface. Figure 10 shows the gravitationalacceleration magnitude from the polyhedron shape model. Figure 11 shows the acceleration due to thecentripetal acceleration.

From Figures 9 through 11, it is seen that the generalized and shape model gravity are in generalagreement, as expected. The maximum generalized acceleration is 5.6×10−3 m/s2, comparable to 5.3×10−3m/s2 for the shape model. The minimum generalized acceleration is 2.5×10−3 m/s2, compared to 3.7×10−3m/s2 for the shape model. The centripetal is smaller overall, ranging from zero for points on the z-axis upto 2× 10−3 m/s2.

The accelerations shown are for the surface of Eros only. The gravitational acceleration decreases withdistance, while the centripetal acceleration increases with distance. When the gravitational and centripetalaccelerations add to zero, we have the well-known curves of zero velocity.17,19,20

As the spacecraft moves further away from the asteroid, it is less practical to use the body-fixed frame,as most of the acceleration is apparent acceleration due to the rotation of the asteroid. Further away, thespacecraft’s motion is better analyzed as motion about a rotating body.

3. Soft landing on asteroid Eros

Landing site selection is one of the most important subjects for a practical asteroid landing missions. Factorssuch as surface conditions, communications, relative position of the Sun and the Earth, and science valuemust all be taken into consideration. Most of these issues are not dealt with in this paper. Instead, twodifferent landing sites are chosen, representative of two different general landing scenarios. The first is alanding from a circular 35-km polar orbit to a point near Eros’s north pole. The second is a landing from a35-km prograde equatorial orbit to a point on Eros’s equator, near the intermediate inertial axis. Because thespacecraft is operating near the asteroid’s surface, the polyhedron gravitational model is used for simulations.

The polar orbit is in the inertial XZ-plane, about the inertial Y -axis. At mission time t = 0, thespacecraft is at a latitude of 60◦, and the two frames are co-aligned. In the body frame, the initial velocity

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-1.5

-1

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1

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-5000

0

5000

-5000

0

5000

xy

z

2.5

3

3.5

4

4.5

5

5.5

x 10-3

Figure 9. Generalized gravitational acceleration.

-1.5

-1

-0.5

0

0.5

1

1.5

x 104

-5000

0

5000

-5000

0

5000

xy

z

3.8

4

4.2

4.4

4.6

4.8

5

5.2

x 10-3

Figure 10. Gravitational acceleration from polyhedron shape model.

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-1

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0

5000

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0

5000

xy

z

2

4

6

8

10

12

14

16

18

x 10-4

Figure 11. Centripetal acceleration term.

is in the negative x- and positive z-directions due to the orbital velocity. There is also a component in thenegative y-direction due to the rotation of the body frame relative to the inertial frame. The mission timeis selected to be 2400 s. The initial velocity is given as v0 = [−3.093, −5.796, 1.7855]T m/s. The initialposition is r0 = [17.5, 0, 30.311]

Tkm. The landing site is chosen as rf = [−3.2267, 0.3934, 5.5764]T km.

For a soft landing, the final velocity is vf = 0 m/s.Figures 12 through 15 show the trajectories and acceleration histories for the landing mission from a polar

orbit. Figure 16 shows the performance index histories for the different ZEM/ZEV algorithms. It is seenthat the simplest algorithm yields the highest performance index, while the full predictive algorithm achievesthe minimum performance index. Including an approximated gravity term does not improve performance.

For the ZEM/ZEV-p algorithm, Figure 17 compares the approximated generalized gravity term to theactual gravity term from the polyhedron model. There are significant differences in all three components.Figure 18 shows the acceleration terms due to the kinematics. These are seen to contribute accelerations ofthe same order as the gravity terms. For a real mission, the true gravity field will not be known, but therotation rate of the body can be known fairly accurately. A real guidance algorithm would not benefit fromadding gravity terms.

The prograde orbit is in the inertial XY -plane, about the inertial Z-axis. At mission time t = 0, thespacecraft is at a longitude of -60◦, and the two frames are co-aligned. In the body frame, the initial velocityis in the positive x- and positive y-directions due to the orbital velocity. There are also components in thenegative x- and y-directions due to the rotation of the body frame relative to the inertial frame. At a distanceof 35 km these terms dominate, so the total body-frame initial velocity is in the negative x- and y-directions.The mission time is again selected to be 2400 s. The initial velocity is given as v0 = [−6.946, −4.010, 0]Tm/s. The initial position is r0 = [17.5, −30.311, 0]T km. The landing site is chosen on the positive x sideas rf = [−8.166, −7.643, 1.487]T km. Landing sites on the middle or negative x side are challenging due tothe kinematic accelerations. For a soft landing, the final velocity is again vf = 0 m/s.

Figures 19 through 22 show the trajectories and acceleration histories for the landing mission from apolar orbit. Figure 23 shows the performance index histories for the different ZEM/ZEV algorithms. Thesimplest algorithm clearly yields the highest performance index, while the other three ares similar, withthe full predictive algorithm again achieving the minimum performance index. Including an approximatedgravity term still does not improve performance.

For the ZEM/ZEV-p algorithm, Figure 24 compares the approximated generalized gravity term to theactual gravity term from the polyhedron model. There are significant differences in all three components.Figure 25 shows the acceleration terms due to the kinematics. These are seen to contribute accelerations ofthe same order as the gravity terms, and are overall more significant than they were for the polar landing.

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x(km)y(km)

z(km

)

(a) Trajectory.

0 500 1000 1500 2000 2500-0.03

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-0.01

0

0.01

0.02

0.03

0.04

a (m

/s2 )

Time(s)

x y z

(b) Acceleration history.

Figure 12. Trajectory and acceleration history, ZEM/ZEV-z.

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010

20

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10-10

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5

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30

35

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z(km

)

(a) Trajectory.

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a (m

/s2 )

Time(s)

x y z


Figure 13. Trajectory and acceleration history, ZEM/ZEV-a.

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35

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z(km

)

(a) Trajectory.

0 500 1000 1500 2000 2500-0.04

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0

0.01

0.02

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a (m

/s2 )

Time(s)

x y z


Figure 14. Trajectory and acceleration history, ZEM/ZEV-g.

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x(km)y(km)

z(km

)

(a) Trajectory.

0 500 1000 1500 2000 2500-0.03

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/s2 )

Time(s)

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Figure 15. Trajectory and acceleration history, ZEM/ZEV-p.

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ZEM/ZEV-zZEM/ZEV-aZEM-ZEV-gZEM/ZEV-p


0 500 1000 1500 2000 2500-10

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0

2

4

6x 10

-3

appr

oxim

ated

g(r)

(m/s

2 )

Time(s)

x y z

(a) Approximate gravitational acceleration.

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-4

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0

1

2x 10

-3

actu

al g

(r) (m

/s2 )

Time(s)

x y z

(b) Actual gravitational acceleration.

Figure 17. Approximate and actual gravitational accelerations.

0 500 1000 1500 2000 2500-4

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0

2

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8x 10

-3

-2v

(m/s

2 )

Time(s)

x y z

(a) Coriolis acceleration.

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5

10

15

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-4

-r (

m/s

2 )

Time(s)

x y z

(b) Centripetal acceleration.

Figure 18. Coriolis and centripetal acceleration terms.

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z(km

)

(a) Trajectory.

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a (m

/s2 )

Time(s)

x y z


Figure 19. Trajectory and acceleration history, ZEM/ZEV-z.

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1015

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z(km

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(a) Trajectory.

0 500 1000 1500 2000 2500-0.03

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0.01

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0.03

a (m

/s2 )

Time(s)

x y z


Figure 20. Trajectory and acceleration history, ZEM/ZEV-a.

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1015

20

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-30

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10

x(km)y(km)

z(km

)

(a) Trajectory.

0 500 1000 1500 2000 2500-0.03

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0.03

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/s2 )

Time(s)

x y z


Figure 21. Trajectory and acceleration history, ZEM/ZEV-g.

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z(km

)

(a) Trajectory.

0 500 1000 1500 2000 2500-0.04

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-0.01

0

0.01

0.02

0.03

a (m

/s2 )

Time(s)

x y z


Figure 22. Trajectory and acceleration history, ZEM/ZEV-p.

0 500 1000 1500 2000 25000

0.1

0.2

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0.7

J (m

2 /s3

)

Time(s)

ZEM/ZEV-zZEM/ZEV-aZEM/ZEV-gZEM/ZEV-p


0 500 1000 1500 2000 2500-3

-2

-1

0

1

2

3x 10

-3

appr

oxim

ated

g(r)

(m/s

2 )

Time(s)

x y z

(a) Approximate gravitational acceleration.

0 500 1000 1500 2000 2500-2

-1

0

1

2

3

4

5x 10

-3

actu

al g

(r) (m

/s2 )

Time(s)

x y z

(b) Actual gravitational acceleration.

Figure 24. Approximate and actual gravitational accelerations.

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0

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12x 10

-3

-2v

(m/s

2 )

Time(s)

x y z

(a) Coriolis acceleration.

0 500 1000 1500 2000 2500-4

-3

-2

-1

0

1

2x 10

-3

-r (

m/s

2 )

Time(s)

x y z

(b) Centripetal acceleration.

Figure 25. Coriolis and centripetal acceleration terms.

V. Conclusion

In this paper, several subjects pertaining to asteroid proximity operations were investigated. Two dif-ferent gravitational models, a spherical harmonic expansion and a constant-density polyhedron, were con-sidered. The spherical harmonic model is suitable for orbital operations away from the asteroid’s surface.For operations near the surface, the polyhedron model must be used. Two phases of proximity operationswere examined. Orbital transfer from a high-altitude orbit to a low-altitude orbit was considered, followedby a soft landing mission starting from the low orbit. Orbital transfer was simulated with two differentZEM/ZEV algorithms, and landing was simulated with four different ZEM/ZEV algorithms. Numericalsimulations demonstrate that the ZEM/ZEV feedback guidance scheme performs well for a realistic landingmission.

Acknowledgments

This research work was supported by a research grant from the Iowa Space Grant Consortium (ISGC)awarded to the Asteroid Deflection Research Center at Iowa State University. The authors would like tothank Dr. Ramanathan Sugumaran (Director of the ISGC) for his support of this research project.

References

1Antreasian, P. G., Chesley, S. R., Miller, J. K., Bordi, J. J., and Williams, B. G., “The Design and Navigation of TheNEAR Shoemaker Landing on Eros,” AAS 01-372 , 2001.

2Park, R. S., Werner, R. A., and Bhaskaran, S., “Estimating Small-Body Gravity Field from Shape Model and NavigationData,” Journal of Guidance, Control, and Dynamics, Vol. 33, No. 1, 2010, pp. 212–221.

3Bryson, A. E. and Ho, Y.-C., Applied Optimal Control , New York: Wiley, 1975.4Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education

Series, Reston, VA, 1999.5D’Souza, C. N., “An Optimal Guidance Law for Planetary Landing,” AIAA 1997-3709 , 1997.6Ebrahimi, B., Bahrami, M., and Roshanian, J., “Optimal Sliding-mode Guidance with Terminal Velocity Constraint for

Fixed-interval Propulsive Maneuvers,” Acta Astronautica, Vol. 62, No. 10, 2008.7Furfaro, R., Selnick, S., Cupples, M. L., and Cribb, M. W., “Non-linear Sliding Guidance Algorithms for Precision Lunar

Landing,” AAS 11-167 , 2011.8Guo, Y., Hawkins, M., and Wie, B., “Optimal Feedback Guidance Algorithms for Planetary Landing and Asteroid

Intercept,” AAS 11-588 , 2011.9Guelman, M. and Harel, D., “Power Limited Soft Landing on an Asteroid,” Journal of Guidance, Control, and Dynamics,

Vol. 17, No. 1, 1994, pp. 15–20.10Hawkins, M., Guo, Y., and Wie, B., “Guidance Algorithms for Asteroid Intercept Missions with Precision Targeting

Requirements,” AAS 11-531 , 2011.

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11Winkler, T., Hawkins, M., Lyzhoft, J., and Wie, B., “Fuel-Efficient Feedback Control of Orbital Motions around anIrregular-Shaped Asteroid,” AIAA GNC Conference, 2012.

12Werner, R. A. and Scheeres, D. J., “Exterior Gravitation of a Polyhedron Derived and Compared with Harmonic andMascon Gravitation Representations of Asteroid 4769 Castalia,” Celestial Mechanics and Dynamical Astronomy, Vol. 65, No. 3,1996, pp. 313–344.

13Miller, J. K. et al., “Determination of Shape, Gravity, and Rotational State of Asteroid 433 Eros,” Icarus, Vol. 155, 2002,pp. 3–17.

14“NEAR-A-5-COLLECTED-MODELS-V1.0,” URL: http://sbn.psi.edu/pds/resource/nearbrowse.html.15Hawkins, M., Guo, Y., and Wie, B., “Spacecraft Guidance Algorithms for Asteroid Intercept and Rendezvous Missions:

A Review,” International Journal of Aeronautical and Space Sciences, Vol. 13, 2012, pp. 345–360.16Guo, Y., Hawkins, M., and Wie, B., “Applications of Generalized Zero-effort-miss/Zero-effort-velocity Feedback Guidance

Algorithm,” AAS 12-197 , 2012, to appear in the Journal of Guidance, Control, and Dynamics.17Scheeres, D. J., “Orbital Mechanics about Small Bodies,” Acta Astronautica, Vol. 72, 2012, pp. 1–14.18Rao, A. V., “User’s Manual for GPOPS Version 4.x: A MATLAB Software for Solving Multiple-Phase Optimal Control

Problems Using hp-Adaptive Pseudospectral Methods,” URL: http://www.gpops.org/gpopsManual.pdf.19Scheeres, D. J., Williams, B. G., and Miller, J. K., “Evaluation of the Dynamic Environment of an Asteroid: Applications

to 433 Eros,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 3, 2000, pp. 466–475.20Scheeres, D. J., Miller, J. K., and Yeomans, D. K., “The Orbital Dynamics Environment of 433 Eros,” AAS 01-373 ,

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zem/zev feedback guidance applicationto fuel-eﬃcient ...the gravitational potential of a constant...

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