ZEM/ZEV Feedback Guidance Application to Fuel-Efficient Orbital Maneuvers Around an Irregular-Shaped Asteroid

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,, USA The new vision for advanced missions to asteroids, including soft landing, presents many challenges that have essential differences from previous experiences with planetary landing. This paper focuses on two subjects pertaining to asteroid proximity operations: highaccuracy modeling of the gravitational environment and fuel-efficient guidance and control algorithm design. Both a spherical harmonic expansion method and a polyhedron shape model 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 an optimal control scheme, however it is conceptually simple and easy to implement, and in many cases it approaches optimality. Two mission phases, orbital transfer between observational orbits and soft landing, are numerically simulated using different implementations of the ZEM/ZEV algorithm. These simulations show that the ZEM/ZEV algorithm is suitable for asteroid proximity operations, and important considerations for using the algorithm are discussed. Nomenclature C nm P S nm ω λ A a D d f G g n R r V v ZEM ZEV Zonal spherical harmonic coefficients (normalized) Normalized associated Legendre function Tesseral spherical harmonic coefficients (normalized) Asteroid angular velocity vector Longitude angle Control acceleration vector expressed in an inertial frame Control acceleration vector expressed in a body frame Disturbance acceleration vector expressed in an inertial frame Disturbance acceleration vector expressed in a body frame Generalized acceleration vector expressed in a body frame Gravitational acceleration vector expressed in an inertial frame Gravitational acceleration vector expressed in a body frame Face-normal vector Position vector expressed in an inertial frame Position vector expressed in a body frame Velocity vector expressed in an inertial frame Velocity vector expressed in a body frame Zero-effort miss distance Zero-effort velocity error Ph.D. Candidate, Department of Aerospace Engineering, Iowa State University, 7 Howe Hall. AIAA Student Member. Visiting Student (-). Ph.D. Candidate, Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Peoples Republic of China Vance Coffman Endowed Chair Professor, Department of Aerospace Engineering, Iowa State University, 7 Howe Hall. AIAA Associate Fellow. of4

e E e F f p r e r f μ ω f φ ψ ρ τ A x A y d G H J L e M r R s t t go U u v V x V y X x Y y Unit vector along a coordinate direction Edge matrix Face matrix Co-state vector field point-to-edge position vector field point-to-face position vector Gravitational parameter Gradient operator Solid angle subtended by polyhedron face Partial derivative operator Latitude angle Asteroid attitude angle Asteroid density Dummy variable for integration Inertial X acceleration component Inertial Y acceleration component Signed distance Universal gravitational constant Hamiltonian function Performance index Potential of a -D wire Asteroid mass Radial coordinate Nominal asteroid radius Path length Time Time-to-go Gravitational potential Radial velocity component Transverse velocity component Inertial X velocity component Inertial Y velocity component Inertial X coordinate Body-fixed x coordinate Inertial Y coordinate 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. Scientific interest in these objects mainly focuses on what they can tell us about the evolution and diversity of our solar 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, detailed images of the surface can be taken as well as readings on the gravitational forces present. Such data can tell us more about the physical makeup of the body, which would help determine the most efficient deflection method. In order to evaluate the overall performance of the guidance and control system designed for the asteroid s proximity operations, one must establish a high-fidelity dynamic model to simulate the asteroid gravitational environment, where the gravitational acceleration and the centrifugal acceleration due to asteroid s rotation must be taken into account. A common method to describe the gravitational potential is the spherical harmonic expansion method, from which the gravitational acceleration can be determined. Previous studies have shown that it is the best model to simulate the asteroid s gravity field outside a circumscribing sphere about the asteroid. When the spacecraft completes proximity operations like soft landing or close flyby, the polyhedron shape model is more desirable to provide correct estimates of the asteroid s gravity field., of4

Bryson and Ho discussed optimal control laws for a simple rendezvous problem, considering both free terminal velocity and constrained terminal velocity. 3 They also discussed the relationship between optimal control and proportional navigation guidance. Battin also discussed an optimal terminal state vector control for 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 a computational method to determine the optimal time-to-go. Ebrahimi et al. proposed a robust optimal sliding mode guidance law for an exoatmospheric interceptor, using fixed-interval propulsive maneuvers. 6 In this paper, gravity was considered to be an explicit function of time. One major contribution of Ebrahimi et al. 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 accelerations are imparted. Furfaro et al. later employed the ZEM/ZEV concept to construct two classes of non-linear guidance algorithms for a lunar precision landing mission. 7 Guo et al. 8 showed that in a uniform gravitational field, the ZEM/ZEV logic is basically a generalized form of various well-known optimal feedback guidance solutions such as soft landing on an asteroid, 9 intercept or rendezvous, 3 terminal guidance, 4 and planetary landing. The performance of the ZEM/ZEV logic for an asteroid intercept mission with precision targeting requirements was evaluated by Hawkins et al., and compared with the performances of classical missile guidance methods like proportional navigation guidance (PNG) and augmented proportional navigation guidance (APNG). Two proximity operations are considered in this paper. Optimal orbital transfer from a high-altitude orbit to a low-altitude orbit is considered first. A soft landing on the surface of the asteroid starting from the low-altitude orbit is then considered. A fuel-efficient way to maintain the low-altitude orbit is described by Winkler et al. For the asteroid soft landing mission, the gravitational acceleration is neither constant nor an explicit function of time, but is instead a function of position, velocity and the rotation of the asteroid. The ZEM/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 the generalized 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 generalized optimal control problem and the ZEM/ZEV algorithm, two different ZEM/ZEV algorithms for orbital transfer are presented, followed by four different types of ZEM/ZEV optimal feedback control algorithms for the asteroid soft landing mission. These control laws are used to simulate proximity operations about the asteroid 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 the inertial frame (X, Y, Z). Typically the body-fixed frame is used for landing, and the inertial frame is used for 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 the spherical 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 for the asteroid considered over the time span considered. An illustrative figure using the body-fixed coordinate system is given in Figure. The inertial reference frame shares its origin with the body-fixed frame. The inertial axes are denoted as X, Y,andZ. The two frames are co-aligned at time t =. Unless otherwise specified, the asteroid is assumed to have an inertial orientation angle ψ =att =. 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 will have rotated during the time shown. Due to the asteroid s rotation, the spacecraft s angular position in the inertial frame is ψ + λ. Figure illustrates the inertial and body-fixed axes for the asteroid 433 Eros, considered in this study. 3of4

Spacecraft Figure. Coordinate system definition. y Y x X Figure. Inertial and body-fixed frames. 4of4

B. Dynamic Equation The equations of motion of the spacecraft are expressed in the asteroid s body-fixed frame as ṙ = v v = a + g (r) ω v ω (ω r)+d () where r is the spacecraft position, v is the spacecraft velocity, a is the applied control acceleration on the spacecraft, g (r) is the gravitational acceleration due to the asteroid, d includes disturbances, such as modeling uncertainties and perturbations from solar effects, and ω is the angular velocity vector of the asteroid. 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 acceleration term, ω v, and the centripetal acceleration, ω (ω r), into a generalized acceleration term, f (r, v, ω). Equation then becomes ṙ = v v = a + f (r, v, ω) () where r =[x, y, z] T. The equations of motion of the spacecraft in the inertial frame can also be expressed as Ṙ = V V = 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 defines the 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 cos (ωt) sin(ωt) r = CR where C = sin (ωt) cos (ωt) () C. Asteroid Shape and Gravity Model. Spherical Harmonic Expansion Method The gravitational potential of an asteroid can be modeled with a spherical harmonic expansion as U = GM n ( ) n R P m n (sin φ) [ Cnm cos (mλ)+ r r S nm sin (mλ) ] (6) n= m= where n and m are the degree and order, respectively, of the polynomials, G is the universal gravitational constant, M is the asteroid mass, R is the nominal radius, and r, φ, and λ are the radius, latitude, m and longitude, respectively. P n ( ) is the normalized associated Legendre function, and C nm and S nm are normalized spherical gravitational coefficients. The gravitational acceleration, needed for simulation of asteroid proximity operations, can be obtained by computing the gradient of the gravitational potential g (r, φ, λ) = U (r, φ, λ) = U r e r + U r φ e φ + U r cos φ λ e λ (7) of4

where Substituting the expressions for the unit vectors e r, e φ and e λ into Equation 7 gives U r = GM r U φ = GM r U λ = GM r cos φ cos λ sin φ cos λ sin λ g (r, φ, λ) = cos φ sin λ sin φ sin λ cos λ sin φ cos φ n { (n +) n= m= n {( R r n= m= n {( R n= m=. Polyhedron Shape Model r ( R U r U r φ r cos φ U λ ) n P m n (sin φ) [ Cnm cos (mλ)+ S nm sin (mλ) ]} (8) r ) n [ P m+ n (sin φ) m tan φ P n m (sin φ) ][ Cnm cos (mλ)+ S nm sin (mλ) ]} ) n m P m n (sin φ) [ Snm cos (mλ)+ C nm sin (mλ) ]} Werner and Scheeres give a comprehensive treatment of the polyhedron shape model, which will now be briefly described. The gravitational potential of a constant density polyhedron at a field point r can be described as U = Gρ e edges L e r T e E e r e Gρ f faces ω f r T f F f r f (9) where ρ is the asteroid s density, r e and r f are column vectors denoting the position from the field point to any point on the edge and face, respectively, and E e and F f are defined as E e = ˆn A (ˆn A ) T + ˆnB (ˆn B ) T F f = ˆn f ˆn T f where the face-normal vector n f and the edge-normal vector on face f, n f ij are defined as n f =(r r ) (r 3 r )=r r + r r 3 + r 3 r n f ij =(r j r i ) n f Figure 3 shows the face-normal and edge-normal vectors for a section of the polyhedron. Let r i be the vector from the field point to vertex P i,withlengthgivenby r i = r i,ande ij the length of the edge between vertices P i and P j. The potential of a -D straight wire is L e = e r ds ln r i + r j + e ij r i + r j e ij (i, j =,,...,N) () Let ω f be the signed solid angle subtended by each polyhedron face, viewed from the field point. It is found as det ([r i r j r k ]) ω f = r i r j r k + r i r T j r k + r j r T k r i + r k r T i r (i, j, k =,,...,N) () j The gravitational acceleration at the field point r can be found as g (r) = U (r) = Gρ L e E e r e + Gρ ω f F f r f () e edges f faces 6of4

Point Facet B Point Facet A 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 (ω f d f L ij d eij L jk d ejk L ki d eki ) ˆn f (3) f faces where d f is the signed distance between the field point and the face, L ij, L jk,andl ki are the edge potentials, and d eij, d ejk,andd eki represent the signed distances from the projection of the field point in each face plane to each of the three edges. Each d parameter is positive if it lies in the outward-pointing side of the corresponding edge, otherwise it is negative. The d parameters are computed as d f = ˆn T f r f d eij = ˆn T ijr i d ejk = ˆn T jkr j d eki = ˆn T kir k (4) Also of interest when computing the gravitational attraction is the Laplacian. It can be shown that Equation 6 can be differentiated to give U = Gρ ω f () f faces The sum ω f is zero when the field point is inside the polyhedron, and equals 4π inside the polyhedron. This term can be computed for almost no cost, since ω f is already found at every step, and provides a 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 and physical characteristics have been extensively studied, 3 and high-fidelity models exist for both the spherical harmonic expansion and the polyhedron. 4 These models allow comparisons between the two approaches. There are advantages and drawbacks to using either the spherical harmonic expansion model or the polyhedron 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 6 th degree and order involves 3 terms. The main drawback of the harmonic model is that it is only valid outside the smallest 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 of the model. The lowest-resolution model for asteroid Eros has 78 faces. The polyhedron model also assumes constant density, which is a reasonable approximation for Eros, but may not be for other asteroids. Adjusting for this would require introducing one or more additional polyhedrons, while it requires only modifying the 7of4

coefficients of the harmonic expansion. The polyhedron model is more accurate near bounding sphere, and is 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 the asteroid. Both models are useable in either the body-fixed or the inertial frame. The gravitational acceleration can 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 as the 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 algorithms can be found. Lowercase variables will be used in this discussion, as it is independent of the reference frame. For a mission from time t to t f, the optimal control acceleration is determined by minimizing the classical performance index of the form tf J = a T a dt (6) t subject to Equations or 3 and the following given boundary conditions: The Hamiltonian function for this problem is r (t )=r r (t f )=r f v (t )=v v (t f )=v f (7) H = at a + p T r v + p T v (g (t)+a) (8) where p r and p v are the co-state vectors associated with the position and velocity vectors, respectively. The co-state equations say that the optimal control solution can be expressed as a linear combination of the terminal values of the co-state vectors. Defining the time-to-go, t go,as t go = t f t (9) the optimal acceleration command at any time t is a = t go p r (t f ) p v (t f ) () By substituting the above expression into the dynamic equations to solve for p r (t f )andp v (t f ), the optimal control solution with the specified r f, v f,andt go is finally obtained as a = 6[r f (r + t go v)] t go (v f v) + 6 tf (τ t) g (τ) dτ t t go t go 4 t f t g (τ) dτ t go () The zero-effort-miss (ZEM) distance and zero-effort-velocity (ZEV) error denote, respectively, the differences between the desired final position and velocity and the predicted final position and velocity if not additional control is commanded after the current time. For the assumed gravitational acceleration g (t), the ZEM and ZEV have the following expressions [ tgo ZEM = r f r + t go v + [ tf ] ZEV = v f v + g (τ) dτ t t ] (t f τ) g (τ) dτ () (3) 8of4

The optimal control law, Equation, can be expressed as a = 6 t ZEM ZEV (4) go t go For certain missions where the terminal velocity is not specified, the optimal control law, in terms of ZEM only, can be obtained as a = 3 t ZEM () go The optimal control law to regulate only the terminal velocity, which will be used for orbital transfer, in terms of ZEV only, can also be obtained as a = ZEV (6) t go Since the gravitational acceleration cannot be simply modeled as a function of time, the ZEM and ZEV must be found by some means. Hawkins et al. describes how the ZEM and ZEV can be found by numerically 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-optimal level while maintaining acceptable computational complexity. For highly nonlinear systems, predicting the future states is prone to errors. Another alternative form of the ZEM/ZEV algorithm can be adopted for this situation. Rather than predicting the effect of the nonlinear terms, the effects of these terms are directly compensated for at all times. The algorithm thus approaches feedback linearization behavior. The control algorithm, Equation, then simply becomes the following form suggested by Battin 4 a = 6[r f (r + t go v)] t go B. ZEM/ZEV guidance for orbital transfer (v f v) t go g (r) (7) There are two implementations of ZEM/ZEV guidance to achieve orbital transfer. For proximity operations near an asteroid, it is assumed that the angular position at orbit insertion is unimportant. The longitude of the spacecraft would be relevant for spacecraft in a resonant orbit, but for this paper resonant orbits are avoided. The first orbital transfer formulation uses polar coordinates, and does not specify the angular position at orbit insertion. The second formulation uses Cartesian coordinates, and although no particular angular position is needed, one must be chosen to for the guidance law to work. For the short mission time of the orbital transfer, the asteroid can be modeled as a point mass, and differences in the gravitational force due to the point-mass model become disturbances for the control system to overcome. The two guidance schemes are described next.. ZEM/ZEV guidance in polar coordinates The objective of the orbital transfer problem here is to transfer a spacecraft from one circular orbit to another circular orbit. The terminal constraints are that the spacecraft should be placed at a specified distance from the asteroid with corresponding circular orbital velocity. The final radial velocity is zero, and the angular position is free. Due to the nature of the constraints, polar coordinates are used. The standard dynamical models for this type of orbit raising problem are described by ṙ = u u = v r μ r + a r (8) v = uv r + a t where r, u, andv represent the distance of the spacecraft from the center of mass of the asteroid, the radial velocity, and the transverse velocity, respectively, and a r and a t are the control accelerations in the radial 9of4

and transverse directions, respectively. The gravitational parameter for the asteroid is given as μ. Recalling the harmonic expansion model, we see that μ = GM. The required terminal states, as described above, are μ r (t f )=r f u (t f )= v (t f )= (9) r f The orbit raising problem is somewhat unusual in that the control requirements are different along the radial and tangential axes. In the radial direction, there are position and velocity requirements as usual. In the tangential direction, we have the rare case where only the velocity is specified. For asteroid proximity operations, the gravitational environment is highly nonlinear, so the ZEM/ZEV algorithm with direct compensation of gravitational terms, Equation 7, can be used as: a r = 6 t (r f (r + t go u)) ( v (u f u) go t go r μ r ( ) a t = t go (v f v) uv r ) (3) It is seen that the law is a combination of the ZEM/ZEV law, Equation 4, in the radial direction, and the ZEV law, Equation 6, in the tangential direction. For the orbital transfer problem between circular orbits, recall also that u f =.. ZEM/ZEV guidance in Cartesian coordinates Despite being posed in such a way as to ignore the final angular position, the compensating ZEM/ZEV law suffers from the nonlinear effects of the gravitational field. Guo et al. 6 shows this for a Mars orbital transfer example, and suggests ways to overcome this. The method suggested there is to use an offline optimal solution to generate a series of waypoints to track. This is less practical for the asteroid orbit problem, as the optimal 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 and using ZEM/ZEV guidance in Cartesian coordinates, is suggested. For the ZEM/ZEV orbital transfer problem in Cartesian coordinates, consider the following spacecraft dynamic equations Ẋ = V x Ẏ = V y (3) X V x = G (X, Y, ψ)+a x μ (X + Y ) 3 Y V y = G (X, Y, ψ)+a y μ (X + Y ) 3 + A x + A y (3) where (X, Y )and(v x,v y ) denote the position and velocity components in the inertial frame, and (A x,a y ) are control accelerations along the (X, Y ) axes. Although the true gravitational field is represented by the harmonic expansion signified by G (X, Y, ψ), the nonlinearities are such that it is better to use the approximate values on the right hand side of Equation 3 for numerical propagation of the orbit. The terminal conditions for the asteroid orbital transfer problem are as follows: X (t f )=X c Y (t f )=Y c V x (t f )=V xc V y (t f )=V yc (33) The equations of motion are strongly coupled, and an analytic optimal control algorithm does not exist. The ZEM/ZEV algorithm, Equation 4, can control the terminal position and velocity at a specified final time. These encompass all of the requirements of the orbital transfer problem, making it a good candidate of 4

for solution with ZEM/ZEV guidance. Expressed in the X- andy -coordinates, the proposed ZEM/ZEV law becomes [ ] [ ] [ ] A x = 6 ZEM x A y t ZEV x (34) go ZEM y t go ZEV y where the ZEM and ZEV are obtained by subtracting the predicted terminal states (with no further control accelerations) from the required terminal states, as follows: [ ] [ ZEM x X = c X ] f ZEM y Y c [ ] [ Ỹf ] ZEV x V = xc Ṽxf ZEV y V yc Ṽyf (3) (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 any practical asteroid proximity operation, the actual gravitational acceleration of the target asteroid will not be exactly known a priori, and its magnitude is often small in comparison to other accelerations and disturbances. 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 apparent accelerations, treating them as disturbances for the ZEM/ZEV feedback law to overcome. The ZEM/ZEV law with zero gravitational acceleration is called ZEM/ZEV-z. The next option is to directly compensate for the Coriolis and centripetal accelerations. This law makes use of the spin rate of the asteroid without getting into the details of the gravitational field. The ZEM/ZEV law 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 from the true field, it is still possible to account for gravity in the ZEM/ZEV law. The generalized gravitational acceleration is the acceleration due to a point mass. This generalized gravitational acceleration can be directly compensated 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 to consider the gravitational model. The predicted final states are found by propagating the equations of motion with point-mass gravity. The predictive ZEM/ZEV law is called ZEM/ZEV-p. We now have the four ZEM/ZEV laws as follows:. ZEM/ZEV-z:. ZEM/ZEV-a: a = 6 t (r f r)+ 4 v (37) go t go a = 6 t (r f r)+ 4 v +ω v + ω (ω r) (38) go t go 3. ZEM/ZEV-g: 4. ZEM/ZEV-p: a = 6 t (r f r)+ 4 v + GM r +ω v + ω (ω r) (39) go t go r 3 a = 6 t (r f r f )+ 4 (v ṽ f ) (4) go t go of 4

IV. Simulations and Results A. Low-Thrust Orbital transfer A low-thrust (non-impulsive) orbital transfer mission from a -km orbit to a 3-km orbit around asteroid Eros was simulated using both the polar and the Cartesian forms of the ZEM/ZEV algorithm. The spherical harmonic expansion gravitational model was used. The orbits are prograde in the XY -plane. In general, prograde planar orbits are the least stable, 7 so guidance that works for such orbits will be able to work with any other inclination. Once the spacecraft reaches the 3-km orbit, further proximity operations can begin. A detailed description of an algorithm to maintain a 3-km orbit via feedback control can be found in Winkler et al. The orbital transfer mission starts with the spacecraft on the positive X-axis, and the asteroid at an attitude angle of zero. The mission time is 4,34 seconds, corresponding to one half of the period of a Hohmann transfer between the two circular orbits. The exact mission time is not critical, as the true optimal solution would depend on angular positions in addition to transfer time. For such a short mission phase, a range of transfer times gives reasonable performance. The polar coordinate ZEM/ZEV algorithm does not require a final anomaly angle. For the Cartesian ZEM/ZEV algorithm, angular change is chosen as 3. The Cartesian version is limited to angular changes of less than 8. Figure 4 shows the transfer orbit trajectories for the two different algorithms. Also shown is the asteroid at its initial attitude. The normalized acceleration vector is shown every / th of the total time. Both algorithms command mostly radial acceleration at the beginning. Later on the Cartesian algorithm is able to find smaller commands than the polar algorithm. Figure shows these acceleration histories. Finally, Figure 6 shows the performance index histories. It is seen that the Cartesian algorithm has a lower performance index. 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 of greater than 8 and reduce the performance index. There is a practical limit to this, as for longer missions the polar algorithm will command the spacecraft to travel past the lower orbit, risking collision. 9 8 6 -km Orbit 3-km Orbit Transfer Orbit 9 8 6 -km Orbit 3-km Orbit Transfer Orbit 6 3 6 3 4 4 8 8 33 33 4 3 4 3 7 7 (a) Polar ZEM/ZEV. (b) Cartesian ZEM/ZEV. Figure 4. Transfer orbit trajectories. of 4

x -4 x -4 Radial acceleration Tangential acceleration Radial acceleration Tangential acceleration control acceleration, m/s - control acceleration, m/s - - - 3 4 Time (s) x 4 (a) Polar ZEM/ZEV. 3 4 Time (s) x 4 (b) Cartesian ZEM/ZEV. Figure. Transfer orbit acceleration histories. 6 x -4 Polar Cartesian 4 J (m /s 3 ) 3 3 4 Time (s) x 4 Figure 6. Performance index comparison. 3 of 4

B. Soft landing on an asteroid. Ideal spherical asteroid In order to demonstrate the effectiveness and optimality of the ZEM/ZEV guidance algorithm, a landing on an ideal spherical asteroid is first analyzed. The asteroid is assumed to have a diameter of 4 meters, and a total mass of kilograms. The rotation period of the asteroid is hours. Since the asteroid is a uniform sphere, the gravitational field is exactly known, as it is simply the gravitational field due to a point mass. The optimal solution is found using GPOPS, an open-source optimization software package. For more information on GPOPS, the reader is referred to Reference 8. An asteroid landing problem using the ZEM/ZEV-p law, Equation 4, was numerically simulated. The lander is initially on an equatorial orbit, with initial velocity of v =[,.49, ] T m/s, and an initial position of r = [3,, ] T m. The final conditions for the soft landing are: t f = s, r f =[, 4.3, 4.4] T m, v f =[,, ] T m/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 histories and performance index for the ZEM/ZEV law and the open-loop optimal solution. It is seen that the acceleration histories are nearly identical along the x- andz-axes, with small deviations along the y-axis. The performance index comparison verifies that the ZEM/ZEV approach is indeed nearly optimal. Trajectories of vehicle 3 z (m) - - -3-3 - - y (m) 3 3 - - -3 Figure 7. Soft-landing trajectories on a spherical asteroid using ZEM/ZEV and GPOPS.. Gravity field analysis for asteroid Eros The asteroid 433 Eros considered in this study has an irregular shape. The longest dimension is more than twice as long as the shortest dimension. There are also significant concavities. The harmonic expansion is only valid outside the smallest bounding sphere, so the polyhedron model must be used for operations near the surface. 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 78 to 7. 4 Here we choose 779 faces to generate results with acceptable accuracy and computational efficiency. Eros s density is considered to be a uniform value of 76 kg/m 3. The rotation period is.7 hours, giving an angular velocity of 3.3 4 rad/s. Three acceleration terms will be considered. The generalized gravitational acceleration is the gravitational force from a point mass on the surface of Eros. This is the gravitational acceleration used by the control laws. Using Equation 3, the acceleration from the polyhedron shape model can be found. Finally, given the angular velocity, the contribution from centripetal acceleration can be found. The Coriolis acceleration term does not allow for a similar analysis as it is a function of the spacecraft s velocity. In any case, for soft 4 of 4

6 x -4 4 Optimal ZEM/ZEV x -4. Optimal ZEM/ZEV a x (m/s ) a y (m/s ) - -. -4 x (m) (a) a x. - x (m) (b) a y. a z (m/s ) 4 x -4 3 Optimal ZEM/ZEV J (m /s 3 ) 6 x - 4 3 Peformance Index Optimal, J=.e- ZEM/ZEV, J=.e- - x (m) (c) a z. Time (s) (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 Coriolis acceleration at the surface cannot be compared with the other terms. Figure 9 shows the generalized gravitational magnitude at the surface. Figure shows the gravitational acceleration magnitude from the polyhedron shape model. Figure shows the acceleration due to the centripetal acceleration. From Figures 9 through, it is seen that the generalized and shape model gravity are in general agreement, as expected. The maximum generalized acceleration is.6 3 m/s, comparable to.3 3 m/s for the shape model. The minimum generalized acceleration is. 3 m/s,comparedto3.7 3 m/s for the shape model. The centripetal is smaller overall, ranging from zero for points on the z-axis up to 3 m/s. The accelerations shown are for the surface of Eros only. The gravitational acceleration decreases with distance, while the centripetal acceleration increases with distance. When the gravitational and centripetal 7, 9, accelerations add to zero, we have the well-known curves of zero velocity. 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, the spacecraft 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. Factors such as surface conditions, communications, relative position of the Sun and the Earth, and science value must all be taken into consideration. Most of these issues are not dealt with in this paper. Instead, two different landing sites are chosen, representative of two different general landing scenarios. The first is a landing from a circular 3-km polar orbit to a point near Eros s north pole. The second is a landing from a 3-km prograde equatorial orbit to a point on Eros s equator, near the intermediate inertial axis. Because the spacecraft 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 =, the spacecraft is at a latitude of 6, and the two frames are co-aligned. In the body frame, the initial velocity of 4

x -3. 4. z 4 - -. 3. - - -. x 4 3 y.. x. Figure 9. Generalized gravitational acceleration. x -3. 4.8 z - 4.6 4.4 - -. 4.. -. - x 4 4 3.8 y. x Figure. Gravitational acceleration from polyhedron shape model. 6 of 4

x -4 8 6 4 z - 8 - -. 6. -. - x 4 4 y. x Figure. Centripetal acceleration term. is in the negative x- andpositivez-directions due to the orbital velocity. There is also a component in the negative y-direction due to the rotation of the body frame relative to the inertial frame. The mission time is selected to be 4 s. The initial velocity is given as v =[ 3.93,.796,.78] T m/s. The initial position is r =[7.,, 3.3] T km. The landing site is chosen as r f =[ 3.67,.3934,.764] T km. For a soft landing, the final velocity is v f =m/s. Figures through show the trajectories and acceleration histories for the landing mission from a polar orbit. Figure 6 shows the performance index histories for the different ZEM/ZEV algorithms. It is seen that the simplest algorithm yields the highest performance index, while the full predictive algorithm achieves the minimum performance index. Including an approximated gravity term does not improve performance. For the ZEM/ZEV-p algorithm, Figure 7 compares the approximated generalized gravity term to the actual gravity term from the polyhedron model. There are significant differences in all three components. Figure 8 shows the acceleration terms due to the kinematics. These are seen to contribute accelerations of the same order as the gravity terms. For a real mission, the true gravity field will not be known, but the rotation rate of the body can be known fairly accurately. A real guidance algorithm would not benefit from adding gravity terms. The prograde orbit is in the inertial XY -plane, about the inertial Z-axis. At mission time t =,the spacecraft is at a longitude of -6, and the two frames are co-aligned. In the body frame, the initial velocity is in the positive x- andpositivey-directions due to the orbital velocity. There are also components in the negative x- andy-directions due to the rotation of the body frame relative to the inertial frame. At a distance of 3 km these terms dominate, so the total body-frame initial velocity is in the negative x- andy-directions. The mission time is again selected to be 4 s. The initial velocity is given as v =[ 6.946, 4., ] T m/s. The initial position is r =[7., 3.3, ] T km. The landing site is chosen on the positive x side as r f =[ 8.66, 7.643,.487] T km. Landing sites on the middle or negative x side are challenging due to the kinematic accelerations. For a soft landing, the final velocity is again v f =m/s. Figures 9 through show the trajectories and acceleration histories for the landing mission from a polar orbit. Figure 3 shows the performance index histories for the different ZEM/ZEV algorithms. The simplest algorithm clearly yields the highest performance index, while the other three ares similar, with the full predictive algorithm again achieving the minimum performance index. Including an approximated gravity term still does not improve performance. For the ZEM/ZEV-p algorithm, Figure 4 compares the approximated generalized gravity term to the actual gravity term from the polyhedron model. There are significant differences in all three components. Figure shows the acceleration terms due to the kinematics. These are seen to contribute accelerations of the same order as the gravity terms, and are overall more significant than they were for the polar landing. 7 of 4

3 3.4.3. z(km) a (m/s ). -. -. - - y(km) - - - x(km) -.3 (a) Trajectory. (b) Acceleration history. Figure. Trajectory and acceleration history, ZEM/ZEV-z. 3 3.4.3. z(km) a (m/s ). -. -. - - y(km) - - - x(km) -.3 (a) Trajectory. (b) Acceleration history. Figure 3. Trajectory and acceleration history, ZEM/ZEV-a. 8 of 4

3 3.4.3 z(km) a (m/s ).. -. -. -.3 - - y(km) - - - x(km) -.4 (a) Trajectory. (b) Acceleration history. Figure 4. Trajectory and acceleration history, ZEM/ZEV-g. 3 3.4.3. z(km) a (m/s ). -. -. - - y(km) - - - x(km) -.3 (a) Trajectory. (b) Acceleration history. Figure. Trajectory and acceleration history, ZEM/ZEV-p. 9 of 4

.7.6 ZEM/ZEV-z ZEM/ZEV-a ZEM-ZEV-g ZEM/ZEV-p. J (m /s 3 ).4.3.. Figure 6. Performance index comparison. 6 x -3 x -3 4 approximated g(r) (m/s ) - -4 actual g(r) (m/s ) - - -3-6 -4-8 - - (a) Approximate gravitational acceleration. -6 (b) Actual gravitational acceleration. Figure 7. Approximate and actual gravitational accelerations. 8 x -3 x -4 6 4 - v (m/s ) - r (m/s ) - -4 (a) Coriolis acceleration. - (b) Centripetal acceleration. Figure 8. Coriolis and centripetal acceleration terms. of 4

.4.3.. z(km) - - - y(km) -3-4 - - - - x(km) a (m/s ) -. -. -.3 -.4 (a) Trajectory. (b) Acceleration history. Figure 9. Trajectory and acceleration history, ZEM/ZEV-z..3.. z(km) - a (m/s ) - - y(km) -3-4 - - - - x(km) -. -. -.3 (a) Trajectory. (b) Acceleration history. Figure. Trajectory and acceleration history, ZEM/ZEV-a..3.. z(km) - a (m/s ) - - y(km) -3-4 - - - - x(km) -. -. -.3 (a) Trajectory. (b) Acceleration history. Figure. Trajectory and acceleration history, ZEM/ZEV-g. of 4

.3.. z(km) - a (m/s ) -. - - y(km) -3-4 - - - - x(km) -. -.3 -.4 (a) Trajectory. (b) Acceleration history. Figure. Trajectory and acceleration history, ZEM/ZEV-p..7.6 ZEM/ZEV-z ZEM/ZEV-a ZEM/ZEV-g ZEM/ZEV-p. J (m /s 3 ).4.3.. Figure 3. Performance index comparison. 3 x -3 x -3 4 approximated g(r) (m/s ) - actual g(r) (m/s ) 3 - - -3 (a) Approximate gravitational acceleration. - (b) Actual gravitational acceleration. Figure 4. Approximate and actual gravitational accelerations. of 4

x -3 x -3 8 - v (m/s ) 6 4 - r (m/s ) - - - -3-4 (a) Coriolis acceleration. -4 (b) Centripetal acceleration. Figure. Coriolis and centripetal acceleration terms. V. Conclusion In this paper, several subjects pertaining to asteroid proximity operations were investigated. Two different gravitational models, a spherical harmonic expansion and a constant-density polyhedron, were considered. 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 operations were examined. Orbital transfer from a high-altitude orbit to a low-altitude orbit was considered, followed by a soft landing mission starting from the low orbit. Orbital transfer was simulated with two different ZEM/ZEV algorithms, and landing was simulated with four different ZEM/ZEV algorithms. Numerical simulations demonstrate that the ZEM/ZEV feedback guidance scheme performs well for a realistic landing mission. 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 to thank Dr. Ramanathan Sugumaran (Director of the ISGC) for his support of this research project. References Antreasian, P. G., Chesley, S. R., Miller, J. K., Bordi, J. J., and Williams, B. G., The Design and Navigation of The NEAR Shoemaker Landing on Eros, AAS -37,. Park, R. S., Werner, R. A., and Bhaskaran, S., Estimating Small-Body Gravity Field from Shape Model and Navigation Data, Journal of Guidance, Control, and Dynamics, Vol. 33, No.,, pp.. 3 Bryson, A. E. and Ho, Y.-C., Applied Optimal Control, New York: Wiley, 97. 4 Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series, Reston, VA, 999. D Souza, C. N., An Optimal Guidance Law for Planetary Landing, AIAA 997-379, 997. 6 Ebrahimi, B., Bahrami, M., and Roshanian, J., Optimal Sliding-mode Guidance with Terminal Velocity Constraint for Fixed-interval Propulsive Maneuvers, Acta Astronautica, Vol. 6, No., 8. 7 Furfaro, R., Selnick, S., Cupples, M. L., and Cribb, M. W., Non-linear Sliding Guidance Algorithms for Precision Lunar Landing, AAS -67,. 8 Guo, Y., Hawkins, M., and Wie, B., Optimal Feedback Guidance Algorithms for Planetary Landing and Asteroid Intercept, AAS -88,. 9 Guelman, M. and Harel, D., Power Limited Soft Landing on an Asteroid, Journal of Guidance, Control, and Dynamics, Vol. 7, No., 994, pp.. Hawkins, M., Guo, Y., and Wie, B., Guidance Algorithms for Asteroid Intercept Missions with Precision Targeting Requirements, AAS -3,. 3 of 4

Winkler, T., Hawkins, M., Lyzhoft, J., and Wie, B., Fuel-Efficient Feedback Control of Orbital Motions around an Irregular-Shaped Asteroid, AIAA GNC Conference,. Werner, R. A. and Scheeres, D. J., Exterior Gravitation of a Polyhedron Derived and Compared with Harmonic and Mascon Gravitation Representations of Asteroid 4769 Castalia, Celestial Mechanics and Dynamical Astronomy, Vol. 6, No. 3, 996, pp. 33 344. 3 Miller, J. K. et al., Determination of Shape, Gravity, and Rotational State of Asteroid 433 Eros, Icarus, Vol.,, pp. 3 7. 4 NEAR-A--COLLECTED-MODELS-V., URL: http://sbn.psi.edu/pds/resource/nearbrowse.html. Hawkins, M., Guo, Y., and Wie, B., Spacecraft Guidance Algorithms for Asteroid Intercept and Rendezvous Missions: AReview, International Journal of Aeronautical and Space Sciences, Vol. 3,, pp. 34 36. 6 Guo, Y., Hawkins, M., and Wie, B., Applications of Generalized Zero-effort-miss/Zero-effort-velocity Feedback Guidance Algorithm, AAS -97,, to appear in the Journal of Guidance, Control, and Dynamics. 7 Scheeres, D. J., Orbital Mechanics about Small Bodies, Acta Astronautica, Vol. 7,, pp. 4. 8 Rao, 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. 9 Scheeres, 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. 3, No. 3,, pp. 466 47. Scheeres, D. J., Miller, J. K., and Yeomans, D. K., The Orbital Dynamics Environment of 433 Eros, AAS -373,. 4 of 4