ALGORITHMS FOR SAFE SPACECRAFT PROXIMITY OPERATIONS

Transcription

1 AAS ALGORITHMS FOR SAFE SPACECRAFT PROXIMITY OPERATIONS INTRODUCTION David E. Gaylor *, Brent William Barbee Emergent Spae Tehnologies, In., Greenbelt, MD, Future missions involving in-spae serviing, repair, inspetion, or rendezvous and doking require algorithms for safe, autonomous proximity operations. Algorithms for relative navigation, safe separation, and irumnavigation trajetory design are presented. The algorithms rely on safe, natural motion trajetories and ovariane information from relative navigation to minimize the probability of spaeraft ollision. The algorithms are desribed in the ontext of a spae inspetion mission where two attahed spaeraft separate and one irumnavigates the other. These algorithms an also be applied to a variety of rendezvous or other proximity operations missions, whether autonomous or manually operated. Future spaeraft missions requiring in-spae serviing, repair, inspetion, or rendezvous and doking need algorithms for safe, autonomous proximity operations. Emergent Spae Tehnologies, In. has developed algorithms for safe, autonomous proximity operations, inluding aurate relative navigation, spaeraft separation trajetory design, and irumnavigation trajetory design. The algorithms rely on safe natural motion trajetories and ovariane information from the relative navigation system to minimize the probability of ollision while meeting mission objetives. In this paper, we desribe the algorithms in the ontext of a hypothetial spae inspetion mission where two initially attahed spaeraft safely separate and then one spaeraft irumnavigates the other. The algorithms an also be applied to a variety of rendezvous missions or other missions requiring spaeraft to safely operate near eah other, whether autonomous or manually operated. An example is presented in whih safe natural motion trajetories are utilized to design a passively safe rendezvous trajetory for a spaeraft visiting the International Spae Station (ISS). SPACECRAFT MISSION SCENARIO The primary and inspetor spaeraft begin in a joined onfiguration in geosynhronous Earth orbit (GEO), with the inspetor attahed to the primary s outer hull. It is assumed that the primary spaeraft dos not ommuniate with the inspetor. However, it may be possible for the inspetor spaeraft to reeive state information for the primary spaeraft via ground station updates. The inspetor spaeraft may also be observing the primary using relative sensors, suh as radar, LIDAR, or optis. Suh relative measurements may inlude range, range-rate, and relative loation angles (e.g., azimuth and elevation). If the primary spaeraft is ompletely non-ooperative, only relative measurements, possibly ombined with absolute state updates from the ground, will be available to the inspetor. * Vie President, Aerospae Systems, dave.gaylor@emergentspae.om. Aerospae Engineer, brent.barbee@emergentspae.om. 1

2 The mission objetives are for the inspetor to safely separate from the primary, irumnavigate the primary and perform inspetion by traveling to six waypoints that offer views of the primary from all six faes of a virtual ube entered on the primary, and then enter into passively safe relative motion from whih further inspetions or other operations an be initiated, ultimately onluding with safe disposal for the inspetor at the end of its mission life, whih is beyond the sope of this study. COLLISION PROBABILITY Muh work has been done to address the omputing of ollision probability for neighboring spae objets.(ref. 1-9) Typially, one determines if and when a seondary objet will enter a user-defined safety zone. The unertainties assoiated with position are represented by three-dimensional Gaussian probability densities. These densities take the form of ovariane matries whih an be obtained from a sequential or bath orbit estimation proess. Positions and ovariane matries are propagated to the time of losest approah and the probability of ollision is usually estimated at this point. As will be detailed in a subsequent setion, the ovariane information an be used to define an error ellipsoid entered at the origin of the relative motion frame, whih oinides with the nominal loation of the primary spaeraft s enter of mass. Error ellipsoids are defined suh that there is a known probability that the primary spaeraft s enter of mass is within or on the surfae of the error ellipsoid. If the inspetor remains outside this ellipsoid then the probability of ollision is less than or equal to a ertain value. Keeping the inspetor spaeraft outside the hosen error ellipsoid is the safety strategy employed in this paper. RELATIVE NAVIGATION Aurate relative navigation is a key requirement for safe spaeraft proximity operations. An extended Kalman filter (EKF) an be used onboard one or both spaeraft to estimate the absolute position and veloity of both spaeraft based on GPS, differential GPS, state updates from the ground, or relative range, range-rate, and angle sensors. The relative state estimate an be formed by taking the differene between the estimated absolute position and veloity vetors for both spaeraft. Navigation is typially performed in the Earth-Centered Inertial (ECI) referene frame beause high fidelity equations of motion are available. If relative navigation is performed in the relative motion or Hill s frame, referred to as the Radial-In-trak-Cross-trak (RIC) frame in this paper, only simplified equations of motion an be used. One of the key outputs of the relative navigation system is the ovariane matrix, whih haraterizes the unertainty of the spaeraft position and veloity estimates. If we assume that the position errors are unorrelated, the 3 3 sub-matries of the ovariane matrix assoiated with the inertial position vetors of eah spaeraft an be added together, resulting in the Joint Covariane Matrix (JCM). The JCM is used to generate an error ellipsoid entered on the primary spaeraft. This error ellipsoid is omputed suh that the enter of mass of the primary spaeraft is within or on the surfae of the ellipsoid with probability1 P, where P is the desired imum probability of ollision. Hene, if the desired P is 2 % then the error ellipsoid about the primary spaeraft is omputed suh that the primary is known to be within or on the surfae of the ellipsoid with 98% ertainty. Figure 1 shows the dimensions of an error ellipsoid in its prinipal frame and the same ellipsoid in its orientation in the RIC frame. The error ellipsoid is always entered at the origin of the RIC frame, where the enter of mass of the primary spaeraft is nominally loated. 2

3 z PRIN Ĉ a y PRIN Î x PRIN b Figure 1 Error Ellipsoid in the Prinipal Frame and in the RIC Frame The parameter σ is the standard deviation of the error ellipsoid. This value must be hosen to orrespond to P aording to Eq. (1). (Ref. 9) 1 2 σ σ 2 2 ( 1 P ) = erf σ e (1) 2 π Thus if P is 2% or 0.2, 1 P is equal to 0.98, and Eq. (1) is solved for this value. Sine Eq. (1) annot be solved analytially, an effiient adaptive solver algorithm was written that brakets the interval on whih σ lies and then samples the interval at an adaptive resolution until σ is found within a speified tolerane. This value of σ is then used to ompute the error ellipsoid dimensions. For a P of 2%, σ is SAFE SPACECRAFT SEPARATION The inspetor spaeraft must safely separate from the primary in order to begin mission operations. The separation may be aomplished using a spring or other mehanism to provide the inspetor spaeraft with its initial relative veloity for separation, Δv SEP. Figure 2 illustrates the onfiguration at the time of separation. Rˆ X BODY p?? Primary Spaeraft r ATT? Δ v SEP Free Drift Trajetory Figure 2 Spaeraft Configuration at the Time of Separation The inspetor spaeraft is attahed to the outer hull of the primary at some attahment point and the primary spaeraft s body frame is at some orientation with respet to the RIC frame and possibly rotating as well, as shown in Figure 2,. The separation impulse, Δv SEP, is therefore a funtion of the primary s orientation, the harateristis of the deployment mehanism (e.g., spring onstant), and the primary s attitude and attitude rates. 3

4 The magnitude of Δv SEP is determined by the harateristis of the deployment mehanism and the mass of the inspetor spaeraft. The orientation of Δv SEP is ditated at any instant in time by the orientation of the primary. The angular veloity of the primary, if any, also ontributes to Δv SEP. We assume that it is possible to speify an orientation and angular veloity for the primary at the time of separation and thus are able to hoose the diretion of Δv SEP. We further assume in this study that the primary has zero angular veloity, though the analysis that follows is easily modified to inlude a non-zero angular veloity for the primary, as shown in Eq. (3). Figure 3 illustrates the orientation of Δv SEP in terms of two spherial oordinate angles, α (azimuth) and δ (elevation). The objetive of the safe separation study is to hoose the azimuth and elevation that imize the losest approah distane between the inspetor and the primary over 7 days. Rˆ Ĉ α Δ v SEP Figure 3 Separation Maneuver Parameterization Evaluation of the objetive funtion given in Eq. (4) requires propagation of the equations of motion for both spaeraft. The inertial initial position vetor for the inspetor spaeraft is given in Eq. (2) and the initial veloity vetor, inluding a non-zero angular veloity vetor for the primary spaeraft and the separation maneuver, is speified in Eq. (3). ( ) ( ) δ r t r t T r BODY i 0 = p 0 + ECI ATT ECI ECI BODY Î (2) ( ) ( ) ( 0 0 ω / ) v t v t r T v RIC i = p + BODY ECI ATT + ECI Δ SEP ECI ECI BODY RIC min ( () () ) peci ieci P = r t r t (3) (4) Sine evaluation of P requires the numerial integration of the orbital equations of motion for both spaeraft using dynamis models of adequate fidelity, the problem of imizing P does not lend itself to analytial optimization methods. While numerial optimization methods exist that may be able to α, δ in less time than is required to map out the entire solution spae determine the optimal ( ) methodially, knowing the sensitivity of the optimal solution to hanges in ( α, δ ), as well as the ranges of ( α, δ ) that meet the probability of ollision requirements, is important sine in pratie the exat optimal values will never be perfetly realized and there may be limitations on the available orientations for the primary spaeraft. Therefore the entire solution spae for P is mapped out as a funtion of ( α, δ ) and presented in Figure 4. From Figure 5 it is lear that there is a globally optimal solution and a spread of solutions that are still advantageous. 4

5 Figure 4 Three Dimensional Performane Index Surfae in Azimuth-Elevation Spae The primary and inspetor spaeraft begin olloated in GEO. The optimal solution for a Δ vsep of 0.1 m/s is α = 90, δ = 0, and the assoiated imum minimum approah distane over 7 days is km. SAFETY ELLIPSE THEORY A safety ellipse is an out-of-plane elliptial periodi relative motion trajetory around the primary spaeraft suh that the trajetory never rosses the veloity vetor of the primary. In a safety ellipse trajetory, drift of the two spaeraft, due to relative state estimation errors or other problems, will not result in ollision, so the trajetory is onsidered passively safe. (Ref. 10) A safety ellipse may be entered on the primary spaeraft (the origin of the RIC frame) or another point along the primary spaeraft in-trak diretion. A safety ellipse may remain stationary (stati) or it may drift along the in-trak diretion. A safety ellipse that drifts along the in-trak axis is alled a walking safety ellipse (WSE). Examples of entered and offset stati safety ellipses are shown in Figure 5. Note that safety ellipses are always rotated out of the primary spaeraft s orbit plane and are always rotated solely about the radial axis. By definition a safety ellipse may never have a rotation angle of 0 or 90 about this axis. Z RIC Offset Safety Ellipse Safety Ellipse Centered on Primary Spaeraft y Y RIC Inspetor Spaeraft X RIC Orbit Plane Crossing Node Figure 5 Primary-Centered and Offset Stati Safety Ellipses A dynamial derivation of the safety ellipse equations of motion is found in Ref. 10. However, we present a geometrial approah to illustrate key features of the safety ellipse relevant to safe irumnavigation trajetory design. This geometrial approah permits derivation of the equations of 5

6 motion for entered and offset stati safety ellipses but does not lead diretly to the full equations of motion that also desribe WSEs. To begin, we define a safety ellipse referene frame where the X E and Y E axes lie in the plane of the safety ellipse. This referene frame is rotated by an angle θ with respet to the RIC frame and this is depited in Figure 6a. Figure 6b shows the plane of the safety ellipse motion and introdues a polar angle in this plane, referened to the X E axis, denoted by χ. This polar angle speifies the loation of the inspetor spaeraft on the safety ellipse at any time. Z RIC X E Y E Y E X RIC Z E θ Y RIC Safety Ellipse Plane (a) (b) Figure 6 (a) Safety Ellipse Referene Frame and (b) Safety Ellipse Plane with Polar Angle Noting also the semi-axis lengths of the safety ellipse, a SE and b SE, it is possible to write the position vetor of the inspetor in the safety ellipse frame, given in Eq. (5). r E χ a SE b SE X E ( χ ) ( χ ) xe ase os re = y E = bse sin z E 0 (5) Sine the Y E axis is aligned with the the safety ellipse is the imum distane of the inspetor spaeraft along the as x. This ondition is expressed in Eq. (6). X RIC axis, as seen in Figure 6a, it is lear that the semi-minor axis of X RIC axis, denoted bse = x (6) To express the RIC position vetor of the inspetor spaeraft along the safety ellipse, it is neessary to develop the transformation between the RIC and safety ellipse frames. Figure 9 illustrates the neessary rotation about the X RIC axis by θ and shows the basis vetor orrespondenes. Examination of Figure 7 leads to the matrix that transforms vetors between the RIC frame and the safety ellipse frame, presented in Eq. (7) E TRIC = os( θ) 0 sin ( θ) sin ( θ ) 0 os( θ ) (7) 6

7 Substituting Eq. (6) into Eq. (5) and then multiplying by Eq. (7) leads to the RIC position vetor for the inspetor spaeraft, given in Eq. (8). Z RIC X E X, Y RIC E θ Y RIC θ Figure 7 Rotation between the RIC Frame and the Safety Ellipse Frame Note that the RIC position vetor for the inspetor spaeraft in Eq. (16) is parameterized by the safety ellipse polar angle, the angle by whih the safety ellipse is rotated out of the primary s orbit plane, the imum radial axis extent of the safety ellipse, and the semi-major axis of the safety ellipse. It is desirable to find a means to remove the dependene on the safety ellipse semi-major axis so that the position vetor onsists solely of terms that are related diretly to the RIC frame. Z E ( χ ) ( χ ) os( θ ) ( χ ) sin ( θ ) x sin E rric = TRICrE = ase os ase os (8) This is aomplished by examining a view of the safety ellipse edge-on in the In-trak-Cross-Trak plane of the RIC frame, shown in Figure 8. Z RIC 2x Δz a SE θ z Y RIC Δy Safety Ellipse (Edge View) Figure 8 In-trak-Cross-Trak View of the Safety Ellipse, Edge-On In Figure 8 the rotation angle θ appears, along with a new quantity, z, whih is the imum extent of the safety ellipse motion along the Z RIC axis. Figure 10 illustrates the desired relationships between the safety ellipse semi-major axis, x, and z, shown in Eqs. (9). 7

8 a a SE SE ( θ ) ( θ ) os = 2x sin = z (9) Substituting Eqs. (9) into Eq. (8) yields Eqs. (10), whih speify the position of the inspetor spaeraft in the RIC frame solely as a funtion of the safety ellipse motion plane polar angle, χ, noting that x and z are onstants. Note also that the term y is added to the y RIC oordinate to allow the onstrution of the offset stati safety ellipses mentioned previously. A non-zero y allows the enter of the safety ellipse to be positioned anywhere along the In-trak axis. ( χ) = x sin ( χ) ( χ) 2 os( χ) ( χ) = z os( χ) x y = x + y z The RIC veloity of the inspetor spaeraft on the safety ellipse is omputed by taking the first time derivative of Eqs. (10), but first the time dependene of the polar angle must first be speified. Sine the period of the safety ellipse motion is equal to the period of the primary spaeraft s orbit, the time dependene of the polar angle depends on the mean motion, n, of the primary, as shown in Eq. (11) (Ref. 2). χ t = n t t χ = 0 (11) () ( ) Thus, taking the first time derivative of Eqs. (10) yields the inspetor spaeraft RIC veloity, given in Eqs. (12). x ( χ) = xnos( χ) y ( χ) = 2xnsin( χ) + y (12) z χ = z nsin χ ( ) ( ) Eqs. (10) and (12) are the equations of motion for entered and offset stati safety ellipses. To arrive at the omplete safety ellipse equations of motion that also aount for WSEs, period-mathing onstraints and phase spae analyses are invoked (Ref. 2) and this derivation is omitted here for brevity. The full safety ellipse equations of motion are presented in Eqs. (13) (Ref. 6). (10) x 2y = 3n ( χ) x sin ( χ) ( χ) = z os( χ) ( χ) = os( χ) ( χ) 2 sin( χ) ( χ) = sin ( χ) ( χ π ) y y( χ) = 2x os( χ) + n 2 + y z x x n y = x n + y z z n (13) It is of interest to note that the safety ellipse motion plane polar angle, χ, also arises in the dynamial derivation of safety ellipse motion as the X RIC phase spae angle. The dynamial derivation also referenes the Z RIC phase spae angle, denoted as γ, and Eq. (14), whih is the ore safety ellipse riterion, makes it lear that γ is also a safety ellipse motion plane polar angle. 8

9 π ψ = χ γ = (14) 2 Thus χ (and γ ) have meaning in both phase spae and Eulidean spae. In terms of the safety ellipse equations of motion presented in Eqs. (13), we find it most instrutive to think of χ as the safety ellipse motion plane polar angle and this is the motivation for the presentation of the geometrial derivation of the equations of motion in lieu of the dynamial derivation. WSE motion is shown from a perspetive view in Figure 8 and in the RIC planes in Figure 9. Figure 8- Perspetive View of Walking Safety Ellipse Motion Figure 9 (a) In-trak-Cross-Trak View of a Walking Safety Ellipse and (b) Radial-Cross-Trak View of a Walking Safety Ellipse Note that the WSE forms a tubular manifold around the in-trak axis. This fat will be utilized in the safe irumnavigation trajetory design presented in a subsequent setion. Injetion onto a safety ellipse is possible in one of two modes. In the first mode the position vetor of the inspetor spaeraft already orresponds to a safety ellipse and a simple veloity hange maneuver aomplishes the injetion. In the seond mode the inspetor spaeraft s position does not satisfy the safety ellipse position equations and so the inspetor must perform a maneuver to transfer to a viable injetion point and then perform a seond veloity hange maneuver to ahieve injetion. One means of ahieving injetion is summarized by Eqs. (15) and Eq. (16). 2y Δ x = x x sin ( χ ) 3n Δ z = z z ( os( χ )) ( χ π ) y 2 Δ y = y 2x os( χ ) + + y n (15) 9

10 2 2 2 Δ r = Δ x +Δ y +Δ z (16) In Eqs. (15), x, y, and z are the RIC position oordinates of the inspetor spaeraft at the time at whih it is desired to injetion onto a safety ellipse. Eq. (16) expresses the distane between the inspetor spaeraft and the nearest viable injetion point and this equation is numerially minimized to determine the losest available injetion point. In the ase where the inspetor is already at a viable injetion point, the minimum of Eq. (16) is zero. If the minimum Δ r is greater than zero, time of flight along the trajetory to the injetion must be hosen and a maneuver omputed. Upon arrival at the injetion point, the injetion maneuver itself is performed. A numerial optimizer an be used to solve for the set of safety ellipse parameters { x, z, χ, y, y } that minimizes Eq. (16). These parameters may all be left free, or some may be speified a priori. In either ase, one the safety ellipse parameters have been omputed, the position of the injetion point is speified by the position portion of Eqs. (13) if it does not oinide with the inspetor s urrent position and the required veloity at the injetion point is given by the veloity portion of Eqs. (13). If v CURR is the inspetor s veloity vetor at the time of arrival at the injetion point, then the injetion maneuver is given by Eq. (17). Δ v = v v (17) CIRCUMNAVIGATION TRAJECTORY DESIGN INJECT SE CURR Cirumnavigation of the primary by the inspetor requires the inspetor to travel to six waypoints. Eah of the six waypoints lies on the fae of a virtual ube entered on the nominal position of the primary. These six points are also the verties of an otahedron entered on the primary suh that the otahedron is irumsribed by the virtual ube. Although attitude dynamis are not modeled, it is assumed that the inspetor ontinually performs attitude maneuvers that keep the inspetion sensor pointing inward and diretly at the primary while the irumnavigation is performed as the inspetor visits eah of the six waypoints in turn. The design of the irumnavigation trajetory segments inludes onsiderations for safety as well as the total Δ v required to omplete the irumnavigation. In partiular, a waypoint range, R WP, and Time of Flight (TOF) per trajetory segment will be hosen. It is assumed in this study that the TOF is the same for all trajetory segments. R WP is hosen to satisfy safety requirements and an algorithm is desribed that allows the seletion of the TOF to satisfy total Δ v requirements. Figure 10 depits the irumnavigation otahedron for whih the waypoints all lie on the RIC axes. Cˆ Waypoints Primary Spaeraft Î Rˆ Figure 10 Prototypial Cirumnavigation Otahedron 10

11 The otahedron is transformed into a Safety Otahedron suh that none of the otahedron s edges (possible trajetory segments) ross the primary spaeraft s veloity vetor. The SO, shown in Figure 11, also has the property that all the waypoints lie on the surfae of a WSE manifold. Ĉ + Î Rˆ Figure 11 Radial-Cross-Trak View of a Safety Otahedron WAYPOINT TRAVERSAL ORDERING This study determined that there is a minimum total Δ v waypoint traversal order and that this order is independent of both the waypoint range and the TOF per trajetory segment. However it was found that the minimum Δ v order does depend on the orientation of the SO, meaning that the minimum Δ v waypoint order analysis must be performed eah time the orientation of the SO hanges. This also means that the minimum Δ v waypoint order an be seleted prior to the final determination of waypoint range or TOF per trajetory segment. The parameter to be minimized is the total Δ v for the irumnavigation sequene, given in Eq. (18), where N is the number of waypoints and the subsript i referenes a partiular waypoint. N 1 Δ vtotal = Δv i + i 1 (18) i= 1 The minimum Δ v waypoint order analysis onsists of omputing the total Δ v required for eah permutation of waypoint order. The set of permutations onsidered is formed subjet to the following onstraints: first, no trajetory segment is allowed to pass through the origin and seond, in the ase where the inspetor spaeraft returns to the first waypoint (a losed trajetory sequene), waypoints are not allowed to repeat. Results for the non-losed trajetory sequene are presented in this setion and the results have the same harater for the losed sequene. In the non-losed ase the inspetor simply visits all six waypoints and does not return to the first waypoint. The results of sanning all waypoint order permutations are presented in Figure 12. Figure 12 Total Cirumnavigation Δv for eah Viable Waypoint Order Permutation 11

12 While the number of viable permutations is large, on the order of hundred or thousands, a typial modern desktop omputer an sample them all in a matter of seonds. Although it is not neessary to ompute the waypoint range and TOF prior to performing the optimal waypoint analysis, doing so illustrates the differene between the imum and minimum total Δ v, quantifying the benefit of using the optimal order. For this example, a waypoint range of m and a TOF per trajetory segment of 1200 seonds were used. Figure 16 shows that the total Δ v for the worst waypoint order is 8.43 m/s and the Δ v for the optimal waypoint order is 5.56 m/s, a redution of approximately 34% from the worst ase. The optimal waypoint order is given in Table 1 along with the original waypoint order on the prototypial otahedron. Note that transformation of the prototypial otahedron to a SO does not hange the waypoint order. Table 1 Optimal Waypoint Order and Original Waypoint Order for Prototypial Otahedron Optimal Waypoint Order Original Waypoint Order Waypoint Loations on Prototypial Otahdron x, 0, x, 0, , + y, 0 WAYPOINT RANGE SELECTION 2 4 0, y, , 0, + z 6 6 0, 0, z The seletion of waypoint range is typially a trade-off between safety and sensor performane. The waypoint range also affets the total Δ v required to omplete the irumnavigation sequene. We will examine the seletion of waypoint range solely to ensure safety. Total Δ v requirements will be met by seleting the TOF per trajetory segment. The waypoint range is shown graphially in Figure 13a. In order to meet a safety requirement that P is kept to a imum of 2%, we irumsribe a sphere, termed the Safety Sphere (SS) around the σ error ellipsoid, realling that this error ellipsoid is derived suh that there is a 98% ertainty that the primary spaeraft s enter of mass is within or on the surfae of this ellipsoid. We then irumsribe the SO about the SS, whih means that none of the possible trajetory segments (otahedron edges) will penetrate the SS and hene never penetrate the 98% error ellipsoid, thus satisfying the requirement that P is 2 %. Ĉ Safety Sphere Boundary Ĉ R WP Î R SS a 45 Î a a 2 2 Rˆ (a) (b) Figure 13 (a) Safety Sphere Cirumsribing the Error Ellipsoid About the Primary Spaeraft and (b) Planar View of Safety Otahedron Cirumsribing the Safety Sphere 12

13 Figure 13b shows a planar view of the SO irumsribed about the SS and shows the omputation of the waypoint range as a funtion of the radius of the SS, whih is equal to the longest axis, a, of the error ellipsoid. The relationship between the waypoint range and a is illustrated in Figure 13b and expressed in Eq. (30). RWP = a 2 (19) The waypoint range is modified suh that it aounts for the extent of the physial struture of both spaeraft as well as a seletable margin of safety, as shown in Figure 14. y SS Boundary Primary Spaeraft Strutural Bounding Sphere Error Ellipsoid Boundary a R p M SAFE Nominal Trajetory x Inspetor Spaeraft Strutural Bounding Sphere R i Margin of Safety Figure 14 Waypoint Range Augmentation Eq. (20) expresses the waypoint range as a funtion of the longest error ellipsoid axis, the strutural bounding spheres around both spaeraft, and the seleted margin of safety. R = a + R + M + R (20) ( ) WP p SAFE i 2 For the example irumnavigation results that are presented in a subsequent setion, a sample ovariane matrix was formed and values of 3, 0.5, and 10 m were hosen for R p, R i, M SAFE, respetively. The resulting waypoint range is m. It is of note that when irumsribing a sphere around the error ellipsoid, empty volume remains between the ellipsoid and the sphere that ould theoretially be utilized but isn t. If the error ellipsoid is too oblong this volume of unused but theoretially safe spae between the ellipsoid and the sphere beomes rather large and it may be desirable to take another approah to safety that the does not fore the inspetor spaeraft unneessarily far from the nominal loation of the primary. One possible approah is to rotate the SO suh that the oblong ellipsoid passes through the SO without rossing any possible trajetory segments, as shown in Figure 15. This reorientation of the SO may ause one of the possible trajetory segments to ross or nearly ross the primary s veloity vetor, whih is ontrary to the definition of the SO, so this possibility must be balaned against the benefits of reorienting the SO to pass around the very oblong ellipsoid. The primary benefit is that the inspetor spaeraft an get muh loser than it ould otherwise to the primary while still maintaining the desired probability of ollision. (a) (b) Figure 15 (a) Perspetive View and (b) Planar View of Safety Otahedron Reoriented to Pass around a Very Oblong Error Ellipsoid 13

14 TIME OF FLIGHT (TOF) SELECTION An algorithm has been developed to determine the TOF and the waypoint range that satisfy a given total Δ v requirement for the irumnavigation sequene. This is useful for keeping the total fuel usage for irumnavigation at an ahievable level and for providing a means to selet TOF. The only other relevant effet of TOF is the urvature of eah trajetory segment; the longer the TOF, the more eah trajetory segment deviates from a straight line between waypoints. The two main inputs to the algorithm are a vetor of TOFs and a vetor of waypoint ranges. In the ase where the waypoint range has already been hosen for safety, this single hosen waypoint range replaes the vetor of waypoint ranges, reduing the design spae from two dimensions to one. Additionally the target total Δ v is provided, along with a tolerane. The final inputs are a desired waypoint range (if the waypoint range is left free) and a desired TOF. The algorithm forms the design spae and adaptively sans it until finding at least one solution that meets the target total Δ v within tolerane. If more than one solution is found, the system indiates whih solution is losest to the desired TOF and waypoint range. Total Δ v is omputed via Eq. (18), as previously disussed in the setion on optimal waypoint ordering. Example system output is shown in Figures 16a and b. Figure 16a shows the output when the design spae only onsists of TOF (waypoint range is already hosen) and Figure 16b shows the output when the design spae onsists of both TOF and waypoint range. (a) (b) Figure 16 Algorithm Output for (a) TOF Only and (b) for TOF and Waypoint Range In Figure 16a the waypoint range has been fixed at m and the urve shows total Δ v as a funtion of TOF in seonds. The algorithm was given a target total Δ v of 5 ± 1 m/s and determined that the losest solution is a TOF of 1200 se, whih yields a total Δ v of 5.75 m/s. This Δ v is redued to 5.56 m/s one the optimal waypoint order is determined, as desribed in a previous setion. The omputed TOF of 1200 seonds per trajetory segment will be utilized in the subsequent setion in whih the simulation results for an example irumnavigation mission are presented. In Figure 16b the same parameters are used exept that a vetor of waypoint ranges is provided. The resulting solution surfae is a slightly warped plane of total Δ v values. FITTING SAFETY OCTAHEDRON WAYPOINTS TO A WALKING SAFETY ELLIPSE Given the orientation of a safety otahedron, it is desirable to plae the waypoints on a WSE manifold. From Figure 8 it is lear that WSE motion forms a tubular manifold around the in-trak axis with an elliptial ross-setion. It is possible to ompute WSE parameters and slightly modify the waypoint range suh that, for the partiular safety otahedron orientation, all the waypoints lie on a WSE manifold. This allows the inspetor spaeraft to diretly injet onto the WSE from any waypoint. Figure 17 (not to sale) illustrates the relevant geometries. 14

15 Rˆ Walking Safety Ellipse Manifold Boundary Ĉ x x 2y + 3n 2y 3n Î R WP Waypoint R os( 45 ) 45 WP Î Figure 17 Geometri Relationships to Walking Safety Ellipse Manifold Boundary One of the key onepts illustrated in Figure 17 stems diretly from the x position equation in Eqs. (13), whih is that y is not zero beause a WSE is being onsidered and this introdues a bias in the x diretion that must be aounted for. First the waypoint range is extended by a small but omfortable margin in antiipation of the entire assembly of waypoints shifting in the radial diretion, and this is shown in Eq. (21). Note that the waypoint range omputed in previous setions, m, inluded the small effet shown in Eq. (21). 4y R WP = RWP + (21) n This requires that the walk rate of the WSE must be hosen ahead of time, prior to waypoint set onstrution. One the waypoint range has been modified aording to Eq. (21), the x parameter for the WSE is set to the negative of the waypoint range as shown in Eq. (22). x = R (22) WP Note that the geometry in Figure 21 indiates how to ompute the appropriate z in terms of the waypoint range, given in Eq. (23). z = R WP 2 (23) Finally, the bias shown in Figure 17, whih is due to the influene of the walk rate term in the x position equation in Eqs. (13), is applied to all waypoint oordinates as shown in Eq. (24). 2y WPi = WPi 0 0 3n T (24) The results of these alulations are a set of waypoint oordinates that are fitted to a WSE and a partial set of parameters for the WSE, these being { x, z, y }. Thus, to injet onto the WSE from any waypoint, Eq. (16) is minimized using Eqs. (15) with the { x, z, y } parameters fixed to the values desribed in this setion. The result is that the minimum of Eq. (16) is zero, meaning that the waypoint position orresponds to a loation on the WSE as intended. The minimum solution onsists of the χ and y that orrespond to the waypoint that is being injeted from. This χ and y, along with the pre- x, z are used so that diret WSE injetion maneuver an be omputed using Eq. (17). omputed { } 15

16 It is important to note that the radial bias introdued by the non-zero y is small and manageable when y is small ompared to the magnitude of mean motion, n, of the primary spaeraft s orbit. Therefore are must be exerised to selet a y that is small enough to not pull the WSE manifold s enter too far off of the in-trak axis sine the goal is to have motion that is nominally entered about the primary spaeraft. WAYPOINT DRIFT-OUT ANALYSIS One possible failure mode of the inspetor spaeraft is to perform a maneuver to embark upon a trajetory segment to get to the next waypoint in the sequene and afterwards lose the ability to make any subsequent maneuvers. In this ase the inspetor will nominally arrive at the targeted waypoint and then depart that waypoint on a free drift trajetory. While beyond the sope of this paper, waypoint drift-out analyses have been performed using the full non-linear equations of orbital motion for the safety otahedron irumnavigation desribed herein. It is interesting to note that beause all the waypoints lie on a WSE manifold, the inspetor spaeraft s state spae at these points is always near WSE state spae, further evidened by how small in magnitude the WSE injetion maneuvers are from these points. Figure 18 demonstrates the WSE-like nature of drift-out motion from a waypoint; this nature is present in drift-out from all waypoints. (a) (b) Figure 18 (a) Perspetive View of Relative Motion and (b) Distane versus Time for 7-day Drift of Inspetor from Waypoint 5 of Non-Closed Trajetory Sequene EXAMPLE SAFE CIRCUMNAVIGATION MISSION RESULTS A sample irumnavigation mission using a hosen ovariane matrix was simulated in software using all the algorithms and parameters desribed thus far, along with standard Clohessy-Wiltshire (CW) targeting algorithms to demonstrate the effiay of the tehniques developed in this paper. The numerial values of the relevant parameters have been presented in their respetive setions throughout this paper. Figures 19a and b present the results of the example irumnavigation mission, showing all the waypoints, all the trajetory segments, and the result of the inspetor spaeraft being ommanded to injet onto the WSE at waypoint 3. Additionally, a portion of the resulting WSE motion is plotted. WP 6 WP 5 WP 4 WP 2 WP 3 (a) WP 1 (b) Figure 19 (a) Perspetive View and (b) Radial-Cross-Trak View of Example Cirumnavigation Mission Results 16

17 From Figure 19b it is lear that all six waypoints lie on the WSE manifold as desired. Figures 19a and b together show the trajetory segments just grazing the SS, rendered in yan, and show the SS irumsribing the error ellipsoid, rendered in blue. The WSE motion resulting from the injetion at waypoint 3 result in passively safe relative motion that remains outside the SS and gradually arries the inspetor away from the primary. The probability of ollision, P, is maintained at or below the desired value of 2% sine the inspetor spaeraft s motion remains outside the 98% error ellipsoid. APPLICATION OF SAFETY ELLIPSE THEORY TO RENDEZVOUS The safety ellipse theory presented previously an also be applied to rendezvous. By approahing the destination spaeraft on a walking safety ellipse spiral, the visiting spaeraft is naturally on a passively safe approah trajetory that an be designed to respet keep-out zones around the destination. The appliation presented here is rendezvous with the ISS. The visiting spaeraft begins on the In-trak axis, ahead of the ISS by 15 km. The visiting spaeraft is to be transferred from that relative loation to an insertion point for a -R-bar (Radial axis) final approah to the ISS in whih it hops towards the ISS along the -R-bar. The ISS has two keep-out zones defined around it. The outermost is the Approah Ellipsoid (AE), whih is a 2 km 4 km 2 km ellipsoid entered on the ISS with the long axis of the ellipsoid aligned with the In-trak axis. The innermost zone is the Keep-Out Sphere (KOS) whih has a 200 m radius and is entered on the ISS. The visiting spaeraft must remain outside the AE while it travels to the -R-bar approah insertion point. The visiting spaeraft will then penetrate the AE during the -R-bar approah and penetrate the KOS during the final 200 m of the -Rbar approah. The spaeraft positions at the time of rendezvous initiation are shown in Figure 20, along with the AE, KOS. For referene, the ISS orbit normal diretion is oming diretly out of the page. Walking Safety Ellipse Injetion Point V-bar, In-Trak Visiting Spaeraft y 3000 m Keep-Out Sphere m Walking Safety Ellipse Manifold Boundary ISS 4000 m 400 m -R-bar, Radial -R-bar Approah Initiation Point Approah Ellipsoid 2000 m Figure 20 Spaeraft Configuration at Time of Rendezvous Initiation (not to sale) From Figure 20 it is lear that the WSE offset, y, at the WSE injetion point should be equal to m. Referring bak to Figures 6a and b, it is lear that the safety ellipse plane polar angle, χ, is equal to 90 at the WSE injetion point. It is also lear from the x position equation in Eqs. (13) that x must be y / 3n m, where n is equal to the mean motion of the ISS, in order for the WSE motion to interept the -R-bar approah initiation point, whih is 3000 m out along the -R-bar. The remaining safety ellipse variables to speify are either θ or z and the In-trak approah rate y. For this example, z is hosen to be 2000 m and hene θ is , as speified by Eq. (17). Note that z is seleted to be larger than the extent of the AE by a omfortable margin so that the WSE motion 17

18 is always well outside the AE. This is also true for x though x is primarily driven by the loation of the -R-bar approah initiation point (whih is well outside the AE). The In-trak approah rate is seleted aording to the initial In-trak offset distane and the desired TOF for the WSE rendezvous trajetory. In this example the desired rendezvous flight time is 3 ISS orbital periods, the In-trak offset is m, as speified above, and hene the In-trak approah rate is omputed aording to Eq. (25). y y = (25) TOF Note that the sign of Eq. (25) is determined by whether the visiting spaeraft begins ahead of or behind the ISS. Sine the visiting spaeraft begins ahead of the ISS in this example ( y > 0), y must be < 0 for rendezvous. At this point all the safety ellipse parameters orresponding to the WSE injetion point are speified, and so the RIC position oordinates of this point an be omputed with the position equations in Eqs. (13). The first maneuver in the rendezvous sequene is thus a maneuver omputed using CW targeting that takes the visiting spaeraft from the initial position on the V-bar to the WSE injetion point. For this initial transfer, the TOF hosen for this example is ¼ of an ISS orbit. The seond maneuver injets the visiting spaeraft onto the WSE, where it will passively remain safely outside the AE as it spirals around the V-bar, bringing it to the -R-bar approah initiation point 3 ISS orbits later. The maneuver hanges the visiting spaeraft s RIC veloity vetor from its magnitude and diretion at the WSE injetion point to the required safety ellipse veloity vetor speified by Eqs. (16) and (17). Upon arrival at the -R-bar approah initiation point, a series of small maneuvers are made to gently guide the visiting spaeraft down the -R-bar towards the ISS. The advantage of this rendezvous strategy is that the trajetory always passively remains outside the AE. Thus if the visiting spaeraft suffers a system failure that renders it inapable of making subsequent maneuvers after injeting onto the WSE, it will ontinue to remain outside the AE, preluding a ollision with the ISS at any future time. The omplete set of rendezvous trajetories in simulation are shown in Figures 21 and 22a and b. WSE -R-bar Approah Initiation Point Figure 21 Walking Safety Ellipse Rendezvous Simulation Results, Perspetive View 18

19 AE KOS (a) (b) Figure 22 Walking Safety Ellipse Rendezvous in (a) the Radial-Cross-Trak Plane and (b) the Intrak-Cross-Trak Plane The typial -R-bar rendezvous senario has the visiting spaeraft flying trajetories solely in the orbit plane and requires a total Δ v of 29.4 m/s. The walking safety ellipse rendezvous presented here requires a total Δ v of 21.7 m/s (26.2% less) and the total flight time is longer than the typial rendezvous by 4.9%. CONCLUSIONS Algorithms and operations onepts have been presented for ensuring safety during a spaeraft irumnavigation mission. These tehniques are also appliable to a wide range of other spaeraft proximity operations missions, inluding rendezvous. The inspetion mission desribed in this paper is very useful for spae situational awareness and spaeraft health verifiation. Additionally, any mission requiring one spaeraft to servie another will first require the serviing spaeraft to safely lose distane and operate for a period of time in proximity to the spaeraft to be servied. Furthermore, a wide variety of spaeraft siene missions have been proposed that rely on small spaeraft operating in lose proximity to eah other, and any suh mission an benefit from the algorithms and operations onepts presented. Ongoing safe spaeraft proximity researh inludes the simulation of the irumnavigation mission in a high-fidelity hardware-in-the-loop simulation environment that inludes the effets of non-linearity, spae environment perturbations, and realisti navigation errors. Additionally, design efforts are underway, with promising initial results, for a ontroller apable of ahieving all the trajetories desribed in this paper in the presene of non-linearity, perturbations, and navigation error. REFERENCES 1. Foster, J. L. and Estes, H. S. "A Parametri Analysis of Orbital Debris Collision Probability and Maneuver Rate for Spae Vehiles," NASA JSC 25898, August Khutorovsky, Z.N., Boikov, V., and Kamensky, S.Y. "Diret Method for the Analysis of Collision Probability of Artifiial Spae Objets in LEO: Tehniques, Results, and Appliations," Proeedings of the First European Conferene on Spae Debris, ESA SD-01, 1993, pp Carlton-Wippern, K. C. "Analysis of Satellite Collision Probabilities Due to Trajetory and Unertainties in the Position/Momentum Vetors," Journal of Spae Power; Vol. 12, No. 4, Chan, K. F. "Collision Probability Analyses for Earth Orbiting Satellites," Advanes in the Astronautial Sienes, Vol. 96, 1997, pp Brend, N. "Estimation of the Probability of Collision Between Two Catalogued Orbiting Objets," Advanes in Spae Researh, Vol. 23, No. 1, 1999, pp

20 6. Oltrogge, D. and GIST, R. "Collision Vision Situational Awareness for Safe and Reliable Spae Operations," 50th International Astronautial Congress, Otober 4-8, 1999, Amsterdam, The Netherlands, IAA-99-IAA Akella, M. R. and Alfriend, K. T. "Probability of Collision Between Spae Objets," Journal of Guidane, Control, and Dynamis, Vol. 23, No. 5, September-Otober 2000, pp Chan, K.F. "Analytial Expressions for Computing Spaeraft Collision Probabilities," AAS/AIAA Spae Flight Mehanis Meeting, Santa Barbara, California, February , Patera, R. P. "General Method for Calulating Satellite Collision Probability," Journal of Guidane, Control, and Dynamis, Volume 24, Number 4, July-August 2001, pp Bryson and Ho, Applied Optimal Control, Taylor & Franis Publishing, 1st Ed., 1975, p Naasz, Bo, Safety Ellipse Motion with Coarse Sun Angle Optimization, NASA GSFC Flight Mehanis Symposium, Greenbelt, MD, Otober 18-20,