* I I NAVIGATION ACCURACY GUIDELINES FOR ORBITAL FORMATION FLYING J. Russell Carpenter* NASA Goddard Space Flight Cente‘r; Greenbelt, MD 20771 and Kyle T. &friend1 Tam A M U nisssity, College Station, TX 77843 Abstract in the next few years separations in the range of 100 m to Some simple guidelines based on the accuracy in deter- 10 km are expected to be attempted. Ref. 1 describes some mining a satellite formation’s semi-major axis differences relevant aspects of one such recent mission. are useful in making preliminary assessments of the nav- Many perturbations affect the accuracy of maneuvers re- igation accuracy needed to support such missions. These quired to maintain these formations, but in principle the guidelines are valid for any elliptical orbit, regardless of maneuver planning process can accommodate all known eccentricity. Although maneuvers required for formation perturbations. However, the navigation errors at the time of establishment, reconfiguration, and station-keeping require the maneuver computation will form a lower bound on the accurate prediction of the state estimate to the maneuver accuracy of the maneuvers, since even a perfectly executed we, and hence are directly affected by errors in all the or- maneuver will “lock in” the navigation errors. Indeed, bital elements, experience has shown that determination of Ref. 2 suggests that velocity uncertainty may be the lim- orbit plane orientation and orbit shape to acceptable levels iting technology for formation flying. is less challenging than the determination of orbital period Although the errors in all the states will affect the maneu- or semi-major axis. Furthennore, any differences among ver, as Ref. 3 discusses, semi-major axis error is the most the member’s semi-major axes are undesirable for a satel- difficult to estimate. An essential point of Ref. 3 is that lite formation, since it will lead to differential along-track (for circular orbits) semi-major axis uncertainty depends on drift due to period differences. Since inevitable navigation three quantities: the radial position error, the along-track errors prevent these differences from ever being zero, one velocity error, and the correlation between these errors, may use the guidelines this paper presents to determine which arises due to the the conservation of energy. The how much drift will result from a given relative naviga- most notable consequence of semi-major axis differences tion accuracy, or conversely what navigabon accuracy is for a formation of satellites is relative drift in the direc- required to limit drift to a given rate. Since the guidelines tion of the orbital path, which Figure 2 illustrates. For do not account for non-two-body perturbations, they may the relatively high-thrust, short burning propulsion systems be viewed as useful preliminary design tools, rather than flown on current missions, the result of this drift will be as the basis for mission navigation requirements, which more frequent stationkeepingm aneuvers. For “continuous” should be based on detailed analysis of the mission con- low-thrust systems proposed for many upcoming missions, figuration, including all relevant sources of uncertainty. this drift will be random error that must be counteracted, limiieed by the accuracy of the navigation feedback signal. Introduction Figure 1 illustrates the effect of the various contributors One of the most significant differences between many for- to along-track drift in a circular orbit, based on the rela- mation flying satellite missions that are currently of inter- tionships of Ref. 3. To interpret Figure 1, consider a nav- est, and the intentionally close approaches that past mis- igation system that can produce radial accuracy of 10 cm, sions have performed - e.g. rendezvous, docking, and prox- and speed accuracy of 0.1 mm/sec. Figure 1 indicates that imity operations of the Space Shuffle - is the need for the corresponding along-track drift may be on the order of long-term and efficient maintenance of relatively close for- 1 mlorbit, and could be as poor as about 3 m/orbit, if the mations. As of 2002, separations of a few hundred kilorne- correlation between radius and speed, prv, is poor. ters have been achieved for long-duration formations, and Ref. 4 describes a necessary condition for obtaining a no-drift solution in eccentric orbits in a deterministic set- *Aerospace Engineer, Flight Dynamics Analysis Branch, Guidance, Navigation, and Control Division Senior Member, AL4A. ting. Ref. 1 describes requirements on relative semi-major t Wisenbaker II Professor, .4erospace Engineering Dept Fellow. axis knowledge and control for relative drift of the GRACE AIA.4. mission due to differential drag. The contribution of the Copyright @ 2003 by the American Institute of Aeronautics and Actronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The development below is to describe and quantify the relative U.S. Government has aroyalty-frcc license to exercise all rights under the copyri&t drift issue in tenns of the @on-deterministic)n avigation er- claimed herem for Governmental Purposes. All other rights are reserved by the rors for elliptical orbits, generalizing the results of Ref. 3. copyii&t owner. I OF7 a- (In-track DriftlOrbit Uncertainty) [meters] for orbit period = 90 rnin 1o o .- 1o -~ 1o -2 lo-' 1o o 10' 1o 2 a, (Radius Uncertainty) [meters] Fig. 1 Along-track drift due to semi-major axis error for a typical low earth orbit3 Each family of contours is based on a constant semi-major axis error, resulting from various combinations of radial position error, along-track velocity error, and the correlation between these. Relative motion due to &=1 rn Ifo r e=0.8 over 15 orbits where fi is the initial true anomaly. It is not hard to show (viz. Ref. 3) that 2- ti; - 6Tp=37i -6a 1.5. E .Ez. 1- - where a is the semi-major axis and p is the gravitational 0.5 4 h..F constant, GM,s o that E 0- 2.- -0.5. 6S(fi + 2T) = -3TV(fi + 2%)ficZ (3) 3 - -1 - At periapse, -1.5 E/= - -2 -2.5-" 5 -4.5 -4' -3'.5 -3I -2 .5 -2 -1.5* -1 -0.'5 01 v(0) = V(0,2%, . . .) = 1-e (4) Tangential Separation Frn] where e is the eccentricity, and so the in-track growth per Fig. 2 Along-track drift due to semi-major axis differences orbit at periapse is between two spacecraft in a highly elliptical orbit. The paper concludes with examples in low circular and high apogee elliptical Earth orbit missions. while at apoapse, Development J'd- 1-e For every completer evolutioni n any two-body elliptical or- V(T) = V(T, 3%,. . .) = a l+e bit, an error in knowledge ofthe orbit period, 6Tp,r esults in an along-track error growth proportional to the speed (ve- and hence the in-track growth per orbit at apoapse is locity magnitude), t~, I 2OF7 Since according to Eq. (5) or (7) the along-track error any point in the orbit, the following relationshipsa re useful: growth per orbit is linear in the semi-major axis error, their error distributions share this linear relationship. For exam- w, = E e s i n j (15) ple their standard deviations are related by -a&) = 3 ? i p - a a l+e Ref. 3 gives the following relation, valid for circular orbits, -- -un among semi-major axis error, radius error, speed error, and rl the correlationb etween radius and speed errors, prv: ;a;.n(P2) 2 = where n is the mean motion. Note that the velocity terms where 77 = d = F.rom the definition of Eq. (12), ba = in Eq. (9) are with respect to an inertial Game. Relative to A(xref)bx,w hich may be written in terms of radial and a frame rotating at the (constant) orbital angular velocity, along-track components, z and y, respectively, as such as Hill's Game, = -2a2 [w rbw, + wy6w, + -P7 %,] (21) + 4 + 1 P ffa = 2 4 4 -pry-ar-aQ --a? (10) J n n2 Y Using Eqs. (15) - (20) in Eq. (21), it is not hard to show that where y in Eq. (1 0) refers to the along-track component of velocity relative to the rotating Game. [esin(f)bv, +(1 +ecos f)bv,] The general form of the semi-major &vis variance, also given in Ref. 3, is Taking the expectation of the square of Eq. (22), and as- suming a zero mean, results in the following expression for where the state is xT = [rT,v T], r is the position, v is the the semi-major axis variance at any point in an elliptical velocity, Px is the state error covariance, xref is given, and orbit: - - 4 Applying Eq. (1 1) at apoapse, and using Eq. (7), it is not hard to show that Eq. (9) may be generalized to + (1 ecos f)~rv,-argv,] + + 2e(sin f e sin f cos f) Pu,v, UU% ffv, (23) (n42 If the correlations between radius and radial velocity, prv, , and between radial velocity and along-track velocity, or similarly at periapse to P ~ , ~,c ,a n be assumed to be insignificant, then the follow- ing simpler expression results: _-a: - (1+ec0sf)~ - ar 4 vs 1 l f e + 7 (G) -a: (14) To get an expression for the semi-major axis variance at r. e = 0.8,a = 42098 km, or = 10 m, a = 10 mmls, uw= 1 mmls Eq. (25) gives the speed error that produces the same size w error in semi-major ayis as the given radius error. Table 1 p =-.5 gives some numerical examples of Eq. (28). If Eq. (28) p, = 0 (Poor Nav Fitter) :\ Table 1 Speed uncertainty, uf[ds],“ balancing” radius un- certainty for a range of eccentricities, according to JZq. (28). 1.2e-05 0.00022 0.0039 0.070 holds, then in-track drift per orbit may be viewed as a func- eri : kl ‘ A e e : i ;Perige tion of the radius error only. O,b seeq’5 ti5 %I 2;5 270 315 True Anomaly [deg] If there are no relative measurements, but instead each satellite’s absolute state is estimated separately, e.g. from Fig. 3 Semi-major axis uncertainty as a function of true GPS pseudoranges, pij may be quite small. If the filter anomaly for a particular high Earth orbit example. processes relative measurements such as cross-link ranges Figure 3 illustrates Eq. (24) for a particular example. Note and/or GPS meausurement differences, pij 0.9 may be that different relationships among ar, a,,,, and u,,= than a reasonable assumption. Based on the assumptions that Figure 3 shows produce quite different results. = 0.9, that the approximate velocity constraint of pjj In most satellite formations, it is desirable for the satel- Eq. (28) is valid, and that the radius and speed are well- lites to minimize their semi-major axis differences so as to correlated (prv = -.9), Figure 4(a) gives some examples minimize relative along-track drift. The general fonn of of relative drift due to semi-major axis error for various ec- the variance of the relative semi-major axis between any centricities, for Earth-orbiting formations. The left subplot two satellties i and j, also given in Ref. 3, is illustrates drift rates at apogee, and the right subplot drift & + rates at perigee. From the figure, it is clear that as eccen- = A(x,)fiA(xi)T A(xj)PjA(~j)~ tricity increases, the formation will drift apart more slowly - A(x)PcjA(~-j )A~( xj)PzA(~i()2~5) at apogee, and more quickly at perigee, for a given radius or semi-major &,is error. where Pi, = PxCxjT.o approximate and simplify Eq. (25), Figure 40) shows similar results, instead assuming no assume that the satellites in the formation have approxi- correlation between the two satellites’ state estimates. Fig- mately the same a and e, that their navigation errors have ure 4(c) shows some additional results, in which a “fudge the same distributions, and that Pij = pijPi. Then, the factor” of 50% is applied to Eq. (28), and zero correlations version of Eqs. (13) - (14) and (23) - (24) corresponding assumed. This is intended to capture the type of “poor” ve- to relative semi-major axis errors is locity estimation that Ref. 3 describes may occur with some GPS-based orbit determination systems. C7Aa = d-’fla (26) Finally, it may be of interest to determine the time to Therefore, as in Eq. (8), the standard deviation of the in- drift a given distance due to a semi-major axis error. Since track drift per orbit between any two satellites in a foFa- time-to-drift depends inversely on the drift rate, a nonlinear tion, evaluated~at~eacahpo apse, is relakd to their relative mapping of the statistical moments of the drift rate per orbit semi-major axis standard deviation by is required to find the statistics of the time to drift a given I distance. Let D be the deadband size, i.e. the size of the UAs(3T) = -3T “control box” in which the satellite must remain, and 7 be the time to reach the deadband. Assume that the relative In a good orbital navigation filter, -1 < prv 5 -.9, navigation errors are zero-mean and Gaussian, then due to due to the dynamical constraint imposed on the estimates their linear relationship, the relative drift rate also has a by conservation of energy’. A consequence of the high Gaussian probability density, correlation between radius and speed errors is that speed (-$) error may be viewed as a dependent variable of radius error. - Lear5 gives the following approximation of the speed mor AS) = d2E1 zc exp (29) 2UA* required to “balance” a corresponding radius error: Since “Ref. 5 shows that one should not always assume that GPS-based or- bital navigation systems properly account for this constraint is a nonlinear function, its probability density may be found 4 OF7 in terms of Eq. 29 via6 c Peigee n ml fYh) = fi (xi> (31) * \ q=g-l(y) where the 2; are the values of z where y = g(2). Using Eqs. (29) and (30) in Eq. (3 1) results in 6- (--) fT(7)= 2 D e xp 2 D42 3 ’ (32) 5i U&T2 where the fact that the deadband is reached uith a positive or negative drift rate has been used. Figures S(a) and S(b) show the probability density and the probability of not reaching the deadband in time T for M various values of D/uA,. Figure 5(b) can be used for finding the minimum accept- a) pij = 0.9. able value of the deadband For example, if one wants there to be a 75% probability that the deadband will not be reached in four orbits, then D/ah, > 5, keeping in - - le*W mind that this analysis does not include perturbations such 5- E le+07 as differentiald rag or thrust errors. le+W om le+M Examples loo0 0 5ib?, loo s.2t_ llee+4045 Se u=p=p 0o.s8e, t hfoerrme iast isoonm fely iinntger mesits isnio fnly, iinng wa hhiicghh -tehcec esantterilcliittyes, E B 5 are supposed to be 10 km apart at apogee. If the separa- 5 KJoo tion varies by more than 10%a t apogee, or if there is some lo 2 :j2: 1 -:ea: 11000 dduanragteior no, fr eal actoivlleilsyio hnig, hth-teh rsuastte slltiatetiso nmkuesetp ipnegr fmoramne ushvoerrts-. -f 0.1 -b&rn I rIant ioornd, esru tcoh m maaxniemuivzeer sth seh oscuiledn ncoet r eotcucrunr amnodr em oisfstieonn t hdaun- , 0.01 ‘ 0.1 le*M every four weeks. The orbit period is approximately one 0‘01 Radius uncertai’n0ly°, ~Jrn$*0~4 0’01 Radius uncertainly, mJm] day, so the relative driR at apogee must be less than one kilometer per 28 orbits, or about 36 meters per orbit. Con- b) pij = 0. sulting Eq. (27), the relative semi-major axis error should therefore have a “one si,ma’’ value of 36/x = 11 me- ters. If relative navigation system is highly correlated, i.e. pTV = -.9 and pij = 0, Figure 4(a) indicates that a “one sigma” radius error of about 55 meters at apogee should be suficient to meet the 10% requirement in a “one sigma” sense. If the navigation system is not well-correlated, i.e. pr, = pi.j = 0, and the relative velocity errors dominate as with many of the GPS systems described in Ref. 3, then Figure 4(c) indicates that a “one sigma” radius error of about 2 meters would be necessary to meet the 10% re- quirement. Note however that at perigee, the 11 meter relative semi- major axis error would produce a drift of about 3 10 meters per orbit, which may be found using Eq. (5). From Fig- ure 4(a), this drift corresponds to a radius error of 0.6 m for a well-correlated system. For a poorly correlated sys- e) ‘‘Poor“v elocity accuracy and zero correlations. tem like Figure 4(c) illustrates, the corresponding radius error that would be at perigee is about O.I5 m. Fig. 4 Relative drift per orbit due to semi-major axis error for various eccentricities For most elliptical orbit formations, the separation will in- crease at perigee, so a few kilometer change in relative position may amount to much less than 10%. However, 5oFl In-track displacement Probability density 130, , I 0.7 1 0.6 .... ......... ........... 0.5 ........... 0.4 ......... ........... i 0.3 ......... ............. ......... 1 ........ I 1I I I 1I 1 74 io io io 40 I 100 Tit%- $$its 70 0 10 20 30 40 50 60 Number of orbits until drift to control box boundary Fig. 6 Along-track control error time history for LEO leader- a) Probability density function for time to reach deadband, for var- follower formation with semi-major axis errors. ious ratios of deadband size to drift rate such that the value of the correlation coefficient has a re- duced effect, since the dominating velocity noise places the Probability of not reaching control box boundary positidvelocity noise combination above and to the left of the main diagonal of the fiewe, which is where the curves split based on correlation coefficient (this is consistent with the findings of Ref. 2). It therefore seems appropriate to use the poorly correlated example of Figure 4(c) to assess the drift rate per orbit, from which one can determine the drift to be about 5 in per orbit. For a 20 m deadband, one should therefore interpolate between the D/oas = 3 and D/aas = 5 curves of Figure 5(b), from which it is possible to determine that there is about an 80% probabilityt hat the deadband will not be reached in less than about 5 orbits, or equivalently that there is a 20% probability that every 5 orbits, the deadband will be reached. Finally, Figure 6 demonstrates the effect that relative navigation error has on a formation. The figure shows the 0 IO 20 30 40 50 60 along-track relative motion time history for one week (10 5 Number of orbits orbits) for a leader-follower formation with a desired sepa- b) Probabilitg of not reaching the deadband in given number of or- ration of 100 m with the 2 mm’s relative velocity and 6 cm bits, for various ratios of deadband size to drift rate relative position relative navigation errors described above. Fig. 5 Driftprobabilities. The relative state control strategy is a minimum fuel in- track impulsive strategy in which each maneuver consists there do exist elliptical orbit formations for which separa- of three impulses separated by 0.5 orbits. A maneuver is tion at perigee decreases relative to apogee, in which case initiated when 75% (15 m) of the deadband of 20 in is a collision avoidance maneuver might be required much reached.’ To initiate relative motion there is differential more o htha n desired unless the navigation error could drag, but after the first maneuver the differentiald rag is set be further reduced. to zero and the only error is the relative navigation error. Next, suppose a low-Earth orbiting (550 km altitude), As Figure 6 shows, when there is a high drift rate due to leader-follower mission is proposed in which the separa- the semi-major axis error the deadband is exceeded. tion distance is 60 m, and this must be maintained within Conclusion f20 m. The mission will use a highly accurate differen- tial Global Positioning System relative navigator, which This paper has presented some guidelines on relative nav- can determine relative position to within 6 cm, and relative igation that may be useful for conceptual analysis of for- velocity to within 2 mm/sec, “one sigma,” per axis. From mation flying missions. These guidelines generalize pre- Figure 1, one can see that the corresponding drift uncer- viously reported formulae to the case of elliptical orbits. tainty is about 3 m, that the velocity noise is the limiting Since the guidelines do not account for non-two-body per- error, and that the positiodvelocity noise combination is turbations, they may be viewed as useful preliminary de- sign tools, rather than as the basis for mission navigation requirements, which should be based on detailed analysis of the mission confi,wation, including all relevant sources of uncertainty. Acknowledgment The development of this paper benefited from conversa- tions with David Folta and Jon How. References ‘Kirschner, M., Montenbrk O., and Bettadpur, S., “Flight Dynamics Aspects of the GRACE Formation Flying,” 2nd International Workshop on Satellite Constellations and Formation Flyins, CNES, Haifa, Israel, February 2001. ‘How,J . P. and Tillenon. M., “Analysis of the Impact of Sensor Noise on Formation Flying Con&$ Procffidings of thl American Contml Conference, 2001,pp. 3986-3991. 3Carpenter, J. R. and Schiesscr, E. R., “Semimajor Axis Knowledge and GPS Orbit Determination,” N.WIGATION Journal of The Institute VoL 4S, No. 1, Spring 2001, pp. 57-68! also .US Paper 99-190, Feb., 1999. 41nalhan. G., Tillerson, M., and How, J. P, “Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbib:’ Journal of Guidance, Control and Dynamics, Vol. 25, No. 1,2002, pp. 48-59. 5Lear, W. M., “Orbital Elements including the 12 Harmonic,” Tcch Rep. 86-FM-1 8, JSC-22213, Rev. 1, Mission Planning and Analysis Di- vison, NASA Johnson Space Center, Houston, TX, 1987. GPapoulis, A,, Probabilitl: Random Variables. and Stochastic Processes, McGraw-Hill, 1984. ’Alfriend. K. T. and Lovell, T. A, Tormation Maintenance for Low Earth Near-Circular Orbits,” American .4stronautical Society, Univelt, San Diego, CA, 2003.