Voyager Orbit Determination at Jupiter 259

256 IEEE TRANSACTIONS ONCONTROL, AUTOMATIC VOL. AC-28,NO. 3, MARCH 1983 Voyager Orbit Determinationat Jupiter

JAMES K. CAMPBELL, STEPHEN P. SYNNOTT, AND GERALD J. BIERMAN, SENIOR MEMBER, IEEE

Abstract--This paper summarizes the Voyager 1 and Voyager 2 orbit determinationactivity extending fromencounter minus 60 days to the Jupiter encounter, and includes quantitative results andconclusions derived from mission experience. The major topics covered include an identifica- tion and quantificationof the major orbit determination errorsources and a review of salient orbit determination results from encounter, with emphasis on the Jupiter approach phase orbit determination.Special attention is paid to the use of combinedspacecraft-based optical observations and Earth-based radiometric observations to achieve accurate orbit determination during the Jupiter encounter approach phase.

I. INTRODUCTION

N March 5, 1979 after a journey of 546 days and 0slightly more than 1 billionkm, the Voyager 1 spacecraft passed within 0.3 Jupiter radii of the innermost Galilean satellite Io. Four months later, on July 8, Voyager 2 flewby the third Galilean satellite Ganymede at 0.8 Jupiter radii, and then by Jupiter the next day, at 10 Jupiter radii. Fig. 1 shows the Voyager heliocentric trajectory, and Figs. 2 and 3 show the near Jupiter trajectory for each mission. These Voyagerencounters with Jupiter proved to beboth spectacular and historic, witheach mission returning voluminous data from 11 scientific instruments, including some 15 000 high resolution pictures of Jupiter VOYAGER 2 and five of its satellites. After passing Jupiter, both spacecraft flew on to Saturn. Voyager 1 had encounters \ with several Saturn satellites, including a close flyby of the massive satellite Titan, on November 12, 1980. Voyager 2 encountered Saturn and its satellites on August 26, 1981 ; this spacecraft has undertaken the long journey to Uranus Fig. I. Voyager 1 and 2 heliocentric trajectories and will encounter that planet in January 1986, providing the first closeup view of that planet and its satellites. If the spacecraftremains healthy, it willthen embarkon its determined by the orbit estimation accuracies available at fourth interplanetary cruise, amving at the planet Neptune the time of maneuverspecification. Frompostencounter in 1989. The orbit determination discussed in this paper is reconstruction of eachflyby trajectory it has been de- confined to the Jupiter encounter phase. termined that the final orbit control was within the accu- The spacecraftflight path had to beaccurately con- racy predicted by the filter/smoother covariance analyses. trolledto achieve its scientificobjectives. Since the final The accurate instrument sequences required to obtain the trajectory correction maneuver and its associated execution near encounter science data were highly dependent upon errors weresmall, the Jupiter deliveryaccuracies were accurate postcontrol knowledge of the spacecraft trajectory and satellite orbits. The results from the reconstructed orbits indicate that the near encounter spacecraft orbits Manuscript received March 5. 1982:re\%ed September 7.1982 and September 20. 1982. This work was supported by NASA under Contract were predicted to withn 50 km and that spacecraft-satellite NAS7- 100 and Factorized Estimation Applications. Inc. pointing was predicted to within 3 mrad, even for the close J. K. Campbell and S. P. Synnott are with the Jet Propulsion Labora- tory. California Institute of Technology. Pasadena, CA 91 109. (20 000 km) Voyager 1 Io flyby. G. J. Bierman was Xvith the Navigation Systems Section, Jet Propulsion This paper brieflydescribes the navigational measure- Laboraton.. California Institute of Technology. Pasadena. CA 91 109. He ment system and error source modeling used to produce is now with Factorized Estimation Applications. Inc.. Canoga Park. CA 9 1307. accurate Jupiter-relative orbit determination. We will focus

00 18-9286/83/0300-0256SO 1.OO 1983 IEEE CAMPBELL et ul.: VOYAGER ORBIT DETERXIINATIO~U’AT JUPITER 251

readers are urged to consult this reference and also [lo], which documents the Voyager Saturn encounter orbit de-

ID ENCOUNTER AND termination.

11. NAVIGATIONFILTER DESIGN RATIONALE There were two principal reasons for using a sequential stochastic filtering algorithm to process the Voyager tracking data. The first reason has to do withmodeling of nongravitational accelerations and the second has to do with modeling of the optical data to account for pointing errors. Letus first focus on the nongravitational force

\ I / problem. Fig. 2. Voyager I Jupiter flyby geometry. The Voyager spacecraft, shown in Fig. 4, are three-axis stabilized vehicles, and remain in anEarth-pointed orientation relative to the Sun and a star, usually Canopus, for long periods of time. Notable features of the spacecraft are the large antenna dish, the science scan platform, and the groups of thrusters. The thrusters are unbalanced, since they do not fire in pairs, and are separated from the center of mass on opposite moment arms. A consequence of this configuration is that each time that a thruster is fired to either maintain or change the spacecraft attitude, there is a net translational velocity imparted to the spacecraft. Mo- tion of the scan platform puts torque on the spacecraft, requiring thruster firings to maintain alignment. The effective center of solar pressure does not coincide with the spacecraft center of mass, and this can causeone-sided thruster firings. A designflaw of the spacecraft is that the exhaust plumes from the positive and negative pitch attitude thrusters, which have velocity components along the radiometric measurement direction, strike the spacecraft structure. This fact was known before launch, but it was believed that the effect would be neghgible. The conclusion of a postlaunch / study was that, in fact, the plume impingement effect is Fig. 3. Voyager 2 Jupiter flyby geometry. significant. Spacecraft outgassing as a byproduct of attitude control was considered to be basically a dynamicstochastic process on the particular problem of combining the orbit informa- which imparted AV velocity impulses to the spacecraft. The tion content from two distinct sources; Earth-based radio- nature of these pulses was suchthat over a daily period the metric observations (range and Doppler) and spacecraft- net translational effect on the spacecraft was zero. How- based optical observations toachieve accurate orbit de- ever, the Doppler tracking data weresignificantly cor- termination during the planetary approach. Becausethis rupted by this attitude control pulsing. It has been known paper is an application of modern estimation, the reader is for some time [9] that estimate accuracy severely degrades assumed to be familiar with the estimation concepts that when such disturbances are not accounted for in the filter are employed. The algorithmic formulations based on ma- model, and it was demonstrated early in the flight that the trix factorization that are used to compute the orbit de- OD estimates produced from radiometric datawithout termination estimates, estimate error sensitivities, and taking into account a dynamic stochastic process propa- estimate error covariances (filter/smoother and consider gated poorly and gave inaccurate orbit predictions. Dop- filter/smoother) are described (in great detail) in [l], [2], pler residuals could be more accurately predicted from a [6], and [7]. The latter part of Section I1 contains a discus- previous fit which assumed a stochastic process than from sion of the merits of the SRIF/SRIS and U- D covariance a fit to the same data where stochastic effects were ignored. factorized estimation algorithms that are used throughout These spacecraft generated forces are commonly termed this application. spacecraft nongravitational forces. For Voyager, small atti- The research reported hereis extracted inlarge part tude control impulses were averaged over a daily period from the Voyagernavigation team report [8]. Interested and treated as piecewise constant stochastic accelerations 258 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-28, NO. 3, MARCH 1983

'.-- RADIOISOTOPE THERMOELECTRIC GENERATOR Fig. 4. Voyagerspacecraft configuration. along the respective spacecraft axes. Short term larger tude. The optical data for orbit estimate solutions that are magnitude velocity impulses resulting from spacecraft turns in agreement with the chosen Voyager 1 and 2 reconstruc- wereexplicitly modeled as velocityimpulses and were tion solutions were fit to 0.25 pixels rms and essentially included in the list of parameters to be solvedfor (see zero mean. No remaining systematic trends were detected a Table I). Because of observability considerations, the sto- posteriori, indicating that there were no pixel level biasesor chastic effect was only significant in the axis which was center-finding errors inherent in the image extraction pro- aligned with the Doppler measurement direction, i.e., the cess. Earth-spacecraft direction. Being able to eliminate unnec- The major navigational error source to be accounted for essaryfilter model state variableswas doubly important after extracting the star locations and satellite center loca- because storage limitations imposed by the JPL Voyager tions isassociated with inertial camera pointing. These ODP allows at most 70 parameters to be included in the errors were accounted for on a frame-by-frame basis, i.e., filter model and, as a perusal of Table I shows, there are a camera pointing errors were modeled as (frame dependent) considerable number of items to beestimated. whitenoise, and known inertial positions of the back- Letus turnattention now to the second reason for ground starsin each frame wereutilized. This type of utilizing a sequential stochastic model,which hasto do stochastic model structure invitesthe use of a Kalman with the processing of optical measurements. These ap- filter.Processing of theseimages produced accurate proach measurements were required to allow accurate orbit Jupiter-relative spacecraft orbit estimates that were used to controlat the final approach maneuvertime. Doppler construct the final control maneuvers. In addition, accurate measurements which are sufficiently sensitive to Jupiter's satellite ephemerides were also determined [15] and these gravity to allow determination of accurate planet-relative were used to define science instrument pointing. orbits occur too late for the information to be used for the Since this paper is a Kalmanfilter application. it is final control maneuvers. important to discuss the estimation processing techniques The optical data consisted of about 95 pictures processed that wereemployed. It is a standard practice at JPL to for Voyager 1, and 113 pictures processed for Voyager 2. implement navigationfilters using square-root factoriza- Galilean satellites (130 Voyager 1 satellite images, 143 tion techniques; even unweighted least-squares parameter Voyager 2 satellite images)were imaged against a star fits are generally implemented using orthogonal transfor- background with the narrow-angle imaging science instru- mation triangularization. Our experience with such things ment, which has a resolution of 10 prad. The exposure time is that there are so manymodeling and mission-related of the frames was 960 ms, sufficiently long to ensure the problems to worry about that we prefer to minimize detection of dim stars up to a 9.5 effective visual magni- numerical error effects as even a possible cause of poor CAMPBELL ef Ol.: VOYAGER ORBIT DETERMINATION AT JUPITER 259

TABLE I 67 STATE VOYAGER NAVIGATION MODELWITH ESTIhiATION RESULTS BASEDON ENCOUNTERDATA FROM FEBRUARY9-MARCH 18,1979

A Posteriori state A priori Sigm (smoothed siscas) RWkB

Cartesianpositions (3) 500.0 Ian 88.0 Ian Smwthed values are FXS Cartesian velocities (31 .5 ds 0.06 ds over the 108-step time m.

Line-of-sight (s/c - EaFth) mn-gravitational accelerations bias (1) 5.G~10-~~kds2 1.5~10-l~Ws2 piecewiseconstant random (1) 5.0~10-~~Ws2 2.5x10-12 Ws2 Smmthed value is F34S overthe lobstep time arc.

o.a& Ws 1.GX10y ws only 6 cross-track (mobsemable) .:V ccmpollents included.

50.0 lua RHs of the 19 pararmeters.

50.0 d/s2 5.0 &/s2

600.0 d/s2 40.0 &/s2

200.0 !an

3 tracking stations. 0.4 m 0.5 m 1.0 lua Range measureaent biases, one for each station.

camera pointing angles (3) Independent from frame to Prame (white) clock, cone 0.3 deg 1 .OxlO~ hthed values AHS aver 87 fremes. Mst 1.0 dep 0.2 deE

A priori Smoothed Estimate Uncertaintyresiduals B Data Points

Radimtric:

Doppler (60 seo. sample average) 1.0 ds 0.4 mm/s 2400

Range 1.0 Ian 0.1 lua n opticalpixels 0.25 pixels 0.5 139 stars 101 satellite iuages

filter performance. Ourconcerns are illustrated in [ 113 efficient for larger data sets than is the Kalman filter. It is whichsummarizes some of the peculiar (and wrong) of interest to note that, despite this, whenthe other ancillary numerical results that were obtained in premission simula- aspects of the problem are included (such as integration of tions of the Voyager Jupiter approach using conventional the variational equations and computation of the measure- Kalman filter covariance formulations. ment observables and differential correction estimate par- The Earth-based radiometric data are processed using a tial derivatives), the difference in computational cost (as batch sequential, square-root information filter/smoother will be shown in the following paragraphs) turns out to be [l], [2],and ["I, and the spacecraft-based optical data are relatively unimportant. processed with a U-D covariance factorized Kalman filter To give an idea of thecomputational burden that is [2], [6].The radiometric observation set is much largerthan involved, consider a typical radiometric SRIF/SRIS solu- the set of optical observations; for the encounter application with 67 state variables (Table I). This model contains tion reported here the radiometric data consisted of 2400 only 4 process noise states (line-of-sight acceleration and 3 Doppler and 27 range measurements and the optical data camera pointing errors); there are 3500 data points and set consisted of 139 stars and 101 satellite images. The data 132 time propagation steps. The problem run on a UNI- types and associated statistics are summarized in Table 11. VAC 1110, in double precision,used 275 CPU s for As is pointed out in [2] the SRIF is computationally more filtering; smoothed solutions and covariance computation 260 IEEE TR4NSACTIONS ONCONTROL, AUTObMTIC NO.VOL. AC-28. 3, MARCH 1983 used 265 CPU s. The entire run scenario including trajec- deep space navigation problem it is most important that tory variational equation integration, observable partials the estimates and covariances be computed correctly. generation, solution mapping, and generation of smoothed 2) Computational efficiency: The algorithms are for- residuals used 4320 CPU s. Thus, the filter and smoother mulated so as to exploit the structure of the orbit de- portion eachinvolved little more than 6 percent of the termination problem. In particular. there are many mea- CPU time. Two points of note, in regard to these sample surements to be processed per time propagation step, the run times are as follows. dynamical model involves only a small number of stochas- 1) We generallyiterate and generate several tic variables, there are a preponderance of bias parameters, filter/smoother solution sets for a given nominal trajectory and the position-velocity states are cast in a pseudoepoch and file of observation partials. Smoothed covariances use state formulation. Early tests demonstrated that the SFUF, the lion’s share of the smoother computationand these for this structure, was more efficient than a conventional need only be computed for the last iteration. (and less reliable) Kalman filter mechanization. 2) The program inputs were not configured to minimize Using the SRIF it is relatively easy to take the processed filter/smoother CPU requirements. results and generate estimates that correspond to models For this dynamic state configuration optimally arranged with a smaller number of bias parameters. This feature is smooth code should require less than 25 percent of the especially useful for confirming that parameters thought to CPU time required for the filter phase. The point we are be of little significance turn out, in fact, to have little effect aiming at is that, based onour experience in [ll], a on the key estimate state vector components. conventional Kalman filter covariance formulation would The U- D covariance factorization was chosen to mecha- execute in essentially the same amount of CPU time (& 15 nize the optical navigation filter for many of the same percent), and evidently the filter CPU cost is small com- reasons (numericalreliability and computational effi- pared with total run CPU time requirements. ciency), except in this case the measurement set per time As pointed out in [l] the pseudoepoch state formulation propagation issmall, and forsuch problems the U- D modelis an evolutionary outgrowth of the least-squares formulation ismore efficient than the SRIF. In [ll] the initial condition estimator [3]. If X,? represents current time U- D formulation and the Kalmanfilter both conventional position and velocity differential correction estimates, then (optimal) and stabilized (Joseph/suboptimal) forms are the pseudoepoch state x/”is defined by the equation compared for a Jupiter approach simulation with radiometric data. In the tests reported there the covariance mecha- x,’=@.~~(tJ~to)xJ+@~Xy(cJ’t~)~+~~Xp(tJ’tJ~~)P~~~nized filters performed very poorly; they gave results that ranged from inaccurate, but whch might be thought cor- where the components of y are bias parameters and the rect (20-50 percent errors) to impossible(negative vari- components of vector pJ are piecewise constant stochas- -, ances and estimates that were absurd). It happensthat tic model parameters. The transition matrices @.xx( tJ,to). optical navigation data are not nearly as ill-conditioned as (t,: to1 and are the radiometric data,and in fact, the early optical navigation studies successfullycarried outin [5]used a @xXp~~j~~,-l~=@~’~~,~~o~@.~~~~~~~o~ conventional Kalman filter mechanization. The decision to -~~‘(fj-l,fo)@~~(~j-,,to)use a U- D factorization in place of the conventional mechanization was based on the following facts: are obtained by integrating variational differential equa- 1)Comparisons (operation counts, actualCPU, and tions for storage requirements) show that optimally coded U-D and conventional covariance mechanizations are nearly indis- [@x.x(t7to>9@x,,(L to),@& to)] tinguishable in terms of storage and computationalrequire- from an epoch time to. In Table I the filter state vector x, ments. y, and p components are defined. 2) U-D factor mechanization has accuracy that is com- Orbit determination problems with radiometric data are parable with the SRIF. On the other hand, one cannot be ill-conditioned. This is due, in themain, to poor observabil- certain when the covariance mechanization will degrade or ity. and large dynamicranges of the variables that are fail (e.g., when the a priori uncertainties are too large. the involved(viz. acceleration errors - 10 -I2 km/s’, range measurement uncertainties are too small, the data geome- values - lo8 km, etc). The observability problem is aggra- try is near linearly dependent, etc., one can expect stability vated by the inclusion of largenumbers of parameters and accuracy problems). (such as ephemerides) that are very weakly coupled to the It is the conviction of (one of) the authors(who is spacecraft observables. The two features of the SRIF/SRIS believedby the others!) that covariancemechanized algorithmic formulation that are most important for this Kalman filters should newer be computer implemented. application are as follows. Further, it is believed that if the Kalman filter applications 1) Numerical reliability: The SRIF/SRIS algorithms are community had more experience with efficiently and relia- the most (numerically) accurate and reliable formulation of bly mechanized factorization alternatives, there would be the Kalman filter/smoother known. Because of the com- few instances where a covariance mechanized Kalman filter plexity and difficulty inherent inthe formulation of the would find application. We note in closing this factoriza- CAMPBELL et (I/.: VOYAGER ORBIT DETERhiINATION AT JUPITER 26 1

TABLE111 SUMMARY OF JUPITER APPROACH ORBIT CONTROL

Voyager 1 All, mlsec 58, KJr TCM Execution Time Designed Achieved CB. R 8-T LT

3 E - 35 days 4.146 4.208 -1725. +!3125. toh 14m

4 E - 12.5 days 0.586 0.594 +7iO. -100. -Oh Om 14’

Voyaqer 2 TCM

3 E - 45 days 1.442 1.386 +3330. t5040. -Oh 4m 42’ 4 E - 12 days 0.576 0.574 -895. +?lo. +Oh Om 03’ tion algorithm discussion that the SRIF and U-D algo- range) wasvirtually continuous for the last 60 days of rithms that were used in this application have been refined approach. For each encounter, there were several critical and generalized, andare commerciallyavailable in the events for which orbit estimate deliveries to certain ele- form of portable Fortran subroutines [12]. ments of the project were required. The events consisted of two trajectory correction maneuvers, at about E -40 days and E - 12 days, and several updates to the onboard 111. OD PERFORMANCE instrument pointing, a critical element in the near encoun- This section will summarize the near-encounter OD re- ter science return. The detailed summary of the delivery sults for both spacecraft. These results are 1) the orbit events for each spacecraft is shown in Table 111. determination that was done to deliver the Voyager Based on a preliminary data acquisition schedule, covari- spacecraft to theirrequired final target points met the ance analysis estimates were computed preflight of the required accuracies; and 2) that the final postdelivery orbit orbit determination errors whichwould result at these determination needed to accurately point science instru- various delivery times. The primary purpose ofthis section ments in the near and postencounter phases also exceeded is to compare the near real-time filter and the current best specification accuracies. A limitation of the Voyager orbit reconstructed smoothed orbit estimates, and also to com- determination program is that one can include at most 70 pare both of these to the expected covariance performance filter states, and this includes the sum of both estimated computed preflight. It will be shown that all orbit estimates and consider filter states. For completeness we remind the but one fell well within the expected preflight capability reader that consider parameters, cf. [ 11 and [2], play no role (1 - u) even though those early statistics were derived with in the estimation except to provide a deweighting to the unrealistic, very optimistic, assumptions about the “small” confidence one would otherwise put on the estimates. As nongravitational forcescaused by spacecraft attitude noted earlier, the size limitation does not allow us to changes. include all the known error sources in the filter model. More specifically, as discussed in Section 11, the Table I lists one of the several filter models that were used, spacecraft’s attitude control system did not employ cou- and quantifies the results obtained. Observe that although pled thrusters, and hence there were many small velocity there are 12maneuvers only six cross-trackmaneuver impulses imparted to the spacecraft. Thesevelocity im- correction terms are included, and theseterms have a pulses are impossible to model individually, and affect neghgible effect on estimator performance. This conclusion (to a sensible level)the meter level ranging and submillime- is based both on estimate comparisons with and without ter/second Doppler observables if they are imparted along such terms included and on the incremental change in the the Earth-line. In addition, whenever the spacecraft orien- computed covariance due to the addition of consider tation was changed to allow an engineering calibration, or parameter sensitivity effects. The discussion that follows is a science scan of the Jovian system, velocity impulses of an elaboration of the results summarized in Table I. tens of millimeters per second were imparted. These larger impulses were not accounted for in the preflight analysis, Jupiter Approach Solutions and the more continuous smaller impulses resulted effectively in accelerations of the order of 5- 10 x 10 - l2 km/s2 For approximately the last 60 days of each approach, along all three spacecraft axes. The larger impulsive veloc- both radio and optical data were continuously acquired to ity changes have two effects. When they occur within the allow orbit estimates to be revised every few days. Optical data arc, an estimate of their components becomes con- data in the time period beginning 60 days from encounter fusedwith the estimates of the dynamically important (E) to E - 30 days were acquired at the rate of approxi- parameters, and the result is an incorrect orbit; when they matelyone picture per day, and this rate increased to occur past the end of the data arc they cause incorrect about four or five pictures per day in the last few days mapping to the encounter time, and the result is an incor- before encounter. Radio tracking (coherent Doppler and rect prediction of encounter conditions. Based on covari- 262 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-28, NO. 3, MARCH 1983 ance analyses, it did not appear possible to accurately VOYAGER I predict the velocity components at the maneuverevent RECONSTRUCTED APPROACH SOLUTIONS RELATLVE times from a priori information. TO FINAL RECONSTRUCTED TRAJECTORY

Data Weighting

In the vicinity of a planetary encounter, the radio and optical data are geometrically complementary. The Earth- spacecraft line-of-sight range and velocitymeasurements indirectly provide the spacecraft-planet distance through Y \ observable dynamic effects, and the optical data effectively \ measure the instantaneous spacecraft-satellite crossline- of-sight position relative to a particular satellite. The radio and optical data were planned to allow determination of both the satellite orbits and the spacecraft trajectory relative to the planet. In the estimation process using both radio and optical observations, the solutions at certain stages were found to be sensitive to the relative weighting of the two data types. Based on the image extraction analysis, the optical data were thought to measure the star and satellite center locations to 0.5 pixels or less in a random measurement noise sense. In addition, if there were any systematic optical image extraction errors, theywere thought to be small ( < 0.5 pixels) and to behave as biases in the 2-D picture space, or as long term slowly varying functions, i.e., nearly biases. Using these arguments the optical data usually were 01 I I I I I assumed to have 0.5 pixelmeasurement noise; 0.5 pixel -30 -25 -20 -15 -1 0 -5 optical biases were used as consider parameters to compute DAYS FROM ENCOUNTER the expected uncertainty in the orbit estimates. Inciden- Fig. 5. Reconstructed history of Voyager 1 B.R solutions relative to tally, these two consider parameters, together with the 67 postencounter reconstructed orbit. estimated parameters listed in Table I, essentially exhausts the 70 parameter ODP storage limitation. After calibrating for troposphere, ionosphere, and space AVs; these are the 12AV state vector components included plasma effects, the measurement errors for radio data are in Table I. equivalent to only a few meters in range and0.3-0.5 mm/s One of the best ways to measure the real limit to the for a 60 s count time for Doppler. However, because of the orbit determination capability as a function of time to known level of unmodelable nongravitational accelerations, encounter is to examine the variation in solutions along a range was more usually weighted at 100 m, and Doppler at trajectory which has already passed through several estima- 1 m/s, for a 60 s count time. These values, together with tion iterations for the magnitude of the impulsive compo- the postfit rms residual results are displayed in Table 11. nents. With as detailed a treatment of the larger impulsive For the radio data, the mostlikely sources of error are events as ispossible, and with stochastic and constant transmissionmedia calibration errors andEarth station acceleration components solved for in the sequential filter location errors, both of which are usually combined into an to remove any excess nongravitational effects, the relative assumed systematic error uith a diurnal signature. Data time history B-plane’ solutions for the last 25 days before noisefor Voyager 2 waslarger because its view periods Voyager 1 encounter were computed and are displayed in occurred during localdaylight hours, when transmission Figs. 5 and 6 along with the 1 - (I uncertainties in these media effects were larger. The Voyager 2 encounter also solutions. These uncertainties were calculated using no coincides with a more active solar period. assumed systematic errors in either radio or optical data, such as optical biases, and therefore represent lower limits Reconstruction of Encounter Orbits to the error that could be expected. In general, the solutions fall within even this optimistic uncertainty level. The Since the encounters, it has been possible to do a de- 55 km AB .R solution at E - 3.5 days is the only anomaly tailedanalysis of thelarger nongravitational impulsive events for Voyager 1, using data from approximately 60 days after encounter. In addition to errors associated with ‘Planetary targeting is usually expressed in terms of the E-plane, a plane passing through the center of a target planet and perpendcular to the two-approach TCMs, there occurred 10 preencounter the incoming approach hyperbola asymptote of a spacecraft. ‘‘B‘T” is the intersection of the B-plane with the ecliptic and ‘‘B.R” is a vector in events whoseeffects were hrectly observable along the the B-plane that is perpendicular to B.T and making a right-handed Earth-line and whch therefore were estimated as impulsive system R,S. T, where S is the incoming asymptote. CAMPBELL ef o/.: VOYAGER ORBIT DETER\fINATION AT JUPITER 263

2w - VOYAGER 1 Real Time Solutions RECONSTRUCTED APPROACH SOLUTIONS RELATIVE TO FINAL RECONSTRUCTED In this subsection we compare the near-real time trajec- TRAJECTORY tory estimates for both spacecraft that were delivered to the Voyager project at certain stages of each approach, to the final reconstruction solutions, and compare the accuracies of these deliveries to the attainable accuracies. The Voyager 1 approach solutions relative to reconstruction are shown in Figs. 7 and 8. The lump, or flattening, in the Voyager 1 uncertainty curves reflects the fact that the data arc for this and subsequentsolutions was shortened to include data only within 30 days of encounter. In general, near-real time orbit determination accuracies for both spacecraft fell well within the 1 - u uncertainties, which are taken from the computer run that was made in near-real time. The errors (withsmoothed estimates regarded as truth) also generally fell well within the preflight covariance predictions. The only point on approach atwhich the near-real time solutions were not substantially below the 1 - a uncertainty leveloccurred on Voyager 1 at the final delivery for the last approach trajectory correction maneuver TCM4. After considerable study it became ap- parent that the relative weighting of the radio and optical dataat this epoch wasresponsible for this anomalous

01 I I I I I estimate. -30 -25 -20 -15 -10 -5 The variability of the Voyager 1 B-plane solutions lead- DAYS FROM ENCOUNTER ing up to the finalTCM4 delivery at E - 15 days are Fig. 6. Reconstructed history of Voyager I B .T solutions relative to postencounter reconstructed orbit. shown in Figs. 9 and 10 as functions of time, relative data weightings, and estimated parameter list. The correct (best postfit) answers are indicated in each figure. The solutions and it occurs at a time when significant ( > 1 - o) tran- plotted by the smallcircles in Fig. 9 were generated by sients are occumng in several important dynamic parame- solving for the state vector whose components are listed in ter estimate components, suchas, for example, planet Table I; the estimate list includes spacecraft state, stochas- mass.Significant parameter estimate variations in static tic and constant nongravitational accelerations, several im- components such as this indicate possibly a station loca- pulsive AVs, planet mass, and planet and satellite tion, media, or nongravitational mismodeling problem. The ephemerides, as well as the three camera orientation angles, behavior in the B. R estimate also suggests the possibility which were always estimated. The solutions of Fig. 10 also of an optical center-finding error whichwould become included Earth station locations in the estimate list. In significant only when the satellite diameters in the picture both figures there are significant sensitivities for all three become very large (approximately 100 pixels). However, an curves to both data arc length and relative radio/optical analysis of the center-finding at this time period for all the weightings.Comparison of the results of Figs. 9 and 10 different satellites indicates that this is an unlikely possibil- indicates that there is a large sensitivity (of the order of ity. A center finding error wouldalso imply that earlier several hundred kilometers for the radio onlycase) to satellite residuals wouldshow strong systematic effects. station location, or station location-like, errors. This type This was not, however,observed when the entire arc of of sensitivity typically is an indication of mismodeling of optical residuals was plotted from the current best recon- the radio observables. struction run. (The orientation of the Voyager camera in The orbit determination performance for the last ap- space was such that to within about 20°, pixels represent a proach TCM and the pointing update for Voyager 2 are measurement in the ecliptic or trajectory plane, and lines summarized in Fig. 11 in which the B-plane conditions represent measurements that are normal to this plane.) predicted using data to E - 17 days and E - 9.5 days and A slightslope in the smoothed estimate line residuals their uncertainty ellipses are compared to the final recon- seems to indicate that there is someremaining out-of- struction orbit estimate. Because of the Voyager 1 experi- trajectory-plane mismodeling, but the error at encounter ence and because of the difficulty encountered with a long caused by this slope should be no more than 10-15 km. arc of Voyager 2 radio data, many different arc lengths and Ths same error behaviorwas observed to occur in the combinations of data andparameter estimate listswere near-real time approach solutions. The general results and used for the second spacecraft. By thus experimenting we conclusions stated herewere derived mostly from the were able to eliminate the larger effect of systematic mis- Voyager 1 orbit reconstruction experience; they are appli- modeling and arrived at solutions that fell well within allof cable directly, however, to Voyager 2. the uncertainty ellipses. 264 IEEE TRANSACTIONSON AUTOMATIC CONTROL, VOL. AC-28, NO. 3, MARCH 1983

1- VOYAGER 1 APPROACH SOLUTIONS RELATIVE TO FINAL RECONSTRUCTED * TRAJECTORY

\ .\ 1- \

\ I-UUNCERTAINW \\\ I- X \ \ \ I- \ PREFLIGHT I-U‘UNCERTAINW /

I-

X l-

) I I I I I I 1 XI -45 -40 -35 -30 -25 -20 -15 -10 -5 DAYS FROM ENCOUNTER Fig. 7. History of Voyager 1 real-time B.R solutions relative to postencounter reconstructed orbit.

tion of the overall capability andperformance of the Voyager radiometric data. Fig.12 shows, in geocentric angular coordinates, the shift in Jupiter position coordinates obtained from a set of orbit estimates based on \ radiometric data arcs extending from E - 60 to E + 30 days. This data arc allows for an accurate Jupiter-relative orbit estimate and somesensitivity to Jupiter ephemeris error. One solution estimates only ephemeris error; the second solution estimates both ephemeris and station location errors, and essentially trades off these twoerror sources to give the samenet geocentric angular offset as the ephemeris-only solution. This plot indicates the general inability of Earth-based radiometric datato fully dis- tinguish station location error from ephemeris error, and sets a lower bound of about 0.25 pradto the Voyager radiometric performance. Fig. 13 plots in geocentric angular coorchnates the radio-based OD that was done for the final Voyager 1 delivery, and compares this real-time result @An FWM EKWNIE1 with the best reconstructed orbit at the delivery point, Fig. 8. History of Voyager I real-time B.T solutions relative to postencounter reconstructed orbit. obtained using a combined radio plus optical data set. This plot shows that the radiometric OD for Voyager 1 performed to about the 0.5 prad level.

Tables IV and V indicate the OD delivery performance V. SATELLITESTATE ESTIMATES for each spacecraft. The last columns of each table show that the delivery OD performance for each spacecraft A natural fallout of using the optical and radio data in easily met the established accuracy criteria. In the case of the spacecraft trajectory determination process is the im- Voyager 2, the delivery requirements were less stringent, provementin the knowledge of the orbits of the four and the achieveddelivery was nearly equivalent to 0.5 Galilean satellites. The optical data are directly sensitive to pixel, Jupiter relative. Figs. 12 and 13give a brief indica- satellite positionchanges. Near the satellite close ap- CAMPBELL el a/.:VOYAGER ORBIT DETERhlINATlON AT JUPITER 265

59,jOo END OF DATA ARC END OF DATA ARC -23.5 DAYS @ -23.5 DAYS @ @ -21.5 DAYS @ -21.5 DAYS -19 DAYS @ -19 DAYS @ @) -17 DAYS @ -17 DAYS -15.5 DAYS @ -15.5 DAYS 0

RADIO PLUS OPTICAL RADIO PLUS OPTICAL WEAKER OPTICAL WEAKER OPTICAL DATA WEIGHTING DATA WEIGHTING

‘sJ23 c,

STRONGER OPTICAL 2 DATA WEIGHTING

RADIO ONLY

54JxGRAm; ESTIMATE ESTIMATE SATELLITE, PLANET EPHEMERIDES; NON-GRAVS; EQUIVALENT STATION SATELLITE, PLANET EPHEMERIDES LOCATION ERRORS I I I I I 937,400 937,Mo 937,600937,700937,800937,400937,Mo I I I I I 6. T, km 937,400 937,500937,400 537,600 937.800937.700 6-T, h Fig. IO. Variation in Voyager 1 real-timeB-plane solutions; station locations estimated. Fig, 9. Variation in Voyager 1 real-time B-plane solutions; station locations not estimated.

proaches, the radio data are sensitive normal to the line- of-sight to the dynamiceffects on spacecraft motion due to satellite mass or position errors in certain directions. The combination of both data types from both Voyageren- counters results in a Galilean satellite ephemerissigmfi- cantly improvedover that available to Voyager from Earth-based observations [15]. A briefsummary of the orbit estimates based on the Voyager 1 data isbriefly discussed here. A dynamical theory for the Galilean satellites developed by Lieske [ 131, [ 141 was used to compute bothresiduals and partials for the optical data and todevelop the ephemerides which were the starting point for the Voyager analyses.The adjustable Lieske parameters (included as satellite posi- tional errors in Table I) ‘consist of three constants for each satellite which are fractional errors in the mean motion, the eccentricity and the sine of the inclination and three angular parameters which are essentially the initial longitude, the argument of periapse, and the nodal position. The changes to the estimated Lieske parameters, andthe a priori and a posteriori errors for these parameters are shown in Table VI. In addition, the masses of the satellites, and Jupiter, and the right ascension and declination of the north pole of Jupiter are also adjustable. 1,911,6001,911,500 1,911,400 1,911,300 In the analysis described here the arguments of periapse B.T, km were not estimated because their effect on the data was Fig. 11. Voyager 2 approach OD performance. 266 IEEE TRANSACTIONS ON AUTOhlATIC CONTROL, VOL. AC-28, NO. 3, MARCH 1983

TABLE IV SUXMARY OF VOYAGER 1 NAMGATIONDELIVERY PERFOFNANCE

Co ntrol Parameter Rationale Target Value Achieved Allowable ActualAllowableAchieved Value Target RationaleParameter Control Va lue Error Error Error Value

Geocentricocculation Timing of JupiterMarch 5, March1979 5, 301979 +].4 entry at scan limb 15:35:20.715:45:22.1 occulation

Distance off center-lineGuarantee sampling -7.4 km +lo1 km 1000 km 108 km of fluxtube ofcurrent in Io model, km tube flux

Io closestapproach Io mosaicking Rarch 5, 1979Parch 5, 1979 20 +1.8secsec time sequencecontrol15:13:18.9 15:13:20.7

Finaldelivery maneuver was executed12.5 days prior to Jupiter closest approach.

TABLE V SU~~MARYOF VOYAGER 2 NAVIGATIONDELIVERY PERFORMANCE

C on trol Parameter Rationale Target Value Acbieved Expected 1-0 ActualAcbieved Expected1-0Parameter ValueControlRationale Target !r,l ue PerformanceError

JupiterB-plane Minimizemagnitude B.R = 134,806 km 134,730 km 223 km -74 km coordinates of post-Jupiter B.T = 1,911,454 km 1,311,337 km 161 km -67 km TCM5

Jupiterclosest Pointing control 79 79 approachtime of Earthline TCM5 2Zh2$& GMT 22$19679 29 1.6 s GMT 22 sec+1.6 set

~~~ ~

Finaldelivery maneuver was executed 12 daysprior to Jupiter closest approach.

“I I I I I NOMINAL REFERENCE l, ! - -- ESTIMATE EPHEMERISFOR EPHEMERIS AND i ONLY LOCATIONSSTATION ,’/ or---

~ - ESTIMTE EPHEMERISa 0.1 - STATION LOCATIONS 0.5 - REALTIME RADIO-ONLY . - .. .. - , . DELIVERY ORBIT LOCATKIN MAPPED TO UNCERTAINTY 0e ENCOUNTER v 0.2 - a 0.4 - e i L 0 2 STATION 2 - 2 SOLUTION 2 2 0.3- AA = 0.3m 0.3 - -I Ars = 0.3m 4

Q 2 c u z i? 0.4- ;0.2 - $ !2 U 0 P 0 0.5 -

0.6 -

GEOCENTRIC ART ASCENSION, grad Fig. 13. Voyager 1 radiometricperformance atdelivery 0.7 I I I I -0.5 -0.4 -0.3 -0.2 -0.1 GEOCENTRIC ART ASCENSION, grad shown in Table VI1 in which the entries represent the RSS Fig. 12. Indication of Voyager 1 radiometric capability. of the sigmas and the changes in the three components of position of each satellite at the epoch of Voyager 1 Jupiter know to be small (less than about 15 km). Mean motions, closest approach. The Voyager 1 a posteriori errors are very whose values were extremely well-detemined from many similar for all the satellites with the slightdifferences years of Earth-based data were also not included. explainable by the variability of optical data distribution A comparison of the formal standard errors in Cartesian and quantity, and by the ratio of the satellite periods which coordinates determined from the Earth-based and Voyager affects the “average quality” of the data around the ob- 1 data (using the Earth-based estimates as a priori) is served orbits. The Cartesian changes at the encounter CAMPBELL et d:VOYAGER ORBIT DETERMINATIONAT JUPITER 267

TABLE VI CHANGES TO ESTIMATED LIESKEPARA?%lE.TERS

Lieske Parameter Lieske A Earth-baseda posteriori 0 a priori G

SEPS 16 - .175 .213 .42 5EPS 17 - ,0022 .179 .26 eccentricity SEPS 18 - .013 .0071 .019 SEPS 19 .0024 .0013 .0036

SEPS 21 -.134 .174 .41 SEPS 22 - .0082 .0066 .025 sine i 5EPS 23 ,0032 .020 .050 SEPS 24 -.0114 .018 .ll

5BET 01 - .0014 .0028 .017 long 5BET 02 - .0045 ,0011 .004 SBET 04 - .0054 ,0005 .005

5BET 11 7.5 5.7 18.0 node 5BET 12 .512 .575 1.05 5BET 13 ,407 1.16 2.5 SBET 14 -.551 ,817 2.4

!?A ZACPL!?A 5 ,0064 .0084 .05 pol e DECZOEPL 5 - .0003 .0049 .026

Satellite, 501 l.24 29.4 4.6 56 Jupiter 503 GU 3.6 3.5 76 masses 504 GM 7.5 1.4 47 m5 657.5 24.7 1200

TABLE VI1 estimation software performed so well that most of the COMPARISONOF CARTESIAN FORMAL STANDARD ERRORS AND time it was taken for granted by the navigation team, and CORRECTIONS that is the ultimate compliment. Earth Based Voyager As a result of having actually used satellite images to a priori a posteriori G G ziz:; perform Jupiter and satellite-relative navigation, it has been determined that several prelaunch hypotheses regard- 87 20 Io 87 35 ing error sources for optical measurements were not cor- Europa 137 28 43 rect. Specifically, before the encounter experience it was

Ganymede34 161 BO assumed that center-finding errors for large satellite images would scale with the size of the image. From a detailed Call isto 448 35 180 analysis of hundreds of images it was found that the center Computed fromVoyager 1 EncounterData. finding errors were more likely to either decreasewith Numbersshown are RSS of x, y, z components. increasing image size, or to remain essentially constant. As Epoch ofCorrections = March5, 1979 12:05 GYT indicated inTable I1 the rms noise associated with the optical measurement postfit residuals were found to be epoch are well within the Earth-based a priori, but may about 0.25 pixel as compared with the 1.0 pixel error vary somewhat at other points of the orbit. Only changes assumed prelaunch. It is to be expected that the postfit to the longitudes of Europa and Callisto were of the order residuals should have a smaller rms than the data noise (I of an a priori sigma. It is apparentthat the a priori because it is known that the postfit residual z, - Hxj, has ephemeris was an excellent product. VarianCe - HP/,,H? VI. CONCLUSIONS In addition, we have learned that the combination of We have shown that the orbit determination performed radiometric with optical measurements must be done care- to deliver each spacecraft to its target at Jupiter was within fully, with regard to the information content of each data thepredicted capability of theradiometric and type. This was especially true in the case of Voyager, which optical-based navigation system for Voyager, and in fact represented a distinct extreme in the level of dynamical improved for Voyager 2 as a result of the Voyager 1 corruption of the radiometric signal by small spacecraft- experience. The postdelivery OD knowledge solutions done generated velocitypulses, that did not affect the corre- to refine scan platform pointing were well within the stated sponding optical data. requirement, even though the knowledge epochs were earlier than anticipated prelaunch. Numerous tests were made throughout the mission to ACKNOWLEDGMENT test estimate consistency and accuracy of the SRIF/SRIS and U-D algorithm implementations. The navigation J. E. Reidel provided much of the postencounter com- estimation software performedflawlessly. Indeed, the puter support for this paper. 268 3E TRANSACTIONS ON AUTOMATIC CONTROL., VOL. AC-28, NO. 3, hiARCH 1983

REFERENCES tion analysis for the Viking missions to Mars. Most recently, he par- ticipated in the navigation of the Voyager spacecrafts to Jupiter and Saturn as leader of the Orbit Determination Group. G. J. Bierman. “Sequential least-squares using orthogonal transfor- mations.” Jet Propulsion Lab., Pasadena. CA, Aug. I. 1975. Tech. Mr. Campbell is a member of the American Geophysical Union and the Memorandum 33-735. American Institute of Aeronautics and Astronautics. G. J. Bierman, Factorization Methods For Discrete Sequential Esri- mation. New York: Academic, 1977. T. D. Moyer, “Mathematical formulation,,of the double-precision orbit determination program (DPODP). Jet Propulsion Lab., Pasadena, CA, May 15, 1971, Tech. Rep. 32-1527. J. Ellis, “Voyager software requirements document for theorbit determination program (ODP) Jupiter encounter version,” Jet Prop- ulsion Lab.. Pasadena, CA, May IO, 1977, Rep. 618-772. N. Jerath. “Interplanetary approach optical navigation with appli- Stephen P. Synnott was born in NJ in 1946. He cations,” Jet Propulsion Lab.. Pasadena, CA. June I, 1978. Pub. received the B.S. degree in aeronautical engineer- 78-40. ing from Rensselaer Polytechnic Institute, Troy, S. P. Synnott, “Voyager software requirements document,” Jet NY, in 1968 and the M.S. and Ph.D. degrees in Propulsion Lab.. Pasadena, CA. July IO. 1977, Pub. 6 I 8-729. astrodynamics from Massachusetts Institute of G. J. Bierman. “Square-root information filtering and smoothing Technology, Cambridge, in 1970 and 1974,re- for precision orbit determination.” Mathematical Programming Str& 18. dlgorirhn2s and Theon in Filterrngand Control. 1982, pp. 61-75. spectively. J. K. Campbell, S. P. Synnott. J. E. Reidel, S. Mandel, L. A. Since 1975 he has been employed at the Jet Morabito. and G. C. Rinker. “Voyager 1 and Voyager2 Jupiter Propulsion Laboratory, California Institute of encounter orbit determination.” presented at the AIAA 18th Technology. Pasadena, where he specializes in Aerospace Sci. Meet., Pasadena, CA. Jan. 14- 16. 1980, paper AIAA spacecraft navigation, image processing, and cel- 80 0241. estial mechanics. C. S. Christensen. “Performance of the square-root information filter for navigation of the Mariner 10 spacecraft,” Jet Propulsion Lab., Pasadena, CA, Jan. 1976, Tech. Memorandum 33-757. J. K. Cambell, R. A. Jacobson, J. E. Reidel, S. P. Synnott. and A. H. Taylor, “Voyager I and Voyager I1 Saturn encounters,” presented at the 20th Aerospace Sci. Meet., Jan. 1982, paper AIAA 8-82-0419, G. J. Bierman and C. L. Thornton,“Numerical comparison of Kalman filter algorithms; Orbit determination case study,” Auto- matica. vol. 13. pp. 23-35, 1977. Gerald J. Bierman (h4’68-SW77) received the G. J. Bierman and K. H. Bierman, “Estimation subroutine library. Ph.D. degree in applied mathematics from the directory and preliminaq user’s guide.” Factorized Estimation Ap- Polytechnic Institute of Near York, Brooklyn, plications, Inc., Sept. 1982. NY, in 1967, and the MS. degrees in electrical J. H. Lieske. “Improved ephemeris of the Galilean satellites.” engineering and mathematics from the Courant Asrrophxsics. vol. 82. pp. 340-348, 1980. Institute and School of Engineering, New York I. H. Lieske. “Theory of motion of the Galilean satellites.” Astron. University, New York, respectively. and .Isrrophys.. vol. 56. pp. 333-352. 1977. S. P. Synnott and J. K. Campbell, “Orbits and masses of the Jovian He worked at Litton Systems Guidance and system from Voyager data: Preliminary.” presented at the IAU-Col- Controls Division, developing and analyzing sub- loquium 57, Institute for Astronomy, Univ. Hawaii. May 1980. optimal Kalman filter designs, andat the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, where he was responsible for creating high precision orbit determination and reliable parameter estimation software. h 1979 he served as a NATO consultant and soon after created Factorized James K. Campbell was born in Brooklqn, NY, Estimation Applications, a small consulting firm specializing in high-qual- on July I. 1942. He received the B.S. and M.S. ity numerically reliable estimation software. He is known for hs contri- degrees from the School of Engineering. Univer- butions related to the square-root information filter/smoother and the sity of California, Los Angeles, in 1964 and 1966, L:- D covariance factorized Kalman filter. He hasconsiderable navigation respectively. system applications experience. over fifty refereed publications, and a He has rvorked at the Jet Propulsion Labora- research monograph. Factorization Methods for Discrete Sequential Estima- ~. tory,California Institute of Technology-. tion (New York: Academic, 1977) that is devoted to the computational - Pasadena. since 1966. He contributed to the en- aspects of estimation. His publications and applications experiences have counter trajectory design of the Mariner 6 and 7 sensitized both estimation researchers and control applications engineers missions to Mars and the Pioneer IO and 11 to the importance of estimation algorithms that are both efficient and missions to Jupiter, and to the orbit determina- numerically reliable.