The Multi-Stage-Lambert Scheme for Steering a Satellite-Launch Vehicle (SLV)

Transcription

1 The Multi-Stage-Lambert Scheme or Steering a Satellite-Launch Vehicle (SLV) Syed Ari Kamal* University o Karachi, Karachi, Pakistan; prodrakamal@gmail.com Abstract The determination o an orbit, having a speciied transer time (time-olight) and connecting two position vectors, requently reerred to as the Lambert problem, is undamental in astrodynamics. O the many techniques existing or solving this twobody, two-point, time-constrained orbital boundary-value problem, Gauss' and Lagrange's methods were combined to obtain an elegant algorithm based on Battin's work. This algorithm included detection o crossrange error. A variable TYP, introduced in the transer-time equation, was lipped, to generate the inverse-lambert scheme. In this paper, an innovative adaptive scheme was pre-sented, which was called the Multi- Stage-Lambert Scheme. This scheme proposed a design o autopilot, which achieved the pre-decided destination position and velocity vectors or a multi-stage rocket, when each stage was detached rom the main vehicle ater it burned out, completely. Keywords Lambert scheme, inverse-lambert scheme, multi-stage Lambert scheme, two-body problem, transer-time equation, orbital boundaryvalue problem c e G m M n p r r r r r t t t TYP Length o cord o the arc connecting the points corresponding to radial ccentricity o the ellipse ccentric anomaly ccentric anomaly corresponding to the launch point ccentric anomaly corresponding to the inal destination True anomaly Universal constant o gravitation Mass o spacecrat Mass o earth Unit vector, indicating normal to the trajectory plane Parameter o the orbit Flight-path angle Radial coördinate Radial coördinate o launch point Radial coördinate o inal destination Radius vector in the inertial coördinate system Radius vector o inal destination Universal time Launch time Time o reaching inal destination Variable expressing direction o motion o spacecrat relative to earth s rotation NOMNCLATUR v Velocity vector in the inertial coördinate System A. Symbols (in alphabetical order) Product o universal constant o gravitation and sum o masses o a Semi-major axis o the ellipse spacecrat and earth a m Semi-major axis o the minimum- Time o pericenter passage energy orbit Transer angle *PhD; MA, Johns Hopkins, United States; Interdepartmental Faculty, Institute o Space and Planetary Astrophysics (ISPA), University o Karachi; Visiting Faculty, Department o Aeronautics and Astronautics, Institute o Space Technology, Islamabad paper mail: Proessor, Department o Mathematics, University o Karachi, PO Box,84,Karachi 7570, Pakistan; telephone: ext. 9 homepage: member AIAA Angle o inclination o velocity vector Astrodynamical terminologies and relationships are given in Appendices A and B, respectively. Appendix C contains a proo o the relationship connecting true and eccentric anomalies, providing a justiication o positive sign in ront o radical /08/$ I 94 Proceedings o the th I International Multitopic Conerence, dited by Anis MK, Khan MK, Zaidi SJH, December, 4,008, Bahria University, Karachi, Pakistan, pp 94-00, I Catalog Number: CFP0859-PRT, ISBN: , Library o Congress:

2 TH MULTI-STAG-LAMBRT SCHM FOR STRING AN SLV 95 B. Compact Notations In order to simpliy the entries, = e, = e, = G (m + M), are used in the expressions e and equations. I. INTRODUCTION Determination o trajectory is an important problem in astrodynamics. For a spacecrat moving under the inluence o gravitational ield o earth in ree space (no air drag) the trajectory is an ellipse with the center o earth lying at one o the oci o the ellipse. This constitutes a standard two-body-centralorce problem, which has been treated, in detail, in many standard text-books [, ]. The trick is to irst reduce the problem to two dimensions by showing that the trajectory always lies in a plane perpendicular to the angular-momentum vector. Then the problem is set up in plane-polar coördinates. Angular mo-mentum is conserved and the problem, eectively, reduces to one-dimensional problem involving only the variable r []. A problem amous in astrodynamics, called the Lambert Problem, is based on the Lambert theorem [4, 5]. According to this theorem the orbital-transer time depends only upon the semi-major axis, the sum o the distances o the initial and the inal points o the arc rom the center o orce as well as length o the line segment join-ing these points. Based on this theorem a problem called the Lambert problem is ormulated. This problem deals with determination o an orbit having a speciied lighttime and connecting the two position vectors. Battin [6, 7] has set up the Lambert problem involving computation o a single hypergeometric unction. Since transer-time (time-o-light) computation is done on-board, it is desirable to use an algorithm employ-ing as ew computation steps as possible. The use o polynomials instead o actual expression and reduction o the number o degrees o reedom contribute towards the same goal. An elegant Lambert algorithm, presented by Battin, was scrutinized and omissions/oversights in his calculations pointed out [8]. Battin s ormulation, which highlighted the main principles involved, was developed and expanded to a set o ormulae suitable or coding in the assembly language. These ormulae could be used as a practical scheme outside the atmosphere or steering a satellite-launch vehicle (SLV). This scheme computes velocity and light-path angle required at any intermediate time to be compared with the actual velocity and the actual light path angle o the spacecrat, as reported by the on-board computing system. A spacecrat cannot reach the desired location i cross-range error is present. Battin s original work does not address this issue. A mathematical ormulation was given by the author to detect crossrange error [8]. Algorithms were developed and tested, which indicated cross-range error. In order to correct cross-range error velocity vector should be perpendicular to normal to the desired trajectory plane (i. e., the velocity must lie, entirely, in the desired trajectory plane). A variable TYP was introduced in the transer-time equation to incorporate direction o motion o the spacecrat [9]. This variable can take on two values, + (or spacecrats moving in the direction o rotation o earth) and (or spacecrats moving opposite to the direction o rotation o earth). In the inverse- Lambert scheme, TYP was lipped, whereas all other parameters remained the same [8]. For an eicient trajectory choice, a transer time close to minimumenergy orbital transer time was selected. A procedure or inding the minimum-energy orbital transer time is included in this work. Additionally, ormulae are given to compute the orbital parameters in which the SLV must be locked in at a certain position, at a time, t, based on the Lambert scheme. In this paper, the Multi-Stage-Lambert Scheme is presented. In this ormulation, section-wise corrections are achieved, where destination point o the irst stage is initiating point o the second stage and so on. In this way, position and rate satu-rations (out o range deviations) are avoided. A similar ormulation was, earlier, given or the Q System [9]. II. STATMNT OF TH PROBLM In order to choose a particular trajectory on which the spacecrat could be locked so as to reach a certain point one must select a certain parameter to ix this trajectory out o the many possible ones connecting the two points. One is, thereore, interested to put the Kepler equation in such a orm so as to make it computationally eicient utilizing hypergeometric unctions or quadratic unctions instead o circular unctions (sine or co-sine, etc.). This equation should express transer time between these two points in terms o a series or a polynomial, and another ormula should be available to compute light-path angle, correspon-ding to this transer time. Velocity desirable or a particular trajectory may, then, be computed on- board using this ormula and compared with velocity o the spacecrat obtained rom integration o acceleration inormation, which is available rom on-board accelerometers and rate velocity board /08/$ I Proc. o th I INMIC, Dec., 4,008

3 96 KAMAL gyroscopes. III. TH LAMBRT THORM In 76 Johann Heinrich Lambert, using a geometrical argument, demonstrated that the time taken to traverse any arc (now called transer time), t, is a unction, only, o the major axis, a, the t ( r r arc, ), and length o chord o the arc, c, or elliptical orbits, i. e., t = ( r r, c, ) t a The symbol,, is used to express unctional relationship in the above equation (do not conuse with true anomaly). Thereore, one notes that the transer time does not depend on true or eccentric anomalies o the launch point or the inal destination. Fig. illustrates geometry o the problem. Fig. Geometry o the boundary-value problem Mathematically, transer time or an elliptical trajectory may be shown to be [6] ( t t ) ( sin ) ( sin ) a r r c r r c where, sin, sin. 4a 4a In the case at hand, G( m M) GM, because m M. Based on this theorem a ormu-lation to calculate transer time and velocity vector at any instant during the boost phase is developed. This ormulation is termed as the Lambert scheme. IV. TH LAMBRT SCHM Suppose a particular elliptical trajectory is connecting the points P and P. Let t and t be the times, when the spacecrat passes the points P and P, respectively, the radial coördinates being r and r. The standard Kepler equation ( is time o pericenter passage) / ( t ) a ( esin ) () may be expressed as ( t t ) a ( sin ) () where ( ), e ( ). This equation may be used to calculate transer time between two points by iterative procedure. However, it is unsuitable or on-board compu-tation, because computing time is large owing to the presence o circular unctions. In order to put this in a orm involving power series, one intro-duces a m s ( r r c) () s r r (4) where a m is the semi-major axis o the minimumenergy orbit and the transer angle. From the geometry, one has c r r r (5) r y ( x ) (6) Where x ( ). Further, introducing y x (7) S ( x) (8) and a Q unction 4 5 Q F(,; ; S ) (9) expressible in terms o a hypergeometric unction, 5 F (,; ; S), instead o a circular unction. This is needed to reduce onboard computation time. Transer time may be expressed as ( t t) Q 4 (0) a m The magnitude o velocity, v, and light-path angle,, may be evaluated using the ollowing expressions [4, 6] am r v [ ( x)] sin (a) a r r m r ( ) (b) r v The hypergeometric unction in (9) is given by the continued-raction expression (this expression is needed to reduce on-board computing time) /08/$ I Proc. o th I INMIC, Dec., 4,008

4 TH MULTI-STAG-LAMBRT SCHM FOR STRING AN SLV 97 5 F(,; ; S) 8S 6S 5 40S 7 4S 9 70S 8S 08S About 00 terms are needed to get an accuracy o 0-4. To compute the transer time corresponding to the minimum energy orbit connecting the current position and the inal destination one uses the transertime equation (), with x 0 substituted in (8), corresponding to a am and solves it using Newton-Raphson method. Transer time in the Lambert algorithm must be set close to this time. Fig. shows the low chart o Lambert algorithm. jectory plane). In other words, the velocity vector must make an angle o 90 0 with normal to the trajectory plane. This is equivalent to [8] v.( r r ) 0 () which says that v (current velocity), r (current position) and r (position o destination) are coplanar. For no cross-range error, the angle o inclination,, between the velocity vector, v, and normal to the trajectory, n, must be In the Lambert scheme a subroutine computes devi-ation o rom xtended-cross-product steering [0], dot-product steering [] and ellipse-orientation steering [] could be used to eliminate cross-range error. VI. TH INVRS-LAMBRT SCHM Transer-time equation between two points having eccentric anomalies and, (corresponding to times t and t, respectively) may be expressed as [8] t t () a [( esin ) ( esin )][ TYP] The variable TYP has to be introduced because the Kepler equation is derived on the assumption that t increases with the increase in. Thereore, the dierence [( esin ) ( esin )] Fig. Flow chart o the Lambert scheme V. CROSS-RANG-RROR DTCTION (MATHMATICAL MODL) For no cross-range error, velocity o the spacecrat must lie in the plane containing r r (the tra- shall come out to be negative or spacecrats orbiting in a sense opposite to rotation o earth. The variable TYP ensures that the transer time (which is the physical time) remains positive in all situations by adapting the convention that TYP =+ or spacecrats moving in the direc-tion o earth s rotation, whereas, TYP = or spacecrats moving opposite to the direction o earth s rotation. This becomes important in computing correct light-path angles in the Lambert scheme. I one wants to put another spacecrat in the same orbit as the original spacecrat, but moving in the opposite direction, one can use the inverse- Lambert scheme. This can be accomplished by lipping sign o the variable, TYP, whereas all the other parameters remain the same. Burns and Sherock [] try to accomplish the same objective by a three-degree-oreedom interceptor simulation designed to rendezvous with a ballistic target: position and velocity matching, no lipping o TYP. The strategy put or /08/$ I Proc. o th I INMIC, Dec., 4,008

5 98 KAMAL ward by this author was simpler [8]: orbit matching, lipping o TYP. The inverse-q system has, also, been proposed by the author to accomplish the same objective [9]. Do not conuse Q system with the symbol Q introduced in (9). VII. TH MULTI-STAG-LAMBRT SCHM Let us consider a three-stage rocket. Destination point o the irst (second) stage is the initiating point o the second (third) stage. Mathematically, v, inal v, initial ; v, inal v, initial (4a) inal, initial ;, inal, initial (4b) quations (a, b) can be used to compute velocities and light-path angles at the initiating and destination points o various stages. By using these section-wise corrections, one can avoid position saturation and rate saturation. VIII. DISCUSSION AND CONCLUSIONS The Lambert problem is a ixed-transer-timeboundary-value problem. The Lambert-scheme ormulations available in literature run into problems in terms o computing correct light-path angles and velocities, in particular, or spacecrats moving opposite to earth rotation because o deinition o time in the Kepler equation. This problem was resolved by introducing a variable, TYP, in the transer-time equation. This, also, led to a natural ormulation o the inverse-lambert scheme. The Lambert scheme is applicable in ree space, in the absence o atmospheric drags, or burnout times large as compared to on-board computation time (or example, i the burnout time or a given light is 8 second and the computation time is second, there may not be enough time to utilize this scheme). This is needed to allow suicient time or the control decisions to be taken and implemented beore the rocket runs out o uel. Detection o cross-range error is incorporated in this ormulation. It is assumed that rocket is ired in the vertical position so as to get out o the atmosphere with minimum expenditure o uel. Later, in ree space this scheme is applied to correct the path o rocket. Since the rocket remains in ree space or most o the time, this method may be useul in calculating the desired trajectory. The Lambert scheme is an explicit scheme, which generates a suitable trajectory under the inluence o an inverse-square-central-orce law (gravitational ield o earth) provided one knows the latitudes and the longitudes o launch point and o destination as well as the transer time (time spent by the spacecrat to reach destination). The Lambert scheme is an explicit scheme, which generates a suitable trajectory under the inluence o an inverse-square-central-orce law (gravitational ield o earth) provided one knows the latitudes and the longitudes o launch point and o destination as well as the transer time. ACKNOWLDGMNT This work was made possible, in part, by Dean's Research Grant awarded by University o Karachi, which is, grateully, acknowledged. APPNDIX A: ASTRODYNAMICAL TRMINOLOGIS Down-range error is the error in the range assuming that the vehicle is in the correct plane; crossrange error is the oset o the trajectory rom the desired plane. An unwanted pitch movement shall produce down-range error; an unwanted yaw movement shall produce cross-range error. For an elliptical orbit true anomaly,, is the polar angle measured rom the major axis ( PFX in Fig. ). Through the point P (current pos Fig. Justiication o the positive sign in position o spacecrat, mpf r, the radial coördinate) erect a perpendicular on the major axis. Q is the intersection o this perpendicular with a circumscribed auxiliary circle about the orbital path. The angle, QOF (c. Fig.) is called the eccentric anomaly, /08/$ I Proc. o th I INMIC, Dec., 4,008

6 TH MULTI-STAG-LAMBRT SCHM FOR STRING AN SLV 99 For this orbit, pericenter, the point on the major axis, which is closest to the orce center (point A in Fig. ), is chosen as the point at which = 0. Apocenter is the opposite point on the major axis, which is arthest rom the orce center (point A in Fig. ). The line joining the pericenter and the apocenter is called the line o apsides. APPNDIX B: ASTRODYNAMICAL RLATIONHIPS Some useul relationships among radial coördinate, eccentric anomaly, eccentricity and semi-major axis or an elliptical orbit are listed below: r a( e) (Ba) r a( e) (Bb) rsin asin (Bc) r a( e) (Bd) r sin a( e) sin (Be) The ollowing may be useul in converting circular unctions involving true anomalies to those involving eccentric anomalies and vice versa. e (Ba) e e (Bb) e sin sin e (Bc) sin sin e (Bd) tan tan (Be) In Appendix C, the last relation is proved and a justi-ication is given or the positive sign taken in ront o the square root appearing in the expression or. APPNDIX C: RLATION CONNCTING CCNTRIC ANOMALY TO TRU ANOMALY In Fig., semi-minor axis o the ellipse, b, is related to a by b a. Do not conuse the point Q in Fig. with the quantity Q deined in (9). Using the p relations r e and p a( e ), one may write, r( e ) a( e ). Rearranging er a( e ) r Adding er to both sides and using (Ba) on the righthand side, one gets r( ) a( e)( ) (C) Subtracting er rom both sides and using (Ba) on the right-hand side, one gets r( ) a( e)( ) (C) Dividing (C) by (C) Using the identities ( e)( ) ( e)( ) sin with similar results or the expressions ( ) and ( ), one sees that the above equation reduces to e tan tan e which implies e tan tan e Below, it is justiied that only positive sign with the radical gives the correct answer. Consider ORF (c. Fig. ). One notes that, Further, 0 0 ; 0 0. Thereore, and have the same sign. When 0, tan 0, tan 0, which implies that positive sign with the radical should be chosen. Similarly, 0, tan 0, tan 0 and, hence, positive sign with the radical is the correct choice. Two-body problem can be handled elegantly by using the elliptic-astrodynamical-coördinate mesh [4, 5]. One discovers additional constants o motion i the problem is set up using this ormulation [6] /08/$ I Proc. o th I INMIC, Dec., 4,008

7 00 KAMAL RFRNCS ormulation o extended-cross-product steering, Proceedings o the First International [] H. Goldstein, Classical Mechanics, nd Bhurban Conerence on Applied Sciences and ed., Reading, MA: Addison Wesley, Technologies (IBCAST 00), vol., Advanced 98, pp Materials, Computational Fluid Dynamics and [] J. B. Marion, Classical Dynamics o Control ngineering, ed. H. R. Hoorani, A. Particles and Systems, nd ed., New Munir, R. Samar and S. Zahir, National Center or York: Academic, 970, pp Physics, Bhurban, 00, pp [] J. T. Wu, Orbit determination by sol- [] S. A. Kamal, Dot-product steering: a new conving or gravity parameters with multi- trol law or satellites and spacecrats, Proceedple arc data, Journal o Guidance, ings o the First International Bhurban Coner- Control and Dynamics, vol. 5, pp. 04 ence on Applied Sciences and Technologies -, February 99 (IBCAST 00), vol., Advanced Materials, [4] R. H. Battin, An Introduction to the Computational Fluid Dynamics and Control ng- Mathematics and the Methods o Astro- ineering, ed. H. R. Hoorani, A. Munir, R. Samar dynamics, st ed., Washington, DC: and S. Zahir, National Center or Physics, Bhur- AIAA, 987, pp α ban, 00, pp [5] R. Deusch, Orbital Dynamics o Space [] S. A. Kamal, llipse-orientation steering: a Vehicles, nglewood Clis, NJ: Pren- control law or spacecrats and satellite-launch tice Hall, 96, pp. 0- vehicles (SLV), Space Science and Challenges [6] R. H. Battin, Lambert s problem re- o the Twenty-First Century, ISPA-SUPARCO visited, AIAA Journal, vol. 5, pp. 707 Collaborative Seminar (in connection with -7, May 977 World Space Week), University o Karachi, [7] R. H. Battin and R. M. Vaughan, An 005 (invited lecture) elegant Lambert algorithm, Journal o [] S. P. Burns and J. J. Scherock, Lambert-gui- Guidance and Control, vol. 7, pp. 66- dance routine designed to match position and 670, June 984 velocity o ballistic target, Journal o Gui- [8] S. A. Kamal, Incorporating cross-range dance, Control and Dynamics, vol. 7, pp error in the Lambert scheme, the Tenth 996, June 004 National Aeronautical Conerence, Coll- [4] S. A. Kamal, Planetary-orbit modeling based on ege o Aeronautical ngineering, PAF astrodynamical coördinates, the Pakistan Insti- Academy, Risalpur, 006, pp µ tute o Physics International Conerence, Lahore, [9] S. A. Kamal and A. Mirza, The multi- 997, pp. 8-9 stage-q system and the inverse-q sys- [5] S. A. Kamal, Use o astrodynamical coördinates tem or possible application in satellite- to study bounded-keplarian motion, the Fith Intlaunch vehicle (SLV), Proceedings o ernational Pure Mathematics Conerence, the the Fourth International Bhurban Con- Quaid-é-Azam University and the Pakistan Matherence on Applied Sciences and Tech- ematical Society, Islamabad, 004 Technologies (IBCAST 005), vol., [6] S. A. Kamal, Strong Noether s theorem: applica- Control and Simulation, ed. S. I. Hussain, ions in astrodynamics, the Second Symposium on A. Munir, J. Kayani, R. Samar and M. A. Cosmology, Astrophysics and Relativity, Center Khan, National Center or Physics, Bhur- or Advanced Mathematics and Physics, NUST ban, 006, pp. 7- and National Center or Physics, Rawalpindi, [0] S. A. Kamal, Incompleteness o cross- 004, p. Product steering and a mathematical α Review: µ Full text: Full text: Full text: Full text: Abstract:hhttp:// Abstract: Abstract: Abstract: Web address o this document (author s homepage): Abstract: /08/$ I Proc. o th I INMIC, Dec., 4,008