Cross-Range Error in the Lambert Scheme

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1 Proceedings o the Tenth National Aeronautical Conerence, Edited by Sheikh SR, Khan AM, College o Aeronautical Engineering, PAF Academy, Risalpur, NWFP, Pakistan, April -, 006, pp 55-6 Cross-Range Error in the Lambert Scheme Syed Ari Kamal* University o Karachi, Karachi, Pakistan Abstract The determination o an orbit, having a speciied transer time (time-o-light) and connecting two position vectors, requently reerred to as the Lambert problem, is undamental in astrodynamics. O the many techniques existing or solving this two-body, two-point, time-constrained orbital boundary-value problem, Gauss' and Lagrange's methods are combined to obtain an elegant algorithm based on Battin's work. This algorithm includes detection o cross-range error. A variable TYPE, introduced in the transer-time equation, is lipped, to generate the inverse-lambert scheme. The Lambert scheme could be useul in steering a satellite-launch vehicle (SLV) as well as constructing the control system o a passenger crat traveling in a ballistic trajectory. Keywords: Lambert scheme, inverse-lambert scheme, two-body problem, transer-time equation, orbital boundaryvalue problem Nomenclature r Radius vector o inal destination t Universal time a) Symbols (in alphabetical order) t Launch time t Time o reaching inal destination a Semi-major axis o the ellipse TYPE Variable expressing direction o motion a m Semi-major axis o the minimum- o spacecrat relative to earth s rotation energy orbit v Velocity vector in the inertial coördinate c Length o cord o the arc connecting System the points corresponding to radial Product o universal constant o coördinates r and r gravitation and sum o masses o e Eccentricity o the ellipse spacecrat and earth E Eccentric anomaly Time o pericenter passage E Eccentric anomaly corresponding to the Transer angle launch point Angle o inclination o velocity vector E Eccentric anomaly corresponding to the and normal to the plane inal destination True anomaly Astrodynamical terminologies and relationships G Universal constant o gravitation are given in Appendices A and B, respectively. m Mass o spacecrat Appendix C contains a proo o the relationship M Mass o earth connecting true and eccentric anomalies, providing n Unit vector, indicating normal to the trajectory plane a justiication o positive sign in ront o the radical. p Parameter o the orbit Flight-path angle b) Compact Notations r Radial coordinate In order to simpliy the entries, = e r Radial coördinate o launch point, = r Radial coördinate o inal destination e, = G (m + M), are used in the expressions r Radius vector in the inertial coördinate e system and equations. *Pro. Dr. Syed Ari Kamal ( Convener, National Curriculum Revision Committee (Mathematics), Higher Education Commission; Proessor and Chairman, Department o Mathematics ( and Inderdepartmental Faculty, Departments o Physics and Computer Science, Institute o Space and Planetary Astrophysics (ISPA), University o Karachi ( Visiting Faculty, Department o Aeronautics and Astronautics, Institute o Space Technology, Islamabad; paper mail: Department o Mathematics, University o Karachi, Karachi 7570; sakamal@uok.edu.pk; Member AIAA 55

2 56 KAMAL. 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-central-orce problem, which has been treated, in detail, in many standard textbooks [, ]. The trick is to irst reduce the problem to two dimensions by showing that the trajectory always lies in a plane perpendicular to the angularmomentum vector. Then the problem is set up in plane-polar coordinates. Angular momentum 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 semimajor 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 joining 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 light-time 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 onboard, it is desirable to use an algorithm involving 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. In this paper an elegant Lambert algorithm, presented by Battin, is scrutinized and omissions/oversights in his calculations pointed out. Battin s ormulation, which highlighted the main principles involved, was developed and expanded to present a set o ormulae suitable or coding in the assembly language to be used as a practical scheme outside the atmosphere or steering the satellite-launch vehicle (SLV). These ormulae may be used to compute the velocity and the light-path angle required at any intermediate time to be compared with the initial velocity and light-path angle o the spacecrat. A spacecrat cannot reach the desired location i cross-range error is present. Battin s original work does not address this issue. In this paper a mathematical ormulation is given to detect cross-range error. Algorithms have been developed and tested, which indicate cross-range error. In order to correct cross-range error velocity vector should be perpendicular to normal to the desired trajectory (i. e., the velocity must lie, entirely, in the desired trajectory plane). A variable TYPE is introduced in the transertime equation to incorporate direction o motion o the spacecrat. This variable can take on 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, TYPE is lipped, whereas all other parameters remain the same. For an eicient trajectory choice, a transer time close to minimum-energy orbital transer time should be selected. The paper highlights a procedure or inding the minimumenergy orbital transer time. Additionally, ormulae are given to compute the orbital para-meters in which the SLV must be locked in at a certain position, at a time, t, based on the Lambert scheme.. Statement o the Problem 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. The parameter to be chosen in the Lambert problem is transer time, which is ixed between 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 cosine, etc.). This equation should express the transer time between these two points in terms o a series or a polynomial, and another ormula should be available to compute velocity vector (speed and light-path angle), corresponding 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 gyroscopes.. The Lambert Theorem 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

3 CROSS-RANGE ERROR IN THE LAMBERT SCHEME 57 sum o radii vectors to the end o the speciied arc, ( r r ), 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. (page number 6) illustrates geometry o the 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 our case, G( m M) GM, because m M. Based on this theorem a ormulation to calculate transer time and velocity vector at any instant during the boost phase is developed. This ormulation is termed as the Lambert scheme. 4. The Lambert Scheme 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 ( E esin E) () may be expressed as ( t t ) a ( sin cos ) () Where ( E E ), cos ecos ( E E). This equation may be used to calculate transer time between points by iterative procedure. However, it is unsuitable or on-board computation, because computing time is large owing to the presence o circular unctions. In order to put this in a orm involving power series, one introduces a m s ( r r c) () s r r cos (4) where a m is the semi-major axis o the minimum- energy orbit and the transer angle. From the geometry, one has c r r r cos (5) r y ( x ) (6) where x cos ( ). Further, introducing y x (7) S ( x) (8) and a Q unction expressible in terms o a hypergeometric unction 4 5 Q F(,; ; S ) (9) Transer time may be expressed as ( t t) Q 4 (0) a m This involves a hypergeometric unction instead o a circular unction, and hence it is easy to evaluate. The magnitude o velocity, v, and the light-path angle,, may be evaluated using the expressions [4, 6] am r v [ ( x)] sin (a) a r r m r ( cos ) cos (b) r v The hypergeometric unction in equation (9) is given by the continued-raction expression (this expression is needed to reduce on-board computing time) 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 transer-time equation, with x 0, 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. (page number 6) shows the low chart o Lambert algorithm.

4 58 KAMAL 5. Cross-Range-Error Detection (Mathematical Model) For no cross-range error, velocity o the spacecrat must lie in the plane containing r r (the trajectory 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 Extended-cross-product steering [9] and dot-product steering [0] could be used to eliminate cross-range error. 6. The Inverse-Lambert Scheme Transer-time equation between two points having eccentric anomalies E and E, (corresponding to times t and t, respectively) may be expressed as [-] t t (4) a [( E esin E) ( E esin E)][ TYPE] The variable TYPE has to be introduced because the Kepler equation is derived on the assumption that t increases with the increase in. Thereore, the dierence [( E esin E) ( E esin E )] shall come out to be negative or spacecrats orbiting in a sense opposite to rotation o earth. The variable TYPE ensures that the transer time (which is the physical time) remains positive in all situations by adapting the convention that TYPE = + or spacecrats moving in the direction o earth s rotation, whereas, TYPE = or spacecrats moving opposite to the direction o earth s rotation. This becomes important in computing correct light-path angles in 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 lipping sign o the variable, TYPE, whereas all the other parameters remain the same. Burns and Sherock [4] try to accomplish the same objective by a three-degree-o-reedom interceptor simulation designed to rendezvous with a ballistic target: position and velocity matching, no lipping o TYPE. The strategy presented in this paper is simpler: orbit matching, lipping o TYPE. The inverse-q system has, also, been proposed by the author to accomplish the same objective [5]. Do not conuse Q system with the symbol Q introduced in eq. (9). 7. Conclusions 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). Acknowledgement This work was made possible, in part, by Dean's Research Grant awarded by University o Karachi, which is, grateully, acknowledged. Appendix A: Astrodynamical Terminologies Down-range error is the error in the range assuming that the vehicle is in the correct plane; cross-range error is the oset o the trajectory rom the desired plane. An unwanted pitch movement shall produce down-range error; an unwanted yaw

5 CROSS-RANGE ERROR IN THE LAMBERT SCHEME 59 accomplished by movement shall produce cross-range error. For an elliptical orbit true anomaly,, is the polar angle measured rom the major axis ( PFX in Fig. page number 6). Through the point P (current 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. QOF (c. Fig.) is called the eccentric anomaly, E. 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. Appendix B: Astrodynamical Relationhips Some useul relationships among radial coördinate, eccentric anomaly, eccentricity and semi-major axis or an elliptical orbit are listed below: r a( ecose) (Ba) r cos a(cose e) (Bb) rsin asin E (Bc) E r cos a( e) cos (Bd) E 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. cos cos E e e cos E (Ba) cos e cos E ecos (Bb) sin E sin ecos E (Bc) sin sin E ecos (Bd) E tan tan (Be) In Appendix C, the last relation is proved and a justiication is given or the positive sign taken in ront o the square root appearing in the expression or. Appendix C: Relation Connecting Eccentric Anomaly to True 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 eq. (9). p Using r ecos and p a( e ), one may write, r( ecos ) a( e ). This may, in turn, be written as er cos a( e ) r Adding er to both sides and using (Ba) on the right-hand side, one gets r( cos ) a( e)( cose) (C) Subtracting er rom both sides and using (Ba) on the right-hand side, one gets r( cos ) a( e)( cose) (C) Dividing (C) by (C) cos cos Using the identities ( e)( cos E) ( e)( cos E) cos sin cos cos with similar results or ( cos E) and ( cos E), one sees that the above equation reduces to e E tan tan e which implies e E tan tan e Below, it is justiied that only positive sign with the radical gives the correct answer. Consider ORF (Fig. ). One notes that, E E Further, E 0 0 ; E 0 0. E Thereore, and have the same sign. When E E 0, tan 0, tan 0, which implies that positive sign with the radical should be

6 60 KAMAL chosen. Similarly, 0 E E ormulation o extended-cross-product, tan 0, tan 0 steering. Proceedings o the First and, hence, positive sign with the radical is the International Bhurban Conerence on correct choice. Applied Sciences and Technologies, National Center or Physics, Bhurban, 00; pp [0] Kamal SA. Dot-product steering: a new Reerences control law or satellites and spacecrats. Proceedings o the First International [] Goldstein H. Classical Mechanics, nd ed., Addison Wesley, Reading, Ma., 98; pp Bhurban Conerence on Applied Sciences and Technologies, National Center or Physics, Bhurban, 00; pp [] Marion JB. Classical Dynamics o Particles and Systems, nd ed., Academic Press, New York, 970; pp [] Kamal SA. Planetary-orbit modeling based on astrodynamical coördinates. The Pakistan Institute o Physics International [] Wu JT. Orbit determination by solving or Conerence, Lahore, 997; pp 8-9 gravity parameters with multiple arc data. Journal o Guidance, Control and Dynamics 99; 5 (): 04- [] Kamal SA. Use o astrodynamical coördinates to study bounded-keplarian motion. The Fith International Pure [4] Battin RH. An Introduction to the Mathematics and the Methods o Astrodynamics, AIAA, Washington, DC, 987; Mathematics Conerence, the Quaid-é- Azam University & the Pakistan Mathematical Society, Islamabad, 004 pp [] Kamal SA. Strong Noether s theorem: [5] Deusch R. Orbital Dynamics o Space Vehicles, Prentice Hall, Englewood Clis, New Jersey, USA, 96; pp. 0- applications in astrodynamics. The Second Symposium on Cosmology, Astrophysics and Relativity, Center or Advanced [6] Battin RH. Lambert s problem revisited. AIAA Journal, 977; 5 (5): Mathematics and Physics, NUST & National Center or Physics, Rawalpindi, [7] Battin RH, Vaughan RM. An elegant 004; p Lambert algorithm. Journal o Guidance and Control, 984; 7 (6): [4] Burns SP, Scherock JJ. Lambert-guidance routine designed to match position and [8] Kamal SA. Ellipse-orientation steering: a control law or spacecrats and satellitelaunch vehicles (SLV). Space Science and velocity o ballistic target. Journal o Guidance, Control and Dynamics 004; 7 (6): Challenges o the Twenty-First Century, ISPA-SUPARCO Collaborative Seminar (in connection with World Space Week), University o Karachi, 005 (invited lecture) [9] Kamal SA. Incompleteness o crossproduct steering and a mathematical [5] Kamal SA, Mirza A. The multi-stage-q System and the inverse-q system or possible application in satellite-launch vehicle (SLV). The Fourth International Bhurban Conerence on Applied Sciences and Technologies, National Center or Physics, Bhurban, 005; pp 5

7 CROSS-RANGE ERROR IN THE LAMBERT SCHEME 6 Fig. Geometry o the boundary-value problem

8 6 KAMAL Fig. Flow chart o the Lambert scheme

9 CROSS-RANGE ERROR IN THE LAMBERT SCHEME 6 Fig. Justiication o the positive sign in Web address o this document (author s homepage): Abstract: