Space Guidance Analysis Memo #28-65

Transcription

1 Spce Guidnce Anlysis Mem #28-65 TO FROM. Msschusetts Institute f Technlgy Instrumenttin Lbrtry Cmbridge, Msschusetts SGA Distributin Rbert Birns fther DATE: Nvember 19, 1965 SUBJECT Cnic Time f Flight Clcultins t Rdius f Specified Length Intrductry Cmments We wish t clculte the time t trverse n elliptic rc tht strts t n rbitrry rdius nd ges in the directin f velcity t rdius f specified length rh. Bsed upn requirements f Flight 202, sme grund rules fr n AGC prgrm cpble f cmputing time f flight re:. the prgrm shuld be generl, nd nt limited by the size f the trnsfer ngle. b. the entire AGC clcultin shuld require less thn 100 ms. c. ccurcies n the rder f ne secnd re cceptble. d. the resulting time f flight shuld hve the sme binry scle fctr s the T3 register; tht is, ne lw rder bit equls 1 cs. Pint is esily stisfied by slutin f Kepler's equtin fr the time f flight tf except tht Pint b is f equl imprtnce nd might preclude s generl slutin fr t f. Hwever, the speed requirement suggests tht single precisin rithmetic might be used where prcticl, suggestin tht mkes clcultin f n rc tngent n ttrctive pssibility. In single pre- / cisin, the errr in the ngle Efir s determined by the rctn is t best ne bit, 2-14, nd cuses n errr in t tht is 2-14 times the dimensinl fctr, f IT NI ht, where p. estblishes the time units. Thus the cmputtinl ccurcy t which t f is knwn is the sme s the ccurcy t which the rctn is determined. But the vlue f the lest significnt bit in the t f register de- pends n the mximum vlue tht the register is clled upn t cntin, nd is prprtinl t 4 Cmputtin shuld be crried ut t binry scle level cnsistent with bth the mximum t f nd ny relevnt numericl ppr:, prgrm. -u-15 nd shuld be independent f the scle level wnte, by the user 1

2 Equtins fr Elliptic Orbits The equtins t fllw summrize the results f the Appendix. The time f flight fr the elliptic rbit in the directin f velcity '70 frm rdius 70 t rdius f specified length r h is prvided by Kepler's equtin. (See Ref. 1) tf = E - E -e (sin E - sin E )] (1) where either f the fllwing frms is vlid h ' - r v 6 h = sgn (i-h) p _ i -h (2 - - Ni pi (2. 1) (2. 2) nd where the quntity (E - E )is btined frm tn E - E l e (sin E - sin E ) 2 r r h (3) The tw cnic prmeters tht pper re supplied by H= 7 x P TI" (4) 1 2 r - v v nd re used t give p/, r /, rh/. Fr negtive t f, see Appendix, Summry I nd IL Equtins (4) ssume tht r = IT i is vilble in AGC memry. Equtin (2. 1) is preferred fr reentry n Flight 202 becuse in tht pplictin bth sgn( 0) nd sgn(i-h) re knwn nd re negtive nd permit fster cmputtin, prticulrly if single precisin squre rts re used. In pplictins where the rdil velcity f T cn hve either sign, (2. 2) is the better chice. Blck I AGC Time Estimte If we ssume tht nne f the quntities in (4) re lredy vilble in

3 AGC memry, lmst ll f the time used in clculting t f,77 ms ut f 93 ms, is invested in the AGC Interpreter lnguge prgrm fr evluting (4) nd in estblishing r /, rh/, nd p/ in the push dwn list. Evlutin f (1) nd (2) is dne in bsic AGC lnguge nd requires '7 ms. A finl 9 ms (Interpreter) is expended in t f test t recgnize if T hs pssed rh. It my be pssible t trim few ms here nd there, but mjr gin is unrelizble unless either p r li is lredy vilble in memry, r unless the prducts invlving 1/ re dne in bsic AGC lnguge. Arc Tngent The rctngent pprximnt given by the Hstings plynmil (see Ref. 2) f fur terms is cmptible with single precisin rithmetic, nd hs mximum errr between the pprximnt nd the functin f rdins, The plynmil is scled t E/ir, s the errr is but r less thn ne bit f single precisin wrd. Fr ngles in the rnge + 45, the pprximnt is plynmil in x: y = f(x), where x = N/D nd is in the rnge (-1, 1). T cver the full first qudrnt, the pprximnt becmes y = r/4 f(z), where z = (x - 1)/(x + 1) = (N - D)/(N +D) nd x is in the rnge (0,00). The ltter reltin vids the prblem sscited with zer denmintrs tht wuld ccur if x = N/D were used. Hwever, the indeterminte cse, crrespnding t 90, cn rise nd must be checked fr. Thrughut mst f the first qudrnt, the rctn prgrm uses the plynmil in z. This plynmil hs the disdvntge tht it prduces nnzer errr in y t the extremes, 0 nd 90. The plynmil in x hs n errr t 0 nd is used fr smll ngles. The trnsitin frm x t z is mde t but 15, crrespnding t ne f the severl vlues f y fr which bth plynmils exhibit the sme eirr. The clcultin f the rctn s detiled belw is bsed n the fur term plynmil in Ref. 2. The prgrm ccepts N nd D, where tn E = N/D, nd gives E in the rnge (0, r). E = r/4 N1 = bs(n), D1 = bs(d) If (N1 + Di)/e neg, E = 2E, return. (indet. cse) z = (N1 - D1)/(N1 + If z neg, z = N1 /D1, E = 0 (trnsitin cse) z 2 (C3 z 2 (C5 z 2 C7))) E = E + z (C 1 + 3

4 If sgn (N D) neg, E= return nd C = , C , C5 3 = E = , C = ' 7 Numericl Results Equtins (1) thrugh (4) were tested by MAC prgrm n the MH1800. In the test the terminl pint r h crrespnds t 400, 000 feet ltitude nd true nmly f = 330. Time f flight t f is clculted fr nd Ti crrespnding t selected vlues f f rnging frm 90 t 327 fr ech f the fllwing vlues f e: 0. 3, 0. 2, 0, 1, 0, 05, 0, 01. These cnditins yield vlues f t f tht rnge frm nerly 7000 secnds t 40 secnds. The errr between tf nd crefully evluted check slutin in ll cses is ttributble t the rctn pprximnt nd is 0. 2 secnds r less. The errr in tf due t errr 6E in rctn is t f = 2 6 E nd 6 E is rdins. Equtins fr Hyperblic Orbits The time f flight equtins fr hyperblic rbits re similr t thse fr elliptic rbits nd re summrized belw. The derivtin is nt given, but it prllels the ne given in the Appendix fr elliptic rbits. The time f flight t f required t trverse hyperblic rc frm n rbitrry rdius T t rdius f specified length r h is prvided by Kepler's equtin (see Ref. 1). tf = jµ - (H - H ) + e (sinh H - sinh H )] (5) where either f the fllwing cn be used r h p /r r e (sinh H - sinh H) = sgn (rh ) N h (2 + -T. ) sgn (i- ) V--2 (2 + L) ),- ± r h) _ p _ T. Tr = sgn (rh)(,\i m (6.1) (6. 2) nd the quntity (H H ) cmes frm 4

5 H - H e (sinh H - sinh H) tnh ( 2 ) r r h 2+ + (7) In these equtins, 1 / is psitive number. The inverse hyperblic tngent is clculted frm y = tnh 1 (x) = 2 ln(10) lg_ ( 1 + u 1 - where the lg term cn be pprximted by Hstings plynmil (see Ref. 2) f(z) = lg(z) ver the rnge 1 < z < 10 r equivlently 0 < x < Fr reference, the three term pprximtin is suitble in single precisin nd is 1 lg (z) = 2 + C Q + C Q 3 + C5 Q where Q z - C +x z Cx nd C = , C 1 = , C 2 = O , C 3 = Using the signed 1 / btined in (4), the equtins fr hyperblic nd elliptic rbits cn be cmbined int single set, except fr tking the inverse functin; the prper inverse is selected by the sign f 1/.

6 REFERENCES 1. Astrnuticl Guidnce, R. H. Bttin, generl reference, Chpter 2. Apprximtins fr Digitl Cmputers, C. Hstings, Jr. 6

7 APPENDIX Elliptic Time f Flight t Rdius f Specified Length The time required t trverse the elliptic rc frm n initil rdius F0 t terminl rdius f specified mgnitude r h nd in the directin f velcity 70 is given by Kepler's equtin *(equtins s mrked re fund in Chpter 2 f Astrnuticl Guidnce, by R. H. Bttin) t = [E-E -e(sine-sin En (A-1) where E is eccentric nmly crrespnding t T- E is eccentric nmly crrespnding t r h (rh is rdius crrespnding t desired ltitude) is eccentricity is semi mjr xis is grvittinl cnstnt Methd I - One pprch is t clculte independently bth E 0 nd E. The frmer is btined frm initil psitin nd velcity using the identity* tn sin f / /Nri _ - e 1 - e (A -2) e tn 2 1+e l+csf r (1 + e - ) since* e r sin f = FE--) 0 /.4 e r cs f -r0 A-2. 1) nd ls e r (1 + cs f ) = (1 - ) [ 1 + e - Fr cnvenience in the fllwing derivtins, let (A-3) Fr E we btin

8 tn E sin f (A -4) -2 + e 1 + cs f r h 1 + e - since, squring the cs f reltin f (A-2. 1) nd using (A-3), there fllws rh e r h sin f = Q ( A-5) In clculting the E's, the rctn uses the cnventin tht if tn -1 ( ) hs 1 negtive vlue, replce tn ( ) by r + tn -1 ( ). It is pssible t this pint t slve (A-1), since bth E nd E re knwn, by cmputing the tw sines. Such ctin is unnecessry becuse bth re expressible in terms f quntities lredy derived: rh. ) = Q e (sin E - sin E N/ pi (A-6) This reltinship rises in the fllwing mnner. The quntity E is expressed in terms f the identity * belw nd (A-2.1) r (A-5) sin E N/1 - e 2 sin f r sin f e cs f 'V p (A-6.1) sin E = r h sin f r - (- 1 ) Q( rh e ) (A-6,2) Summry: Methd I The time required t trverse the elliptic rc frm n initil rdius r t terminl rdius f specified length r h nd in the directin f velcity 70 is given by Kepler's equtin in the frm t=. LE - E - sgn h ) r h r, (2 - P )- + (A-7.1) where E nd E cme frm tn r 17' 0 (1 e - 2 pi (A-7.2)

9 E h tn = 3gn(r 2 h - r p h ) - (1 + e - ) (A-7. 3) These require three prmeters, p, e. Sgn(ic h ) designtes which f the tw vectrs hving length r h is t be used. During reentry, sgn(i" h) is negtive. Bth f the tngent equtins re indeterminte when the true nmly is 180 but crrespnd t n eccentric nmly f 180. This situtin rises nly fr (A-7. 2) nd is recgnized by the rctn. Equtin (A-7. 3) exists in ll useful cses since r h lies n the ellipse. When this is nt true, negtive rgument f the rdicl btins nd indictes tht r h is less thn perigee rdius (r greter thn pgee). Fr scling purpses, the rnge f vlues in the denmintr f the tngent functin is (0, 2e). The numertr cn lie in the rnge (-e, e) s seen frm (A-6. 1) nd (A-6. 2). When rh is chsen fr reentry, t in (A-7. 1) will be psitive fr ll tht mve twrd rh in the directin f T, nd will becme negtive when r hs pssed rh. This ssumes 0 < f < f, where bth f nd f re mesured psitively. Hwever, in generl (A-7. 1) gives the time f flight frm F t rh nd negtive vlue f t mens tht the perigee rdius lies n the rc t be trversed nd s 27r is missing frm E. Cnsidered nther wy, negtive vlue fr t represents the time f flight frm 1--0 t rh in the ppsite directin f the velcity nd must be dded t ne rbitl perid t btin the time f flight in the desired directin. In either cse, t get the prper vlue fr flight time when t is negtive, replce t by t + 27r The frmultin (A-7) fr slving Kepler's equtin is smewht undesirble becuse () it requires evluting tw rctngents, nd (b) it my exhibit lss f precisin fr smll vlues f the difference E - E 0. Methd II - The frmultin f (A-7) cn be chnged int mre desirble slutin tht hs imprved ccurcy nd speed. The chnge is ccmplished by slving directly fr the quntity (E - Ed. T enble lgebric mnipultin, (A-6. 1) is expressed fter the mnner f (A-6. 2). e sin E '\/ pt (2-9-- ) - 12 Q( (3 ) (A-8) Nw (A-2) nd (A-4) becme similr, nd re regruped

10 E tn = tn 2 r 1 + e - r sgn j e 2 1 )2 r e -( - 1) (A-9. 1) rh E Q ( ) sgn(r ie 2 - ( h h) - 1) 2 tn = e - r r h h e - ( - 1) Define the fllwing intermedite sets f vribles (A-9. 2) E A tn 2 = sgn(ṟ ) e c t + \ B =tn 2 = sgn(r n, ) Ne - (A-10) where c = r0/ -1 nd d s r h / -1. The definitins f (A-10) re intrduced int the trignmetric identity fr the tngent f difference nd the numertr rtinlized by (B + A). tn E - tn E E - E _ B 2 - A 2 tn - 2 E 1 + tn -tn --0- A (1 + B 2) + B (1 + A 2 ) 2 2 Substitute (A-10), perfrm the indicted pertins, nd simplify tn E - E 0 _ 2 d - c sgn(i-) "1e 2 - c 2 + sgn(i-h) Je t - d 2 Eliminte c, d nd restre s in (A-9) tn r r h E - E - _ 2 Q(-11. -)± Q( 1112-) (A-11) Tke the rctn f (A-11) t get the desired eccentric nmly difference. As befre, negtive tngents re cnsidered s hving rguments in the rnge (r/2, 7T). The sine term (A-6) in Kepler's equtin with the id f (A-8) nw is e (sin E - sin E ) = Q( ) - Q( (A-12)

11 Further simplifictin is indicted: (A-11) cn be put in mre cnvenient frm by bserving tht the denmintr f (A-11) differs frm (A-12) by single minus sign. Rtinlize (A-11) rh E- E Q( ) - Q(-2.) tn (A - 13) 2 r r h r equivlently tn E - E _ e(sine - sin E 0 ) 2 r r h (A -13.1) Summry - Methd II The time f flight fr the elliptic rc in the directin f the velcity V t rdius f specified length r h is given by Kepler's equtin (A-1) frm T / t = v - [E - E -e (sine - sine where either f the fllwing my be used e (sine - sine ) = sgn (rh) _ - sgn(r r () tr.) - _ P,, = sgn(rh) 1-L1 ( ) r q (A-14) nd E - E cmes frm tn E - E e (sine - sin E ) 2 r h 2 - -r This requires tw prmeters, p. Sgn(i. h) designtes which f the tw pssible vectrs f length rh is wnted nd is negtive fr reentry. The rdicls hve vlues in the rnge (-e, e) nd vnish t bth pgee nd perigee. Negtive rguments fr the rdicl cntining r h signifies tht the rdius r h des nt lie n the ellipse. A negtive rgument fr the rdicl cntining r cn nly rise frm numericl effects nd indictes tht the rgument shuld be zer. Shuld this situtin ccur, set the rdicl equl t zer.

12 Bth the numertr nd denmintr f the tngent cn tke n vlues in the rnge (-2e, 2e). The tngent is indeterminte nly in the specil cse when the rdii re pgee nd perigee, nd crrespnds t E - E = 7r. The rctn will recgnize this fct. The usefulness f the tngent equtin is nt gretly limited by e, prvided e 0 nd the rnge f e is knwn in dvnce fr prper scling. The selectin f ne f the tw equtins fr e(sin E - sin E ) depends n the pplictin. The first ne is useful when bth signum functins re knwn in dvnce. The secnd is preferred fr generl pplictin. As discussed under Summry - Methd I, negtive vlues fr t require tht t be replced by t Ni