Dot-Product Steering A New Control Law for Satellites and Spacecrafts

Transcription

1 The First International Bhurban Conerence on Alied Sciences and Technologies Bhurban, Pakistan. June 1-15,, Dot-Product Steering A New Control Law or Satellites and Sacecrats Sed Ari Kamal 1 Deartments o Mathematics and Comuter Science Uniersit o Karachi Karachi, Pakistan. Abstract A control law is ormulated, which emlos dot roducts o elocit and time rate o change o elocit. Mathematical reresentation using ellitic-astrodnamical-coördinate mesh is resented. Introduction The normal-comonent-cross-roduct steering control law [1] ut orward in the other aer ma be used to derie another law, which can be used to derie normal comonents o elocit to ero. This law, termed as, dot-roduct steering, is urther deeloed into another control law, ellise-orientation steering, and conditions are deried to determine and, eentuall, eliminate down-range and cross-range errors. This aer is a continuation o [1], and, hence, the list o smbols, comact notations and coördinate sstems collected in the Nomenclature section alies to calculations resented in this aer, as well. A good oeriew o orbital dnamics, needed to understand these calculations, ma be ound in []. The ellitic-clindrical-coördinate mesh [3, 4] is adated to deal with the bounded kelarian orbits, as ellitic-astrodnamical-coördinate mesh. Reerences [3, 4] 1 PhD; MA, Johns Hokins, United States; Project Director, the NGDS Pilot Project (htt://ngds-ku.org); aer mail: Proessor, Deartment o Mathematics, Uniersit o Karachi, Karachi 757, Pakistan; telehone: ext. 93; rodrakamal@gmail.com; homeage: htt:// 178

2 Kamal illustrate another adatation o these coördinates the cardiac-coördinate mesh, which is used to model surace anatom o the human heart. In this aer mathematical ormulations o dot-roduct steering, normal-comonent-dotroduct steering (as the secial case o dot-roduct steering) and ellise-orientation steering are resented. In Aendix A, trajector comuted under the assumtion o constant g (arabola) is shown to be the limiting case o ellitical trajector, when its semi-minor axis, b, is er small as comared to its semi-major axis, a. In Aendix B, equation o ellise written in the orm, r 1 ecos ; 18 is shown to be equialent to the orm, traditionall, recognied: ( x h) ( k) 1 a b Dot-Product Steering Dot-roduct steering is a control law, which inoles dot roducts o the ector, and its time rate o change. In order to derie a ector A to ero, one needs to derie the actor A ( 1 cos ) to ero, where is the angle between A and. In other words. (1) A. A In the trajector roblems, it is customar to require the normal comonent o elocit to anish. Hence, one ma deelo a secial case o dot-roduct steering. Normal-Comonent-Dot-Product Steering In order to bring a ehicle to the desired trajector one needs to derie the actor d d ( 1 cos) to ero, where is the angle between and. The aboe condition ma be exressed, mathematicall, as d d (). Proo: From normal-comonent-cross-roduct steering, the condition to drie a ector to ero is anishing o the roduct d 179

3 This means that the direction o change. The ector d makes an angle o Dot-Product Steering ma not change. Howeer, its magnitude should d should go to ero i. This is ossible onl i 18 with, that is d. d cos18 1 which reduces to (3) d d. This comletes the roo o normal-comonent-dot-roduct steering. xamles: xamles o rectilinear, circular and ellitical trajectories are worked out to illustrate this control law. a) Straight Line: To simli the calculations, x axis is chosen along the trajector. ara x xê x, ê ê. The conditions or normal-comonentdot-roduct steering become d d d d. ;. b) Circle: x lane is chosen so that the circular trajector lies entirel in it, with center o circle coinciding with the origin o the coördinate sstem. In clindricalcoördinate mesh, (,, ), the comonents o elocit are ara ê, ê ê takes the orm. Thereore, normal-comonent-dot-roduct-steering law d d d d. ;. c) llise: In order to write this condition or ellitic-astrodnamical-coördinate mesh, one notes that ê, ê ê. One notes that, ara P ê shall contribute to down-range error and N error. Thereore, the steering law ma be exressed as d d d d.,. ê to cross-range 18

4 Kamal llise-orientation Steering The condition or no down-range error is (4a) ( â.r ae )( â.)( e 1) ( â.r)( â.) ara ara and the condition or no cross-range error is (4b). â N These equations describe another control law, termed as ellise-orientation steering. Proo: xressing in terms o x and x sinh( ae ) cos cosh( ae ) sin and substituting the alues o x ae sinh( ae ), cosh( ae ), tan 1 aesin aecos ( x ae) one gets, or no down-range error x ( x ae) ae which can be, immediatel, generalied to (4a). Similarl, or no cross-range error,, which is generalied to (4b). Cross-Range rror Detection For no cross-range error elocit o the sacecrat must lie in the lane containing r r. In other words (5).r r In order to detect cross-range error resent, is irst exressed in terms o x,, (bodcoördinate mesh), and then in terms o inertial sstem to be able to obtain a condition to eliminate down-range error. Similarl, is exressed in terms o inertial-coördinate mesh so that conditions ma be obtained to eliminate cross-range error. Down-Range rror and Cross-Range rror limination To eliminate down-range error, the ollowing condition should hold x (6a) tan and to eliminate cross-range error 181

5 Dot-Product Steering (6b) This can be easil roed using the exressions or, and q. (5). Conclusions Normal-comonent-dot-roduct steering, inoles, dot roducts o normal comonent o elocit and its time rate o change. It is deried rom extended-cross-roduct steering. The ormulation resented in [1] and this aer ma be useul in satellite dnamics. Using this ormulation a satellite-launch ehicle (SLV) ma be constructed, which can inject satellites into the desired orbits. Suitable autoilots ma be designed or attitude control o the satellites. It is imeratie to deelo similar ormulations or arabolic and herbolic orbits, and comare the results to this ormulation. Acknowledgements The author would like to exress his heartelt thanks to National Center or Phsics, in articular, the Organiing Committee o IBCAST or making m sta at Bhurban enjoable. This work was made ossible, in art, b Dean's Research Grant awarded b Uniersit o Karachi. Aendix A: quialence o a Projectile Trajector in Constant Grait (Parabola) and a Trajector Comuted From Keler's quation (llise) A rojectile is a reel alling bod. Its trajector, as determined in elementar hsics under the assumtion o constant grait, comes out to be a arabola in ree sace. The rojectile is bound to the earth's graitational ield and, thereore, comes back to the surace o earth. Table 1 shows that the trajector o a bound rojectile (the otential energ larger than the kinetic energ) must be either an ellise or a circle. A arabolic Table 1. Orbits or Two-Bod, Central-Force Motion nerg ( ) ccentricit (e) Shae o the Orbit Te o the Orbit Sstem (Bound/Free) Number o Turning Points < min e< Not allowed = min e = Circle Closed Bound min < < < e< 1 llise Closed Bound = e = 1 Parabola Oen Free 1 > e > 1 Herbola Oen Free 1 18

6 Kamal Table. Features o Orbits nerg ( ) Features < min min = V ( r ) min = min r = eerwhere, and r = constant min < < r min = ericenter, r max = aocenter, which are the turning oints = rmin is the turning oint > r is the turning oint min orbit is ossible onl when total energ o the rojectile is ero. In other words, the otential energ must be numericall equal to the kinetic energ (the hae oosite signs). I this condition is satisied at the surace o earth or a ertical lunch, elocit o the rojectile becomes equal to the escae elocit to get out o the earth's graitational ield. This contradiction ma be resoled i one looks at the arabolic trajector as the limiting case o an ellitical trajector the semi-major axis being er large as comared to the semi-minor axis. In this case the eccentricit, e, shall aroach unit. Mathematicall, which gies, b a (1 e ) e b lim e lim 1 a a a b 1 a 1 In act, this is, exactl, what haens in the case o constant g. The assumtion o constant g (acceleration due to grait) imlies that the range is er small as comared to the circumerence o earth (altitude is small). Since one o the oci lies at the center o earth the semi-major axis is o the order o radius o earth. The semi-minor axis, howeer, is o the order o range. Thereore, arabolic trajector is, actuall, the limiting case o ellitical trajector. It is interesting to note that or a bound sstem the number o turning oints is greater than one, the energ is negatie and the orbit is closed. Table list some eatures o orbits. Aendix B: quialence o llise quations An equation o ellise in the cartesian-coördinate mesh with center at (h, k) ( x h) ( k) (B1) 1 a b is shown to be equialent to an equation o ellise in olar coördinates: 183

7 Dot-Product Steering (B) r 1 ecos ; 18 To do so one notes that (B) reers to an ellise with origin at one o the oci. The coördinates o center must, thereore, be taken as ( ae, ) in (B1), that is h = ae and k =. The semi-major axis, a, and the semi-minor axis, b, are related b b a (1 e ). Thereore, erimeter o the orbit (semi latus-rectum o the ellise),, ma be ound b noting that it is the alue o r, when = 9 (or, the alue o, when x = ). Substituting, x =, =, in (B1) and rearranging, one gets (B3) a(1 e ) Substituting, b a (1 e ), x r cos, r sin, h ae, k, in (B1), and using (B3), one obtains a quadratic equation: (B4) (1 e cos ) r (e cos ) r whose roots are r 1 1 ecos, r 1 ecos The solution r 1 is unhsical because it gies negatie alues o r (r, being the radius ector, is alwas ositie). The solution r is the required orm. Reerences 1. S. A. Kamal, "Incomleteness o Cross-Product Steering and a Mathematical Formulation o xtended-cross-product Steering", Proceedings o the First International Bhurban Conerence on Alied Sciences and Technologies, Bhurban, KP, Pakistan, June 1-15,, , ull text: htt:// R. Deusch, Orbital Dnamics o Sace Vehicles, Prentice Hall, nglewood Clis, New Jerse, United States (1963) 3. S. A. Kamal & K. A. Siddiqui, "Modeling o Heart Function", Proceedings o Smosium on Trends in Phsics, Deartment o Phsics, Uniersit o Karachi, Karachi, Pakistan, ed. B K. A. Siddiqui, 199, , ull text: htt:// 4. S. A. Kamal & K. A. Siddiqui, "The Human Heart as a Sstem o Standing Waes", Kar. U. J. Sc. (to be ublished), ull text: htt:// Web address o this document (author s homeage): htt:// Abstract: htt:// 184