The Multi-Stage-Q System and the Inverse-Q System for Possible Application in Satellite-Launch Vehicle (SLV)

Transcription

1 The Multi-Stae-Q System and the Inerse-Q System for Possible Application in Satellite-Launch Vehicle (SLV) Syed Arif Kamal 1, Ashab Mira 2 Department of Mathematics Uniersity of Karachi Karachi, Pakistan Abstract Steerin a Satellite-Launch Vehicle (SLV) to strictly follow a predefined trajectory imposes unnecessary load on the control loop, and may, possibly, saturate seros. This If may introduce a permanent error in the ehicle-destination position and elocity ectors. Consequently, the payload (the satellite) would be deployed in a wron orbit. The orbitalerror correction utilies onboard enery, which reduces the operational life of the satellite. Therefore, it is desirable that SLV is capable of alterin its trajectory accordin to the new operatin conditions, in order to achiee the required destination position and elocity ectors. In this paper, an innoatie adaptie scheme is presented, which is based on the Multi-Stae-Q System. Usin the control laws expressed in the elliptic-astrodynamicalcoördinate mesh (the normal-component-cross-product steerin and the normal-componentdot-product steerin) this scheme proposes a desin of autopilot, which achiees the predecided destination position and elocity ectors for a multi-stae rocket, when each stae is detached from the main ehicle after it burns out, completely. In the Inerse-Q System, one applies the extended-cross-product steerin to the ector sum of elocity ectors of spacecraft and interceptor. Keywords: Q system, extended-cross-product steerin, dot-product steerin, ellipticastrodynamical-coördinate mesh Introduction Fliht dynamics of satellite-launch ehicle (SLV) is an example of unstable, non-autonomous, multiariable and nonlinear system. Seeral modern control techniques exist that can be effectiely employed to desin a robust controller for attitude control of aerospace ehicles. 1 PhD (Mathematical Neuroscience); MA, Johns Hopkins, United States; MS, Indiana, Bloominton, United States; SSO, Control Systems Laboratories, SUPARCO (Plant), Pakistan Space and Upper Atmosphere Research Commission ( ); Interdepartmental Faculty, Institute of Space and Planetary Astrophysics (ISPA), Departments of Physics and Computer Science, Uniersity of Karachi paper mail: Professor and Chairman, Department of Mathematics, Uniersity of Karachi, PO Box 8423, Karachi 75270, Pakistan telephone: ext profdrakamal@mail.com homepae: member AIAA 2 PCSIR-Institute of Industrial and Electronics Enineerin, Karachi (Pakistan); ashab@ieee.or member IEEE 27

2 In the absence of an efficient uidance scheme, een a state-of-art control system cannot uarantee steerin of ehicle to its required destination, under constrained conditions. Saturation and limited response rate of seromechanism are the uniersal nonlinear constraints of any control loop, which make impossible to implement all the control actions required to moe a system from one state to any other state throuh any trajectory. Sometimes, it is almost impossible to reach the desired position in the state space. Saturation can ie rise to inaccurate or een unstable response of a control loop. Q system has been used for SLV by different authors [2, 12]. In this paper, we are presentin the Multi-Stae-Q System. In this formulation, sectionwise corrections are achieed, where destination point of first stae is the initiatin point of the second stae and so on. In this way position and rate saturations (out of rane deiations) are aoided. In the Inerse-Q System, one applies the extended-cross-product steerin [4] to the ector sum of elocity ectors of spacecraft and interceptor, expressed in the elliptic-astrodynamical coordinate mesh in order to derie them to ero. Use of normal-component-cross-product steerin [4] and normal-component-dot-product steerin [5] is recommended to achiee this objectie. This paper includes a brief description of these control laws. The attitude-control problem can be tackled by a number of ways. Real-time computer processin requires that the control law should be simple, and these techniques, enerally, lead to hiher hiher-order controllers. On the other hand a nonlinear system, which can be linearied with continuous-time feedback, cannot necessarily be linearied with sampled feedback. A simpler attitude control loop can be desined for the attitude control of SLV based on effectie-interal-control (EIC) [9, 10], or by Selectie-State Feedback methodoloy [11]. Other schemes, which could be used to steer a satellite-launch ehicle include the Lambert scheme [1, 8]. The Q System The Q system is described briefly in [1]. Let r be the radius ector representin the position of the spacecraft at an arbitrary time, t, after launch, represented by M in Fi. 1, the correlated elocity ector, C, is defined as the elocity required by the spacecraft at the position r(t) in order that it miht trael thereafter by free-fall in acuum to the desired terminal condition. One defines a correlated spacecraft located at point M. The correlated spacecraft is assumed to experience only raity acceleration and moes with elocity C. The actual spacecraft elocity is m, and is affected by both raity and enine thrust acceleration a T. 28

3 Fi. 1. Correlated trajectory and elocity-to-be-ained The elocity to be ained,, may be expressed as: C m Computation of this elocity is only one element of the Q system. Of equal importance is a method to control the spacecraft in pitch and yaw, in order that the thrust acceleration causes all the three components of to anish, simultaneously. This may be accomplished by extended-cross-product steerin [4] or dot-product steerin [5]. Battin remarks in his book [1], paes 10-11: If you want to drie a ector to ero, it is sufficient to alin the time rate of chane of the ector with the ector itself. Therefore, components of the ector-cross product could be used as the basic autopilot rate sinals a technique that became known as cross-product steerin". Howeer, this definition has a condition missin. The complete definition follows. Extended-Cross-Product Steerin In order to drie a ector to ero, it is sufficient to alin the time rate of chane of the ector with the ector itself proided the time rate of chane of the manitude of this ector is a monotonically decreasin function [4]. This law may be termed as extended-cross-product steerin. Let A be a ector, which needs to be drien to ero. Then, we must hae d A A 0, 0 (1) This is the basis of the followin control law. Normal-Component-Cross-Product Steerin In order to brin a ehicle to the desired trajectory one needs to alin the normal component of elocity with its time rate of chane and make its manitude a monotonically decreasin function of time [4]. By normal component one means the component of elocity in the plane normal to reference trajectory. This plane passes throuh a point on the reference trajectory, which is closest to current location of center-of-mass of spacecraft. Mathematically, 29

4 d 0, 0 (2) Therefore, components of the ector should be used as the basic autopilot rate sinals. For elliptic-astrodynamical-coördinate formulation [3, 4, 6, 7] the endicular component of elocity may be expressed as: ê ê (3) To correct for down-rane error, one must hae d 0, 0 (4a) To correct for cross-rane error, the followin could be used as autopilot rate sinals d 0, 0 (4b) Dot-Product Steerin Dot-product steerin is a control law, which inoles dot products of the ector, and its time rate of chane [5]. In order to derie a ector A to ero, one needs to derie the factor A ( 1 cos ) to ero, where is the anle between A and. In other words A. A 0 (5) In the trajectory problems, it is customary to require the normal component of elocity to anish. Hence, one may deelop a special case of dot-product steerin. Normal-Component-Dot-Product Steerin In order to brin a ehicle to the desired trajectory one needs to derie the factor ( 1 cos) to ero, where is the anle between and. The aboe condition may be expressed, mathematically, as. 0 (6) The steerin law may be expressed in elliptic-astrodynamical coordinate mesh as:. 0,. 0 (7) The Multi-Stae-Q System Let us consider a 3-stae rocket. Destination point of the first (second) stae is the initiatin point of the second (third) stae. Mathematically, 30

5 1, final 2, initial ; 2, final 3, initial (8) By usin these section-wise corrections, one can aoid position saturation and rate saturations. This can be effectiely, established if the difference elocity ector is expressed in ellipticastrodynamical-coördinate mesh and conditions expressed in terms of normal components of elocity, which hae to be drien to ero. The Inerse-Q System Let us define a elocity ector,, representin the ector sum of elocity of the spacecraft and elocity of the interceptor. Mathematically, (9) spacecraft int erceptor This elocity has to be deried to ero. Extended-cross-product steerin [4] may be used to desin basic autopilot, with dot-product steerin [5] incorporated in the computation loop to check whether the control action has taken place or not, makin it a closed-loop control system. Conclusions Control enery aailable to SLV is, mainly, used (or should, mainly, be used) for stabiliation of ehicle and for disturbance-rejection action. Forcin the ehicle follow an unnatural fliht path can saturate the control loop, leain be no room for disturbance-rejection action. This situation makes the whole uidance and control desin inefficient. Sometimes, the saturation can make the loop unstable or, at least, a permanent error is introduced due to failure of implementation of control laws. Error in the position and the elocity ectors at the payload-ejection point can deploy the payload (satellite or an experimental payload) in a wron orbit (or trajectory). The multi-stae-q system attempts to address this problem usin corrections achieed when one stae is separated after burnout. The inerse-q system may be used to remoe an unwanted debris or out-of-serice satellite from the orbit. Acknowledement This work was made possible, in part, by Dean's Research Grant awarded by Uniersity of Karachi. Appendix: SAK s Reiew of An Introduction to the Mathematics and the Methods of Astrodynamics by Richard H. Battin This reiew appears in as well as on author s homepae full text of the book: [Book Ratin: ] The followin comments refer to the 1987 edition. Some of these comments were communicated to Professor Battin, who, ery kindly, acknowleded them. 31

6 The book by Richard Horace Battin, Senior Lecturer in Aeronautics and Astronautics, Massachusetts Institute of Technoloy, United States, coers essential mathematical backround needed to work with astrodynamical problems. Topics coered include hypereometric functions, elliptic interals, continued fractions, coördinate transformations as well as essentials of two-body-centralforce motion. The author's way of discussin these topics with historical introduction and personal narratie makes the book interestin to read. There are minimal typoraphical errors, probably, because the author, personally, typeset this book. Howeer, there are a few omissions and oersihts. For example, on pae 172 captions are ien for Fi and Fi. 4.16, whereas the actual fiures are missin (The author has rectified this omission in the 1999 edition). In addition: a) On pae 7 it is stated: s Δ r where s. In this equation, a scalar on the left-hand side is equated to a ector on the riht-hand side. The equation should be modified as: s b) On paes it is stated: "If you want to drie a ector to ero, it is sufficient to alin the time rate of chane of the ector with the ector itself." This is not true, in eneral, but only if time rate of chane is neatie (cf. [4]). c) On pae 13 the author tries to show that curl of c in the equation: c constant anishes by the followin arument. "The demonstration concludes with an arument that the fluid is conerin on the taret point r T so that the density in the icinity of r T is becomin infinite. Hence, the constant is ero, implyin that the curl is eerywhere ero." There are 2 problems in this line of arument: (i) The statement hence, the constant (see note at the end of manuscript) is ero is true, only if the numerator is finite. B implies A B 0, only if A. Otherwise, one has to apply l'hospital rule; (ii) Een if the constant is supposed to be ero, this does not imply that the curl is eerywhere ero. A B 0, where B does not imply that A 0. In fact, A could hae any finite alue. d) On pae 109 equation of motion in a frame of reference moin with acceleration a1 is written as: m m Gm m r m a a ( 2 1) m 2 1 r r Since the frame is noninertial (accelerated) Newton's second law, F ma, is not applicable in this frame. e) On pae 223 it is stated: "When we compare Eqns. (5.57) and (5.58), it is clear that we must hae: 6( E sine) sine sine." This is not the only choice for sine, which reduces (5.58) to (5.57) in the limit E 0. The word "must" is inappropriately used here. 32

7 I would recommend this book ery stronly to any one inoled in astrodynamical research. References 1. R. H. Battin, An Introduction to the Mathematics and (the) Methods of Astrodynamics, AIAA Education Series, New York, United States (1987) pp the fiure inserted in the paper as Fi.1 appears as Fi. 3 on p M. S. Bhat & S. K. Shriastaa, An optimal Q-uidance scheme for satellite-launch Vehicles, Journal of Guidance, Control and Dynamics, 10 (1), (1987) 3. S. A. Kamal, "Planetary-orbit modelin usin astrodynamical coördinates", the Pakistan Institute of Physics International Conference, Lahore, Pakistan, March 30 - April 3, 1997, pp 28-29, abstract: 4. S. A. Kamal, "Incompleteness of cross-product steerin and a mathematical formulation of extendedcross-product steerin", Proceedins of the First International Bhurban Conference on Applied Sciences and Technoloies, Bhurban, KP, Pakistan, June 10-15, 2002, pp , full text: 5. S. A. Kamal, "Dot-product steerin: a new control law for spacecrafts and satellites", Proceedins of the First International Bhurban Conference on Applied Sciences and Technoloies, Bhurban, KP, Pakistan, June 10-15, 2002, pp , full text: 6. S. A. Kamal, Use of astrodynamical coördinates to study bounded-keplarian motion, the Fifth International Pure Mathematics Conference, Quaid-é-Aam Uniersity & Pakistan Mathematical Society, Islamabad, Pakistan, 2004, abstract: 7. S. A. Kamal, Stron Noether s theorem: applications in astrodynamics, the Second Symposium on Cosmoloy, Astrophysics and Relatiity, Center for Aanced Mathematics and Physics, NUST and National Center for Physics, Rawalpindi, Pakistan, 2004, p. 2, abstract: 8. S. A. Kamal, Incorporatin cross-rane error in the Lambert scheme, the Tenth National Aeronautical Conference, PAF Academy, Risalpur, KP, Pakistan, April 20,21, 2006, full text: 9. A. Mira & S. Hussain, Robust controller for nonlinear and unstable system: inerted pendulum, AMSE Journal of Aances in Modelin & Analysis, 55 (4), (2000) 10. A. Mira & S. Hussain, Flexible broom balancin, AMSE Journal of Aances in Modelin & Analysis, 56 (1&2), (2001) 11. A. Mira & S. Hussain, DC-motors speed asynchroniation in nonlinear process by selectie-state feedback and interal control, AMSE Journal of Aances in Modelin & Analysis, 57, (2002) 12. A. Mira & S. Hussain, Steerin solid fuel satellite-launch ehicles by estimated-impulse response based adaptie Q uidance, Proceedins of the Fourth International Conference on Modelin and Simulation, Victoria Uniersity, Melbourne, Australia, Noember 11-13, 2002, pp ADDITONAL NOTE (related to this paper, but not part of the published manuscript) The constant on pae 32, line 2 should, actually, be a constant ector Noor Fatima Siddiqui, Lecturer, Department of Mathematics, Uniersity of Karachi commented on March 25, Web address of this document (first author s homepae): Abstract: 33