Canonical Transformations for Space Trajectory Optimization

Optimal space trajectory problems and the necessaly and sufficient conditions that define their solutions are stated most compactly in terms of position and velocity vectors. To obtain analytical or numerical solutions, however, the problems are expressed using a particular set of coordinates. Each set of coordinates has advantages and disadvantages depending on the application. Thus, it may be useful to be able to transfo~m from one set of ccn>rdinates to another during the course of solving an optimization problem. If the problem has been formulated using adjoint coordinates, the transformation requires not only transfonnation of the state coordinates, which are well-known, but also transformation of the adjoint coordinates. This combined transformation of the state and adjoint must be a canonical trcmsforntotion for the extremal trajecto~y generated with the new coordinates to be the same as the extremal trajectory generated with the old coordinates. In this paper, the canonical transfonnations between four common sets of coordinates used in space trajectory optimization-trajectory variables, orbit elements, equinoctial elements, Cartesian coordinates-are developed for the coplanar case. Applications of the canonical transfonnations in numerical optimization and the development of the necessary conditions for optilnality are discussed. Introduction Optimal space trajectory problems, such as the minimumfuel transfer problem for a spacecraft in a central inverse square gravity field, and the necessary and sufficient conditions that define their solutions are stated most compactly in terms of position and velocity vectors [Mar79]. To obtain analytical or numerical solutions, the problems are expressed using a particular set of coordinates. The Cartesian coordinates, trajectory variables, orbital elements, and equinoctial elements are four common sets of coordinates for space tryiectory problems. Cartesian coordinates have been used for the analytical solution of optimal high thrust orbit transfer problems [Law63, Haz701 and the numerical solution of optimal low thrust transfer problems [BarXX, Enr90, RetlX41. Orbital elements have been used to develop approximate solutions for optimal low thrust problems, using averaging [Ede65, Mar77, * Graduate Student, Memher ** Assistant Professor. Associate Fellow t Professor, Meniher Copyright @ 1992 by the Anierical Institute of Aerorlaulics nritl Astro~~aulics. Inc. All riglrts reserved. MarRI] and linearization [Edeh4, Edehh]. To avoid the singularities of the classic orbital elements, the equinoctial elements [Bro72, Bat871 have been used to develop a solution to the linearized rendezvous problem [GobhS]. Trajectory variables have been used for the approximate and numerical solution of optimal low thrust transfer problems [Rosbl, Bro91, Mea901. Each of these sets of coordinates has both attractive and unattractive features for analysis and computation. The coordinates that are best for analytical solutions such as averaging may not be best for numerical optimization. Therefore, it is advantageous to convert between sets of coordinates when solving optimal space trajectory problems. This conversion requires not only transfo~mations of the state variables, which are well-known, but also transformations of the adjoint or costate variables. The combined transfonnation of the state and adjoint must be a rrir~oriirril r~msfortiicition for the extrernal trajectoiy generated in the new coordinates to be the same as the extremal trajectory generated in the old coordinates. One common purpose for using a canonical transfoimation is to increase the number of ignorable coordinates in a Flainiltonian system, which makes the differential equations easier to solve. Frae.jis de Veubekel used a canonical lransformatiol~ i n the minimum-fuel transfer problem for a thrust-limited rocket to transform the optimal steeling control to a state variable when investigating the case of intermediate-thrust extremals [Fra65]. Marec and Vinh used a canonical transfonnation to change the independent variable from time to characteristic velocity when solving the minimum-fuel, impulsive thrust problem [Vin70]. Marec developed the general minimum-fuel transfer problem using orbit elements with a three-dimensional canonical transfonnation from Cartesian coordinates to orbit elements [Mar79]. In this paper, canonical tlansfonnations between the four sets of coordinates commonly used in coplanar trajectory optimization4artesian coordinates, trqjecto~y variables, orbital elements, and equinoctial elements-are developed. These canonical transformations are composed of state and adjoint coordinate transfo~mations. The state transfonnations. which are intlependent of the adjoint transfonnations, are well-known, so the focus is on determining the corresponding adjoint transformations such that the combined state and adjoint transformation is canonical. The canonical transfonnations M m c credits Fra$;ijis de Veuheke as the first person 10 apply cill~o~i~cal tm~~st 'or~r~i l ion syslzr~ulically to t l ~ e prohle~n of optmal transfers in [Mnr79]. pp. 46. between the orbit elements and trajectory variables [HaiQI] and between the Cartesian coordinates and orbital elements [Mar791 and have been developed previously but are considered here for completeness. l%llo&ng the derivation of the canonical transformations, specific applications of these canonical transformations for space trajectory optimization problems are presented. Hamiltonian System for an Optimal Trajectory The Mayer form of the general optimal control problem is to detennine the m-dimensional piecewise-continuous control function u on the time interval [to, tf] that drives the ndimensional state x from an initial value xo to a final value xf, or more generally to a final manifold defined by the k-dimensional (k 5 11) constraint vector ~ ( x ( t f ) , tf) = 0, and minimizes the performance index subject to the state equations (differential constraints) and the control constraint These conditions along with the state equations (2) and the hxmlary conditions constitute a two-point boundary value problem, the solutions of which are candidates for locally minimizing the performance index. The state trajectory corresponding to a solution of the two-point boundary-value problem is called an extreniul rrtrjectory. Equations (7) and (8) are known as the trunsversn~ify corztfitions. Assuming an explicit solution of eq. (6) of the fonn the optimal I-Iamiltonian H* is defined by (3 ) H* = pT(t) f(t. x4'(t), g(t, x4:, p)) for t E [to, tf] and U(t) a closed subset of S"'. 'The state and adjoint differential equations can be expressed as a In space trajectory optimization [Mar79], the control is canonical Htrt~dloriitrn sys/errr, the thrust or thrust accelwaticm vector, and the state is comprised of the position vector r, velocity vector v, and the mass m. For the minimum-fuel problem, the performance index to be dH* X * = (T) (1.1) minimized is J = m(tf). The state equations are where g is the gravitational acceleration and g, is its magnitude at sea level, T is the thrust vector and T is i h magnitude, and Isr is the specific impulse. In special cases, it is advantagous to replace the mass with another coordinate. If the propulsion system is assumed to have constant e.jcction velocity, the characteristic velocity, or the time integral of the thrust acceleration magnitude, is used in place of mass; if the propulsion system is assumed to have limited power, the time integral of the thrust acceleration magnitude squared is used in place of mass. For the planar problem, the state x = (r, v, ~ n ) ~ is five-dimensional, since r and v each have dimension 2. If u* is the optimal control and x* the conesponding optimal trajectcxy and assuming the problem is normal, it is necessary [Fle75] that a nonzero k-dimensicma1 constant vector p and a nonzero n-dimensional acljoirit vector function p(t) exist such that In the case of space trajectory design, p = (prT, p,T, p,,JT. The qualifier " * " is suppressed from here on. Canonical Transformation Theory Because the state and adjoint equations of the optimal control problem comprise a Hamiltonian system, the powerful rtmonird trcrn.Ffor7~1rrtion themy of mechanics can be applied. A canonical transformation introduces no new physics to a problem, but it may facilitate analysis or physical interpretation of the motion, whose underlying properties are the same in the old and new coordinates [Lic87]. One use of canonical transformations is to increase the number of ignorable cocmlinates, thus simplifying the integration of the Hamiltonian system. In the optimal control context, canonical transformations allow one to translate the necessary conditions, integrals of the motion, approximate analytical solutions, etc., tierivcd in tenns of one set ol' coordinates into the equivalent results in tenns of another bet of coordinates, avoiding the need (5' for re-derivation. Let x* and p* denote the state and adjoint coordinate u*(t) = arg ~ n a x (pT(t) f(t, x*(t),u(t))) (6) vectors for coordinate set A, and xB and pB denote those for coordinate set B. In the transfmmations of interest here, the aa, 3w transfo~mation between the state coordinate vectors are timepT(tf) = 5;; (x*(tf). tf) + p T 5; (x*(tf). tf) ( 7 ) independent and do not involve the adjoint coordinate vectors: thus they are of the form and correspond to time-independent Lagrangian point transfosmations of mechanics [Lan49]. In the case of a Lagrangian point transfosmation, the Hamiltonian is an invariant of the associated canonical transformation from A coordinate vectors (xA; pA) to B coordinate vectors (xB; pB) [Lan49], It follows that the adjoint trruisfo~mation and the transformation of eq. (15) form a canonical B tsa~isfortnation CA = [ $: , T:] (xA; PA)+ (xB; pB). A set of transfo~mations together with the composition operation constitute a group if the following four properties are satisfied [Gol59]: (i) The set contains the identity transfo~mation. (ii) The inverse of each element of the set exists and is a member of the set. (iii) The set is closed under composition. (iv) The composition operation is associative. The canonical transformations associated with point transformations form a group on regions of the abstract state space whae the point transformations are one-to-one and onto (injective). Since our interest is transformations between any two of the four sets of state coordinates, there are twelve canonical transfonnations to be determined. G~ven the group structure it is sufficient to develop the three canonical transfo~mations from the Cartesian coordinates to the trajectory variables, the orbital elements to the trajectory variables, and the equinoctial elements to the orbital elements; the others can be obtained from composition and inversion of these. Smce the state coordinate transformations are known, it is the adjo~nt coordinate transformations that are of interest here. Explicitly, if