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Simple Control Laws for Continuous-Thrust Escape Or Capture and Their Use in Optimisation

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Simple Control Laws for Continuous-Thrust Escape Or Capture and Their Use in Optimisation SIMPLE CONTROL LAWS FOR CONTINUOUS-THRUST ESCAPE OR CAPTURE AND THEIR USE IN OPTIMISATION Anastassios E. Petropoulos,* Gregory J. Whiffen? and Jon A. Simst Jet Propulsion Laboratory California Institute of Technology Pasadena, California 91 109-8099 Interplanetary missions which use low-thrust, high specific impulse propulsion can further capitalise on the capabilities of the propulsion system by using it to effect escape from the launch body or capture at a target body. For multi-revolution escape or capture, where optimisation becomes increasingly laborious the larger the number of revolutions, we investi- gate simple control laws which not only offer reasonable estimates of the propellant-optimal performance, but also provide initial guesses for optimisation. We find that quite different control laws can often yield similar performance. The utility of these control laws as initial guesses for optimisation is also assessed. Introduction cess, but also lead to different local minima. We opti- mise each of these initial guesses, as well as the initial The problem of low-thrust escape from an initially guess of tangential thrust, to achieve the minimum circular orbit around a body with a l/r2 gravity field escape time. The optimisation software is based on has been studied by numerous researchers in the last the static/dynamic control (SDC) alg~rithm.~-~SDC few decades. In the late 1950s, Lawden1p2 found that best fits into the direct method category, although, in the case of continuous, constant thrust accelera- unlike other direct methods, the explicit time depen- tion, thrust in the tangential direction yields nearly dence of the optimisation problem is not removed by the minimum characteristic velocity (and hence min- parametrisation. We compare the number of itera- imum propellant consumption and minimum escape tions needed for convergence and the optimal solu- time, assuming constant propellant mass flow rate). tions obtained from each initial guess. In this paper we study the same problem, but with In all numerical computations, the following pa- constant thrust and eccentric initial orbits, ultimately rameters are assumed: The thrust is 465mN, the focusing on the geostationary transfer orbit (GTO). specific impulse is 3100s, the initial spacecraft mass Using numerical integration of the equations of mo- is 1500kg, the radius of the central body (Earth) is tion, we first determine the performance of tangential 6378.14km1the gravitational parameter of the Earth thrust over a wide range of initial eccentricities and is 398600.48504296km3/s2 , and thrust commences at periapse altitudes. Then, for the case of a nominal periapsis. The nominal GTO has a 200km periapsis GTO initial orbit, we present four control laws for altitude and an eccentricity of 0.7306. Shadowing is the thrust direction that out-perform the tangential- not considered. thrust law. The four laws have different functional forms which were selected from over fifteen functional Tangential Thrust Initial Guess forms that were studied in detail, each having its own set of parameters to adjust. The laws were selected We examine the time needed to escape from initial based on their performance and on their perceived Earth orbits of varying eccentricity, but of initial peri- potential to serve in optimisation initial guesses as apsis altitude fixed at 200km. For eccentricities near that might not only speed-up the optimisation pro- that of GTO, the escape time is found to oscillate sig- 'Senior Member of the Engineering Staff; Member AIAA; nificantly with initial eccentricity. This oscillation is Member AAS. depicted in Fig. 1, where the equations of motion are tSenior Member of the Engineering Staff; Member AAS. integrated using two approaches. The first approach $Senior Member of the Engineering Staff; Member AIAA; uses the Runge-Kutta integrator of MATLAB, with Member AAS. the equations of motion expressed in polar coordi- Copyright @ 2002 by Anaatassios E. Petropoulos, Gregory J. Whiffen, and Jon A. Sims. Published by the American Institute nates and the thrust direction aligned with the veloc- of Aeronautics and Astronautics, Inc., with permission. ity at each instant - that is, a continuously varying 1 American Institute of Aeronautics and Astronautics thrust direction. The second approach, which is also to be of a certain functional form, with parameters employed in the optimisation software, uses Cartesian in the functions serving as secondary controls that coordinates to express the equations of motion, divid- determine the precise thrust angle. The second ap- ing the trajectory into a large number of management proach, which might be termed a “guided trial and periods during each of which the thrust direction is error approach,” involves maximising weighted aver- inertially fixed. For tangential thrust, the inertial di- ages of the rates of change of the osculating orbital rection is that of the velocity vector at the start of elements. The exact weightings, or functional forms the management period. The slight phase and up- of the weightings, then become the secondary con- ward shifts of the curve computed with the second trols. This approach has been taken in the past, for approach are partly due to the higher integration ac- example, by Kluever6 and GeferL7 Thrust along the curacy used, and partly due to the stepped nature of velocity vector, which maximises the rate of increase the thrust direction. of orbital energy (or, equivalently, semimajor axis), The oscillation belies the effect of phasing. The is the simplest example of this approach. local minima in escape time are all seen to occur for The thrust direction is specified in terms of the those initial orbits where the osculating eccentricity thrust angle, a, which is measured from the circum- of the spiral trajectory almost reaches zero shortly be- ferential direction, positive away from the gravita- fore escape, subsequently increasing roughly linearly tional centre (as for the flight path angle, 7). Three with time to unity. In effect, L‘escapeoccurs from of the more successful trial-and-error functional forms apoapsis,” that is, at the point when the spacecraft for a, with numerical parameters tailored for the has nearly reached the last apoapsis before escape, nominal GTO, are given by enough thrust is available to circularise the orbit, ef- fectively turning apoapsis to periapsis and continuing with a positive and increasing flight path angle up to escape. Whether the correct timing is obtained to effect these conditions depends on the initial orbit, since the thrust and gravitational parameters are as- sumed hed. The amplitude of the oscillations is greater, the (3) greater the eccentricity. The local minima and max- ima may be thought of as providing an envelope for the escape times. Fig. 2 shows these envelopes for where 0 E (-T,T] is the osculating true anomaly, T is various initial periapsis altitudes and initial eccentric- the radius, v is the velocity, and p is the gravitational ities varying from zero to about 0.85. As expected, parameter of the central body. It is evident that these the oscillations are of zero amplitude at zero eccen- three equations provide variations from the tangential tricity, and grow steadily with increasing eccentricity. thrust law, a = y. In the last equation, for example, The envelopes were produced with the Runge-Kutta represents a simple phase shift which gives the integrator. y(@++) thrust angle as the flight path angle not at the cur- The nominal GTO orbit was deliberately chosen rent position, but on the osculating conic at a true to provide an escape time near a local maximum for anomaly advanced by 4 from the current osculating the tangential thrust control law (using the stepped- true anomaly. The tailoring of the numerical param- thrust integrator). With this orbit and thrust law, eters was based on escape time computed with the we try to tax the optimisation process as much as same integrator used by the optimisation program. possible, in a sense obtaining an upper bound on the An examination of the escape spiral for each of the difficulty of the optimisation. control laws given by Eqs. 1-3 reveals that the escape Having established the nominal initial orbit, we from apoapsis characteristic is present in each case. may instead adjust the thrust control law to obtain With this observation in mind, we develop a thrust the correct phasing for escape from apoapsis. We now law based on maximising the rate of decrease of the present four control laws which accomplish this. impulsive AV needed to escape from the osculating apoapsis radius. The thrust angle giving the maximal rate of decrease is found using the variational equa- Control Laws tions for the semi-major axis, a, and the eccentricity, In developing the control laws, two basic ap- e, along with elementary conic relations. A weighting proaches are taken. The first is a simple trial and factor, c,, is included for added flexibility. The value error approach, where the thrust angle is assumed c, = 1 gives the maximal rate. An adjustment away 2 American Institute of Aeronautics and Astronautics 13! 1342 134 g 133.5 0m m0 13: m -0 E F 132.5 132 131.5 Nominal iv j Initial DeriaDsis altitude = .. 200kmI I I I 131 0.727 0.728 0.729 0.73 0.731 0.732 0.733 0.734 Initial eccentricity Fig. 1 Oscillation in escape time using tangential thrust for various initial Earth orbits, with thrust starting at periapsis. I I I I I I I I 0' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Initial eccentricity Fig. 2 Upper and lower bounds in escape time using tangential thrust for various initial Earth orbits, with thrust starting at periapsis. 3 American Institute of Aeronautics and Astronautics from unity will typically be needed to give the min- the escape time for the afll-based optimum when imum escape time.
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