Interplanetary Flight Control With Electric Engine In View Of Thrust Errors

The optimal space flight to the planet with using electric propulsion system (EPS) is found on the assumption of nominal operating conditions of EPS as cruise engine. However on account of possible errors of program execution of engine thrust, the actual trajectory can deviate from the optimal nominal one and cannot provide performance of certain goal conditions of the space flight. The problem of definition of necessary correction of trajectory for optimal achievement by spacecraft (SC) of the final condition is investigated. Correction is realized by means of renewal of electric propulsion program during the space flight. Aprioristic estimations of possible schemes of electric propulsion program renewal are carried out. The following scheme of trajectory correction is offered. The trajectory is divided into equal time intervals. It is supposed that in the beginning of each time interval the position and velocity of spacecraft are known practically precisely. Errors of EPS operation are given as errors of value and direction of thrust. Because of these errors the spacecraft condition at the end of the time interval cannot coincide with nominal optimal one. It is assumed that the actual trajectory not strongly deviates from the optimal trajectory and the estimations of possible dispersion of spacecraft condition at the end of time interval are fulfilled in linear statement. Then the electric engine thrust program is changed in such manner that the spacecraft reaches the set final vector of condition at further optimal space flight without errors. The minimal duration of the interval is conditioned by following demands. During each of these intervals it is necessary: 1) to receive and process telemetric information about system operation for support current thrust program, 2) to carry out and process trajectory measuring, 3) to carry out ballistic computation for more precise definition of parameters of spacecraft motion and EPS operation, 4) to calculate new further thrust program of EPS, 5) to transmit new thrust program to spacecraft. The time interval of the thrust program renewal varies in admissible boundaries and for each of them the maximal mass expenses for the trajectory correction realization are estimated. The numerical results are carried out for space flight to Mars within the framework of Russian “Phobos-samplereturn” project. It is supposed to apply the Russian plasma engine SPT-140 to this flight with solar batteries as an energy source. It is assumed that the errors of thrust can reach 5 percent from nominal value and 1 degree on direction. Dependence of mass expenses on time interval value for thrust program renewal was found. This dependence was used for search of the optimal time interval and on the basis of this result the scheme of this trajectory correction was chosen and the maximal mass expenses were appreciated. 1. PRELIMINARY REMARKS Forthcoming flight of spacecraft with EPS on interplanetary trajectory on time interval ] , [ F N t t T = is investigated. EPS operation is carried out under the socalled program of thrust. It is supposed, that nominal thrust program is realized from the moment of time tN and provides (in case of the absolutely exact thrust program execution) the SC flight along the optimal nominal trajectory from the point of view of the SC mass expenses. The kinamatic parameters vector T T T )) t ( ), t ( ( ) t ( V R X = of this trajectory at the moments tN and tF coincides with the given vectors T T N T N N ) , ( V R X = and T T F T F F ) , ( V R X = . The position R and velocity V vectors are determined in heliocentric non-rotating coordinate system Oxyz at J2000.0. According to [1,2], it is supposed, that the value of the EPS thrust P depends on the electrical power supplied for electric engine, this power in turn depends on the SC position in the coordinate system Oxyz. Generally speaking, the program thrust can be executed with errors. Therefore flight should be planned in view of aprioristic errors of the thrust program performance and opportunity to change the thrust program at some fixed moments ti∈T for i = 2,3,...,k-1, k>2. The moments of time t1 and tk are equal to tN and tk respectively. The thrust program starting from the moment ti and the trajectory corresponding to this program on interval [ti, tk] have number i, i=1,2,...,k-1. It is obvious, that the first trajectory corresponds to the nominal thrust program: N 1 ) t ( X X = , (1) F k ) t ( X X = . (2) The estimation of additional expenses ∆M of the SC mass for realization of target flight at the account possible of deviation δP and α from the magnitude P and direction e of thrust at each current time has practical significance. The deviation δP is characterized by relative value P / P δ ν = . The random values ν and α can take on values, absolute values of which are not surpassing the given small values νl and αl (for “Phobos-Ground” project 06 . 0 l < ν , o 1 l < α ): l ν ν ≤ , l α α ≤ . (3) The values νl and αl in domain Oxyz for each current moment determine domain Ω of the possibility increment δP of the thrust vector P=Pe. The additional expenses ∆M calculation with known values of the SC initial mass MN, parameters of EPS ,νl and αl is carried out at following assumptions: The SC flight on all interval T is realized in small vicinity of the nominal (first) trajectory; Each program of thrust in case of EPS ideal operation provides performance of a terminal condition with minimal losses of the SC mass; At each moment the random vectorδP has the uniformly distribution low [2] in the three-dimensional domain Ω; The SC trajectory kinematic parameters at the initial moment tN is known with neglected errors; Deviations of kinematic parameters of the SC trajectory take place only because of the mentioned above errors of the thrust program execution. 2. ALGORITHM OF THE PROBLEM DECISION Before to state algorithm of calculation of additional expenditure of the mass necessary for performance of a target task of flight, it is necessary to present mathematical model of the SC motion on the optimum trajectory. 2.1 Mathematical model of the SC motion The SC interplanetary flight with EPS on the optimum trajectory is described in coordinates system Oxyz by fourteen differential equations system:        