Research article Open Access
Special Aspects in Attitude Control of a Spacecraft, Equipped With Inertial Actuators
Levskii MV*
Research Institute of Space systems, Khrunichev State Space research-and-production Center, Korolev, Moscow region, Russia
*Corresponding author: Levskii MV, Research Institute of Space systems, Khrunichev State Space research-and-production Center, Korolev, Moscow region, Russia, E-mail: @
Received: September 19, 2017; Accepted: October 09, 2017; Published:October 31, 2017
Citation: Levskii MV (2017) Special Aspects in Attitude Control of a Spacecraft, Equipped With Inertial Actuators. J Comp Sci Appl Inform Technol. 2(4): 1-9.
Abstract
The problem of correct choosing the time of optimal rotation from an arbitrary initial position into prescribed final angular position is investigated. The case, when slew maneuver is carried out with minimal magnitude of spacecraft angular momentum, is considered. Calculated dependences for constructing optimal program of reorientation (in sense of minimum angular momentum) are written as the law of variation of the angular velocity vector. Optimal control is in the class of regular motions. Dynamics of spacecraft rotation during the slew maneuver is similar to the known method of control, designed earlier, which includes as much as possible fast acceleration of a spacecraft, rotation with constant modulus of angular momentum and as much as possible fast cancellation of angular momentum. The formalized equations are presented, and computational expressions for determination of optimal duration of reorientation maneuver are obtained for known mass-inertial characteristics of a spacecraft, if attitude control is made by inertial actuators (system of control moment gyroscopes, gyrodynes). The knowledge of fit range of preferable duration of a slew maneuver helps to plan correctly the flight program of spacecraft controlled by the powered gyroscopes. The condition for determining the moment of the beginning of deceleration which uses current parameters of motion (information on angular position of a spacecraft and measurements of angular velocity) was given, what considerably increases accuracy of spacecraft move into a required position. Example of optimal spacecraft rotation and results of mathematical simulation adds the made theoretical descriptions, and illustrates reorientation process in visual form.

Keywords: Optimal control; Spatial orientation; Spacecraft attitude; Optimized functions; Control moment gyroscopes;
Introduction
Below we solve the problem of a choosing the duration of maneuver of spacecraft transfer into position of the required orientation (reorientation maneuver of a spacecraft). As spatial reorientation we mean transfer of axes ОXYZ bounded to the spacecraft body from one known angular position into another known (typically prescribed) angular position in a finite amount of time T. In this case, parameters of a slew maneuver (for example, components of slew quaternion) are known in advance, even before the beginning of maneuver; any initial angular mismatches can be possible (from a few degrees up to 180°). The angular orientation of the right-hand coordinate system ОXYZ (as well as its initial and final positions, ОXst Yst Zst and ОXf Yf Zf) is determined relative to a chosen coordinate system (reference basis I). In most cases the reference basis is inertial coordinate system ОXin Yin Zin (ICS).

Many papers are dedicated to questions of controlled rotations of solid body around its centre of mass [1-13]. In particular, the kinematic problem of a slew maneuver is studied in detail where solution was resulted for variant when angular velocity vector is limited by modulus [3]. In [4], the questions of optimal rotation of a spacecraft by criteria of fast response and minimum of energy expense for a case when the region of admissible values of the controlling moment is limited by sphere are considered, and itself spacecraft rotates around the vector of finite slew. Interesting results are presented in [5] where control is constructed by method of the combined synthesis by criterion of the generalized work. Later works [6,7] are known also (including decisions on the basis of algorithms of indistinct logic [7]) which also describe the constructing the controls stabilizing rotation of a spacecraft abound the instantaneous Euler axis (we will notice, that the majority of decisions corresponds to spacecraft rotation around motionless axis) though the rotation in the plane of the least slew angle is not always optimum in practice (but only in individual special cases), however exactly it was not executed [3-8].

In this paper, it is supposed that inertial actuators, in particular, powered gyroscopes (or gyrodynes), are used for control of spacecraft attitude, and size of angular momentum of spacecraft becomes essential-critical, therefore we must take into account this size at control of spacecraft rotation [9]. Earlier, the control method for spacecraft reorientation has been designed, which takes into account restriction of angular momentum of a spacecraft [1]. This method is accepted as model motion (as reference). From theoretical studies is known that level of spacecraft’s angular momentum and time of a slew maneuver T are interdependent; if angular momentum is small (i.e. less) then duration of maneuver is more large [1,2,12]. However at presence of perturbations acting upon the spacecraft, the choice of time T is not so obvious (further we will show that available stock of the angular momentum of gyro-system has significant role at timing T). This investigation is dedicated to finding of optimal duration of spacecraft reorientation (within the mode [1]) at which the reserved stock of angular momentum of gyroscopes system would be maximum.
Equations of motion and statement of the control problem
It is assumed that the spacecraft attitude control is realized using actuators that generate torques relative to all three principal central axis of inertia of the spacecraft. Angular motion of the spacecraft as a solid body we describe by the equations written down in quaternion variables [3].
2 Λ ˙ =Λω      (1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiqbfU 5amzaacaGaeyypa0Jaeu4MdWKaeSigI8MaeqyYdCNaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGPaaaaa@438E@
where ω is the vector of absolute angular velocity of a spacecraft; Λ is the quaternion reflected spacecraft orientation relative to inertial basis I, and the symbol MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSigI8gaaa@372F@ denotes the multiplication of quaternion’s. The equation (1) has the boundary conditions Λ(0) = Λin and Λ(T) = Λf. The quaternions Λin and Λf that specify directions of spacecraft axes at initial and final moments of time have any arbitrary a priori given values satisfying the condition || Λin ||= || Λf ||= 1. Spacecraft motion is considered optimum if the value
max 0<t<T J 1 2 ω 1 2 + J 2 2 ω 2 2 + J 3 2 ω 3 2       (2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaaicdacqGH8aapcaWG0bGaeyipaWJa amivaaqabaGcdaGcaaqaaiaadQeadaqhaaWcbaGaaGymaaqaaiaaik daaaGccqaHjpWDdaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWk caWGkbWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaeqyYdC3aa0baaS qaaiaaikdaaeaacaaIYaaaaOGaey4kaSIaamOsamaaDaaaleaacaaI ZaaabaGaaGOmaaaakiabeM8a3naaDaaaleaacaaIZaaabaGaaGOmaa aaaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeik aiaabkdacaqGPaaaaa@5740@
is minimal (T is the time of termination of the slew maneuver), where ωi are the projections of the absolute angular velocity on axes of the body’s coordinate system, Ji are the spacecraft central principal moments of inertia. Optimal control of a spatial rotation consists in transfer of a spacecraft from the position Ʌin into the position Ʌf according to the equation (1) with the minimum value of the index (2) which corresponds to maximal magnitude of angular momentum L with the components Li = Ji ωi.

For the spacecrafts, controlled by powered gyroscopes (gyrodynes or flywheels), minimization of angular momentum of a spacecraft body is very important. Now powered gyroscopes (gyrodynes) have received good application as executive mechanisms of spacecraft attitude system. Their use in regime of a slew maneuver demands that the summary angular momentum of the gyro-system would not exceed the admissible value. At control of spacecraft attitude with use of inertial actuators (control moment gyroscopes), the vector of angular momentum of the gyro-system should be within preset limited region S, and the extending of this vector beyond the region S leads to loss of controllability of a spacecraft; the size of available angular momentum determines the controlling possibilities of gyrosystem. Control of spacecraft rotation is achieved by redistribution of the angular momentum between the system of gyrodynes and spacecraft body; total angular momentum of a spacecraft as solid body with rotating masses is equal or close to zero [9]. When designing, analyzing, elaborating, and modeling the algorithms of attitude control of a spacecraft with powered gyroscopes it is assumed that region S of available angular momentum of the system of powered gyroscopes is limited by a sphere with radius R0. Such statement is used by many developers, designers and researchers [14-17]; it is valid for a large number (if not for the majority) of spacecraft. Obviously, for the guaranteed presence of angular momentum G of powered gyroscopes’s system within region S it is necessary that | G | was as small as possible. Hence, it is necessary, that during operated rotation of a spacecraft from position Ʌin in position Ʌf the modulus of angular momentum of spacecraft body was minimum possible (since L + G ≈ 0, where L is the angular momentum of the spacecraft body, and G is the angular momentum of the system of powered gyroscopes). From here we understand a sense of the formulated problem of control with minimization of the index (2), because the stock of the angular momentum of the gyro-system, determined by difference R0 – | L | will be maximal in this case only.
Formalization of optimal control of spacecraft’s slew maneuver
Let us shortly write down basic relationships and equations describing optimal motion of a spacecraft. The accepted functional (2) does not contain (in an explicit form) components Мi of the moment of forces. Therefore the projections of angular velocity ωi ( i= 1,3 ¯ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9maanaaabaGaaGymaiaacYcacaaIZaaaaaaa@3A22@ ) are considered as the controlling variables (controls).

It is known the following. Minimal value of index (2) is no less than quantity F / T, where
F = 0 T J 1 2 ω 1 2 + J 2 2 ω 2 2 + J 3 2 ω 3 2 dt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbGaaiiOaiabg2da98aadaWdXbqaamaakaaabaGaamOsamaa DaaaleaacaaIXaaabaGaaGOmaaaakiabeM8a3naaDaaaleaacaaIXa aabaGaaGOmaaaakiabgUcaRiaadQeadaqhaaWcbaGaaGOmaaqaaiaa ikdaaaGccqaHjpWDdaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqGHRa WkcaWGkbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaeqyYdC3aa0ba aSqaaiaaiodaaeaacaaIYaaaaaqabaGccaWGKbGaamiDaaWcbaGaaG imaaqaaiaadsfaa0Gaey4kIipaaaa@52B9@
But index (2) is equal to F / T if angular velocity ω is piecewise continuous function (i.e. when the jump-like changes in the angular velocity are possible and the situation ω(0+) ≠ 0, ω(T) ≠ 00 is admissible) and | L |= const within interval of time 0 < t < T. We must minimize the integral F above all. For minimization of the integral F, the Hamilton function H=0.5 r 0 ( ω 1 p 1 + ω 2 p 2 + ω 3 p 3 ) J 1 2 ω 1 2 + J 2 2 ω 2 2 + J 3 2 ω 3 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9iaaicdacaGGUaGaaGynaiaadkhadaWgaaWcbaGaaGimaaqabaGc caGGOaGaeqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaamiCamaaBaaale aacaaIXaaabeaakiabgUcaRiabeM8a3naaBaaaleaacaaIYaaabeaa kiaadchadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcqaHjpWDdaWgaa WcbaGaaG4maaqabaGccaWGWbWaaSbaaSqaaiaaiodaaeqaaOGaaiyk aiabgkHiTmaakaaabaGaamOsamaaDaaaleaacaaIXaaabaGaaGOmaa aakiabeM8a3naaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaa dQeadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqaHjpWDdaqhaaWcba GaaGOmaaqaaiaaikdaaaGccqGHRaWkcaWGkbWaa0baaSqaaiaaioda aeaacaaIYaaaaOGaeqyYdC3aa0baaSqaaiaaiodaaeaacaaIYaaaaa qabaaaaa@618E@ May be used together with the equations [3]
p ˙ 1 = ω 3 p 2 ω 2 p 3 ,  p ˙ 2 = ω 1 p 3 ω 3 p 1 ,  p ˙ 3 = ω 2 p 1 ω 1 p 2      (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaca WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeqyYdC3aaSbaaSqaaiaa iodaaeqaaOGaamiCamaaBaaaleaacaaIYaaabeaakiabgkHiTiabeM 8a3naaBaaaleaacaaIYaaabeaakiaadchadaWgaaWcbaGaaG4maaqa baGccaGGSaGaaeiiaiqadchagaGaamaaBaaaleaacaaIYaaabeaaki abg2da9iabeM8a3naaBaaaleaacaaIXaaabeaakiaadchadaWgaaWc baGaaG4maaqabaGccqGHsislcqaHjpWDdaWgaaWcbaGaaG4maaqaba GccaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaabccaceWGWbGb aiaadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcqaHjpWDdaWgaaWcba GaaGOmaaqabaGccaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Ia eqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaamiCamaaBaaaleaacaaIYa aabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaaboda caqGPaaaaa@6564@
Where р1 , р2 , р3 are projections of certain vector р onto axes related with spacecraft, and | P |= 1; r0 = const > 0 (the equations (3) show that the vector р is immovable relative to inertial basis I).

If there were no restrictions of the moment of forces, then optimal rotation of a spacecraft (in sense of criterion (2)) satisfies the equations (3) and the equations [2]
ω i  =  b   p i   /   J i 2       (4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaabaaaaaaaaapeGaamyAaaWdaeqaaOWdbiaacckacqGH9aqp caGGGcGaamOya8aadaWgaaWcbaWdbiaacckaa8aabeaak8qacaWGWb WdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaacckacaGGVaWdamaa BaaaleaapeGaaiiOaaWdaeqaaOWdbiaadQeapaWaaSbaaSqaa8qaca WGPbaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOWdamaaBaaaleaa peGaaiiOaaWdaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabIcaca qG0aGaaeykaaaa@4DFE@
Where b > 0 is scalar value. Note that T ≠ 0 (the time T satisfies the condition 0 < T < ∞). It is obvious that the equations (4) provide maximum of the Hamiltonian H for any b. Optimal value is
r 0 =1/ p 1 2 / J 1 2 + p 2 2 / J 2 2 + p 3 2 / J 3 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaIWaaabeaakiabg2da9maalyaabaGaaGymaaqaamaakaaa baWaaSGbaeaacaWGWbWaa0baaSqaaiaaigdaaeaacaaIYaaaaaGcba GaamOsamaaDaaaleaacaaIXaaabaGaaGOmaaaaaaGccqGHRaWkdaWc gaqaaiaadchadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaakeaacaWGkb Waa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgUcaRmaalyaabaGa amiCamaaDaaaleaacaaIZaaabaGaaGOmaaaaaOqaaiaadQeadaqhaa WcbaGaaG4maaqaaiaaikdaaaaaaaqabaaaaaaa@4B25@
Since the turn is terminated at time T, i.e. the terminal condition Λ(T) = Λf must be satisfied at given time T). The problem of determining the optimal control consists in solving the system of equations of angular motion (1) and the equations (3) under the condition that optimal vector-function ω (t) is chosen basing on the requirements (4) and this solution satisfies the boundary conditions Λ(0) = Λin , Λ(T) = Λf. The angular momentum L and the magnitude b are connected by the formula:
L 2   =   b 2  ( p 1 2 / J 1 2 + p 2 2 / J 2 2 + p 3 2 / J 3 2 )   MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHmbWdamaaCaaaleqabaWdbiaaikdaaaGccaGGGcGaeyypa0Zd amaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaadkgapaWaaWbaaSqabe aapeGaaGOmaiaacckaaaGcpaGaaiikamaalyaabaGaamiCamaaDaaa leaacaaIXaaabaGaaGOmaaaaaOqaaiaadQeadaqhaaWcbaGaaGymaa qaaiaaikdaaaaaaOGaey4kaSYaaSGbaeaacaWGWbWaa0baaSqaaiaa ikdaaeaacaaIYaaaaaGcbaGaamOsamaaDaaaleaacaaIYaaabaGaaG OmaaaaaaGccqGHRaWkdaWcgaqaaiaadchadaqhaaWcbaGaaG4maaqa aiaaikdaaaaakeaacaWGkbWaa0baaSqaaiaaiodaaeaacaaIYaaaaa aakiaacMcadaWgaaWcbaWdbiaacckaa8aabeaaaaa@531E@
Notice that p 1 2 / J 1 2 + p 2 2 / J 2 2 + p 3 2 / J 3 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGWbWaa0baaSqaaiaaigdaaeaacaaIYaaaaaGcbaGaamOsamaaDaaa leaacaaIXaaabaGaaGOmaaaaaaGccqGHRaWkdaWcgaqaaiaadchada qhaaWcbaGaaGOmaaqaaiaaikdaaaaakeaacaWGkbWaa0baaSqaaiaa ikdaaeaacaaIYaaaaaaakiabgUcaRmaalyaabaGaamiCamaaDaaale aacaaIZaaabaGaaGOmaaaaaOqaaiaadQeadaqhaaWcbaGaaG4maaqa aiaaikdaaaaaaaaa@4757@ = const for system of the equations (3), (4) (since p 1 p ˙ 1 / J 1 2 + p 2 p ˙ 2 / J 2 2 + p 3 p ˙ 3 / J 3 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaaIXaaabeaakiqadchagaGaamaaBaaaleaacaaIXaaabeaa kiaac+cacaWGkbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaS IaamiCamaaBaaaleaacaaIYaaabeaakiqadchagaGaamaaBaaaleaa caaIYaaabeaakiaac+cacaWGkbWaa0baaSqaaiaaikdaaeaacaaIYa aaaOGaey4kaSIaamiCamaaBaaaleaacaaIZaaabeaakiqadchagaGa amaaBaaaleaacaaIZaaabeaakiaac+cacaWGkbWaa0baaSqaaiaaio daaeaacaaIYaaaaaaa@4CC7@ = 0 for values p ˙ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaca WaaSbaaSqaaiaadMgaaeqaaaaa@380D@ calculated using the equations (3) with ωi calculated according to (4)). In case of unlimited moments Мi, spacecraft rotates with | L |= const = F / T at entire interval of motion 0 < t < T; optimal rotation of a spacecraft is carried out with constant modulus of angular momentum Lm. Concrete value Lm is unequivocally determined by time of maneuver termination T. The optimal vectors ω and р are connected by relation [2]
ω i = L m p i J i 2 p 1 2 / J 1 2 + p 2 2 / J 2 2 + p 3 2 / J 3 2  (i= 1,3 ¯ )      (5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacaWGmbWaaSbaaSqa aiaad2gaaeqaaOGaamiCamaaBaaaleaacaWGPbaabeaaaOqaaiaadQ eadaqhaaWcbaGaamyAaaqaaiaaikdaaaGcdaGcaaqaaiaadchadaqh aaWcbaGaaGymaaqaaiaaikdaaaGccaGGVaGaamOsamaaDaaaleaaca aIXaaabaGaaGOmaaaakiabgUcaRiaadchadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccaGGVaGaamOsamaaDaaaleaacaaIYaaabaGaaGOmaa aakiabgUcaRiaadchadaqhaaWcbaGaaG4maaqaaiaaikdaaaGccaGG VaGaamOsamaaDaaaleaacaaIZaaabaGaaGOmaaaaaeqaaaaakiaayk W7caqGGaGaaiikaiaadMgacqGH9aqpdaqdaaqaaiaaigdacaGGSaGa aGjbVlaaiodaaaGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabIcacaqG1aGaaeykaaaa@6333@
At satisfying the conditions Λ(0) = Λin , Λ(T) = Λf for solution to equations (1) Λ(t), where Lm > 0 is the magnitude (modulus) of angular momentum with which the slew maneuver of a spacecraft is made. The equations for angular velocities ωi can be formalized in following form:
ω ˙ 1 = ω 2 ω 3 ( J 2 2 J 3 2 ) / J 1 2 ,  ω ˙ 2 = ω 1 ω 3 ( J 3 2 J 1 2 ) / J 2 2 ,  ω ˙ 3 = ω 1 ω 2 ( J 1 2 J 2 2 ) / J 3 2       (6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyYdCNbai aadaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaWcgaqaaiabeM8a3naa BaaaleaacaaIYaaabeaakiabeM8a3naaBaaaleaacaaIZaaabeaaki aacIcacaWGkbWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaeyOeI0Ia amOsamaaDaaaleaacaaIZaaabaGaaGOmaaaakiaacMcaaeaacaWGkb Waa0baaSqaaiaaigdaaeaacaaIYaaaaaaakiaacYcacaqGGaGafqyY dCNbaiaadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcgaqaaiabeM 8a3naaBaaaleaacaaIXaaabeaakiabeM8a3naaBaaaleaacaaIZaaa beaakiaacIcacaWGkbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaey OeI0IaamOsamaaDaaaleaacaaIXaaabaGaaGOmaaaakiaacMcaaeaa caWGkbWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiaacYcacaqGGa GafqyYdCNbaiaadaWgaaWcbaGaaG4maaqabaGccqGH9aqpdaWcgaqa aiabeM8a3naaBaaaleaacaaIXaaabeaakiabeM8a3naaBaaaleaaca aIYaaabeaakiaacIcacaWGkbWaa0baaSqaaiaaigdaaeaacaaIYaaa aOGaeyOeI0IaamOsamaaDaaaleaacaaIYaaabaGaaGOmaaaakiaacM caaeaacaWGkbWaa0baaSqaaiaaiodaaeaacaaIYaaaaaaakiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeykaa aa@77E3@
During optimal maneuver (without restrictions for Мi ), the desired solution ω(t) has the following properties
J 1 2 ω 1 2 + J 2 2 ω 2 2 + J 3 2 ω 3 2 =   R =   const,  J 1 4 ω 1 2 + J 2 4 ω 2 2 + J 3 4 ω 3 2 =   D =   const (7) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGkbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWaaWbaaSqabeaa peGaaGOmaaaak8aacqaHjpWDdaWgaaWcbaWdbiaaigdaa8aabeaakm aaCaaaleqabaWdbiaaikdaaaGccqGHRaWkcaWGkbWdamaaBaaaleaa peGaaGOmaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaak8aacqaHjp WDdaWgaaWcbaWdbiaaikdaa8aabeaakmaaCaaaleqabaWdbiaaikda aaGccqGHRaWkcaWGkbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWaaW baaSqabeaapeGaaGOmaaaak8aacqaHjpWDdaWgaaWcbaWdbiaaioda a8aabeaakmaaCaaaleqabaWdbiaaikdaaaGccqGH9aqppaWaaSbaaS qaa8qacaGGGcaapaqabaGcpeGaamOuaiabg2da98aadaWgaaWcbaWd biaacckaa8aabeaak8qacaWGJbGaam4Baiaad6gacaWGZbGaamiDai aacYcacaGGGcGaamOsa8aadaWgaaWcbaWdbiaaigdaa8aabeaakmaa CaaaleqabaWdbiaaisdaaaGcpaGaeqyYdC3aaSbaaSqaa8qacaaIXa aapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaey4kaSIaamOsa8aa daWgaaWcbaWdbiaaikdaa8aabeaakmaaCaaaleqabaWdbiaaisdaaa GcpaGaeqyYdC3aaSbaaSqaa8qacaaIYaaapaqabaGcdaahaaWcbeqa a8qacaaIYaaaaOGaey4kaSIaamOsa8aadaWgaaWcbaWdbiaaiodaa8 aabeaakmaaCaaaleqabaWdbiaaisdaaaGcpaGaeqyYdC3aaSbaaSqa a8qacaaIZaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaeyypa0 ZdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaadseacqGH9aqppaWa aSbaaSqaa8qacaGGGcaapaqabaGcpeGaam4yaiaad+gacaWGUbGaam 4CaiaadshacaqGGaGaaeikaiaabEdacaqGPaaaaa@7D0F@
Optimal control of spatial rotation consists of a short-term imparting the initial conditions of motion (the calculated angular velocity) to the spacecraft at the beginning of a slew maneuver, maintenance of spacecraft rotation with demanded (programmed) angular velocity ω(t) at which the modulus of angular momentum of a spacecraft has constant value | L |= const, and a short-term suppressing of available angular velocity to zero at the moment of time t = T, when Ʌ(t) = Ʌf (at achievement by spacecraft of final position Λf ). Key problem is the finding the law of variation of the vector p(t) and value Lm that at result of solving the system of the equations (1), (3), (5) with the initial condition Λ(0) = Λin , the boundary condition Ʌ(T) = Ʌf was satisfied at right endpoint (the determination of the vector p(0) is a separate and rather complicated problem).

The problems in which the boundary values ω(0) = ω(T) = 0 (such conditions of spacecraft rotation are most typical) are of practical importance. Of course, at the times t = 0 and t = T the angular velocity for a nominal program of spacecraft rotation, determined by equations (5), are not equal to zero. Consequently, transferring segments are unavoidable: acceleration of rotation as a transfer from the state of rest (when ω = 0) to the regime of rotation with an angular momentum of maximal magnitude H0, and braking, i.e., reduction of the spacecraft’s angular momentum down to zero; and for that to finish a maneuver during given time T should be H0 > Lm. Between acceleration of rotation and braking, the equations (3) and
ω i = H 0 p i J i 2 p 1 2 / J 1 2 + p 2 2 / J 2 2 + p 3 2 / J 3 2 (i= 1,3 ¯ , H 0 =const)     (8) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacaWGibWaaSbaaSqa aiaaicdaaeqaaOGaamiCamaaBaaaleaacaWGPbaabeaaaOqaaiaadQ eadaqhaaWcbaGaamyAaaqaaiaaikdaaaGcdaGcaaqaaiaadchadaqh aaWcbaGaaGymaaqaaiaaikdaaaGccaGGVaGaamOsamaaDaaaleaaca aIXaaabaGaaGOmaaaakiabgUcaRiaadchadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccaGGVaGaamOsamaaDaaaleaacaaIYaaabaGaaGOmaa aakiabgUcaRiaadchadaqhaaWcbaGaaG4maaqaaiaaikdaaaGccaGG VaGaamOsamaaDaaaleaacaaIZaaabaGaaGOmaaaaaeqaaaaakiaacI cacaWGPbGaeyypa0Zaa0aaaeaacaaIXaGaaiilaiaaiodaaaGaaiil aiaadIeadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGJbGaam4Bai aad6gacaWGZbGaamiDaiaacMcacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqG4aGaaeykaaaa@66CF@
are satisfied.

If the conditions of slew maneuver Λin , Λf , and the time T are such that times of acceleration and braking are very small (in comparison with the total time of a maneuver T) and we may to neglect them, then one can consider as impulsive processes both imparting necessary angular momentum H0 to the spacecraft and reducing available angular momentum down to zero, and almost during the entire maneuver (between acceleration and braking) | L (t) |= const = H0 with satisfaction of the equations (3), (8). When finding optimal solutions p(t), ω(t) the value of vector р at the moment of instant t = 0 is determinative.

If the controlling moment М is limited, then a boost of spacecraft angular momentum to the required level | L |= H0 at the beginning of the slew maneuver and damping of available angular momentum to zero at the end of reorientation maneuver occupy some finite (distinct from zero) time. Of interest is the general case, when conditions of slew maneuver Λin and Λf are such that one cannot neglect the transition segments (acceleration and braking). Let the vector М satisfy the condition М 1 2 + М 2 2  + М 3 2 m 0 2        (9) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGCqWdamaaBaaaleaapeGaaGymaaWdaeqaaOWaaWbaaSqabeaa peGaaGOmaaaakiabgUcaRiaadYbbpaWaaSbaaSqaa8qacaaIYaaapa qabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaaiiOaiabgUcaRiaadYbb paWaaSbaaSqaa8qacaaIZaaapaqabaGcdaahaaWcbeqaa8qacaaIYa aaaOWdaiabgsMiJ+qacaWGTbWdamaaBaaaleaapeGaaGimaaWdaeqa aOWaaWbaaSqabeaapeGaaGOmaaaak8aacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeikaiaabMdacaqGPaaaaa@4CD3@ Where Мi are the projections of the moment of forces М on the central principal axes of the spacecraft; m0 is the maximum admissible magnitude of the controlling moment. Dynamics of spacecraft rotation is described by the equations
J 1 ω ˙ 1 +( J 3 J 2 ) ω 2 ω 3 = M 1 ,  J 2 ω ˙ 2 +( J 1 J 3 ) ω 1 ω 3 = M 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaaIXaaabeaakiqbeM8a3zaacaWaaSbaaSqaaiaaigdaaeqa aOGaey4kaSYaaeWaaeaacaWGkbWaaSbaaSqaaiaaiodaaeqaaOGaey OeI0IaamOsamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiab eM8a3naaBaaaleaacaaIYaaabeaakiabeM8a3naaBaaaleaacaaIZa aabeaakiabg2da9iaad2eadaWgaaWcbaGaaGymaaqabaGccaGGSaGa aeiiaiaadQeadaWgaaWcbaGaaGOmaaqabaGccuaHjpWDgaGaamaaBa aaleaacaaIYaaabeaakiabgUcaRmaabmaabaGaamOsamaaBaaaleaa caaIXaaabeaakiabgkHiTiaadQeadaWgaaWcbaGaaG4maaqabaaaki aawIcacaGLPaaacqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqaHjpWD daWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWGnbWaaSbaaSqaaiaaik daaeqaaaaa@5E93@ J 3 ω ˙ 3 +( J 2 J 1 ) ω 1 ω 2 = M 3      (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaaIZaaabeaakiqbeM8a3zaacaWaaSbaaSqaaiaaiodaaeqa aOGaey4kaSYaaeWaaeaacaWGkbWaaSbaaSqaaiaaikdaaeqaaOGaey OeI0IaamOsamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiab eM8a3naaBaaaleaacaaIXaaabeaakiabeM8a3naaBaaaleaacaaIYa aabeaakiabg2da9iaad2eadaWgaaWcbaGaaG4maaqabaGccaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeimaiaabMcaaa a@4F8E@
The laws of the fastest imparting and reduction of the angular velocity under the constraint (9) are known [10]. At the segment of acceleration, optimal control has the form
М   = m 0 J.ω/| J.ω |,     (11) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHCqWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiabg2da9iaa d2gapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaamOsaiaac6caca WHjpGaai4lamaaemaabaGaamOsaiaac6cacaWHjpaacaGLhWUaayjc SdGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabg dacaqGPaaaaa@4AEB@
Where J = diag ( J1 , J2 , J3 ) is the inertia tensor of spacecraft. At optimal motion, angular momentum of a spacecraft does not change the direction in inertial coordinate system, because the controlling moment М and angular momentum L have same direction. The magnitude of angular momentum varies according to the law | L |= m0 t. At the segment of braking, optimal control we write in following form
М   = m 0 J.ω/| J.ω |     (12) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHCqWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiabg2da9iab gkHiTiaad2gapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaamOsai aac6cacaWHjpGaai4lamaaemaabaGaamOsaiaac6cacaWHjpaacaGL hWUaayjcSdGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae ymaiaabkdacaqGPaaaaa@4BD9@
At optimal motion, angular momentum of a spacecraft does not change the direction in inertial coordinate system, and the controlling moment М makes with the angular momentum an angle of 180 degree. The angular momentum varies according to the law L = L br − m0 (t − t0) , where L br = | J·ω (t0) |; t0 is time of beginning of rotation damping. For both acceleration and braking, optimal control (as fast response) is control under which the controlling moment is parallel to angular momentum of a spacecraft at any moment of time.

At time instant t = 0 the angular momentum of a spacecraft L = 0, and the control (11) is necessary for the fastest reaching the preset level L= H0. Until | J ω(t) |< H0 , the controlling moment М = m0 L / | L | is optimum. Since the instant tac when the equality | J ω(tac) |= H0 becomes valid, optimum will be the motion (3), (8) at which the equality | J ω(t) |= H0 take place. It is obvious that because of existence of the boundary condition ω(T) = 0 such a time instant tbr (tbr < T ) should exist starting from which the angular momentum is begun to be reduced at a maximum controlling moment М = −m0 L / | L | (the instant tbr is chosen so that by the moment of a absolute stop ω = 0 the spacecraft would occupy the desired angular position Ʌf ). At the intervals of rotation acceleration and braking, the controlling moment М has ultimate maximum, and condition (9) is a strict equality, and within interval between acceleration and braking, the equations (3), (8) and equality | L |= const = H0 are valid. As result, the trajectory of spacecraft rotation Ʌ(t) is partitioned in three components: Ʌ(0) − Ʌ(tac) , Ʌ(tac) − Ʌ(tbr), and Ʌ(tbr) − Ʌ(T). Let us represent the quaternion of a slew in the form:
Λ s = Λ ˜ in Λ f =Δ Λ ac Δ Λ nom Δ Λ br , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaabohaaeqaaOGaeyypa0Jafu4MdWKbaGaadaWgaaWcbaGa aeyAaiaab6gaaeqaaOGaeSigI8Maeu4MdW0aaSbaaSqaaiaabAgaae qaaOGaeyypa0JaeuiLdqKaeu4MdW0aaSbaaSqaaiaabggacaqGJbaa beaakiablIHiVjabfs5aejabfU5amnaaBaaaleaacaqGUbGaae4Bai aab2gaaeqaaOGaeSigI8MaeuiLdqKaeu4MdW0aaSbaaSqaaiaabkga caqGYbaabeaakiaacYcaaaa@54D5@
Where Δ Λ ac = Λ ˜ in Λ( t ac ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeu 4MdW0aaSbaaSqaaiaabggacaqGJbaabeaakiabg2da9iqbfU5amzaa iaWaaSbaaSqaaiaabMgacaqGUbaabeaakiablIHiVjabfU5amjaacI cacaWG0bWaaSbaaSqaaiaabggacaqGJbaabeaakiaacMcaaaa@466E@ is the quaternion of spacecraft rotation during the acceleration segment; Δ Λ br = Λ ˜ ( t br ) Λ f MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeu 4MdW0aaSbaaSqaaiaabkgacaqGYbaabeaakiabg2da9iqbfU5amzaa iaGaaiikaiaadshadaWgaaWcbaGaaeOyaiaabkhaaeqaaOGaaiykai ablIHiVjabfU5amnaaBaaaleaacaqGMbaabeaaaaa@4590@ is the quaternion of spacecraft rotation during braking; Δ Λ nom = Λ ˜ ( t ac )Λ( t br ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeu 4MdW0aaSbaaSqaaiaab6gacaqGVbGaaeyBaaqabaGccqGH9aqpcuqH BoatgaacaiaacIcacaWG0bWaaSbaaSqaaiaabggacaqGJbaabeaaki aacMcacqWIyiYBcqqHBoatcaGGOaGaamiDamaaBaaaleaacaqGIbGa aeOCaaqabaGccaGGPaaaaa@49C6@ is the quaternion of rotation during spacecraft rotation with maximum angular momentum H0 ( Λ ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafu4MdWKbaG aaaaa@3779@ is the conjugate of the quaternion Λ, and Λ ˜ in MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafu4MdWKbaG aadaWgaaWcbaGaaeyAaiaab6gaaeqaaaaa@3982@ in is the conjugate of the quaternion Λin).

The optimal motion of a spacecraft consists of segments on which the controlling moment maximum in magnitude acts (segments of acceleration and braking), and of a segment of rotation with constant (in modulus) angular momentum, equal to the designed value H0 supporting the reaching of final position Ʌf in the given time T. On the segment of maximal controlling moment the angular momentum vector L has a permanent direction in the inertial space, but it is variable in magnitude (increase up to present value on the acceleration segment, and decrease to zero on the braking segment), while the controlling moment М is immovable with respect to the reference basis I (the vectors М and L are parallel); angular motion is determined by equations (1), (10). During the spacecraft rotation with maximum angular momentum the parameters of motion are determined by equations (3), (8). In this case, the angular momentum vector L has a constant magnitude H0, but its direction varies from a position preset at spacecraft acceleration to a position required during spacecraft braking. The spacecraft motion during its slew maneuver proceeds according to the following program of angular momentum variation: an increase of the magnitude of vector L from zero to H0 with a maximum velocity (with maximum moment М) at the invariable direction relative to the reference basis I; next follows rotation of vector L with a constant modulus of H0 according to the optimal law determined by equations (3), (8); and, finally, a decrease of the modulus of vector L down to zero with maximum possible velocity (with maximum moment М) at the invariable direction relative to the reference basis I. This program fully determines the spacecraft motion in the process of transfer from the state Ʌ= Ʌin , ω = 0 to the state Ʌ=Ʌf , ω = 0, since the equations (1) and the equality ω(t) = J -1 L(t) take place.

Since initial and final angular velocities are equal zero and the magnitude of controlling moment is constant | M |= const = m0 , duration of stages of acceleration and braking will be identical. The optimal solution ω(t) during segment of nominal motion (between acceleration and braking) possesses the properties (7), the vectors М and L are orthogonal, the magnitude of angular momentum is maximum and constant | L |= const = H0. Dynamics of motion is fully similar to solution of problem of spacecraft’s optimal rotation which was obtained in [1]. If the magnitude H0 of maximum possible angular momentum of a spacecraft is known, then after solving the problem of maximally fast response we shall find the time of a maneuver T; if the time T of termination of a maneuver is given, then value of parameter H0 in control law is subject to definition.

The optimal control of angular position of a spacecraft can be realized by the method which is similar to the known mode [1]. Since at spacecraft breaking the torque М is directed strictly against angular momentum L, the instant when braking begins can be predicted with a good accuracy. Duration of rotation damping is τ = | L |/ m0. The moment of the beginning of braking segment is determined by the condition:
4arcsin K q 2 2 + q 3 2 ( J 2 ω 2 ) 2 + ( J 3 ω 3 ) 2 = K 2 ω 2 2 + ω 3 2 m 0 ( J 2 ω 2 ) 2 + ( J 3 ω 3 ) 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeinaiaabg gacaqGYbGaae4yaiaabohacaqGPbGaaeOBamaalaaabaGaam4samaa kaaabaGaamyCamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgUcaRi aadghadaqhaaWcbaGaaG4maaqaaiaaikdaaaaabeaaaOqaamaakaaa baGaaiikaiaadQeadaWgaaWcbaGaaGOmaaqabaGccqaHjpWDdaWgaa WcbaGaaGOmaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaiikaiaadQeadaWgaaWcbaGaaG4maaqabaGccqaHjpWDdaWgaa WcbaGaaG4maaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaa aOGaeyypa0ZaaSaaaeaacaWGlbWaaWbaaSqabeaacaaIYaaaaOWaaO aaaeaacqaHjpWDdaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqGHRaWk cqaHjpWDdaqhaaWcbaGaaG4maaqaaiaaikdaaaaabeaaaOqaaiaad2 gadaWgaaWcbaGaaGimaaqabaGcdaGcaaqaaiaacIcacaWGkbWaaSba aSqaaiaaikdaaeqaaOGaeqyYdC3aaSbaaSqaaiaaikdaaeqaaOGaai ykamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaacIcacaWGkbWaaSba aSqaaiaaiodaaeqaaOGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaai ykamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaaa@6D0F@
Where q j are the components of quaternion of mismatch Λ ˜ (t) Λ f MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafu4MdWKbaG aacaGGOaGaamiDaiaacMcacqWIyiYBcqqHBoatdaWgaaWcbaGaaeOz aaqabaaaaa@3D8F@ ( j = 0, 1, 2, 3 ); К=| J.ω | MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGAqGaeyypa0ZaaqWaaeaacaWGkbGaaiOlaiaahM8aaiaawEa7 caGLiWoaaaa@3DB6@ is the magnitude of spacecraft’s angular momentum. At braking segment, cancellation of angular momentum is make according to linear law | L(t) |= H0- m0 (t-tbr) , where tbr is the instant of the beginning of braking.

Determination of the time instant tbr according to actual (measured values) kinematic parameters of motion (angular mismatch and angular velocity) improves the accuracy of guiding the spacecraft to the required state Ʌ=Ʌf , ω = 0.
Example of mathematical simulation of optimal rotation of a spacecraft
The aspects of optimization of the terminal control that matches, in the prescribed time T, the right-handed coordinate system OXYZ fixed to the spacecraft body with the basis whose attitude in the inertial space is given are above investigated. Now, in this section, we describe the numerical finding of the optimal control of the spacecraft slewing maneuver. By way of example, consider the slew maneuver of the spacecraft by 150° from the initial angular position Λin in which the spacecraft axes coincide with the axes of the reference basis I to the prescribed angular position Λf. It was assumed in this case that the initial and final angular velocities are zero: ω(0) = ω(T) = 0. The elements of the quaternion Λf describing the desired attitude of the spacecraft after its slewing are assumed to be λ0 = 0.258819, λ1 = 0.723196, λ2 = 0.5, and λ3 = 0.4. We determine the optimal angular velocity control program ω(t) for the spacecraft maneuver from the attitude Λ(0) = Λin to the attitude Λ(T) = Λf . We assume that the central principal moments of inertia of the spacecraft are J1 = 217.3 kg-m2, J2 = 440.6 kg-m2, and J3 = 416.1 kg-m2; maximum magnitude of the controlling moment is m0 = 0.4 N-m; and maneuver duration is assumed equal T = 150 s. Notice that the slew quaternion Λs corresponds to the case when the final rotation vector (Euler’s axis) makes approximately the same angle with the axis OX as with the plane perpendicular to OX. This is a fairly difficult case of reorientation of a solid body [4].

As a result of solving the boundary value problem corresponding to the transfer of the spacecraft from the attitude Λ(0) = Λin to the attitude Λ(T) = Λf, we obtained the rated value for vector р(0)={0.20548, -0.15337, 0.96657} (for possibility of solving the system of the equations (3), (8)). The desired accuracy (approximately 2 angular minutes) is attained at the fifth approximation step of the vector р(0) to the solution (method of solving the boundary value problem is identical to the algorithm described in [2]).

Key problem is to find of the values M(0) and H0 at which result of integration of the equations of motion with the initial conditions Λ(0) = Λin, ω(0) = 0 satisfies terminal conditions Λ(T) = Λf, ω(T) = 0 at the instant of time Т taking into account that the entire interval of motion consists of three stages - in the beginning of a maneuver (at acceleration) the trajectory of motion satisfies the equations (1), (10), (11); during spacecraft rotation with constant modulus of angular momentum (between acceleration and braking) the trajectory of motion satisfies the equations (1), (6); in the end of a maneuver (at braking) the trajectory of motion satisfies the equations (1), (10), (12).

The results of mathematical simulation for the slew maneuver under the optimal control are presented in Figure 1-4. Figure 1 shows the dynamics of the optimal angular velocities ω1(t), ω2(t), and ω3(t) in the body-fixed coordinate system in the course of the slew maneuver. One clearly see partition of the entire maneuver into three characteristic stages: acceleration of a spacecraft (gain of angular velocity), spacecraft motion with constant module of angular momentum (segment of nominal rotation), and spacecraft braking (reduction of angular velocity to zero). The law of forming the control also confirms this. The behavior of the rate of changing the angular momentum value is of a relay character, the duration of the segments of acceleration τac (when К ˙ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOgeyaaca aaaa@36A1@ > 0) and braking τbr (when К ˙ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOgeyaaca aaaa@36A1@ < 0) is identical and equals τ= 14.6 s. The maximum magnitude of spacecraft angular momentum during optimal rotation (the programmed level) is H0 = 5.83 N-m-s. The variation of the components λ0(t), λ1(t), λ2(t), and λ3(t) of the quaternion Λ(t), which determines the current attitude of the spacecraft in the course of the slew maneuver, is illustrated in Figure 2. Figure 3 shows the plots of the components l1(t), l2(t), and l3(t) of the ort of angular momentum L as functions of time. A distinctive feature is that the projection l1 varies insignificantly (its variation is much less than the variation of l2 and l3). This indicates that OX is the longitudinal axis of the spacecraft. It is also worth noting that the variables ω1(t) and l1(t) corresponding to the longitudinal axis of the spacecraft have a constant sign (for all combinations of the boundary values Λin and Λf)
Figure 1: Variation of angular velocities during optimal maneuver
Figure 2: Variation of components of the quaternion Λ
Figure 3: Variation of the variables l1, l2, l3 under optimal control
Figure 4: Character of modulus of the controlling moment M during rotation.
Finally, Figure 4 shows the variation of magnitude of controlling moment in the course of the slew maneuver. All three stages of the maneuver are well pronounced in it. During acceleration and braking processes the controlling moment is maximum possible. In the time interval between acceleration and braking, the controlling moment magnitude is substantially less than during gain and reduction of the spacecraft angular velocity; controlling moment M is determined according to dynamic Euler’s equations in which the time functions of angular velocities ωi (t) are a solution to system of equations (3) with expressions (8). At the stages of acceleration and braking, the controlling moment module is constant; during spacecraft rotation with maximum angular momentum the controlling moment magnitude varies slightly (for a dynamically symmetric spacecraft it is constant).

At numerical modelling, the trajectory Ʌ(t) and functions ωi corresponding to the optimal maneuver are obtained by integration of the equations of motion (1) and dynamic equations in which the derivatives ω ˙ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyYdCNbai aadaWgaaWcbaGaamyAaaqabaaaaa@38E5@ are calculated according to the equations: (10), (11) if t < T / 2 and | L | < H0; or (6) if tac ≤ t < tbr; or (10), (12) if t ≥ tbr, taking into account the initial conditions Λ(0) = Λin, ω(0) = 0 and M(0) = Mst, where tac is the instant of termination of acceleration (the instant of achieving the equality | L | = H0 in the beginning of maneuver), tbr is the instant of start of braking; T is the given time of maneuver (the known fixed value); H0 is the calculated constant magnitude of spacecraft’s angular momentum between acceleration and braking; tac ≤ T / 2, tbrT / 2in is the quaternion of attitude before maneuver, Mst is the calculated value of the controlling moment to start acceleration). The moment of forces M within the interval between acceleration and braking is calculated by the equations (10) in which angular accelerations ω ˙ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyYdCNbai aadaWgaaWcbaGaamyAaaqabaaaaa@38E5@ are calculated using the equations (6). In the presented example of simulation, the constants T, H0, tac, tbr and Mst have the following values: T = 150 s, H0 = 5.83 N-m-s, tac = 14.6 s, tbr = 135.4 s, and Mst = {0.156484, -0.060996, 0.363032 } N-m (as stated above Λin = 1). The values H0 , tac , tbr and Mst are calculated so that they obtained rotation ω(t) and the corresponding trajectory Ʌ(t) satisfy the equalities Λ(T) = Λf, ω(T) = 0 with account for the initial conditions Λ(0) = Λin, ω(0) = 0 and M(0) = Mst.

Accuracy of computations may be improved if to use the law M= Λ ˜ Λ ˜ in M st Λ ˜ in Λ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCytaiaab2 dacuqHBoatgaacaiablIHiVjqbfU5amzaaiaWaaSbaaSqaaiaabMga caqGUbaabeaakiablIHiVjaah2eadaWgaaWcbaGaae4Caiaabshaae qaaOGaeSigI8Mafu4MdWKbaGaadaWgaaWcbaGaaeyAaiaab6gaaeqa aOGaeSigI8Maeu4MdWeaaa@4993@ for mathematical simulation of spacecraft acceleration.
Problem of choosing the optimal time of the slew maneuver
For a spacecrafts with inertial actuators (powered gyroscopes), is important not only to minimize the angular momentum during the slew maneuver, but also to preset properly (maximum correctly) time T of maneuver ending. The problem consists in determination of such duration of a slew T (and the value of the parameter H0), that during the spacecraft motion around its center of mass, evolution of vector G of the total angular momentum of the system of gyrodynes will not bring this vector beyond the limits of region S of admissible values (no saturating of the system of gyrodynes will occur), and no necessity will emerge in unloading, i.e., in removal of accumulated angular momentum of the gyrodyne system by applying the moment of forces of another nature (magnetic [4], produced by switching the attitude control jet engines, etc.). Such motions of spacecraft are considered as allowable (in the sense of controlling the spacecraft orientation without unloading the system of gyrodyne).

At solving the formulated problem (choice of optimal time of slew maneuver) two moments are initial - (a) slew maneuver of a spacecraft occurs according to the method [1], and (b) the stock of the angular momentum of the gyrodynes system should be maximum (at the found parameters of slew maneuver H0 and T ); last requirement allows to reduce probability of involving any other (except the gyrodynes) means of attitude control (for example, jet engines). For the case with zero boundary conditions ω(0) = ω(T)=0 only a single type of motion is realized: the first phase - acceleration of spacecraft rotation with maximal controlling moment | M |= m0 up to | L |= H0 , next the site of spacecraft motion with constant in magnitude of angular momentum | L |= H0 (with satisfaction of equalities (7)), and then the symmetric segment of spacecraft braking with maximal controlling moment | M |= m0 up to a full stop of the spacecraft ( М||L ). Changing the magnitude of angular momentum G of the system of powered gyroscopes during the slew maneuver is such that the condition d|G|/dt|const MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaacY hacaWHhbGaaCiFaiaac+cacaWGKbGaamiDaiaacYhacqGHijYUcaWG JbGaam4Baiaad6gacaWGZbGaamiDaaaa@43B9@ is true on the segments of acceleration and braking (since the moment M of the controlling forces is much more perturbations moment Мper). And, in most cases, it is possible to assume |  d | G |  /dt  | t<tac =|  d | G |  /dt  | t>tbr MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaqa aaaaaaaaWdbiaabccacaWGKbaapaGaay5bSlaawIa7a8qacaqGGaGa aC4ra8aadaabdaqaa8qacaqGGaGaai4laiaadsgacaWG0bGaaeiiaa WdaiaawEa7caGLiWoadaWgaaWcbaWdbiaadshacqGH8aapcaWG0bGa amyyaiaadogaa8aabeaak8qacqGH9aqppaWaaqWaaeaapeGaaeiiai aadsgaa8aacaGLhWUaayjcSdWdbiaabccacaWHhbWdamaaemaabaWd biaabccacaGGVaGaamizaiaadshacaqGGaaapaGaay5bSlaawIa7am aaBaaaleaapeGaamiDaiabg6da+iaadshacaWGIbGaamOCaaWdaeqa aaaa@5C5C@ ,where tac is the instant of termination of acceleration; tbr is the instant of the beginning of braking. For the hypothetical case when Мper = 0, acceleration of spacecraft rotation can be carried out up to situation | L |= R0 , because in this ideal case | L |= | G |, and within the interval between acceleration and braking we have d | G |/dt= 0 (here R0 is radius of the sphere entered in region S of available values of the angular momentum G of powered gyroscopes system).

In real conditions of flight Мper ≠ 0, and therefore L + G ≠ 0 , and hence in general case | d | G | /dt |≠ 0 on the segment of nominal rotation (when | L(t) |= const ). At the presence of perturbations moments Мper ≠ 0 there is a problem - what value should be time of a maneuver T (or level H0) that up to termination of maneuver the possible increase in size | G | was less R0- H0 (dependence H0 from T is the monotonously decreasing function). Though perturbations Мper can “help” the rotation of a spacecraft (in this case d | G |/dt < 0 ), but to guarantee such evolution of events throughout all interval of time [tac, tbr] is impossible. Therefore at constructing the designed program of control, we must take into account the worst scenario - to assume the perturbations Мper greatest possible in magnitude and directed against angular momentum L of spacecraft body. Then d|G|/dt=| M per  and max t ac <t< t br d| G |/dt= M P cal MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeivaiaabI gacaqGLbGaaeOBaiaabccacaWGKbGaaiiFaiaadEeacaGG8bGaai4l aiaadsgacaWG0bGaeyypa0JaaiiFaiaah2eadaWgaaWcbaGaaeiCai aabwgacaqGYbaabeaakiaabccacaqGHbGaaeOBaiaabsgadaWfqaqa aiGac2gacaGGHbGaaiiEaaWcbaGaamiDamaaBaaameaacaqGHbGaae 4yaaqabaWccqGH8aapcaWG0bGaeyipaWJaamiDamaaBaaameaacaqG IbGaaeOCaaqabaaaleqaaOGaamizamaaemaabaGaaC4raaGaay5bSl aawIa7aiaac+cacaWGKbGaamiDaiabg2da9iaah2eadaWgaaWcbaGa aeiuaiaabogacaqGHbGaaeiBaaqabaaaaa@62E0@ is the calculated (the greatest possible) magnitude of perturbation moment Мper (i.e. | Мper |< МP cal ). The increment of magnitude of gyro-system angular momentum G during spacecraft rotation with | L |= const will following
ΔG = t ac t br | М per |dt (   t br t ac  )   М P cal  =  (T τ ac   τ br  )   М P cal , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHuoarcaWGhbGaaiiOaiabg2da98aadaWdXbqaamaaemaabaGa aCihemaaBaaaleaacaqGWbGaaeyzaiaabkhaaeqaaaGccaGLhWUaay jcSdGaamizaiaadshaaSqaaiaadshadaWgaaadbaGaaeyyaiaaboga aeqaaaWcbaGaamiDamaaBaaameaacaqGIbGaaeOCaaqabaaaniabgU IiYdGccqGHKjYOdaqadaqaamaaBaaaleaapeGaaiiOaaWdaeqaaOWd biaadshapaWaaSbaaSqaa8qacaWGIbGaamOCaaWdaeqaaOWdbiaaco bicaWG0bWdamaaBaaaleaapeGaamyyaiaadogacaGGGcaapaqabaaa kiaawIcacaGLPaaadaWgaaWcbaWdbiaacckaa8aabeaak8qacaWGCq WdamaaBaaaleaapeGaamiuaiaacckacaWGJbGaamyyaiaadYgaa8aa beaak8qacaGGGcGaeyypa0JaaiiOa8aacaGGOaWdbiaadsfacaGGta IaeqiXdq3damaaBaaaleaapeGaamyyaiaadogaa8aabeaak8qacaGG taYdamaaBaaaleaapeGaaiiOaaWdaeqaaOGaeqiXdq3aaSbaaSqaa8 qacaWGIbGaamOCaiaacckaa8aabeaakiaacMcadaWgaaWcbaWdbiaa cckaa8aabeaak8qacaWGCqWdamaaBaaaleaapeGaamiuaiaacckaca WGJbGaamyyaiaadYgaa8aabeaakiaacYcaaaa@79C6@
Where τac and τbr are the durations of acceleration and braking; G = | G. Thus, the relation G   ( t ac )   +   (T τ ac   τ br  )   М P cal  R 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGhbWdamaaBaaaleaapeGaaiiOaaWdaeqaaOGaaiika8qacaWG 0bWdamaaBaaaleaapeGaamyyaiaadogaa8aabeaakiaacMcadaWgaa WcbaWdbiaacckaa8aabeaak8qacqGHRaWkpaWaaSbaaSqaa8qacaGG GcaapaqabaGccaGGOaWdbiaadsfacaGGtaIaeqiXdq3damaaBaaale aapeGaamyyaiaadogaa8aabeaak8qacaGGtaYdamaaBaaaleaapeGa aiiOaaWdaeqaaOGaeqiXdq3aaSbaaSqaa8qacaWGIbGaamOCaiaacc kaa8aabeaakiaacMcadaWgaaWcbaWdbiaacckaa8aabeaak8qacaWG CqWdamaaBaaaleaapeGaamiuaiaacckacaWGJbGaamyyaiaadYgaca GGGcaapaqabaGccqGHKjYOpeGaamOua8aadaWgaaWcbaWdbiaaicda a8aabeaaaaa@5B69@ should be satisfied, where R0 is the known value beforehand preset.

At the moment of the termination of acceleration G ac) =Lac) (since | Мper | < < m0). “Saturation” of system of powered gyroscopes can take place in the limiting case, if the equality | L   ( τ ac )| =   R 0    (T    τ ac     τ br  )   М Pcal MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGG8bGaaCita8aadaWgaaWcbaWdbiaacckaa8aabeaakiaacIca cqaHepaDdaWgaaWcbaWdbiaadggacaWGJbaapaqabaGccaGGPaGaai iFa8qacqGH9aqppaWaaSbaaSqaa8qacaGGGcaapaqabaGcpeGaamOu a8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGGcGaeyOeI0Iaai iOa8aacaGGOaWdbiaadsfacaGGGcGaai4eG8aadaWgaaWcbaWdbiaa cckaa8aabeaak8qacqaHepaDpaWaaSbaaSqaa8qacaWGHbGaam4yaa WdaeqaaOWdbiaacckacaGGtaYdamaaBaaaleaapeGaaiiOaaWdaeqa aOWdbiabes8a09aadaWgaaWcbaWdbiaadkgacaWGYbGaaiiOaaWdae qaaOGaaiykamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaadYbbpaWa aSbaaSqaa8qacaWGqbGaam4yaiaadggacaWGSbaapaqabaaaaa@6020@ cal is valid. It is necessary to note, that if the time of slew maneuver T less, then probability of achievement by the moment tbr of the braking beginning of the threshold level G (t)= R0 will less, it would seem. However with reduction of time T, the required angular momentum Lac) increases, and the stock ΔR = R0Lac) decreases, that raises probability of presence of gyro-system’s “saturation” in the course of slew maneuver (probability of satisfaction of equality G= R0 increases). Here the problem of spacecraft rotation with minimum angular momentum L during given time T is very important (for that to increase the stock of angular momentum R0L for its use for indemnification of prospective perturbation moments Мper).

At construction of the optimum program of control, we can vary only two parameters – time of a maneuver T or magnitude of spacecraft’s angular momentum Lac) (other characteristics are preset by conditions of slew maneuver and cannot be changed). We will write down the equation setting connection between time of rotation T, an estimation of the maximum magnitude of perturbation moments МP cal , and the computed value of spacecraft’s angular momentum Lac). For the rotation of solid body along the trajectory which satisfies the equations (3)-(4), the equality
0 T | L |dt =   cons t   =   S L   MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada abdaqaaiaahYeaaiaawEa7caGLiWoacaWGKbGaamiDaaWcbaGaaGim aaqaaiaadsfaa0Gaey4kIipakabaaaaaaaaapeGaeyypa0ZdamaaBa aaleaapeGaaiiOaaWdaeqaaOWdbiaadogacaWGVbGaamOBaiaadoha caWG0bWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiabg2da98aada WgaaWcbaWdbiaacckaa8aabeaak8qacaWGtbWdamaaBaaaleaapeGa amitaaWdaeqaaOWaaSbaaSqaa8qacaGGGcaapaqabaaaaa@4F04@
is true, where the value SL is determined exclusively by quaternion of a slew and inertial characteristics of a spacecraft J1, J2, J3 [11]. Having designated L0 = | L (τac) |, and if we believe that on the segments of acceleration and braking, the modulus of angular momentum varies under linear law | d | L | /dt |= m , where m is the maximum speed of varying the modulus of angular momentum, we will receive following relationships:
2 0 T | L |dt = L 0  (2T τ ac τ br  ) or L 0  ( T L 0 /   m   )= S L (since  t br t ac  = L 0 /m). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWdamaapehabaWaaqWaaeaacaWHmbaacaGLhWUaayjcSdGa amizaiaadshaaSqaaiaaicdaaeaacaWGubaaniabgUIiYdGcpeGaey ypa0Jaamita8aadaWgaaWcbaWdbiaaicdacaGGGcaapaqabaGccaGG OaWdbiaaikdacaWGubGaai4eG8aacqaHepaDdaWgaaWcbaWdbiaadg gacaWGJbaapaqabaGcpeGaai4eG8aacqaHepaDdaWgaaWcbaWdbiaa dkgacaWGYbGaaiiOaaWdaeqaaOGaaiyka8qacaGGGcGaam4Baiaadk hacaWGmbWdamaaBaaaleaapeGaaGimaiaacckaa8aabeaakmaabmaa baWdbiaadsfacaGGtaIaamita8aadaWgaaWcbaWdbiaaicdaa8aabe aak8qacaGGVaWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaad2ga paWaaSbaaSqaa8qacaGGGcaapaqabaaakiaawIcacaGLPaaacqGH9a qppeGaam4ua8aadaWgaaWcbaWdbiaadYeaa8aabeaakiaacIcapeGa am4CaiaadMgacaWGUbGaam4yaiaadwgacaqGGaGaamiDa8aadaWgaa WcbaWdbiaadkgacaWGYbaapaqabaGccqGHijYUpeGaamiDa8aadaWg aaWcbaWdbiaadggacaWGJbGaaiiOaaWdaeqaaOGaeyypa0ZdbiaadY eapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaai4laiaad2gapaGa aiyka8qacaGGUaaaaa@7B55@
Two statements of a problem are possible: (a) at the presence of constraint T   T set MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiabgsMiJkaa dsfapaWaaSbaaSqaa8qacaWGZbGaamyzaiaadshaa8aabeaaaaa@3E49@ we use criterion L0 → min or МP cal T →max ; and (b) if right endpoint is free, then we receive the system of two equations in which the time of a maneuver T is specified from the condition МP cal → max. These equations have the following form L 0  +   М P cal  T  =   R 0 ,  L 0  ( T    L 0 /   m   )   =   S L     or MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGmbWdamaaBaaaleaapeGaaGimaiaacckaa8aabeaak8qacqGH RaWkpaWaaSbaaSqaa8qacaGGGcaapaqabaGcpeGaamihe8aadaWgaa WcbaWdbiaadcfacaGGGcGaam4yaiaadggacaWGSbGaaiiOaaWdaeqa aOWdbiaadsfacaGGGcGaeyypa0ZdamaaBaaaleaapeGaaiiOaaWdae qaaOWdbiaadkfapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiil aiaacckacaWGmbWdamaaBaaaleaapeGaaGimaiaacckaa8aabeaakm aabmaabaWdbiaadsfacaGGGcGaai4eG8aadaWgaaWcbaWdbiaaccka a8aabeaak8qacaWGmbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbi aac+capaWaaSbaaSqaa8qacaGGGcaapaqabaGcpeGaamyBa8aadaWg aaWcbaWdbiaacckaa8aabeaaaOGaayjkaiaawMcaamaaBaaaleaape GaaiiOaaWdaeqaaOWdbiabg2da98aadaWgaaWcbaWdbiaacckaa8aa beaak8qacaWGtbWdamaaBaaaleaapeGaamitaaWdaeqaaOWaaSbaaS qaa8qacaGGGcaapaqabaGcpeGaaiiOaiaacckacaWGVbGaamOCaaaa @66C7@ T   =  ( R 0    L 0  )   /   М P cal  ,  L 0 2 /   m     L 0  T   +   S L   =   0 ; MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiabg2da9iaa cckapaWaaeWaaeaapeGaamOua8aadaWgaaWcbaWdbiaaicdacaGGGc aapaqabaGcpeGaai4eG8aadaWgaaWcbaWdbiaacckaa8aabeaak8qa caWGmbWdamaaBaaaleaapeGaaGimaiaacckaa8aabeaaaOGaayjkai aawMcaamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaac+capaWaaSba aSqaa8qacaGGGcaapaqabaGcpeGaamihe8aadaWgaaWcbaWdbiaadc facaGGGcGaam4yaiaadggacaWGSbGaaiiOaaWdaeqaaOWdbiaacYca caGGGcGaamita8aadaWgaaWcbaWdbiaaicdaa8aabeaakmaaCaaale qabaWdbiaaikdaaaGccaGGVaWdamaaBaaaleaapeGaaiiOaaWdaeqa aOWdbiaad2gapaWaaSbaaSqaa8qacaGGGcaapaqabaGcpeGaai4eG8 aadaWgaaWcbaWdbiaacckaa8aabeaak8qacaWGmbWdamaaBaaaleaa peGaaGimaiaacckaa8aabeaak8qacaWGubWdamaaBaaaleaapeGaai iOaaWdaeqaaOWdbiabgUcaR8aadaWgaaWcbaWdbiaacckaa8aabeaa k8qacaWGtbWdamaaBaaaleaapeGaamitaaWdaeqaaOWdbiaacckacq GH9aqppaWaaSbaaSqaa8qacaGGGcaapaqabaGcpeGaaGimaiaabcca caGG7aaaaa@6B9F@ Whence L 0 2  ( 1   /   М P cal   +   1   /   m   )  L 0  R 0  /   М P cal  +  S L   =   0   . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGmbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabeaa peGaaGOmaiaacckaaaGcpaGaaiika8qacaaIXaWdamaaBaaaleaape GaaiiOaaWdaeqaaOWdbiaac+capaWaaSbaaSqaa8qacaGGGcaapaqa baGcpeGaamihe8aadaWgaaWcbaWdbiaadcfacaGGGcGaam4yaiaadg gacaWGSbGaaiiOaiaacckaa8aabeaak8qacqGHRaWkpaWaaSbaaSqa a8qacaGGGcaapaqabaGcpeGaaGyma8aadaWgaaWcbaWdbiaacckaa8 aabeaak8qacaGGVaWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaa d2gapaWaaSbaaSqaa8qacaGGGcaapaqabaGccaGGPaWdbiaabccaca GGtaIaamita8aadaWgaaWcbaWdbiaaicdacaGGGcaapaqabaGcpeGa amOua8aadaWgaaWcbaWdbiaaicdacaGGGcaapaqabaGcpeGaai4la8 aadaWgaaWcbaWdbiaacckaa8aabeaak8qacaWGCqWdamaaBaaaleaa peGaamiuaiaacckacaWGJbGaamyyaiaadYgaa8aabeaak8qacaGGGc Gaey4kaSIaaiiOaiaadofapaWaaSbaaSqaa8qacaWGmbaapaqabaGc peGaaiiOaiabg2da98aadaWgaaWcbaWdbiaacckaa8aabeaak8qaca aIWaWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaac6caaaa@6DCB@ The solution of last equation is L 0  =   0.5 ( R 0 m± R 0 2 m 2 4 S L m M Pcal (m+ M Pcal ) ) / ( m+ M Pcal ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGmbWdamaaBaaaleaapeGaaGimaiaacckaa8aabeaak8qacqGH 9aqppaWaaSbaaSqaa8qacaGGGcaapaqabaGcpeGaaGimaiaac6caca aI1aWdamaalyaabaWaaeWaaeaacaWGsbWaaSbaaSqaaiaaicdaaeqa aOGaamyBaiabgglaXoaakaaabaGaamOuamaaDaaaleaacaaIWaaaba GaaGOmaaaakiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI 0aGaam4uamaaBaaaleaacaWGmbaabeaakiaad2gacaWGnbWaaSbaaS qaaiaabcfacaaMe8Uaae4yaiaabggacaqGSbaabeaakiaacIcacaWG TbGaey4kaSIaamytamaaBaaaleaacaqGqbGaaGjbVlaabogacaqGHb GaaeiBaaqabaGccaGGPaaaleqaaaGccaGLOaGaayzkaaaabaWaaeWa aeaacaWGTbGaey4kaSIaamytamaaBaaaleaacaqGqbGaaGjbVlaabo gacaqGHbGaaeiBaaqabaaakiaawIcacaGLPaaaaaaaaa@664D@ If m → ∞ , then values τac → 0, τbr → 0 and SL = L0 T; therefore R0 - L0 = МP cal T = МP cal SL / L0, and hence L 0  =( R 0 ± R 0 2 4 S L M Pcal )/2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGmbWdamaaBaaaleaapeGaaGimaiaacckaa8aabeaak8qacqGH 9aqppaGaaiikaiaadkfadaWgaaWcbaGaaGimaaqabaGccqGHXcqSda GcaaqaaiaadkfadaqhaaWcbaGaaGimaaqaaiaaikdaaaGccqGHsisl caaI0aGaam4uamaaBaaaleaacaWGmbaabeaakiaad2eadaWgaaWcba GaaeiuaiaaysW7caqGJbGaaeyyaiaabYgaaeqaaaqabaGccaGGPaGa ai4laiaaikdaaaa@4D09@ (It is obvious that L0 < R0).

It is obvious that m > МP cal , and there are two solutions of quadratic equation (in both L0 > 0); smaller value L0 corresponds to higher border T (Tmax), more big value L0 corresponds to the lower border T (Tmin, a slew maneuver for minimum time). Time T we appoint in range Tmin < T < Tmax . Outside of interval borders [Tmin, Tmax] the rotation without “unloading” of gyro-system is not guaranteed, as “saturation” can occur or because of big L0 which was imparted after acceleration of rotation (it is close to R0), owing to small time T, or because of the big duration of a stage of nominal rotation (between acceleration and braking), and, as a consequence, the perturbations accumulated up during the slew maneuver “will eat” all stock R0 – L0 of angular momentum of the gyro-system. Taking into account that m >>| Мper |, we will give the approximate estimations (precisely enough reflecting true required levels L0 and T ). Since R 0 2 m 2 4 S L m M Pcal (m+ M Pcal ) R 0 m2 S L m M Pcal / R 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGsbWaa0baaSqaaiaaicdaaeaacaaIYaaaaOGaamyBamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaaisdacaWGtbWaaSbaaSqaaiaadYeaae qaaOGaamyBaiaad2eadaWgaaWcbaGaaeiuaiaaysW7caqGJbGaaeyy aiaabYgaaeqaaOGaaiikaiaad2gacqGHRaWkcaWGnbWaaSbaaSqaai aabcfacaaMe8Uaae4yaiaabggacaqGSbaabeaakiaacMcaaSqabaGc cqGHijYUcaWGsbWaaSbaaSqaaiaaicdaaeqaaOGaamyBaiabgkHiTi aaikdacaWGtbWaaSbaaSqaaiaadYeaaeqaaOGaamyBaiaad2eadaWg aaWcbaGaaeiuaiaaysW7caqGJbGaaeyyaiaabYgaaeqaaOGaai4lai aadkfadaWgaaWcbaGaaGimaaqabaaaaa@5FE3@ We can assume as optimum L 0 R 0 /2 S L M Pcal / R 0  and T2 S L / R 0 + R 0 /2m. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaaIWaaabeaakiabgIKi7kaadkfadaWgaaWcbaGaaGimaaqa baGccaGGVaGaaGOmaiabgkHiTiaadofadaWgaaWcbaGaamitaaqaba GccaWGnbWaaSbaaSqaaiaabcfacaaMe8Uaae4yaiaabggacaqGSbaa beaakiaac+cacaWGsbWaaSbaaSqaaiaaicdaaeqaaOGaaeiiaiaabg gacaqGUbGaaeizaiaabccacaWGubGaeyisISRaaGOmaiaadofadaWg aaWcbaGaamitaaqabaGccaGGVaGaamOuamaaBaaaleaacaaIWaaabe aakiabgUcaRiaadkfadaWgaaWcbaGaaGimaaqabaGccaGGVaGaaGOm aiaad2gacaGGUaaaaa@5961@ If the magnitude МP cal is not known or can vary its value over a wide range, then we must appoint time of a maneuver T using the condition of maximum of parameter МP cal for which the inequality R 0 2  m   >   4   S L    М P cal  ( m   +    М P cal  ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabeaa peGaaGOmaiaacckaaaGccaWGTbWdamaaBaaaleaapeGaaiiOaaWdae qaaOWdbiabg6da+8aadaWgaaWcbaWdbiaacckaa8aabeaak8qacaaI 0aWdamaaBaaaleaapeGaaiiOaaWdaeqaaOWdbiaadofapaWaaSbaaS qaa8qacaWGmbaapaqabaGcdaWgaaWcbaWdbiaacckacaGGGcaapaqa baGcpeGaamihe8aadaWgaaWcbaWdbiaadcfacaGGGcGaam4yaiaadg gacaWGSbGaaiiOaaWdaeqaaOGaaiika8qacaWGTbWdamaaBaaaleaa peGaaiiOaaWdaeqaaOWdbiabgUcaR8aadaWgaaWcbaWdbiaacckaca GGGcaapaqabaGcpeGaamihe8aadaWgaaWcbaWdbiaadcfacaGGGcGa am4yaiaadggacaWGSbGaaiiOaaWdaeqaaOGaaiykaaaa@5BC6@ is satisfied. The function S L    М P cal  ( m   +    М P cal  ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbWdamaaBaaaleaapeGaamitaaWdaeqaaOWaaSbaaSqaa8qa caGGGcGaaiiOaaWdaeqaaOWdbiaadYbbpaWaaSbaaSqaa8qacaWGqb GaaiiOaiaadogacaWGHbGaamiBaiaacckaa8aabeaakiaacIcapeGa amyBa8aadaWgaaWcbaWdbiaacckaa8aabeaak8qacqGHRaWkpaWaaS baaSqaa8qacaGGGcGaaiiOaaWdaeqaaOWdbiaadYbbpaWaaSbaaSqa a8qacaWGqbGaaiiOaiaadogacaWGHbGaamiBaiaacckaa8aabeaaki aacMcaaaa@502A@ increases with increase of МP cal . Hence, it is possible to assume that the roots of quadratic equation in which L0 is single unknown value, are converged nearer to each other with increase of the parameter МP cal, and the range [Tmin , Tmax] concerning preferable values of the slew maneuver time T will be narrowed. Let us find such critical value of parameter МP cal (we will designate it Мcrit) at which the rotation from position Λin into position Λf is still possible without infringement of requirement | G | ≤ R0 . For this purpose, it is necessary to solve the equation R 0 2  m   4   S L    М P cal  ( m   +    М P cal  )   =   0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabeaa peGaaGOmaiaacckaaaGccaWGTbGaaiiOaiaacobicaGGGcGaaGina8 aadaWgaaWcbaWdbiaacckaa8aabeaak8qacaWGtbWdamaaBaaaleaa peGaamitaaWdaeqaaOWaaSbaaSqaa8qacaGGGcGaaiiOaaWdaeqaaO WdbiaadYbbpaWaaSbaaSqaa8qacaWGqbGaaiiOaiaadogacaWGHbGa amiBaiaacckaa8aabeaakiaacIcapeGaamyBa8aadaWgaaWcbaWdbi aacckaa8aabeaak8qacqGHRaWkpaWaaSbaaSqaa8qacaGGGcGaaiiO aaWdaeqaaOWdbiaadYbbpaWaaSbaaSqaa8qacaWGqbGaaiiOaiaado gacaWGHbGaamiBaiaacckaa8aabeaakiaacMcadaWgaaWcbaWdbiaa cckaa8aabeaak8qacqGH9aqppaWaaSbaaSqaa8qacaGGGcaapaqaba GcpeGaaGimaaaa@5F6E@ With respect to МP cal (all other values in this equation are known, they were preset by conditions of the problem of optimal maneuver). As result, we have М crit =(m+ m 2 + R 0 2 m/ S L )/2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamihemaaBa aaleaacaqGJbGaaeOCaiaabMgacaqG0baabeaakiabg2da9iaacIca cqGHsislcaWGTbGaey4kaSYaaOaaaeaacaWGTbWaaWbaaSqabeaaca aIYaaaaOGaey4kaSIaamOuamaaDaaaleaacaaIWaaabaGaaGOmaaaa kiaad2gacaGGVaGaam4uamaaBaaaleaacaWGmbaabeaaaeqaaOGaai ykaiaac+cacaaIYaaaaa@49FC@ (Here, Мcrit = МP max is maximum possible perturbation); usually, max | Мper |< МP max (i.e. МpcalМP max). With account for action of perturbations Мper having a priori unknown magnitude, by optimal values will be L 0 = R 0 / ( 1+ 1+ R 0 2 / S L m )  and T= 2 S L 1+ R 0 2 / S L m / R 0 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGmbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaamOuamaaBaaaleaa caaIWaaabeaaaOqaamaabmaabaGaaGymaiabgUcaRmaakaaabaGaaG ymaiabgUcaRiaadkfadaqhaaWcbaGaaGimaaqaaiaaikdaaaGccaGG VaGaam4uamaaBaaaleaacaWGmbaabeaakiaad2gaaSqabaaakiaawI cacaGLPaaaaaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaWGubGa eyypa0ZaaSGbaeaacaaIYaGaam4uamaaBaaaleaacaWGmbaabeaakm aakaaabaGaaGymaiabgUcaRiaadkfadaqhaaWcbaGaaGimaaqaaiaa ikdaaaGccaGGVaGaam4uamaaBaaaleaacaWGmbaabeaakiaad2gaaS qabaaakeaacaWGsbWaaSbaaSqaaiaaicdaaeqaaaaakiaac6caaaa@585E@ As a rule, R 0 2 / S L m<<1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaDa aaleaacaaIWaaabaGaaGOmaaaakiaac+cacaWGtbWaaSbaaSqaaiaa dYeaaeqaaOGaamyBaiabgYda8iabgYda8iaaigdaaaa@3EC0@ . Therefore, as optimum we can assign Topt ≈ 2 SL / R0.

Have come to conclusion that Topt = 2Tfast, where Tfast is theoretically achievable lowest level-border of time of spacecraft rotation at the presence of the constraint | J ω | ≤ R0 (in these formulas SL = КС tpr, where tpr is the predicted time of the slew maneuver from the position Λin into the position Λf obtained by the simulating the spacecraft motion according to the equations (1), (3), (5) in which Lm = КС > 0).

We could expect that in optimal case the time of the slew maneuver should be twice more the time of ideal rotation (when the modulus of spacecraft angular momentum L is equal to as much as possible admissible value R0 and the perturbation moments are absent).
Conclusions
Control of spacecraft attitude using inertial actuators (in particular, by system of powered gyroscopes) is carried out often enough. Efficiency of use of inertial devices (control moment gyroscopes) in regime of spatial rotation is defined by not only the accepted algorithm of spacecraft attitude control but by time of the ending the reorientations maneuver. For the accepting the saturation of gyro-system its total angular momentum should not exceed admissible value (the slew maneuver is carried out due to redistribution of the angular momentum between the system of gyrodynes and spacecraft body). For situation that rotation will occur without the unloading of gyro-system, the modulus of angular momentum of spacecraft body should be obviously less radius of the sphere entered in region of available values of the angular momentum of gyro-system. The reorientation duration in many cases can become critical factor. We have formulated the following problem: it is necessary to define such duration of slew maneuver that during spacecraft motion around the centre of mass the angular momentum vector of gyrodynes system was within region of admissible values, this fact will exclude necessity of “unloading” of gyro- system and will provide executing of slew maneuver without the applying of other means for attitude control (for example, jet engines). The solving the specified problem allows to make spatial turns of a spacecraft using only the control moment gyroscopes. The knowledge of fit range of preferable duration of the slew maneuver helps to plan correctly the flight program of spacecraft controlled by the powered gyroscopes, and to appoint correctly time characteristics of main operations (including dynamic operations).

It is known that the method of orientation [1] is optimum at control with use inertial actuators. In the most general case the slew maneuver is partitioned in three characteristic phases: acceleration of the spacecraft (the imparting to it the angular velocity) up to a given magnitude of angular momentum, rotation of the spacecraft with constant modulus of angular momentum, and cancellation of angular velocity down to zero. The problem of attitude control is reduced to solving the three problems: the fastest imparting to a spacecraft the required angular momentum, rotation of the spacecraft with a calculated velocity of motion, and maximum possible deceleration (damping) of the spacecraft. At the segments of acceleration and braking, the controlling moment is maximum possible and parallel to the angular momentum vector, which ensures a minimum time of reaching the present value of the spacecraft angular momentum (or reduction of available angular momentum down to zero). At the stage between acceleration and braking, the spacecraft rotates with angular momentum having constant modulus; the controlling moment is formed using the condition that the spacecraft attitude motion should proceed along the prescribed trajectory of rotation determined by the rated vector of a turn and preset value of the angular momentum modulus. During stages of an intensive gaining and suppressing the angular momentum, the modulus of controlling moment remains to constants. This circumstance allows us to use the described control method of slew maneuver in a case when the controlling moment is limited by sphere. Basing on properties of optimal rotation, the described control method of spacecraft maneuver is exclusively important to apply in cases when the controlling moment is limited by sphere [1].

Optimization of spacecraft reorientation when motion is controlled by inertial actuators (for example, gyrodynes) includes two main tasks. First task is traditional optimization problem which consists in finding the mode of rotation with minimal magnitude of angular momentum. Second problem is new; it consists in calculating the most correct duration of slew maneuver under which necessity of “unloading” of gyro-system is excluded. In this paper, both problems were successfully solved. Numerical example and results of mathematical simulation of optimal spacecraft rotation, as illustration in visual form, are given. They add the made theoretical descriptions and confirm the efficiency and practical feasibility of the control method developed.
Acknowledgements
The author would like to express greatest thanks to colleagues and specialists from Korolev Rocket Space Corporation (in Russia), and to all participants of the seminars and International conferences on optimization problems.
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