Satellite attitude acquisition by momentum transfer-the controlled wheel speed method

An implementation of the momentum transfer method for spacecraft attitude acquisition of momentum wheel stabilized geostationary satellites is presented, in which the wheel speed is varied in a predeterminable manner to reduce the nutation usually associated with the method. The implementation is found to be capable of achieving the transfer to the desired zero nutation end point with 5° to 20° of residual nutation in practical situations without additional nutation damping. The transfer time is typically 10–30 min. The implementation is described in terms of the momentum sphere and energy ellipsoid. The detailed functional dynamics and parametric relationships are given in terms of phase planes and elliptic integral solutions. Feasibility of the implementation is shown to be dependent on the moment of inertia configuration and the degree to which tolerances on moments of inertia, satellite spin rate, and wheel speed are predeterminable.Nomenclature A, B, C Principal moments of inertia of the total spacecraft - b₁,b₂,b₃ Unit vectors parallel to principal axes - E, E(t) Energy of the spacecraft body (excludes the spin energy of momentum wheel) - E₀ Initial energy of the spacecraft (wheel not spinning) - E_c A constant arbitrary value ofE(t) - E_i Energy after an impulse which causes to equal zero, Equation (13) - E₂ The constant value ofE during Stage 2 - E₃ Energy at the end of Stage 3 - E_s The separatrix value ofE, i.e.E when =0 - H₀,H₀ Angular momentum vector of the total spacecraft - H_x,H_y,H_z Variable components ofH₀ resolved with respect tob₁,b₂, andb₃ - J Moment of inertia of the momentum wheel - L The slope of the wheel speed profile, Equation (31) - L_y Momentum wheel torque - m Direction cosine of the angle, , betweenH₀ andb₂ - m₁,m₂ Constant values ofm - O The centre of the momentum sphere - Oxyz Spacecraft coordinate axes - P Represents a state on the momentum sphere and energy ellipsoid - s Momentum wheel speed - s₂ Constant value ofs during Stage 2 - s_i Wheel speed after an impulse which causes to be zero - t Time - t₁, t₂, t₃ Elapsed time for Stages 1, 2, and 3 respectively - ₁ to₄ Roots of Equation (19) (poles on the phase plane) - Parameter defined in Equation (9) - cos^–1m - ₂ The value of at the termination of Stage 2 - _min The minimum of the periodic function (t) (during Stage 2) - _x, _y, _z Angular rates of spacecraft body - ₀ Initial value of _z - K_m Curvature of the momentum sphere - K_xy,K_yz Curvatures of the principal lines of the energy ellipsoid at (O,H₀,O), in theOH_xH_y) and (OH_yH_z) planes respectively