The spacecraft angular velocity estimation in the orbital stabilization mode by the results of the local vertical sensor measurements

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described by a system of the Euler dynamic equations:
Jx
dωx
dt
+ (Jz − Jy)ωyωz = Mx ;
Jy
dωy
dt
+ (Jx − Jz )ωxωz = My ;
Jz
dωz
dt
+ (Jy − J x)ωxωy = Mz ,
(4)
where Jx , Jy, Jz are the main moments of SC inertion; ωx, ωy , ωz are the
projectors of SC angular velocity at the axis of coordinate systems, rigidly
connected to the craft; Mx , My, Mz are external forces moments.
To describe SC movement in the reference coordinates system it is
necessary to know kinematic correlations demonstrating the dependence
of angular velocity projections ωx , ωy, ωz on the position of a connected
coordinate system relative to the reference one. This dependence is set
up with the help of three Euler angles: roll angle γ ; yaw angle ψ ; pitch
angle ϑ.
For craft orientation in the orbital coordinates system:
ωx = ˙γ + (Ω − ˙
ϑ ) sin ψ;
ωy = ˙ψ cos γ − (Ω − ˙
ϑ) sin γ cos ψ;
ωz = −(Ω − ˙
ϑ) cos γ cos ψ − ˙
ψ sin γ.
(5)
We differentiate the kinematic equations (5):
˙ ωx = ¨γ + ( ˙Ω − ¨
ϑ) sin ψ + (Ω − ˙
ϑ) ˙ψ cos ψ;
˙ ωy = ¨ψ cos γ − ˙
ψ ˙ γ sin γ − ( ˙Ω − ¨
ϑ ) sin γ cos ψ−
−(Ω − ˙
ϑ)( ˙γ cos γ cos ψ − ˙
ψ sin γ sin ψ);
˙ ωz = −( ˙Ω − ¨
ϑ ) cos γ cos ψ + (Ω − ˙
ϑ)( ˙γ sin γ cos ψ+
+ ˙ψ cos γ sin ψ) − ¨
ψ sin γ − ˙
ψ ˙ γ cos γ.
(6)
After the linear approximation the systems (5) and (6) will gain the
following appearance
ωx = ˙γ + Ωψ; ωy = ˙ψ − Ω γ; ωz = ˙ϑ − Ω; (7)
˙ ωx = ¨γ + Ω ˙ψ; ˙ωy = ¨ψ − Ω ˙γ; ˙ωz = ¨ϑ. (8)
Inserting (7) and (8) into (4) we obtain:
J x¨γ + Ω2( Jz − Jy )γ + Ω(Jx + Jy − Jz) ˙ψ = Mx ;
J y
¨
ψ + Ω2(Jz − Jx) ψ − Ω(Jx + Jy − Jz ) ˙γ = My;
J z
¨
ϑ = Mz .
(9)
6 ISSN 0236-3933. HERALD of the BMSTU. Series “Instrument Engineering”. 2014. No. 5