Guidance Stabilization of Satellites Using the Geomagnetic Field

Algorithm 1

START;

CALCULATION of (Ω,𝐵)(𝑡₀);

FOR 𝑘=0 TO 𝑛−1

   CALCULATION of:

   – the guide trajectory (Ω,𝐵) and the guide stabilizer 𝑊 on [𝑡₀+𝑘𝜀,𝑡₀+(𝑘+1)𝜀];

   – the satellite stabilizer

       𝑈=(𝑈(𝑡₀+𝑘𝜀),𝑈(𝑡₀+(𝑘+1/𝑚)𝜀),…,𝑈(𝑡₀+(𝑘+(𝑚−1)/𝑚)𝜀))

   using a numerical algorithm to MINIMIZE the max_{𝑡∈[𝑡0+𝑘𝜀,𝑡0+(𝑘+1)𝜀]}|(Ω,𝐵)(𝑡)| in 𝑚

   STEPS, where

      (Ω̇,𝐵̇)(𝑡)=𝐴(𝑡)(Ω,𝐵)(𝑡)+(𝐽⁻¹(𝑈×𝐹(𝑡,Ω,𝐵))0)^∗−(𝐽⁻¹𝑊0)^∗,

              𝐴(𝑡)=∇_(Ω,𝐵)𝑓(𝑡,Ω,𝐵),

         𝑓(𝑡,Ω,𝐵)=^{𝐽−1(−Ω×𝐽Ω+3𝜔2𝑏3×𝐽𝑏3)}_{𝑏𝑖×(Ω−𝜔𝑏2)},𝑖=1,2,3;

   FOR 𝑗=1 TO 𝑚

         CALCULATION of the satellite trajectory (Ω,𝐵) on

              [𝑡₀+(𝑘+(𝑗−1)/𝑚)𝜀,𝑡₀+(𝑘+𝑗/𝑚)𝜀]

         using the stabilizer 𝑈(𝑡₀+(𝑘+(𝑗−1)/𝑚)𝜀);

   END

END