The Draper Technology Digest - Draper Laboratory

where μ iis the mean of the Gaussian function and σ isthe variance. This is arranged into a network of basisfunctions:g(p) = [g 1(p) g 2(p) ... g N(p)] T (5)The value of the means and the common variance for thisnetwork can be chosen a priori to approximate a givenfunction to a given accuracy [10]. With an increasingnumber of basis functions, the approximation error, εtends to zero [11]. Figure 14 shows an example of six basisfunctions in a GRNN, and Figure 15 shows an exampleof a weighted output of this GRNN. This shows how anarbitrary function can be approximated by determiningthe appropriate weights, a, for each basis function.AmplitudeAmplitude10.80.60.40.20-10 -8 -6 -4 -2 0 2 4 6 8 10PFIGURE 14. Basis functions forming a GRNN.10864WeightsWeighted Output of GRNNg 1g 2g 3g 4g 5g 6g 7g 8g 9g 10g 11These networks, however, cannot accurately approximatethe reaction wheel vibrations. This is because they aremissing a sinusoidally varying time dependence. TheGRNN can easily be modified to beg( RW, RW) =g 1( RW)sin( RW) Tg 1( RW)cos( RW)...g N( RW)sin( RW)g N( RW)cos( RW)By adding in sine and cosine terms, the magnitude andphase of these sinusoids can effectively be estimatedthrough adaptation of the weights of these basis functions.Estimation LawAn estimation law can now be developed. Considering onlya single-axis model of spacecraft attitude, the dynamics arelinear and can therefore be written as(6)ẋ = Ax + B uu + B dd (7)where x is the state of the system, A is the dynamics matrix,B uis the control input matrix, and B dis the disturbanceinputmatrix. The sensor output, y, and adaptation output,z, is given byy = C yxz = C zxFor the satellite model, C ywas chosen to be the identitymatrix since there are sensors to measure both the attitudeand angular rate of the spacecraft. The C zmatrix, however,was chosen to select only the spacecraft angular rate. Thisis because the z variable will be used in the adaptationlaw, and only the gyros will have a high enough samplingfrequency to capture the reaction wheel vibrations. TheCMOS detectors that measure the spacecraft attitude havea sampling frequency that is too low to capture the reactionwheel vibrations.(8)20-10 -8 -6 -4 -2 0 2 4 6 8 10PFIGURE 15. Weighted output of a GRNN.48 One-Arcsecond Line-of-Sight Pointing Control on ExoplanetSat, a Three-Unit CubeSat