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Fig. 18-15. Position-Plus-Rate for Pitch Angle Control

Fig. 18-15. Position-Plus-Rate for Pitch Angle Control

The rate gain, Kpsa0.8 N-m-s, provides damping of the pitch response with time constant r = 2Iy/A^«67 s and the position gain, Kpm0.012 N-m, provides a restoring torque with frequency fm^Kp/Iy «0.021 s_I. The pitch loop response to a 5-deg initial error is shown in Fig. 18-16.

Substituting Eq. (18-82) into Eq. (18-53b) and taking the Laplace transform gives

{//+ V + [3^(7,- 7,)+ K,]}£,(*)= ) and the roots of the characteristic equation are

The gains, Kp and Kp are chosen to provide near-critical damping and minimize the overshoot, as discussed in Section 18.1. For critical damping,

Thus, the design value, X^ = 0.8 N-m-s corresponds a damping ratio of pswl/v^ (see Section 18.1), which results in one overshoot and no undershoots.

Roll and yaw control are achieved by commanding the y axis electromagnet based on magnetometer and horizon scanner roll angle data [Stickler, el al1976].

Fig. 18-16. HCMM Pitch Loop Response to a 5-Deg Error The electromagnet strength is commanded according to the control law

Fig. 18-16. HCMM Pitch Loop Response to a 5-Deg Error The electromagnet strength is commanded according to the control law

where B is the measured magnetic field in the body and KP and KN are the precession and nutation gain, respectively. Although the magnitude of Dy is limited to 10 A-m2 (10,000 pole-cm) by hardware constraints, we will ignore this complication in the subsequent analysis. Substitution of Eqs. (18-48) and (18-63) into Eq. (18-86) gives the control torque

7Vc=DxB={^[sinX(2i+) - cos Aß, - ) ] + kp cos A£,}

where the gains, magnetic field strength, and unit conversions have been absorbed into the constants kn and kp. Substituting Eq. (18-87) into Eq. (18-53) and taking the Laplace transform leads to the coupled roll and yaw equations in matrix notation as

where

l^-ik, jsin2X

0 0

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