## Attitude Prediction

17.1 Attitude Propagation

General Techniques, Integration Methods 172 Environmental Torques

Gravity-Gradient Torque, Solar Radiation Torque, Aerodynamic Torque, Magnetic Disturbance Torque 17 J Modeling Internal Torques 17.4 Modeling Torques Due to Orbit Maneuvers

Thrust Vector Collinear With the Spin Axis, Thrust Vector Not Collinear With Spin Axis but Nominally Passing Through Spacecraft Center of Mass

To meet spacecraft attitude determination and control requirements, we must frequently predict the attitude motion for a given set of initial conditions. This requires specifying the differential equations governing the attitude motion and a method of solution. The general methods used for attitude prediction, given appropriate torque models, are discussed in Section 17.1. The necessary modeling of the environmental and internal torques is described in Sections 17.2 and 17.3. Torque modeling during orbit maneuvers is discussed in Section 17.4.

17.1 Attitude Propagation

17.1.1 General Techniques

To model or predict the time evolution of the attitude, two basic methods are used: dynamic modeling and gyro modeling. Dynamic modeling consists of integrating both the dynamic and the kinematic equations of motion (see Section 16.1) using analytical or numerical models of the torque. Gyro modeling consists of using rate sensors or gyroscopes to replace the dynamic model such that only the kinematic equations need be integrated.

The dynamic equations of motion of a rigid spacecraft are given by Euler's equations as

where I is the moment of inertia tensor and to is the spacecraft angular velocity vector. The time derivative is taken and the vectors are resolved in a body-fixed coordinate system. The terms ND/ST and ^control the disturbance and control torques, respectively, acting on the spacecraft The kinematic equations can be written in differential form using the quaternion representation of the attitude (see

Section 12.1 and Appendix D) as where dq At

0 0