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The quaternion representation is generally prefered to the Euler angle representation because of its analytical characteristics.

As the dynamic complexity of the spacecraft increases, each degree of freedom must be represented by its appropriate equation of motion. For example, incorporating momentum or reaction wheels for attitude stability and maneuvering adds additional degrees of freedom. Momentum wheel dynamics can be included as an additional term in Euler's equations and an additional equation of motion for the wheels themselves. For this case, Eq. (17-la) is rewritten as

^ (/<o) = NDlST+^control - « X /« - [« X -h + N WHEEL ]

where h is the total angular momentum of the reaction wheels and ^ wheel is net torque applied to the momentum wheels, which is a function of bearing friction, wheel speed, and applied wheel motor voltage. The equation of motion of the wheels is

The dynamic and kinematic equations of motion are taken as a set of coupled differential equations and integrated using one of the methods described in Section 17.1.2. The integration state vector consists of the three angular velocity body rates or angular momentum components, the attitude quaternion, and any additional degrees of freedom due to nonrigidity (wheels, movable and flexible appendages, rastering instruments, etc.) Alternatively, for spacecraft which have a set of gyros as part of their attitude determination hardware (Section 6.5), the gyro assembly performs a mechanical integration of Euler's equations (irrespective of whether the spacecraft is rigid or flexible), and consequently only the kinematic equations require numerical integration. The gyro package flown aboard a spacecraft usually consists of three or more gyros which are capable of measuring the spacecraft's angular rates. The discussion of the gyro model used to compute the spacecraft's angular velocity from the gyro measurements is described in Section 7.8.2. The attitude propagation problem is diminished for spacecraft which fly a gyro package; in many cases, however, the calibration of the gyro model (see Section 6.5) can be a significant part of the attitude determination problem.

11.12 Integration Methods

Once the appropriate differential equations for attitude propagation have been established, it is necessary to choose a method for solving them. Because exact closed-form solutions of the complete equations to be integrated are almost never available, an approximation method is needed. Two methods are discussed in this section: direct integration using standard methods of numerical analysis, and a method for the kinematic equations using a closed-form solution of the equations with constant body rates.

Direct Integration. The equations of motion of attitude dynamics are a set of first-order coupled differential equations of the form

where f is a known vector function of the scalar t and the vector y. In this section, we will consider for simplicity the single differential equation

The extension to coupled equations is straightforward, with a few exceptions that will be pointed out.

Numerical algorithms will not give the continuous solution y(t), but rather a discrete set of valuesyn, n = 1,2,..., that are approximations toy(t) at the discrete times /„ = t0+ nh. Values of y(t) for arbitrary times can be obtained by interpolation. (For interpolation procedures, see, for example, Carnahan, et al., [1969]; Hamming [1962]; Hildebrand [1956]; Ralston [1965]; or Henrici [1964].) The parameter h is called the step size of the numerical integration. A minimum requirement on any algorithm is that it converge to the exact solution as the step size is decreased, i.e., that jm^-^C.) 07-8)

where the number of steps, n, is increased during the limiting procedure in such a way that nh = tn —10 remains constant.

Three important considerations in choosing an integration method are truncation error, roundoff error, and stability. Truncation error, or discretization error, is the difference between the approximate and exact solutions y„—y(t„), assuming that the calculations in the algorithm are performed exactly. If the truncation error introduced in any step is of order hp+l, the integration method is said to be of order p. Roundoff error is the additional error resulting from the finite accuracy of computer calculations due to fixed word length. An algorithm is unstable if errors introduced at some stage in the calculation (from truncation, roundoff, or inexact initial conditions) propagate without bound as the integration proceeds.

Truncation error is generally the limiting factor on the accuracy of numerical integration; it can be decreased by increasing the order of the method or by decreasing the step size. It is often useful to vary the step size during the integration, particularly, if the characteristic frequencies of the problem change significantly; the ease with which this can be done depends on the integration method used. The computation time required is usually proportional to the number of Junction evaluations, i.e., evaluations of J„=J(tH,y„) that are required. It is clear that decreasing the step size increases the number of function evaluations for any fixed integration algorithm.

Two families of integration methods are commonly employed. In one-step methods, the evaluation of yn+1 requires knowledge of only yn and /„. Multistep methods, on the other hand, require knowledge of back valuesy} or J, for some j<n as well. One-step methods are relatively easy to apply, because only y0 and /„ are needed as initial conditions. The step size can be changed, as necessary, without any additional computations. For these reasons, one-step methods are widely used. The most common one-step methods are the classical R-stage Runge-Kutta methods [Lambert, 1973]

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