# Aerospace Control Systems

## Spacecraft GNC Systems Engineering

Article Index
Pointing Error Analysis Tool
Pointing Error Definitions
Concepts from Optics
Standard Pointing Error Metrics
Computational Techniques for Evaluating the Pointing Error Metrics
Why Not Use the Peak Stability or Other Metric?
Pointing Error Analysis
All Pages

### Standard Pointing Error Metrics

The pointing error metrics are called the Accuracy Metric, Displacement Metric, Jitter Metric, Stability Metric, and the Windowed Stability Metric. Time domain expressions for the pointing error metrics can be written directly from the corresponding definitions. The time-domain equations for the pointing error metrics are not used as a computational tool because they are computationally burdensome, and because the time-domain data does not usually fit a time window exactly. The pointing error metrics in the time domain are converted to the frequency domain by using Parseval's equality. The time-domain and frequency-domain expressions are, mathematically, exactly equivalent.

In the frequency domain, the pointing error metrics are given by the following expressions,

Accuracy Metric σa2 = S(ω) dω

Displacement Metric σj2 = S(ω) Wj(ωTd) dω

Jitter Metric σj2 = S(ω) Wj(ωTj) dω

Stability Metric σs2 = S(ω) Ws(ωTs) dω

Windowed Stability Metric σsw2 = S(ω) Wsw(ωTd, ωTs) dω

where S(ω) is the power spectral density of the attitude error, and Td, Tj, and Ts are the window times introduced in the pointing error definitions. The weighting functions Wd, Wj, Ws, and Wsw are given by

Displacement Weighting Function Wd(ν) = 2(1 – cos ν)/ν2

Jitter Weighting Function Wj(ν) = 1 – 2(1 – cos ν)/ν2

Stability Weighting Function Ws(ν) = 2(1 – cos ν)

Windowed Stability Weighting Function Wsw1, ν2) = Ws1)Wd2)

The expressions for the pointing error metrics may appear to be difficult to evaluate, but a very simple computational method is described in the next section. In the case of a few sinusoidal disturbances, these metrics are particularly easy to evaluate. An example for the case of reaction wheel imbalance is given in the references (see the Further Reading page of this article).