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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 timedomain equations for the pointing error metrics are not used as a computational tool because they are computationally burdensome, and because the timedomain 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 timedomain and frequencydomain expressions are, mathematically, exactly equivalent.
In the frequency domain, the pointing error metrics are given by the following expressions,
Accuracy Metric σ_{a}^{2} = ∫ S(ω) dω
Displacement Metric σ_{j}^{2} = ∫ S(ω) W_{j}(ωT_{d}) dω
Jitter Metric σ_{j}^{2} = ∫ S(ω) W_{j}(ωT_{j}) dω
Stability Metric σ_{s}^{2} = ∫ S(ω) W_{s}(ωT_{s}) dω
Windowed Stability Metric σ_{sw}^{2} = ∫ S(ω) W_{sw}(ωT_{d}, ωT_{s}) dω
where S(ω) is the power spectral density of the attitude error, and T_{d}, T_{j}, and T_{s} are the window times introduced in the pointing error definitions. The weighting functions W_{d}, W_{j}, W_{s}, and W_{sw} are given by
Displacement Weighting Function W_{d}(ν) = 2(1 – cos ν)/ν^{2}
Jitter Weighting Function W_{j}(ν) = 1 – 2(1 – cos ν)/ν^{2}
Stability Weighting Function W_{s}(ν) = 2(1 – cos ν)
Windowed Stability Weighting Function W_{sw}(ν_{1}, ν_{2}) = W_{s}(ν_{1})W_{d}(ν_{2})
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).
