Uncertainty Mapping
Mathematical methods for determining the uncertainty of a calculated result involve mapping the individual variances of input variables through a functional relationship. Variance propagation allows engineers to estimate the total error in a system by combining the noise profiles of every sensor in the chain. The process is fundamental for establishing the confidence intervals of navigation and control solutions.
Jacobian Matrix
Calculating the sensitivity of the output to each input requires the derivation of partial derivatives. In a complex system, the variance propagation uses a jacobian matrix to transform the input covariance into the output space. The linear approximation works well when the underlying functions are relatively smooth and the input errors are small.
If the system is highly non-linear, higher order terms or Monte Carlo simulations may be needed to achieve the required accuracy. The resulting output variance provides a metric for the precision of the entire measurement process.
Error Chain
Components from the sensing element to the final software output contribute to the total noise. Identifying the largest contributor to variance propagation allows designers to focus their improvement efforts on the most impactful stage of the hardware.
Linearity Limit
Valid results depend on the assumption that the transformation remains linear within the range of the input errors. When this condition fails, the propagated variance may substantially underestimate the actual uncertainty of the final result.
