Covariance Analysis
Analytical uncertainty covariance matrices for both points and instrument base locations can be generated as *.csv file output.
Methods employed are those as described in the paper “On the Representation and Estimation of Spatial Uncertainty” by Randall C. Smith and Peter Cheeseman published in “The International Journal of Robotics Research”, Vol. 5, No. 4, Winter 1986.
x^2 is the Chi-Squared error
M = number of observations (number of rows in a matrix)
n = degrees of freedom (number of variables to solve – number of columns in a matrix)
nominally, (M – n) = x^2 when expected error is consistent with observed error.
Uncertainty covariance report includes:
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Composite point uncertainty covariance matrix data relative to working frame
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Instrument base uncertainty covariance matrix data relative to world frame
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Chi-square “Goodness of Fit” assessment (values << 1 indicate data too good to be true, values >> 1 indicate measurement errors much greater than expected)
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Robustness: Solution solver SVD ((min(singular values) / max(singular values)) -- negative values reflect the input data does not provide enough information to resolve all of the degrees of freedom available to the model. Note that model degrees of freedom does not mean a one-to-one mapping to solution parameters. Eigen vectors are a composition of solution parameters similar to the I, J, K vectors of a normal vector. This only defines a single degree of freedom, but it requires input for the displacement along each of the coordinate system XYZ axes.
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Singular value tolerance threshold (minimum value an singular value is permitted to assume to be considered non-zero)
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Number of input equations
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Number of ignored equations
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Number of active equations
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Number of variables
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Number of independent variables
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Working frame
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Composite instrument base parameter covariance matrix
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Composite instrument base parameter correlation matrix
The input points used to develop a composite point are used to generate an error covariance matrix. This data may be helpful in terms of evaluating the data spread of the input points about the composite point. This data will not be generated for any composite point with less than three input points.
R-Click on “CoVar” button will allow the user to experiment with different values of singular value tolerances (default is 0.01).