than that of the UT estimate, suggesting that linearization has likely produced an underestimate of the actual error in its mean. In this example there is no way to determine the absolute accuracy of the UT and linearized estimates without ground truth in the form of the actual probability distribution associated with the original estimate and the mean and covariance of that distribution after application of the nonlinear transformation (e.g., as determined analytically or through numerical integration). Such analyses have been performed for coordinate transformations under the assumption of Gaussianity for the underlying distributions, and the UT estimates tend to be significantly more accurate than those obtained from linearization. Empirical analysis has shown that the use of the minimal simplex set of sigma points is significantly less accurate than the use of the symmetric set of points when the underlying distribution is Gaussian. This suggests that the use of the simplex set in the above example would not be the best choice if the underlying distribution associated with is symmetric. Even if the underlying distribution is not symmetric, the simplex set is still likely to be less accurate than the symmetric set because the asymmetry of the simplex set is not matched to the asymmetry of the actual distribution. Returning to the example, the minimal symmetric set of sigma points can be obtained from the covariance matrix simply as the mean vector, plus and minus the columns of : This construction guarantees that the mean and covariance of the above four sigma points is , which is directly verifiable. Applying the nonlinear function to each of the sigma points gives: The mean of these four transformed sigma points, , is the UT estimate of the mean in polar coordinates: The UT estimate of the covariance is: where the each squared term in the sum is a vector outer product. This gives: The difference between the UT and linearized mean estimates gives a measure of the effect of the nonlinearity of the transformation. When the transformation is linear, for instance, the UT and linearized estimates will be identical. This motivates the use of the square of this difference to be added to the UT covariance to guard against underestimating of the actual error in the mean. This approach does not improve the accuracy of the mean but can significantly improve the accuracy of a filter over time by reducing the likelihood