then be computed as the estimate of the nonlinear transformation of the original distribution. More generally, the application of a given nonlinear transformation to a discrete distribution of points, computed so as to capture a set of known statistics of an unknown distribution, is referred to as an unscented transformation. In other words, the given mean and covariance information can be exactly encoded in a set of points, referred to as sigma points, which if treated as elements of a discrete probability distribution has mean and covariance equal to the given mean and covariance. This distribution can be propagated exactly by applying the nonlinear function to each point. The mean and covariance of the transformed set of points then represents the desired transformed estimate. The principal advantage of the approach is that the nonlinear function is fully exploited, as opposed to the EKF which replaces it with a linear one. Eliminating the need for linearization also provides advantages independent of any improvement in estimation quality. One immediate advantage is that the UT can be applied with any given function whereas linearization may not be possible for functions that are not differentiable. A practical advantage is that the UT can be easier to implement because it avoids the need to derive and implement a linearizing Jacobian matrix. Sigma points To compute the unscented transform, one first has to choose a set of sigma points. Since the seminal work of Uhlmann, many different sets of sigma points have been proposed in the literature. A thoroughgoing review of these variants can be found in the work of Menegaz et al. In general, sigma points are necessary and sufficient to define a discrete distribution having a given mean and covariance in dimensions. A canonical set of sigma points is the symmetric set originally proposed by Uhlmann. Consider the vertices of an equilateral triangle centered on origin in two dimensions: It can be verified that the above set of points has mean and covariance (the identity matrix). Given any 2-dimensional mean and covariance, , the desired sigma points can be obtained by multiplying each point by the matrix square root of and adding . A similar canonical set of sigma points can be generated in any number of dimensions by taking the zero vector and the points comprising the rows of the identity matrix, computing the mean of the set of points, subtracting the mean