Unscented Transformation The unscented transformation (UT) is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation . Consider propagating a random variable x (dimension L) through a nonlinear function, y β f (x). Assume x has mean x and covariance Px. To calculate the statistics of y, we form a matrix X of 2L + 1 sigma vectors X, according to the following:
X 0 β x;
Xi β x + (V(L + 1)Px)f, i β 1, ..., L, (7.30)
X i β x β (P(L + A)Px)iβL, i β L + 1, ..., 2L,
3While ββ second-orderββ versions of the EKF exist, their increased implementation and computational complexity tend to prohibit their use.
4A popular technique to improve the ββfirst-orderββ approach is the iterated EKF, which effectively iterates the EKF equations at the current time step by redefining the nominal state estimate and re-linearizing the measurement equations. It is capable of providing better performance than the basic EKF, especially in the case of significant nonlinearity in the measurement function . We have not performed a comparison to the UKF at this time, though a similar procedure may also be adapted to iterate the UKF.
7.3 THE UNSCENTED KALMAN FILTER 229
where l β a2(L + k) β L is a scaling parameter. The constant a determines the spread of the sigma points around x, and is usually set to a small positive value (e.g., 1 < a < 10β4). The constant k is a secondary scaling parameter, which is usually set to 3 β L (see  for details), and b is used to incorporate prior knowledge of tine; distribution of x (for Gaussian distributions, b β 2 is optimal). (y/(L + i)Px)( is the ith column of the matrix square root (e.g., lower-triangular Cholesky factorization). These sigma vectors are propagated through the nonlinear function
Yi β f(X) i β 0; ...; 2L; (7.31)
and the mean and covariance for y are approximated using a weighted sample mean and covariance of the posterior sigma points,
y Wfn)yi ;
Py ~Β£ Wi(e)(y. - y)(Y - y)
with weights Wt given by
W(m) _ 1
Wo L + 1 β
+ 1 β a2 + b
W(m) = W(c) -
i β 1 ; ... ; 2L.
A block diagram illustrating the steps in performing the UT is shown in Figure 7.2. Note that this method differs substantially from general Monte Carlo sampling methods, which require orders of magnitude more sample points in an attempt to propagate an accurate (possibly non-Gaussian) distribution of the state. The deceptively simple approach taken with the UT results in approximations that are accurate to the third order for Gaussian inputs for all nonlinearities. For non-Gaussian inputs, approximations are accurate to at least the second order, with the accuracy of third- and higher-order moments being determined by the choice of a and b. The proof of this is provided in Appendix A. Valuable insight into the UT can also be gained by relating it to a numerical technique called
230 7 THE UNSCENTED KALMAN FILTER
* y = yl(L+X)
fe}=[x x + yVP* x-y-v/P,]
Figure 7.2 Block diagram of the UT.
Gaussian quadrature numerical evaluation of integrals. Ito and Xiong  recently showed the relation between the UT and the Gauss-Hermite quadrature rule5 in the context of state estimation. A close similarity also exists between the UT and the central difference interpolation filtering (CDF) techniques developed separately by Ito and Xiong  and N0rgaard, Poulsen, and Ravn . In  van der Merwe and Wan show how the UKF and CDF can be unified in a general family of derivativefree Kalman filters for nonlinear estimation.
A simple example is shown in Figure 7.3 for a two-dimensional system: Figure 7.3a shows the true mean and covariance propagation using Monte Carlo sampling; Figure 7.3b shows the results using a linearization approach as would be done in the EKF; Figure 7.3c shows the performance of the UT (note that only five sigma points are required). The superior performance of the UT is clear.
5In the scalar case, the Gauss-Hermite rule is given by J1 f (x)(2n)_1=2e_x2 dx β 1 wf (x 'j), where the equality holds for all polynomials, f (β’), of degree up to 2m _ 1 and the quadrature points xi and weights wi are determined according to the rule type (see  for details). For higher dimensions, the Gauss-Hermite rule requires on the order of mL functional evaluations, where L is the dimension of the state. For the scalar case, the UT with a β 1, b β 0, and k β 2 coincides with the three-point Gauss-Hermite quadrature rule.