For the state estimation of nonlinear systems, shortly after Rudolf Kalman’s original publication, the Extended Kalman Filter (EKF) was developed to extend the Kalman filter to nonlinear systems. In the EKF, the nonlinear system is linearized around the current state estimate, which requires analytical derivation of the system equations. For many years, this approach was the standard solution for nonlinear systems, which are common in practice.

However, a major drawback of the EKF is its susceptibility to divergence. Since the linearization is performed around the estimated state rather than the true state, any estimation error causes the linearization to occur at an incorrect point. In systems with strong nonlinearities, these errors can compound quickly, leading to rapid divergence of the filter.

A breakthrough came with the seminal work of Simon J. Julier and Jeffrey K. Uhlmann [1], who proposed a fundamentally different approach to nonlinear estimation. Instead of linearizing the nonlinear functions—which are typically known exactly—they suggested selecting a set of deterministically chosen sample points (called sigma points) that capture the mean and covariance of the state distribution. These points are propagated through the nonlinear functions, and the resulting transformed points are then recombined to approximate the new mean and covariance.

This method became known as the Unscented Transformation, and the corresponding filter—built upon it—became famous as the Unscented Kalman Filter (UKF).

Unscented Transformation

It results in the following calculation rules for the sigma points. For \( i=0,1,…,n_x \) with \( \text{dim}( \mathbf{x} ) = n_x \) the sigma points are

\[ \mathcal{X}^{(0)}(k-1) = \mathbf{x}(k-1|k-1) ,\]

\[ \mathcal{X}^{(i)}(k-1) = \mathbf{x}(k-1|k-1) + \sqrt{\frac{n_x}{1 - W_0}} \bigl(\mathbf{P}^{\frac{1}{2}}(k-1|k-1)\bigl)_i , \]

and

\[ \mathcal{X}^{(i+n_x)}(k-1) = \mathbf{x}(k-1|k-1) - \sqrt{\frac{n_x}{1 - W_0}} \bigl(\mathbf{P}^{\frac{1}{2}}(k-1|k-1)\bigl)_i \]

where the parenthesis \( \bigl( \cdot \bigl)_i \) indicates that the i-column of the matrix is to be used. The corresponding weight is computed with

\[ W_i = \frac{1-W_0}{2n_x} .\]

Subsequently, the sigma points \( \mathcal{X} \) are used to be sent through the (nonlinear) function \( \mathbf{f} \) and by means of the weights \( W_i \) to calculate the first central moment

\[ \mathbf{x}(k|k-1) = \sum^{2n_x}_{i=1} \mathbf{f}(\mathcal{X}^{(i)}(k-1)) W_i .\]

The predicted covariance matrix of the predicted state results insignificantly different with additive addition of the process noise

\[ \mathbf{P}(k|k-1) = \sum^{2n_x}_{i=1} \bigl( \mathbf{f}(\mathcal{X}^{(i)}(k-1))-\mathcal{x}(k|k-1) \bigl) \bigl( \mathbf{f}(\mathcal{X}^{(i)}(k-1))-\mathcal{x}(k|k-1) \bigl)^T W_i + \mathbf{Q}_d(k-1) .\]

For the correction step, another set of sigma points must be calculated from the predicted state \( \mathbf{x}(k|k-1) \) and the predicted covariance \( \mathbf{P}(k|k-1) \),

\[ \mathcal{X}^{(0)}(k) = \mathbf{x}(k|k-1) ,\]

\[ \mathcal{X}^{(i)}(k) = \mathbf{x}(k|k-1) + \sqrt{\frac{n_x}{1-W_0}} \bigl(\mathbf{P}^{\frac{1}{2}}(k|k-1)\bigl)_i \]

and

\[ \mathcal{X}^{(i+n_x)}(k) = \mathbf{x}(k|k-1) - \sqrt{\frac{n_x}{1-W_0}} \bigl(\mathbf{P}^{\frac{1}{2}}(k|k-1)\bigl)_i .\]

To perform the correction step, the measurement vector must be predicted

\[ \mathbf{y}(k|k-1) = \sum^{2n_x}_{i=1} \mathbf{h}(\mathcal{X}^{(i)}(k)) W_i .\]

With the help of the intermediate variables, cross covariance

\[ \mathbf{P}_{xy}(k|k-1) = \sum^{2n_x}_{i=1} \bigl( \mathcal{X}^{(i)}(k) - \mathbf{x}(k|k-1) \bigl) \bigl( \mathbf{h}(\mathcal{X}^{(i)}(k)) - \mathbf{y}(k|k-1) \bigl)^T W_i \]

and innovation

\[ \mathbf{S}(k|k-1) = \sum^{2n_x}_{i=1} W_i \bigl( \mathbf{h}(\mathcal{X}^{(i)}(k))- \mathbf{y}(k|k-1) \bigl) \bigl( \mathbf{h}(\mathcal{X}^{(i)}(k))- \mathbf{y}(k|k-1) \bigl)^T + \mathbf{R}(k) \]

the state

\[ \mathbf{x}(k|k) = \mathbf{x}(k|k-1) + \mathbf{P}_{xy}(k|k-1) \mathbf{S}(k|k-1)^{-1} (\mathbf{y}(k) - \mathbf{y}(k|k-1) ) \]

and covariance

\[ \mathbf{P}(k) = \mathbf{P}(k|k-1) - \mathbf{P}_{xy}(k|k-1) \mathbf{S}(k|k-1)^{-1} \mathbf{P}_{xy}(k|k-1)^T \]

can be updated.

Conclusion

The Unscented Transformation is fascinating and is a fundamental contribution to the use of deterministic methods for state estimation of nonlinear systems. A major disadvantage of the UKF that there is a tuning parameter with W.

References

[1] Julier, Simon J.; Uhlmann, Jeffrey K. (1997). “New extension of the Kalman filter to nonlinear systems” (PDF). In Kadar, Ivan (ed.). Signal Processing, Sensor Fusion, and Target Recognition VI. Proceedings of SPIE. Vol. 3. pp. 182–193.

[2] Chandra, Kumar Pakki & Gu, Da-Wei. (2019). Nonlinear Filtering: Methods and Applications. 10.1007/978-3-030-01797-2.

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