UNSCENTED KALMAN FILTER UNSCENTED KALMAN FILTER MATTHIEU BLOCH - PowerPoint PPT Presentation

UNSCENTED KALMAN FILTER UNSCENTED KALMAN FILTER MATTHIEU BLOCH April 21, 2020 1 / 9

RECAP: EXTENDED KALMAN FILTER (EKF) RECAP: EXTENDED KALMAN FILTER (EKF) EKF can be applied to non-linear models of the form EKF is based on Taylor series expansions of and around current state estimate Advantages: simple, intuitive, computationally efficient Disadvantages: local approximation, differentiability requirements, only for Gaussian noises Linearization assumes that all second and higher order terms in Taylor series expansion are negligible In some situations, higher order terms have negligible effects In others, they can significantly degrade estimator performance. Possible to extend EKF approach to higher order terms Second-order Gaussian filter Assumes piece-wise quadratic model and truncates Taylor series expansion a�er second term Hessian required Need for a method that is: 1. Provably more accurate than lineariza-tion; 2. Does not incur computational costs of other higher order filtering schemes 2 / 9

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UNSCENTED KALMAN FILTER UNSCENTED KALMAN FILTER Key observation: Easier to approximate a pdf than an arbitrary nonlinear transformation From Julier and Uhlmann, Proc. IEEE, 2004 Unscented transform Choose a set of points ( sigma points) Nonlinear function is applied to each point to yield a cloud of transformed points statistics of transformed points calculated to form an estimate of the nonlinearly transformed mean and covariance 4 / 9

UNSCENTED KALMAN FILTER UNSCENTED KALMAN FILTER Superficial resemblance to particle filters Sigma points are not drawn at random High-order information about the distribution captured with a fixed small number of points Sigma points can be weighted in ways that are inconsistent with the distribution interpretation of sample points in a particle filter. General Sigma-point selection framework No practical filter can maintain the full distribution of the state Simpler distribution of the form may be heuristically assumed, e.g., Gaussian Goal is to match different properties of a given distribution, e.g., moments In general, find sigma points that solve 5 / 9

UNSCENTED KALMAN FILTER UNSCENTED KALMAN FILTER Matching mean and covariance of a Gaussian Solution: set of points that lie on the covariance contour , For and For and with -th column Propagate sigma points through non-linear transform 6 / 9

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