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Research Papers

Variational Integrators for Structure-Preserving Filtering

[+] Author and Article Information
Jarvis Schultz

Mem. ASME
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: jschultz@northwestern.edu

Kathrin Flaßkamp

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: kathrin.flasskamp@northwestern.edu

Todd D. Murphey

Mem. ASME
Professor
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: t-murphey@northwestern.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 27, 2015; final manuscript received September 1, 2016; published online December 2, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 12(2), 021005 (Dec 02, 2016) (10 pages) Paper No: CND-15-1347; doi: 10.1115/1.4034728 History: Received October 27, 2015; Revised September 01, 2016

Estimation and filtering are important tasks in most modern control systems. These methods rely on accurate discrete-time approximations of the system dynamics. We present filtering algorithms that are based on discrete mechanics techniques (variational integrators), which are known to preserve system structures (momentum, symplecticity, and constraints, for instance) and have stable long-term energy behavior. These filtering methods show increased performance in simulations and experiments on a real digital control system. The particle filter as well as the extended Kalman filter benefits from the statistics-preserving properties of a variational integrator discretization, especially in low bandwidth applications. Moreover, it is shown how the optimality of the Kalman filter can be preserved through discretization by means of modified discrete-time Riccati equations for the covariance updates. This leads to further improvement in filter accuracy, even in a simple test example.

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Figures

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Fig. 1

Approximations of the stochastic harmonic oscillator flow for a set of initial conditions. All integrators use the same step-size h = 0.03125 and an identical sample path of the Wiener process with σ=γ=1.0. The results from the Euler-Maruyama integrator are depicted in blue with circular markers, the VI midpoint in red with diamond markers, and the symplectic Euler in green with square markers. The analytic solution from Ref. [10] is shown in black with no markers for comparison. It is difficult to distinguish the VI midpoint, the symplectic Euler, and the analytic solution as they lie on top of each other (see online figure for color).

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Fig. 2

Image of the particle filter covariance propagation for the harmonic oscillator without resampling. Note that the system's covariance matrix has two eigenvalues, thus there are two eigenvalues plotted for each integrator.

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Fig. 3

Schematic of planar crane system including relevant geometric parameters

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Fig. 4

Plot illustrating variations in EKF filter performance using two different discrete representations of the continuous dynamics. Both curves are simulated from 1000 trials, with Gaussian noise added to produce “measurements”.

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Fig. 5

This set of figures illustrates the results of utilizing a particle filter (PF) and an EKF to estimate the dynamic configuration variables of the system shown in Fig. 3. For each filter, both VI and RK1 integrators are used. In the left plots, measurements, controls, and integrations happen at 30 Hz, while in the right plots they all occur at 6 Hz. To generate the data for a given frequency, an experimental system was sent a set of commands at the desired frequency; the same commands were used to step the integrators for predictions. A Microsoft Kinect® configured to provide data at the target frequency was used to measure both of the dynamic configuration variables (x, y). For the 30 Hz data, only some of the measurements/estimator updates are shown to avoid overcrowding the figure. The particle filters used 1000 particles and the “low variance sampler” algorithm from page 110 of [3] is used to resample. The ellipses shown represent the local covariance estimates for each filter. The eigenvalues and eigenvectors of the covariance are used to define the size and orientation of the ellipses. Note that the VI-based filters outperform the RK1-based filters at both frequencies, and that the VI filter performance is similar for both frequencies. Additionally, note that the VI covariance estimates are in excellent agreement between the filters at both frequencies; this is not true for the RK1 filters.

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Fig. 6

This figure shows the time evolution of the eigenvalues of the covariance prediction for particle filters (PF) and EKFs applied to the planar crane problem of Fig. 3 with both a 30 Hz (left plots) and a 6 Hz (right plots) measurement, estimator and integrator update rate. These eigenvalues are the same eigenvalues used to define the major and minor axes of the uncertainty ellipses plotted in Fig. 5. Thus, there are two eigenvalues for each filter and integrator. Note that the VI-based PF and EKF estimates are in excellent agreement with each other even when compared across frequencies. The two 6 Hz RK1 filters predict very different covariance propagation, and they both disagree with the corresponding 30 Hz filters.

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Fig. 7

Illustration of the VI improving particle deprivation. The red circles represent the 6 Hz VI particle filter, and the noisy, dashed blue line is the 6 Hz RK1 particle filter.

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Fig. 8

Comparison of the Kalman filter variants for the harmonic oscillator. Analytic deterministic solution is given in black, standard discrete covariance updates with explicit Euler state updates in blue, VI midpoint state updates with standard discrete covariance updates in green, and symplectic Euler state update with symplectic covariance update in red. The top left figure shows the filtered states and some covariance ellipses in the phase plane. The top right figure is a zoomed-in view of the data in the top left figure. The symplectic covariance updates increase the filter performance. The bottom left figure shows the time evolution of the eigenvalues of the covariance matrices (as in Fig. 2 there are two eigenvalues per filter). The symplectic method leads to eigenvalues that are an order of magnitude smaller. The bottom right figure shows the mechanical energy of the oscillator's filtered states for each of the filter variants; the symplectic covariance update yields a better approximation that the two other filters with standard covariance updates.

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