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For that reason, implementations are often restricted to first-order methods. While this paper does not explicitly consider results related to estimation for constrained systems, we note that an additional interesting feature of the midpoint VI integrator is that it exactly satisfies holonomic constraints at each time-step while still admitting the explicit, structured linearization. This section presents experimental and simulated examples demonstrating how the choice of discretization scheme may dramatically affect the performance of standard particle filters and extended Kalman filters.
Further, we demonstrate improved performance of both filters when leveraging the structure-preserving properties of variational integrators. In Secs. This system and corresponding modeling strategy have been discussed by the authors in Refs. In those works, a modeling strategy was presented where the position of the winch system x r and the length of the string r were treated as kinematic inputs, i. With this modeling strategy, the Lagrangian for the system is only a function of the dynamic configuration variables x , y and is given by.
As a result of the kinematic input assumption, all damping in the crane portion of the system is assumed to be negligible. Moreover, through experimentation, we found that the damping in the swing dynamics of the payload was low enough to also be considered negligible. Thus, the deterministic models of the dynamics of this system have no damping.
For all results in this section, the noise models are assumed to be additive Gaussian noise with zero mean. The covariances parameterizing these distributions are all assumed to be diagonal, and the experimental system only measures relevant configuration variables, i. Figure 4 shows simulated effects of the choice of integration scheme on performance of the EKF algorithm at a range of time-steps. Each point on the plot was generated by running trials of the filter on a nominal feasible trajectory for the system described in Fig. Measurements were simulated by adding Gaussian noise to the nominal trajectory.
For each trial, the L 2 error between the filtered signal and the reference signal was determined at each time-step.
The error norms were then averaged, and their standard deviations calculated to produce the points and error bars in Fig. For the upper, solid curve, the local linearization was performed by evaluating the infinitesimal derivatives of the continuous dynamics about the current best estimate, and then using an explicit Euler approximation to convert this into a discrete linearization.
For the bottom curve, the structured linearization of the midpoint variational integrator discussed in Sec. It is easily seen that as the filter frequency decreases, the performance of the discrete-mechanics-based variational integrator representation significantly outperforms the explicit Euler approximation. In this section, we use experimental data from the robotic planar crane system described in Refs. This system utilizes digital encoders to close control loops around x r and r. Thus, the kinematic-input modeling strategy, mentioned in Sec. Since these inputs are assumed to be perfectly controlled, the dynamic configuration variables x , y are the only state variables estimated.
Figure 5 shows the parametric evolution of estimates of the dynamic configuration variables using particle filters and EKFs at two different frequencies using both a midpoint VI and an RK1 integrator to represent the system. The strength of the VI integrator can be seen by noting that the particle filter and the EKF estimates of the system's uncertainty are nearly identical even at frequencies as low as 6 Hz. The two filter estimates are not only in agreement with each other, but they are also in agreement between the frequencies.
With this large time-step, the RK1 particle filter is useless. Figure 6 shows the time evolution of the eigenvalues of the covariances from each of the estimators in Fig. This figure further demonstrates that the VI covariance predictions are stable even at large time-steps. Additionally, it demonstrates that, in this particular system, the structured linearization allows the EKF to perform nearly identically to the particle filter.
In this case, the increased accuracy of the structured linearization avoids the increased computational expense of the particle filter while achieving similar performance. It was explained that the VI reduces the likelihood of this issue by removing artificial noise injected by a traditional integrator; this is especially true at low frequencies. In several cases, the particle distribution from the RK1 integrator is artificially spread to the point that resampling produces only two unique particles out of which implies that uncertainty estimates from the filter are essentially useless.
However, the Kalman filter provides further structure due to its optimality property. In this section, we derive a modification to the previously used Kalman filter that preserves this optimality structure through discretization and thereby further improves the filter performance. To this aim, we start with the continuous-time Kalman filter, as introduced in Ref. The continuous-time Kalman filtering problem is dual to a linear quadratic regulator LQR problem: While a linear quadratic regulator controller minimizes a quadratic cost functional, the Kalman filter determines state estimates that minimize the expected squared error to the process' true state cf.
In particular, both problems exhibit an additional Hamiltonian structure independent of the mechanical Hamiltonian structure. This can be seen by examining the state-adjoint system that belongs to the respective problem's optimal solutions. Our approach aims at additionally preserving this property through numerical discretization.
Variational Methods in Theoretical Mechanics
Therefore, we set up a symplectic discretization of the state-adjoint system for the Kalman filtering problem. It is not necessary to explicitly solve for the adjoint equations, though. Instead, we directly derive modified discrete-time Riccati equations which are independent of the states and adjoints as previously presented in Ref. Thus, our Kalman filter algorithm only consists of a single covariance update step and an all-in-one state update afterward see Algorithm 6. The performance of this algorithm is studied in Example 6. L emma 6. So we have a linear Hamiltonian system, for which the flow is defined by the transition matrix.
In our discrete-time approximation of the Riccati equation for the covariance matrix, we derive discrete-time covariance updates that exactly preserve the following two properties: the symplectic structure of the state-adjoint differential equations and their linear relationship defined by P t. As in the continuous-time Kalman filter, the update equation for the covariance matrix P k does not depend on the state or on the adjoint. If the system and measurement model matrices are time-varying though, this step would have to be performed in every iteration.
For estimation problems of mechanical systems, it is desirable to also preserve the mechanical structure i. Therefore, we combine the structure-preserving covariance update of Theorem 6. This leads to the following algorithm that resembles the continuous-time filter see Proposition 6. Proposition 4. Algorithm 6. For the implementation of the structure-preserving Kalman filter with symplectic state and covariance updates using symplectic Euler integration, we modify the continuous-time Kalman filter algorithm of Ref.
Step 4 is again an implicit, but linear update, so it can be analytically inverted beforehand and then directly evaluated during filtering. While Algorithm 6. In general, the covariance update from Algorithm 6.
The previous results specifically use a first-order symplectic scheme for two reasons: We want to give practitioners a good idea how to implement the filter and we want to have a fair comparison to the standard approach that is based on nonsymplectic, first-order, explicit Euler discretizations see Example 6. However, if higher-order symplectic discretization schemes, e. Example 6. We revisit the example of a stochastic harmonic oscillator as introduced in Example 2. The structure-preserving Kalman filter method as presented in Algorithm 6.
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Figure 8 shows the resulting filter behavior in state space top row , as well as the covariance matrix eigenvalues bottom left and the mechanical energy of the oscillator bottom right. Unlike the examples presented in Sec. However, the novel variant of a structure-preserving filter has smaller covariances, as can be seen from the ellipses in the top plot and the covariance eigenvalues.
In particular, the mechanical energy of the filtered state, which would ideally—in the absence of noise and without numerical errors—be a constant, is much more accurately preserved than in the other two cases. This paper presents three structure-preserving filtering methods for mechanical control systems: 1 Particle filter with symplectic state predictions, 2 Extended Kalman filter with symplectic state predictions and structured linearizations of variational integrators, and 3 Kalman filter with symplectic state update and additional symplectic covariance update.
The filtering techniques have been tested in simulated examples and in a real-time experiment with a planar crane system. Symplectic discrete-time predictions of the states increased the performance of both the particle and the extended Kalman filter compared to the standard choice of explicit Euler updates. Replacing the covariance updates in the linear Kalman filter can improve its performance even further.
This novel filter variant can be extended to nonlinear systems by linearizing the discretized state-adjoint system of the optimality conditions. The one-step map of the symplectic state-adjoint scheme, which is used for the covariance Riccati equations, will then become time dependent.
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Thus, one would expect an increased computational effort. However, this will be balanced by the observed higher accuracy of the method which allows a coarser time-grid than the nonsymplectic first-order methods. Further theoretical studies of the symplectic filtering approach can be performed by backwards error analysis [ 7 ] applied to the state-adjoint system. We also note that recent work [ 18 ] has shown that it may be possible to construct arbitrarily high-order variational integrators that are still linearizable one-step maps.
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These integrators are developed through backward error analysis [ 7 ] by constructing surrogate Lagrangians. High-order VI-based filtering techniques are another area for future investigation. In many real-world applications, the filtering methods presented herein would be combined with optimal control techniques.
First results of a combination with the presented filters are given in Ref. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author s and do not necessarily reflect the views of the National Science Foundation. In the conclusion of this document we mention some recent work [ 18 ] that describes the generation arbitrary-order VIs that are still one-step maps.
These VIs may provide another choice of integrator that satisfies the aforementioned requirements while also yielding a higher order method. In general, a map is symplectic if it preserves a symplectic form [ 6 , 7 ]. To get an intuition for symplecticity, it is helpful to consider the fact that symplectic transformations are area-preserving. The slot derivative D n L A 1 , A 2 , … represents the derivative of the function L with respect to the nth argument, An.
At times, the arguments to the function L may be dropped for compactness — in that case the slot derivative still applies to the argument order from the function definition. By statistics-preserving, we mean that these symplectic integrators are capable of both accurately predicting statistical quantities such as temperature [ 19 ] and of almost surely preserving symmetries of stochastic systems [ 11 ]. Note that the states of the Hamiltonian system do not coincide with the filter state. Since the Hamiltonian structure of the mechanical system matrix A suggests a splitting of the state and adjoint variables, though, we choose a symplectic Euler that treats half of the states and adjoints as implicit and the other half as explicit.
However, we point out that other splittings between the set of coordinates define a first-order symplectic integration scheme, as well. Approximations of the stochastic harmonic oscillator flow for a set of initial conditions. 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.