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Multibody dynamic systems as Bayesian networks: Applications to robust state estimation of mechanisms

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Abstract

This article addresses the problem of robustly estimating the dynamic state of a mechanism from a set of noisy sensor measurements. We start with a rigorous treatment of the problem from the perspective of graphical models, a popular formalism in the fields of statistical inference and machine learning. The modeling power of such a formalism is demonstrated by showing how the sequential estimation of a mechanism state with an extended Kalman filter (EKF), often used in previous works, becomes just one of the possible solutions. As an interesting alternative, we derive the formulation of a sequential Monte Carlo (SMC) filter, also known as a particle filter (PF), suitable for online tracking the state of a rigid mechanism. We validate our ideas with both simulated and real datasets. Moreover, we prove the usefulness of the particle filtering solution for real-work applications due to its unmatched capability of automatically inferring the initial states of the mechanism along with its “assembly configuration” or “branch” if several ones are possible, a feature not matched by any previously proposed state observer in the multibody literature.

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Notes

  1. The three types are: (i) directed acyclic graphs (or Bayesian networks) [24], (ii) undirected graphs (Markov random fields) [25], and (iii) factor graphs [26].

  2. The rule for removing variables from a DBN determines that new edges must be created between the neighbors of removed nodes.

  3. From p(a|b)=p(a,b)/p(b) it follows that p(a,b)=p(a|b)p(b), which can be generalized to p(a,b|c)=p(a|b,c)p(b|c) via, for example, the multiplication rule for p(a,b,c).

  4. The Bayes rule establishes how the observation of b affects the prior knowledge regarding another variable p(a). It says that p(a|b)∝p(b|a)p(a), with p(b|a) customarily called the likelihood of b. In the text, we apply the rule extension for conditional probabilities, where all terms can be conditioned to another variable or variable set c, that is, p(a|b,c)∝p(b|a,c)p(a|c).

  5. This law states that any marginal distribution p(a) can be expressed as the summation or integration (for discrete or continuous domains, respectively) of the conditional p(a|b) for all possible values of b, that is, \(p(a)=\int_{-\infty}^{_{\infty}} p(a|b)p(b)\, db\). It can be trivially extended for conditioned probabilities, as required in our derivation, that is, \(p(a|c)=\int_{-\infty}^{_{\infty}} p(a|b,c)p(b|c)\,db\). For discrete variables, as in the case of the branch B, integrals are replaced by summations over all the potential values.

  6. For planar rotations in SO(2), weighted averaging can be performed by weighting the two-dimensional coordinates of points at each given angle over the unit circle and then projecting the average point back to the unit circle. For SO(3) rotations, several averaging methods exist depending on the desired matrix metric over the manifold [42].

  7. The continuous uniform distribution \(p(x)=\mathcal{U}(a,b)\) assigns the same probability density to every point x∈(a,b), whereas the discrete version \(P(x)=\mathcal{U}(S)\) models the case where each event in the set S has exactly the same probability of occurrence.

  8. See http://www.youtube.com/watch?v=7Zru0oiz36g.

  9. Hosted at https://github.com/jlblancoc/mbde.

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Acknowledgements

This work has been partially funded by the Spanish “Ministerio de Ciencia e Innovación” under the contract “DAVARBOT” (DPI 2011-22513), the grant program JDC-MICINN 2011, and the Andalusian Regional Government grant program FPDU 2009, cofinanced by the European Union through the European Regional Development Fund (ERDF).

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  1. University of Almería, Almeria, Spain

    J. L. Blanco-Claraco, J. L. Torres-Moreno & A. Giménez-Fernández

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  1. J. L. Blanco-Claraco
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Correspondence to J. L. Blanco-Claraco.

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Blanco-Claraco, J.L., Torres-Moreno, J.L. & Giménez-Fernández, A. Multibody dynamic systems as Bayesian networks: Applications to robust state estimation of mechanisms. Multibody Syst Dyn 34, 103–128 (2015). https://doi.org/10.1007/s11044-014-9440-9

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