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Investigation of a method for the three-dimensional rigid body dynamics inverse problem

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Abstract

The three-dimensional rigid body dynamics inverse problem is cast as a discrete optimal control problem and solved using dynamic programming. The optimal control problem uses a forward dynamic model to provide estimates of unknown forces while matching noisy measurement histories. An L curve analysis is used to objectively select the amount of smoothing by trading off the magnitude of the estimated unknown forces with the fit to the noisy measurements. The forward dynamic model is derived using finite-element methodology and accounts for the inertia and mass properties of rigid bodies with the use of natural coordinates. The advantages of this forward model are that the large displacements’ nonlinearities can be routinely represented in the inverse problem and that it also allows the use of an exact energy conserving method to numerically integrate the equations of motion. A numerical example of a large displacement three body model is included to demonstrate the performance of the methodology.

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Authors and Affiliations

  1. TRUCOMP, Fountain Valley, CA, USA

    David M. Trujillo

  2. The Ohio State University, Columbus, OH, USA

    Henry R. Busby

Authors
  1. David M. Trujillo
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  2. Henry R. Busby
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Correspondence to David M. Trujillo.

Appendix: Inertia matrix C

Appendix: Inertia matrix C

The matrix C (equation (1)) can be expressed in terms of the I body matrix:

$$\mathbf{I}_{\mathrm{BODY}} = \left[ \begin{array}{c@{\quad}c@{\quad}c}I_{11} & I_{12} & I_{13} \\& I_{22} & I_{23} \\\mathit{Sym} & & I_{33} \\\end{array} \right]$$

Let

The 9×9 C matrix is then

$$\mathbf{C} = \left[ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}a & 0 & 0 & - I_{12} & 0 & 0 & - I_{13}& 0 & 0 \\ & a & 0 & 0 & - I_{12} & 0 & 0 & - I_{13} & 0 \\ & & a & 0 &0 & - I_{12} & 0 & 0 & - I_{13} \\ & & & b & 0 & 0 & - I_{23} & 0 & 0 \\& & & & b & 0 & 0 & - I_{23} & 0 \\ & & & & & b & 0 & 0 & - I_{23} \\ && \mathit{Sym} & & & & c & 0 & 0 \\ & & & & & & & c & 0 \\ & & & & & & & & c\end{array} \right]$$

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Trujillo, D.M., Busby, H.R. Investigation of a method for the three-dimensional rigid body dynamics inverse problem. Multibody Syst Dyn 27, 423–435 (2012). https://doi.org/10.1007/s11044-011-9290-7

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