Abstract
In musculoskeletal simulations, it is standard practice to model muscles or strands wrapping on a surface as thin massless lines. When the strand has non-negligible and in particular non-constant cross section thickness, the assumptions of infinitely thin lines does not apply anymore, and current methods rely on discretizations of the strands as chains of spherical beads, which however are discontinuous and thus produce jerky motions which lead to unrealistic forces when coupled to dynamic muscle models. In this work, we present a novel continuous approach which solves the problem using smooth differential equations as the limit of infinitesimally close beads. The present contribution is an extension of our previous work in which the equations were first derived for the case of a planar conical piece of strand lying on a given arc of a surface. The paper further develops these equations for the case of varying strand thickness and prescribed motion of the free ends under the constraint of constant strand length. The results are compared with the bead method, showing their superiority both in terms of smoothness and computational efficiency. Based on this approach, the 3D and stretchable-strand case can be tackled.
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Notes
- 1.
This equation results from the limit of pair-wise velocity transmission between beads as firstly derived in [5]. If \(\rho _{i}\) is the surface curvature at bead “i”, \(r_i\) is its radius, and \(\varDelta \rho _{i}\) is the difference of surface curvature from bead “i” to bead “i+1”, then the total velocity transmission from first to last bead is the product of these pair-wise transmissions. However, in [5], for the limit of infinitesimally close beads, in [5] we erroneously neglected higher-order terms of infinitesimal quantities. This is wrong and the correct limit condition is
$$\begin{aligned} \lim \limits _{\varDelta \rho _{i} \rightarrow 0}{ \left( 1 + \eta _1 \, \varDelta \rho _{1} \right) \ldots \left( 1 + \eta _{N-1} \, \varDelta \rho _{N-1} \right) } = \lim \limits _{\varDelta \rho _{i} \rightarrow 0} \exp \left( {\sum _{i=1}^{N-1} \eta _i \, \varDelta \rho _{i}}\right) \quad \text {with}\ \eta _i = \frac{r_{i}}{\rho _{i} \left( \rho _{i} + r_{i} \right) }. \end{aligned}$$which for \(\varDelta \rho \rightarrow \mathrm{d}\rho = (\partial \rho / \partial \sigma )\, \mathrm{d}\sigma = \rho ' \, \mathrm{d}\sigma \) yields the integral form. Unfortunately we cannot include the proof here due to lack of space, so we refer here to this equation as a correction of [5].
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Müller, K., Kecskemethy, A. (2021). A New Approach for Continuous Wrapping of a Thick Strand on a Surface — The Planar Case with Constant Length and Free Ends. In: Lenarčič, J., Siciliano, B. (eds) Advances in Robot Kinematics 2020. ARK 2020. Springer Proceedings in Advanced Robotics, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-50975-0_36
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