Multibody System Dynamics

, Volume 36, Issue 2, pp 195–219 | Cite as

A fast multi-obstacle muscle wrapping method using natural geodesic variations

  • Andreas ScholzEmail author
  • Michael Sherman
  • Ian Stavness
  • Scott Delp
  • Andrés Kecskeméthy


Musculoskeletal simulation has become an essential tool for understanding human locomotion and movement disorders. Muscle-actuated simulations require methods that continuously compute musculotendon paths, their lengths, and their rates of length change to determine muscle forces, moment arms, and the resulting body and joint loads. Musculotendon paths are often modeled as locally length minimizing curves that wrap frictionlessly over moving obstacle surfaces representing bone and tissue. Biologically accurate wrapping surfaces are complex, and a single muscle path may wrap around many obstacles. However, state-of-the-art muscle wrapping methods are either limited to analytical results for a pair of simple surfaces, or they are computationally expensive. In this paper, we introduce the Natural Geodesic Variation (NGV) method for the fast and accurate computation of a musculotendon’s shortest path across an arbitrary number of general smooth wrapping surfaces, and an explicit formula for the path’s exact rate of length change. The total path is regarded as a concatenation of straight-line segments between local surface geodesics, where each geodesic is naturally parameterized by its starting point, direction, and length. The shortest path is computed by finding the root of a global path-error constraint equation that enforces that the geodesics connect collinearly with adjacent straight-line segments. High computational speed is achieved using Newton’s method to zero the path error, with an explicit, banded Jacobian that maps natural variations of the geodesic parameters to path-error variations. Three simulation benchmarks demonstrate that the NGV method computes high-precision solutions for path length and rate of length change, allows for wrapping over biologically accurate surfaces, and is capable of simulating muscle paths over hundreds of surfaces in real time. We thus believe the NGV method will facilitate the development of more accurate yet very efficient musculoskeletal models.


Muscle wrapping Musculotendon path Shortest path Geodesic Geodesic variation Jacobi field 



The authors gratefully acknowledge the support of Leonidas Guibas, Adrian Butscher, and Justin Solomon when discussing Jacobi fields; Matthew Millard for his valuable feedback on the method, the manuscript and the figures; and Francisco Geu Flores for his support in benchmarking.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Andreas Scholz
    • 1
    Email author
  • Michael Sherman
    • 2
  • Ian Stavness
    • 3
  • Scott Delp
    • 4
  • Andrés Kecskeméthy
    • 5
  1. 1.Department of Mechanical EngineeringUniversity of Duisburg-EssenDuisburgGermany
  2. 2.Department of BioengineeringStanford UniversityStanfordUSA
  3. 3.Department of Computer ScienceUniversity of SaskatchewanSaskatoonCanada
  4. 4.Departments of Bioengineering and Mechanical EngineeringStanford UniversityStanfordUSA
  5. 5.Chair of Mechanics and RoboticsUniversity of Duisburg-EssenDuisburgGermany

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