Annals of Biomedical Engineering

, Volume 39, Issue 7, pp 1925–1934 | Cite as

An Optimization-Based Simultaneous Approach to the Determination of Muscular, Ligamentous, and Joint Contact Forces Provides Insight into Musculoligamentous Interaction

  • Daniel J. CleatherEmail author
  • Anthony M. J. Bull


Typical inverse dynamics approaches to the calculation of muscle, ligament, and joint contact forces are based on a step-wise solution of the equations of motion. This approach is therefore limited in its ability to provide insight as to the muscular, ligamentous, and articular interactions that create joint stability. In this study, a new musculoskeletal model of the lower limb is described, in which the equations of motion describing the force and moment equilibrium at the joints of the lower limb are solved simultaneously using optimization techniques. The new model was employed to analyze vertical jumping using a variety of different optimization cost functions and the results were compared to more traditional approaches. The new model was able to find a solution with lower muscular force upper bounds due to the ability of the ligaments to contribute to moment equilibrium at the ankle and knee joints. Equally, the new model produced lower joint contact forces than traditional approaches for cases which also included a consideration as to ligament or joint contact forces within the cost function. This study demonstrates the possibility of solving the inverse dynamic equations of motion simultaneously using contemporary technology, and further suggests that this might be important due to the complementary function of the muscles and ligaments in creating joint stability.


Musculoskeletal modeling Muscle force Joint contact force Ligament force Inverse dynamics 

List of Symbols

\( \hat{a}_{i} \)

Linear acceleration of the center of mass

\( \hat{c}_{i} \)

Vector from the proximal joint to the segment COM

\( \hat{d}_{i} \)

Vector from the proximal to the distal joint


Cost function


Magnitude of force in muscle

\( \hat{g} \)

Acceleration due to gravity


Segment/joint number (numbering from distal to proximal)

\( \hat{I}_{i} \)

Inertia tensor


Muscle or ligament number

\( \hat{J}_{i} \)

Joint contact force at proximal end of segment

k1, k2, k3

Cost function coefficients


Magnitude of force in ligament

\( \hat{m}_{i} \)

Mass of segment

\( \hat{M}_{i} \)

Inter-segmental moment at proximal end of segment

n1, n2, n3

Cost function exponents

\( \hat{o}_{ij} \)

Line of action of biarticular muscle j about segment i

\( \hat{p}_{ij} \)

Line of action of muscle j about joint i

\( \hat{q}_{ij} \)

Line of action of ligament j about joint i

\( \hat{r}_{ij} \)

Moment arm of muscle j about joint i

\( \hat{S}_{i} \)

Inter-segmental force at proximal end of segment

\( \hat{s}_{ij} \)

Moment arm of ligament j about joint i


Total number of muscles


Total number of ligaments


Total number of joints

\( \dot{\hat{\theta }}_{i} \)

Angular velocity of segment

\( \ddot{\hat{\theta }}_{i} \)

Angular acceleration of segment


Conflict of interest

There are no conflicts of interest.


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Copyright information

© Biomedical Engineering Society 2011

Authors and Affiliations

  1. 1.St. Mary’s University CollegeLondonUK
  2. 2.Department of BioengineeringImperial College LondonLondonUK

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