Optimization and Engineering

, 9:213 | Cite as

Dynamic simulation of human motion: numerically efficient inclusion of muscle physiology by convex optimization

  • Goele Pipeleers
  • Bram Demeulenaere
  • Ilse Jonkers
  • Pieter Spaepen
  • Georges Van der Perre
  • Arthur Spaepen
  • Jan Swevers
  • Joris De Schutter


Determining the muscle forces that underlie some experimentally observed human motion, is a challenging biomechanical problem, both from an experimental and a computational point of view. No non-invasive method is currently available for experimentally measuring muscle forces. The alternative of computing them from the observed motion is complicated by the inherent overactuation of the human body: it has many more muscles than strictly needed for driving all the degrees of freedom of the skeleton. As a result, the skeleton’s equations of motion do not suffice to determine the muscle forces unambiguously. Therefore, muscle force determination is often reformulated as a (large-scale) optimization problem.

Generally, the optimization approaches are classified according to the formalism, inverse or forward, adopted for solving the skeleton’s equations of motion. Classical inverse approaches are fast but do not take into account the constraints imposed by muscle physiology. Classical forward approaches, on the other hand, do take the muscle physiology into account but are extremely costly from a computational point of view.

The present paper makes a double contribution. First, it proposes a novel inverse approach that results from including muscle physiology (both activation and contraction dynamics) in the inverse dynamic formalism. Second, the efficiency with which the corresponding optimization problem is solved is increased by using convex optimization techniques. That is, an approximate convex program is formulated and solved in order to provide a hot-start for the exact nonconvex program. The key element in this approximation is a (global) linearization of muscle physiology based on techniques from experimental system identification. This approach is applied to the study of muscle forces during gait. Although the results for gait are promising, experimental study of faster motions is needed to demonstrate the full power and advantages of the proposed methodology, and therefore is the subject of subsequent research.


Convex optimization Biomechanics Motion analysis Musculoskeletal modelling Dynamic simulation 



Activation of muscle j, [–]


Pennation angle of muscle j, [°]


Sample frequency of the simulation, [Hz]


Musculotendon force of muscle j, [N]


Index indicating the muscles of the musculoskeletal model; j={1,…,J}, [–]


Total number of muscles included in the musculoskeletal model, [–]


Index indicating the time instants of the simulation; k={1,…,K}, [–]


Total number of time instants of the simulation, [–]


Muscle fiber length of muscle j, [m]


Musculotendon length of muscle j, [m]


Tendon length of muscle j, [m]


Index indicating the degrees of freedom of the musculoskeletal model; n={1,…,N}, [–]


Total number of degrees of freedom of the musculoskeletal model, [–]


Sample period of the simulation, [s]


Activation time constant of muscle j, [s]


Deactivation time constant of muscle j, [s]


Excitation of muscle j, [–]


Muscle fibre lengthening speed of muscle j, [m/s]


∈ℝ N , vector of generalized coriolis and centrifugal forces, [N, N m]


∈ℝ J , vector of the musculotendon forces, [N]


∈ℝ N , vector of generalized gravitational forces, [N, N m]


∈ℝ N×N , generalized inertia matrix, [kg, kg m2]


∈ℝ N , vector of generalized coordinates of the skeleton, [m, rad]


∈ℝ N×J , geometric transformation matrix of F mt to generalized joint forces, [m, –]


∈ℝ N×12, geometric transformation matrix of W ext to generalized joint forces, [m, –]


∈ℝ N , vector of generalized joint forces delivered by the generalized external forces, [N, N m]


∈ℝ N , vector of generalized joint forces delivered by the musculotendon forces, [N, N m]


∈ℝ N , vector of generalized passive forces, [N, N m]


∈ℝ12, vector of generalized external forces acting on the skeleton, [N, N m]


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Goele Pipeleers
    • 1
  • Bram Demeulenaere
    • 1
  • Ilse Jonkers
    • 2
  • Pieter Spaepen
    • 3
  • Georges Van der Perre
    • 3
  • Arthur Spaepen
    • 4
  • Jan Swevers
    • 1
  • Joris De Schutter
    • 1
  1. 1.Div. PMA, Dept. of Mechanical EngineeringKatholieke Universiteit LeuvenHeverleeBelgium
  2. 2.Dept. of KinesiologyKatholieke Universiteit LeuvenHeverleeBelgium
  3. 3.Div. BMGO, Dept. of Mechanical EngineeringKatholieke Universiteit LeuvenHeverleeBelgium
  4. 4.Lab. for Ergonomics, Dept. of KinesiologyKatholieke Universiteit LeuvenHeverleeBelgium

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