### Abstract

We present a novel method for deriving the governing equations of the musculoskeletal system, a new class of multibody systems in which the constituent components are connected together via anatomical joints which behave differently compared with traditional mechanical joints. In such systems, the kinematics of the joints and the corresponding constraints are characterized experimentally. We generate the equations of motion of these complex systems in which the homogeneous transformation matrices become matrix-valued functions of the generalized coordinate vector due to the empirical expression of body coordinates as smooth functions of generalized coordinates. The detailed mathematical procedure is provided to derive each term of the equations of motion using the novel calculus for the efficient evaluation of the partial derivatives of matrix-valued functions with respect to a vector. The governing equations obtained using the presented technique are expressed with ordinary differential equations rather than algebraic differential equations while not suffering from any simplification in experimental data describing the kinematics of the system. We then apply this method to derive the equations of motion of the “Andrews’ squeezer mechanism” for the validation. Furthermore, we successfully use this technique to model the shoulder rhythm with empirically-derived constraints in a trajectory tracking problem.

Original language | English (US) |
---|---|

Pages (from-to) | 673-690 |

Number of pages | 18 |

Journal | Mechanism and Machine Theory |

Volume | 133 |

DOIs | |

State | Published - Mar 1 2019 |

### Fingerprint

### Keywords

- Anatomical joints
- Coupled motion
- Empirically-derived constraints
- Matrix calculus
- Musculoskeletal system
- Shoulder rhythm

### ASJC Scopus subject areas

- Bioengineering
- Mechanics of Materials
- Mechanical Engineering
- Computer Science Applications

### Cite this

*Mechanism and Machine Theory*,

*133*, 673-690. https://doi.org/10.1016/j.mechmachtheory.2018.11.016

**Efficient embedding of empirically-derived constraints in the ODE formulation of multibody systems : Application to the human body musculoskeletal system.** / Ehsani, Hossein; Poursina, Mohammad; Rostami, Mostafa; Mousavi, Azin; Parnianpour, Mohamad; Khalaf, Kinda.

Research output: Contribution to journal › Article

*Mechanism and Machine Theory*, vol. 133, pp. 673-690. https://doi.org/10.1016/j.mechmachtheory.2018.11.016

}

TY - JOUR

T1 - Efficient embedding of empirically-derived constraints in the ODE formulation of multibody systems

T2 - Application to the human body musculoskeletal system

AU - Ehsani, Hossein

AU - Poursina, Mohammad

AU - Rostami, Mostafa

AU - Mousavi, Azin

AU - Parnianpour, Mohamad

AU - Khalaf, Kinda

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We present a novel method for deriving the governing equations of the musculoskeletal system, a new class of multibody systems in which the constituent components are connected together via anatomical joints which behave differently compared with traditional mechanical joints. In such systems, the kinematics of the joints and the corresponding constraints are characterized experimentally. We generate the equations of motion of these complex systems in which the homogeneous transformation matrices become matrix-valued functions of the generalized coordinate vector due to the empirical expression of body coordinates as smooth functions of generalized coordinates. The detailed mathematical procedure is provided to derive each term of the equations of motion using the novel calculus for the efficient evaluation of the partial derivatives of matrix-valued functions with respect to a vector. The governing equations obtained using the presented technique are expressed with ordinary differential equations rather than algebraic differential equations while not suffering from any simplification in experimental data describing the kinematics of the system. We then apply this method to derive the equations of motion of the “Andrews’ squeezer mechanism” for the validation. Furthermore, we successfully use this technique to model the shoulder rhythm with empirically-derived constraints in a trajectory tracking problem.

AB - We present a novel method for deriving the governing equations of the musculoskeletal system, a new class of multibody systems in which the constituent components are connected together via anatomical joints which behave differently compared with traditional mechanical joints. In such systems, the kinematics of the joints and the corresponding constraints are characterized experimentally. We generate the equations of motion of these complex systems in which the homogeneous transformation matrices become matrix-valued functions of the generalized coordinate vector due to the empirical expression of body coordinates as smooth functions of generalized coordinates. The detailed mathematical procedure is provided to derive each term of the equations of motion using the novel calculus for the efficient evaluation of the partial derivatives of matrix-valued functions with respect to a vector. The governing equations obtained using the presented technique are expressed with ordinary differential equations rather than algebraic differential equations while not suffering from any simplification in experimental data describing the kinematics of the system. We then apply this method to derive the equations of motion of the “Andrews’ squeezer mechanism” for the validation. Furthermore, we successfully use this technique to model the shoulder rhythm with empirically-derived constraints in a trajectory tracking problem.

KW - Anatomical joints

KW - Coupled motion

KW - Empirically-derived constraints

KW - Matrix calculus

KW - Musculoskeletal system

KW - Shoulder rhythm

UR - http://www.scopus.com/inward/record.url?scp=85058928234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85058928234&partnerID=8YFLogxK

U2 - 10.1016/j.mechmachtheory.2018.11.016

DO - 10.1016/j.mechmachtheory.2018.11.016

M3 - Article

AN - SCOPUS:85058928234

VL - 133

SP - 673

EP - 690

JO - Mechanism and Machine Theory

JF - Mechanism and Machine Theory

SN - 0374-1052

ER -