# A Novel Explicit Analytical Multibody Approach for the Analysis of Upper Limb Dynamics and Joint Reactions Calculation Considering Muscle Wrapping

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## Abstract

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## Featured Application

**Explicit biomechanical multibody model for the inverse dynamics of upper limb musculoskeletal systems considering muscle wrapping.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Musculoskeletal System

#### 2.2. Kinematical Analysis Based on the Constraints

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- ${n}_{dof}$ rheonomic constraints which drive the degrees of freedom to move with the known trajectories (${\mathit{C}}_{r}$);
- -
- ${n}_{c}$ scleronomic constraints due to the relative motions blocked by the joints kinematical behaviour (${\mathit{C}}_{s}$);
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- $\left({n}_{B}-1\right)$ constraints which guarantee that the bodies’ quaternions keep the unitary norm (${\mathit{C}}_{b})$.

- -
- the joint position calculated starting from the body $i$, ${\mathit{r}}_{J,i}$, has to be equal to the one seen by the body $j$, ${\mathit{r}}_{J,j}$;
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- a vector fixed to body $j$, ${\overline{\mathit{s}}}_{j}$, parallel to the rotation axis $\widehat{\mathit{v}}$, has to be orthogonal with respect to others two vectors fixed to the body $i$, ${\overline{\mathit{s}}}_{i1}$ and ${\overline{\mathit{s}}}_{i2}$, both orthogonal to the rotation axis $\widehat{\mathit{v}}$ and to each other.

#### 2.3. Inverse Dynamics

#### 2.4. Hill Muscle Model

- -
- according to the force–length relation ${f}_{l}$, increasing the muscle fibre length, the active force increases until it reaches a peak (the maximum isometric force ${F}_{0}$) in correspondence with the optimal fibre length ${l}_{0}$, then it decreases;
- -
- the muscle generates a greater force than the maximum isometric one (in correspondence of zero deformation velocity) when it lengthens, with an asymptotic behaviour, and a lower force until it reaches the maximum contraction velocity ${v}_{max}$, beyond which it is not able to produce actuation, following the force–velocity relation ${f}_{v}$;
- -
- the passive element opposes a growing resistance only if the muscle length is greater than the optimal fibre length ${l}_{0}$, until it reaches a maximum value following the relation ${f}_{PE}$.

#### 2.5. Wrapping Muscles

#### 2.6. Static Optimization

## 3. Results and Discussion

## 4. Conclusions

- -
- despite the kinematical analysis, exactness is noticeable, the only difference found was a small overestimation of the long biceps muscle fibre length, probably due to the different wrapping algorithm used by OpenSim;
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- the inverse dynamics in terms of driving force results were almost completely matched;
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- the static optimization was characterised by a general underestimation of the muscles’ activations in all the cases considered (both physiological and non-physiological criterion in absence of passive muscles’ forces and physiological criterion considering the passive muscle actions compared with the Computed Muscle Control tool of OpenSim). The underestimation was regained by comparing the muscle forces, the differences were probably due to the different approaches in the minimization techniques and, certainly, to the different approach to the inverse dynamics of the Computed Muscle Control;
- -
- the good outcome of the comparison was also confirmed by the matching of the residual actuators.

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- a full validation with more upper limb kinematics;
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- the application of the model in the framework of the lower limb musculoskeletal system, in order to compare the simulated joint reactions with the in vivo measurements during the gait analysis;
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- the coupling of the model with a lubrication one, in order to estimate the wear of an artificial joint or to evaluate the joints friction forces or torques to include in the equation of motion as non-conservative actions;
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- the upgrade of the model by including the forward dynamics, in order to make a detailed comparison with the Computed Muscle Control tool of the OpenSim software;
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- the deepening of the Hill muscle model, in order to include the tendon dynamics;
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- the possible applications of the model in the framework of the biomechatronics and robotics fields.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

$\mathit{q}$ | Lagrangian coordinates |

$\mathit{t}$ | Translation vector |

$\mathit{\theta}$ | Unit quaternion vector |

$\mathit{R}$ | Rotation matrix |

$\overline{\mathit{u}}$ | Local position vector |

$\mathit{r},\mathit{v},\mathit{a}$ | Position, velocity and acceleration vector in the ground reference frame |

${n}_{B},{n}_{dof},{n}_{c}$ | Number of bodies, degrees of freedom and constraints |

$\mathit{C}$ | Constraint equations |

${\mathit{C}}_{\mathit{q}}$ | Constraint (Jacobian) |

$\widehat{\mathit{v}}$ | Rotation axis unit vector |

${\mathit{q}}_{dof}$ | Degrees of freedom vector |

$\mathit{M}$ | Mass matrix |

$\mathit{Q}$ | Generalised force vector |

$\mathit{\lambda}$ | Lagrange multipliers |

${\mathit{J}}_{q}$ | Coordinate change matrix |

${l}_{t}$ | Tendon length |

${l}_{m}$ | Muscle fibre length |

${v}_{m}$ | Muscle fibre deformation velocity |

${w}_{m}$ | Muscle width |

${\alpha}_{p}$ | Muscle fibre pennation angle |

${F}_{CE}$ | Muscle contractile element force |

${F}_{PE}$ | Muscle passive element force |

${F}_{m}$ | Muscle force |

${l}_{0},{\alpha}_{0}$ | Optimal muscle fibre length and the related pennation angle |

${F}_{0}$ | Maximum isometric force |

${l}_{mt},{v}_{mt}$ | Musculotendon length and deformation velocity |

$\mathit{x}\left(u,v\right),f\left(\mathit{x}\right)$ | Surfaces’ parametric and implicit equations |

$s$ | Curvilinear coordinate |

$\mathit{c}$ | Curve |

$\widehat{\mathit{t}},\widehat{\mathit{n}},\widehat{\mathit{b}}$ | Tangent, normal and binormal unit vectors |

${\Phi}_{\mathit{q}}$ | Muscle (Jacobian) |

$\mathit{a}$ | Muscle activation vector |

${\mathit{\lambda}}_{r},{\mathit{\lambda}}_{s}$ | Lagrange multipliers related to the rheonomic constraints (or residual actuators) and to the scleronomic constraints (or joint reactions) |

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**Figure 14.**Muscles’ activations with non-physiological criterion and no passive forces: (

**a**) Matlab and (

**b**) OpenSim.

**Figure 15.**Muscles’ activations with physiological criterion and no passive forces: (

**a**) Matlab and (

**b**) OpenSim.

**Figure 16.**Muscles’ forces with physiological criterion and no passive forces: (

**a**) Matlab and (

**b**) OpenSim.

**Figure 17.**Muscles’ activations with physiological criterion and passive forces: (

**a**) Matlab and (

**b**) OpenSim.

**Figure 18.**Muscles’ forces with physiological criterion and passive forces: (

**a**) Matlab and (

**b**) OpenSim.

**Figure 20.**Upper limb joint reactions: (

**a**) shoulder forces, and (

**b**) torques, (

**c**) elbow forces and (

**d**) torques.

Body | $\mathbf{Mass}\mathit{m}\left[\mathrm{kg}\right]$ | Inertia Tensor Entries | |||||
---|---|---|---|---|---|---|---|

${\overline{\mathit{I}}}_{\mathit{x}\mathit{x}}\left[g{m}^{2}\right]$ | ${\overline{\mathit{I}}}_{\mathit{y}\mathit{y}}\left[g{m}^{2}\right]$ | ${\overline{\mathit{I}}}_{\mathit{z}\mathit{z}}\left[g{m}^{2}\right]$ | ${\overline{\mathit{I}}}_{\mathit{x}\mathit{y}}\left[g{m}^{2}\right]$ | ${\overline{\mathit{I}}}_{\mathit{y}\mathit{z}}\left[g{m}^{2}\right]$ | ${\overline{\mathit{I}}}_{\mathit{x}\mathit{z}}\left[g{m}^{2}\right]$ | ||

Humerus | 1.86 | 14.81 | 4.55 | 13.19 | 0 | 0 | 0 |

Forearm | 1.53 | 19.28 | 1.57 | 20.06 | 0 | 0 | 0 |

Body | Local Position Vector | ||
---|---|---|---|

${\overline{\mathit{u}}}_{\mathit{G}\mathit{x}}\left[\mathit{m}\right]$ | ${\overline{\mathit{u}}}_{\mathit{G}\mathit{y}}\left[\mathit{m}\right]$ | ${\overline{\mathit{u}}}_{\mathit{G}\mathit{z}}\left[\mathit{m}\right]$ | |

Humerus | 0 | −0.180 | 0 |

Forearm | 0 | −0.181 | 0 |

Joint | Local Position Vector | ||
---|---|---|---|

${\overline{\mathit{u}}}_{\mathit{J}\mathit{x}}\left[\mathit{m}\right]$ | ${\overline{\mathit{u}}}_{\mathit{J}\mathit{y}}\left[\mathit{m}\right]$ | ${\overline{\mathit{u}}}_{\mathit{J}\mathit{z}}\left[\mathit{m}\right]$ | |

Shoulder | −0.0175 | −0.0070 | 0.1700 |

Elbow | 0.0061 | −0.2904 | −0.0123 |

Joint | Unit Vector Axis | ||
---|---|---|---|

${\widehat{\mathit{v}}}_{\mathit{x}}$ | ${\widehat{\mathit{v}}}_{\mathit{y}}$ | ${\widehat{\mathit{v}}}_{\mathit{z}}$ | |

Shoulder | −0.0589 | 0.0023 | 0.9983 |

Elbow | 0.0494 | 0.0366 | 0.9981 |

Muscle | Maximum Isometric Force ${\mathit{F}}_{0}\left[\mathit{N}\right]$ | Optimal Muscle Fibre Length ${\mathit{l}}_{0}\left[\mathit{m}\right]$ | Slack Tendon Length ${\mathit{l}}_{\mathit{t}}\left[\mathit{m}\right]$ | Optimal Pennation Angle ${\mathit{\alpha}}_{0}\left[\mathit{r}\mathit{a}\mathit{d}\right]$ | Maximum Contraction Velocity ${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}\left[{\mathit{l}}_{0}/\mathit{s}\right]$ |
---|---|---|---|---|---|

Long triceps | 798.52 | 0.1340 | 0.1430 | 0.209 | 10 |

Lateral triceps | 624.30 | 0.1138 | 0.0980 | 0.157 | 10 |

Medial triceps | 624.30 | 0.1138 | 0.0908 | 0.157 | 10 |

Long biceps | 624.30 | 0.1157 | 0.2723 | 0 | 10 |

Short biceps | 435.56 | 0.1321 | 0.1923 | 0 | 10 |

Brachialis | 987.26 | 0.0858 | 0.0535 | 0 | 10 |

**Table 6.**Wrapping surfaces location and orientation with respect to the reference joints and geometry.

Wrapping Surface | Local Position Vector | Local Orientation | Geometry | ||||
---|---|---|---|---|---|---|---|

${\overline{\mathit{u}}}_{\mathit{w},\mathit{x}}\left[\mathit{m}\right]$ | ${\overline{\mathit{u}}}_{\mathit{w},\mathit{y}}\left[\mathit{m}\right]$ | ${\overline{\mathit{u}}}_{\mathit{w},\mathit{z}}\left[\mathit{m}\right]$ | ${\mathit{\theta}}_{\mathit{w},\mathit{x}}\left[\mathit{r}\mathit{a}\mathit{d}\right]$ | ${\mathit{\theta}}_{\mathit{w},\mathit{y}}\left[\mathit{r}\mathit{a}\mathit{d}\right]$ | ${\mathit{\theta}}_{\mathit{w},\mathit{z}}\left[\mathit{r}\mathit{a}\mathit{d}\right]$ | $\left[\mathit{m}\mathit{m}\right]$ | |

Ground cylinder | −0.0439 | −0.0039 | 0.1478 | 1.3753 | −0.2946 | 2.4360 | $\begin{array}{c}r=3\\ h=30\end{array}$ |

Shoulder ellipsoid 1 | −0.0078 | −0.0041 | −0.0014 | 3.0016 | −0.8535 | 2.5742 | $\begin{array}{c}{s}_{x}=35\\ {s}_{y}=20\\ {s}_{z}=20\end{array}$ |

Shoulder ellipsoid 2 | 0.0033 | 0.0073 | 0.0003 | −2.0043 | −1.0016 | 0.9755 | $\begin{array}{c}{s}_{x}=25\\ {s}_{y}=20\\ {s}_{z}=20\end{array}$ |

Elbow cylinder | 0.0028 | −0.2919 | −0.0069 | −0.1402 | −0.0063 | 0.1550 | $\begin{array}{c}r=16\\ h=50\end{array}$ |

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**MDPI and ACS Style**

Ruggiero, A.; Sicilia, A.
A Novel Explicit Analytical Multibody Approach for the Analysis of Upper Limb Dynamics and Joint Reactions Calculation Considering Muscle Wrapping. *Appl. Sci.* **2020**, *10*, 7760.
https://doi.org/10.3390/app10217760

**AMA Style**

Ruggiero A, Sicilia A.
A Novel Explicit Analytical Multibody Approach for the Analysis of Upper Limb Dynamics and Joint Reactions Calculation Considering Muscle Wrapping. *Applied Sciences*. 2020; 10(21):7760.
https://doi.org/10.3390/app10217760

**Chicago/Turabian Style**

Ruggiero, Alessandro, and Alessandro Sicilia.
2020. "A Novel Explicit Analytical Multibody Approach for the Analysis of Upper Limb Dynamics and Joint Reactions Calculation Considering Muscle Wrapping" *Applied Sciences* 10, no. 21: 7760.
https://doi.org/10.3390/app10217760