## 3 Simulation Modeling  
OpenSim 3.0 [33] biomechanics simulation and analysis environment was used to predict the forces and ranges of motion. The _Gait2345_Simbody_ model was adapted by altering the file and removing components (e.g., bones, muscles, etc.) to a suspended human leg to reproduce a target hinged gait pattern with similar constraints as the physical model. Each configuration and marker set file was imported into the Scaling Tool (Figure 4) to easily adapt to different sized users and models (e.g., varying lengths, masses, etc.). The Scaling Tool matched the physical prototype and the participant according to the reflective infrared markers placed on both subjects with mm precision. The calibration process required a static file with markers that match the same placement on the simulated model and the physical subject. The Forward Dynamics (FD) tool was used to solve the system's governing dynamic equations that integrate the model's musculoskeletal dynamical equations, which could be expressed for each muscle as:  
* Moments due to muscle forces: \(\tau_{m}=[R(q)]f(a,l,l)\)
* Muscle contraction rate: \(l=\Lambda(a,l,q,q)\)
* Muscle activation rate: \(\dot{a}=A(a,x)\)  
where \(\tau_{m}\) is the net muscle moment, \([R(q)]\) is the moment arm for the position \(q\), \(f\) is the muscle force, \(a\) is muscle activation value, \(l\) is the muscle fiber length, \(x\) is the model controls parameter. FD demonstrated that with any experimental set of motions, the resulting muscle activations yield similar kinematic results to other available tools as well as control techniques for robotic manipulators. The equation of motion is:  
\[\ddot{q}=[M^{-1}(q)]\left\{\tau-C(q,\dot{q})-G(q)\right\}] \tag{1}\]  
where \(q\), \(\dot{q}\) and \(\ddot{q}\) are the positions, velocities and accelerations. \([M^{-1}(q)]\) represents the inverse of the mass matrix, \(\tau\) is the vector of generalized forces, \(C(q,\dot{q})\) is the vector of Coriolis and centrifugal forces, and \(G(q)\) is the vector of gravitational forces [38]. A fifth-order Runge-Kutta-Feldberg integrator was used to solve for the coordinate trajectories over a user-defined time interval. Those trajectories are imported into the Forward Dynamics tool to create a series of muscle activation patterns [38]. The \"active\" elements capable ofactuation were modeled as muscles using OpenSim 3.0's _Thelen_ muscle model [39], and the \"passive\" tensile elements as ligaments. This is a Hill-type muscular-tendon contraction mechanics model that consist of several elastic elements connected in series and in parallel, where the active force-length relationship is assumed to be Gaussian. The equilibrium muscle model [38] consists of three main components: contractile elements \(F_{CE}\), parallel elements \(F_{PE}\) and series elements \(F_{SE}\). The active, \(F_{A}\), and passive, \(F_{P}\), components follow the muscle-tension ratios to represent varying levels of force with length shown rely on the equation given from the _Thelen2003 Muscle_ model [39]:  
\[l^{M}=F_{v}^{-1}\Big{(}\frac{\frac{F_{P}(l^{T})}{cos(\alpha)}-F_{P}(l^{M})}{a( t)\cdot F_{A}(l^{M})}\Big{)} \tag{2}\]  
\[l^{M}=F_{v}^{-1}\Big{(}\frac{F_{P}(l^{T})-F_{P}(l^{M})cos(\alpha)}{F_{A}(l^{M} )a(t)}\Big{)}\]  
\[l^{M}=\frac{F_{P}}{F_{v}F_{A}(l^{M})}\Big{(}\frac{l^{T}-l^{M}cos(\alpha)}{a( t)}\Big{)} \tag{3}\]  
To predict the force velocity curve using the equation above, we used data available in the published literature (Table 1 [38; 39]):  
* Muscle activation value \(a(t)\);
* Normalized length of the unit muscle \(l^{M}\) and tendon \(l^{T}\);
* Normalized velocity of the unit muscle \(F_{v}\);
* Maximum Pennation Angle \(\alpha\). The proposed model is actuated by three groups of virtual muscles (equivalent to the quadriceps, the hamstring, and hip flex muscle groups) that activate both hip's and knee's flexion and extension. Activation values \(a(t)\) change the output of the forces generated from the muscles.