## 2 Materials and Methods
### Leg NMS Dynamic Model  
The admittance \(H_{adm}\) can be parameterized by a linear NMS model which represents the dynamics of the ankle-foot interaction with the gas pedal in Figure 1. The inputs of the NMS model are the disturbing torque \(T_{dis}\) and the anticipated gas pedal position \(\hat{\theta}_{ped}\). The measurable model outputs are \(T_{c}\) and \(\theta_{c}\)[21]. The model structure is an extension of NMS models proposed in previous research [22].  
The inertia component can be described by the endpoint inertia of the limb \(I_{seg}\) in the form of a transfer function as follows:  
\[H_{seg}(s)=\frac{1}{I_{seg}s^{2}} \tag{6}\]  
The muscle contact dynamic is described as:  
\[H_{c}(s)=k_{con}+b_{con}s \tag{7}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline J\({}_{\text{st}}\) & Steering wheel inertia & 0.1395 (kg m\({}^{2}\)) \\ B\({}_{\text{st}}\) & Steering wheel damper & 1.56 (Nm s/rad) \\ K\({}_{\text{st}}\) & Steering wheel spring & 2.29 (Nm/rad) \\ K\({}_{\text{a}}\) & Active stiffness & \(-\)1920 (Nm/rad) \\ J\({}_{\text{dr}}\) & Arm inertia & 0.172 (kg m\({}^{2}\)) \\ B\({}_{\text{dr}}\) & Arm damping & 1.032 (Nm s/rad) \\ K\({}_{\text{dr}}\) & Arm spring & 60.707 (Nm/rad) \\ K\({}_{\text{r}}\) & Arm reflection stiffness & 3.423 (Nm s/rad) \\ B\({}_{\text{r}}\) & Arm reflection damping & 1.69 (Nm/rad) \\ \(\tau_{\text{del}}\) & Inherent time delay & 0.04 (s) \\ K\({}_{\text{pos}}\) & Stretch feedback gain & 0.52 (Nm/rad) \\ K\({}_{\text{tend}}\) & Tendon stiffness & 2799 (Nm/rad) \\ f\({}_{0}\) & Eigen-frequency & 1.1 (Hz) \\ I\({}_{\text{seg}}\) & Motion of the inertia & 105.5 (g/m\({}^{2}\)) \\ K\({}_{\text{f}}\) & GTO feedback gain & 1.18 (Nm/rad) \\ K\({}_{\text{vel}}\) & Stretch velocity feedback gain & 40.4 (Nm/rad) \\ K\({}_{\text{int}}\) & Co-contraction stiffness & 334 (Nm/rad) \\ B\({}_{\text{int}}\) & Co-contraction damping & 19.4 (Nm/rad) \\ K\({}_{\text{con}}\) & Contact elasticity & 1033 (Nm/rad) \\ B\({}_{\text{con}}\) & Contact viscosity & 11.2 (Nm /rad) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of driver neuromuscular (NMS) model.  
The intrinsic component is defined by the activated muscle stiffness \(k_{\text{int}}\) and damping \(b_{\text{int}}\) in the form of transfer function:  
\[H_{\text{int}}(s)=k_{\text{int}}+b_{\text{int}}s \tag{8}\]  
The reflexive component can be deduced as follows:  
\[H_{\text{{act}}}(s)=\frac{1}{(\frac{1}{a_{0}^{2}})s^{2}+\frac{2\beta}{a_{0}}s+1} \tag{9}\]  
The muscle golgi tendon organs (GTO) dynamic can be demonstrated as:  
\[H_{\text{{got}}}(s)=k_{f}e^{-\tau_{\text{{dd}}}s} \tag{10}\]  
The muscle spindle dynamic is expressed as:  
\[H_{\text{{ms}}}(s)=(K_{\text{{pos}}}+K_{\text{{vel}}}s)e^{-s\cdot\tau_{\text{{ dd}}}} \tag{11}\]  
where the \(K_{\text{{pos}}}\) and \(K_{\text{{vel}}}\) represent the gains of the monosynaptic stretch and stretch velocity feedback respectively.  
As a serial elastic element to the muscles, tendons dynamic can be expressed as:  
\[H_{\text{{tend}}}(s)=\frac{1}{-k_{\text{{tend}}}} \tag{12}\]  
where \(k_{\text{{tend}}}\) represents the stiffness of the tendon.  
Noteworthily, the motion of the inertia \(I_{\text{{seg}}}\) is the result of the sum of \(T_{c}(s)\) and \(T_{\text{{mus}}}(s)\). The pedal-torque can be combined with the muscles, partially. Then, the motion of the limb is deduced as:  
\[\theta_{\text{{limb}}}(s)=H_{\text{{seg}}}(s)[T_{c}(s)+T_{\text{{mus}}}(s)] \tag{13}\]  
Pedal rotations do not only cause joint rotations but also small displacements of the skin or soft tissue. This effect is described by contact dynamics as follows:  
\[T_{c}(s)=H_{c}(s)[\theta_{c}(s)-\theta_{\text{{limb}}}(s)] \tag{14}\]