## 2 Specifications and Modeling of Power Assisted Cart: RT.1
Notations \(r\) and \(R\) express the displacements between one handle and the rotational center and between the rotational center and one of the wheels, respectively. Figure 2b shows the resultant force and moment applied to the cart. Figure 2: (**a**) Model of the power-assisted cart and (**b**) resultant force and moment applied to the cart. Assistance ratios \(\alpha_{T}\) and \(\alpha_{R}\) are normally predefined as constants and their default values are set to be 1.5 and 0, respectively, for RT.1. In this study, the motors assist both forward and rotational movements, and for simplicity, the ratio for forward movement is equivalent to the one for rotational movement, i.e., \(\alpha_{T}=\alpha_{R}=\alpha\). First, we introduce the equations of motion for the translational movement of the cart. We define \(M\), \(K_{v}\) and \(g\) as the total mass of the cart and load, the equivalent viscous friction coefficient related to the translational movement, and gravitational acceleration, respectively. The translational velocities of both rear wheels are \(v_{L}\) and \(v_{R}\), respectively, and we define \(v\) and \(\omega\) as the translational and rotational velocities of the cart, respectively. It should be noted that  
\[\left[ \begin{array}{c}v\\ \omega\end{array} \right]=\mathbf{J}_{4}\left[ \begin{array}{c}v_{L}\\ v_{R}\end{array} \right]\,, \tag{3}\]  
where \(\mathbf{J}_{4}=\left[ \begin{array}{cc}\frac{1}{2}&\frac{1}{2}\\ -\frac{1}{2K}&\frac{1}{2K}\end{array} \right]=\mathbf{J}_{3}^{-T}\). When the cart moves on a slope as shown in Figure 3, the dynamics of the translational motion can be described as  
\[F_{T}=F_{\text{Human}}+F_{\text{Assist}}=M\frac{dv}{dt}+F_{v}+d_{T}\, \tag{4}\]  
where  
\[F_{v} =K_{v}v\, \tag{5}\] \[d_{T} =Mg(c\beta s\theta s\gamma+s\beta c\gamma)+\mu_{c}(F_{NL}+F_{NR}) c\beta c\theta\,\]  
\(\mu_{c}\) is the coefficient of kinetic friction in the translational direction, and \(d_{T}\) expresses the disturbance including gravity and Coulomb friction. Notations \(F_{NL}\) and \(F_{NR}\) are for normal forces on the wheels, where \(F_{NL}+F_{NR}=Mg\); \(c*\) and \(s*\) mean \(\cos(*)\) and \(\sin(*)\), and \((\beta,\theta,\gamma)\) correspond to the Euler angles. It should be noted that \(d_{T}\) is constant while moving on a flat surface or going straight on a slope whereas it will vary rapidly if the cart rotates on a slope. Let \(I\) be the rotational inertia of the cart and load with respect to the rotational center and \(K_{\omega}\) be the coefficient of viscosity in rotational movement. Similarly, the dynamics of the cart related to rotational motion can be written as  
\[M_{T}=M_{\text{Human}}+M_{\text{Assist}}=I\frac{d\omega}{dt}+M_{w}+d_{R}\, \tag{6}\]  
where  
\[M_{w} =K_{\omega}\omega\, \tag{7}\] \[d_{R} =IMg(-c\beta s\theta c\gamma+s\beta s\gamma)+\mu_{c}(F_{NR}-F_{NL })Rc\beta c\theta\,\]  
Figure 3: Movement of the cart on a slope. and \(d_{R}\) represents the disturbance related to the rotational movement. If the disturbance can be ignored, the transfer functions related to translational and rotational movements are expressed as  
\[ \begin{split}\frac{v(s)}{F_{T}(s)}=&\frac{1}{Ms+K_{v}} \,\\ \frac{\omega(s)}{M_{T}(s)}=&\frac{1}{Is+K_{\omega}} \.\end{split}  \tag{8}\]