## 2 Specifications and Modeling of Power Assisted Cart: RT.1  
The experimental device, RT.1, is produced by Funai Electric Co. and RT.WORKS Co. (Figure 1a) and is now on sale in Japan [31]. The power-assisted cart can sustain the weight of the user and carry heavy loads during walking; the former decreases muscular activity, while the latter reduces the load-carrying burden. The number of cart users is dramatically increasing in Japan. The specifications of the cart are reported in Table 1. There are force sensors inside the handles for measuring the force from the user, along the forward/backward direction only. The measured forces at the left and right handles are expressed as \(F_{HL}\) and \(F_{HR}\), respectively. Besides, the acceleration and angular velocity of the cart can be obtained via a built-in inertial measurement unit (IMU; MPU6050, TDK Invensense, San Jose, CA, USA). When the user pushes the handles of the power-assisted cart during walking (Figure 1b), two brushless DC motors at the rear wheels provide torque (driving force) for moving the cart. The rotational velocities of the two motors are acquired via hall effect sensors. The signal processing and control strategy are implemented by a microcontroller unit (RX63T, Renesas, Japan), of which the maximum operating frequency is up to 100 MHz. The frequency for the control loop was set as 500 Hz. \begin{table}
\begin{tabular}{c c} \hline Characteristic & RT.1 \\ \hline Dimensions & Width 510 mm \(\times\) length 601 mm \(\times\) height 819–1019 mm \\ \hline Weight & 14.8 kg \\ \hline Power & Lithium battery \\ \hline Mounted sensors & Force sensor, inertial measurement unit \\ \hline Microcontroller unit & RX63T, Renesas \\ \hline \end{tabular}
\end{table} 
Table 1: Specifications of RT.1. Figure 1: (**a**) Appearance and (**b**) experimental setup of the power-assisted cart RT. 1 provided by RT. WORKS [31]. As previously mentioned, the basic idea of such power-assisted devices for practical use can be summarized as the output for assistance is proportional to the input. For example, the cart enables the amount of assistance required to proceed and varies the steering depending on the input from the user. In this subsection, we describe the dynamic model of the cart with respect to the rotational center (the center of the rear wheels). Figure 2a presents the model for the power-assisted cart. The front passive wheels can be considered as one caster for simplicity. The terms \(F_{\text{Human}}\) and \(M_{\text{Human}}\) are the resultant force and moment from the user applied to the rotational center, respectively; \(F_{\text{Assist}}\) and \(M_{\text{Assist}}\) are the resultant force and moment generated by two DC brushless motors to drive the rear wheels, respectively; and the forces generated by the left and right motors are represented by \(F_{ML}\) and \(F_{MR}\), respectively. The distance between the mass center and the rotational center is denoted as \(l\). The relationship between the forces from the user and the two motors can be written as  
\[\left[ \begin{array}{c}F_{\text{Assist}}\\ M_{\text{Assist}}\end{array} \right]=\mathbf{J}_{1}\left[ \begin{array}{c}F_{\text {Human}}\\ M_{\text{Human}}\end{array} \right]\,, \tag{1}\]  
where the matrix \(\mathbf{J}_{1}\) is equivalent to \(\left[ \begin{array}{cc}\alpha_{T}&0\\ 0&\alpha_{R}\end{array} \right]\) and \(\alpha_{T}\) and \(\alpha_{R}\) are the ratios of the output to the input for translational and rotational movements, respectively. These forces and moments satisfy the following conditions:  
\[\left[ \begin{array}{c}F_{\text{Human}}\\ M_{\text{Human}}\end{array} \right] =\mathbf{J}_{2}\left[ \begin{array}{c}F_{HL}\\ F_{HR}\end{array} \right]\,, \tag{2}\] \[\left[ \begin{array}{c}F_{\text{Assist}}\\ M_{\text{Assist}}\end{array} \right] =\mathbf{J}_{3}\left[ \begin{array}{c}F_{ML}\\ F_{MR}\end{array} \right]\,,\]  
where \(\mathbf{J}_{2}=\left[ \begin{array}{cc}1&1\\ -r&r\end{array} \right]\) and \(\mathbf{J}_{3}=\left[ \begin{array}{cc}1&1\\ -R&R\end{array} \right]\).