## 2 Materials and Methods
### Motion Analysis  
The muscle exertion patterns differ between stair-climbing and level walking. Stair-climbing requires more muscle activity in the quadriceps and vastus lateralis muscles than level walking [19]. By placing a greater physical load on the lower extremities during level walking, we can generate a muscle exertion pattern similar to that of stair-climbing. For this purpose, we attached the PGMs above the knee of the user, as shown in Figure 2a. The contraction force to be exerted by the PGMs was designed from inverse dynamics calculations in a body model. The aim was to provide a muscle sensation corresponding to the muscle exertion pattern during stair-climbing and level walking. The motions were recorded by a motion analysis system (Optitrack, Acuity Co., Ltd., Tokyo, Japan) and a floor-reaction force meter (Bertec Co., Ltd., Columbus, OH, USA). The desired knee joint torque was calculated based on the two-dimensional skeleton model. Joint torque and muscle activity pattern were calculated by the musculoskeletal model, which more accurately reflects the human musculoskeletal system. By these, it can be expected to increase the type of the kinesthesia that can be presented and to increase the accuracy. The inverse dynamics were computed using SKYCOM analysis software. The body torque was determined from the three-dimensional position coordinates, acceleration, and angular acceleration obtained by the motion capture system, and the external force obtained by the floor-reaction force meter. When wearing the PGM, the knee joint torque in the bending direction was defined as \(\tau\), the torque predicted by PGM was \(\tau_{M}\), and the actual knee joint torque obtained by the three-dimensional motion analysis was \(\tau_{W}\). These three torques are related as  
\[\tau=\tau_{M}+\tau_{W}. \tag{1}\]  
Figure 8a presents the kinematic model for calculating \(\tau\). Here, the force generated by the PGM is \(F\) and the knee joint angle is \(\theta\). From the rotation center, the distances to the knee joint and PGM mounting position are denoted as \(L_{1}\) and \(L_{2}\), respectively. The length \(L\) of the PGM is calculated from \(\theta\) as follows:  
Figure 7: Force characteristics results of PGMs: (**a**) force versus length of a PGM with a natural length of 300 mm under different applied air pressures (50 to 300 kPa) and (**b**) contraction force versus air pressure in PGMs with natural lengths of 260, 280, and 300 mm. \[L=\sqrt{l_{1}^{2}+l_{2}^{2}-2l_{1}l_{2}\cos\theta}. \tag{2}\]  
The contraction-force characteristics of the PGM are controlled by the air pressure \(P\). Figure 7 plots the relationship between the generated force and the provided air pressure. The knee joint torque \(\tau_{M}\) accompanying the force from the PGM is a function of the lengths, the contraction force \(F_{p}\), and the number of PGMs \(k\):  
\[\tau_{M}=\frac{kF_{p}\sqrt{4L^{2}l_{1}^{2}-(L^{2}+l_{1}^{2}-l_{2}^{2})^{2}}}{2L}. \tag{3}\]  
The torques \(\tau_{W}\) and \(\tau_{M}\) during the walking cycle are plotted in Figure 8b,c, respectively. Substituting \(\tau_{W}\) and \(\tau_{M}\) into Equation (3), we get the total knee joint torque \(\tau\). Figure 8d shows the total knee joint torque during the walking cycle when the PGMs are operated at different air pressures (0-300 kPa). The knee joint torque peaked at approximately 40% of the walking cycle. Figure 9a,b plot the peak knee-joint torques as functions of air pressure during level walking and of stair height during stair-climbing (also calculated by inverse dynamics analysis), respectively. Meanwhile, Figure 9c relates the air pressure \(P\) to the stair height \(H\), derived from the linear regression lines of Figure 9a,b. From this figure, the linear relationship between the air pressure in the PGM and the stair height was determined as  
\[P=920H-21. \tag{4}\]  
Equation (4) gives the air pressure at which the PGM can generate the sense of stair-climbing in the virtual environment.