## 2 Materials and Methods
In (Equation (9)), \(\underline{x}\), \(\underline{v}\), and \(t\) represents state, control variables, and time, respectively, and \(f\) is the time derivatives of the state variables. The subscript \(i\) represents the \(i\)th node and its maximum value is \(N-1\):  
\[\underline{\Delta}_{i}\equiv\Big{(}\underline{x}_{i+1}-\underline{x}_{i}\Big{)} -\Big{(}\underline{t}_{i+1}-\underline{t}_{j}\Big{)}\bigg{(}\frac{\underline{ x}_{i}+\underline{x}_{i+1}}{2},\frac{\underline{v}_{i}+\underline{v}_{i+1}}{2}, \frac{\underline{t}_{i}+\underline{t}_{i+1}}{2}\bigg{)},i=1,\cdots,(N-1) \tag{9}\]  
The state variable \(\underline{x}\) in (Equation (9)) means \(q\), \(\dot{q}\), \(a\), \(l\) mentioned in (Equations (2)-(5)), and \(v\) denotes \(u\), \(\tau\), \(f\) used in (Equations (6)-(8)). Since human body walking is periodic, the actuator input should also be input periodically. To satisfy this, additional periodicity constraints were imposed so that the actuator input matches at the first and last node of the walk (Equations (10) and (11)):  
Figure 1:0  
Figure 1:1where the \(\tau_{i,1}\) means \(i\)th torque actuator input at 1 st node, and the \(\tau_{i,N}\) means input at \(N\)th node. The \(f\) can be interpreted in the same manner. The relationship between the forces acting on the human body due to the muscle, GRF, and external actuator, and the motion caused thereby is expressed in (Equation (12)) [26; 36]:  
Figure 1:2  
where \(M\Big{(}\underline{q}\big{)}\) is the system mass matrix, \(\underline{C}\big{(}\underline{q},\underline{\dot{q}}\big{)}\) is Coriolis and centripetal term, \(\underline{G}\big{(}\underline{q}\big{)}\) is gravity term, \(R\Big{(}\underline{q}\big{)}\) is moment arm matrix, \(F_{MT}\) is a vector of muscle forces, \(\underline{E}\big{(}\underline{q},\underline{\dot{q}}\big{)}\) is external force term, and \(\underline{q},\ \underline{\dot{q}},\underline{\dot{q}}\) is the vector of generalized coordinates, velocities, and accelerations. In the microgravity environment, the gravitational force is reduced, resulting in a decrease in body weight (BW). This causes a reduction in the GRF, which is an external force generated in the stance phase. Thus, when compared to a 1 g gait, kinematically different gait patterns and joint moments are generated. The GRF and BW correspond to the \(E\big{(}\underline{q},\underline{\dot{q}}\big{)}\) and \(G\big{(}\underline{q}\big{)}\) terms in (Equation (12)). If external actuators are used to compensate for the effects caused by the change in the \(E\big{(}\underline{q},\underline{\dot{q}}\big{)}\) and \(G\big{(}\underline{q}\big{)}\) terms in (Equation (12)), then the same governing equation as that in the case of 1 g normal walking can be established, and the tracking can be achieved. For the model modified for microgravity simulation in this study, (Equation (12)) can be simplified as (Equation (13)), and the goal of the simulation in the view of the musculoskeletal system can be summarized as finding \(L\) to satisfy the desired \(\underline{q},\ \underline{\dot{q}}\), and \(F_{MT}\). Figure 1:3  
where \(BL\) represents a vector of six externally applied loads (force/moment on each foot, or six exoskeleton torques), \(L\) is the magnitudes of the actuator inputs, and \(B\) is an invertible \((6\times 6)\) matrix. Muscle activation dynamics (Equation (14)) and musculendon dynamics (Equation (15)) were used to obtain \(F_{MT}\) in (Equation (13)) [37; 38]:  
Figure 1:4  
Figure 1:5  
A simulation was performed for Type 1, in which the actuators are attached to the position similar to that when the GRF acts on the human body during walking (Figure 2a). In the simulation for Type 2, the rotary actuator was attached to the generalized coordinates of the model (Figure 2b). Figure 1:6 Actuation system configuration. (**a**) Type 1: two forces and one torque actuators acting on the midpoint of the sole (**b**) Type 2: rotary actuators attached to the HKA joints.