## 3 Method of Generating Running Motion
### Method of Simulation to Generate Human Running Motion
#### 3.2.2 Motion Tactics Model Applied to the SLIP Model  
A motion tactics model was introduced to the SLIP model to generate motion based on the motion strategy. This model enables dynamic optimization of running by adjusting parameters and applied forces using feedback from the current motion state. In this study, three tactics (\(T_{k},T_{\theta},T_{f}\)) are empirically designed using the SLIP model parameters. Each tactic adjusts the parameters for the next step based on the feedback from the previous step's motion state. Notably, the tactics are designed with stability, speed, and energy as the primary evaluation criterion without considering time-series motion changes.  
Figure 2: SLIP model and motion tactics model. The stiffness and landing posture are adjusted at each step according to the motion strategies \(T_{k}\) and \(T_{\theta}\). In the force control introduced in this study, a force indicated by the orange line is applied. The magnitude of this force is adjusted by the strategy \(T_{f}\) allowing energy to be injected into the system.  
\[T_{k} \text{Speed tactic with stiffness control}\] This tactic \[T_{k}\] was designed to run at reference speed as a strategy. The stiffness was adjusted so that the running speed approached the reference speed. The deviation \[e_{\text{\tiny{$\mathrm{I}$}}}^{(i)}(=v_{\text{\tiny{ref}}}-v_{\text{\tiny{ave}} }^{(i)})\] between the reference speed \[v_{\text{\tiny{ref}}}\] and the running speed at \[i\] step \[v_{\text{\tiny{ave}}}^{(i)}\] was fed back. The stiffness at \[i+1\] step, \[k^{(i+1)}\], was adjusted as in Equation (5). \[u_{k}:k^{(i+1)}=k^{(i)}+K_{k}e_{\text{\tiny{$\mathrm{I}$}}}^{(i)} \tag{5}\] \[T_{\theta} \text{Stabilizing tactic by landing posture control}\] This tactic ensured stable running without falling or backward motion. If the landing posture deviated significantly from the desired range, the SLIP model failed to maintain stability. To counteract this, landing posture was adjusted at each step to eliminate acceleration changes. The angle between the legs of the SLIP model and the vertical direction at the landing of the \(i\) step was denoted as \[\theta^{(i)}\]. \[u_{\theta}:\theta^{(i+1)}=\theta^{(i)}+K_{\theta}(e_{\text{\tiny{$\mathrm{I}$} }}^{(i)}-e_{\text{\tiny{$\mathrm{I}$}}}^{(i-1)})\] (6) \[T_{f} \text{Energy tactic by force control}\] The tactic \[T_{f}\] was designed as a strategy to maintain system energy at the reference state during running. The total mechanical energy was regulated by controlling a jumping force applied along the axial direction of the SLIP model from the point of maximum compression to takeoff. The force magnitude remained constant during each step. The applied force at \[i+1\] step, \[f_{\text{\tiny{in}}}^{(i+1)}\], was updated based on the difference between the current energy \[E^{(i)}\] and the reference energy \[E_{\text{\tiny{ref}}}\], as described in Equation ( 7 ). The \[E^{(i)}\] was calculated as the energy at the flight phase of the \[i\] step. \[u_{f_{\text{\tiny{in}}}}:f_{\text{\tiny{in}}}^{(i+1)}=f_{\text{\tiny{in}}}^{(i) }+K_{f}(E_{\text{\tiny{ref}}}-E^{(i)}) \tag{7}\]