# The Analysis of Mechanical Structure of a Robotic Leg in Running for Impact Mitigation
## 3 Simulations
### Actuator Configuration  
The actuator configuration can be divided into serial and parallel configurations, as shown in Figure 7. The serial configuration takes the structure of hip and knee actuators connected in series as in Figure 7a such that the torque generated by the knee actuator affects the torque generated by the hip actuator while the motion generated by the knee actuator is affected by the motion generated by the hip actuator. The parallel configuration, on the other hand, features a structure different from the serial configuration. Hip and knee actuators are mounted on the body to independently operate the upper and lower limbs as in Figure 7b. For this reason, the motions and torques generated by the hip and knee actuators are not affected by each other.  
To simulate CIEs of each actuator configuration, the moments of inertia and COMs for each limb should be provided. According to the second assumption, the moments of inertia of upper and lower limb in parallel configuration are:  
\[H_{1(p)}=\tfrac{1}{12}m_{1}l_{1}^{2},\ \ \ H_{2(p)}=\tfrac{1}{12}m_{2}l_{2}^{2}\, \tag{20}\]  
and COM of upper and lower limb is \(r_{1(p)}=r_{2(p)}=0.5\). In serial configuration, the mass of the knee actuator, i.e., \(M\), along with the mass of the link, i.e., \(m_{1}\), is included in the mass of the upper limb. Therefore, the moment of inertia and COM for the upper limb are:  
\[H_{1(s)}=\left[\frac{1}{12}+\left(\frac{1}{2}-r_{1}\right)^{2}\right]m_{1}l_{1 }^{2}+Mr_{1}^{2}l_{1}^{2} \tag{21}\]  
and:  
\[r_{1(s)}=\frac{m_{1}}{2\left(m_{1}+M\right)}. \tag{22}\]  
The moment of inertia and COM for the lower limb are equal to the parallel configuration, i.e., \(H_{2(s)}=H_{2(p)}=H_{2}\) and \(r_{2(s)}=r_{2(p)}=r_{2}\). Using the moments of inertia and COMs, CIEs of each actuator configuration could be simulated by (19). Note that the Jacobian matrix of the parallel configuration is different from (A6) of the serial configuration, i.e.,  
\[\mathbf{J}(\mathbf{q})=\left[\tfrac{\partial\mathbf{R}(\theta_{1})}{\partial \theta_{1}}\hat{l}_{1}\ \ \ \tfrac{\partial\mathbf{R}(\theta_{12})}{\partial\theta_{12}}\hat{l}_{2}\right] \in\mathbb{R}^{2\times 2}, \tag{23}\]  
where \(\theta_{1}\) and \(\theta_{2}\) denote the relative joint angles of upper and lower limbs, respectively, as shown in Figure A1; \(\theta_{12}\) is abbreviated as \(\theta_{1}+\theta_{2}\). \(\hat{l}_{1}\) and \(\hat{l}_{2}\) represent neutral length vectors of the upper and lower limbs in the same manner as in \(\hat{l}\) of (4).  
CIEs of both actuator configurations were exactly the same with regard to \(l_{n}\) and \(\theta\), as shown in Figure 8. \(l_{n}\) and \(\theta\) denote the extension ratio and attack angle, respectively. When the leg fully extended, for example, \(l_{n}\) became one. The reason why both CIEs of serial and parallel configurations resulted in being the same is because both moments of inertia about the upper joint axis for the upper  
Figure 7: Actuator configuration: (**a**) serial; (**b**) parallel.  
limb were consequently the same. It can be interpreted that the mass of the knee actuator, i.e., \(M\), made COM drift away from the point of contact, which eventually compensated the increased mass of the upper limb in the serial configuration. In this regard, a detailed mathematical proof is given in Appendix B.