# Analysis Method of Articulated Torque of Heavy-Duty Six-Legged Robot under Its Quadrangular Gait
## 3 Static Torque Analysis of Hip Joint and Knee Joint
### Analysis of Static Articulated Torques under Working Condition III
#### 3.3.2 Analysis of Static Articulated Torques under Working Condition III-2  
Based on Figure 6c, the static torques are solved for the hip joint and knee joint. The equivalent leg 1\({}_{\text{III-2}}\) is obtained, when the equivalences are respectively executed through legs 1 and 3 to the straight line \(l_{1}\). The equivalent leg 5\({}_{\text{III-2}}\) is acquired, when the equivalence is implemented through leg 4 to leg 5. The equivalent model is shown in Figure 19. The moments of couples that were brought by the equivalence can be ignored. The reason is that their turnings are perpendicular to the turnings of torques of the hip joint and knee joint. They have no effect on solving the static torques of the hip joint and knee joint. The relations of the included angles are acquired: \(\beta^{\prime}1_{,3}=\beta^{\prime}5_{,4}=\beta_{1}^{\prime}=\beta_{3}^{ \prime}=\beta_{4}^{\prime}=\beta_{5}^{\prime}\) and \(\beta_{1,3}=\beta_{5,4}=\beta_{1}=\beta_{3}=\beta_{4}=\beta_{5}\). Based on Figures 6c and 19, the inequality is obtained: \(\boldsymbol{F}_{\text{eIII-2}\text{5}}\leq\boldsymbol{F}_{\text{eIII-2}\text{ 1}}\). Referring to the analysis of static articulated torques under working condition I, the normal contact force \(F_{\text{eIII-2}\text{1}}\) can be solved. The mathematical expression can be written for the static torque \(M_{\text{III-2}\text{1}}\) of hip joint and static torque \(M_{\text{III-2}\text{k1}}\) of knee joint of leg 1. According to the constrained condition of the working condition III-2: \(G\text{cos}\ \alpha_{i}/4\leq F_{\text{III-2}\text{1}}\leq G\text{cos}\ \alpha_{i}/2\) and \(0^{\circ}\leq\beta_{i}^{\prime}\leq\beta_{i}\leq 90^{\circ}\), the MATLAB software is used to solve the normal contact force \(F_{\text{III-2}\text{1}}\), static torque \(M_{\text{III-2}\text{1}}\) of the hip joint, and static torque \(M_{\text{III-2}\text{k1}}\) of the knee  
Figure 19: Equivalent model of support phase under working condition III-2. Figure 20 shows that the maximum value of \(\mathbf{F}_{\text{III-2z1}}\) is 1204.00 N when the value of included angle \(\beta_{i}\)' is 62\({}^{\circ}\) and the value of included angle \(\beta_{i}\) is 85\({}^{\circ}\). The minimum value of \(F_{\text{III-2z1}}\) is 642.80 N when the values of included angles \(\beta_{i}\)' and \(\beta_{i}\) are zero. Figure 21 shows that the maximum value of positive static articulated torque \(M_{\text{III-2h1+}}\) is 589.50 N\(\cdot\)m when the values of included angles are zero for \(\beta_{i}\)' and \(\beta_{i}\). The maximum value of negative static articulated torque \(M_{\text{III-2h1-}}\) is 47.89 N\(\cdot\)m when the values of included angles \(\beta_{i}\)' and \(\beta_{i}\) are 57\({}^{\circ}\) and 90\({}^{\circ}\), respectively. Figure 22 shows that the maximum value of positive static articulated torque \(M_{\text{III-2k1+}}\) is 305.30 N\(\cdot\)m when the values of included angles are zero for \(\beta_{i}\)' and \(\beta_{i}\). The maximum value of negative articulated torque \(M_{\text{III-2k1-}}\) is 199.50 N\(\cdot\)m when the value of included angle \(\beta_{i}\) is 90\({}^{\circ}\) and the value of included angle \(\beta_{i}\)' can range from 0\({}^{\circ}\) to 57\({}^{\circ}\). According to Figures 20-22, it can be concluded that the included angle \(\beta_{i}\)' of the hip joint and the included angle \(\beta_{i}\) of the knee joint are limited. Under the wording condition III-2 to pass the maximum slope, the included angle \(\beta_{i}\)' of the hip joint and the included angle \(\beta_{i}\) of the knee joint should meet the flowing requirements: 0\({}^{\circ}\)\(\leq\)\(\beta_{i}\)' \(\leq\)\(\beta_{i}\)\(\leq\) 90\({}^{\circ}\) and 0\({}^{\circ}\)\(\leq\)\(\beta_{i}\)' + \(\beta_{i}\)\(\leq\) 147\({}^{\circ}\). According to the above research, the maximum static articulated torques of the hip joint and knee joint are respectively obtained. The relevant poses of the robot are also acquired when the crab-type quadrangular gait is employed to pass the maximum slope for the heavy-duty six-legged robot.