# Research on the Posture Control Method of Hexapod Robot for Rugged Terrain
## 3 Posture Maintenance Strategy for Mildly Rugged Terrain
### Virtual Suspension Dynamic Model  
#### 3.2.1 Linear Model of Virtual Suspension Dynamic Model  
According to the characteristics of the FBM mechanism of the hexapod robot, a virtual suspension dynamic model (VSDM) is established to suppress the change of posture during walking, in the vertical, pitch, and roll directions of the torso, as shown in Figure 3. A one-dimensional virtual spring damper is set up, and the vertical direction fluctuation and the pitch angle and roll angle disturbance are suppressed by controlling the spring damper. Here, the ground and the body are assumed rigid, and the mathematical expression obtained according to the model is as follows:  
\[\left\{\begin{array}{c}M_{g}\ddot{z}=-K_{z}z-C_{z}\dot{z}\\ I_{p}\ddot{\theta}_{p}=-K_{p}\theta_{P}-C_{P}\dot{\theta}_{P}\\ I_{r}\ddot{\theta}_{r}=-K_{r}\theta_{r}-C_{r}\dot{\theta}_{r}\end{array}\right. \tag{8}\]  
where \(K_{z}\), \(K_{p}\), and \(K_{r}\) are the virtual spring stiffness coefficients in the vertical, pitch, and roll directions respectively, \(C_{z}\), \(C_{p}\), and \(C_{r}\) are the virtual damping coefficient for vertical, pitch, and roll directions respectively, \(I_{p}\) and \(I_{r}\) are the moment of inertia in the pitch and roll directions respectively, z, \(\theta_{p}\), and \(\theta_{r}\) are trunk height, pitch angle, and roll angle respectively, and \(M_{g}\) is the robot weight.  
After Equation (8), taking the Laplace transform, the characteristic equation of the virtual model in three directions is expressed as follows:  
\[M\cdot s^{2}+C\cdot s+K=0 \tag{9}\]  
where \(M=\left[m,I_{p},I_{r}\right]^{T}\), \(C=\left[C_{z},C_{p},C_{r}\right]^{T}\), and \(K=\left[K_{z},K_{p},K_{r}\right]^{T}\). The frequency characteristic equation is as follows:  
\[s^{2}+2\xi_{i}\omega_{ni}\cdot s+\omega_{ni}{}^{2}=0\ (i=g,p,r) \tag{10}\]  
The characteristic frequency and damping ratio of the system are expressed as follows:  
\[\omega_{ni}=\sqrt{\frac{K_{i}}{M_{i}}},\ \xi_{i}=\frac{C_{i}}{(2\omega_{ni} \cdot M_{i})} \tag{11}\]  
The damping ratio is selected between 0.7 and 1.0 to improve the system response speed and reduce the overshoot. In addition, the interference caused by the pose control amount has a very low-frequency characteristic during the execution of the controller due to the collision and sliding of the foot with the ground and the change of the robot's own motion state. The system's characteristic frequency, \(\omega_{n}\), has a large value to avoid resonance. When the robot is in a static state, no disturbance is caused by the external and its own state changes. In this state, the corner frequency and damping ratio are \(\omega_{n}=\left[\omega_{nz},\omega_{np},\omega_{nr}\right]=\left[240,\,240, \,60\right]\) and \(\xi_{n}=\left[\xi_{nz},\xi_{np},\xi_{nr}\right]=\left[0.85,\,0.85,\,0.85\right]\), respectively. The mass of the robot is 3.2 kg, and the moments of inertia in the pitch and roll directions in the initial pose are respectively Substituting \(\mathrm{I_{P}}=8.98\times 10^{-2}\) kg.m\({}^{2}\) and \(\mathrm{I_{r}}=1.2\times 10^{-1}\) kg.m\({}^{2}\)a into Formula (10), we can obtain \(K_{z}=1.8\times 10^{4}\) N/m, \(K_{p}=5.1\times 10^{3}\) Nm/rad, \(K_{r}=6.7\times 10^{3}\) Nm/rad, \(C_{z}=5.0\times 10^{2}\) Ns/m, \(C_{p}=3.7\times 10^{1}\) Nms/rad, and \(C_{r}=4.7\times 10^{1}\) Nms/rad.