# Design and Demonstration of a Flying-Squirrel-Inspired Jumping Robot with Two Modes
## 5 Pitching Requirement and Adjustment
### Pitching Requirement  
According to the locomotion conceiving of flexible wing-limb blending platform, the robot needs to unfold the main membrane at the top after takeoff and then to glide. The ratio of the horizontal distance to the vertical height in the whole gliding process is called the glide ratio, which is one of the key indicators of gliding performance [39]. The larger the value, the better the gliding performance. Therefore, the aerial posture of the robot at the top should meet the requirements for the robot to obtain the maximum glide ratio. In order to facilitate the analysis, the influence of different pitching posture on the glide ratio is only considered in this paper. As shown in Figure 17, the rectangular coordinate frames \(\{G\}\) is built. Assuming that the robot is located at a point on the gliding trajectory at time t, its position coordinate is \((x(t),y(t))\); its speed is \(v(t)\); the angle between its speed and horizontal line is \(\theta(t)\); the angle between its body axis \(BC\) and horizontal line is \(\phi_{B}(t)\); the angle of attack is \(\alpha(t)\). The forces on the robot are the gravity \(mg\), drag force \(F_{D}(t)\) and lift force \(F_{L}(t)\). The direction of drag force is opposite to the direction of speed,  
Figure 16: Trajectory of the end of the hind limbs when loading. and the direction of lift force is perpendicular to the direction of speed. According to the D'Alembert's principle, the equilibrium equations of forces can be expressed as  
\[\left\{ \begin{array}{l}m\ddot{x}(t)=-F_{D}(t)\cos\theta(t)+F_{L}(t)\sin\theta(t )\\ m\ddot{y}(t)=-mg+F_{D}(t)\sin\theta(t)+F_{L}(t)\cos\theta(t)\end{array} \right. \tag{13}\]  
According to the theory of aerodynamics, \(F_{D}(t)\) and \(F_{L}(t)\) can be acquired as  
\[\left\{ \begin{array}{l}F_{D}(t)=\frac{1}{2}C_{D}S\rho v^{2}(t)\\ F_{L}(t)=\frac{1}{2}C_{L}S\rho v^{2}(t)\end{array} \right. \tag{14}\]  
where \(C_{D}\) and \(C_{L}\) represent drag coefficient and lift coefficient, respectively; \(S\) and \(\rho\) denote the area of the wing membranes and the density of air, respectively. The relationship between drag, lift coefficients, and the attack angle can be expressed as [41]  
\[\left\{ \begin{array}{l}C_{D}=2\sin^{2}\alpha(t)\\ C_{L}=2\sin\alpha(t)\cos\alpha(t)\end{array} \right. \tag{15}\]  
Assuming that the robot can use its tail membrane to achieve pitching moment balance like a flying squirrel during gliding [42], the pitch angle \(\phi_{B}(t)\) remains constant, and  
\[\phi_{B}=\alpha(t)-\theta(t) \tag{16}\]  
Since,  
\[\left\{ \begin{array}{l}\cos\theta(t)=\frac{\dot{x}(t)}{\sqrt{\dot{x}^{2}(t) +\dot{y}^{2}(t)}}\\ \sin\theta(t)=\frac{\dot{y}(t)}{\sqrt{\dot{x}^{2}(t)+\dot{y}^{2}(t)}}\end{array} \right. \tag{17}\]  
Equation (13) can be further arranged as  
\[\left\{ \begin{array}{l}\ddot{x}(t)=\frac{C_{0}C_{1}\dot{x}^{3}(t)+C_{0}(C_{2 }-C_{4})\dot{x}(t)\dot{y}^{2}(t)}{\sqrt{\dot{x}^{2}(t)+\dot{y}^{2}(t)}}\\ -\frac{1}{2}C_{0}C_{3}\frac{\dot{x}^{2}(t)\dot{y}(t)+\dot{y}^{2}(t)}{\sqrt{ \dot{x}^{2}(t)+\dot{y}^{2}(t)}}\\ \ddot{y}(t)=-g+\frac{C_{0}C_{2}\dot{y}^{3}(t)+C_{0}(C_{1}+C_{4})\dot{x}^{2}(t) \dot{y}(t)}{\sqrt{\dot{x}^{2}(t)+\dot{y}^{2}(t)}}\\ -\frac{1}{2}C_{0}C_{3}\frac{\dot{x}^{2}(t)+\dot{x}(t)\dot{y}^{2}(t)}{\sqrt{ \dot{x}^{2}(t)+\dot{y}^{2}(t)}}\end{array} \right. \tag{18}\]  
Figure 17: Forces on the robot when gliding. where \(C_{0}=-\frac{\rho S}{m}\), \(C_{1}=\sin^{2}\phi_{B}\), \(C_{2}=\cos^{2}\phi_{B}\), \(C_{3}=\sin(2\phi_{B})\), and \(C_{4}=\cos(2\phi_{B})\). It is the second-order ordinary differential equation set of \(x(t)\) and \(y(t)\).