# Design and Demonstration of a Flying-Squirrel-Inspired Jumping Robot with Two Modes
## 4 Design of Jumping Robot  
### Hind Limbs  
According to the above analysis of the movement characteristics of flying squirrels' hind limbs, a parallel four-bar linkage mechanism with springs is used to mimic the leg structure of flying squirrels. As shown in Figure 9, the linkage rod \(ac\), \(bd\), \(de\), \(ae\) are connected by hinges successively, where the length \(l_{ab}\) and \(l_{de}\) are equal; the length \(l_{ae}\) and \(l_{bd}\) are equal; the ratio of the length \(l_{de}\), \(l_{bd}\), and \(l_{bc}\) is designed according to the length ratio of the flying squirrel's femur, tibiofibrular and tarsal, approximately \(l_{de}:l_{bd}:l_{bc}=1:1:1\). The angles \(\beta_{2}\) (corresponding to the angle of ankle joint) and \(\beta_{3}\) (corresponding to the angle of knee joint) are set to have a constant difference, namely \(\beta_{2}=\beta_{3}+20^{\circ}\). Using the linkage rod \(ae\) as a frame, the parallel four-bar linkage mechanism \(abde\) can ensure the angles \(\beta_{2}\) and \(\beta_{3}\) have the same rate of change. A spring with a stiffness coefficient of \(k\) is connected between points \(a\) and \(d\), whose initial length and stretched length are \(x_{0}\) and \(x_{ad}\), respectively. Ignoring the mass of each rod and spring, we can assume that the takeoff force \(F_{T}\) passes through the center of mass (point \(O\)) of the entire robot. In order to determine the structural parameters of the hind limbs, we assume that the elastic potential energy is all converted into kinetic energy without dissipative losses during the takeoff process [13]. Although this assumption does not reflect real-life conditions, it allows for a simple and clear view of the relationship between structural parameters and energy, which is convenient for the initial design. The energy equation is thus  
\[\frac{1}{2}kn(x_{ad}-x_{0})^{2}=\frac{1}{2}mV_{0}^{2} \tag{1}\]  
where \(n\) represents the number of springs connected in parallel, \(m\) represents the mass of entire robot, and \(V_{0}\) represents the initial velocity after takeoff. According to the geometric relationship, \(x_{ad}\) can be acquired as  
\[x_{ad}=2l_{de}\cos\frac{\beta_{3}}{2} \tag{2}\]  
where \(\beta_{3}\) is set to minimum value \(30^{\circ}\) and maximum value \(120^{\circ}\), referring to the variation range of angle of the flying squirrel's knee joint. When \(\beta_{3}\) is \(30^{\circ}\), the spring is stretched to its maximum length. In this case, the corresponding \(V_{0}\) is the initial velocity in the mode of jumping for gliding. According to the average initial velocity of flying squirrels as they glide [35], the corresponding \(V_{0}\) is set to 3 m/s, and then the length of linkages can be obtained from (1) and (2). The resulting structure of hind limbs is shown in Figure 10.  
Figure 9: Schematic of structure of hind limbs. Takeoff angle \(\theta_{T}\) (see Figure 9) is defined as the angle between the takeoff force \(F_{T}\) and the horizontal line. In order to ensure that the hind limbs do not skid during takeoff, the horizontal component of the takeoff force \(F_{T}\) should be less than or equal to the maximum static sliding friction between hind limbs and the ground, namely  
\[F_{T}\cos\theta_{T}\leq\mu_{0}F_{T}\sin\theta_{T} \tag{3}\]  
where \(\mu_{0}\) is coefficient of static friction. Equation (3) is further organized to obtain  
\[\mu_{0}\geq\frac{1}{\tan\theta_{T}} \tag{4}\]  
Equation (4) shows that the coefficient of static friction between hind limbs and the ground and the takeoff angle will affect whether the robot can successfully jump. Note that the direction of the takeoff force is perpendicular to the linkage rod \(ac\), and according to the geometric relationship, the takeoff angle can be acquired as  
\[\theta_{T}=260^{\circ}-\beta_{3}-\gamma \tag{5}\]  
where \(\gamma\) (see Figure 9) is the angle between the linkage rod \(ae\) and the horizontal line. It can be seen from (5) that the takeoff angle \(\theta_{T}\) is related to \(\beta_{3}\) and \(\gamma\).