# Omnidirectional Jump of a Legged Robot Based on the Behavior Mechanism of a Jumping Spider
## 3 Methods
### Kinematics with Redundant DoF
#### 3.1.2 Inverse Kinematics Analysis
Figure 5 provides a sketch of the leg flexion and the extension of the spider. The attitude angle of the foot \(\theta_{t}\) is required to solve the leg joint angles. Thus, the attitude angle of the foot can be obtained by using the relationships between the changes of joint angles and the foot posture during jumping. In the jumping process of the spider, the rapid extension of the leg makes the effective leg length increase rapidly to complete the fast jumping. Simultaneously, the joint angle of the spider leg changes regularly with the effective length, and the attitude angle of the foot vary regularly with the joint angle. Figure 5 shows a sketch of the leg flexion and the extension of the spider. Figure 6 shows the angle curve for the duration of the acceleration phase. During the jump process, the spider completes the jump in a fixed pattern. The effective leg length, the body attitude angle, the joint angle and the foot stance angle determine the maximum values of the take-off form, direction,  
Figure 5: Schematic of the leg at defined instances. Black circles indicate the joint and the red circle indicates the foot. The arrows indicate the movements of the joints. The red line represents the axis of the spider’s body, the blue line represents the leg of spider, and black line represents the ground. height and distance. The leg extends in the jumping direction during its contact phase, which allows the spider to quickly accumulate speed, while the posture of the spider's body changes depending on the jumping direction, and the attitude angle varies with different jumping forms. When a spider jumps upward, the effective stretch length of each leg is the same, and the body posture hardly changes. When the spider jumps sideways, the body has a certain initial roll angle. At the same time, the length of the hind leg is rapidly stretched along the back of the jumping direction and the effective length of the leg increases rapidly. Then, the forelegs stretch in the jumping direction to achieve the sideways jump. In the process of the sideways jump, the length of the hind legs is longer than that of the forelegs of the spider in the jumping direction and, as a result, the rolling angle of the spider is larger. When the spider jumps forward, it has a certain initial pitch angle in the jump direction. At the same time, the hind legs extend rapidly at first; the effective length of the legs increases rapidly, and then the forelegs rapidly extend to achieve the forward jump. In the whole process of the forward jump, the pitch angle of the body posture becomes bigger. In this paper, we study the kinematic of the jumping spider [7] by observing and studying the relationships between the changes of joint angles and effective leg length during the jumping process. Using the experimental data and the curve of the jumping process of the spider, some groups of jumping motion curves are analyzed, and then several sets of the joint angle curve are fitted out in the interval angle of each joint. According to the D-H kinematic model in the last section and the definition of joint angles, the mathematical relationship between the joint angles of the jumping spider and the joint angles of the robot can be obtained. A suitable angle curve and the attitude angle of the foot are used for the spider-inspired robot. Figure 6 shows the angle curve of the femur-patella joint, femur-patella joint, the metatarsus-tarsus joint and the attitude angle of the foot for the robot during the acceleration phase. Similarly, in the robot's acceleration phase during take-off, the angles of each joint change with the extension of the leg, including the angle of the femoral-patella joint which increases gradually by approximately \(90^{\circ}\). The tibia-metatarsal angle decreases gradually by approximately \(55^{\circ}\); the metatarsus-tarsus angle decreases gradually and the angle varies by \(35^{\circ}\); and the attitude angle of the feet is kept within \(70\)-\(80^{\circ}\).