# Omnidirectional Jump of a Legged Robot Based on the Behavior Mechanism of a Jumping Spider
## 3 Methods
### Locomotion Planning  
#### 3.2.1 Path Planning  
There are three phases in the whole jumping process: take-off, flying and touchdown. In this paper, it is mainly the take-off phase of the robot that is studied. The take-off phase refers to the whole movement process of the robot, from a stationary state to jumping away from the ground. In other words, from the moment the robot foot has been subjected to the ground force \(F_{S}\) in the take-off phase until the moment the robot's feet leave the ground. The take-off process of jumping consists of flexing the leg to store energy and extending to release energy. In the first phase, the legs flex and the CoG drops. Then, the initial attitude angle is adjusted with stored energy to take off. In the phase of releasing energy, the robot legs extend, the body mass center rises, and the robot adjusts to reach a proper take-off stance. Then, the legs continue to extend, hit the ground violently and cause impact force, which makes the robot accelerate and realize the jumping movement. Therefore, the trajectory of the mass center decreases slowly and then rises rapidly. When the robot jumps, the speed of the CoG is an important index of the take-off performance of a robot, as it determines the jumping height and the jumping distance when the robot jumps off the ground. Suppose that the moment the robot takes off from the ground is \(t_{f}\), the velocity of the CoG is \([\dot{X}(t)\dot{Y}(t)\dot{Z}(t)]^{\mathrm{T}}\), and the acceleration is \([\ddot{X}(t)\ddot{Y}(t)\ddot{Z}(t)]^{\mathrm{T}}\). The foot of the robot generates the impact force via continuous contact with the ground so that the robot will accumulate acceleration and speed and finally realize the jumping task. Figure 6: Angle curve for the duration of the acceleration phase: (**a**) femur–patella angle; (**b**) femur–patella angle; (**c**) metatarsus–tarsus angle; and (**d**) attitude angle of the foot. The red dashed line indicates the optimal curve, and the shadow area indicates the range of the angle of the joint indicated by the Quartile Range (IQR) method. The black solid line indicates the optimal curve of the attitude angle of the foot. The other solid line indicates the angle curve for the duration of the acceleration phase. In this paper, three different types of jumping--vertical jumping, side jumping and forward jumping--are discussed. The position vector of the robot foot relative to the base coordinate is \([\mathrm{X}\,\mathrm{Y}\,\mathrm{Z}]^{\mathrm{T}}\). When the robot jumps vertically, the whole body has an upward acceleration and velocity relative to the ground. The robot has acceleration and velocity in the Z direction. When the robot jumps sideways, the robot has acceleration and velocity in the X and Z direction. When the robot jumps forward, the robot has acceleration and velocity only in the Y and Z direction. Thus, the constraint conditions of the motion of the CoG in these jumping forms of the robot must be discussed. At first, the reacting force of the ground to the foot of the robot gradually decreases to zero when the foot of the robot leaves the ground. At this instant, the contact force of the foot fulfills the condition  
\[\mathrm{F_{gx}}=\mathrm{F_{gy}}=\mathrm{F_{gz}}=0. \tag{13}\]  
The acceleration of CoG fulfills the following constraints:  
\[\left. \begin{array}{c}\ddot{\mathrm{X}}_{\mathrm{f}}=0\\ \ddot{\mathrm{Y}}_{\mathrm{f}}=0\\ \ddot{\mathrm{Z}}_{\mathrm{f}}=-\mathrm{g}\end{array} \right\}. \tag{14}\]  
At the take-off moment \(\mathrm{t_{f}}\), the robot accumulates sufficient velocity to achieve three types of jumps and the robot leaves the ground. The velocity constraints are as shown in Table 2. Here, \(\mathrm{V_{xuf}}\), \(\mathrm{V_{yuf}}\), and \(\mathrm{V_{zuf}}\) denote the velocity of the vertical jump. \(\mathrm{V_{xsf}}\), \(\mathrm{V_{ysf}}\), and \(\mathrm{V_{zsf}}\) denote the velocity of the sideways jump. \(\mathrm{V_{xff}}\), \(\mathrm{V_{yff}}\), and \(\mathrm{V_{zff}}\) denote the velocity of the forward jump. Therefore,  
\[\left.