# Controllable Height Hopping of a Parallel Legged Robot
## 4 Control Strategy
### Height Control Strategy  
An ideal spring-mass model can realize continuous hopping directly without energy consumption. However, as mentioned above, our robot leg with its internal damping does not have this capacity without extra energy input. In this case, we used different desired parameters in VMC at different moments to make the robot hop to a targeted summit height. As shown in Figure 6, our controller includes two VMC modes: fall-down VMC and target-height VMC.  
In the flight and touchdown phase, we used the VMC law with a constant virtual stiffness, damping, and initial leg length to maintain the stability and compliance of the robot. The initial kinetic energy and gravitational potential energy were transformed into damping energy consumption during this touchdown process. However, this process in experimental situation is not linear as it is in simulation-based calculations. Some tiny chattering will occur during this impact process that may even cause vibration in the velocity in the following movement. Therefore, we left out the analysis of the detailed touchdown procedure at this stage and only measured the height at the bottom position \(h_{B}\) as the prior value for subsequent control.  
After the bottom position, the robot enters the takeoff phase. In this phase, we changed VMC into target-height module. Specifically, we proposed a height mapping function based on the modeling in Section 3 and parameter calibration in following simulation and experiment. The prospective hopping height of the next jump can be expressed as:  
\[\hat{h}_{smt}=\mathbf{f}(h_{B},k_{vl}) \tag{14}\]  
where \(\hat{h}_{smt}\) is the estimated hopping height in the next period, a nonlinear function of the bottom height \(h_{B}\) in the previous period and the desired virtual stiffness \(k_{vl}\) for the next takeoff motion. If the reduction ratio is small enough to enable linear representation of the system, the differential equation for the dynamics may be solved and \(\mathbf{f}\) could thus be expressed analytically. However, because of the complexity of the model in (8), it is difficult  
Figure 6: State estimation diagram in three different states in one hopping period.  
to obtain an analytical solution for this mapping relation. To avoid this difficulty, we used simulations with approximate parameters to calculate numerical solutions and fitted this binary function in a mathematic model. The results of the simulation and fitting processes are shown in Section 5. After using mathematical model to fit function \(f\) as in (15) and deducing its analytical solution in (16), we can use the given values bottom height \(h_{B}\) and target summit height \(h_{d}\) as arguments to calculate the required stiffness \(k_{vl}\) in the controller.  
\[\hat{h}_{smt}=\mathbf{f}(h_{B},k_{vl})=p_{0}+p_{1}k_{vl}+p_{2}h_{B}+{p_{3}}{k_{vl}}^ {2}+...+\epsilon \tag{15}\]  
\[k_{vl}=\mathbf{S}(h_{d},h_{B}) \tag{16}\]  
where \(p_{0}\)-\(p_{5}\) are fitting coefficients, \(\epsilon\) is a high-order infinitesimal, \(\mathbf{S}\) is the solution of function \(f\), and \(h_{d}\) is the desired hopping height in the next period. Thus, we get the control law of desired summit height according to (14).