## 2 The Principle Analysis and Modeling for ALS-EHA
### Modeling Investigation
#### 2.2.2 Load-Sensitive Loop
According to the [21], the formula for calculating the stiffness of the back pressure spring of the variable mechanism is shown below:  
\[K=\frac{P_{\text{max}}-P_{0}}{D_{\text{max}}-D_{\text{min}}}\frac{A_{v}K_{p}}{L_{ f}}=G\frac{P_{\text{max}}-P_{0}}{D_{\text{max}}-D_{\text{min}}} \tag{16}\]  
where \(G=\frac{A_{v}K_{p}}{L_{f}}\). By combining formulas (15) and (16), the following equation can be derived. \[D=\left\{ \begin{array}{ll}D_{p\text{max}}&for\quad 0\leq P_{l}\leq P_{0}\\ D_{p\text{max}}-\frac{G}{K}(P_{l}-P_{0})&for\quad P_{0}\leq P_{l}\leq P_{\text {max}}\end{array} \right. \tag{17}\]  
As shown in Equation (16), the stiffness of the compressed spring of the variable mechanism for DLS-EHA must be selected according to the maximum load pressure, therefore, K should be selected large enough to ensure the stability of the system. According to Figure 6 and Equation (17), it can be concluded that the gradient of displacement to load pressure is a constant and cannot be adjusted which cannot guarantee that under all load conditions, the system performance such as efficiency and dynamics are optimal. the load-sensitive circuit has passive attribute. PFV has the same principle as the proportional pressure reducing valve, of which output pressure can be linearly controlled by actively controlling valve input current. the PFV characteristics are shown below Figure 7. So, it can be modeled as follows  
\[P_{A}=P_{l}-K_{v}i_{v} \tag{18}\]  
where \(K_{v}\), \(i_{v}\) are pressure gain and input current of the valve, respectively. Load sensitivity is expressed below:  
\[\alpha=\frac{P_{A}}{P_{l}} \tag{19}\]  
Let \(k_{a}\) be equal to \(1-\alpha\). Figure 6: Relationship between displacement and load pressure for DLS-EHA. Figure 7: The PFV characteristics curve. Equation (14) can be written as:  
\[P_{A}=P_{l}(1-k_{\alpha}) \tag{20}\]  
**Remark 2**. : _When the input pressure \(P_{l}\) is constant, the output pressure is a linear function of the input current. As the input current grows, the output pressure declines. \(P_{l}\) affects the gradient value of the output._  
Based on the ALS principle combined with the characteristics of the DLS variables, we can draw the relationship of the characteristics of the ALS variables as shown follows:  
\[D=\left\{ \begin{array}{ll}D_{\text{pmax}}&for\quad 0\leq\alpha P_{l}\leq P_{0} \\ D_{p\text{max}}-\frac{G}{K}(\alpha P_{l}-P_{0})&for\quad P_{0}\leq\alpha P_{l} \cap P_{l}\leq P_{\text{max}}\end{array} \right. \tag{21}\]  
As shown in Figure 8, a PFV capable of active control is added to the original DLS loop. the stiffness of the spring is still selected according to the DLS. It can be concluded that the gradient of displacement to load pressure varies with the load sensitive, that is, it can be actively adjusted. the greater the load sensitivity is, the greater the gradient is. We can also see that the EHA-FPVM and the DLS-EHA are two boundaries of active load sensitivity. the former has a load sensitivity of 0 and the latter is 1. In the ALS loop, we choose the stiffness of the back pressure spring according to the following formula (22). \[K=\frac{\alpha P_{\text{lmax}}-P_{0}}{D_{\text{max}}-D_{\text{min}}}\frac{A_{ v}K_{p}}{L_{f}}=G\frac{\alpha P_{\text{lmax}}-P_{0}}{D_{\text{max}}-D_{\text{min}}} \tag{22}\]  
It can be known from the formula 22 that the spring stiffness for ALS-EHA can be less than that for DLS-EHA which is conducive to increasing the adjustment range and is beneficial to the improvement of system performance and the flexibility of control. After the compressed spring stiffness is corrected, the characteristic curve for ALS-EHA is shown in Figure 9 below. Figure 8: Load-sensitive characteristics for ASL-EHA.