## 2 The Principle Analysis and Modeling for ALS-EHA
### Modeling Investigation  
#### 2.2.1 Motor Pump Main Loop  
For a brushless DC motor, the torque dynamics balance equation is:  
\[T_{m}=K_{t}i_{m}=J_{m}\dot{\omega}_{m}+B_{m}\omega_{m}+T_{l} \tag{1}\]  
where \(K_{t}\) is the motor torque constant; \(J_{m},B_{m}\) is the total inertia moment and damping coefficient equivalent to the motor shaft, respectively; \(T_{l}\) is the motor output torque (pump input torque), which drives the pump to rotate; \(i_{m}\), \(T_{m}\), \(\omega_{m}\) represent motor current, electromagnetic torque, and rotational speed, respectively.  
Without considering leakage, the system flow equation is shown as follow:  
\[Q_{l}=A_{a}\cdot v_{a}=D_{p}\cdot\omega_{m} \tag{2}\]  
where \(Q_{l}\) is load flow of the system; \(A_{a},v_{a}\) represent the piston area and speed of the cylinder, respectively; \(D_{p}\) is pump displacement. the equation of pump input torque is obtained from load pressure and displacement:  
\[T_{l}=D_{p}P_{l} \tag{3}\]  
where \(P_{l}\) is load pressure of the system.  
(1) In the stalling condition, combining formulas (1) and (3), the follow equation can be gotten  
\[K_{t}i_{m}=D_{p}P_{l} \tag{4}\]  
At this time, the motor current is proportional to the pump displacement, that is, the larger the displacement, the more heat the motor generates in the stall condition.  
(2) In non-stalling mode, for purpose of simplifying the calculation, it is assumed that the angular acceleration of the motor is known. Equation (5) can be obtained by combining formulas (1) and (3).  
\[K_{t}i_{m}=-J_{m}\frac{A_{a}\dot{x}_{a}\dot{D}_{p}}{D_{p}^{2}}+\left(\frac{J_{ m}A_{a}\ddot{x}_{a}+B_{m}A_{a}\dot{x}_{a}}{D_{p}}+D_{p}P_{l}\right) \tag{5}\]  
Figure 4: ALS-EHA schematic.  
where \(x_{a}\) is actuator displacement. Observing formula (5), the part in parentheses shows a logarithmic function relationship. It can be seen that, in a certain range under this operating condition, the motor current and the pump displacement are approximately a logarithmic function relationship, i.e., the current gradually decreases to minimum first and then increases as the displacement goes down.  
The pump in EHA is equivalent to a reducer without considering pump loss, and the hydraulic reduction ratio is defined as \(k_{h}=v_{a}/\omega_{m}\). Under the law of conservation of power, \(T_{l}\omega_{m}=F_{l}v_{a}\), combined \(T_{l}=D_{p}P_{l}\) with \(F_{l}=P_{l}A_{a}\), we get \(k_{h}=v_{a}/\omega_{m}=T_{l}/F_{l}=D_{p}P_{l}/P_{l}A_{a}=D_{p}/A_{a}\) which is the ratio of the pump displacement to the actuator area. Formula (5) can be changed as follows:  
\[K_{li_{m}}=-J_{m}\frac{\dot{x}_{a}\dot{k}_{h}}{k_{h}^{2}}+\left(\frac{I_{m} \ddot{x}_{a}+B_{m}\dot{x}_{a}}{k_{h}}+k_{h}A_{a}P_{l}\right) \tag{6}\]  
where \(F_{l}\) is the output force (load force) of the actuator. Formula (6) shows that the hydraulic reduction ratio can be controlled by adjusting pump displacement to minimize motor current.  
In addition, if the iron loss of the motor and various efficiency losses of the pump are comprehensively considered, the relationship between the total heat generation of the motor and the displacement of the pump is complicated, and further modeling research is needed. For two control variable ALS-EHA, motor velocity and pump displacement can be actively controlled. Therefore, by designing a reasonable control law, e.g., the fuzzy neural control method of lifting system dynamics as proposed in Ref. [24], it is possible to effectively reduce the motor current and lower the motor heating. This is the basic idea of power matching design.