## 3 Dynamic Model of Incremental Step Shift of DIHPS
### HA Dynamic Model  
Double-acting HAs with a double-piston rod are called symmetric actuators because they provide equal forces in both directions. In this HA, the actuator piston areas were identical; therefore, the piston speeds in both directions were the same. Applying [PERSON]'s second law of motion, the equation for the balance of forces in an HA can be written as follows:  
\[m\,\frac{d^{2}s}{dt^{2}}+b\,\frac{ds}{dt}=A\,(p_{1}-p_{2}) \tag{33}\]  
where \(s\) is the step shift of the actuator piston, \(m\) is the load mass, \(b\) is the viscosity coefficient, \(A\) is the effective action area of the actuator piston, and \(p_{1}\) and \(p_{2}\) are the pressure in the actuator chambers. In the dynamic model, the first-order differential equation is used, and then (33) is as follows:  
\[\left\{ \begin{array}{l}v=\frac{ds}{dt}\\ \frac{dv}{dt}=\frac{m}{m}[-b\,v+A\,(p_{1}-p_{2})]\end{array} \right. \tag{34}\]  
where \(v\) is the speed of the actuator piston. Two states will be considered in the simulation modeling, the instantaneous state according to Figure 4a, and the steady state according to Figure 4c. For these states, the balance of the flow rates for both chambers of the actuator can be expressed as follows:  
\[\left\{ \begin{array}{l}q_{v1}=A\,v+C_{1}\,\frac{dp_{1}}{dt}\\ q_{v2}=-A\,v+C_{2}\,\frac{dp_{2}}{dt}+q_{v21}+q_{v22}\end{array} \right.\text{ for}s_{s}\leq s\leq s_{a} \tag{35}\]  
\[\left\{ \begin{array}{l}q_{v1}=A\,v+C_{1}\,\frac{dp_{1}}{dt}+q_{v11}\\ q_{v2}=-A\,v+C_{2}\,\frac{dp_{2}}{dt}+q_{v22}\end{array} \right.\text{ for}s_{a}\leq s\leq s_{c} \tag{36}\]  
where \(p_{1},p_{2}\) are the pressure in the actuator chambers, \(C_{1}\) and \(C_{2}\) are the hydraulic capacitances of the actuator chambers,  
\[C_{1}=\frac{A\,(s_{in}+s)}{K},\,C_{2}=\frac{A\,(S-s_{in}-s)}{K} \tag{37}\]  
where \(s_{in}\) is the initial position of the actuator piston, \(S\) is the stroke of the actuator piston, and \(K\) is the bulk modulus of the hydraulic fluid. The system of Equations (35) and (36) has been converted to the form,  
\[\left\{ \begin{array}{l}\frac{dp_{1}}{dt}=\frac{1}{C_{1}}(q_{v1}-A\,v)\\ \frac{dp_{2}}{dt}=\frac{1}{C_{2}}(q_{v2}+A\,v-q_{v21}-q_{v22})\end{array} \right. \text{ for}s_{s}\leq s\leq s_{a} \tag{38}\]  
\[\  
during the step shift of the actuator piston. Before the simulation test, the selection of the hydraulic resistance at the stop position of the DIHPS was verified based on the hydraulic resistance circuits shown in Figure 8.  
In the stop position of the DIHPS, there is a balance of forces acting on the actuator piston and symmetrical hydraulic resistance on both sides of the piston,  
\[R_{1}+R_{11}=R_{2}+R_{22}=R_{s} \tag{40}\]  
and  
\[q_{v1}\approx q_{v2}\ q_{v}=q_{v1}+q_{v2} \tag{41}\]  
The selection of unknown pressure \(p_{v}\) results from the equivalent hydraulic resistance,  
\[\frac{1}{\sqrt{\frac{R_{v}}{2}}}\sqrt{p_{s}-p_{v}}=\frac{1}{\sqrt{R_{v}}}\sqrt {p_{v}}\Rightarrow\ p_{\rm v}=\left(\frac{2R_{v}}{R_{s}+2R_{v}}\right)p_{s} \tag{42}\]  
In the simulation tests performed in the Matlab&Simulink R2020a software of MathWorks, the following basic parameters of constant values were introduced: \(p_{s}=10\) MPa, \(m=50\) kg, \(b=2500\) Nsm\({}^{-1}\), \(D=0.05\) m, \(d=0.025\) m, \(S=0.1\) m, \(K=1800\) MPa, \(\rho=860\) kg\({}^{-3}\), \(R_{1}=R_{2}=8.26\times 10^{11}\) Pa s\({}^{2}\) m\({}^{-8}\), \(R_{\rm v}=3.9\times 10^{11}\) Pa s\({}^{2}\) m\({}^{-8}\), \(R_{11}=R_{22}=1.80\times 10^{13}\) Pa s\({}^{2}\) m\({}^{-8}\), and the actuator piston step by \(s=0.02\) m, and the piston step shift is \(s_{i}=0.02\), \(0.04\), \(0.06\), \(0.08\), and \(0.1\) m. The full list of simulation parameters is provided in Appendix A Table 1.  
Figure 9 shows the dynamic response parameters of the DIHPS during step incremental positioning, the step shift \(s\) of the actuator piston, the speed \(v\) of the actuator piston, and the pressure \(p_{1}\) and \(p_{2}\) in the actuator chambers.