The system below is used to modulate the angular position of the foot of a transtibial prosthesis. The foot is connected to a hydraulic actuator through a series elastic element $\backslash (\backslash$left$(\backslash$mathrm$\{$k$\}\_\{\backslash$mathrm$\{$sc$\}\}\backslash$right$)\backslash ).$ Fluid in the hydraulic actuator is driven by a positive displacement pump. You can assume the following:

$-$ Torque on the pump shaft is input to the system.

$-$ Gravitational effects on the fluid inertia are negligible.

$-$ The piston and rod of the hydraulic actuator have significant inertia and friction due to seals$/$bearings$.$

$-$ The foot has significant rotational inertia.

$-$ Lump the bearing friction at the pivot between the foot and shank $($between the foot and "ground"$).$

The gear pump can be characterized by a total entrapped fluid volume per revolution of $\backslash ($ V$\_\{$e$\}\backslash ),$ and thus the gear pump constitutive behavior can be characterized by $\backslash ($ Q$=$k$\_\{$p$\}\backslash$omega $\backslash ),$ where $\backslash ($ k$\_\{$p$\}=$V$\_\{$e$\}/2\backslash$pi $\backslash ).$ The hydraulic actuator has a net piston area given by $\backslash ($ A$\_\{\backslash$mathrm$\{$p$\}\}\backslash ).$

$($a$)$ Draw a bond graph of the system. Leave parameters in general terms $($i$.$e$.,$ fluid resistance can be represented by $\backslash (\backslash$mathrm$\{$R$\}\_\{\backslash$mathrm$\{$f$\}\}\backslash ),$ etc.$)$

$($b$)$ Assuming all elements are linear $($including the kinematic transformation between cylinder motion and foot angle$),$ derive the state equations $($utilizing power variables$)$ and put in state space form.

$($c$)$ Reflect the entire bond graph into the domain of the torque input.

$($d$)$ You would like to conduct an analog simulation of the positioning system with an electrical circuit. Draw a schematic of the equivalent electrical circuit, identify all circuit elements in terms of the parameters of part $($a$),$ and identify what the current in each circuit branch represents.