Problem $4-$ State$-$space representations and graphs $-$ systems with $2-$port

elements

The system shown below is a schematic representation of the dynamics of a powered

lower extremity exoskeleton for rehabilitation after stroke. From right to left, the system

includes an actuator whose rotation is converted into linear displacement of a rack,

which is then transferred into the rotation of a joint $($not shown in the schematic$).$ The

motion of the joint is then transferred to a cuff with inertia $m_c$ via a connection with

stiffness $k_r,$ and the cuff is then secured to the participant leg $($inertia $m_u).$ The connection

between the cuff and the participant's leg is compliant due to the soft tissue of the

participant's calf$/$thigh$,$ and the intrinsic compliance of the cuff, as modeled by spring $k_c$

and damper $b_c.$

The robot applies a driving torque $T_r,$ the associated rotary speed is ${}_m.$ As a

consequence, the rack translates at a velocity dependent on the rack$/$pinion

transmission constant $r=\frac{\text{r}\text{a}\text{c}\text{k}\text{t}\text{r}\text{a}\text{n}\text{s}\text{l}\text{a}\text{t}\text{i}\text{o}\text{n}}{\text{p}\text{i}\text{n}\text{i}\text{o}\text{n}\text{r}\text{o}\text{t}\text{a}\text{t}\text{i}\text{o}\text{n}}.$ During the swing phase, the participant applies

force $f_u$ to contribute to his own walking, while during the stance phase, the input from

the participant can be modeled as a velocity source, with $v_u=0.$ For the stance phase

$($participant velocity is equal to zero$),$ please complete the steps below

Draw a linear graph of the model, marking all the variables of interest

Write the constitutive equations for all elements, make the appropriate

simplifications for elements in parallel, and write node equations for the three

nodes with unknown velocity

Write a state space model for the system, choosing $v_c$ and $f_u$ as outputs $($output

velocity and force$),$ and $T_r$ as input.

What is the order of the resulting system? Provide a rationale for your response.

Simulate the model using matlab function Isim, for a sinusoidal input torque $T_r$

$($amplitude: $10$ Nm $,$ frequency: $2$ Hz $).$ For the numerical simulation, assume $J_r=$

$0\mathrm{.}1kg*m^2,b_r=0\mathrm{.}01N*\frac{s}{m},r=0\mathrm{.}1\frac{m}{r}ad,k_r=200\frac{N}{m},k_c=2\frac{N}{m},b_h=0\mathrm{.}1$

$N*\frac{s}{m},b_c=0\mathrm{.}1N*\frac{s}{m}{m}_u=2kg,m_c=0\mathrm{.}1kg.$

Calculate numerically the Bode plot of the transfer function $H(s)=\frac{v_c(s)}{T_r(s)}$

Using Matlab's root locus function, calculate the maximum gain that can be used

for a proportional controller that updates input $T_r$ based on the measurement of

$v_c$ resulting in stability.