Functional Electrical Stimulation is a techeology reed

to re$-$animate the muscles of a paralyzed limb$.$ We can

model the horixontal plane $(2$D$,$ no gravity$)$ movemest

of the endpoint of an anm ne$-$animaied by FES by

considering the activity of the bioeps and triceps. This

model can be appeosimuted as a sinuple

mass'spring$/$damper model. Each muscle is modeled as

a spring and a damper in parallel. In this model the

springs act as the force generating elemeat and the

'activation' of the muscle can be changed by varying

the spring or damping constants, and the resting length of the spring. The mass is may also be subjected to a

perturtation Ferensil which is external to the am$.$ The resulting equation is:

Afler some manipulation, we get the differential equation for the position $.$

$\frac{d^2}{dt}=-\frac{(c_D+c_T)}{m}+\frac{d}{dt}-\frac{(k_a+k_T)}{m}++\frac{k_T+L_{\text{v}\text{e}\text{a}\text{t}}+k_a+L_{\text{v}\text{e}\text{n}\text{t}}}{m}+\frac{F_{\text{e}\text{x}\text{i}\text{e}\text{n}\text{a}\text{i}\text{l}\text{i}\text{n}\text{g}\text{u}\text{t}}}{m}$

Where the system vesting length is $L_{\text{r}\text{e}\text{s}\text{t}}=18*(Ac{t}_y-Ac{t}_T),k$ t and $kg$ are the biceps and triceps spring

constants and $c_1$ and $c_9$ are the damping constancs, ghen by

$k_T=Ac{t}_T+{40}_{\text{;}}{k}_B=Ac{t}_a+50$

$c_T=1+\frac{k_V}{6}$;$c_S=1+\frac{k_a}{7}$

Wrike a fusctions FES.m that can be used by an ODE solver to numerically solve this equation w should

be passed through the odexr solver to your fusction. Remember, anything that depends oet tiene MUST

be defined insile your tale funetien. $(3$ pey$).$ Aseame

$F_{zxt}(t)=80a(t-8)e^{-\frac{tt-0|}{}},Ac{t}_z=0\mathrm{.}15,Ac{t}_D=\{\begin{array}{ccc}0,& t,& 10\\ 0\mathrm{.}35,& t>10\end{array}$

You'll also want to gather the actaal time points that the ode solver passes into your ode fanction. You'll

plot these values $($vs iteration number$)$ ia a question belox. One way to do this is to insert code imo your

main function that will save solver times into a vector:

And is eoupled with the following code inerted into your ode function $($assuming you call your

if iserpty $($evalin $(\text{'}$base$\text{'},$ "Eolvertines' $1)$

evalin$("$base$\text{'},[\text{'}$golverTines $|1|={?}^{\text{'}}$ num$2$str$($t$)\text{'}$;$\text{'}])$;

else

end

Another way to do this would be to simply output your independent variable $($the $L^t$ input$)$ into your ode

function to the comenand window and then copy and passe them into a variable once you're done running

the ode solver.

Solve this sysiem for $m-3.$ Assume $(0)=-1\mathrm{.}75,\frac{d}{dx}(0)=0,$ and a daration of $25$ seconds. On a $12$

figure plot: $)$

a$.$ In position $1,$ plot the resulting locarion when the system was solved with odel $50.$

b$.$ In position $2$ plot the resulting locarion when the system was solved with ode $15$ s $0.$

How moch RAM was requied for the cole$450$ solutions? For the ode$15$s$()$ solutions? How much time

was recuired to solve the equation using ode$450$ and edel$5$s$0?$ What was the sixe of the selutions

resulting from the two differem methods? Diseuse which solver was best and why. $(3$ pos$)$

Ples the time values that were passel frem each solver versus iterazion number. Discuss what is happening

urang & $(3$ psy$)$