The biceps and triceps can be modeled by

the spring mass damper system shown in the figure

to the left and the differential equation below.

$\frac{d^{}}{dt}=-\frac{(c_B+c_T)}{m}**\frac{d}{dt}-\frac{(k_B+k_T)}{m}**+\frac{k_T**L_{\text{r}\text{e}\text{s}\text{t}}+k_B**L_{\text{r}\text{e}\text{s}\text{t}}}{m}+\frac{F_{\text{e}\text{x}\text{t}\text{e}\text{r}\text{n}\text{a}\text{l}\text{i}\text{n}\text{p}\text{u}\text{t}}}{m}$

where the system resting length is $L_{\text{r}\text{e}\text{s}\text{t}}=12**(Ac{t}_B-Ac{t}_T),$

$kT$ and $kB$ are the biceps and triceps spring constants and $CT$ and $CB$ are the damping constants, given by

$k_T=Ac{t}_T**40$;$k_B=Ac{t}_B**20$

$c_T=1+\frac{k_T}{5}$;$c_B=1+\frac{k_B}{5}$

We were previosusly given activations $Ac{t}_T$ and $Ac{t}_B$ and asked to find the response. In this lab, we will observe

experimental data of the response of the arm to stimulation, and use lsqcurvefit and odeXX to estimate the mass

of the arm $m,$ as well as the activation levels that resulted in the observed response.

In MATLAB: Load FESdata.mat. This should create four variables in the workspace.

Model system using armPosition$1.(3$ pts$)$

a$.$ Estimate values for $m,Ac{t}_T$ and $Ac{t}_B$ that would result in the observed position response

armPositionI. Do this using a for loop that iterates $25$ times. Remember:

i$.$ Assume the external perturbation that created this response was $f_{\text{e}\text{x}\text{t}}=80u(t)e^{-\frac{t}{3}}.$

ii$.$ Also assume zero initial conditions for position and velocity.

iii. When passing the initial guesses to lsqcurvefit, pass new random values each iteration.

iv$.$ Save all results from Isqcurvefit from every iteration.

v$.$ MAKE SURE YOUR BOUNDS, INITIAL GUESSES, AND PARAMETERS ARE IN THE

SAME ORDER!

b$.$ Find the iteration that produced the results with the least error, that is$,$ with a resnorm as close to $0$ as

possible. This is your best model. Then, in Figure $1,$ a $3$ row $x2$ column subplot:

i$.$ In position $1$: Plot the experimental data armPosition I vs time as blue dots, and the response

of your BEST model vs time as a solid black line on the same axis.

ii$.$ In position $2$: Plot your residuals $($not your resnorm$)$ for your BEST model, also vs time.

iii. For both plots, be sure to label your axes. Create a title that has the values of $m,Ac{t}_T$ and

$Ac{t}_B$ for the left plot and a title with the amount of error in the mode $($resnorm$)$l for the right

plot.

c$.$ Create a second figure, a $31$ subplot:

i$.$ Plot the amount of error $($resnorm$)$ versus each call to lsqcurvefit.

Redo for armPosition$2.$ Use subplot $2$ of Figure $2$ and subplots $3$ and $4$ of Figure $1.(1pt)$

Redo for armPosition $3.$ Use subplot $3$ of Figure $2$ and subplots $5$ and $6$ of Figure $1.(1$ pt$)$