An autonomous robot needs to get from vec$(x{)}_0=(3\mathrm{.}25,3\mathrm{.}75)$ to vec$(x{)}_{N+1}=(1\mathrm{.}25,-3\mathrm{.}75)$ avoiding three ponds. The

ponds are contained within the regions

$(x-4{)}^2+(y-3{)}^2=1,(x-2{)}^2+(-3-y{)}^2=1,$ and $,\frac{x-1}{5^2}+(y+\frac{1}{2}{)}^2=1.$

Construct a barrier to keep the robot out of the pond and use any unconstrained method $($example like fminunc in

MATLAB$)$ to plan the best route such as approximate the route as a chain connecting the points xi to xi$+1.$ You can

implement this by using Python.

Then, with barrier $B(x)$ the objective function is

$f(x)={\displaystyle \underset{i=1}{\overset{N+1}{}}}(x_i-x_{i-1}{)}^2+(y_i-y_{i-1}{)}^2+B(x),$ where $x=[x_i,y_i{]}_1^N.$

You will need to think about how to set up the problem, how to initialize and then ramp up the barrier and

how to initialize the path. Here is the set up: