$"""$

Please use the class structure below, and make sure your code can run the

two tests included. You may not use any kinematics libraries in your solution,

although you may find it helpful to compare the outputs you calculate with the

results from those libraries.

$"""$

import numpy as np

class Robot:

def $\_\_$init$\_\_($self$)$:

$\text{'}\text{'}\text{'}$TODO: write code to initialize your robot

$\text{'}\text{'}\text{'}$

pass

def forward$\_$kinematics$($self$,$joints$)$:

$\text{'}\text{'}\text{'}$TODO: define this method to calculate the end effector position with

respect to the reference frame of the base. It accepts a list $($or numpy

array$)$ of joint angles $[$theta$1,$ theta$2,$ theta$3]($as drawn above$)$ as input,

and returns the $($x$,$ y$,$ z$)$ position of the end effector in $3-$D space'$\text{'}\text{'}$

pass

def inverse$\_$kinematics$($self$,$ pos$)$:

$\text{'}\text{'}\text{'}$TODO: define this method to calculate the robot's joint angles that yield

a desired position pos $=($x$,$ y$,$ z$)$ in $3-$D space. This method should

return an array of joint angles, $[$theta$1,$ theta$2,$ theta$3].$ You may use

either of the numerical methods for IK introduced in the course in your

implementation. Tune the number of update steps and step sizes $($i$.$e$.$ values

of lr or alpha$)$ so that your code passes the test cases.$\text{'}\text{'}\text{'}$

pass

Describe any limitations of LQR and its extensions. For which classes of control problems are LQR$-$based methods useful, and for which might we need to look to another family of methods?

Suppose that you are presented with a robot manipulator and asked to write a numerical LQR solver from scratch for a balancing problem with this manipulator. Describe all of the steps that you will need to take to accomplish this task.