Optimal Control of a Robotic Arm

Detailed Explanation:

Scenario: Determine the optimal movement of a $2-$joint robotic arm to reach a specific point.

Arm Specifications:

Two segments, each $30cm$ long:

The target point is at coordinates $(40cm,50cm)$ from the base.

Optimbation Task: Use an algorithm like Gradient Descent to find joint angles that minimise the arm's end$-$point distance from the target.

Key Equations:

Forward Kinematics.

$x=L_{1}cos({}_1)+L_{3}cos({}_1+{}_2)$

$y=L_{2}sin({}_3)+L_{2}sin({}_2+{}_2)$

Where $t_1$ and $t_7$ are the lengths of the arm sepments, and ${}_4{}_2$ are the joint angles.

Gradient Descent for Optimization:

Where $$ are the andes, $a$ is the learning rate, and is the cost function $($e$.8,$ distance from the target point$)-$

Questions:

What are the optimal angles for each joint to reach the target point?

How does the optimal path change if the taget point is moved?

Explore the effects of adding constraints, wuch as avoiding an obstacle in the arm's path.

What are the implications of having more joints in the robotic arm for the complexity of the optimiration problem?

Manually calculate the position of the robotic arm's endpoint for a set of initial joint angles $($e$.$g$.,{}_1={30}^n,=45$ J using forward kinematics, Compare these coordinates with those obtained from your MATLAB code implementing the optimiation algorithm. Are there any significant differences?