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.

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

Key Equations:

Forward Kinematics:

$x=L_{1}cos({}_1)+L_{2}cos({}_1+{}_2)$

$y=L_{1}sin({}_1)+L_{2}sin({}_1+{}_2)$

Where $L_1$ and $L_2$ are the lengths of the arm segments, and ${}_1,{}_2$ are the joint angles.

Gradient Descent for Optimization:

${}_{\text{n}\text{e}\text{w}}={}_{\text{o}\text{l}\text{d}}-gradJ()$

Where $$ are the angles, $$ is the learning rate, and is the cost function $($e$.$g$.,$ 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 target point is moved?

Explore the effects of adding constraints, such 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 optimization problem?

Manually calculate the position of the robotic arm's endpoint for a set of initial joint angles $($e$.$g$.,{}_1=30,{}_2=45)$ using forward kinematics. Compare these coordinates with those obtained from your MATLAB code implementing the optimization algorithm. Are there any significant differences?