Robotics Control Numerical Solutions

I have a batch of robotics-control numericals that span trajectory planning, the Jacobian matrix, and both forward and inverse kinematics. Every problem needs a clear, step-by-step derivation of the governing equations followed by the final numeric answer.

My priority is to see the complete chain of reasoning—clean algebra, neatly laid-out matrices, and concise explanations of each assumption—so that I can follow the logic and reuse the method later. To validate the work, include a short MATLAB or Python snippet (your choice) that recreates the calculation and prints the same result. A simple script or Jupyter cell is enough; no elaborate GUI is necessary.

Deliverables
• PDF or Word file with the worked solutions, formatted for easy reading
• Accompanying MATLAB or Python code that reproduces every answer
• Brief comments in the code explaining key lines (e.g., Jacobian build, inverse kinematic solver)

I will supply any specific numeric values or boundary conditions as soon as we start; otherwise, use symbolic variables and note where substitutions occur. Please keep notation consistent throughout (θ for joint angles, q̇ for velocities, etc.).

I’m ready to review the first solved problem as a quick milestone before you proceed with the remainder.