Robotics Arms Kinematics Class II (2D 3R links and 3D links)

Useful Links:

Resources, Slides and Guides: https://github.com/Bakel-Bakel/aurora...
Extra Study Material: https://github.com/Bakel-Bakel/aurora...
Workshop Practicals: https://github.com/Bakel-Bakel/aurora...
Workshop Practical Solutions: https://github.com/Bakel-Bakel/task-4...
Workshop Docs: https://github.com/Bakel-Bakel/roboti...

About Video:

The video discusses forward kinematics for a 3D robotic arm, starting with a review of a 2D two-link (L1, L2) system from a previous class.

The presenter shows how to determine the final position (X, Y) for a two-link arm, where:
$Y = L1 \sin(\theta_1) + L2 \sin(\theta_1 + \theta_2)$
$X = L1 \cos(\theta_1) + L2 \cos(\theta_1 + \theta_2)$

The video then explores adding a third link (L3). By breaking the final position into components (X1, X2, X3 and Y1, Y2, Y3) using trigonometry and corresponding angles, the following equations for the three-link arm's position are derived:

Big X $= L1 \cos(\theta_1) + L2 \cos(\theta_1 + \theta_2) + L3 \cos(\theta_1 + \theta_2 + \theta_3)$

Big Y is expressed similarly with sine functions:
$Y = L1 \sin(\theta_1) + L2 \sin(\theta_1 + \theta_2) + L3 \sin(\theta_1 + \theta_2 + \theta_3)$.

(Note: The speaker says "cos" instead of "sin" near the end of the Y formula's derivation, but the overall structure aligns with using sine for the Y-component in forward kinematics.)

The process is described as a "very repetitive process", and a solution for a four-link arm follows the same pattern. The speaker notes that while the analytical or geometrical method (tracing lines and using trigonometry) works for 2D, it becomes "super complicated real quick" for 3D space due to the need to trace variables across multiple planes (X, Y, Z axis).

To address the complexity of 3D, the video introduces a new mathematical approach based on frames and perspective.
Frames: A frame is described as an axis or reference point. The position of a point (like a ball, P) is relative to the observer's frame.

Transformation: To move from one frame (the initial, fixed frame) to another (a rotated frame), two things must occur: translation (linear movement) and rotation.

The video focuses first on the mathematics of pure rotation. By analyzing the coordinates of a point P (PX, PY) in a non-rotated frame and P' (PX', PY') in a frame rotated by angle $\alpha$, the speaker derives two fundamental equations relating the coordinates of the same point in the two different frames:

$P_X = P'_X \cos(\alpha) - P'_Y \sin(\alpha)$
$P_Y = P'_X \sin(\alpha) + P'_Y \cos(\alpha)$

These two equations are then converted into a single rotation matrix equation:

$$\begin{bmatrix} P_X \ P_Y \end{bmatrix} = \begin{bmatrix} \cos(\alpha) & -\sin(\alpha) \ \sin(\alpha) & \cos(\alpha) \end{bmatrix} \begin{bmatrix} P'_X \ P'_Y \end{bmatrix}$$

The speaker explains that this matrix is used to move from geometry into matrix mathematics, which simplifies handling the variables. This rotation matrix handles the rotational aspect, while the position matrix on the left handles the translational (X and Y) aspect. The goal is to build upon this matrix method to solve the 3D problem.