From Angles to Position
Forward kinematics is the foundational calculation in robotics: given the angle of every joint, where is the end-effector? For a serial robot arm, each joint adds a rotation (and possibly a translation) to the kinematic chain. The end-effector's position and orientation are the cumulative effect of all joints. This simulation visualizes a 3-link planar arm where you directly control each joint angle.
The Kinematic Chain
A planar arm with N revolute joints and equal link lengths L has a beautifully simple FK equation. The end-effector x-coordinate is the sum of L·cos(cumulative angle) for each link, and similarly for y. Each link's orientation is the running sum of all joint angles up to that point. Despite this simplicity, the resulting workspace — the set of all reachable positions — has complex boundaries determined by joint limits.
Transformation Matrices
In 3D robotics, FK is computed by multiplying 4×4 homogeneous transformation matrices — one per joint. Each matrix encodes the rotation and translation introduced by that joint. The Denavit-Hartenberg convention provides a systematic way to define these matrices using just four parameters per joint. The product of all matrices gives the end-effector's pose in the base frame.
Workspace Visualization
The workspace of a robot arm — the volume its end-effector can reach — depends on link lengths, joint limits, and the number of DOF. For a 2D arm, the workspace is the swept area as all joints rotate through their full range. Understanding workspace geometry is critical for robot placement in manufacturing cells, ensuring the arm can reach all required work points without singularity or collision issues.