Forward Kinematics Formula

Key Notes

• Definition: Forward kinematics maps a set of joint angles (θ₁, θ₂, …, θₙ) to the position and orientation of the end-effector in 3D space. Given the joint configuration, the end-effector pose is uniquely determined — there is always exactly one answer.
• Homogeneous transformation matrices: Each joint contributes a 4×4 transformation matrix combining rotation and translation. The final end-effector pose is the product: T = T₁ × T₂ × … × Tₙ. Matrix multiplication order matters — T₁T₂ ≠ T₂T₁.
• Denavit-Hartenberg (DH) convention: A standard 4-parameter representation (link length a, link offset d, link twist α, joint angle θ) per joint. DH convention reduces each joint's transformation to a single standardized 4×4 matrix, making systematic robot modeling possible.
• Contrast with inverse kinematics: Inverse kinematics (IK) goes the other direction — given a desired end-effector pose, find the joint angles. IK is much harder: it may have multiple solutions, no solution, or require iterative numerical methods. Forward kinematics always has a unique, direct solution.
• Applications: Forward kinematics is used in robot arm simulation and control, animation rigs in film/games (character joint hierarchies), CNC machine position calculation, and exoskeleton position tracking.