Forward Kinematics (FK)

Computing the position and orientation of an arm's end effector in space, given all the joint angles.

Forward kinematics uses a robot's geometry — link lengths, joint axis directions — to compute the end-effector pose from the joint angles. The standard method starts at the base and multiplies a 4×4 transformation matrix for each joint and link in turn, all the way out to the tip; the result is the end-effector's pose in the base frame. The classical way to describe each of these per-joint transforms is the set of DH (Denavit-Hartenberg) parameters, introduced by Jacques Denavit and Richard Hartenberg in 1955, which uses 4 parameters per joint; the product-of-exponentials formula is another common approach. For a serial-chain arm, forward kinematics has a unique solution and takes only a handful of matrix multiplications to compute. It's used to display the end-effector's position from encoder readings, to render a robot in simulation, to convert recorded joint data into end-effector pose labels, and it's also the foundation that inverse kinematics builds on.

ExampleFor a planar two-link arm with link lengths l₁ and l₂ and joint angles θ₁ and θ₂, the end-effector coordinates are x = l₁cosθ₁ + l₂cos(θ₁+θ₂), y = l₁sinθ₁ + l₂sin(θ₁+θ₂).

Also called

FK

Related

Inverse Kinematics (IK) · Denavit-Hartenberg (DH) Parameters · Homogeneous Transformation Matrix · Kinematic Chain · Product of Exponentials Formula · Joint Space

Sources

Wikipedia: Forward kinematics
Modern Robotics（Lynch & Park）Ch.4 Forward Kinematics