Product of Exponentials Formula

Treats each joint as motion about a screw axis, and multiplies each joint's matrix exponential together to get the end-effector pose.

The product of exponentials formula was introduced by Roger Brockett in 1984, and is a way of describing a serial manipulator's forward kinematics; Lynch and Park's textbook Modern Robotics builds its whole treatment around it. The space form is written T(θ) = e^[S₁]θ₁ ⋯ e^[Sₙ]θₙ M: M is the end-effector's pose when every joint is at zero, Sᵢ is joint i's screw axis expressed in the fixed base frame (6-dimensional, combining a rotation direction and a linear-velocity part), θᵢ is the joint's angle or displacement, and e^[S]θ denotes the rigid-body transformation produced by moving θ along that screw axis. There's also a body form with the screw axes expressed in the end-effector frame, T = M e^[B₁]θ₁ ⋯. Compared with DH parameters, it only needs two frames — base and end effector — treats revolute and prismatic joints uniformly, and has a directly intuitive geometric meaning, at the cost of not using the fewest possible parameters. Jacobians, inverse kinematics, and kinematic calibration can all be derived on top of it.

ExampleA single-joint planar arm: a link of length L rotating about the base's z-axis. At the zero configuration the end effector is at (L, 0, 0), which is M; the screw axis is S = (0, 0, 1, 0, 0, 0). T(θ) = e^[S]θ M then gives the end-effector position (L·cosθ, L·sinθ, 0).

Also called

PoE, Product of Exponentials

Related

Screw Theory · Denavit-Hartenberg (DH) Parameters · Forward Kinematics (FK) · Exponential Map · Lie Group · Kinematic Calibration

Sources

Wikipedia: Product of exponentials formula
Modern Robotics (Lynch & Park), Sec. 4.1 Product of Exponentials Formula