01
Forward Kinematics
Forward kinematics determines the pose of a robot's end-effector from its joint variables.
T(θ) = T_1(θ_1)T_2(θ_2)...T_n(θ_n)
For serial manipulators, the transformation chain expresses how each joint contributes to the final rigid-body pose.
02
Inverse Kinematics
Inverse kinematics solves the reverse problem: determining joint configurations capable of producing a desired end-effector pose.
- Analytical solutions can be exact and computationally efficient when available.
- Numerical methods iteratively reduce pose error.
- Redundant robots can have multiple valid solutions.
- Joint limits, collisions, and singularities constrain feasible solutions.
03
Jacobians and Singularities
V = J(θ) θ̇
The Jacobian maps joint velocities to end-effector velocity. It also relates endpoint forces and torques to joint-level quantities.
At a singular configuration, the Jacobian loses rank and certain Cartesian motions become difficult or impossible to produce.