A robot's purpose is to move its end-effector to a specific position and orientation in space. But we don't control the end-effector directly. We control the individual joints. The central question is: how do the speeds of the individual joints combine to produce the final linear and angular velocity of the end-effector? The answer lies in the Geometric Jacobian

Think of it as a linear map. It takes a vector of joint velocities from the robot's configuration space and transforms it into a six-dimensional vector representing the end-effector's velocity in task space. This velocity vector is known as a twist, comprising three linear velocity components and three angular velocity components. This isn't just a simple multiplication; it's a mapping between the tangent space of the configuration space manifold and the tangent space of the special Euclidean group SE(3), which represents all possible rigid body motions.

For a serial manipulator, the end-effector's velocity is the sum of the velocities induced by each joint moving up the kinematic chain. We can build the Jacobian column by column, where each column represents the contribution of a single joint to the final twist. The structure of the column depends on whether the joint is revolute (rotating) or prismatic (sliding).

For a revolute joint that rotates about an axis, its motion contributes both linear and angular velocity to the end-effector. The angular velocity contribution is straightforward: it's simply the axis of rotation, scaled by the joint's angular velocity. The linear velocity contribution at the end-effector is caused by this rotation. It's calculated as the cross product of the joint's rotation axis and the position vector pointing from the joint to the end-effector. This is the tangential velocity you'd expect from basic physics.