Which statement best describes the role of the Jacobian matrix in robotics?

• A. It computes the end effector pose from joint positions.
• B. It estimates joint torques from external forces.
• C. It maps joint velocities to end effector spatial velocities.
• D. It defines the robot's geometric dimensions.

Answer: (C)

The main idea is that the Jacobian translates how fast the joints are moving into how fast the end effector is moving in space. Mathematically, the end effector’s spatial velocity (its linear and angular velocity, i.e., its twist) is obtained by multiplying the Jacobian by the vector of joint rates: v_end = J(q) qdot. This shows exactly why the Jacobian is used to map joint velocities to end-effector velocities in the world frame.

This concept is central because it explains instantaneous motion: if you know how fast you want the end effector to move, you can compute the necessary joint rates (often via a pseudoinverse) to achieve that motion, at least locally. The Jacobian depends on the current joint angles, so its accuracy is tied to the robot’s configuration and it can become singular when certain motions are not feasible.

Other statements don’t fit as well because they describe different ideas: computing the end effector pose from joint positions is forward kinematics, which deals with pose itself rather than velocity. Estimating joint torques from external forces involves dynamics and the Jacobian transpose or related equations, not the basic mapping of velocities. Defining the robot’s geometric dimensions is just describing geometry, not the velocity relationship captured by the Jacobian.