What is the Jacobian in robotics, and what is its role in velocity kinematics?

• A. The Jacobian maps joint velocities to end-effector linear and angular velocity; J * q_dot = v_end_effector.
• B. The Jacobian maps end-effector velocity to joint accelerations.
• C. The Jacobian measures the stiffness of joints.
• D. The Jacobian is the inverse of the transformation matrix between frames.

Answer: (A)

In velocity kinematics, the key idea is that joint motions produce motion at the end-effector, and the Jacobian captures exactly how fast that happens. The Jacobian is the matrix that relates joint rates to the end-effector’s velocity. Put simply, the end-effector’s linear and angular velocity, collected as a 6-element twist, equals the Jacobian times the vector of joint velocities: [v; ω] = J(q) q_dot. This relationship shows how each joint’s speed contributes to both translation and rotation of the end-effector, with J depending on the current joint positions q.

Why this matters: for revolute joints, rotating about an axis causes a contribution to the end-effector’s velocity proportional to the distance from the axis; for prismatic joints, joint translation directly affects linear velocity. The Jacobian therefore encodes the geometry and configuration of the manipulator to give a complete velocity description.

The other statements don’t fit because the Jacobian itself is not a tool for mapping end-effector velocity to joint accelerations (that involves J_dot q_dot plus J q_double_dot), it doesn’t measure stiffness, and it isn’t simply the inverse of a frame transformation. Its fundamental meaning is the differential relationship between joint rates and end-effector velocities.