Hardware & control
Robot Jacobian
A robot Jacobian is a configuration-dependent matrix that maps joint velocities to a specified task velocity, such as the linear and angular velocity of an end effector. It describes instantaneous motion, rather than the end effector's finite displacement or the forces needed to accelerate the robot.
Also known as: Jacobian, Jacobian matrix, Manipulator Jacobian
Updated
From joint speeds to hand motion
Forward kinematics maps a robot configuration to a pose. The Jacobian answers a different question: how does motion at each joint contribute to the hand's velocity at the current configuration?
In the relation v = J(q) q_dot, q is the configuration, q_dot contains joint velocities, and v is the chosen task velocity. Each column describes the contribution from one joint moving at unit speed while the other joints remain still. Modern Robotics' space-Jacobian derivation makes this interpretation explicit for an open-chain robot.
The matrix must match the velocity representation. A position-only task and a full spatial-motion task have different rows. A spatial twist also depends on the coordinate frame in which it is expressed; a matrix labelled only “Jacobian” is not a complete interface specification.
The transpose maps a wrench to joint efforts
A wrench combines force and moment. With compatible frame and sign conventions, tau = J(q)^T F gives the joint efforts associated with an end-effector wrench. Modern Robotics derives this relationship from mechanical power.
That contribution is not the robot's entire torque command. Gravity, inertia and other effects belong to the dynamics calculation. Nor is transposing the Jacobian the same operation as inverting it to solve a velocity task.
Preserve the matrix's meaning in robot data
For a humanoid hand-velocity command, record the robot model, joint order, configuration, controlled point, reference frame and ordering of linear and angular components. If the calculation includes torso or floating-base motion, identify those velocity coordinates too.
These details let a reader distinguish a model-derived velocity from a measured one. A stored Jacobian without its configuration and conventions cannot reliably explain why the same hand command produced different joint motion in two demonstrations.
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Related terms
Hardware & control
Robot kinematics
Robot kinematics describes the geometric relationship between a robot’s joint configuration and the position, orientation and velocity of its links or end-effector, without modelling the forces that cause the motion. Forward kinematics computes pose from joint values; inverse kinematics searches for joint values that achieve a requested pose.
Hardware & control
Inverse kinematics
Inverse kinematics (IK) finds robot configurations that satisfy a desired position, orientation or other geometric constraints. It reverses the question asked by forward kinematics: instead of computing where a hand or foot is from the joint values, it searches for joint values that place it at a target. A target can have multiple solutions or no feasible solution.
Hardware & control
Kinematic singularity
A kinematic singularity is a robot configuration where the task Jacobian has lower rank than the maximum it can attain, removing one or more instantaneous motion directions. Singularity is a property of the mechanism, configuration and chosen task, not simply another name for a joint limit or an unreachable target.
Hardware & control
Coordinate frame
A coordinate frame is a defined origin and set of oriented axes used to express positions, orientations, motions, forces, or other spatial quantities. A value has no complete geometric meaning until its frame and convention are known. Transformations relate measurements expressed in frames such as world, robot base, camera, end effector, object, or sensor.
Hardware & control
Torque control
Torque control is a control mode in which desired joint torques, or equivalent linear forces, are the command variables sent to a robot’s low-level actuators. Unlike position control, it does not directly command where each joint should be. Motion and contact instead emerge from the applied efforts, robot dynamics, feedback, constraints and any higher-level controller.