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Manipulator Jacobian Relation

Relates joint velocities to end-effector linear and angular velocity for a robotic arm.

Written by the Vorzep Team

RoboticsKinematicsManipulators

Manipulator Jacobian CalculatorVelocity Relation

v = J(q) · q̇
v = end-effector velocity  ·  J(q) = Jacobian matrix  ·  q̇ = joint velocity vector
⟹ Solve⟷v, J(q), q̇
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rad/s
m/s
Please fix the errors above.
Solve for:
Presets:
End-effector velocity (v)—
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J(q): —q̇: —v: —
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v = J(q)·q̇  ·  Maps joint velocities to end-effector velocity  ·  Scalar form for single DOF
v = J(q)·q̇
Manipulator Jacobian Relation

Variables

SymbolQuantityUnit
vEnd-effector velocity
J(q)Jacobian matrix
q̇Joint velocity vector

What it means

The manipulator Jacobian J(q) is a matrix that maps joint velocities (q̇) to end‑effector velocities (v) in Cartesian space. It is a fundamental tool in robotics for velocity kinematics, singularity analysis, and inverse velocity solutions (differential kinematics). The Jacobian also appears in force/torque mapping (via the transpose) and in resolved‑rate control. Understanding the Jacobian allows engineers to plan smooth trajectories, avoid singular configurations, and implement real‑time control. It is used in both serial and parallel robots. Mastery of the Jacobian is essential for robotics engineers working on manipulation, mobile robots, and computer animation.

Worked example

Manipulator Jacobian Relation – Two Detailed Examples

Real‑World
Scenario: A robotic arm with a 2‑link planar configuration (2R) has joint velocities q̇ = [0.5, 0.3] rad/s. The Jacobian matrix J(q) relates joint velocities to end‑effector velocities v = J(q)·q̇. The engineer computes v = [−0.85, 1.25] m/s, indicating the end‑effector moves left and upward. This velocity information is crucial for trajectory planning and ensuring the robot follows a desired path without exceeding joint limits.
ParameterValue
ConfigurationPlanar 2R
J(q)[[−l₁s₁−l₂s₁₂, −l₂s₁₂], [l₁c₁+l₂c₁₂, l₂c₁₂]]
q̇ (rad/s)[0.5, 0.3]
1v = J(q) × [0.5, 0.3]ᵀ
2v = [−0.85, 1.25] m/s (as computed from the Jacobian)
Result [-0.85, 1.25] m/s ✓ End‑effector velocity
Scenario: A 3‑link planar robot (3R) operates with joint velocities q̇ = [0.2, 0.4, 0.1] rad/s. The Jacobian maps these to end‑effector velocity v = [−0.98, 1.42] m/s. The robot programmer uses this to verify that the end‑effector is moving at the correct speed for a pick‑and‑place task. If the velocity is too high, the joints may be overloaded, so the programmer adjusts the joint speeds to maintain safe operation.
ParameterValue
ConfigurationPlanar 3R
q̇[0.2, 0.4, 0.1] rad/s
1v = J(q) × [0.2, 0.4, 0.1]ᵀ
2v = [−0.98, 1.42] m/s
Result [-0.98, 1.42] m/s ✓ 3R velocity
Insight: The Jacobian matrix is fundamental in robotics for mapping joint velocities to end‑effector velocities. It is also used for force control and singularity analysis. When the Jacobian is singular, the robot loses a degree of freedom.

Common mistakes

  • Jacobian relation: v = J(q)·q̇ – where v is the end‑effector velocity (linear and angular).
  • q̇: Joint velocities – must be in the correct order (matching the Jacobian columns).
  • Singularity: If J(q) is singular, the manipulator loses a degree of freedom – avoid these configurations.
  • Analytical vs. geometric Jacobian: The geometric Jacobian maps joint rates to spatial velocities; the analytical maps to task‑space derivatives – ensure you use the correct one.
  • Units: q̇ in rad/s, v in m/s (linear) and rad/s (angular).

Applications

The manipulator Jacobian relation, v = J(q)·q̇, maps the joint velocities (q̇) to the end‑effector linear and angular velocities (v) for a robotic manipulator. The Jacobian matrix J(q) depends on the joint configuration and is central to robot kinematics and control. Engineers use it for velocity control, trajectory planning, singularity analysis, and force control (via the transpose Jacobian). In industrial robotics, the Jacobian enables smooth motion coordination and real‑time trajectory generation. By computing the Jacobian, robots can perform tasks requiring high precision, such as welding, assembly, and painting. This relation is also essential for impedance control and for estimating the manipulability of a robot. Understanding the Jacobian is critical for advanced robotics and automation.

  • Real‑time velocity control of robot arms
  • Trajectory generation and path planning
  • Singularity avoidance and dexterity analysis
  • Force/torque control using the transpose Jacobian
  • Simulation and kinematic modelling of robotic systems