What is the Jacobian matrix in robotics?

The Jacobian is a matrix that relates joint velocities (d-theta/dt) to end-effector Cartesian velocities (dx/dt, dy/dt). For a 2-DOF planar arm, it is a 2x2 matrix whose entries depend on link lengths and current joint angles.

What is a kinematic singularity?

A singularity occurs when the Jacobian determinant equals zero, meaning the robot loses one or more degrees of freedom in Cartesian space. For a 2-DOF arm this happens when theta2 = 0 or 180 degrees (fully extended or folded).

What units should I use for angles?

This tool accepts angles in degrees and converts internally to radians for trigonometric computation. Link lengths can be any consistent unit (meters, centimeters, millimeters) — the Jacobian entries share the same unit.

What do the Jacobian entries represent?

J[1][1] = partial x / partial theta1, J[1][2] = partial x / partial theta2, J[2][1] = partial y / partial theta1, J[2][2] = partial y / partial theta2. Each entry shows how sensitive the end-effector position is to a change in that joint angle.

Can I use this for a 3-DOF or 6-DOF robot?

This tool is specific to planar 2-DOF arms. For higher-DOF robots the Jacobian has more rows and columns and requires DH-parameter-based computation beyond the scope of this tool.

How is the end-effector position computed?

Using standard forward kinematics: x = L1*cos(theta1) + L2*cos(theta1+theta2), y = L1*sin(theta1) + L2*sin(theta1+theta2).

Robot Jacobian Calculator

Name: Robot Jacobian Calculator
Availability: InStock
Author: Nham Vu

Enter two link lengths and joint angles to compute the 2x2 Jacobian matrix and end-effector position for a planar 2-DOF robot arm.

Robot Parameters

Link Length L1

Link Length L2

Joint Angle θ1 (base)

deg

Joint Angle θ2 (elbow)

deg

End-Effector Position (Forward Kinematics)

—

L1·cos(θ1) + L2·cos(θ1+θ2)

—

L1·sin(θ1) + L2·sin(θ1+θ2)

Jacobian Matrix J

[

J₁₁

—

-L1s1 - L2s12

J₁₂

—

-L2s12

J₂₁

—

L1c1 + L2c12

J₂₂

—

L2c12

]

det(J) = —

|det(J)| = —

Manipulability: —

Velocity Mapping Formula

[ dx/dt ] = J · [ dθ1/dt ]

[ dy/dt ] [ dθ2/dt ]

Where s1 = sin(θ1), c1 = cos(θ1), s12 = sin(θ1+θ2), c12 = cos(θ1+θ2). The determinant equals L1 · L2 · sin(θ2).

Arm Visualization

Summary

Enter two link lengths and joint angles to compute the 2x2 Jacobian matrix and end-effector position for a planar 2-DOF robot arm.

How it works

Enter the length of link 1 (L1) and link 2 (L2) in any consistent unit.
Enter joint angle theta1 (base joint) and theta2 (elbow joint) in degrees.
The tool computes forward kinematics: end-effector position (x, y).
It then builds the 2x2 Jacobian matrix relating joint velocities to Cartesian velocities.
The determinant is computed to detect singularities (det = 0 means singular configuration).
Results update in real time as you adjust any input.

Use cases

Verify Jacobian matrix computations for robotics coursework.
Detect singularity configurations before commanding robot motion.
Understand velocity mapping from joint space to Cartesian space.
Prototype inverse-kinematics control strategies for planar arms.
Validate hand calculations against a reliable reference.
Teach kinematics concepts interactively in robotics labs.
Check workspace reachability and manipulability at given poses.
Quick reference tool for robotics engineers during design reviews.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu