What does the Jacobian determinant tell you for a 2R manipulator?

For a planar 2R manipulator, the determinant of the Jacobian measures how effectively joint velocities map to end-effector velocities. When the determinant is zero, the robot is in a singular configuration, meaning it loses one degree of freedom in Cartesian space and cannot move in some directions.

How do you compute the Jacobian of a 2R robotic arm?

For a 2R planar arm with link lengths L1, L2 and joint angles θ1, θ2, the end-effector position is x = L1 cos θ1 + L2 cos(θ1+θ2), y = L1 sin θ1 + L2 sin(θ1+θ2). The Jacobian is the 2×2 matrix of partial derivatives of (x, y) with respect to (θ1, θ2).

When is a 2R manipulator in a singular configuration?

A 2R planar manipulator is singular when its two links are collinear, i.e., when θ2 = 0 or θ2 = π (mod 2π). In these configurations the Jacobian determinant is zero and the arm cannot generate velocity in some Cartesian directions.

What is the difference between the Jacobian matrix and its determinant?

The Jacobian matrix describes the full linear mapping from input velocities to output velocities. Its determinant is a single scalar that measures local scaling and orientation change. A non-zero determinant means the mapping is locally invertible; a zero determinant indicates a singularity.

Jacobian Matrix Calculator (2R Manipulator & General Jacobian)

Q: What is the Jacobian matrix?

The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. For a function f: R^n -> R^m, the Jacobian is an m×n matrix whose (i,j) entry is the partial derivative of the i-th output with respect to the j-th input.

Compute the Jacobian matrix, determinant, and inverse for a planar 2R robotic arm, plus a general 2D Jacobian for custom functions. Visualize singularities and see the formulas used.

Planar 2R Manipulator Jacobian

Model a simple 2‑link planar robot arm with link lengths \(L_1, L_2\) and joint angles \(\theta_1, \theta_2\) (in radians or degrees). The end‑effector position is:

\[ x = L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2), \quad y = L_1 \sin\theta_1 + L_2 \sin(\theta_1 + \theta_2) \] \[ J(\theta_1,\theta_2) = \begin{bmatrix} \frac{\partial x}{\partial \theta_1} & \frac{\partial x}{\partial \theta_2} \\ \frac{\partial y}{\partial \theta_1} & \frac{\partial y}{\partial \theta_2} \end{bmatrix} \]

Link length L₁

Link length L₂

Angle units: Radians Degrees

Joint angle θ₁

Joint angle θ₂

General 2×2 Jacobian Matrix

Compute the Jacobian of a 2D vector function \( \mathbf{f}(x,y) = (f_1(x,y), f_2(x,y)) \) numerically at a point \((x_0,y_0)\). Enter expressions using JavaScript syntax (e.g. Math.sin(x), x*y, Math.exp(x)).

Function components

f₁(x, y)

f₂(x, y)

Supported: Math.sin, Math.cos, Math.tan, Math.exp, Math.log, Math.sqrt, Math.pow, etc.

Point x₀

Point y₀

What is the Jacobian matrix?

The Jacobian matrix generalizes the derivative to multivariable vector functions. For a mapping \( \mathbf{f} : \mathbb{R}^n \rightarrow \mathbb{R}^m \) with \( \mathbf{f}(x_1,\dots,x_n) = (f_1,\dots,f_m) \), the Jacobian is the \( m \times n \) matrix of first‑order partial derivatives:

\[ J(\mathbf{x}) = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}. \]

Intuitively, the Jacobian describes how small changes in the input variables produce changes in the outputs. It is the linear approximation of the function near a point.

Jacobian of a planar 2R manipulator

For a 2‑link planar robot arm with joint angles \( \theta_1, \theta_2 \) and link lengths \( L_1, L_2 \), the end‑effector position in the plane is:

\[ x(\theta_1,\theta_2) = L_1 \cos\theta_1 + L_2 \cos(\theta_1+\theta_2) \] \[ y(\theta_1,\theta_2) = L_1 \sin\theta_1 + L_2 \sin(\theta_1+\theta_2) \]

The Jacobian maps joint velocities \( \dot{\boldsymbol{\theta}} = [\dot{\theta}_1,\dot{\theta}_2]^T \) to end‑effector linear velocity \( \dot{\mathbf{p}} = [\dot{x},\dot{y}]^T \):

\[ \dot{\mathbf{p}} = J(\theta_1,\theta_2)\,\dot{\boldsymbol{\theta}} \] \[ J(\theta_1,\theta_2) = \begin{bmatrix} \dfrac{\partial x}{\partial \theta_1} & \dfrac{\partial x}{\partial \theta_2} \\ \dfrac{\partial y}{\partial \theta_1} & \dfrac{\partial y}{\partial \theta_2} \end{bmatrix} = \begin{bmatrix} -L_1 \sin\theta_1 - L_2 \sin(\theta_1+\theta_2) & -L_2 \sin(\theta_1+\theta_2) \\ L_1 \cos\theta_1 + L_2 \cos(\theta_1+\theta_2) & L_2 \cos(\theta_1+\theta_2) \end{bmatrix}. \]

Determinant and singularities

For the 2R arm, the determinant of the Jacobian simplifies to:

\[ \det(J) = L_1 L_2 \sin\theta_2. \]

If \( \det(J) \neq 0 \), the mapping from joint velocities to Cartesian velocities is locally invertible.
If \( \det(J) = 0 \), the robot is in a singular configuration and loses one degree of freedom in Cartesian space.

For a 2R planar manipulator, singularities occur when the links are collinear: \( \theta_2 = 0 \) (fully stretched) or \( \theta_2 = \pi \) (folded back).

How this calculator computes the general Jacobian

In the “General 2×2 Jacobian” tab, the calculator evaluates the Jacobian numerically using central finite differences. For example, for \( f_1(x,y) \):

\[ \frac{\partial f_1}{\partial x}(x_0,y_0) \approx \frac{f_1(x_0+h,y_0) - f_1(x_0-h,y_0)}{2h}, \quad \frac{\partial f_1}{\partial y}(x_0,y_0) \approx \frac{f_1(x_0,y_0+h) - f_1(x_0,y_0-h)}{2h}, \] with a small step \( h \).

This approach works for many smooth functions and is useful when an analytical derivative is tedious or unavailable. For high‑precision or symbolic work, tools like CAS systems (e.g. MATLAB, Mathematica, SymPy) are recommended.

Typical applications of the Jacobian matrix

Robotics: mapping joint velocities to end‑effector velocities, singularity analysis, force mapping.
Optimization & machine learning: gradients and backpropagation in neural networks.
Change of variables: transforming integrals and probability densities between coordinate systems.
Differential equations: linearization of nonlinear systems around equilibrium points.

Jacobian Matrix Calculator (2R Manipulator & General Jacobian)

Planar 2R Manipulator Jacobian

End‑Effector Position

Jacobian Matrix \(J(\theta_1,\theta_2)\)

Determinant det(J)

Singularity Check

Inverse \(J^{-1}\)

Analytical Formulas Used

General 2×2 Jacobian Matrix

Function Value at (x₀, y₀)

Jacobian Matrix \(J(x_0,y_0)\)