This interactive tutorial visualizes the Jacobian matrix for a 2-DOF planar robotic arm—the “Aha!” moment where the abstract calculus of partial derivatives meets the physical reality of why can’t my robot arm move forward anymore? The focus is the mapping between Joint Velocity Space (θ̇₁, θ̇₂) and Task Space Velocity (ẋ, ẏ).

 

Mathematical foundation

1. Two-link planar arm and forward kinematics

The arm has two links: length L₁ (shoulder–elbow) and L₂ (elbow–end-effector). Joint angles are θ₁ (base) and θ₂ (elbow, relative to link 1). In a math frame with x right and y up, the end-effector position is:

x = L₁ cos(θ₁) + L₂ cos(θ₁ + θ₂)
y = L₁ sin(θ₁) + L₂ sin(θ₁ + θ₂)

2. The Jacobian matrix

The Jacobian J is the matrix of partial derivatives of end-effector position (x, y) with respect to joint angles (θ₁, θ₂): J = ∂[x; y]/∂[θ₁; θ₂]. For the 2-link arm, differentiating the FK expressions gives:

J = [ −L₁ sin θ₁ − L₂ sin(θ₁+θ₂), −L₂ sin(θ₁+θ₂) ]
[ L₁ cos θ₁ + L₂ cos(θ₁+θ₂), L₂ cos(θ₁+θ₂) ]

3. Velocity mapping

J maps joint velocities to end-effector velocity: [ẋ; ẏ] = J [θ̇₁; θ̇₂]. So ẋ = j₁₁θ̇₁ + j₁₂θ̇₂ and ẏ = j₂₁θ̇₁ + j₂₂θ̇₂. The green vector at the end-effector in the simulation is this instantaneous linear velocity.

4. Determinant and singularity

det(J) = j₁₁j₂₂ − j₁₂j₂₁. When det(J) ≈ 0, the Jacobian loses rank: the arm is at a singularity (e.g. fully extended or fully folded). At that configuration, certain task-space directions require infinite joint velocity—the arm “loses” a degree of freedom. The simulation turns the arm red and shows “SINGULARITY WARNING” when |det(J)| is below a threshold.

5. Manipulability ellipsoid

The velocity mapping v = J q̇ sends a unit circle in joint-velocity space (‖q̇‖ = 1) to an ellipse in task space. That ellipse is the manipulability ellipse: it shows how “easy” it is to move in each task-space direction. Let B = J J^T. B is symmetric and positive semi-definite. Its entries are b₁₁ = j₁₁²+j₁₂², b₁₂ = j₁₁j₂₁+j₁₂j₂₂, b₂₂ = j₂₁²+j₂₂².

The eigenvalues λ₁ ≥ λ₂ of B satisfy λ² − (trace B)λ + det(B) = 0, so λ_1,2 = (trace B)/2 ± √[(trace B)²/4 − det(B)]. The semi-axis lengths of the ellipse are the singular values of J: a = √λ₁ (semi-major), b = √λ₂ (semi-minor). The major-axis direction (eigenvector of B for λ₁) has angle θ = (1/2) atan2(2b₁₂, b₁₁−b₂₂) in the task (x,y) frame. The major axis is the direction of highest velocity capability; near a singularity λ₂ → 0 and the ellipse flattens. The dashed circle is the reach limit (L₁ + L₂): at full extension the arm cannot move radially outward.

 

Worked example (with numbers)

Use L₁ = 150, L₂ = 120, θ₁ = θ₂ = 45° (π/4 rad), and joint velocities θ̇₁ = θ̇₂ = 0.5 rad/s. The 0.5 rad/s values are the simulator’s default joint-velocity settings—they are arbitrary choices for this example so that the numbers below match what you see in the “Jacobian J” and “Status” panels when the sliders are at θ₁=45°, θ₂=45°, θ̇₁=0.5, θ̇₂=0.5. Then θ₁ + θ₂ = 90°, so cos(45°) = sin(45°) ≈ 0.707, cos(90°) = 0, sin(90°) = 1.

1. End-effector position

x = 150×0.707 + 120×0 ≈ 106.1,    y = 150×0.707 + 120×1 ≈ 226.1.

2. Jacobian J (∂x/∂θ₁ = −L₁sin θ₁−L₂sin(θ₁+θ₂), etc.)

J =
−226.1	−120.0
106.1	0.0

3. Determinant

det(J) = j₁₁j₂₂ − j₁₂j₂₁ = (−226.1)(0) − (−120)(106.1) = 12727.9.

4. Task-space velocity v = J [θ̇₁; θ̇₂]

ẋ = (−226.1)(0.5) + (−120)(0.5) = −173.0,    ẏ = (106.1)(0.5) + (0)(0.5) = 53.0.

5. Manipulability ellipse (B = J J^T)

b₁₁ = j₁₁²+j₁₂² = (−226.1)²+(−120)² = 65518,   b₁₂ = j₁₁j₂₁+j₁₂j₂₂ = (−226.1)(106.1)+0 = −24008,   b₂₂ = j₂₁²+j₂₂² = (106.1)²+0 = 11257. Trace B = 76775, det(B) = b₁₁b₂₂−b₁₂² ≈ 1.61×10⁸. Eigenvalues λ_1,2 = (trace/2) ± √[(trace/2)²−det] give λ₁ ≈ 74605, λ₂ ≈ 2170. Semi-axes a = √λ₁ ≈ 273.1, b = √λ₂ ≈ 46.6. Major-axis angle θ = (1/2) atan2(2b₁₂, b₁₁−b₂₂) = (1/2) atan2(−48016, 54261) ≈ −20.7°. The green ellipse in the sim at the end-effector has these semi-axes and orientation (scaled for display).

So v = [ẋ; ẏ] = [−173.0; 53.0]. Setting the sim to θ₁=45°, θ₂=45°, θ̇₁=0.5, θ̇₂=0.5 reproduces these values in the “Jacobian J” and “Status” panels, and the manipulability ellipse matches the formulas above.

 

Usage

Choose a Velocity mode and explore:

1. Joint angles (θ₁, θ₂): Use the sliders (−180° to 180°) to set the arm configuration. The arm, velocity vector (in Forward/Inverse), and manipulability ellipse (in Forward/Inverse) update in real time.
2. Forward mode: Set joint velocities θ̇₁, θ̇₂ (−2 to 2 rad/s). The green vector at the end-effector is v = J [θ̇₁; θ̇₂]. Ghost arrows show the two columns of J. The manipulability ellipse is shown.
3. Inverse mode: Set desired task velocity (ẋ, ẏ). Joint velocities are q̇ = J⁻¹ v. The green vector and manipulability ellipse are shown.
4. Position (mouse) mode: Click and drag the pink target to set desired end-effector position (x*, y*). The arm snaps to the target via inverse kinematics (IK)—no Jacobian is used for the joint solution. The velocity vector and manipulability ellipse are hidden in this mode; the Jacobian panel still updates for the current posture.
5. Ghost velocities (Forward/Inverse): Faint arrows show the contributions of θ̇₁ and θ̇₂ (columns of J); they add to form the green total velocity.
6. Manipulability ellipsoid (Forward/Inverse only): The green ellipse at the end-effector comes from J J^T. The major axis is the direction of highest velocity capability. Near full extension (θ₂ ≈ 0) the ellipse flattens.
7. Reach-limit circle: The dashed circle is the maximum reach (L₁ + L₂). In Position mode, the target is clamped to this circle.
8. Singularity: When |det(J)| is small, the arm turns red and “SINGULARITY WARNING” appears. The ellipse becomes extremely thin in Forward/Inverse.
9. Jacobian and Status: The 2×2 matrix J, det(J), and the current velocity relation update live in all modes.

Tips: In Forward or Inverse, set θ₂ ≈ 0 to see the extended singularity and ellipse flattening. In Position (mouse) mode, grab the pink target (click near it) and drag to move the end-effector; the arm uses IK, not the Jacobian, to compute joint angles.

Parameters

• θ₁, θ₂: Joint angles in degrees (−180 to 180). Defaults 45°, 45°.
• θ̇₁, θ̇₂ (Forward): Joint velocities in rad/s (−2 to 2). Defaults 0.5, 0.5.
• ẋ, ẏ (Inverse): Desired task-space velocity; q̇ = J⁻¹ v.
• Position (mouse): Desired end-effector position (x*, y*) set by dragging the pink target. Joint angles come from closed-form IK; velocity vector and manipulability ellipse are not displayed.
• L₁, L₂: Link lengths (fixed at 150 and 120 pixels).
• det(J): Determinant of the Jacobian. Near zero indicates singularity.
• v (ẋ, ẏ): End-effector velocity from v = J q̇ (Forward) or from sliders (Inverse), in math-frame units (scaled for display).