This interactive simulation visualizes Forward Kinematics (FK) and Inverse Kinematics (IK) for a planar 2-link robot arm. You can drive the arm by angles (FK) or by dragging the end-effector target (IK), and see the underlying math update in real time.

 

Math behind the Simulation

1. Two-link planar arm

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). The end-effector position (in a math frame with x right, y up) is:

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

2. Forward kinematics (FK)

Given θ₁, θ₂, L₁, L₂, we compute x, y directly from the equations above. The elbow is at (L₁ cos θ₁, L₁ sin θ₁).

3. Inverse kinematics (IK)

Given a target (x, y), we solve for θ₁, θ₂. Let d = √(x² + y²). If d > L₁ + L₂ we clamp to max reach; if d < |L₁ − L₂| we clamp to min reach. Using the Law of Cosines:

cos θ₂ = (d² − L₁² − L₂²) / (2 L₁ L₂), so θ₂ = ±acos(...).
“Elbow up” uses θ₂ < 0; “elbow down” uses θ₂ > 0. Then θ₁ = atan2(y, x) − atan2(L₂ sin θ₂, L₁ + L₂ cos θ₂).

4. Workspace and singularities

The reachable set is an annulus (donut): outer radius L₁ + L₂, inner radius |L₁ − L₂|. Singularities occur when the arm is fully extended or fully folded (θ₂ = 0 or ±π): the Jacobian becomes singular and small Cartesian motions require large joint motions.

5. Jacobian and manipulability ellipsoid

Definition. 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]. The entries are j₁₁ = ∂x/∂θ₁, j₁₂ = ∂x/∂θ₂, j₂₁ = ∂y/∂θ₁, j₂₂ = ∂y/∂θ₂, which you can derive from the FK expressions above.

Velocity mapping. J maps joint velocities to end-effector velocity: [ẋ; ẏ] = J [θ̇₁; θ̇₂], i.e. ẋ = j₁₁θ̇₁ + j₁₂θ̇₂ and ẏ = j₂₁θ̇₁ + j₂₂θ̇₂.

Use in this simulation. We compute J from the current (θ₁, θ₂, L₁, L₂) and show its entries live in the Jacobian J panel. The manipulability ellipsoid (cyan ellipse at the end-effector) is derived from JJ^T: its axes indicate directions of higher velocity capability for unit joint velocity. The ẋ, ẏ from J panel compares end-effector velocity from J·[θ̇₁; θ̇₂] with measured velocities (from finite differences), and the velocity subplots show θ̇₁, θ̇₂ and ẋ, ẏ over time.

 

Usage

Follow these steps to explore 2D robot arm kinematics:

1. Mode: Forward (FK) — you control θ₁, θ₂ via sliders; the end-effector position is computed. Inverse (IK) — you click and drag the pink target; the angles are computed from the target. If the target is outside the reachable annulus, a warning overlay appears and the arm clamps to the workspace; the pink dot remains at the requested position.
2. Sliders: In FK mode, use θ₁ and θ₂ to move the arm. The FK math panel updates with the current x, y and the formula values.
3. L₁, L₂: Link lengths (pixels). Changing them updates the workspace annulus and FK/IK.
4. Elbow Up / Down: In IK mode, toggles between the two IK solutions. The ghost arm (dashed) shows the other configuration.
5. Trace: Check the Trace checkbox to show a trail behind the end-effector; uncheck to clear it. The checkbox is on its own row; Step Fwd, Step Bwd, Run, and Reset are in a single line below.
6. Step Fwd / Step Bwd: Nudge angles (FK) or target (IK) by a small step. Step Fwd stops any running animation first.
7. Run / Stop: Animates the arm (FK: angles change; IK: target moves on a circle).
8. Reset: Restore default angles, clear the trace, and uncheck Trace.
9. Velocity subplots: Two plots at the top-right of the canvas show joint velocities (θ̇₁, θ̇₂) and end-effector velocities (ẋ, ẏ) over a rolling time window.
10. Info panels: Below the simulation, three panels show live joint velocities, the Jacobian J (formulas and values), and ẋ, ẏ from J compared to measured velocities.
11. Manipulability ellipsoid: The cyan ellipse at the end-effector indicates directions of higher velocity capability (from the Jacobian).

Tips: Switch to IK and drag the target outside the annulus to see it clamp to max/min reach. Use Elbow Up/Down to compare both IK solutions. Check Trace and Run to see the path sweeps.

Parameters

• θ₁, θ₂: Joint angles (degrees). θ₁ is the base angle; θ₂ is the elbow angle relative to link 1.
• L₁, L₂: Link lengths in pixels (20–250). Defaults 150, 120.
• Workspace: Annulus with outer radius L₁ + L₂, inner radius |L₁ − L₂|.
• Velocities: Joint velocities (θ̇₁, θ̇₂) in rad/s; end-effector velocities (ẋ, ẏ) in px/s. Subplot time window and layout are configurable via robotarm.js Config.