Inverse Kinematics (2 Link) Calculator

Work out inverse kinematics (2 link) instantly with clear inputs, formula shown and shareable results.

About Inverse Kinematics (2 Link) Calculator

For a planar two-link arm the elbow angle comes from the cosine rule, cos(theta2) = (d squared - L1 squared - L2 squared)/(2 L1 L2), and the shoulder angle from the target bearing minus the internal triangle angle. Two solutions always exist — elbow up and elbow down — and this returns the elbow-up branch.

Formula

Two-link IK

cos(theta2) = (x^2 + y^2 - L1^2 - L2^2) / (2 L1 L2); theta1 = atan2(y,x) - atan2(L2 sin t2, L1 + L2 cos t2)

Worked example

With the default values below, Inverse Kinematics (2 Link) Calculator returns Shoulder angle of -14.43. Change any input above and every figure here is recalculated in your browser.

Inputs used

Link 1 length: 300 mm
Link 2 length: 250 mm
Target X: 400 mm
Target Y: 150 mm

Results

Shoulder angle: -14.43 °
Elbow angle: 78.46 °
Distance to target: 427.20 mm
Reachability: Reachable (elbow-up solution shown)

Step by step

1Values used
2Two-link IK
3Shoulder angle= -14.43 °
4Elbow angle= 78.46 °
5Distance to target= 427.20 mm
6Reachability= Reachable (elbow-up solution shown)

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 2 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Why are there two solutions?

The triangle formed by the links and the target can fold either way. Real controllers pick the branch that avoids collisions and stays away from the singularity.

What happens at full extension?

The two solutions merge and the arm is singular: the Jacobian loses rank, so no joint motion can move the tool radially outward.