Why are there two solutions for many goals?

The law of cosines for the elbow angle has a ± choice, giving mirrored elbow-up and elbow-down postures that place the hand at the same point. At the workspace boundary those solutions coalesce.

What happens outside the reachable workspace?

If the goal is farther than L₁+L₂ or closer than |L₁−L₂|, no real elbow angle exists. The arm is fully stretched or folded and cannot reach; the simulator shows the geometric failure clearly.

Is this the same as the 3-link 3D CCD arm?

No. The 2R page uses analytic planar IK with explicit elbow branches. The 3-link 3D lab uses iterative CCD in space with joint limits and trajectory tracking — a different solution method for a harder chain.

Two-Link Arm IK (2R)

Planar 2R manipulator: mouse goal, elbow-up / elbow-down inverse kinematics; joint angles live.

About this model

A planar 2R manipulator has two revolute joints and link lengths that map joint angles (θ₁, θ₂) to end-effector position via straightforward forward kinematics. Inverse kinematics for a mouse-specified goal yields up to two real solutions — elbow-up and elbow-down — when the target lies between |L₁−L₂| and L₁+L₂. This simulator computes those analytic angles live as you move the goal, showing both configurations when reachable. Assumptions include a perfect planar mechanism, no joint limits beyond geometric reach, no dynamics or collisions, and instantaneous IK (not a timed trajectory controller). Drag the target inside and outside the annulus of reachability to see solutions appear, merge, and vanish at singularities.

Who it's for: Introductory robotics, mechatronics, and kinematics courses on analytic inverse kinematics.

Key terms

Two-link arm
Inverse kinematics
Elbow-up
Elbow-down
2R manipulator
Workspace

How it works

Planar two-link arm with inverse kinematics: drag on the canvas to move the goal (magenta). Toggle elbow-up vs elbow-down for the second algebraic branch.

Key equations

c₂ = (x²+y²−L₁²−L₂²)/(2L₁L₂)

θ₂ = atan2(s₂, c₂), θ₁ = atan2(y,x) − atan2(L₂ sin θ₂, L₁+L₂ cos θ₂)

Frequently asked questions

Why are there two solutions for many goals?: The law of cosines for the elbow angle has a ± choice, giving mirrored elbow-up and elbow-down postures that place the hand at the same point. At the workspace boundary those solutions coalesce.
What happens outside the reachable workspace?: If the goal is farther than L₁+L₂ or closer than |L₁−L₂|, no real elbow angle exists. The arm is fully stretched or folded and cannot reach; the simulator shows the geometric failure clearly.
Is this the same as the 3-link 3D CCD arm?: No. The 2R page uses analytic planar IK with explicit elbow branches. The 3-link 3D lab uses iterative CCD in space with joint limits and trajectory tracking — a different solution method for a harder chain.