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COMPUTER SCIENCE & TECHNOLOGY • ROBOTICS

Inverse Kinematics (2-DOF Robot Arm)

Move a target through the planar workspace and solve the two joint angles for elbow-up and elbow-down arm configurations.

● IK SOLVER2-link planar arm
θ₁
θ₂
Target radius
Reachability

Reasoning inspector

live model state
Target radius r—
Inner workspace radius—
Outer workspace radius—
Boundary clearance—
θ₁—
θ₂—
Forward-check error—
Elbow branch—
Reachability—

Solving joint angles from position

Inverse kinematics uses triangle geometry to recover joint angles from the desired end-effector position while respecting the arm’s inner and outer reach limits.

LEARN WITH THE SIMULATION

Understand Inverse Kinematics (2-DOF Robot Arm) by experimenting.

START HERE

Inverse kinematics asks for joint variables that place a robot end effector at a desired Cartesian target. For a two-link planar arm, the geometry is completely visible: reachable targets lie inside an annulus with outer radius L₁+L₂ and inner radius |L₁−L₂|, while most interior targets admit two mirror-related elbow branches. This laboratory solves both angles, forward-checks the endpoint and reports clearance from the singular workspace boundaries.

BUILD THE IDEA

What you should understand first

  • Derive the two-link reachability condition |L₁−L₂| ≤ r ≤ L₁+L₂.
  • Use the law of cosines to solve θ₂ from a target x,y position.
  • Explain why elbow-up and elbow-down branches can reach the same end-effector point.
  • Use forward-kinematics endpoint error as a numerical verification of the inverse solution.

TRY IT YOURSELF

Why can the same reachable target have two joint solutions, yet those solutions merge at workspace boundaries?

  1. Choose a target comfortably inside the workspace annulus and record θ₁ and θ₂.
  2. Switch elbow branch and verify that the endpoint remains at the same target while the intermediate joint moves.
  3. Move toward the outer radius L₁+L₂ and watch the two configurations approach a straight-arm singular state.
  4. Move toward the inner radius |L₁−L₂| and compare the folded-arm boundary.
Read the visual:

Interior geometry can form the link triangle on either side of the target direction, creating two branches. At full extension or full folding the triangle degenerates, branch distinction collapses and the arm approaches a kinematic singularity.

WHY IT WORKS 01

The reachable workspace is an annulus from the triangle inequality

The shoulder-to-target distance is r=√(x²+y²). Two links of lengths L₁ and L₂ can form a triangle with that third side only when |L₁−L₂|≤r≤L₁+L₂. The upper boundary corresponds to full extension, while the lower boundary corresponds to the links folding back against one another.

WHY IT WORKS 02

The law of cosines gives the elbow angle and creates two branches

Rearranging the triangle relation gives cosθ₂=(r²−L₁²−L₂²)/(2L₁L₂). The sine of θ₂ can be positive or negative, producing elbow-down and elbow-up solutions. Once θ₂ is selected, θ₁ follows from the target direction minus the internal link-triangle angle.

WHY IT WORKS 03

Forward kinematics provides an essential consistency check

After solving θ₁ and θ₂, the simulator recomputes the end effector using x=L₁cosθ₁+L₂cos(θ₁+θ₂) and the corresponding y equation. The distance between this reconstructed endpoint and the requested target is the forward-check error. It should be near floating-point precision for reachable targets.

DO NOT MIX THESE UP

Common misconception

Inverse kinematics is not necessarily unique, and a Cartesian target can be mathematically unreachable even when its x and y values look modest. A low forward-check error verifies the computed endpoint but does not measure robustness near a singularity. Real robots also impose joint limits, collisions and 3D orientation constraints omitted here.

PRACTICE CHALLENGE

Can you place a target safely inside the workspace and prove that switching elbow branch changes configuration but not the endpoint?

  • Choose a workspace-clearance target and move the Cartesian point until it is satisfied.
  • Record the two joint angles and forward-check error.
  • Switch elbow branch without moving the target.
  • Compare the new joint position and angles while verifying that reachability, target radius and endpoint error remain effectively unchanged.
Reveal the reasoning

Both branches arise from the ± choice for sinθ₂ after cosθ₂ is fixed by the target triangle. They therefore use different joint angles and elbow locations but reconstruct the same x,y endpoint. Clearance measures distance from the inner or outer workspace boundaries, where these branches approach singular configurations.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The robot is a planar two-link serial arm with ideal revolute joints and fixed link lengths.
  • The target specifies end-effector position only; tool orientation is not constrained.
  • Inverse kinematics uses the geometric two-link solution with explicit elbow-up and elbow-down branches.

Model limitations

  • Joint limits, collisions, dynamics, motor torque and three-dimensional motion are omitted.
  • Near singular workspace boundaries, small target changes can produce large joint-angle changes in real systems.

CONNECT IT

Where this matters

  • Robot manipulators
  • Animation and character rigs
  • Pick-and-place automation
  • Motion planning and workspace design

WORDS THAT MATTER

Key vocabulary

Inverse kinematics
Solving joint variables from a desired end-effector position or pose.
Forward kinematics
Computing end-effector position from known joint variables.
Workspace
Set of Cartesian positions reachable by the mechanism.
Elbow branch
One of the two mirror-related planar joint configurations that can reach many interior targets.
Kinematic singularity
Configuration where available motion directions or inverse-solution behavior becomes degenerate, such as full extension.