Planar two-link inverse kinematics calculator

Two-link planar elbow-up solution; target within reach.

robotics-planar-inverse-kinematics.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenario	cosElbow	elbowrad	shoulderrad
	0.0416667	1.52912	-0.430988
	0.352083	1.211	-0.385778

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

target in reach0.04 ✓ passes

BASELINE INPUTS

linkA200 mmlinkB150 mmx250 mmy50 mm

CALCULATED

cosElbow0.041667

elbow1.5291 rad

shoulder-0.43099 rad

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Planar two-link inverse kinematics
# Scope: Two-link planar elbow-up solution; target within reach.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

linkA = 200 mm
linkB = 150 mm
x = 250 mm
y = 50 mm

cosElbow := (x^2+y^2-linkA^2-linkB^2)/(2*linkA*linkB)
elbow := acos(cosElbow) [rad]
shoulder := atan2(y/linkA,x/linkA)-atan2(linkB*sin(elbow)/linkA,1+linkB*cos(elbow)/linkA) [rad]

check target in reach: abs(cosElbow) <= 1

scenario alternative: linkA = 160 mm

Calculated from your source · not saved

Baseline · Baseline

target in reach0.04 ✓ passes

alternative · alternative

target in reach0.35 ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

cos elbow: cosElbow = (x^2+y^2-linkA^2-linkB^2)/(2*linkA*linkB)
elbow: elbow = acos(cosElbow) [rad]
shoulder: shoulder = atan2(y/linkA,x/linkA)-atan2(linkB*sin(elbow)/linkA,1+linkB*cos(elbow)/linkA) [rad]

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

link a: 200 mm
link b: 150 mm
x: 250 mm
y: 50 mm

Calculated

cos elbow: 0.0416667
elbow: 1.52912 rad
shoulder: -0.430988 rad

Assumptions and limits

Two-link planar elbow-up solution; target within reach.

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: Planar two-link inverse kinematics. Inputs: link lengths; target point; branch. Outputs: joint angles; reachability. Model scope: Rigid links with explicit joints, loads and coordinate frames. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

Inputs: link lengths, target point, branch
Outputs: joint angles, reachability
Modes: calculate, compare, sweep
Drawing: Robot workspace and trajectory

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS