Chapter 02 · Turn the maths into robot motion

Today in the field story

One problem, then the next

The Camera-Crane Recovery finally reconnects its base, shoulder, elbow, and camera mount. Measure both links, declare joint zeros and signs, then predict three camera positions with forward kinematics before moving the cardboard arm. Record the residual instead of correcting the formula until link length, zero offset, backlash, measurement, and frame conventions have each been inspected.

Why now: The mission can calculate its camera pose only after every rigid transform and joint convention is trustworthy.
Ignore today: Ignore inverse solving and obstacle avoidance; today joint values are known inputs.
Unlocks next: A forward model that can verify every inverse-kinematics candidate.

Understand

Build the physical picture first

Forward kinematics is a chain of rulers and hinges. Starting at the bolted base, every hinge turns the direction inherited by all links after it; adding each link's contribution locates the tool.

Forward kinematics, shortened to FK, maps known joint values and known robot geometry to an end-effector pose. Its input is the configuration q plus link and frame definitions. Its output must name the reference frame. FK asks 'where does this configuration put the tool?' It does not choose the configuration for a desired target.

For a two-link flat arm, link 1 points at θ1. Link 2 points at the absolute angle θ1+θ2 because the elbow rotates relative to link 1 and inherits the shoulder turn. Each link contributes a horizontal component l cos(angle) and a vertical component l sin(angle); adding the two contributions gives the tool position.

A longer serial arm follows the same ordered idea with one rigid transform per joint and link. General robotics software composes those transforms rather than writing a custom sine-cosine equation for every mechanism. The calculation remains deterministic: the same model and joint values should return the same pose.

The physical robot can disagree with its mathematical model. Link-length measurement error, joint zero offsets, backlash, flex, or a swapped sign create FK error even when code implements the equation correctly. Credible work compares predicted and measured tool positions in the same frame and unit, then reports the residual instead of hiding it.

Words you need

Name each idea precisely

Forward kinematics: The map from joint configuration and robot geometry to tool pose.
Physical example:
Shoulder and elbow angles predict where a desk-lamp bulb sits.
Serial chain: Links and joints connected one after another from base to tool.
Physical example:
base → shoulder link → elbow link → gripper.
Joint zero offset: The difference between a sensor's reported zero and the model's physical zero.
Physical example:
An encoder reads 0° while the link is actually tilted 3°.
End effector: The robot part whose task pose is being predicted or controlled.
Physical example:
A gripper jaw, welding torch, camera, or suction cup.
Residual error: The measured difference between predicted and observed result.
Physical example:
A predicted tip at (60.6,5.0) cm is measured 8 mm away.

Visual model

See the relationship

Swipe the technical canvas horizontally on a small screen.

Day 13 · Math check — Measured link lengths and joint angles predict the observed tool position. Measured anchor: 0 cm position error.

Math, one line at a time

Work through today’s relationship

Prerequisite rescue · optionalStart at zero: turns, triangles, and pose chains

A joint turn becomes a tool position only after we define the angle, split a link into horizontal and vertical parts, and follow frame order.

θ: theta: the amount a joint has turnedUnit: degrees (°) or radians (rad)
cos θ, sin θ: horizontal and vertical fractions of a turned unit linkUnit: no unit
Tᴬ_B: position and direction of frame B described by frame AUnit: unitless rotation + metres

A full turn is 360° = 2π rad, so 90° × π/180 = π/2 rad. π is about 3.1416.
For a 1 m link at 90°, x = 1 cos 90° = 0 m and y = 1 sin 90° = 1 m.
For several frames, follow the physical path in order. Multiply transforms only when the touching frame names match, then verify by reversing the path.

Programmer analogy

Like nested UI transforms, each child inherits its parent's transform; unlike UI, the order can move metal into an obstacle.

What are x and y for a 2 m link at 0 rad?

x = 2 cos 0 = 2 m; y = 2 sin 0 = 0 m.

For link lengths $l_1,l_2$ and angles $\theta_1,\theta_2$ ,

x=l_1\cos\theta_1+l_2\cos(\theta_1+\theta_2),\qquad y=l_1\sin\theta_1+l_2\sin(\theta_1+\theta_2).

With unit links at $(0^\circ,90^\circ)$ , $(x,y)=(1,1)\,\mathrm m$ .

Predict a two-link tool position

Use l1=0.40 m, l2=0.30 m, θ1=30°, and relative elbow angle θ2=-60°.

Convert angles if coding: θ1=π/6 rad and θ2=-π/3 rad.
Find link 2's absolute angle: θ1+θ2=-30°.
Compute x=0.40 cos30° + 0.30 cos(-30°).
Using cos30°≈0.8660, x≈0.3464+0.2598=0.6062 m.
Compute y=0.40 sin30° + 0.30 sin(-30°)=0.2000-0.1500=0.0500 m.
Check maximum reach: √(x²+y²)≈0.608 m, which is below l1+l2=0.70 m.

Result

The predicted tool point is approximately (0.606,0.050) m in the base frame.

What this proves

Each downstream link inherits upstream joint rotation; use θ1+θ2 for the second link's absolute direction.

Physical examples

Where this appears in real life

Adjustable desk lamp

Two rigid bars with shoulder and elbow hinges position the bulb above a table.

Look for:

The second bar's table direction includes both hinge turns, not only the elbow angle.

Excavator boom and stick

Measured boom and stick angles predict the bucket-pin position relative to the vehicle body.

Look for:

Cylinder play and flex can make the real bucket differ from the ideal rigid-link prediction.

Hands-on exercise

Make the idea observable

Build a two-link cardboard arm with paper fasteners, draw a base frame, and use a protractor and ruler.

Measure both link lengths from joint centre to joint centre or tool mark.
Define angle-zero direction, positive rotation, and base-frame units.
Choose three angle pairs including one negative relative elbow angle.
Calculate each predicted tip coordinate before positioning the arm.
Set the arm with a protractor and mark the measured tip coordinate.
Calculate x error, y error, and Euclidean position residual for every trial.
Change one model parameter only—such as a measured link length or zero offset—and show whether the residual improves.

Observe

Small angle or length errors can grow into visible tool error, especially when links are long or nearly aligned.

Done when

Three trials include predicted pose, measured pose, residual with units, and one justified model correction.

Build today

Extend the browser notebook into a two-link arm visualizer with frame composition, forward kinematics, and a bounded numerical IK trace.

Evidence to save

DONE when a 60–120 second uncut “Forward kinematics for a serial arm” demo links to its command, logs or plots, result count, and honest failure note.

Common mistakes

Catch the wrong mental model

Wrong

Using θ2 as the second link's world angle.

Better

For a relative revolute elbow, use the inherited absolute angle θ1+θ2.

Wrong

Mixing link lengths in centimetres with output expected in metres.

Better

Convert all geometry to one declared unit before calculation.

Wrong

Reporting only the predicted point without a frame.

Better

State that the output is expressed in the chosen base frame and define its axes.

Wrong

Calling a measured mismatch a formula failure immediately.

Better

Inspect calibration, zero offsets, backlash, flex, signs, measurement uncertainty, and frame conventions.

Job connection

How this becomes employable evidence

Robot bring-up and application engineers compare encoder joint states with expected tool poses. They use FK to validate model geometry, joint signs, zero offsets, and frame conventions before motion planning or calibration acceptance.

Relevant target roles

Robotics Deployment, Integration & Validation Engineer
Robotics Application / ROS 2 Integration Engineer
Robotics Software Engineer — ROS 2 / AMR

Chapter 02 interview drill

Interview questions: Forward kinematics for a serial arm

Practise a 60–90 second answer: define the idea, connect it to a physical robot, state assumptions, frames, and units when relevant, then finish with the failure signal or evidence you would inspect.

Primary interview scenario

For a two-link arm, why is the second link angle θ1+θ2? List four reasons measured tool position may disagree with correct FK code.

Answer shape: clarify the situation → trace the physical and software path → test the most likely boundaries → name the evidence that would confirm the result.

Technical follow-up questions

Q1What are FK's inputs and output?

Model interview answer

Joint configuration plus robot geometry/frame conventions map to an end-effector pose in a named frame.

Q2If both links are 1 m and θ1=0°, θ2=90°, where is the tip?

Model interview answer

The tip is at (1,1) m in the base frame: the first link reaches (1,0), then the second link points along positive y.

Q3Can two different joint configurations produce the same tool position?

Model interview answer

Yes. FK is single-valued from q to pose, but the inverse map can have multiple solutions.

Topic reference