Chapter 01 · Meet the robot, then build its mathematical language
Today in the field story
One problem, then the next
The sorter can sense a box, but its cardboard diverter is still described as one vague object. Inspect its base, shoulder strip, elbow strip, and tool flap. The Parcel-Sorting Desk Robot now needs joints that state permitted and forbidden motion, because tomorrow’s possible shapes depend on today’s independent coordinates rather than on the number of motors drawn beside them.
- Why now
Configuration cannot be recorded until links, joint types, limits, and independent freedoms are known.
- Ignore today
Ignore torque, motor selection, and closed-loop joint control; keep the arm unpowered.
- Unlocks next
A defensible count of the coordinates needed to describe every sorter shape.
Understand
Build the physical picture first
Links are the bones of a mechanism; joints are the rules that say how neighboring bones may move.
A link is a body we choose to treat as rigid. A joint connects links and removes some possible relative motions while permitting others. A door hinge permits rotation around one axis but blocks sliding and the other rotations. A drawer runner permits movement along one line but blocks the rest. Robotics calls these common joint types revolute and prismatic. A bolted or glued connection is fixed and permits no relative motion.
One degree of freedom, or DOF, is one independent number needed to describe a mechanism's configuration near an ordinary pose. A single hinge angle is one DOF. A single drawer extension is also one DOF. A free rigid body in ordinary 3D space needs six numbers locally: three for position and three for orientation. DOF counts independent motion, not objects, motors, wires, or the number of directions you can casually name.
Actuation and freedom are separate ideas. A passive elbow hinge still contributes a DOF even if no motor drives it. Two motors can be mechanically coupled so that they control only one independent motion. A joint sitting at its travel stop still belongs to a one-DOF mechanism; the limit removes motion farther in one direction at that instant, but it does not turn the hinge into a fixed connection. If a brake permanently locks it for the model being studied, then the model changes.
Adding joint DOF works directly for a simple open serial chain with independent one-axis joints. Closed loops and coupled mechanisms need more care because constraints can depend on one another. This week uses a two-hinge open cardboard arm, so shoulder angle plus elbow angle gives two DOF. Learn the physical test first: hold all other joints still and ask whether this coordinate can change by itself.
Words you need
Name each idea precisely
- Revolute joint
A joint that permits rotation about an axis.
Physical example:A door hinge or paper-fastener elbow.
- Prismatic joint
A joint that permits sliding along an axis.
Physical example:A drawer runner or telescoping rail.
- Fixed joint
A connection modeled as allowing no relative motion.
Physical example:Two plates bolted tightly together.
- Degree of freedom
One independent coordinate required to describe configuration.
Physical example:The opening angle of one ordinary door hinge.
- Joint limit
The smallest and largest allowed value of a joint coordinate.
Physical example:A servo arm permitted to turn only from 0° to 180°.
Math, one line at a time
Work through today’s relationship
Prerequisite rescue · optionalStart at zero: counting, units, and coordinates
No maths is assumed. First give every robot number a physical meaning, a unit, a zero point, and a positive direction.
- N
- a count, such as number of jointsUnit: no unit
- x
- one position measured from a chosen zeroUnit: centimetres (cm) or metres (m)
- Δx
- change in x; Δ means changeUnit: same unit as x
Draw a number line. Put the origin at 0 cm, choose right as positive, and place the robot at x = 20 cm.
Move 30 cm right, so Δx = +30 cm. Predict x_new = 20 cm + 30 cm = 50 cm.
Measure to check, then convert with 100 cm = 1 m: 50 cm = 0.50 m. Never add centimetres directly to metres.
A variable is like a named field in code, but a robot field must also say its physical unit and what zero means.
A robot starts at x = 40 cm and moves 15 cm left. What is its new x coordinate?
Left is negative, so Δx = -15 cm and x_new = 40 - 15 = 25 cm.
If a cardboard arm has independent shoulder and elbow hinges, then
“DOF” counts independent configuration coordinates, not motors. Revolute joints use an angle; prismatic joints use a sliding distance.
Count the DOF of a simple inspection arm
An open-chain arm has a rotating base, one shoulder hinge, one sliding wrist, and a camera bolted to the wrist.
List each relative connection from ground to camera.
The base revolute joint contributes one independent angle.
The shoulder revolute joint contributes a second independent angle.
The wrist prismatic joint contributes one independent distance.
The bolted camera mount contributes zero relative motion.
Add the independent coordinates: 1 + 1 + 1 + 0 = 3 DOF.
A configuration can be recorded with three ordered values, such as [base angle, shoulder angle, wrist extension].
Count independent coordinates and their constraints, not the visible number of parts or actuators.
Physical examples
Where this appears in real life
Desk lamp
A simple lamp has a base swivel, shoulder hinge, and elbow hinge connected by rigid bars.
Move one joint while holding the others. Each independently changeable angle contributes one DOF.
Drawer and cabinet
The drawer translates along its runners while the cabinet door rotates on hinges.
Both are one-DOF mechanisms even though one coordinate is a distance and the other is an angle.
Hands-on exercise
Make the idea observable
Use two cardboard strips, two paper fasteners, tape, and a ruler.
Join the two strips with one paper fastener and attach the first strip to a cardboard base with the other.
Mark each rigid strip as a link and each fastener as a revolute joint.
Hold the elbow still and move only the shoulder; then reverse the test.
Mark safe angle limits where the cardboard would collide or tear.
Tape the elbow fixed temporarily and recount the independent motions for that altered model.
The untaped arm needs two angles; the temporarily fixed version needs only the remaining shoulder angle.
Your diagram names links, joint axes, coordinates, units, limits, and explains why motor count was never used.
Build today
Build a paper two-link robot and a browser notebook that shows its parts, allowed motion, coordinates, vectors, frames, and matrices.
Evidence to save
DONE when “Joints, allowed motion, and degrees of freedom” runs from one documented command and the nominal plus boundary outputs are attached.
Common mistakes
Catch the wrong mental model
Counting motors as degrees of freedom.
Count independently changeable configuration coordinates; actuation may be absent, redundant, or mechanically coupled.
Calling every connection a joint with one DOF.
Inspect the allowed relative motion; a fixed connection contributes zero, while some joints can contribute more than one.
Saying a hinge loses its DOF at a travel limit.
The configuration lies on a boundary, but the joint type and the mechanism's usual dimensionality remain one DOF.
Job connection
How this becomes employable evidence
Check whether a URDF or commissioning sheet models each physical joint with the correct type, axis, limits, and independent command interface.
Relevant target roles
- Robotics Application / ROS 2 Integration Engineer
- Robotics Software Engineer — ROS 2 / AMR
- Robotics Deployment, Integration & Validation Engineer
Chapter 01 interview drill
Interview questions: Joints, allowed motion, and degrees of freedom
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
A robot has four motors but only three independent motions. Give two physically plausible reasons and describe how you would verify the actual DOF.
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
Q1Does an unpowered revolute joint count as a DOF?
Yes, if its angle can change independently; DOF is about possible configuration, not actuation.
Q2Why does a bolted camera mount contribute zero DOF?
The camera and wrist are modeled with no relative motion, so no independent coordinate is needed between them.
Q3When can simply adding joint freedoms give a wrong result?
In mechanisms with closed loops, coupled motion, or dependent constraints; the assumed independent coordinates must be checked.