Chapter 03 · Dynamics and feedback

Today in the field story

One problem, then the next

The same Cold-Storage Door Tuning run is repeated with a heavier simulated panel and sticky guide. Compare mass, rotational inertia, friction, gravity load, and the drive’s finite effort instead of blaming software immediately. Calculate one simplified acceleration bound, then list every omitted effect. The team must see why raising a gain cannot manufacture torque that the actuator does not possess.

Why now: Controller behavior cannot be judged without the plant’s resistance, losses, and command limits.
Ignore today: Ignore detailed motor curves and structural engineering; retain a declared bounded toy model.
Unlocks next: A defensible distinction between control error and insufficient physical authority.

Understand

Build the physical picture first

A controller asks for motion, but the robot's body negotiates the result. A heavy or widely spread load is harder to accelerate, friction consumes some effort, and the actuator cannot provide more than its physical limit.

Mass measures resistance to linear acceleration. Rotational inertia plays the corresponding role for angular acceleration, but it also depends on where the mass sits relative to the rotation axis. Moving a battery farther from an arm joint can make that joint much harder to accelerate even though total mass is unchanged.

Friction is not one magic constant. Static friction must be overcome before a resting part starts moving; kinetic friction acts while surfaces slide; bearings and gears also add speed-dependent losses and backlash. A low command may therefore produce no motion, while a slightly larger command suddenly moves the mechanism.

An actuator has finite voltage, current, torque, speed, and thermal capacity. In the simplified single-axis relation net torque = inertia × angular acceleration, a requested acceleration is impossible when the needed net torque exceeds the available torque after gravity and friction are accounted for.

A controller cannot repair missing physical authority by endlessly raising its gain. Good engineering separates the requested command, the clamped command actually sent, the measured response, and the reason for any limit. That evidence prevents a software tuning problem from being confused with undersized hardware or a jam.

Words you need

Name each idea precisely

Mass: Resistance to linear acceleration, measured in kilograms.
Physical example:
The same push accelerates an empty trolley more than a loaded trolley.
Rotational inertia: Resistance to angular acceleration about a chosen axis; it depends on mass distribution.
Physical example:
A long bar is harder to spin when weights are fixed near its ends.
Friction: A force or torque that opposes relative motion or the start of motion.
Physical example:
A drawer needs a larger first pull, then a smaller pull to keep sliding.
Actuator limit: A physical or configured bound on torque, force, speed, current, voltage, or temperature.
Physical example:
A servo cannot produce more than its rated torque without stalling, overheating, or violating its limits.
Net torque: The signed sum of driving and resisting torques about an axis.
Physical example:
Motor torque minus friction and gravity torque is what remains to accelerate an arm.

Math, one line at a time

Work through today’s relationship

Prerequisite rescue · optionalChange over time and feedback

Velocity, acceleration, and control error explain whether a robot settles, overshoots, or becomes unsafe.

Δx/Δt: change in position divided by elapsed timeUnit: metres per second (m/s)
e = target − measured: control errorUnit: same unit as the target
u: bounded actuator commandUnit: device-specific

A wheel moves from 1 m to 1.6 m in 0.2 s, so Δx = 0.6 m.
Average velocity is 0.6/0.2 = 3 m/s.
If the target is 2 m and measured position is 1.6 m, error e = 0.4 m; a controller converts that error into a limited command.

Programmer analogy

A feedback loop resembles an event loop that checks state repeatedly, but a missed deadline changes physical motion, not just screen responsiveness.

Position changes by 0.5 m in 0.25 s. What is average velocity?

0.5/0.25 = 2 m/s.

Rotational dynamics gives

\tau=I\alpha=(0.5\ \mathrm{kg\,m^2})(4\ \mathrm{rad/s^2})=2\ \mathrm{N\,m}

Because $2\ \mathrm{N\,m}>1.2\ \mathrm{N\,m}$ , the actuator cannot supply the requested torque.

Check whether a requested acceleration is physically possible

A simplified single-axis arm has rotational inertia I=0.50 kg·m². The controller requests angular acceleration 4 rad/s². The actuator limit is 1.20 N·m, and opposing friction is estimated as 0.20 N·m.

Use the simplified net relation τ_net=Iα for this one-axis example.
Compute needed net torque: 0.50×4=2.00 N·m.
Add the opposing friction that the motor must also overcome: requested motor torque is about 2.20 N·m.
Compare 2.20 N·m with the 1.20 N·m actuator limit; the request is impossible.
At the limit, subtract friction: only 1.20-0.20=1.00 N·m remains for acceleration.
Compute achievable acceleration: 1.00/0.50=2.00 rad/s² under these simplifying assumptions.

Result

The bounded actuator can provide about 2 rad/s², not the requested 4 rad/s², before considering gravity or changing inertia.

What this proves

Always compare required effort with available effort; a larger controller number does not create more motor capability.

Physical examples

Where this appears in real life

Mass near versus far from a pivot

Tape two equal coins near the centre of a cardboard strip, then move them near the ends and gently twist the strip about its centre.

Look for:

The total mass stays the same, but spreading it farther from the axis increases rotational inertia and makes quick angular changes harder.

Loaded spool

Imagine one small motor winding a string that lifts an empty cup and then a cup containing washers.

Look for:

More load needs more torque. Near the motor's limit, acceleration falls; beyond the limit the lift stalls even if software requests more.

Hands-on exercise

Make the idea observable

Use a ruler or cardboard strip, a pencil as a pivot, and two identical coins. Keep the experiment unpowered and clear of fragile objects.

Balance the strip on the pivot and mark the rotation axis.
Attach both coins close to the axis and gently start and stop a small rotation.
Move the same coins near the two ends and repeat with a similar hand motion.
Write what stayed constant: total mass, pivot, and approximate applied motion.
Write what changed: mass distribution and rotational response.
Sketch a motor-command log that separately records request, limit, sent command, and measured angle.

Observe

The far-out coin arrangement resists changes in angular velocity more strongly even though it weighs exactly the same.

Done when

You can explain the result using rotational inertia and can name at least two real actuator limits a robot log should expose.

Build today

Control a simulated pendulum or cart-pole; log setpoint, error, command, saturation, and settling time.

Evidence to save

DONE when “Mass, inertia, friction, and actuator limits” runs from one documented command and the nominal plus boundary outputs are attached.

Common mistakes

Catch the wrong mental model

Wrong

Treating rotational inertia as another name for mass.

Better

Mass is part of inertia, but the distance and distribution of that mass around the chosen axis also matter.

Wrong

Assuming zero measured motion means zero command.

Better

The command may be consumed by static friction, gravity, a hard stop, or saturation; log both command and response.

Wrong

Using τ=Iα without naming the simplifications.

Better

For a real robot also account for gravity, friction, coupling, changing inertia, and external contact forces.

Job connection

How this becomes employable evidence

During commissioning, compare payload location, motor-current limit, joint torque estimate, friction, and tracking error before changing controller gains.

Relevant target roles

Robotics Deployment, Integration & Validation Engineer
Robotics Application / ROS 2 Integration Engineer

Chapter 03 interview drill

Interview questions: Mass, inertia, friction, and actuator limits

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

An arm works empty but stalls with a box held far from the joint. Explain the likely physics and the evidence you would collect before changing software.

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

Q1Why does moving a payload farther from a joint matter?

Model interview answer

It increases rotational inertia and often gravity torque, so more joint torque is needed for the same motion.

Q2What happens when requested torque exceeds the actuator limit?

Model interview answer

The applied torque is bounded by the available limit, so the requested acceleration or tracking cannot be achieved.

Q3Why should requested and applied commands be logged separately?

Model interview answer

Their difference reveals saturation or safety limiting instead of falsely blaming the controller calculation.