Why the derivatives of motion matter in robotics, drones, CNC machines and physical AI

We often describe a machine’s motion using three familiar quantities:

Position → Velocity → Acceleration

But modern machines don’t stop there.

When a robot suddenly changes its acceleration, you feel the resulting jerk. When a drone needs to execute a highly dynamic trajectory, planners may optimize snap. In precision machines, higher-order derivatives of motion can become important for smoothness, vibration, tracking accuracy, and mechanical performance.

The mathematics is simple:$$\mathbf{r}\to {\mathbf{r}}^{\left(1\right)}\to {\mathbf{r}}^{\left(2\right)}\to {\mathbf{r}}^{\left(3\right)}\to {\mathbf{r}}^{\left(4\right)}\to {\mathbf{r}}^{\left(5\right)}\to {\mathbf{r}}^{\left(6\right)}$$

Each step is simply another derivative with respect to time.

---

The hierarchy of motion

Order	Mathematical form	Name	What it describes	Example application
0	$\mathbf r$	Position	Where the object is	Robotics, navigation, CNC
1	$\mathbf r^{(1)} = \frac{d\mathbf r}{dt}$	Velocity	How fast and in what direction it moves	Mobile robots, vehicles, drones
2	$\mathbf r^{(2)} = \frac{d^2\mathbf r}{dt^2}$	Acceleration	How quickly velocity changes	Dynamics, actuator sizing
3	$\mathbf r^{(3)} = \frac{d^3\mathbf r}{dt^3}$	Jerk	How quickly acceleration changes	Elevators, robotics, CNC
4	$\mathbf r^{(4)} = \frac{d^4\mathbf r}{dt^4}$	Snap / Jounce	How quickly jerk changes	Quadrotor trajectory planning
5	$\mathbf r^{(5)} = \frac{d^5\mathbf r}{dt^5}$	Crackle	How quickly snap changes	High-order trajectory optimization
6	$\mathbf r^{(6)} = \frac{d^6\mathbf r}{dt^6}$	Pop	How quickly crackle changes	Specialized high-order motion planning

The names beyond jerk are less standardized in engineering practice, but the mathematical hierarchy itself is straightforward.

1. Position — Where are we?

Position is the starting point.$$\mathbf r(t)$$

For a robot, this could represent the position of its base, end-effector, foot, or any point on its body.

For example, a humanoid robot needs to continuously reason about the position of:

• its feet
• hands
• torso
• center of mass
• joints
• objects it is manipulating

But knowing where something is doesn’t tell us how it is moving.

That brings us to velocity.

2. Velocity — How fast are we moving?

Velocity is the first derivative of position:$$\mathbf v = \dot{\mathbf r}$$

It tells us both the magnitude and direction of motion.

For a mobile robot, velocity determines how quickly it moves.

For a robotic arm, joint velocity determines how quickly each joint moves.

For a drone, velocity describes its translational motion through space.

But velocity alone still doesn’t tell us how quickly that motion is changing.

3. Acceleration — How quickly does velocity change?

Acceleration is the second derivative:$$\mathbf a = \ddot{\mathbf r}$$

It describes the rate of change of velocity.

Acceleration is fundamental to dynamics because force is related to acceleration:$$\mathbf F = m\mathbf a$$

This is where motion starts becoming a physical problem.

Higher acceleration generally means higher forces, higher actuator demands, and greater dynamic loads.

But there is another layer.

Imagine an elevator that goes from stationary to moving upward.

The acceleration cannot instantly jump from zero to a large value without creating an abrupt change in the passenger’s experience.

That abrupt change brings us to jerk.

4. Jerk — How abruptly does acceleration change?

Jerk is the third derivative of position: $$\mathbf{j}={\mathbf{r}}^{\left(3\right)}$$

or$$\mathbf j = \frac{d\mathbf a}{dt}$$

This is one of the most practically important higher-order derivatives in motion control.

Consider two machines with exactly the same:

• position
• velocity
• acceleration

They can still feel completely different if one changes acceleration much more abruptly than the other.

That’s why jerk matters.

Elevators

Elevator systems limit jerk because passengers perceive abrupt changes in acceleration as uncomfortable motion.

Robotics

In robotic arms and humanoid robots, excessive jerk can produce:

• vibration
• mechanical excitation
• tracking errors
• abrupt forces
• undesirable interaction forces

A smooth acceleration profile can therefore be important for both the machine and the environment around it.

CNC machines

CNC trajectory planning also considers jerk because abrupt changes in motion can contribute to vibration and reduce smoothness during high-speed machining.

Jerk is where motion starts becoming noticeably different from simply being “fast.”

5. Snap — How quickly does jerk change?

Take one more derivative:$$\mathbf s = \mathbf r^{(4)}$$

This is commonly called snap or jounce.

Snap becomes particularly interesting in aerial robotics.

Why does a quadrotor care about snap?

A quadrotor has limited thrust and attitude dynamics. It cannot arbitrarily change its motion instantaneously.

When generating aggressive trajectories, planners therefore don’t only ask:

Where should the drone go?

They can also ask:

What trajectory minimizes undesirable higher-order motion?

A well-known approach in quadrotor trajectory generation is minimum-snap trajectory optimization.

Instead of simply connecting waypoints, the planner generates a trajectory whose higher-order derivatives remain controlled.

The result is a trajectory that is smoother and more compatible with the vehicle’s dynamics.

6. Crackle and Pop

The mathematical sequence continues:$$\mathbf r^{(5)}$$

is commonly called crackle, while$$\mathbf r^{(6)}$$

is commonly called pop.

These terms are much less common in everyday engineering than velocity, acceleration, jerk, and snap.

They become relevant mainly when dealing with high-order trajectory generation and optimization, where increasingly smooth motion profiles are mathematically desirable.

At this point, however, an important question appears:

Do we really need to optimize every derivative?

Not necessarily.

The appropriate derivative depends on the physical system, its actuators, its dynamics, its control architecture, and the task it needs to perform.

---

Motion Is More Than Position

This hierarchy reveals something important about robotics.

A robot isn’t simply moving from A to B.

It is executing a continuous trajectory:$$\mathbf r(t)$$

and that trajectory has structure at multiple levels:$$\mathbf{r}\to {\mathbf{r}}^{\left(1\right)}\to {\mathbf{r}}^{\left(2\right)}\to {\mathbf{r}}^{\left(3\right)}\to {\mathbf{r}}^{\left(4\right)}\to {\mathbf{r}}^{\left(5\right)}\to {\mathbf{r}}^{\left(6\right)}$$

Each derivative tells us something different about the motion.

Position tells us where.

Velocity tells us how fast.

Acceleration tells us how quickly velocity changes.

Jerk tells us how abruptly acceleration changes.

Snap tells us how abruptly jerk changes.

And so on.

---

Why This Matters for Physical AI

As robotics moves toward Physical AI, this distinction becomes increasingly important.

A physical AI system doesn’t just predict an action.

It has to execute that action through:

Perception → Planning → Control → Actuation → Physical interaction

The final output is not a digital token.

It is motion in the physical world.

A humanoid robot reaching for an object, a quadruped running across uneven terrain, or a drone navigating through a constrained environment all have to generate trajectories that respect the physical limitations of their bodies.

The question therefore isn’t simply:

Can the robot reach the target?

It is also:

Can it reach the target with a trajectory its body can physically execute?

That is where higher-order motion becomes interesting.

---

The Bigger Picture

We often teach robotics through position, velocity and acceleration.

But as machines become faster, more dynamic and more capable, smoothness becomes a first-class engineering problem.

The derivative that matters depends on the machine:

• Elevators: jerk can determine ride comfort.
• Robotic arms: velocity, acceleration and jerk influence smoothness and mechanical behavior.
• Quadrotors: snap can become important in trajectory generation.
• CNC machines: higher-order motion constraints help manage smoothness and vibration.
• Humanoids and legged robots: higher-order motion must ultimately coexist with contact dynamics, actuator limits and balance.

The deeper lesson is simple:

A machine doesn’t just need to know where to go. It needs to know how to get there.

And sometimes, the difference between a machine that merely moves and one that moves well lies not in its position or velocity, but in the derivatives in between.

Join the conversation on LinkedIn: