ME3120 · Dynamic Modelling
Theme 1 · Foundations
Degrees of freedom & generalized coordinates
Before any equation: count how many ways a system can move, and choose the right variables to describe it.
Source: course notes, Week 1 foundations.
Before you start
What you need first
- What a dynamic model is — a model predicts how a system moves in time (previous topic).
What you'll be able to do
- Count the degrees of freedom (DOF) of a system.
- See how a constraint reduces the DOF.
- Choose good generalized coordinates \(q\).
What is a degree of freedom?
DOF = the number of independent ways a system can move = the fewest numbers needed to say exactly where everything is.
How to count it — ask: "What is the fewest numbers I
must give you so you can draw the system in exactly one position?" That count is the DOF.
🔭 Looking ahead: every joint we add to a robot adds exactly
one DOF. A 6-joint industrial arm has 6 DOF.
Degrees of freedom — examples
| System | DOF | Why |
|---|---|---|
| Bead on a straight wire | 1 | one distance along the wire |
| Point in a flat plane | 2 | \(x\) and \(y\) |
| Point in 3-D space | 3 | \(x,\ y,\ z\) |
| Pendulum (rigid rod, fixed pivot) | 1 | just the angle \(\theta\) |
| 2-link robot arm (flat) | 2 | two joint angles \(\theta_1,\theta_2\) |
| Free rigid body in 3-D | 6 | 3 positions + 3 rotations |
Notice the pendulum: the bob lives in a plane (2 numbers), but the rigid
rod removes one. Result: 1 DOF. The next section shows why.
Constraints cut down the DOF
A pendulum bob looks like 2 numbers \((x,y)\). But the rigid rod forces a rule — a constraint:
$$x^2+y^2=L^2$$
That one rule removes one freedom:
$$\text{DOF}=2-1=1$$
So one angle \(\theta\) fully describes the bob.
Two numbers describe the bob, but the rod ties them together.
🔭 Modeller's habit: choose coordinates that already obey the
constraints (like \(\theta\)). Then the algebra stays short.
Generalized coordinates \(q\)
Generalized coordinates = the independent variables we choose to describe where the system is. We call them \(q_1, q_2, \dots\) (or just \(q\)).
Rules
- They can be angles (radians) or distances (metres).
- We need one coordinate per degree of freedom.
- A good choice → short equations. A bad choice → a mess.
Examples
| System | Natural \(q\) |
|---|---|
| Mass on a spring | \(x\) |
| Pendulum | \(\theta\) |
| 2-link arm | \((\theta_1,\theta_2)\) |
🔭 For a robot, \(q\) is simply the list of joint angles.
Everything we build points at filling in \(q\) for an arm.
✏️ Try it yourself
For each system, give the number of DOF and a sensible choice of generalized coordinate(s) \(q\):
- A train carriage that can only roll along a straight track.
- A boat floating on a lake (ignore tilting — it can move on the surface and turn).
- A double pendulum: a second rod and bob hung from the bottom of the first.
1. 1 DOF —
\(q = s\), the distance along the track.
2. 3 DOF —
\(q = (x, y, \phi)\): position on the surface plus the heading angle.
3. 2 DOF —
\(q = (\theta_1, \theta_2)\), the two rod angles (each rigid rod is one constraint,
like the single pendulum).
Recap — the whole topic on one screen
| Idea | What you own now |
|---|---|
| Degrees of freedom | The fewest numbers to fix the system's position |
| Counting DOF | "How few numbers to draw it in one position?" |
| Constraints | Each rigid link/rule removes one freedom |
| Generalized coordinates \(q\) | One independent variable per DOF (angle or distance) |