🔝​

🔝​

Singularity analysis

Prerequisites

This chapter assumes you are comfortable with:

  • Vectors and matrices
    • 3D vectors, dot product, cross product
    • linear combinations, rank, null space (conceptually)
  • Rigid-body kinematics
    • rotation matrices (or any equivalent representation)
    • the idea that a rigid body has 6 instantaneous degrees of freedom in 3D
  • Basic mechanics vocabulary
    • force, moment (torque), and power (even informally is fine)

General Motivation

Singularities are not just “when a Jacobian determinant becomes zero.”
They are geometric events: certain instantaneous motions become impossible, or certain constraint forces stop being able to oppose motion. In screw theory, these statements become clean linear-algebra facts about subspaces of twists (allowable motions) and subspaces of wrenches (allowable constraints).

This chapter builds that viewpoint step by step: vector spaces → forces and moments → twists and wrenches → Chasles’ theorem → screws (axis and pitch) → Plücker coordinates. Short quizzes after each milestone check understanding and catch common confusions early, while the assignments turn the ideas into concrete calculations and geometric intuition, preparing you to understand degrees of freedom and interpret singularities, especially in parallel robots.

Big picture. Our goal is to use screw theory to answer questions like:

  • What motions can a mechanism produce at a given configuration?
  • What forces/constraints can it transmit?
  • When do these “capabilities” change suddenly (singularity)?

In a serial robot, we typically build a Jacobian column-by-column from joint screws.
In a parallel robot, we often reason from constraint screws (reciprocal to allowable motion screws) using virtual work.

We will proceed slowly and carefully. If you understand the preliminaries in this chapter, the later singularity analysis becomes far less mysterious.

Course Content

Preliminaries

Notations (used throughout the chapter)

  • A point in space is denoted by a capital letter: $O, P, Q$.
  • A vector is bold: $ \mathbf{r}, \mathbf{f}, \boldsymbol{\omega} $.
  • The vector from $O$ to $P$ is $$ \mathbf{r}_{OP} := \mathbf{p} - \mathbf{o}. $$
  • The cross product $\mathbf{a}\times \mathbf{b}$ is a vector orthogonal to both $\mathbf{a}$ and $\mathbf{b}$.
  • A frame $Oxyz$ means origin $O$ with orthonormal axes $\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}$.

Dimensional sanity check.
Forces have units (N), moments have units (N·m), angular velocity is (rad/s), linear velocity is (m/s).
Screw theory mixes these in 6-vectors — so we must be explicit about what each block means.

Vector spaces (why we start here)

A (real) vector space $V$ is a set of objects called “vectors” where you can:

  1. add any two vectors in $V$, producing another vector in $V$
  2. multiply any vector in $V$ by any real number, producing another vector in $V$

and the usual rules (associativity, distributivity, etc.) hold.

Definition (span).
Given vectors $v_1,\dots,v_n \in V$, their span is $$ \operatorname{Span}\lbrace v_1,\dots,v_n\rbrace = \left\lbrace \sum_{i=1}^n \alpha_i v_i \ \middle|\ \alpha_i\in\mathbb{R} \right\rbrace $$ It is the smallest subspace that contains all $v_i$.

Why we care in kinematics.
“Possible instantaneous motions” form a subspace (a twist subspace).
“Possible constraint actions” form a subspace (a wrench subspace).
Singularity analysis becomes: how do these subspaces change with configuration?

Examples vs. counterexamples (closure matters).

  • Forces acting on a particle at a fixed point behave like ordinary vectors → vector space.
  • Pure moments (couples) also form a vector space.
  • “Forces acting on a rigid body at arbitrary points” are not closed under naive addition unless you also track the accompanying moment → this motivates wrenches.
Quick quiz — closure intuition

Question 1: “All forces applied to a rigid body form a vector space under vector addition.”

True
False

Forces, moments, and couples (rigid-body viewpoint)

Force

A force is a 3D vector $\mathbf{f}\in\mathbb{R}^3$. Physically, a force has:

  • magnitude (how strong)
  • direction
  • line of action (where it acts)

When we say “a force applied at point $P$”, we mean:

  • the vector $\mathbf{f}$, and
  • the point $P$ (or equivalently the line of action)
Moment of a force about a point

Let $O$ be a reference point. The moment (torque) of $\mathbf{f}$ applied at point $P$ about $O$ is $$ \mathbf{m}_O := \mathbf{r}_P \times \mathbf{f} $$ This is a 3D vector with units (N·m).

Example (door handle intuition).
A small force $\mathbf{f}$ applied far from the hinge produces a larger $\mathbf{m}_O$ because $\mathbf{r}_P$ is larger.
That is exactly what $\mathbf{m}_O=\mathbf{r}_P\times\mathbf{f}$ encodes.

Changing the reference point

If you change the reference point from $O$ to another point $Q$, the moment changes as $$ \mathbf{m}_Q = \mathbf{m}_O - \mathbf{r}_Q \times \mathbf{f}. $$ So $\mathbf{m}_O$ is not “absolute”: it depends on the chosen reference point.

Couple (pure moment)

A couple is a free moment: a moment that does not depend on the choice of reference point. You can realize a couple by two equal and opposite forces separated by a distance (net force = 0, net moment ≠ 0).

Definition (couple).
A couple is a force system whose resultant force is zero but whose resultant moment is nonzero. It can be represented by a single vector $\mathbf{c}\in\mathbb{R}^3$ (units N·m) that is the same about any point.

Force, moment about a point, and a couple

Figure: (i) force at a point, (ii) moment about $O$, (iii) couple as two equal/opposite forces.
Quick quiz — couple vs moment

Question 2: Which statement best describes a couple?

A couple is the moment of a force about the origin only.
A couple is a pure moment that is independent of the reference point.
A couple is any force whose line of action passes through the origin.
A couple is always equivalent to a single force through the origin.

Linear and angular velocity (instantaneous rigid-body motion)

Consider a rigid body moving in 3D. Pick a reference point $O$ fixed in the body.

  • The angular velocity is a vector $\boldsymbol{\omega}\in\mathbb{R}^3$. Its direction is the instantaneous axis of rotation (right-hand rule), and its magnitude is the rotation rate.
  • The linear velocity of point $O$ is $\mathbf{v}_O\in\mathbb{R}^3$.

A fundamental rigid-body kinematics fact is:

Rigid-body velocity field.
For any point $P$ on the rigid body, $$ \mathbf{v}_P = \mathbf{v}_O + \boldsymbol{\omega}\times \mathbf{r}_P. $$

This formula is the reason rigid-body instantaneous motion is “6-dimensional”: $\boldsymbol{\omega}$ contributes 3 parameters, $\mathbf{v}_O$ contributes 3 parameters.

Twists and wrenches (the 6D objects we actually use)

Now we package the 3D quantities into 6D objects.

Twist (instantaneous motion)

A twist is a pair $$ \boldsymbol{\xi} := \begin{bmatrix} \boldsymbol{\omega}, \mathbf{v}_O \end{bmatrix}^T \in \mathbb{R}^6, $$

where:

  • $\boldsymbol{\omega}$ = angular velocity of the rigid body
  • $\mathbf{v}_O$ = linear velocity of the reference point $O$

Different choices of $O$ change $\mathbf{v}_O$, but the physical motion is the same. Later we will exploit this in “screw axis” representations.

Wrench (force system)

A wrench is a pair $$ \mathbf{w} := \begin{bmatrix} \mathbf{f}, \mathbf{m}_O \end{bmatrix}^T \in \mathbb{R}^6, $$ where:

  • $\mathbf{f}$ = resultant force
  • $\mathbf{m}_O$ = resultant moment about the reference point $O$

Again, $\mathbf{m}_O$ depends on $O$, but the underlying force system is the same.

Why wrenches fix the “closure problem.”
If you add two forces acting at different points on a rigid body, you cannot describe the result by a single 3D force vector unless you also carry the induced moment.
A wrench carries both $\mathbf{f}$ and $\mathbf{m}_O$, so “addition” stays inside the same 6D space.

Power (the pairing between wrench and twist)

When a wrench $\mathbf{w}=(\mathbf{f},\mathbf{m}_O)$ acts on a body moving with twist $\boldsymbol{\xi}=(\boldsymbol{\omega},\mathbf{v}_O)$, the instantaneous power is $$ \mathcal{P} = \mathbf{f}\cdot \mathbf{v}_O + \mathbf{m}_O\cdot \boldsymbol{\omega}. $$

This single scalar is the key bridge to virtual work and reciprocity later.

Quick quiz — power pairing

Question 3: The expression $\mathbf{f}\cdot \mathbf{v}_O + \mathbf{m}_O\cdot \boldsymbol{\omega}$ represents the instantaneous power of a wrench acting on a twist (for a consistent choice of reference point $O$).

True
False

Preview (duality language).
We will later say: wrenches are dual to twists because a wrench maps a twist to a real number (power) by this pairing.

Chasles’ theorem (why “screws” appear in the first place)

Chasles’ theorem (19th century) states that any rigid-body displacement in 3D can be described as:

  • a rotation about some axis, and
  • a translation along that same axis.

This combined motion is called a screw motion.

A bit of history (people worth knowing).

  • Michel Chasles (1793–1880): established the geometric foundation of rigid-body displacements as screw motions.
  • Robert Stawell Ball (1840–1913): developed screw theory into a systematic tool for mechanics and kinematics (often cited as the classical “screw theory” reference).

Chasles screw motion: rotation about an axis + translation along the axis

Figure: a rigid body undergoing a helical motion about a directed axis.

Screws: axis, pitch, and “same line, different amplitude”

A geometric screw is defined by:

  1. a directed line (the screw axis) in 3D
  2. a pitch $h$ (a real number)

Pitch $h$ measures “translation per rotation” along the axis:

  • $h=0$: pure rotation about the axis
  • $h\to\infty$: pure translation along the axis (informally: rotation rate goes to zero)
  • finite nonzero $h$: helical motion (like a screw thread)
Pitch from a twist

If $\boldsymbol{\omega}\neq \mathbf{0}$, one common definition of pitch is: $$ h = \frac{\boldsymbol{\omega}\cdot \mathbf{v}_O}{\boldsymbol{\omega}\cdot \boldsymbol{\omega}}, $$ when $(\boldsymbol{\omega},\mathbf{v}_O)$ is written about a point $O$ on (or consistently related to) the screw axis.

Interpretation: the component of $\mathbf{v}_O$ parallel to $\boldsymbol{\omega}$ encodes translation along the axis, and dividing by $|\boldsymbol{\omega}|^2$ normalizes by rotation rate.

Pitch from a wrench

If $\mathbf{f}\neq \mathbf{0}$, similarly: $$ h = \frac{\mathbf{f}\cdot \mathbf{m}_O}{\mathbf{f}\cdot \mathbf{f}}, $$ for a wrench written in a consistent screw-axis form.

Projective idea (important later).
Scaling a twist by $2$ doubles the speed, but it represents the same screw axis and pitch.
That is why we separate:

  • the geometric screw (axis + pitch) from
  • the twist/wrench vector (axis + pitch + amplitude).
Quick quiz — pitch meaning

Question 4: What does the pitch $h$ of a screw primarily quantify?

The distance from the origin to the screw axis.
The ratio of translation along the axis to rotation about the axis.
The angle between the screw axis and the z-axis.
The magnitude of angular velocity $\|\boldsymbol{\omega}\|$.

Plücker coordinates (a clean way to encode a line)

A directed line $\ell$ in 3D can be represented by Plücker coordinates: $$ (\mathbf{s},\ \mathbf{m}), $$ where:

  • $\mathbf{s}\in\mathbb{R}^3$ is a direction vector along the line (often unit-length),
  • $\mathbf{m}\in\mathbb{R}^3$ is the moment of the line about the origin: $$ \mathbf{m} := \mathbf{p}\times \mathbf{s}, $$ and $\mathbf{p}$ is the position vector of any point on the line.

Key property (Plücker constraint). $$ \mathbf{s}\cdot \mathbf{m} = 0. $$ (Geometrically: $\mathbf{m}$ is perpendicular to $\mathbf{s}$ because it is a cross product with $\mathbf{s}$.)

Example (build a line from two points).
Let the line pass through points $A$ and $B$, and define $\mathbf{s} := \mathbf{r}_{AB}$ (or its unit version).
Pick $\mathbf{p} := \mathbf{r}_A$. Then $\mathbf{m}=\mathbf{p}\times\mathbf{s}$.
If you pick $\mathbf{p}:=\mathbf{r}_B$ instead, you get the same $\mathbf{m}$ (because $\mathbf{r}_B=\mathbf{r}_A+\lambda\mathbf{s}$, and $(\lambda\mathbf{s})\times\mathbf{s}=\mathbf{0}$).

Plücker form of a twist (preview)

A twist associated with a screw axis $(\mathbf{s},\mathbf{m})$ and pitch $h$ can be written (in one common convention) as: $$ \boldsymbol{\xi} = \begin{bmatrix} \boldsymbol{\omega}, \mathbf{v}_O \end{bmatrix}^T = \begin{bmatrix} \dot{\theta}\,\mathbf{s}, \dot{\theta}\,(\mathbf{m} + h\,\mathbf{s}) \end{bmatrix}^T, $$ when $\mathbf{s}$ is unit and $\dot{\theta}$ is the rotational rate about the axis.

  • If $h=0$: $\mathbf{v}_O = \dot{\theta}\,\mathbf{m}$ corresponds to pure rotation about the axis.
  • If $\dot{\theta}=0$ but translation exists: you get a pure translation twist (handled as a limiting case).

Don’t worry if this feels abstract: we will make it concrete with robot joints soon.

Quick quiz — Plücker constraint

Question 5: For Plücker coordinates $(\mathbf{s},\mathbf{m})$ of a directed line, which condition must hold?

$\|\mathbf{s}\| = \|\mathbf{m}\|$.
$\mathbf{s}\cdot \mathbf{m} = 0$.
$\mathbf{s}\times \mathbf{m} = \mathbf{0}$.
$\mathbf{s} + \mathbf{m} = \mathbf{0}$.

Mini-assignment 1 — build intuition with your own numbers (no robot yet)

1) Moment shift practice.
Choose $\mathbf{f}=\begin{bmatrix}2, -1, 3\end{bmatrix}^T$ N applied at point $P$ with $\mathbf{r}_P=\begin{bmatrix}0.2, 0.0, 0.1\end{bmatrix}^T$ m.
Compute $\mathbf{m}_O=\mathbf{r}_P\times\mathbf{f}$.
Now let $Q$ be such that $\mathbf{r}_Q=\begin{bmatrix}0.0, 0.3, 0.0\end{bmatrix}^T$ m. Compute $\mathbf{m}_Q$ using
$\mathbf{m}_Q=\mathbf{m}_O-\mathbf{r}_Q\times\mathbf{f}$.
Explain in one sentence why $\mathbf{m}_Q\neq \mathbf{m}_O$ does not contradict “same force”.

2) Plücker line practice.
Let a line pass through $A=(0.1,0,0)$ m and $B=(0.1,0.2,0.3)$ m.
Compute $\mathbf{s}=\mathbf{r}_{AB}$ and $\mathbf{m}=\mathbf{r}_A\times\mathbf{s}$.
Verify $\mathbf{s}\cdot\mathbf{m}=0$.

3) Explain in words.
Why is it helpful that a line can be encoded by two 3-vectors $(\mathbf{s},\mathbf{m})$ instead of “a point + a direction”?

Up next (next section we will write):
We will connect these preliminaries to joint screws and show how a serial chain builds a twist subspace by “cascading” joint twists to derive the geometric Jacobian.

General Concepts

References and Further Reading (Intro Level)

  • D. S. Zlatanov, Generalized Singularity Analysis of Mechanisms, Ph.D. dissertation, University of Toronto, 1998
  • M. Conconi and M. Carricato, “A New Assessment of Singularities of Parallel Kinematic Chains,” IEEE Transactions on Robotics, vol. 25, no. 4, pp. 757–770, 2009, doi: 10.1109/TRO.2009.2020353.
  • J. Selig, Geometric Fundamentals of Robotics
  • R. Murray, Z. Li, S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC.

Credits

This course page was created by Durgesh Haribhau Salunkhe, EPFL, and funded by IEEE RAS and EPFL.

Resources

Back to Top

$$ . $$