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Vector Spaces, Twists and Wrenches
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Dimiter Zlatanov
DIMEC – University of Genoa
Genoa, Italy
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-2
● Dual Spaces and Scalar Products
● Screw Systems
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-3
● Dual Spaces and Scalar Products
● Screw Systems
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A vector space (or linear space) V over the field
A set of vectors with 2 operations:
● vector additionA1 associative law
A2 commutative law
Definition
R
V ∋ u, v,w, . . .
(u+ v) +w = u+ (v +w)
u+ v = v + u
V × V → V, (u,v) �→ u+ v
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Vector Spaces, Wrenches, TwistsZ-4
A3 zero vector
A4 opposite vector
● scalar multiplicationA5 associative law
A6 distributive law
A7 distributive law
A8 unit scalar
∃o s.t. ∀u o+ u = u
∀u ∃ − u s.t. (−u) + u = o
λ(µu) = (λµ)u
(λ+ µ)u) = λu+ µu
λ(u+ v) = λu+ λv
1u = u
R× V → V, (λ,u) �→ λu
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● Trivial:
● Simple: numbers ; functions
● Fundamental:
● n-tuples
v.a and s.m. : component-wise
Examples and Counterexamples
∅, {o}
Rn ∋ (x1, . . . , xn)
Q, R, C, R−Q {f |f : X → V }
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Vector Spaces, Wrenches, TwistsZ-5
v.a and s.m. : component-wise
● arrows from a point in space, “magnitude and direction”
v.a. : parallelogram rule
s.m.: length dilation
a+ ba
a λa
b
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● Forces acting on a particle,
v.a : resultant force (parallelogram rule)
s.m: proportional change of force intensity
arrow from the particle, magnitude and direction
● Force fields mappings
Examples and Counterexamples
F 3
{f |f : E3 → F 3}f(P )
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Vector Spaces, Wrenches, TwistsZ-6
arrow at every point P
● Velocities of a (free) particle
● Velocity fields
● Velocities and forces ? ?
v.a. must be defined for every pair of vectors
{ | → }
M3
v(P )
f + v =
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● Car traffic through an intersection
arrow with magnitude= average # of cars in a direction
s.m. : clear;
v.a. : parallelogram rule?
Examples and Counterexamples
40 cars/h
50 cars/h ?
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Vector Spaces, Wrenches, TwistsZ-7
v.a. : parallelogram rule?
Not everything with magnitude and direction is a vector
30 cars/h
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● Reaction forces acting on a rigid body with a fixed point
v.a. : resultant force (parallelogram rule)
s.m.: proportional change of force intensity
● Forces acting on a rigid body
Examples and Counterexamples
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Vector Spaces, Wrenches, TwistsZ-8
v.a.: resultant force ?
two forces have a “resultant force” only if
their axes intersect
a vector space must be closed under v.a.
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● Couples of forces acting on a rigid body
v.a. : resultant couple (parallelogram rule)
s.m.: proportional change of the moment magnitude
● Forces and couples acting on a rigid body
Examples and Counterexamples
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Vector Spaces, Wrenches, TwistsZ-9
v.a.: resultant force or couple ?
two forces have a “resultant force or couple” only if
their axes are coplanar
a vector space must be closed under v.a.
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● Instantaneous rotations of a body with a fixed point
v.a. : resultant rotation, (parallelogram rule)
obtained when the body is the end-effector
of an RR chain with intersecting axes
s.m.: proportional change of rotation amplitude
Examples and Counterexamples
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Vector Spaces, Wrenches, TwistsZ-10
● Instantaneous rotations of a rigid body
v.a.: resultant rotation?
two rotation have a “resultant rotation” only if
their axes intersect
a vector space must be closed under v.a.
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● Instantaneous translations of a rigid body
v.a. : resultant translation, (parallelogram rule)
as if the body is the end-effector of a PP chain
s.m.: proportional change of translation speed
● Instantaneous rotations and translations of a rigid body
Examples and Counterexamples
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Vector Spaces, Wrenches, TwistsZ-11
● Instantaneous rotations and translations of a rigid body
v.a.: resultant rotation?
two rotation have a “resultant rotation or translation”
only if their axes are coplanar
a vector space must be closed under v.a.
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-12
● Dual Spaces and Scalar Products
● Screw Systems
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● External action on a system of particles B:
● System of forces at O acting on a rigid body:
force thru O
with intensity and direction
couple
Systems of Forces
{fP |P ∈ B}
f
ΦO = {ϕO,µO}
ϕO
µO
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Vector Spaces, Wrenches, TwistsZ-13
with moment
applied in parallel to the body
described by at O
● Fundamental fact of statics :
all external actions on a rigid body are of this type
[this is an axiom in rigid body dynamics]
µOmO
(f , mO)
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● Instantaneous motion of a particle system B:
● Instantaneous Motion at O :
inst. rotation thru O
with amplitude and direction
inst. translation
Instantaneous Motions
{vP |P ∈ B}
ω
ΥO = {O, τO}
O
τO
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Vector Spaces, Wrenches, TwistsZ-14
with velocity
applied in series to the body
described by at O
● Fundamental fact of rigid body velocity kinematics:
all instantaneous motions are of this type
[this is a theorem in position kinematics]
τOvO
(ω, vO)
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● and
are equivalent iff
and (the shifting law)
● The equivalence class, , is called a wrench
Wrench
ΦO = {ϕO,µO}, (f , mO) Φ′O′ = {ϕ′O′ ,µ′O′}, (f
′
, m′
O′)
f = f′
mO = m′
O′ +−−→OO′ × f
ζ = [Φ]
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Vector Spaces, Wrenches, TwistsZ-15
● The equivalence class, , is called a wrench
● A wrench is a system of forces (reduced at a point)
with equivalent systems identified
● A wrench is an entity invariant of frame choice
● For a given origin, O, it is given by a pair of vectors:
the resultant force and moment at O
ζ = [Φ]
ζ = (f , mO)
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● and
are equivalent iff
and (the shifting law)
● The equivalence class, , is called a twist
Twist
Υ′O′ = {′O′ , τ ′O′}, (ω′, v′O′)
ω = ω′ vO = v′O′ +−−→OO′ × ω
ΥO = {O, τO}, (ω, vO)
ξ = [Υ]
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Vector Spaces, Wrenches, TwistsZ-16
● The equivalence class, , is called a twist
● A twist is an instantaneous motion (reduced at a point)
with equivalent motions identified
● A twist is an entity invariant of frame choice
● For a given origin, O, it is given by a pair of vectors:
the body angular velocity
and the velocity of the point coinciding with O
ξ = [Υ]
ξ = (ω, vO)
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● Wrenches form a vector space, F6, se*(3):
v.a. : resultant wrench: all forces acting in parallel
at O,
s.m. : proportional increase of intensity
at O,
[show that “+” and “.” do not depend on O]
Wrench and Twist Spaces
ζ + ζ′ = (f , mO) + (f ′, m′
O) = (f + f , mO + m′
O)
λζ = λ(f , mO) = (λf , λ mO)
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Vector Spaces, Wrenches, TwistsZ-17
[show that “+” and “.” do not depend on O]
● Twists form a vector space, M6, se(3):
v.a. : resultant motion: all motions acting in series (in any order!)
at O,
s.m. : proportional increase of amplitude
at O,
[show that “+” and “.” do not depend on O]
λξ = λ(ω, vO) = (λω, λvO)
ξ + ξ′ = (ω, vO) + ( ω′, v′O) = (ω + ω, vO + v′O)
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● For any wrench, if then ∀ O
else ∃! line the screw axis
s.t. ∀ P ∈ (force and moment are parallel)
The pitch h and the axis point
Canonical Representative of a Twist/Wrench
ζ = (f , mO)f = o ζ = (o, m)
ℓ(ζ) mP = hf
ℓ(ζ) = {r + λf/|f |}
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Vector Spaces, Wrenches, TwistsZ-18
The pitch h and the axis point
closest to the origin are:
Thus at O,
● Similarly for a twist
h =f · mO
f · f, r =
f × mO
f · f
ζ = (f , hf + r × f)
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● A line l with a pitch h (a metric quantity)
is a geometric element called a screw
● The screw of a couple/translation has no axis,
only a direction: infinite pitch screw, h=∞
Screw
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Vector Spaces, Wrenches, TwistsZ-19
only a direction: infinite pitch screw, h=∞
● A (geometric) screw is not a vector (why?).
Screws form the projective space underlying the
space of twists and wrenches
[The projective space of V is obtained by identifying ] v ∼ λv
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-20
● Dual Spaces and Scalar Products
● Screw Systems
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Definition.
The span of is the set of their linear combinations:
Example. The plane containing in 3D space
Interpretation.
Linear Combinations
v1, . . . ,vn ∈ VSpan (v1, . . . ,vn) = {λ1v1 + . . .+ λnvn|λi ∈ R}
v1, v2
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Vector Spaces, Wrenches, TwistsZ-21
Interpretation.
Span{twists}: all end-effector motions of a serial chain
Span{wrenches} : all end-effector constraints of a parallel chain
Exercise. Consider a spherical RRR chain (concurrent axes).
Find all possible instantaneous motions of
(1) the end-effector (2) the second link
Exercise. Consider a planar RR chain.
Find all instantaneous motions of the end-effector.
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Definition. W is a linear subspace of the vector space V if
(1) W is a subset of V ;
(2) W is a vector space with the v.a. and s.m. of V .
Example. A plane thru the origin. A plane not thru the origin?
Example. Planar motions se(2) ⊂ se(3).
Linear Subspaces
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Vector Spaces, Wrenches, TwistsZ-22
Example. Planar motions se(2) ⊂ se(3).
Theorem. The intersection of two subspaces is a subspace.
The nontrivial union of two subspaces is not a subspace.
The difference of two subspaces is never a subspace (why?).
Example. Impossible motions (e.g. of a planar-chain end-effector).
Theorem. is the smallest subspace with Span (v1, . . . , vn) v1, . . . ,vn
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Definition. The set is linearly dependent if
s.t.
Else it is linearly independent.
Examples. When is l.d.? ?
When are three arrows l.d.?
When are two twists l.d.?
Linear Dependence and Independence
{v1, . . . , vn}
(λ1, . . . , λn) �= (0, . . . , 0) λ1v1 + . . .+ λnvn = 0
{v1} {v1, v2}
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Vector Spaces, Wrenches, TwistsZ-23
When are two twists l.d.?
Facts. (1) If then are l.d.
(2) subset of a l.i. set is l.i.
(3) superset of a l.d. set is l.d.
Exercise. When are three forces l.d.?
{v1, . . . , vn} ∋ o {v1, . . . , vn}
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-24
● Dual Spaces and Scalar Products
● Screw Systems
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Definition. The set is linearly dependent if
s.t.
Else it is linearly independent.
Examples. When is l.d.? ?
When are three arrows l.d.?
When are two twists l.d.?
Linear Dependence and Independence
{v1, . . . , vn}
(λ1, . . . , λn) �= (0, . . . , 0) λ1v1 + . . .+ λnvn = 0
{v1} {v1, v2}
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Vector Spaces, Wrenches, TwistsZ-25
When are two twists l.d.?
A set of forces in equilibrium?
Facts. (1) If then are l.d.
(2) subset of a l.i. set is l.i.
(3) superset of a l.d. set is l.d.
Exercise. When are three forces l.d.?
{v1, . . . , vn} ∋ o {v1, . . . , vn}
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Dimension and Basis
{v1, . . . , vn}
V = Span (v1, . . . , vn)
V = Span (v1, . . . , vn)
Definitions.
(1) dim V < ∞ if
(2) dim V = n if ∃ l.i. (a basis) s.t.
Proposition. dim V = n ⇔ ∃ n-basis ⇔ ∀ basis is an n-basis
⇔ ∀ if l.i. then is l.d. ∀{v1, . . . , vn} {u, v1, . . . ,vn} u
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Vector Spaces, Wrenches, TwistsZ-26
⇔ ∀ if l.i. then is l.d. ∀
Exercises. Find a basis and establish the dimension of
the rotations with axes through a point
the translations
the planar motions
{v1, . . . , vn} {u, v1, . . . ,vn} u
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Dimension and Basis
Example. Plücker bases for a frame Oxyz
The 3 unit rotations about the axes
the 3 unit translations directed as the axes{ρOx,ρOy,ρOz, τx, τ y, τ z}
ξ = (ω, vO) = (ωxi+ ωyj + ωzk, vOxi+ vOyj + vOzk)
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Vector Spaces, Wrenches, TwistsZ-27
The 3 unit forces along the axes
the 3 unit couples directed as the axes
[beware of unit problems! Are fx, mx dimensioned?]
{ϕOx,ϕOy,ϕOz,µx,µy,µz}
ζ = (f , mO) = (fxi + fyj + fzk, mOxi+mOy
j +mOzk)
= fxϕOx + fyϕOy + fzϕOz +mxµx +myµy +mzµz}
= ωxρOx + ωyρOy + ωzρOz + vOxτ x + vOyτ y + vOzτ z}
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Dimension and Basis
Example. Ball bases for a frame Oxyz
6 unit, h= ±1, twists/wrenches along the axes
Example. Joint twists of a 6dof serial manipulator, 1P, 5R joints
6 unit joint translations/rotations in a nonsingular configuration
{ξ+Ox, ξ+
Oy, ξ+
Oz, ξ−
Ox, ξ−
Oy, ξ−
Oz}
{ρ ,ρ , τ ,ρ ,ρ ,ρ }
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Vector Spaces, Wrenches, TwistsZ-28
λi are dimensionless
are the joint speeds, with units
The columns of the Jacobian are not twists
They are not vectors in the same vector space
Treat the Jacobian matrix with care!
{ρ1,ρ2, τ 3,ρ4,ρ5,ρ6}
ξ = λ1ρ1 + λ2ρ2 + λ3τ 3 + λ4ρ4 + λ5ρ5 + λ6ρ6
ξ = ω1
[e1
r1 × e1
]+ω2
[e2
r2 × e2
]+v3
[0e3
]+ω4
[e4
r4 × e4
]+ω5
[e5
r5 × e5
]+ω6
[e6
r6 × e6
]
ξ = Jθ θ = (ω1, ω2, v3, . . . , ω6)T
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Definition. Let U, W ⊂ V (subspaces)
Facts.
Definition. if (1) (2)
Sums and Direct Sums of Subspaces
U +W = {v = u+w|u ∈ U,w ∈ W}
U +W = Span (U ∪W )
dim(U +W ) = dimU + dimW − dim(U ∩W )
V = U ⊕W V = U +W U ∩W = o
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Vector Spaces, Wrenches, TwistsZ-29
Definition. if (1) (2)
Examples. Two planes in 3D
A line and a plane in 3D
Spherical motions and translations
Spherical and planar motions
V = U ⊕W V = U +W U ∩W = o
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-30
● Dual Spaces and Scalar Products
● Screw Systems
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Definition. The dual V* of vector space V ( )
( )
Fact. ∀ basis of V ∃! dual basis
defined by
Example. The space of forces acting on a particle is dual
Dual Spaces and Scalar Products
V ∗ = {f : V → R | f : linear}
dimV = n
{e∗1, . . . , e∗
n}
dim V ∗ = n{e1, . . . , en}
e∗i (ej) = δij
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Vector Spaces, Wrenches, TwistsZ-31
Example. The space of forces acting on a particle is dual
to the space of particle velocities.
Example. The wrenches, se*(3), are dual to the twists, se(3)
ζ(ξ) = ζ ◦ ξ = f · vO + mO · ω
f(v) = f · v
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Interpretation. The application of a wrench on a twist
(also called their reciprocal product) measures the power
exerted by the system of forces for the instantaneous motion
Fact.
For the Plücker twist basis
Dual Spaces and Scalar Products
{ρ ,ρ ,ρ , τ , τ , τ }
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Vector Spaces, Wrenches, TwistsZ-32
For the Plücker twist basis
The dual wrench basis is
Notation.
When dual bases are used,
(interpreting as column coordinate vectors)
Hence, the notation is also used
{ρOx,ρOy,ρOz, τ x, τ y, τ z}
{µx,µy,µz,ϕOx,ϕOy,ϕOz}
ζ ◦ ξ = ζT ξ
ζ · ξ
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Definition . When a dual vector maps a vector into zero,
the two are said to be orthogonal.
Example. When a wrench exerts no power on a twist,
they are orthogonal (also called reciprocal).
Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-33
Exercise. What are the geometric conditions for the following
to be reciprocal:
a translation and a couple
a translation and a wrench/a couple and a twist
a rotation and a force
a twist and a wrench
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Definition . Let U be a subspace of V Then
is the orthogonal annihilator of U
Fact. U⊥ is a subspace. (why?)
Example. In Euclidian spaces V and V* are identified
Dual Spaces and Scalar Products
U⊥ = {w ∈ V ∗ | w(u) = 0, ∀u ∈ U}
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Vector Spaces, Wrenches, TwistsZ-34
Example. In Euclidian spaces V and V* are identified
and we have U ⊕ U⊥ = V
Example. For a twist subspace U, U⊥ is a wrench subspace
composed of all wrenches that exert no power on any
motion in U
Facts. If U⊂ se(3) then dim U + dim U⊥ = 6
Interpretation. U⊥ is a constraint system allowing only motions in U
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Exercises. Describe U⊥ when U is
the planar motions
the system spanned by the rotations in a plane
the system spanned by the rotations shown
Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-35
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Outline
● Vector Spaces
● Wrenches, Twists, and Screws
● Linear Dependence and Independence
● Bases, Coordinates, Dimension
● Dual Spaces and Scalar Products
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Vector Spaces, Wrenches, TwistsZ-36
● Dual Spaces and Scalar Products
● Screw Systems
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Definition . The projective space underlying a twist or wrench subspace
is called a screw system. An n-system underlies an
n-dimensional subspace.
Definition. Two screw systems are reciprocal when any wrench
acting on a screw in one system exerts no power on
any twist on a screw in the other system.
Screw Systems
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Vector Spaces, Wrenches, TwistsZ-37
any twist on a screw in the other system.
Remark. A screw system can be self-reciprocal.
Definition. Two systems are classified as identical when there is a
rigid body displacement that can make them coincide.
Remark. For classification purposes it is sufficient to consider
only systems underlying subspaces of dimension 2 or 3.
(If higher, study the reciprocal.)
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The Gibson-Hunt Classification. Screw systems are labelled by:
2 or 3 Dimension
I or II Contains or not screws of more than one finite pitch
Screw Systems
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Vector Spaces, Wrenches, TwistsZ-38
I or II Contains or not screws of more than one finite pitch
A, ..., D The number from 0 to 3 of the independent
∞-screws in the system
angle, pitch Additional parameters where needed
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Example. 2-IA(hx, hy) the general two system
Screw Systems
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Vector Spaces, Wrenches, TwistsZ-39
Phillips 1984
The screw axes form a self-intersecting surface, the cylindroid
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Example. 3-IA(hx, hy, hz) the general three-systemPhillips 1984
Screw Systems
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Vector Spaces, Wrenches, TwistsZ-40
Screw axes pass through every point in space.
The same-pitch quadrics are hyperboloids of one sheet.
