Lecture Kinematic models of contact Foundations of Statics

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```					                                  Lecture 11.
Kinematic models
of contact
Foundations of
Statics

Kinematic models
Lecture 11.             of contact
Salisbury

Kinematic models of contact     Taxonomy of contacts
Mobility and connectivity of
grasp

Foundations of Statics      Foundations of
statics
Preview of statics.
Foundations.
Equivalence theorems.
Matthew T. Mason         Line of action.
Poinsot’s theorem.
Wrenches.

Mechanics of Manipulation
Spring 2010
Lecture 11.
Today’s outline                            Kinematic models
of contact
Foundations of
Statics

Kinematic models of contact                Kinematic models
of contact
Salisbury                               Salisbury
Taxonomy of contacts
Taxonomy of contacts                  Mobility and connectivity of
grasp
Mobility and connectivity of grasp
Foundations of
statics
Preview of statics.
Foundations.
Foundations of statics                     Equivalence theorems.

Preview of statics.                     Line of action.
Poinsot’s theorem.

Foundations.                            Wrenches.

Equivalence theorems.
Line of action.
Poinsot’s theorem.
Wrenches.
Lecture 11.
Kinematic models of contact   Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
A grasp is like a          Salisbury

kinematic                   Taxonomy of contacts
Mobility and connectivity of
grasp
mechanism.
Foundations of
Assume ﬁngers              statics
Preview of statics.

do not lift or slip.       Foundations.
Equivalence theorems.
Line of action.
Model each                 Poinsot’s theorem.
Wrenches.
contact as a
spherical joint.
Apply Grübler’s
formula!
Lecture 11.
Taxonomy of contact types                                            Kinematic models
of contact
Foundations of
Statics

In previous slide,                                                Kinematic models
No contact            Point contact          of contact
contact was          6 freedoms
without friction
5 freedoms
Salisbury
Taxonomy of contacts
modeled as                                                         Mobility and connectivity of
grasp

spherical joint.                                                  Foundations of
Line contact          Point contact
Are there other         without friction       with friction
statics
Preview of statics.
4 freedoms            3 freedoms
possibilities?                                                    Foundations.
Equivalence theorems.
Line of action.
Salisbury’s PhD                                                   Poinsot’s theorem.
Planar contact        Soft finger            Wrenches.
thesis, 1982,           without friction
3 freedoms            2 freedoms
included a
taxonomy.
Line contact          Planar contact
Terminology was         with friction
1 freedom
with friction
0 freedoms
Lecture 11.
Review of mobility and connectivity                    Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
grasp

Foundations of
statics
Next several slides are repeated from Lecture 4.   Preview of statics.
Foundations.
Equivalence theorems.
Line of action.
Poinsot’s theorem.
Wrenches.
Lecture 11.
Review: Constraint and kinematic                        Kinematic models
of contact
mechanisms                                               Foundations of
Statics

Kinematic models
of contact
Taxonomy of contacts
Mobility and connectivity of

Joint: imposes one or                                    grasp

Planar        Spherical    Foundations of
more constraints on          3 freedoms    3 freedoms   statics
the relative motion of                                  Preview of statics.
Foundations.

Line of action.

Cylindrical   Revolute     Poinsot’s theorem.

Kinematic                    2 freedoms    1 freedom    Wrenches.

mechanism: a bunch
Prismatic     Helical
1 freedom     1 freedom
lower pairs joints
involving positive
contact area.
Lecture 11.
Review: Mobility and connectivity                          Kinematic models
of contact
Foundations of
Statics

Kinematic models
mobility of a mechanism: DOFs                              of contact
Taxonomy of contacts
Mobility and connectivity of
connectivity DOFs of one link                               grasp

Foundations of
relative to another.                             L4        statics

What is the mobility of the ﬁve bar    L3                  Preview of statics.
Foundations.
Equivalence theorems.

linkage at right?                                          Line of action.

L5   Poinsot’s theorem.

What is the connectivity of           L2    L1
Wrenches.

Lecture 11.
Review: Mobility and connectivity                          Kinematic models
of contact
Foundations of
Statics

Kinematic models
mobility of a mechanism: DOFs                              of contact
Taxonomy of contacts
Mobility and connectivity of
connectivity DOFs of one link                               grasp

Foundations of
relative to another.                             L4        statics

What is the mobility of the ﬁve bar    L3                  Preview of statics.
Foundations.
Equivalence theorems.

linkage at right? Two.                                     Line of action.

L5   Poinsot’s theorem.

What is the connectivity of           L2    L1
Wrenches.

Lecture 11.
Review: Mobility and connectivity                          Kinematic models
of contact
Foundations of
Statics

Kinematic models
mobility of a mechanism: DOFs                              of contact
Taxonomy of contacts
Mobility and connectivity of
connectivity DOFs of one link                               grasp

Foundations of
relative to another.                             L4        statics

What is the mobility of the ﬁve bar    L3                  Preview of statics.
Foundations.
Equivalence theorems.

linkage at right? Two.                                     Line of action.

L5   Poinsot’s theorem.

What is the connectivity of           L2    L1
Wrenches.

One.
Lecture 11.
Review: Mobility and connectivity                          Kinematic models
of contact
Foundations of
Statics

Kinematic models
mobility of a mechanism: DOFs                              of contact
Taxonomy of contacts
Mobility and connectivity of
connectivity DOFs of one link                               grasp

Foundations of
relative to another.                             L4        statics

What is the mobility of the ﬁve bar    L3                  Preview of statics.
Foundations.
Equivalence theorems.

linkage at right? Two.                                     Line of action.

L5   Poinsot’s theorem.

What is the connectivity of           L2    L1
Wrenches.

One.
Lecture 11.
Review: Mobility and connectivity                          Kinematic models
of contact
Foundations of
Statics

Kinematic models
mobility of a mechanism: DOFs                              of contact
Taxonomy of contacts
Mobility and connectivity of
connectivity DOFs of one link                               grasp

Foundations of
relative to another.                             L4        statics

What is the mobility of the ﬁve bar    L3                  Preview of statics.
Foundations.
Equivalence theorems.

linkage at right? Two.                                     Line of action.

L5   Poinsot’s theorem.

What is the connectivity of           L2    L1
Wrenches.

One.
Lecture 11.
Review: Grübler’s formula                                    Kinematic models
of contact
Foundations of
Given n links joined by g joints,                                 Statics

with ui constraints and fi freedoms at joint i. (Note that
Kinematic models
ui + fi = 6.)                                                of contact
Salisbury

Assume one link is ﬁxed and constraints are all               Taxonomy of contacts
Mobility and connectivity of
grasp
independent.
Foundations of
statics
The mobility M is                                            Preview of statics.
Foundations.
Equivalence theorems.
M = 6(n − 1) −         ui                    Line of action.
Poinsot’s theorem.
Wrenches.
= 6(n − 1) −     (6 − fi )
= 6(n − g − 1) +        fi

Or, for a planar mechanism:

M = 3(n − 1) −        ui
= 3(n − g − 1) +         fi
Lecture 11.
Review: Grübler: special case for loops                      Kinematic models
of contact
Foundations of
The previous formula works (sort of) for all mechanisms.          Statics

For loops there is a variant.
Kinematic models
One loop: n = g, so                                          of contact
Salisbury
Taxonomy of contacts

M=          fi + 6(−1)                    Mobility and connectivity of
grasp

Foundations of
Two loops: make a second loop by adding k links and          statics
Preview of statics.
k + 1 joints:                                                Foundations.
Equivalence theorems.

M=      fi + 6(−2)                         Line of action.
Poinsot’s theorem.
Wrenches.
Every loop increases excess of joints over links by 1. For
l loops:
M=      fi − 6l
M=        fi − 3l
Lecture 11.
Review: Common sense                                           Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                      of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −        ui =             5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +          fi =   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l =                                      Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                constraints is
a very strong
M = 6(n − 1) −         ui =          assumption.
M = 6(n − g − 1) +        fi =
M=       fi − 6l =

Why?
Lecture 11.
Review: Common sense                                           Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                      of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −        ui = 1           5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +          fi =   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l =                                      Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                constraints is
a very strong
M = 6(n − 1) −         ui =          assumption.
M = 6(n − g − 1) +        fi =
M=       fi − 6l =

Why?
Lecture 11.
Review: Common sense                                             Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                        of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −        ui = 1             5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +          fi = 1   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l =                                        Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                  constraints is
a very strong
M = 6(n − 1) −         ui =            assumption.
M = 6(n − g − 1) +        fi =
M=       fi − 6l =

Why?
Lecture 11.
Review: Common sense                                             Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                        of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −         ui = 1            5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +          fi = 1   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l = 1                                      Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                  constraints is
a very strong
M = 6(n − 1) −         ui =            assumption.
M = 6(n − g − 1) +         fi =
M=       fi − 6l =

Why?
Lecture 11.
Review: Common sense                                            Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                       of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −         ui = 1           5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +         fi = 1   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l = 1                                     Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                 constraints is
a very strong
M = 6(n − 1) −         ui = − 2       assumption.
M = 6(n − g − 1) +         fi =
M=       fi − 6l =

Why?
Lecture 11.
Review: Common sense                                            Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                       of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −         ui = 1           5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +         fi = 1   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l = 1                                     Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                 constraints is
a very strong
M = 6(n − 1) −         ui = − 2       assumption.
M = 6(n − g − 1) +         fi = − 2
M=       fi − 6l =

Why?
Lecture 11.
Review: Common sense                                            Kinematic models
of contact
Foundations of
Statics
Example: what is the mobility of Watt’s
Kinematic models
Planar Grübler’s formula:                                       of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
M = 3(n − 1) −         ui = 1           5
grasp

3            Foundations of
statics
M = 3(n − g − 1) +         fi = 1   3                5
Preview of statics.
10       Foundations.

M=       fi − 3l = 1                                     Equivalence theorems.
Line of action.
Poinsot’s theorem.
Independent          Wrenches.

Spatial Grübler’s formula:                 constraints is
a very strong
M = 6(n − 1) −         ui = − 2       assumption.
M = 6(n − g − 1) +         fi = − 2
M=       fi − 6l = − 2

Why?
Lecture 11.
Applying mobility and connectivity to grasping            Kinematic models
of contact
Foundations of
Statics

Salisbury suggests four measures:                         Kinematic models
of contact
M Mobility of the entire system with the ﬁnger joints    Salisbury
Taxonomy of contacts

free.                                                   Mobility and connectivity of
grasp

M Mobility of the entire system, with the ﬁnger joints   Foundations of
statics
locked.                                                Preview of statics.
Foundations.

C Connectivity of the object relative to a ﬁxed palm,   Equivalence theorems.
Line of action.

with the ﬁnger joints free.                           Poinsot’s theorem.
Wrenches.

C Connectivity of the object relative to a ﬁxed palm,
with the ﬁnger joints locked.
If C = 6 then object can make general motions.
If C ≤ 0 then hand can immobilize object.
Lecture 11.
Example: the Salisbury hand                                   Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
What is C?                                                    Salisbury
Taxonomy of contacts
Mobility and connectivity of

What is C ?                                                    grasp

Foundations of
statics
Preview of statics.

This assumes no ﬁnger is in a singular conﬁguration,     Foundations.
Equivalence theorems.
Line of action.
and contacts are not collinear.                          Poinsot’s theorem.
Wrenches.
This neglects stability of the grasp. You need statics
to even start on grasp stability.
Salisbury’s analysis generalizes nicely: to freely
manipulate an object in the hand with point ﬁngers,
the hand mechanism needs at least nine DOFs.
Lecture 11.
Example: the Salisbury hand                                   Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
What is C? 6                                                  Salisbury
Taxonomy of contacts
Mobility and connectivity of

What is C ?                                                    grasp

Foundations of
statics
Preview of statics.

This assumes no ﬁnger is in a singular conﬁguration,     Foundations.
Equivalence theorems.
Line of action.
and contacts are not collinear.                          Poinsot’s theorem.
Wrenches.
This neglects stability of the grasp. You need statics
to even start on grasp stability.
Salisbury’s analysis generalizes nicely: to freely
manipulate an object in the hand with point ﬁngers,
the hand mechanism needs at least nine DOFs.
Lecture 11.
Example: the Salisbury hand                                   Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
What is C? 6                                                  Salisbury
Taxonomy of contacts
Mobility and connectivity of

What is C ? 0                                                  grasp

Foundations of
statics
Preview of statics.

This assumes no ﬁnger is in a singular conﬁguration,     Foundations.
Equivalence theorems.
Line of action.
and contacts are not collinear.                          Poinsot’s theorem.
Wrenches.
This neglects stability of the grasp. You need statics
to even start on grasp stability.
Salisbury’s analysis generalizes nicely: to freely
manipulate an object in the hand with point ﬁngers,
the hand mechanism needs at least nine DOFs.
Lecture 11.
Preview of statics                                         Kinematic models
of contact
Foundations of
Statics

Kinematic models
We will adopt Newton’s hypothesis that particles       of contact
interact through forces.                               Salisbury
Taxonomy of contacts
Mobility and connectivity of
We can then show that rigid bodies interact through     grasp

Foundations of
wrenches.                                              statics
Preview of statics.
Screw theory applies to wrenches.                      Foundations.
Equivalence theorems.

Wrenches and twists are dual.                          Line of action.
Poinsot’s theorem.

We also get:                                           Wrenches.

Line of force;
Screw coordinates applied to statics;
Reciprocal product of twist and wrench;
Zero Moment Point (ZMP), and its generalization.
Lecture 11.
What is force?                                                  Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
Salisbury
You cannot measure force, only its effects:                  Taxonomy of contacts
Mobility and connectivity of
deformation of structures, acceleration.                     grasp

Foundations of
We could start from Newton’s laws, but instead we           statics
hypothesize:                                                Preview of statics.
Foundations.

A force applied to a particle is a vector.              Equivalence theorems.
Line of action.

The motion of a particle is determined by the vector    Poinsot’s theorem.
Wrenches.
sum of all applied forces.
A particle remains at rest only if that vector sum is
zero.
Lecture 11.
Moment of force about a line                                Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
Salisbury

Deﬁnition                                                    Taxonomy of contacts
Mobility and connectivity of
grasp

Let l be line through origin with direction ˆ
l,          Foundations of
statics
Preview of statics.
Let f act at x.                                         Foundations.
Equivalence theorems.

Then the moment of force (or the torque) of f about l   Line of action.
Poinsot’s theorem.

is given by:                                            Wrenches.

nl = ˆ · (x × f)
l
Lecture 11.
Moment of force about a point                             Kinematic models
of contact
Foundations of
Statics
Deﬁnition
Kinematic models
Let l be line through origin with direction ˆ
l,        of contact
Salisbury

Let f act at x.                                        Taxonomy of contacts
Mobility and connectivity of
grasp

Then the moment of force (or the torque) of f about   Foundations of
O is given by:                                        statics
Preview of statics.
Foundations.
Equivalence theorems.
nO = (x − O) × f                   Line of action.
Poinsot’s theorem.
Wrenches.

If the origin is O this reduces to n = x × f.
If n is moment about the origin, and nl is moment
about l, and l passes through the origin,

nl = ˆ · n
l
Lecture 11.
Total force and moment                                    Kinematic models
of contact
Foundations of
Statics

Consider a rigid body, and a system of forces {fi }
acting at {xi } resp.                                 Kinematic models
of contact
Salisbury
Taxonomy of contacts

Deﬁnition                                                  Mobility and connectivity of
grasp

The total force F is the sum of all external forces.      Foundations of
statics
Preview of statics.
Foundations.
F=       fi                      Equivalence theorems.
Line of action.
Poinsot’s theorem.
Wrenches.

Deﬁnition
The total moment N is the sum of all corresponding
moments.
N=      xi × fi
Lecture 11.
Equivalent systems of forces                               Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
We now develop some equivalence theorems,              Salisbury

comparable to (or dual to) our earlier results in       Taxonomy of contacts
Mobility and connectivity of
grasp
kinematics.
Foundations of
statics
Preview of statics.
Deﬁnition                                                  Foundations.
Equivalence theorems.

Two systems of forces are equivalent if they have equal    Line of action.
Poinsot’s theorem.

total force F and total moment N.                          Wrenches.

Equivalent, speciﬁcally, because they would have the
same effect on a rigid body, according to Newton.
Lecture 11.
Resultant                                                    Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
Salisbury
Taxonomy of contacts

Deﬁnition                                                     Mobility and connectivity of
grasp

Foundations of
The resultant of a system of forces is a system              statics
Preview of statics.
comprising a single force, equivalent to the given system.   Foundations.
Equivalence theorems.
Line of action.
Poinsot’s theorem.
A question: does every system of forces have a           Wrenches.

resultant?
Lecture 11.
Line of action                                                        Kinematic models
of contact
Foundations of
Statics

Consider a force f applied at            x1
f
some point x1 .                                           f        Kinematic models
x2
of contact
Total force: F = f                       F
Salisbury
Taxonomy of contacts
Mobility and connectivity of
Total moment: N = x1 × f.           N             line of action    grasp

Foundations of
statics
Consider line parallel to f through x1 , and a second             Preview of statics.
Foundations.
point x2 on the line.                                             Equivalence theorems.
Line of action.

Force f through x2 is equivalent to force f through x1 .          Poinsot’s theorem.
Wrenches.

So point of application is more than you need to
know . . .

Deﬁnition
The line of action of a force is a line through the point of
application, parallel to the force.
Lecture 11.
Bound and free vectors                                      Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
Salisbury
Taxonomy of contacts
When you ﬁrst learned about vectors (in high             Mobility and connectivity of
grasp

school?) you learned they aren’t attached anywhere.     Foundations of
We refer to those as free vectors.                      statics
Preview of statics.
Foundations.
We can also deﬁne bound vectors, speciﬁcally a          Equivalence theorems.
Line of action.
vector bound to a point, called a point vector, and a   Poinsot’s theorem.
Wrenches.
vector bound to a line, called a line vector.
So a force is a line vector.
Lecture 11.
Resultant of two forces                                          Kinematic models
of contact
Foundations of
Statics

Let f1 and f2 act along L1 and                                Kinematic models
of contact
L2 respectively.                                              Salisbury
Taxonomy of contacts
Slide f1 and f2 along their       L1
Mobility and connectivity of
grasp
f1
respective lines of action to                       f1 + f2   Foundations of
statics
the intersection (if any)              f2                     Preview of statics.
L2                           Foundations.
Resultant: the vector sum                                     Equivalence theorems.
Line of action.
f1 + f2 , acting at the                                       Poinsot’s theorem.
Wrenches.
intersection.
So almost every system of forces in the plane has a
resultant. Sort of like how almost every motion is a
rotation. Can it be extended? Does every system of
forces have a resultant?
Lecture 11.
Change of reference                                   Kinematic models
of contact
Foundations of
Statics

Using reference Q or R, a system is described by      Kinematic models
of contact
Salisbury
FQ =      fi      NQ =      (xi − Q) × fi    Taxonomy of contacts
Mobility and connectivity of
grasp

FR =      fi      NR =      (xi − R) × fi   Foundations of
statics
Preview of statics.

From which it follows                                 Foundations.
Equivalence theorems.
Line of action.
Poinsot’s theorem.
FR =FQ                       Wrenches.

NR − NQ =          (Q − R) × fi

which gives

NR =NQ + (Q − R) × F
Lecture 11.
Couple                                                      Kinematic models
of contact
Is a moment like a force? Can you apply a moment?        Foundations of
Statics
Does it have a line of action?
Kinematic models
Deﬁnition                                                   of contact
Salisbury

A couple is a system of forces whose total force F =   fi    Taxonomy of contacts
Mobility and connectivity of

is zero.                                                     grasp

Foundations of
statics
So a couple is a pure moment.                           Preview of statics.
Foundations.
Notice that the moment N of a couple is independent     Equivalence theorems.
Line of action.
of reference point. N is a free vector.                 Poinsot’s theorem.
Wrenches.
Does a couple have a resultant? No! This answers
the previous question: Not every system of forces
has a resultant.

For an arbitrary couple, can
you construct an equivalent
system of just two forces?
Lecture 11.
Equivalence theorems                                        Kinematic models
of contact
Foundations of
Our goal: to deﬁne a wrench, and show that every             Statics

system of forces is equivalent to a wrench.
Kinematic models
Analogous to the program for kinematics, resulting in   of contact
deﬁnition of twist.                                     Salisbury
Taxonomy of contacts
Mobility and connectivity of
grasp

Theorem                                                     Foundations of
statics
For any reference point Q, any system of forces is          Preview of statics.
Foundations.

equivalent to a single force through Q, plus a couple.      Equivalence theorems.
Line of action.
Poinsot’s theorem.
Wrenches.

Proof.
Let F be the total force;
let NQ be the total moment about Q.
Let new system be F at Q, plus a couple with
moment NQ .
Lecture 11.
Two forces are sufﬁcient                                   Kinematic models
of contact
Foundations of
Statics

Theorem                                                    Kinematic models
of contact
Every system of forces is equivalent to a system of just   Salisbury
Taxonomy of contacts
two forces.                                                 Mobility and connectivity of
grasp

Foundations of
statics
Proof.                                                     Preview of statics.
Foundations.
Equivalence theorems.
Given arbitrary F and N, construct equivalent force    Line of action.
Poinsot’s theorem.
and couple, comprising three forces in total.          Wrenches.

Move couple so that one of its forces acts at same
point as F.
Replace those two forces with their resultant.
Lecture 11.
Planar system with nonzero F has a resultant                    Kinematic models
of contact
Foundations of
Statics
Theorem
A system consisting of a single non-zero force plus a          Kinematic models
of contact
couple in the same plane, i.e. a torque vector                 Salisbury

perpendicular to the force, has a resultant.                    Taxonomy of contacts
Mobility and connectivity of
grasp

Foundations of
Proof.                                                          statics
Preview of statics.
Foundations.
Equivalence theorems.
Let F be the force, acting at P.                            Line of action.
Poinsot’s theorem.

Let N be the moment of the         F
Wrenches.

couple.
Construct an equivalent
couple as in the ﬁgure.                   F          N /F
Translate the couple so −F is
applied at P.
Lecture 11.
Poinsot’s theorem                                           Kinematic models
of contact
Foundations of
Theorem (Poinsot)                                                Statics

Every system of forces is equivalent to a single force,
Kinematic models
plus a couple with moment parallel to the force.            of contact
Salisbury
Taxonomy of contacts
Mobility and connectivity of
Proof.                                                       grasp

Foundations of
statics
Let F and N be the given force and moment. We can       Preview of statics.
Foundations.
assume nonzero F, else the theorem is trivially true.   Equivalence theorems.
Line of action.

Decompose the moment: N parallel to F, and N⊥           Poinsot’s theorem.
Wrenches.

perpendicular to F.
Since planar system with nonzero force has a
resultant, replace F and N⊥ by a single force F
parallel to F.
The desired system is F plus a couple with moment
N .
Lecture 11.
Wrench                                                     Kinematic models
of contact
Foundations of
Statics

Kinematic models
Deﬁnition                                                  of contact
Salisbury
Taxonomy of contacts
A wrench is a screw plus a scalar magnitude, giving a       Mobility and connectivity of
grasp
force along the screw axis plus a moment about the         Foundations of
screw axis.                                                statics
Preview of statics.
Foundations.
Equivalence theorems.

The force magnitude is the wrench magnitude, and       Line of action.
Poinsot’s theorem.

the moment is the twist magnitude times the pitch.     Wrenches.

Thus the pitch is the ratio of moment to force.
Poinsot’s theorem is succinctly stated: every system
forces is equivalent to a wrench along some screw.
Lecture 11.
Screw coordinates for wrenches                              Kinematic models
of contact
Foundations of
Statics

Let f be the magnitude of the force acting along a
Kinematic models
line l,                                                 of contact
Salisbury

Let n be the magnitude of the moment about l.            Taxonomy of contacts
Mobility and connectivity of
grasp
The magnitude of the wrench is f .                      Foundations of
statics
Recall deﬁnition in terms of Plücker coordinates:       Preview of statics.
Foundations.
Equivalence theorems.

w = fq                              Line of action.
Poinsot’s theorem.

w0 = f q0 + fpq                      Wrenches.

where (q, q0 ) are the normalized Plücker coordinates
of the wrench axis l, and p is the pitch, which is
deﬁned to be
p = n/f
Lecture 11.
Screw coordinates for wrenches demystiﬁed                   Kinematic models
of contact
Foundations of
Let r be some point on the wrench axis                       Statics

q0 = r × q
Kinematic models
of contact
With some substitutions . . .                           Salisbury
Taxonomy of contacts
Mobility and connectivity of
grasp
w=f
Foundations of
w0 = r × f + n                      statics
Preview of statics.
Foundations.

which can be written:                                   Equivalence theorems.
Line of action.
Poinsot’s theorem.
Wrenches.
w=f
w0 = n0

where n0 is just the moment of force at the origin.
Screw coordinates of a wrench are actually a familiar
representation (f, n0 ).
Wrenches form a vector space. You can scale and
add them, just as with differential twists.
Lecture 11.
Reciprocal product of twist and wrench                          Kinematic models
of contact
Foundations of
Statics

Kinematic models
of contact
Reciprocal product:                                             Salisbury
Taxonomy of contacts
Mobility and connectivity of
(ω, v0 ) ∗ (f, n0 ) = f · v0 + n0 · ω              grasp

Foundations of
statics
The power produced by the wrench (f, n0 ) and differential      Preview of statics.
Foundations.
twist (ω, v0 ).                                                 Equivalence theorems.
Line of action.
Poinsot’s theorem.
A differential twist is reciprocal to a wrench if and only if   Wrenches.

no power would be produced.
Repelling if and only if positive power.
Contrary if and only if negative power.
Lecture 11.
Force versus motion                                           Kinematic models
of contact
Foundations of
Statics

Wrench coordinates and twist coordinates seem to          Kinematic models
of contact
use different conventions:                                Salisbury
Taxonomy of contacts
For twists, rotation is ﬁrst. For wrenches, the        Mobility and connectivity of
grasp
opposite.                                             Foundations of
For twists, pitch is translation over rotation, the   statics
Preview of statics.
opposite.                                             Foundations.
Equivalence theorems.

But these seeming inconsistencies are not a peculiar      Line of action.
Poinsot’s theorem.

convention. They reﬂect deep differences between          Wrenches.

kinematics and statics. For example, consider the
meaning of screw axis—the line—in kinematics and
in statics. In kinematics, it is a rotation axis. In
statics, it is a line of force.
Lecture 11.
Comparing motion and force                                     Kinematic models
of contact
Foundations of
Motion                          Force                               Statics

A zero-pitch twist is a pure    A zero-pitch wrench is a
Kinematic models
rotation.                       pure force.                    of contact
Salisbury

For a pure translation, the     For a pure moment, the          Taxonomy of contacts
Mobility and connectivity of
grasp
direction of the axis is de-    direction of the axis is de-
Foundations of
termined, but the location      termined, but the location     statics
Preview of statics.
is not.                         is not.                        Foundations.
Equivalence theorems.
Line of action.
A differential translation is   A couple is equivalent to      Poinsot’s theorem.
Wrenches.
equivalent to a rotation        a force along a line at
about an axis at inﬁnity.       inﬁnity.

In the plane, any motion        In the plane, any system
can be described as a ro-       of forces reduces to a sin-
tation about some point,        gle force, possibly at inﬁn-
possibly at inﬁnity.            ity.

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