Lecture Kinematic models of contact Foundations of Statics

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Lecture Kinematic models of contact Foundations of Statics Powered By Docstoc
					                                  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
   widely adopted.
                                                          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
Link: a rigid body;                                     Salisbury
                                                         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.

two links;                                              Equivalence theorems.
                                                        Line of action.

                             Cylindrical   Revolute     Poinsot’s theorem.

Kinematic                    2 freedoms    1 freedom    Wrenches.


mechanism: a bunch
of links joined by joints;
                             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
with one link fixed.                                        Salisbury
                                                            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.




    Link 1 relative to link two?

    Link 3 relative to link 1?
    Link 3 relative to link 4?
                                                              Lecture 11.
Review: Mobility and connectivity                          Kinematic models
                                                               of contact
                                                            Foundations of
                                                                Statics



                                                           Kinematic models
mobility of a mechanism: DOFs                              of contact
with one link fixed.                                        Salisbury
                                                            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.




    Link 1 relative to link two?

    Link 3 relative to link 1?
    Link 3 relative to link 4?
                                                              Lecture 11.
Review: Mobility and connectivity                          Kinematic models
                                                               of contact
                                                            Foundations of
                                                                Statics



                                                           Kinematic models
mobility of a mechanism: DOFs                              of contact
with one link fixed.                                        Salisbury
                                                            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.




    Link 1 relative to link two?
    One.
    Link 3 relative to link 1?
    Link 3 relative to link 4?
                                                              Lecture 11.
Review: Mobility and connectivity                          Kinematic models
                                                               of contact
                                                            Foundations of
                                                                Statics



                                                           Kinematic models
mobility of a mechanism: DOFs                              of contact
with one link fixed.                                        Salisbury
                                                            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.




    Link 1 relative to link two?
    One.
    Link 3 relative to link 1? Two.
    Link 3 relative to link 4?
                                                              Lecture 11.
Review: Mobility and connectivity                          Kinematic models
                                                               of contact
                                                            Foundations of
                                                                Statics



                                                           Kinematic models
mobility of a mechanism: DOFs                              of contact
with one link fixed.                                        Salisbury
                                                            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.




    Link 1 relative to link two?
    One.
    Link 3 relative to link 1? Two.
    Link 3 relative to link 4? 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
for a spatial linkage, and
                        M=        fi − 3l
for a planar linkage.
                                                                  Lecture 11.
Review: Common sense                                           Kinematic models
                                                                   of contact
                                                                Foundations of
                                                                    Statics
Example: what is the mobility of Watt’s
linkage?
                                                               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
linkage?
                                                               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
linkage?
                                                                 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
linkage?
                                                                 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
linkage?
                                                                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
linkage?
                                                                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
linkage?
                                                                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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