NSF
No slip
vs
Sliding with infinite friction
Special for Banff 2014
Comments on
Pulling by Pushing,
Slip With Infinite Friction,
and Perfectly Rough Surfaces
Kevin Lynch, Matt Mason
Int J of Robotics Research, 1995
Matt Kelly,
Andy Ruina,
Gregg Stiesberg,
Tuesday, February 18, 2014
and
Wobbling,
toppling,
and forces of contact
Tad McGeer
Am. J. of Physics, 1989
Mechanical Eng , Cornell U
‘‘
‘‘
‘’
Physics,
‘’
Motivation/inspiration:
1) Mechanics (apparent) paradoxes are interesting
a) friction and b) dynamics
hapter 0.
2) Make a good robot
good simulation
c) deal with these things, or
d) do something worse
Conclusions:
Chapter 0.
3) Slip with
is possible
e) In natural situations
f ) Is a sin
useful model
ter 0.
4) No-slip BCs violates
g) In natural problems
(applied
torque)
h) So
slipsin
is better
Tuesday, February 18, 2014
sin
(PD control)
(or something, sometimes)
Value system (in this talk):
0.
1) Don’t violate
,
symmetries of space,
sinetc.
2) Approximate constitutive laws are OK
(applied
torque)
(like
etc).
(PD control)
sin should
3) The model
have
a
high-precision
meaning.
(Side force)
(horizontal acceleration
of base)
4) No concern for computation
speed.
(applied torque)
5) 2D for simpler pictures etc.
(PD control)
Tuesday, February 18, 2014
sin
time of contact, etc. When contacting bodies are not sliding the meanin
Coulomb/DaVinci/Amonton
friction
equation:
the friction equation 3.4 changes. Friction
still
resists
slip, itare
is theinprese
When two objects
co
of the friction force that prevents slip. But when there is no slip eqn. (
we
call
the
force
which
resi
A
doesn’t describe the force, but rather an upper limit on the possible siz
allymust
assumed
to be
either
the force. That is, the friction force
be less than
or equal
to ‘l
F
magnitude
during
non-sliding
contact.
A
contact they are not in real c
N
N
them. Most of the metal to (
All of the discussion above can be contact
summarized
with
the
following
equat
of
the
car
tires
with
B
B
for the friction force
if
on worn-smooth tires on a
contact are very small com
Partial Free
if
Relative
slip
velocity.
Body
Diagrams
The
friction
force
quick
way
to
estimate
these
⇥⇥
BBN
Figure 3.58: Two bodies in contact. The
tion of lubricated friction f
B
forces between them satisfy the law of Afluid
slip
mechanics. during
For many
on A from B
action and re-action. It is often conve- A Bsin
if
We
now
drop
discussion
of
during stationary contact
nient to decompose theon
force
of interacA from
B
tion into a part tangent to⇥ the surface BM negligible and because esti
B
⇥
sin
if
Dry
friction
of interaction and
a part
perpendicuAn upper
bound
on the forces are
The
magnitude
of the
lar to the surface of interaction.
in mechanics
problems inv
friction
force
friction force
friction forces is called Co
F
Filename:tfigure11-contact
Tuesday, February 18, 2014
Friction
If you don’t like divide When
by zero
etc:
two objects are in contact andFr
on
sin
A
A
F
N
B
N
F
B
Partial Free
Body Diagrams
Filename:tfigure11-contact
if
if
Filename:tfigure2-friction-angle
Figure 3.58: Two bodies in contact. The
tan
forces between them satisfy the law of
action and re-action. It is often convenient to decompose the force of interaction into a part tangent to the surface
of interaction and a part perpendicular to the surface of interaction.
sin
sin
Tuesday, February 18, 2014
we call the force which
if resists this slidi
Som
ally assumed
to be
either ‘lubricated’rath
o
Partial
FBDs
if
contact they are not in real contact at the
all
them. Most of the metal to metal contac
and
contact
is
sin of the car tires
if with the roadbet
onµNworn-smooth tires on a very wet ro
sin
if
N
contact are very small
if compared forces
φ small lubric
quick way to estimate these
Th
ifof characterizing
Figureof3.61:
Two waysfriction
tion
lubricated
forces requir
ma
friction:
the
friction
coefficient
and
fluid mechanics. For many purposescom
lu
friction angle .
We now drop discussion of lubricated
the
negligible(PD
andcontrol)
because estimating them
if are not small an
Dry friction forces
(Side force)
in mechanics problems involving slidi
(horizontal
acceleration
of base)
if Coulomb’s
friction
forces is
called
law
But, use of even this law is full of subtl
tan
sin
if
sin
if
sin
Constitutive
0. relation for friction
Coulomb friction
A surface in the space of
force angle
sliding velocity
sin
force magnitude
hapter 0.
sin
if
if
if
if
sin
Single
constitutive
tan
parameter:
sin
the friction angle (PD control)
(Side force)
(Nothing special about
if
(horizontal acceleration
of base)
)
if
Tuesday, February 18, 2014
1
Chapter 0.
Recall
Conclusions:
Chapter 0.
3) Slip with
is possible
e) In natural situations: 3 examples & demo
f ) Is a sin
useful model
apter 0.
4) No-slip BCs violates
(sometimes)
g) In natural problems
(applied
torque)
sin
h) So
slip is better
(PD control)
sin (Side force)
(applied torque)
(horizontal acceleration of base)
(PD control)
(applied torque)
Tuesday, February 18, 2014
Dragging stick (similar to Lynch/Mason 1995)
Tuesday, February 18, 2014
Constitutive
relation for friction
Coulomb friction
A surface in the space of
sliding velocity v
Chapter 0.
sliding force
F
normal force N
Chapter 0.
Single constitutive parameter: the friction angle
Chapter 0.
Chapter 0.
Nothing special about
quarter planes
(
and one half plane (
Tuesday, February 18, 2014
. Surface becomes two
)
).
sin
Dragging stick (cont’d)
Drag Force
Force per weight
2
1.5
θ = 15 deg
θ = 45 deg
1
0.5
0 −4
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
Coefficient of Friction
F= 0.5 W
Normal Force
Force per weight
0.5
θ = 15 deg
θ = 45 deg
0.4
N=0
0.3
0.2
Chapter 0.
0.1
0 −4
10
−3
10
−2
10
−1
10
0
10
1
10
Coefficient of Friction
Tuesday, February 18, 2014
2
10
3
10
4
10
Nothing special happens
when
.
Chapter 0.
Dragging stick (cont’d)
ple Problem - The Falling Pencil
act Models - Two Old, One New
Summary
Infinite non-slip friction modelA Simple Problem - The Falling Pencil
Coulomb friction model
Contact Models - Two Old, One New
Infinite sliding friction model
Summary
Infinite non-slip friction modelA Simple Problem - The Falling Pencil
Contact Models - Two Old, One New
Coulomb friction model
Summary
Infinite sliding friction model
Infinite non-slip friction A
model
Simple Problem - The Falling Pencil
Coulomb friction model Contact Models - Two Old, One New
Infinite sliding friction model
Summary
A micro-mechanism
that
implements
.
A thought
experiment
A thought experiment
A thought experiment
Infinite non-slip friction model
Coulomb friction model
Infinite sliding friction model
xperiment
pencil
The tip slides up the groove wall
mal force vanishes
G. Stiesberg
pencil
pencil
Simulating Intermittent Contact
G. Stiesberg
Frictionless
vertical
gear teeth
sin
then slides over the top of the groovethen
wall free-falls and collides with the next groov
Simulating Intermittent Contact
Chapter 0.
pencil
G. Stiesberg
Simulating Intermittent Contact
G. Stiesberg
Simulating Intermittent Conta
No vertical force,
some horizontal force,
if
(Demo)if
Tuesday, February 18, 2014
. To find these we could use the friction equation for the sliding
bearing contact
hole in the wheel. Here is shown part of a cart rolling to the right
Infinite
friction
example
2: A wheel
with
a
wheel
rotating
steadily
clockwise.
force balance
sin
cos
cos
or dot prod
with forces
wheel. We neglect the wheels weight because it is ge
smaller than the forces it mediates. To make the situa
picture shows too-large a bearing hole .
ˆ
4.6 Undriven wheels and
tw
sin
ı̂
which could be reduced to 2 scalar equations by takingThe
components
friction angle
or dot products; and moment balance about C, whichwheel
we calculate
(with tan
with forces and perpendicular distances as
describes the friction between the axle and
Of keyfricinter
). The angle describes the effective
C
tion of the wheel. This is not the friction angle forr sliding
between
mathematic
N0.be larger (if not, the wheel
F Chapter
the wheel and ground which is assumed
to
aboveresisfor a
R
would
skid
and
not
roll),
probably
much
larger.
The
specific
θ
Of key interest is finding the force resisting motion
. With some
We follow
tance
or the coefficient of rolling resistance or the specific
cost of a
mathematical manipulation we could solve the 4 scalar
equations
above for any of
and in terms of
, and 0.
Chapter
transport
is. eff
tan . (If there was no wheel, and the cart
or
As
mod
G
We follow a more intuitive approach instead.
F
whatever was just dragged, the specificx resistance would be the fricFilename:tfigure-primitivewheel
As modeled, the wheel is a two-force body so the free body diagram show
tion
between
the
cart
and
ground
.)
F
eff
y
Toagram
figure
forces
involved
wetwodraw
shows out
equal the
and opposite
collinear
forces at the
contacta free body diagram of the
Although we can solve for in terms of or let’s
first conpoints.
points.
wheel. We neglect the wheels weight
because
it is one
generally
much
sider two
extreme cases:
is a frictionless
bearing and the other is
smaller than the forces it mediates.
To make
the situation
clear the and
a bearing
with infinite
friction coefficient
.
Filename:tfigure-primitivewheelFBD
picture shows too-large a bearing hole µ. = 0
ˆ
d
θ
φr
α
ı̂
F
Tuesday, February 18, 2014
r
αR
R
C
Filename:tfigure-primitivewheelFBD2
µ=∞
α
N
r
α
Filename:tfigure-primitivelimits
sin
A wheel
is not
just a
lever.
sin
sin
In
the case that the wheel bearing
(continued...)
sin has no friction we satisfyingly see
Rclearly that there is no ground resistance
tan to motion. The case of in-
if
if
if
if
ms
4.3 Friction and equilib
Infinite friction example 3:
A pulley
M
For allChapter
force
0. at D is vertical
Chapter 0.
g
D
R
r
For
Chapter 0.
r C
fixed ax
le
R
massles
P
G
C
Pulley
g
M
Filename:pfigure-s01-p2-3
Problem 4.3.20
sin
θ
Filename:pfigure2-blue-47-3-a
Problem 4.3.21
ˆ
ı̂
F
A small axle
makes4.3.22
an efficient
pulley
This problem
is similar to
lem 4.3.21.large
A reel of mass
and
for arbitrarily
4.3.21 A reel of mass
and outer radius
is connected by a horizontal string from
sin
point across a pulley to a hanging object
Tuesday, February 18, 2014
radius
is connected by an inext
string from point
across a if
pulle
hanging object of mass . The inne
if
der of the reel has radius
slope has angle . There is no slip b
the reel and the slope. There is grav
terms of , , , and , find:
For a dragging stick and for journal bearings (wheel
and pulley), infinite friction slip seems a reasonable
worst case (biggest friction) model.
Tuesday, February 18, 2014
Chapter 0.
Recall
Conclusions:
Chapter 0.
3) Slip with
is possible
e) In natural situations: 3 examples & demo
f ) Is asin
useful model
pter 0.
4) No-slip BCs violates
g) In natural problems
(applied
torque)
sin
h) So
(sometimes)
slip is better
(PD control)
sin (Side force)
(applied torque)
(horizontal acceleration of base)
(PD control)
(applied torque)
Tuesday, February 18, 2014
Simulation of a falling pencil (McGeer 1989)
Start nearly vertical at rest
Tip is in non-slip infinite
frictional contact with surface,
equivalent to a pin joint
constraint
θ
l
y
x
(Usherwood video)
G. Stiesberg
Tuesday, February 18, 2014
Simulating Intermittent Contact
Constrained dynamics
Simulation of a falling pencil (cont’d)
Integrate the constrained EOM until the normal force
vanishes
1
ft
fn
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0
0.2
0.4
0.6
0.8
1
θ
When the normal force vanishes, switch to free-body EOM
Release constraint when normal force gets to zero.
G. Stiesberg
Tuesday, February 18, 2014
Simulating Intermittent Contact
Simulation of a falling pencil (cont’d)
Released constraint when normal force went to zero.
Surprise!
Tip of pencil immediately accelerates through the floor.
Why?
Tuesday, February 18, 2014
Summary
Why did simulation
Simulation
of a fail?
falling pencil (cont’d)
A graphical proof that the tip must accelerate through the
floor:
JUST BEFORE LIFTOFF
JUST AFTER LIFTOFF
A Simple Problem - The Falling Pencil
Contact Models - Two Old, One New
Summary
Why did simulation fail?
mg
ft
mg
ft
ft
Relaxing the constraint
is equivalent
to adding
G. Stiesberg
Simulating Intermittent
Contacta tangential
force at the tip that causes it to accelerate through the
surface
Conclusion: infinite non-slip friction is not a physically
consistent contact model
Tuesday, February 18, 2014
Simulation of a falling pencil (cont’d)
A Simple Problem - The Falling Pencil
Contact Models - Two Old, One New
Summary
Infinite non-slip friction model
Coulomb friction model
Infinite sliding friction model
Three choices:
Infinite sliding friction model - Advantages
1) Allow slip (with infinite friction or whatever).
2) Allow
No need to check the
theratio
ground
to upon
suck.
of impulses
impact, collisions are
plastic
f
fn
3) Allow interpenetration.
Normal and tangential
ft
contact forces are
continuous
0
t
Zero parameters needed
G. Stiesberg
Tuesday, February 18, 2014
Simulating Intermittent Contact
The root of all evil:
Contact mass matrix relates
Chapter 0.
force and acceleration at contact point
ACC
FORCE
FORCE
ACC
Tuesday, February 18, 2014
(mass matrix video)
Constitutive
relation for friction
Coulomb friction
A surface in the space of
sliding velocity v (a) Chapter 0.
sliding force
F
normal force N
Chapter 0.
Single constitutive parameter: the friction angle
Chapter 0.
Chapter 0.
Nothing special about
quarter planes
(
and one half plane (
Tuesday, February 18, 2014
. Surface becomes two
)
).
sin
Motivation/inspiration:
1) Mechanics (apparent) paradoxes are interesting
a) friction and b) dynamics
hapter 0.
2) Make a good robot
good simulation
c) deal with these things, or
d) do something worse
Conclusions:
Chapter 0.
3) Slip with
is possible
e) In natural situations (e.g., our robot sims)
f ) Is a sin
useful model
ter 0.
4) No-slip BCs violates
g) In natural problems
(applied
torque)
h) So
slipsin
is better
Tuesday, February 18, 2014
sin
(PD control)
(or something else)
Tuesday, February 18, 2014
Tuesday, February 18, 2014
Tuesday, February 18, 2014
Tuesday, February 18, 2014
Tuesday, February 18, 2014
Tuesday, February 18, 2014