Contact Dynamics Modelling for Robots
Contact Dynamics Modelling for Robots Contact Dynamics Modelling for Robots
Introduction Impulse-momentum contact Point contact Friction forces Contact in robot dynamics modelling Conclusions Contact Dynamics Modelling for Robots Mike Boos March 30, 2009 Mike Boos Contact Dynamics Modelling for Robots
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Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong><br />
Mike Boos<br />
March 30, 2009<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Outline<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Outline<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
What is contact dynamics?<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
Ff<br />
N<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Why model contact in robotics?<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
Haptics<br />
Interactions with the<br />
environment<br />
Intentional (gripping,<br />
part placement, etc)<br />
Unintentional<br />
(collisions)<br />
Control and stability<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Examples of contact models<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
f n<br />
kv<br />
Impulse<br />
Momentum<br />
Point<br />
<strong>Contact</strong><br />
Volumetric<br />
<strong>Contact</strong><br />
Finite Element<br />
Methods<br />
Hardwarein-the-loop<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Outline<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Impulse-momentum<br />
Assumptions<br />
vB<br />
vA<br />
x<br />
y - plane of<br />
impact<br />
<strong>Contact</strong> <strong>for</strong>ces >> other<br />
<strong>for</strong>ces<br />
De<strong>for</strong>mation and<br />
restitution is<br />
near-instantaneous<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Impulse-momentum<br />
vB<br />
y - plane of<br />
impact<br />
Conservation of Momentum<br />
m A (v Ay ) 1 = m A (v Ay ) 2<br />
m B (v By ) 1 = m B (v By ) 2<br />
m i (v ix ) 1 = ∑ m i (v ix ) 2<br />
i<br />
∑<br />
i<br />
vA<br />
x<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Coefficient of restitution<br />
vB<br />
y - plane of<br />
impact<br />
Need a second equation to<br />
solve <strong>for</strong> x-velocities<br />
Energy is lost in impact<br />
vA<br />
x<br />
e = (v Ax −v Bx ) 2<br />
(v Bx −v Ax ) 1<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Coefficient of restitution<br />
For low impact velocities and most materials with a linear elastic<br />
range (e.g. metals):<br />
e ≈ 1 − α|(v Bx − v Ax ) 1 |<br />
= 1 − αv 1<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Limitations<br />
Not useful <strong>for</strong> prolonged contact<br />
Cannot solve <strong>for</strong> contact or reaction <strong>for</strong>ces<br />
May not be very useful in robotic or other multibody systems<br />
While simple impulse-momentum models may not be useful, the<br />
concept of a coefficient of restitution may still be important in<br />
other models.<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Outline<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Hunt-Crossley contact<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Point contact<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Hunt-Crossley contact<br />
Common model:<br />
spring-damper system<br />
N = K(δ) + B( ˙δ)<br />
Often, the relationship between<br />
penetration and normal <strong>for</strong>ce is<br />
non-linear (even with<br />
linear-elastic solids)<br />
δ<br />
N<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Hertz theory<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Hunt-Crossley contact<br />
Assumptions:<br />
Non-con<strong>for</strong>ming surfaces<br />
<strong>Contact</strong> patch is small relative to the geometries of the bodies<br />
Bodies are linear-elastic solids<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Hertz theory<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Hunt-Crossley contact<br />
N = kδ 3/2<br />
Two spheres:<br />
k =<br />
4<br />
3π(h m,1 +h m,2 ) ( R 1R 2<br />
R 1 +R 2<br />
) 1/2<br />
Sphere on plane:<br />
k =<br />
4<br />
3π(h m,1 +h m,2 ) R1/2<br />
Ri<br />
δ<br />
N<br />
δ<br />
h m,i = 1−ν2 i<br />
πE i<br />
Rj<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Hunt-Crossley contact<br />
Hunt-Crossley contact<br />
<strong>Contact</strong> <strong>for</strong>ces modelled with a spring in parallel with a non-linear<br />
damper:<br />
N = K(δ) + B( ˙δ, δ)<br />
= kδ 3/2 + (λδ 2/3 ) ˙δ<br />
Consistent with Hertzian contact in the static case (i.e. ˙δ = 0)<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Damping coefficient<br />
Hunt-Crossley contact<br />
If we reintroduce the concept of a coefficient of restitution, where<br />
e ≈ 1 − αv 1 , it can be shown that:<br />
λ ≈ 3 2 αk<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Hunt-Crossley contact<br />
Hunt-Crossley contact<br />
N = kδ 3/2 + (λδ 2/3 ) ˙δ<br />
where<br />
k =<br />
4<br />
3π(h m,1 + h m,2 ) ( R 1R 2<br />
R 1 + R 2<br />
) 1/2<br />
λ ≈ 3 2 αk<br />
Ri<br />
δ<br />
Rj<br />
e ≈ 1 − αv 1<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Outline<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Coulomb friction<br />
Static<br />
F f ≤ −µ S Nû<br />
where û is in the direction of<br />
other <strong>for</strong>ces in the plane of<br />
contact.<br />
c<br />
vcn<br />
Dynamic<br />
F f = −µ k Nˆv ct<br />
Ff<br />
N<br />
vct<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Coulomb friction approximation<br />
μ<br />
μs<br />
μc<br />
vct<br />
vs<br />
vd<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Outline<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
2-link planar manipulator<br />
Example simulation<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
2-link planar manipulator<br />
2-link planar manipulator<br />
Example simulation<br />
y<br />
q2<br />
Dynamic equations<br />
τ 1 = D 1 (q)¨q+C 1 (q, ˙q) ˙q+φ 1 (q)<br />
τ 2 = D 2 (q)¨q+C 2 (q, ˙q) ˙q+φ 2 (q)<br />
q1<br />
x<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
2-link planar manipulator<br />
2-link planar manipulator<br />
Example simulation<br />
What happens to the dynamic equations with contact?<br />
y<br />
x<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Including contact <strong>for</strong>ces<br />
2-link planar manipulator<br />
Example simulation<br />
F<br />
y<br />
<strong>Contact</strong> <strong>for</strong>ces<br />
F = N + F f<br />
Cannot include contact <strong>for</strong>ces<br />
in Lagrangian, since they are<br />
non-conservative (damping and<br />
friction).<br />
x<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Modified dynamic equations<br />
2-link planar manipulator<br />
Example simulation<br />
The torques caused by the contact <strong>for</strong>ces must be included with<br />
the driver torques on the left-hand side of the dynamic equations:<br />
Dynamic equations<br />
τ 1 + T c1 = D 1 (q)¨q + C 1 (q, ˙q) ˙q + φ 1 (q)<br />
τ 2 + T c2 = D 2 (q)¨q + C 2 (q, ˙q) ˙q + φ 2 (q)<br />
What are T c1 and T c2 ?<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Link 2<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
2-link planar manipulator<br />
Example simulation<br />
r2<br />
F<br />
Consider link 2 and the contact<br />
<strong>for</strong>ces independently from the<br />
original system and its <strong>for</strong>ces.<br />
There is an additional reaction<br />
at P from link 1.<br />
-F<br />
P<br />
T c2 = r 2 × F<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Link 1<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
2-link planar manipulator<br />
Example simulation<br />
-F<br />
O<br />
r1<br />
F<br />
Tc2 = r2 x F<br />
Now consider link 1.<br />
We have the reaction at P<br />
from link 2, as well as the<br />
torque from link 2.<br />
T c1 = r 1 × F + T c2<br />
= (r 1 + r 2 ) × F<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Putting it all together<br />
2-link planar manipulator<br />
Example simulation<br />
F<br />
y<br />
r1<br />
r2<br />
Dynamic equations<br />
τ + T c = D(q)¨q + C(q, ˙q) ˙q + φ(q)<br />
where[ ]<br />
((r<br />
T c = 1 + r 2 ) × F ) · ˆk<br />
(r 2 × F ) · ˆk<br />
x<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Example simulation<br />
2-link planar manipulator<br />
Example simulation<br />
Setup<br />
y<br />
45˚<br />
1 m<br />
x<br />
Slender links of mass 1 kg,<br />
length 1 m<br />
Initially at rest<br />
End effector radius: 5 cm<br />
<strong>Contact</strong> properties of end<br />
effector and wall similar to<br />
that of steel <strong>for</strong><br />
Hunt-Crossley<br />
τ 1 = −10Nm, τ 2 = −2Nm<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Joint angles<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
End effector position<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Summary<br />
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
1 Introduction<br />
What is contact dynamics?<br />
Why contact modelling in robotics?<br />
Types of contact models<br />
2 Impulse-momentum contact<br />
3 Point contact<br />
Hunt-Crossley contact<br />
4 Friction <strong>for</strong>ces<br />
5 <strong>Contact</strong> in robot dynamics modelling<br />
2-link planar manipulator<br />
Example simulation<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>
Introduction<br />
Impulse-momentum contact<br />
Point contact<br />
Friction <strong>for</strong>ces<br />
<strong>Contact</strong> in robot dynamics modelling<br />
Conclusions<br />
Questions?<br />
Mike Boos<br />
<strong>Contact</strong> <strong>Dynamics</strong> <strong>Modelling</strong> <strong>for</strong> <strong>Robots</strong>