Download MM5 - CONTROL

Content MM5


Short repetition of mm4
Motions of links
– Jacobians short
– Acceleration of riged bobyes
• Linear
• Angular
F = mv·c
· + ω x Ic ω
N = Ic ω
– Newtons equations
– Eulers equations
– Iterative Newton-Euler dynamic formulation
·
• Outward iteration to get (vc, ω, ·vc, ω)
• Inward iteration to get N
MMS I, Lecture 5
1
Denavit-Hartenberg Frame Attachment
Frame attachment
1.
Identify joint axis
2.
Identify common perpendicular
3.
Assign zi pointing along i-th joint
axis
4.
Assign xi pointing along common
perpendicular
5.
Assign yi to complete frame
6.
Assign frame {0} (base) to match
{1}
MMS I, Lecture 5
Link parameters
ai=dist(zi, zi+1) along xi
ai=ang(zi, zi+1) about xi
di=dist(xi-1, xi ) along zi
θi=ang(xi-1, xi) along zi
2
Denavit-Hartenberg Link Parameters
axis i-1
ai=dist(zi, zi+1) along xi
ai=ang(zi, zi+1) about xi
axis i
link i-1
di=dist(xi-1, xi ) along zi
θi=ang(xi-1, xi) along zi
link i-2
yi
zi-1
zi
yi-1
xi-1
ai-1
MMS I, Lecture 5
xi
ai-1
di
i
3
Example
3
x

L
L1

y0
y2
y1
L2
3
y

x2
3
x1
x0
MMS I, Lecture 5
4
Example on the blackboard
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Get recursive angular velocity iωi+1 and
and
Linear velocity ivi+1 and i+1vi+1
i+1ω
i+1
for i = 0, 1, 2
MMS I, Lecture 5
5
Jacobian for examble
x = l1c1 + l2c12
y = l1c1 + l2s12
·
·
·
x· = -l1s1θ1 - l2s12 θ1 - l2s12 θ2
·
·
y· = l1c1 θ1 + l2c12 θ1 + l2c12 θ· 2
x· =
y· =
·
-l1s1 - l2s12 - l2s12 θ1
·
l1c1 + l2c12 + l2c12 θ2
·
x· = J(θ ) θ
or
-1
·
θ = J(θ ) x·
MMS I, Lecture 5
6
Iterative Newton-Euler dynamics - 1
1.
2.
3.
Compute angular and linear velocities and
accelerations outward from {0}-{N} by iteration
Compute forces and torques acting on each link
Compute forces and torques from {N}-{0} by iteration
MMS I, Lecture 5
7
1) Angular and linear velocities and accelerations
Outwards propagation:
1. Angular velocity and acceleration:
2.
Linear acceleration of frames
3.
Linear acceleration of link CoM
MMS I, Lecture 5
8
2) Force and torque on each link
MMS I, Lecture 5
9
3a) Forces and torques
i+1
i
N
f
i+
1
i
n i+1
i
f
i
F
ni
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Fi: force on link i by link i-1
Ni: torque on link i by i-1
MMS I, Lecture 5
10
3b) Forces and torques iteration
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Force and torque equilibrium

Iteration:
MMS I, Lecture 5
11
Combined Newton-Euler Dynamics - 1
1) Outwards iterations, i: 0-5
MMS I, Lecture 5
12
Combined Newton-Euler Dynamics -2
2) Inward iterations: i: 6-1
MMS I, Lecture 5
13
SCARA robot
TCP
TCP
TCP
MMS I, Lecture 5
14