Planar mechanisms four bar linkage A threelink robot A general hinge 1 I want to focus on constraints still holonomic both simple and nonsimple I can do this in the context of three mechanisms ID: 195167
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Slide1
Lecture 6: Constraints II
Planar mechanisms (four bar linkage)
A three-link robot
A general hinge
1
I want to focus on constraints still holonomic — both simple and nonsimple
I can do this in the context of three mechanisms
and I can put some of this into MathematicaSlide2
planar mechanisms fit into our rubric use the
x =0 plane simple holonomic constraints
2
planar mechanismsSlide3
We have a choice of how to fit this into our existing process
We can preserve
q or we can preserve the idea that the long axes are Ks
If we choose the former, then the long axes become J
s the latter adds π/2 to q
3
I’m going to do the former for todaySlide4
We have the planar picture
4Slide5
This tells us what the connectivity constraints are
5
The system shown (known as a
kinematic chain
) has three degrees of freedom
(The three link robot to come is related to this)Slide6
6
The Lagrangian for this simple three link chain is
You can see that this will lead to some fairly complicated EL equations
Move on to a common planar mechanism, the
four bar linkageSlide7
7
The four bar linkage
crank
coupler
follower
ground link
loop closure equationSlide8
8
Several kinds: crank-rocker: crank can make a full rotation double rocker: neither crank nor follower can make a full rotation
drag link: both crank and follower can make full rotations
The picture on the previous slide is a double rocker.Slide9
9
The loop closure equation has two components we can find two variables
The text discusses finding two angles given all four lengths and the crank angle (the
J4 angle is always π)
If we are doing dynamics, we only need to do that once to give us an initial condition
Even kinematics can be converted to differential equations
The picture we have already seen is of a double rocker linkage
for which the crank cannot make a full circle
I’ll build a crank rocker mechanism for which crank can make a full circleSlide10
10Slide11
11
Differentiate the loop closure equation Solve for two of the rates of change of angle
integrate numerically
The
equations to be integrated
Specify
Integrate to get the other two anglesSlide12
12Slide13
13
??Slide14
14
Three link robot
We’ll look at fancier robots later in the course
but this is enough to locate the end of the robot wherever you want it in the robots work space.
This one will be very simple, made up of three identical cylindersSlide15
15Slide16
16
How does this work? What can it do?
The red link can rotate about the vertical —
y1
The blue link is hinged to the red link — f
2 = y1
The green link is hinged to the blue link —
f
3
=
y
1
The
free
angles
are
y
1
,
q
2
and
q
3
— three degrees of freedomSlide17
17
There are simple orientation constraints
The first link is attached to the ground
also a simple constraint
There are also two vector connectivity constraints
(six altogether) which are nonsimpleSlide18
18
I put numbers into this one: m
= 1, l = 1, and a = 1/20 with g
= 1
We get a Lagrangian and we could go on and set up the differential equations
but they are pretty awfulSlide19
19
In this case we’ll have generalized forces (torques)
from the ground to link one
from link one to link two
from link two to link three
The torques react back on the link imposing them, so we’ll haveSlide20
20
The three external torques
The torques on the three links
The torques that do work when the variables changeSlide21
21
The Euler-Lagrange equations
These are pretty messy, and we don’t know yet how to assign the
Q
s.Slide22
22
??Slide23
23
A more general hinge
We just looked at two hinges, and they were simple
because the first link was anchored.
If no link is anchored, then we really need to exercise our understanding of rotation to figure out how the mechanism will work
I will look at a general hinge that keeps I1
=
I
2Slide24
24
I2 will be equal to
I1 if all three Euler angles are equal for the two links.
That’s the trivial solution but it’s where we need to start
We can add a fourth rotation to model the hingeSlide25
25
Rotation of the inertial coordinates looks like
We need to add a fourth rotation
Rotation of the body coordinates is the inverse of thisSlide26
26
There are four rotation variables
There are three connectivity constraints
There are a total of seven degrees of freedom — seven generalized coordinatesSlide27
27
??
OK, let’s look at some of this in Mathematica