Lecture 6: Constraints II - PowerPoint Presentation

Slide1

Lecture 6: Constraints II

Planar mechanisms (four bar linkage)

A three-link robot

A general hinge

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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

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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

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I’m going to do the former for todaySlide4

We have the planar picture

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This tells us what the connectivity constraints are

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The system shown (known as a

kinematic chain

) has three degrees of freedom

(The three link robot to come is related to this)Slide6

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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

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The four bar linkage

crank

coupler

follower

ground link

loop closure equationSlide8

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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

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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

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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

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??Slide14

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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

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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

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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

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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

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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

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The three external torques

The torques on the three links

The torques that do work when the variables changeSlide21

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The Euler-Lagrange equations

These are pretty messy, and we don’t know yet how to assign the

Q

s.Slide22

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??Slide23

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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

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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

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Rotation of the inertial coordinates looks like

We need to add a fourth rotation

Rotation of the body coordinates is the inverse of thisSlide26

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There are four rotation variables

There are three connectivity constraints

There are a total of seven degrees of freedom — seven generalized coordinatesSlide27

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??

OK, let’s look at some of this in Mathematica