Lecture 26. Control of the Diver - PowerPoint Presentation

Slide1

Lecture 26. Control of the Diver

In order for a diver to do what he or she does the diver applies effective torques at the joints

We want to find a recipe for doing this that will cause the simulated diver to execute the diver

This is a control problem

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Tomorrow’s office hours will be1:30 to 3:30

I have to leave at 3:30Slide3

We have a special case of nonlinear control here and I want to explore that, beginning with some discussion

and then some simpler models3Slide4

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Suppose we want the solution of a second order system to follow some prescription

We can write the second order system

The error

can be made to satisfy the homogeneous equation

by the proper choice of

f

(

t

)Slide5

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choose

then

is the desired error equationSlide6

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This can be extended to a quasilinear system, which is what we typically have

converts the nonlinear equation to the linear form that yields a decaying error

given

this choice for

f

(

t

)

This is called

feedback linearizationSlide7

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What are we doing here?

We are asking that

become

and we choose the forcing function to make this happen

This is cool, but it’s only 1D — what happens in a coupled system?Slide8

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For the class of problems we are considering, which includes divers and some robots, we can choose the generalized coordinates such that each external force appears in only one reduced Hamilton equation

This is not necessary, but it cleans up the algebra

and I’ll assume we’ve taken the trouble to do that, so my simplest coupled case isSlide9

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substitute

into the system and solve for

f

1

and

f

2

When these are substituted back into the original equations we obtainSlide10

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and if the system is not singular

then the identical parenthetical equations must equal zero:

the errors vanish asymptoticallySlide11

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This looks like magic, but it isn’t

To make this work we have a necessary condition: as many forces as controlled variables

There’re more, but I can’t give them all to you.

It works for sufficiently simple systems with only revolute joints —

divers and many robots (not the Stanford arm)

I’d like to look at the inverted double pendulum from this perspective

before moving on to more complicated systemsSlide12

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Planar inverted double pendulum: base fixed to the ground

parameters:

l1, l2

m

1

,

m

2

I treat them as slender

membersSlide13

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We know how the double pendulum goes

Six holonomic constraints to restrict it to the

x = 0 plane

Four connectivity constraints relating

CMs

to the ground

Apply all ten constraints and define a two dimensional

qSlide14

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We can form a LagrangianSlide15

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There are no additional constraints, so we can let qdot equal u

We can find p

and then the pieces of Hamilton’s equationsSlide16

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We have four differential equations and seven algebraic equations

(We can actually look at these in the Mathematica notebook)

The next issue is designing the control

so by assigning

we assign Slide18

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We can look at various desired angles, both steady and time-dependent

Here’s the algorithm for vertical stabilizationSlide19

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and the resultSlide20

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Here we ask both links to follow sinusoidal paths out of phaseSlide21

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and here’s the result of thatSlide22

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Here are the torques required to perform the tracking shown on the previous slideSlide23

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How about a simple diving problem — a three link diver?Slide24

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Three links —> 18 variables

Confine to the x constant plane —> nine variables

Four connectivity constraints —> five variables

Number the links from the bottom to the top

and select the middle link as the reference linkSlide25

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There are two torques — one at each joint — as shown in the sketch and we want to choose our variables accordinglySlide26

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We get the Lagrangian as usualSlide27

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This is a fancy torque, suitable for 3D

but of course only two of the components actually do anything

The important thing is that the choice of

q

has isolated the two torquesSlide28

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Note that we have written this in terms of the desired

qs, not the yi

The next slide shows the development of the

Z

s and their gradientsSlide29

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We need the symbolic versions to build the equations we intend to integrateSlide31

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The two internal angles start at zero, go to a maximum of 2π/3, and back to zero in the time interval tf

Midway through the ends of the two outboard links should touch

We can do this integration and see what the results are

We can start by looking at the closest approach, when the two links are to touchSlide33

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It is now probably worth looking at the Mathematica notebooks