Transcription

Hello and welcome to this unit on robotics fundamentals.

Before we begin,

I wanted to give you an idea of what the basic theme of this module will be about.

Our goal here is to provide you with the mathematical tools that we're gonna need

to talk about where things are in space.

Now, this sounds like a relatively simple task and it is.

But there are few subtleties that we're gonna need to get right.

In order to get our thinking clear and so

that we'll also gonna be able to write code to guide various robots.

To motivate this problem imagine that you're trying to program a humanoid robot

like this one.

To pick up a cup.

You may start by developing a system to locate the cup

with respect to the camera system, which is embedded in the head of the robot.

Now, you may also be interested in figuring out where that cup is

with respect to the gripper that you're going to be using to

actually manipulate the object.

Both of these perspectives are important, and we need to be able to relate them in

our thinking and in the code that we will ultimately write.

Now, one of the basic tools that we're going to using to talk about where

things are goes all the way back to Descartes.

He proposed this idea of a Cartesian coordinate frame of reference.

Here, we're depicting this in the plane.

The basic idea is pretty simple.

We start with a partition plane of reference depicted here by these two axis,

x and y at an implicit origin right here.

Once we've established this coordinate frame of reference, we can talk about

the location of points in the plane, by associating them with tuples.

Denoting their coordinates with respect to this frame of reference.

Note that we could also use this kind of representation to talk about vectors,

corresponding to directions in the plane, or displacements.

Here for instance we can think of a displacement vector.

Which is represented by this tuple.

Indicating a displacement of one unit along the x axis.

And three units along the y axis.

Let's call these tuples of numbers stacked up vertically vectors,

here we're showing an example of vector v associated with this tuple here, 2, 1.

We will say that v is an element of R2 to denote the fact that it's

composed of two real numbers as opposed to say two complex numbers.

We'll have more to say about vectors in this form in subsequent modules.

Given this representation it is natural to imagine simple algebraic operations

like multiplication by a scalar or addition.

It the first example we can imagine a vector.

1, 2, being multiplied by a scalar, in this case five.

The operation proceeds by simply multiplying each

of the elements in that vector in turn by the scale of five.

In this case the result would be a new tubal 5, 10.

Where the first entry is simply 5 times 1 and the second is 5 times 2.

Another operation that we can imagine is addition.

Here we have two vectors, 3, 7 and minus 5, 2 and

the sum of those two vectors corresponds to this third result.

-2, 9, where here again, the first element, -2,

is the sum of 3 and -5, and the second element is the sum of 7 and 2.

Basically, we're simply adding corresponding elements

in the vector together.

Now, we can actually place a kind of geometric

interpretation on these algebraic operations.

Here for instance,

if we think of each of these tuples as corresponding to a vector in space,

scalar multiplication corresponds to scaling of vector by that amount.

So here, we can imagine starting with initial vector v and

scaling it by a factor a which corresponds to scalar multiplication,

I'm getting a new vector a times v.

Similarly, we can imagine two vectors v and

w represented by tuples in the plane consisting of two numbers.

And think of their algebraic sum as corresponding to

a new vector which represents the result of adding these two vectors head to tail.

Note, that there is nothing special About the number 2 here other than the fact that

elements of R2 correspond very naturally to points in the Cartesian plane.

We could equally easily talk about points in R3, like so.

Here we're talking about tuples composed of three real numbers.

Equivalently we could be talking about tuples consisting

of six numbers, in this case this will depicted by R6.

R again corresponding to real numbers and

the exponent six corresponding to the number of elements in the tuple.

In general, when talking about a set Rn

the number n denotes the dimension of the vector.

We will also be interested in talking about the locations of objects

in three dimensional space.

Here the vector space R3 will be particularly helpful.

Here we can imagine a coordinate frame consisting of three orthogonal axis.

X, Y and Z.

And we can talk about the coordinates of a point by associating

them with a triple of numbers denoting the coordinates of that

point along each of these three orthogonal directions.

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