Download ppt - SBEL - University of Wisconsin–Madison

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Matrix multiplication wikipedia, lookup

System of linear equations wikipedia, lookup

Vector space wikipedia, lookup

Euclidean vector wikipedia, lookup

Laplace–Runge–Lenz vector wikipedia, lookup

Covariance and contravariance of vectors wikipedia, lookup

Derivative wikipedia, lookup

Four-vector wikipedia, lookup

Generalizations of the derivative wikipedia, lookup

Matrix calculus wikipedia, lookup

Transcript
ME451
Kinematics and Dynamics
of Machine Systems
Review of Linear Algebra and
Differential Calculus
2.4, 2.5
September 09, 2013
Radu Serban
University of Wisconsin-Madison
Before we get started…

Last time:
 Discussed geometric and algebraic vectors
 Brief review of matrix algebra

Today:



Transformation of coordinates: Rotation Matrix, Rotation + Translation
Vector and Matrix Differentiation
HW 1 Due on Wednesday, September 11



Problems: 2.2.5, 2.2.8. 2.2.10 (from Haug’s book)
Upload a file named “lastName_HW_01.pdf” to the Dropbox Folder
“HW_01” at [email protected]
Dropbox Folder closes at 12:00PM
2
2.4
Transformation of Coordinates
Vectors and Reference Frames (1)

Recall that an algebraic vector is just a representation of a
geometric vector in a particular reference frame (RF)

Question: What if I now want to represent the same geometric
vector in a different RF?
4
Vectors and Reference Frames (2)

Transforming the representation of a vector from one RF to a
different RF is done through (left) multiplication by a so-called
“rotation matrix” A:

Notes




We transform the vector’s representation and not the vector itself.
What changes is the RF used to represent the vector.
As such, the rotation matrix defines a relationship between RFs.
A rotation matrix A is also called “orientation matrix”.
5
6
The Rotation Matrix


Rotation matrices are orthogonal:
Geometric interpretation of a rotation matrix:
Important Relation

Expressing a vector given in one reference frame (local) in a
different reference frame (global):
This is also called a change of base.

Since the rotation matrix is orthogonal, we have

More acronyms:
′ ′
 LRF: local reference frame (𝑂 𝑥 𝑦′)
 GRF: global reference frame (𝑂𝑥𝑦)
7
Example 1
8
Example 2
https://respond.cc
9
Your answers…
10
Example 2 - solution
11
The Kinematics of a Rigid Body:
Handling both Translation and Rotation




What we just discussed was re-expressing a vector from one coordinate
frame (LRF) to another coordinate frame (GRF).
Recall that vectors for us are really “free vectors” and therefore independent
of a translation of the reference frame.
What about position of a point P?
Use the definition of position vector, the vector addition,
and the formula for changing base for vectors:
12
More on Body Kinematics

Much of ME451 is based on the ability to look at the position
of a point P in two different reference frames:

a local reference frame (LRF), typically fixed (rigidly attached) to a
body that is moving in space

a global reference frame (GRF), which is the “world” reference frame
and serves as the universal reference frame

In the LRF, the position of point 𝑃 is described by 𝐬′𝑃
(sometimes, the notation 𝐬 𝑃 is used)

In the GRF, the position of point 𝑃 is described by the
position vector 𝐫 𝑃
13
14
ME451 Important Slide
The position and orientation of a body
(that is, position and orientation of the LRF)
is completely defined by 𝐫 , 𝜙 .
The position of a point P on the body is
specified by:
• 𝐬′𝑃 in the LRF
• 𝐫 𝑃 in the GRF
15
Example
C
2
y1′
1
2222
O′555
5
%
%
%
φ%
Y
1
O
X
2
x1′
D
2.5
Vector and Matrix Differentiation
17
Derivatives of Functions

GOAL: Understand how to

Take time derivatives of vectors and matrices

Take partial derivatives of functions with respect to its arguments



We will use a matrix-vector notation for computing these partial derivs.
Taking partial derivatives might be challenging in the beginning
It will be used a lot in this class
17
Derivative,
Partial Derivative,
Total Derivative

The derivative of a function (of a single variable) is a measure of
how much the function changes due to a change in its argument.

A partial derivative of a function of several variables is the function
derivative with respect to one of its variables when all other
variables are held fixed.

The total derivative of a function of several variables is the derivative
of the function when all variables are allowed to change.
18
Derivatives: Examples

Derivative

Partial derivative

Total derivative
19
Time Derivative of a Vector

20
Consider a vector 𝐫 whose components are functions of time:
which is represented in a fixed (stationary) Cartesian RF.

In other words, the components of r change, but not the reference
frame: the basis vectors 𝑖 and 𝑗 are constant.

Notation:

Then:
20
Time Derivatives
Vector-Related Operations
21
Time derivatives of a matrix
22
Partial Derivatives, Warming Up:
Scalar Function of Two Variables

Consider a scalar function of two variables:

To simplify the notation, collect all variables into an array:

With this, the derivative of f with respect to q is defined as:
23
24
Partial Derivatives, General Case:
Vector Function of Several Variables

You have a set of “m” functions each depending on a
set of “n” variables:

Collect all “m” functions into an array F and collect all
“n” variables into an array q:

So we can write:
Partial Derivatives, General Case:
Vector Function of Several Variables

Then, in the most general case, we have
The result is an m x n matrix!

Example 2.5.2:
25
26
Partial Derivatives
Compact Notation

Let x, y, and  be three
generalized coordinates

Define a (vector) function r of
x, y, and  as

“Verbose” notation

Collect the three generalized
coordinates into the array q

Define the function r of q:

“Terse” notation
27
Example
[handout]
Example (based on Example 2.4.1)

Find the partial derivative of the position of P with respect
to the array of generalized coordinates q
28
Partial Derivatives:
Remember this…

In the most general case, you start with “m” functions in “n” variables,
and end with an (m x n) matrix of partial derivatives.


You start with a column vector of functions and then end up with a matrix
Taking a partial derivative leads to a higher dimension quantity



Scalar Function – leads to row vector
Vector Function – leads to matrix
In this class, taking partial derivatives can lead to one of the following:



A row vector
A full blown matrix
If you see something else chances are you made a mistake…
29
Chain Rule of Differentiation

Formula for computing the derivative(s) of the composition
of two or more functions:




We have a function f of a variable q which is itself a function of x.
Thus, f is a function of x (implicitly through q)
Question: what is the derivative of f with respect to x?
Simplest case: real-valued function of a single real variable:
30
31
Case 1
Scalar Function of Vector Variable

f is a scalar function of “n” variables: q1, …, qn

However, each of these variables qi in turn depends on a
set of “k” other variables x1, …, xk.

The composition of f and q leads to a new function:
Chain Rule
Scalar Function of Vector Variable

Question: how do you compute x ?

Using our notation:

Chain Rule:
32
Assignment
[due 09/16]
33
34
Case 2
Vector Function of Vector Variable



F is a vector function of several variables: q1, …, qn
However, each of these variables qi depends in turn on a
set of k other variables x1, …, xk.
The composition of F and q leads to a new function:
Chain Rule
Vector Function of Vector Variable

Question: how do you compute  x ?

Using our notation:

Chain Rule:
35
Assignment
[due 09/16]
36
37
Case 3
Vector Function of Vector Variables




F is a vector function of 2 vector variables q and p :
Both q and p in turn depend on a set of k other variables
𝐱 = 𝑥1 , ⋯ , 𝑥𝑘 𝑇 :
A new function (x) is defined as:
Example: a force (which is a vector quantity), depends on the generalized
positions and velocities
38
Chain Rule
Vector Function of Vector Variables

Question: how do you compute 𝚽𝐱 ?

Using our notation:

Chain Rule:
[handout]
Example
39
Case 4
Time Derivatives

In the previous slides we talked about functions f depending on y,
where y in turn depends on another variable x.

The most common scenario in ME451 is when the variable x is
actually time, t

You have a function that depends on the generalized coordinates q, and
in turn the generalized coordinates are functions of time (they change in
time, since we are talking about kinematics/dynamics here…)

Case 1: scalar function that depends on an array of m time-dependent
generalized coordinates:

Case 2: vector function (of dimension n) that depends on an array of m
time-dependent generalized coordinates:
40
Chain Rule
Time Derivatives


Question: what are the time derivatives of  and 
Applying the chain rule of differentiation, the results in both cases can
be written formally in the exact same way, except the dimension of
the result will be different

Case 1: scalar function

Case 2: vector function
41
Example
Time Derivatives
42
A Few More Useful Formulas
43