Transcript Document

ME451
Kinematics and Dynamics
of Machine Systems
Review of Matrix Algebra – 2.2
Review of Elements of Calculus – 2.5
Vel and Acc of a Point fixed in a Ref Frame – 2.6
Jan. 27, 2009
© Dan Negrut, 2009
ME451, UW-Madison
Before we get started…

NOTE: Next lecture will be an ADAMS tutorial


Due next Tuesday:



To be held in 2324 Engineering Hall (during regular class hours)
ADAMS assignment, will be emailed to you
Problems*: 2.2.5, 2.2.8. 2.4.4, 2.5.2, 2.5.7
Last time we discussed about:



Reference Frames, and algebraic representation of a vector
Rotation Matrix (for switching from one RF to another RF)
Determining the vector from a global RF to a point in a moving RF
2
Important Relations of Previous Lecture
Y
P
y’
s'
x’
P
f
rP
O’
r
O
X
3
Example (RGC)
Use the array q of generalized coordinates to locate
the point B in the GRF (Global Reference Frame)

Y
O
X
θ1
L
2
22
2
5
55
5 O’1
%
%
%
%
2L
m 1g
E
θ12
L
2222
5555
O’
%
%
% 2
%
B
m2g
4
Example (AGC)

Use array q of generalized coordinates to locate
the point B in the GRF (Global Reference Frame)
y
O
x
θ1
L
y1’
2
22
2
5
55
5 O’1
%
%
%
%
2L
x1’
m 1g
E
L
y2’
2222
O’2 5
555
θ2 %
%
%
%
x2’
B
m2g
5
Relative vs. Absolute
Generalized Coordinates

A consequential question:


Where was it easier to come up with position of point B?
First Approach (Example RGC) – relies on relative coordinates:


Angle 1 uniquely specified both position and orientation of body 1
Angle 12 uniquely specified the position and orientation of body 2 with
respect to body 1
 To locate B wrt global RF, first I position it with respect to body 1 (drawing
on 12), and then locate the latter wrt global RF (based on 1)
 Note that if there were 100 bodies, I would have to position wrt to body 99,
which then I locate wrt body 98, …, and finally position wrt global RF
6
Relative vs. Absolute
Generalized Coordinates

(Cntd)
Second Approach (Example AGC) – relies on absolute (and Cartesian)
generalized coordinates:

x1, y1, 1 position and orient body 1 wrt GRF (global RF)

x2, y2, 2 position and orient body 2 wrt GRF (global RF)


To express the location of B is then very straightforward, use only x2, y2, 2 and
local information (local position of B in body 2)
For AGC, you handle many generalized coordinates

3 for each body in the system (six for this example)
7
Relative vs. Absolute
Generalized Coordinates

Conclusion for AGC and RGC:




(Cntd)
There is no free lunch:

AGC: easy to express locations but many GCs

RGC: few GCs but cumbersome process of locating B
Personally, I prefer AGC, the math is simple…
RGC common in robotics and molecular dynamics
AGC common in multibody dynamics
8
Example 2.4.3: Slider Crank

Based on information provided in figure (b), derive the position
vector associated with point P (that is, find position of point P in
the global reference frame OXY)
O
9
New Topic:
Matrix review

What is a matrix?

Matrix addition, scaling, subtraction


Addition is commutative
Matrix multiplication


Remember the dimension constraints on matrices that can be multiplied: #
columns of first matrix equal to # rows of second matrix
This operation is not commutative

A row-wise perspective on matrices (comes in handy)

A column-wise perspective (comes in handy at times)

Distributivity of matrix multiplication wrt matrix addition:
10
Matrix review, cntd.

Transpose of a matrix

In the context of the sum and product of two matrices

Symmetric matrix

Skew-symmetric matrix

Singular matrix

Inverse of a matrix
11
Matrix-vector review

Linear independence of a set of vectors

Rank of a matrix

For any matrix C,
12
Other Useful Formulas

If A and B are invertible, their product is invertible and

For any two matrices A and B that can be multiplied

For any three matrices A, B, and C that can be multiplied

If A is an orientation matrix (true for any orthogonal matrix)
13
Done with Matrix Review
…
Moving on to Function Derivatives
14
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
15
Taking time derivatives of a time
dependent vector


FRAMEWORK:

Vector r is represented as a function of time, and it has two
components: x(t) and y(t):

Its components change, but the vector is represented in a fixed
reference frame
THEN:
16
Rules for taking time derivatives
of a time dependent vector

You should know how to compute the time derivative of

The sum of two vectors

A scalar product (real number times a vector)

The inner product of two vectors

A vector of constant magnitude
17
Taking time derivatives of
MATRICES

By definition, the time derivative of a matrix is obtained by taking
the time derivative of each entry in the matrix

Otherwise, it’s just a simple extension of what you do for vectors:

The time derivative of the sum of two matrices

The time derivative of a scalar product (matrix scaled by real number)

The time derivative of the the product of two matrices
18
Done with Time Derivatives
…
Moving on to Partial Derivatives
19
What’s the story behind the
concept of partial derivative?

What’s the meaning of a partial derivative?




It captures the “sensitivity” of a function quantity wrt a variable
the function depends upon
Shows how much the function changes when the variable
changes a bit
Simplest case of partial derivative: you have one function that
depends on one variable:
Then,
20
Partial Derivative, Two Variables

Suppose you have one function but it depends on two
variables, say x and y:

To simplify the notation, an array q is introduced:

With this, the partial derivative of f wrt q is defined as
21
…and here is as good as it gets
(vector function)


You have a group of “m” functions that are gathered
together in an array, and they depend on a collection of
“n” variables:
The array that collects all
“m” functions is called F:

The array that collects all
“n” variables is called q:
22
Most general partial derivative
(Vector Function, cntd)

Then, in the most general case, we have F(q), and
This is an m x n matrix!

Example 2.5.2:
23