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 Learn@UW.
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