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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…
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Last time:
 Discussed geometric and algebraic vectors
 Brief review of matrix algebra
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Today:
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Transformation of coordinates: Rotation Matrix, Rotation + Translation
Vector and Matrix Differentiation
HW 1 Due on Wednesday, September 11
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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)
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Recall that an algebraic vector is just a representation of a
geometric vector in a particular reference frame (RF)
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Question: What if I now want to represent the same geometric
vector in a different RF?
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Vectors and Reference Frames (2)
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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:
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Notes
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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”.
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The Rotation Matrix
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Rotation matrices are orthogonal:
Geometric interpretation of a rotation matrix:
Important Relation
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Expressing a vector given in one reference frame (local) in a
different reference frame (global):
This is also called a change of base.
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Since the rotation matrix is orthogonal, we have
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More acronyms:
′ ′
 LRF: local reference frame (𝑂 𝑥 𝑦′)
 GRF: global reference frame (𝑂𝑥𝑦)
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Example 1
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Example 2
https://respond.cc
9
Your answers…
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Example 2 - solution
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The Kinematics of a Rigid Body:
Handling both Translation and Rotation
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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:
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More on Body Kinematics
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Much of ME451 is based on the ability to look at the position
of a point P in two different reference frames:
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a local reference frame (LRF), typically fixed (rigidly attached) to a
body that is moving in space
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a global reference frame (GRF), which is the “world” reference frame
and serves as the universal reference frame
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In the LRF, the position of point 𝑃 is described by 𝐬′𝑃
(sometimes, the notation 𝐬 𝑃 is used)
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In the GRF, the position of point 𝑃 is described by the
position vector 𝐫 𝑃
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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
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Example
C
2
y1′
1
2222
O′555
5
%
%
%
φ%
Y
1
O
X
2
x1′
D
2.5
Vector and Matrix Differentiation
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Derivatives of Functions
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GOAL: Understand how to
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Take time derivatives of vectors and matrices
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Take partial derivatives of functions with respect to its arguments
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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
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Derivative,
Partial Derivative,
Total Derivative
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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.
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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.
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The total derivative of a function of several variables is the derivative
of the function when all variables are allowed to change.
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Derivatives: Examples
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Derivative
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Partial derivative
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Total derivative
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Time Derivative of a Vector
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Consider a vector 𝐫 whose components are functions of time:
which is represented in a fixed (stationary) Cartesian RF.
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In other words, the components of r change, but not the reference
frame: the basis vectors 𝑖 and 𝑗 are constant.
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Notation:
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Then:
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Time Derivatives
Vector-Related Operations
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Time derivatives of a matrix
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Partial Derivatives, Warming Up:
Scalar Function of Two Variables
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Consider a scalar function of two variables:
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To simplify the notation, collect all variables into an array:
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With this, the derivative of f with respect to q is defined as:
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Partial Derivatives, General Case:
Vector Function of Several Variables
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You have a set of “m” functions each depending on a
set of “n” variables:
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Collect all “m” functions into an array F and collect all
“n” variables into an array q:
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So we can write:
Partial Derivatives, General Case:
Vector Function of Several Variables
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Then, in the most general case, we have
The result is an m x n matrix!
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Example 2.5.2:
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Partial Derivatives
Compact Notation
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Let x, y, and  be three
generalized coordinates
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Define a (vector) function r of
x, y, and  as
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“Verbose” notation
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Collect the three generalized
coordinates into the array q
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Define the function r of q:
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“Terse” notation
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Example
[handout]
Example (based on Example 2.4.1)
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Find the partial derivative of the position of P with respect
to the array of generalized coordinates q
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Partial Derivatives:
Remember this…
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In the most general case, you start with “m” functions in “n” variables,
and end with an (m x n) matrix of partial derivatives.
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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
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Scalar Function – leads to row vector
Vector Function – leads to matrix
In this class, taking partial derivatives can lead to one of the following:
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A row vector
A full blown matrix
If you see something else chances are you made a mistake…
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Chain Rule of Differentiation
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Formula for computing the derivative(s) of the composition
of two or more functions:
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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:
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Case 1
Scalar Function of Vector Variable
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f is a scalar function of “n” variables: q1, …, qn
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However, each of these variables qi in turn depends on a
set of “k” other variables x1, …, xk.
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The composition of f and q leads to a new function:
Chain Rule
Scalar Function of Vector Variable
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Question: how do you compute x ?
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Using our notation:
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Chain Rule:
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Assignment
[due 09/16]
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Case 2
Vector Function of Vector Variable
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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
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Question: how do you compute  x ?
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Using our notation:
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Chain Rule:
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Assignment
[due 09/16]
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Case 3
Vector Function of Vector Variables
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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
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Chain Rule
Vector Function of Vector Variables
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Question: how do you compute 𝚽𝐱 ?
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Using our notation:
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Chain Rule:
[handout]
Example
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Case 4
Time Derivatives
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In the previous slides we talked about functions f depending on y,
where y in turn depends on another variable x.
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The most common scenario in ME451 is when the variable x is
actually time, t
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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…)
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Case 1: scalar function that depends on an array of m time-dependent
generalized coordinates:
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Case 2: vector function (of dimension n) that depends on an array of m
time-dependent generalized coordinates:
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Chain Rule
Time Derivatives
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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
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Case 1: scalar function
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Case 2: vector function
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Example
Time Derivatives
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A Few More Useful Formulas
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