ME451 Kinematics and Dynamics

of Machine Systems

Review of Differential Calculus2.5, 2.6

September 11, 2013

Radu SerbanUniversity of Wisconsin-Madison

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Assignments

HW 1 Due today (Dropbox folder closed)

HW 2 Assigned: due September 16 (by 12:00PM) Problems from slides 7 & 10 (PDF available at course website) + 2.4.4,

2.5.2, 2.5.7 Upload a file named “lastName_HW_02.pdf” to the Dropbox Folder

“HW_02” at Learn@UW.

MATLAB 1 Assignment: due September 18 (by 11:59PM) PDF available at course website (includes upload instructions) Hint:

Look at the following Matlab functions (use the help command) inline sym/eval and matlabFunction

Use the forum to post results; OK to discuss possible approaches; not OK to post your entire code.

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Before we get started…

Last time: Discussed vector and matrix differentiation Started chain rule of differentiation

Today: Chain rule; velocity and acceleration of a point in moving frame Discuss absolute vs. relative coordinates

A word on notation: when bold fonts are not available, use underline to indicate vector or matrix quantities (and distinguish them from scalars).

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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:

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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:

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Chain Rule Scalar Function of Vector Variable

Question: how do you compute x ?

Using our notation:

Chain Rule:

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Assignment[due 09/16]

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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:

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Question: how do you compute x ?

Using our notation:

Chain Rule:

Chain Rule Vector Function of Vector Variable

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Assignment[due 09/16]

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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:

A new function (x) is defined as:

Example: a force (which is a vector quantity), depends on the generalized positions and velocities

Case 3 Vector Function of Vector Variables

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Question: how do you compute ?

Using our notation:

Chain Rule:

Chain RuleVector Function of Vector Variables

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Example

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Case 4Time Derivatives

In the previous slides we talked about functions f depending on q, where q 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:

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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

Case 1: scalar function

Case 2: vector function

Chain RuleTime Derivatives

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ExampleTime Derivatives

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A Few More Useful Formulas

Velocity and Acceleration of a Point Fixed in a Moving Frame

2.6

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Velocity and Acceleration of a Point Fixed in a Moving Frame

A moving rigid body and a point P, fixed (rigidly attached) to the body

The position vector of point P, expressed in the GRF is:

and changes in time because both (the body position) and (the body orientation) change.

Questions:

What is the velocity of P?That is: what is ?

What is the acceleration of P?That is: what is ?

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Some preliminaries

Orthogonal Rotation Matrix

Note that, when applied to a vector, this rotation matrix produces a new vector that is perpendicular to the original vector (counterclockwise rotation)

The matrix

The B matrix is always associated with a rotation matrix A.

Important relations (easy to check):

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Velocity of a Point Fixed in a Moving Frame

Something to keep in mind: we’ll manipulate quantities that depend on the generalized coordinates, which in turn depend on time Specifically, the orientation matrix A depends on the generalized

coordinate , which is itself a function of t

This is where the [time and partial] derivatives we discussed before come into play

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Same idea as for velocity, except that you need two time derivatives to get accelerations

Acceleration of a Point Fixed in a Moving Frame

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Example

Absolute (Cartesian) vs. Relative Generalized Coordinates

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Generalized CoordinatesGeneral Comments

What are Generalized Coordinates (GC)? A set of quantities (variables) that uniquely determine the state of the

mechanism the location of each body the orientation of each body (and from these, the position of any point on any body)

These quantities change in time since a mechanism allows motion In other words, the generalized coordinates are functions of time

The rate at which the generalized coordinates change define the set of generalized velocities Most often, obtained as the straight time derivative of the generalized coordinates Sometimes this is not the case though

Example: in 3D Kinematics, there is no generalized coordinate whose time derivative is the angular velocity

Important remark: there are multiple ways of choose the set of generalized coordinates that describe the state of your mechanism

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[handout]

Example (Relative GC)

Use the array q of generalized coordinates to locate the point P in the GRF (Global Reference Frame)

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Example (Absolute GC)

Use the array q of generalized coordinates to locate the point P in the GRF (Global Reference Frame)

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Relative vs. Absolute GCs (1)

A consequential question: Where was it easier to come up with position of point P?

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 point P on body 2 w.r.t. the GRF, we need to first position body 1

w.r.t. the GRF (based on 1), then position body 2 w.r.t. to body 1 (based on 12)

Note that if there were 100 bodies, I would have to position body 1 w.r.t. to GRF, then body 2 w.r.t. body 1, then body 3 w.r.t. body 2, and so on, until we can position body 100 w.r.t. body 99

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Second Approach (Example AGC) – relies on absolute (Cartesian) generalized coordinates:

x1, y1, 1 define the position and orientation of body 1 w.r.t. the GRF

x2, y2, 2 define the position and orientation of body 2 w.r.t. the GRF To express the location of P is then straightforward and uses only x2, y2, 2 and

local information (local position of B in body 2): in other words, use only information associated with body 2.

For AGC, you handle many generalized coordinates 3 for each body in the system (six for this example)

Relative vs. Absolute GCs (2)

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Absolute GC formulation: Straightforward to express the position of a point on a given body (and only

involves the GCs corresponding to the appropriate body and the position of the point in the LRF)…

…but requires many GCs (and therefore many equations) Common in multibody dynamics (major advantage: easy to remove/add

bodies and/or constraints)

Relative GC formulation: Requires a minimal set of GCs …but expressing the position of a point on a given body is complicated (and

involves GCs associated with an entire chain of bodies) Common in robotics, molecular dynamics, real-time applications

We will use AGC: the math is simpler; let the computer keep track of the multitude of GCs…

Relative vs. Absolute GCs:There is no such thing as a free lunch

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Example 2.4.3Slider 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

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What comes next…

Planar Cartesian Kinematics (Chapter 3) Kinematics modeling: deriving the equations that describe motion of

a mechanism, independent of the forces that produce the motion. We will be using an Absolute (Cartesian) Coordinates formulation Goals:

Develop a general library of constraints (the mathematical equations that model a certain physical constraint or joint)

Pose the Position, Velocity and Acceleration Analysis problems

Numerical Methods in Kinematics (Chapter 4) Kinematics simulation: solving the equations that govern position,

velocity and acceleration analysis