ME451 Kinematics and Dynamics

of Machine Systems

Absolute Constraints 3.2

September 20, 2013

Radu SerbanUniversity of Wisconsin-Madison

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Before we get started… Last time:

Set the framework for kinematics analysis: Define basic concepts Pose the kinematics analysis problem

Today Absolute GCs vs. Relative GCs Absolute constraints

Assignments: HW 4

Problems 3.1.3, 3.3.2 Due Wednesday, September 25, in class (12:00pm)

Matlab 2 and ADAMS 1 Due Wednesday, September 25, Learn@UW (11:59pm)

Absolute vs. Relative Generalized Coordinates

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Example (Absolute vs. Relative GC) Position the mechanism in the GRF Position point P on body 2 in the GRF

(a) Absolute GCs (b) Relative GCs

pin joint

pin joint

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Relative position of two adjacent bodies connected through a pin joint

pin joint

i

j

inboard(parent)

outboard(child)

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Back to our mechanism

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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

Kinematic Analysis

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Kinematic Analysis Stages Position Analysis Stage

Challenging

Velocity Analysis Stage Simple

Acceleration Analysis Stage OK

To take care of all these stages, ONE step is critical: Write down the constraint equations associated with the joints

present in your mechanism Once you have the constraints, the rest is boilerplate

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Once you have the constraints…

Each of the three stages of Kinematics Analysis: position analysis, velocity analysis, and acceleration analysis, follow very similar recipes for finding the position, velocity and acceleration, respectively, of every body in the system.

All stages crucially rely on the Jacobian matrix q q – the partial derivative of the constraints w.r.t. the generalized coordinates

All stages require the solution of linear systems of equations of the form:

What is different between the three stages is the expression for the RHS b.

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The Details…

As we pointed out, it all boils down to this: Step 1: Write down the constraint equations associated with the model Step 2: For each stage, construct q and the specific b, then solve for x

So how do you get the position configuration of the mechanism? Kinematic Analysis key observation: The number of constraints (kinematic

and driving) should be equal to the number of generalized coordinates

In other words, NDOF=0 is a prerequisite for Kinematic Analysis

IMPORTANT:

This is a nonlinear systems with:• nc equations

and • nc unknowns

that must be solved for q

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Velocity and Acceleration Analysis

Position analysis: The generalized coordinates (positions) are solution of the nonlinear system:

Take one time derivative of constraints (q,t) to obtain the velocity equation:

Take yet one more time derivative to obtain the acceleration equation:

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Producing the RHS of theAcceleration Equation

The RHS of the acceleration equation was shown to be:

The terms in are pretty tedious to calculate by hand.

Note that the RHS contains (is made up of) everything that does not depend on the generalized accelerations

Implication: When doing small examples in class, don’t bother to compute the RHS using

expression above You will do this in simEngine2D, where you aim for a uniform approach to all problems

Simply take two time derivatives of the (simple) constraints and move everything that does not depend on acceleration to the RHS

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The Drill…

Step 1: Identify a kinematic constraint (revolute, translational, relative distance,

etc., i.e., the physical thing) acting between two components of a mechanism

Step 2: Formulate the algebraic equations that capture that constraint, This is the actual modeling stage in Kinematics

Step 3: Compute the Jacobian

Step 4: Compute , the right-hand side of the velocity equation

Step 5: Compute , the right-hand side of the acceleration equation (messy)

This is what we do almost exclusively in Chapter 3

Absolute Constraints3.2

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Absolute Constraints (1)

Called “Absolute” since they express constraint between a body in a system and the absolute (global, ground) reference frame.

Types of Absolute Constraints:

Absolute position constraints

Absolute orientation constraints

Absolute distance constraints

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Absolute Constraints (2)

Absolute position constraints x-coordinate of Pi

y-coordinate of Pi

Absolute orientation constraint Orientation f of body

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Absolute x-constraint Step 1: the absolute x component of the location of a

point P on body i stays constant and equal to some

known value c1

NOTE: The same approach is used to get the y- and angle-constraints

Step 2: Identify

Step 3:

Step 4:

Step 5:

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Absolute distance-constraint

Step 1: the distance from a point P on body i to a point C defined in the GRF stays constant and equal to some known value c3

Step 2: Identify

Step 3:

Step 4:

Step 5:

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Example 3.2.1

An example using absolute coordinate constraints:simple pendulum

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Example 3.2.2

An example using an absolute angle constraint:slider along x-axis

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Attributes of a Constraint[it’ll be on the exam]

What do you need to specify to completely specify a certain type of constraint? In other words, what are the attributes of a constraint; i.e., the parameters that define it?

For absolute-x constraint: you need to specify the body “i”, the particular point P on that body, and the value that xi

P should assume

For absolute-y constraint: you need to specify the body “i”, the particular point P on that body, and the value that yi

P should assume

For a distance constraint, you need to specify the “distance”, but also the location of point P in the LRF, the body “i” on which the LRF is attached to, as well as the coordinates c1 and c2 of point C (in the GRF).

How about an absolute angle constraint?