ME451
Kinematics and Dynamics
of Machine Systems
Absolute Constraints
3.2
September 20, 2013
Radu Serban
University of Wisconsin-Madison
2
Before we get started…
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Last time:
 Set the framework for kinematics analysis:
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Define basic concepts
Pose the kinematics analysis problem
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Today
 Absolute GCs vs. Relative GCs
 Absolute constraints
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Assignments:
 HW 4
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Problems 3.1.3, 3.3.2
Due Wednesday, September 25, in class (12:00pm)
Matlab 2 and ADAMS 1
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Due Wednesday, September 25, Learn@UW (11:59pm)
Absolute vs. Relative Generalized
Coordinates
4
Example (Absolute vs. Relative GC)
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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
5
Relative position of two adjacent
bodies connected through a pin joint
outboard
(child)
j
pin joint
inboard
(parent)
i
Back to our mechanism
6
7
Relative vs. Absolute GCs:
There is no such thing as a free lunch
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Absolute GC formulation:
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Relative GC formulation:
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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)
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…
Kinematic Analysis
9
Kinematic Analysis Stages
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Position Analysis Stage
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Velocity Analysis Stage
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OK
To take care of all these stages, ONE step is critical:
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Simple
Acceleration Analysis Stage
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Challenging
Write down the constraint equations associated with the joints
present in your mechanism
Once you have the constraints, the rest is boilerplate
10
Once you have the constraints…
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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
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q – the partial derivative of the constraints w.r.t. the generalized coordinates
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All stages require the solution of linear systems of equations of the form:
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What is different between the three stages is the expression for the RHS b.
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The Details…
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As we pointed out, it all boils down to this:
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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?
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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
12
Velocity and Acceleration Analysis
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Position analysis: The generalized coordinates (positions) are solution of the
nonlinear system:
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Take one time derivative of constraints (q,t) to obtain the velocity equation:
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Take yet one more time derivative to obtain the acceleration equation:
Producing the RHS of the
Acceleration Equation
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The RHS of the acceleration equation was shown to be:
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The terms in 𝛾 are pretty tedious to calculate by hand.
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Note that the RHS contains (is made up of) everything that does not depend
on the generalized accelerations
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Implication:
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When doing small examples in class, don’t bother to compute the RHS using
expression above
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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…
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Step 1: Identify a kinematic constraint (revolute, translational, relative distance,
etc., i.e., the physical thing) acting between two components of a mechanism
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Step 2: Formulate the algebraic equations that capture that constraint, 𝚽 𝐪 = 𝟎
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This is the actual modeling stage in Kinematics
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Step 3: Compute the Jacobian 𝚽𝐪
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Step 4: Compute 𝛎(𝐪), the right-hand side of the velocity equation
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Step 5: Compute 𝛄(𝐪, 𝐪), the right-hand side of the acceleration equation (messy)
This is what we do almost exclusively in Chapter 3
3.2
Absolute Constraints
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Absolute Constraints (1)
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Called “Absolute” since they express constraint between a
body in a system and the absolute (global, ground) reference
frame.
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Types of Absolute Constraints:
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Absolute position constraints
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Absolute orientation constraints
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Absolute distance constraints
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Absolute Constraints (2)
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Absolute position constraints
 x-coordinate of Pi
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y-coordinate of Pi
Absolute orientation constraint
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Orientation f of body
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Absolute x-constraint
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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
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Step 2: Identify Φ𝑎𝑥(𝑖,𝑗) = 0
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Step 3: Φ𝐪
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Step 4: 𝜈 𝑎𝑥(𝑖,𝑗) = ?
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Step 5: 𝛾 𝑎𝑥(𝑖,𝑗) = ?
𝑎𝑥(𝑖,𝑗)
=?
NOTE: The same approach is used to get the y- and angle-constraints
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Absolute distance-constraint
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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
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Step 2: Identify Φ𝑎𝑑(𝑖,𝑗) = 0
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Step 3: Φ𝐪
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Step 4: 𝜈 𝑎𝑑(𝑖,𝑗) = ?
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Step 5: 𝛾 𝑎𝑑(𝑖,𝑗) = ?
𝑎𝑑(𝑖,𝑗)
=?
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Example 3.2.1
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An example using absolute coordinate constraints:
simple pendulum
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Example 3.2.2
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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]
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What do you need to specify to completely specify a certain type of constraint?
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In other words, what are the attributes of a constraint; i.e., the parameters that define it?
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For absolute-x constraint: you need to specify the body “i”, the particular point P
on that body, and the value that xiP should assume
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For absolute-y constraint: you need to specify the body “i”, the particular point P
on that body, and the value that yiP should assume
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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).
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How about an absolute angle constraint?