ME451 Kinematics and Dynamics of Machine Systems

1. 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… • 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)

3. Absolute vs. Relative Generalized Coordinates

4. 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

5. Relative position of two adjacent bodies connected through a pin joint outboard (child) j i pin joint inboard (parent)

6. Back to our mechanism

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

8. Kinematic Analysis

9. 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

10. 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 Jacobianmatrix 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.

11. 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 positionconfiguration 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=0is a prerequisite for Kinematic Analysis IMPORTANT: This is a nonlinear systems with: • ncequations • and • ncunknowns • that must be solved for q

12. 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:

13. 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

14. 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

15. 3.2 Absolute Constraints

16. 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

17. Absolute Constraints (2) • Absolute position constraints • x-coordinate of Pi • y-coordinate of Pi • Absolute orientation constraint • Orientation f of body

18. Absolute x-constraint • Step 1: the absolute x component of the location of a point Pon body i stays constant and equal to some known value c1 • Step 2: Identify • Step 3: • Step 4: • Step 5: NOTE: The same approach is used to get the y- and angle-constraints

19. Absolute distance-constraint • Step 1: the distance from a point Pon 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:

20. Example 3.2.1 • An example using absolute coordinate constraints:simple pendulum

21. Example 3.2.2 • An example using an absolute angle constraint:slider along x-axis

22. 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 xiP should assume • For absolute-y constraint: you need to specify the body “i”, the particular point P on that body, and the value that yiPshould 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 c2of point C (in the GRF). • How about an absolute angle constraint?