1. ME451 Kinematics and Dynamics of Machine Systems Absolute Constraints 3.2 September 20, 2013 Radu Serban University of Wisconsin-Madison

2. 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. 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

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

6. 6 Back to our mechanism

7. 7 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

8. Kinematic Analysis

9. 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. 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 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.

11. 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 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

12. 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. 13 Producing the RHS of the Acceleration Equation

14. 14 The Drill… This is what we do almost exclusively in Chapter 3

15. Absolute Constraints 3.2

16. 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. 17 Absolute Constraints (2) Absolute position constraints x -coordinate of P i y -coordinate of P i Absolute orientation constraint Orientation  of body

18. 18 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 c 1 NOTE: The same approach is used to get the y - and angle-constraints

19. 19 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 c 3

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

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

22. 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 x i 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 y i 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 c 1 and c 2 of point C (in the GRF). How about an absolute angle constraint?