ME451 Kinematics and Dynamics of Machine Systems Relative Constraints 3.3 September 23, 2013 Radu Serban University of Wisconsin-Madison

2 Before we get started… Last time: Absolute constraints ( x, y, , distance) Recall the drill: Identify and analyze the physical joint Derive the constraint equations associated with the joint,  (q)=0 Compute constraint Jacobian,  q Get (RHS of velocity equation) Get  (RHS of acceleration equation, this is challenging in some cases) Today Derive  for the absolute distance constraint Relative constraints ( x, y, , distance, revolute, translational) Assignments: HW 4 – due Wednesday, in class (12:00pm) Matlab 2 and ADAMS 1 – due Wednesday, (11:59pm)

3 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

4 Example An example using absolute coordinate constraints: simple pendulum

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

6 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?

7 [handout] Example 3.1.3

8 Example Note that when passing through the origin, the algebraic constraints fail to specify the actual kinematics of the mechanism Translating plain English into equations is not always straightforward Unexpected problem when passing through origin… Translating plain English into the right equations:

Relative Constraints 3.3

10 Loose Ends: Switching representation between two Reference Frames with rotation matrices A i and A j, respectively 10

11 Loose Ends, Continued: Notation (related to changing representation from A j to A i ) 11

12 Loose Ends, Final Slide (related to changing representation from A j to A i ) 12 For later reference, it is useful to recall that, Therefore

13 Vector between P i and P j Something that we’ll use a lot: the expression of the vector from P i to P j in terms of the generalized coordinates q

14 Relative x-constraint Step 1: The difference between the x coordinates of point P j and point P i should stay constant and equal to some known value C 1

15 Relative y-constraint Step 1: The difference between the y coordinates of point P j and point P i should stay constant and equal to some known value C 2

16 Relative angle-constraint Step 1: The difference between the orientation angles of the LRFs associated with bodies i and j should stay constant and equal to some known value C 3

17 Relative distance-constraint Step 1: The distance between the points P j (on body j) and P i (on body i) should stay constant and equal to some known value C 4

18 Revolute Joint

19 Translational Joint

20 Attributes of a Constraint (1) [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? Think about it…

21 Attributes of a Constraint (2) Attributes of a Constraint: That information that you are supposed to know by inspecting the mechanism It represents the parameters associated with the specific constraint that you are considering When you are dealing with a constraint, make sure you understand What the input is What the defining attributes of the constraint are What constitutes the output (the algebraic equation(s), Jacobian, ,, etc.)

22 Attributes of a Constraint (3) Examples of constraint attributes: For a revolute joint: You know where the joint is located, so therefore you know For a translational join: You know what the direction of relative translation is, so therefore you know For a distance constraint: You know the distance C 4

23 [handout] Example / Problem Approach 1: bodies 1, 2, and 3 Approach 2: bodies 1 and 3 Approach 3: bodies 1 and 2 Approach 4: body 2 Four different ways of modeling the same mechanism for Kinematic Analysis

24 Example Consider the slider-crank below. Come up with the set of kinematic constraint equations to kinematically model this mechanism

25 Composite Joints (1) Just a means to eliminate one intermediate body whose kinematics you are not interested in Revolute-Revolute Also called a coupler Practically eliminates need of connecting rod Attributes: Location of points P i and P j Distance d ij of the massless rod Revolute-Translational Eliminates the intermediate body Attributes: Distance c Point P j (location of revolute joint) Axis of translation v i ’

26 Composite Joints (2)