1. ME451 Kinematics and Dynamics of Machine Systems Driving Constraints 3.5 Start Position, Velocity, and Acc. Analysis 3.6 February 26, 2009 © Dan Negrut, 2009 ME451, UW-Madison

2. Before we get started… Today: Continue with relative driving constraints Proper Kinematic Analysis: Position, Velocity, and Acceleration Stages HW due next Th (Mar. 5): 3.5.1, 3.5.4, 3.5.5, 3.5.6, ADAMS 3.5.5: note that the angle  2 is not displayed correctly 3.5.6: get rid of v i, take it unit vector ADAMS problem: posted online Last Time Finished kinematic constraints Point-Follower Started to talk about driving constraints A set of ndof independent drivers need to be specified to “occupy” all the degrees of freedom Driver constraints  Absolute: x, y,   Relative: we’ve only talked about relative distance driver 2

3. Example: Specifying Relative Distance Drivers Generalized coordinates: 3 Motions prescribed: Derive the constraints acting on system Derive linear system whose solution provides velocities

4. Revolute Rotational Driver The framework: at point P we have a revolute joint It boils down to this: you prescribe the time evolution of the angle in the revolute joint 4 Driver constraint formulated as Note that  i and  j are attributes of the constraint

5. Translational Distance Driver The framework: we have a translational joint between two bodies Direction of translational join on “Body i” is defined by the vector v i 5 This driver says that the distance between point P i on “Body i” and point P j on “Body j” measured along in the direction of changes in time according to a user prescribed function C(t):

6. Translational Distance Driver (Cntd.) The book complicates the formulation with no good reason There is nothing to prevent me to specify the direction v i by selecting this quantity to have magnitude 1 Equivalently, the constraint then becomes Keep in mind that the direction of translation is indicated now through a unit vector (you are going to get the wrong motion if you work with a v i that is not unit length) This is the reason why I wanted to discuss this prior to assigning the problem 3.5.6 6

7. Driver Constraints, Departing Thoughts What is after all a driving constraint? You take your kinematic constraint, which indicates that a certain kinematic quantity should stay equal to zero Rather than equating this kinematic quantity to zero, you have it change with time… 7 Or equivalently…

8. Notation used: For Kinematic Constraints:  K (q) For Driver Constraints:  D (q,t) Note the arguments (for K, there is no time dependency) Correcting the RHS… Computing for Driver Constraints the right hand side of the velocity equation and acceleration equation is straightforward Once you know who to compute these quantities for  K (q), when dealing with  D (q,t) is just a matter of correcting… … (RHS of velocity equation) with the first derivative of C(t) …  (RHS of acceleration equation) with second derivative of C(t) Section 3.5.3 discusses these issues 8 Driver Constraints, Departing Thoughts (cntd.)

9. End Driving Constraints Begin Pos Vel Acc Analysis (3.6) 9

10. Mechanism Analysis: Steps Step A: Identify *all* physical joints and drivers present in the system Step B: Identify the corresponding constraint equations  (q,t) Step C: Solve for the Position as a function of time (  q is needed) Step D: Solve for the Velocities as a function of time ( is needed) Step E: Solve for the Accelerations as a function of time (  is needed) 10

11. Position, Velocity, and Acceleration Analysis (3.6) The position analysis [Step C]: It’s the tougher of the three Requires the solution of a system of nonlinear equations What you are after is determining at each time the location of every component (body) of the mechanism The velocity analysis [Step D]: Requires the solution of a linear system of equations Relatively simple Carried out after you are finished with the position analysis The velocity analysis [Step E]: Requires the solution of a linear system of equations What is challenging is generating the RHS of acceleration equation,  Carried out after you are finished with the position and velocity analyses 11

12. Position Analysis Framework: Somebody presents you with a mechanism and you select the set of nc generalized coordinates to position and orient each body of the mechanism: 12 You inspect the mechanism and identify a set of nk kinematic constraints that must be satisfied by your coordinates: Next, you identify the set of nd driver constraints that move the mechanism: NOTE: YOU MUST HAVE nc = nk + nd

13. We end up with this problem: given a time t, find that set of generalized coordinates q that satisfy the equations: What’s the idea here? Set time t=0, and find a solution q by solving above equations Next, set time t=0.001 and find a solution q by solving above equations Next, set time t=0.002 and find a solution q by solving above equations Next, set time t=0.003 and find a solution q by solving above equations … Stop when you reach the end of the interval in which you are interested in the position What you do is find the time evolution on a time grid with step size  t=0.001 You can then plot the solution as a function of time and get the time evolution of your mechanism 13 Position Analysis

14. Two issues with the described methodology for finding the time evolution of the mechanism The equations that we have to solve at each time t are nonlinear, so a first hurdle is being able to solve this nonlinear system Deal with this issue later (next week, Newton-Raphson method) The second issue comes when you start thinking about the solution that you’ve just got using a numerical algorithm How do you know that what you got is a meaningful thing?  Remember, a nonlinear system can have an arbitrary number of solutions Deal with this issue now 14 Position Analysis

15. Is the solution of our nonlinear system well behaved? A sufficient condition is provided by the Implicit Function Theorem In layman’s words, this is what the theorem says: Let’s say that we are at some time t k, and we just found the solution q k and we question the quality of this solution If the constraint Jacobian is nonsingular in this configuration, that is, … then, we can conclude that the solution is unique, and not only at t k, but in a small interval  about time t k. Additionally, in this small time interval, there is an explicit functional dependency of q on t, that is, there is a function f(t) such that: 15 Position Analysis: Implicit Function Theorem

16. End Position Analysis Begin Velocity Analysis 16