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ME451 Kinematics and Dynamics of Machine Systems

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. Before we get started…. Today: Continue with relative driving constraints

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ME451 Kinematics and Dynamics of Machine Systems

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  1. ME451 Kinematics and Dynamics of Machine Systems Driving Constraints 3.5 Start Position, Velocity, and Acc. Analysis 3.6February 26, 2009 © Dan Negrut, 2009ME451, 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 vi , 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: • Motions prescribed: • Derive the constraints acting on system • Derive linear system whose solution provides velocities 3

  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 • Driver constraint formulated as • Note that i and j are attributes of the constraint 4

  5. This driver says that the distance between point Pi on “Body i” and point Pj on “Body j” measured along in the direction of changes in time according to a user prescribed function C(t): 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 vi 5

  6. Translational Distance Driver (Cntd.) • The book complicates the formulation with no good reason • There is nothing to prevent me to specify the direction viby 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 vi 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… • Or equivalently… 7

  8. Driver Constraints, Departing Thoughts(cntd.) • 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

  9. End Driving ConstraintsBegin 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: • 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 12

  13. Position Analysis • 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

  14. Position Analysis • 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

  15. Position Analysis: Implicit Function Theorem • 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 tk, and we just found the solution qk 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 tk, but in a small interval  about time tk. • 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

  16. End Position AnalysisBegin Velocity Analysis 16

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