ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems Position, Velocity, and Acceleration Analysis 3.6 October 02, 2013 Radu Serban University of Wisconsin-Madison

2. Before we get started… • Last Time: • Driving (rheonomic) Constraints • This completes the derivation of Kinematic and Driver Constraints • Today: • Position, Velocity, and Acceleration Analysis • The Implicit Function Theorem • Assignments: • I will continue to enforce the “no late submission accepted” policy • HW 6 – due October 4, in class (12:00pm) • Matlab 4 and ADAMS 2 – due October 9, Learn@UW (11:59pm) • Please follow the instructions on what your submission should contain • ADAMS 2: submit a single PDF with all information asked for • Midterm Exam • Friday, October 11 – regular class hours and place • Closed books, but you can bring one letter-sized cheat sheet • Review session: Wednesday, October 9, 6:30pm in ME1152

3. Carl Gustav Jacob Jacobi (1804 – 1851) 3.6 Position, Velocity, and Acceleration Analysis

4. Kinematic Analysis: Stages • Stage 1: Identify all physical joints and drivers present in the system • Stage 2: Identify the corresponding set of constraint equations • Stage 3: Position AnalysisFind the Generalized Coordinates as functions of timeNeeded: and • Stage 4: Velocity AnalysisFind the Generalized Velocities as functions of timeNeeded: and • Stage 5: Acceleration AnalysisFind the Generalized Accelerations as functions of timeNeeded: and

5. Kinematic Analysis • The position analysis [Stage 3]: • The most difficult of the three • Requires the solution of a system of nonlinear equations • What we are after is determining the location and orientation of each component (body) of the mechanism at any given time • The velocity analysis [Stage 4]: • Requires the solution of a linear system of equations • Relatively simple • Carried out after completing position analysis • The acceleration analysis [Stage 5]: • Requires the solution of a linear system of equations • Challenge: generating the RHS of acceleration equation, • Carried out after completing velocity analysis

6. Position Analysis (1) • 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 end up with nc = nk + nd

7. Position Analysis (2) • We end up with this problem: given a time value 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 • Then advance the time to t=0.001 and find a solution q by solving above equations • Then advance the time to t=0.002 and find a solution q by solving above equations • Then advance the time to 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 (in this example) • You can then plot the solution as a function of time and get the time evolution of your mechanism

8. Position Analysis (3) • Two issues associated with the methodology described on previous slide: • The first issue: related to the fact that you are solving nonlinear equations. • Does a solution exist? Example: x2+4=0 (no real number x will do here) • Is the solution unique? Example: x2-4=0x (both 2 and -2 are solutions) • The second issue: The equations that we have to solve at each time t are nonlinear. How do you actually solve them? • For instance, how do you find the solution (x=-1.2) of this nonlinear equation: • Deal with this issue next lecture (discuss Newton-Raphson method)

9. The Implicit Function Theorem

10. A Simple Example

11. Implicit Function Theorem (IFT)

12. IFT: The Circle Example

13. IFT: Implications for Position Analysis • Informally, this is what the Implicit Function Theorem says: • Assume that, at some time tk we just found asolution q(tk)of . • If the constraint Jacobian is nonsingularin this configuration, that isthen, we can conclude that the solution is unique, and not only at tk, but in a small interval around 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 such that: • Practically, this means that the mechanism is guaranteed to be well behaved in the time interval . That is, the constraint equations are well defined and the mechanism assumes a unique configuration at each time. • Moreover, assuming that is twice differentiable, IFT guarantees that the velocity and acceleration equations hold.

14. Velocity and Acceleration Analysis

15. Velocity Analysis (1) • This is simple. What is the framework? • You just found q at time t, that is, the location and orientation of each component of the mechanism at time t, and now you want to find the velocity of each component (body) of the mechanism • Taking one time derivative of the constraints leads to the velocity equation: • In other words, once you have you can find at time by solving the linear system

16. Velocity Analysis (2) • Observations: • Note that as long as the constraint Jacobian is nonsingular, you can solve this linear system and recover the velocity • The reason velocity analysis is easy is that, unlike for position analysis where you have to solve a nonlinear system, now you are dealing with a linear system, which is easy to solve • Note that the velocity analysis comes after the position analysis is completed, and you are in a new configuration of the mechanism in which you are about to find out its velocity

17. Acceleration Analysis (1) • This is also fairly simple. What is the framework? • We just founda at time , that is, where the mechanism is and what its velocity is at that time • Wenow want to know the acceleration of each component of the model • Taking two time derivatives of the constraints leads to the acceleration equation:

18. Acceleration Analysis (2) • In other words, you find the acceleration (second time derivative of q at time tk) as the solution of a linear system: • Observations: • The equation above illustrates why we have been interested in the expression of , the RHS of the acceleration equation: • Note that you again want the constraint Jacobian to be nonsingular, since then you can solve the acceleration linear system and obtained the acceleration

19. SUMMARY OF CHAPTER 3 • We looked at the KINEMATICS of a mechanism • That is, we are interested in how this mechanism moves in response to a set of kinematic drivers (motions) applied to it • Kinematic Analysis Steps: • Stage 1: Identify all physical joints and drivers present in the system • Stage 2: Identify the corresponding constraint equations • Stage 3: Position Analysis – Find as functions of time • Stage 4: Velocity Analysis – Find as functions of time • Stage 5: Acceleration Analysis – Find as functions of time

20. [handout]Wrecker-Boom Example • We are interested in the KINEMATICS of this mechanism • That is, we are interested in how this mechanism moves in response to a set of *two* kinematic drivers (motions) applied to it • Relatively straight forward to check that this mechanism has NDOF=0 • Recall what we have to do: • Step A: Identify *all* physical joints and drivers present in the system • Step B: Identify the corresponding set of constraint equations • Step C: Compute : needed for the Position Analysis • Step D: Compute : needed for the Velocity Analysis • Step E: Compute : needed for the Accelerations Analysis