ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems Review of Differential Calculus 2.5, 2.6 September 11, 2013 Radu Serban University of Wisconsin-Madison

2. Assignments • HW 1 Duetoday (Dropbox folder closed) • HW 2 Assigned: due September 16 (by 12:00PM) • Problems from slides 7 & 10 (PDF available at course website) + 2.4.4, 2.5.2, 2.5.7 • Upload a file named “lastName_HW_02.pdf” to the Dropbox Folder “HW_02” at Learn@UW. • MATLAB 1 Assignment: due September 18 (by 11:59PM) • PDF available at course website (includes upload instructions) • Hint: Look at the following Matlab functions (use the help command) • inline • sym/eval and matlabFunction • Use the forum to post results; OK to discuss possible approaches; not OK to post your entire code.

3. Before we get started… • Last time: • Discussed vector and matrix differentiation • Started chain rule of differentiation • Today: • Chain rule; velocity and acceleration of a point in moving frame • Discuss absolute vs. relative coordinates • A word on notation: when bold fonts are not available, use underline to indicate vector or matrix quantities (and distinguish them from scalars).

4. Chain Rule of Differentiation • Formula for computing the derivative(s) of the composition of two or more functions: • We have a function f of a variable q which is itself a function of x. • Thus, f is a function of x (implicitly through q) • Question: what is the derivative of f with respect to x? • Simplest case: real-valued function of a single real variable:

5. Case 1Scalar Function of Vector Variable • f is a scalar function of “n” variables: q1, …, qn • However, each of these variables qi in turn depends on a set of “k” other variables x1, …, xk. • The composition of f and q leads to a new function:

6. Chain Rule Scalar Function of Vector Variable • Question: how do you compute x ? • Using our notation: • Chain Rule:

7. Assignment[due 09/16]

8. Case 2Vector Function of Vector Variable • F is a vector function of several variables: q1, …, qn • However, each of these variables qi depends in turn on a set of k other variables x1, …, xk. • The composition of F and q leads to a new function:

9. Chain Rule Vector Function of Vector Variable • Question: how do you compute  x ? • Using our notation: • Chain Rule:

10. Assignment[due 09/16]

11. Case 3Vector Function of Vector Variables • F is a vector function of 2 vector variables q and p: • Both q and p in turn depend on a set of k other variables: • A new function (x) is defined as: • Example: a force (which is a vector quantity), depends on the generalized positions and velocities

12. Chain RuleVector Function of Vector Variables • Question: how do you compute ? • Using our notation: • Chain Rule:

13. [handout]Example

14. Case 4Time Derivatives • In the previous slides we talked about functions f depending on q, where q in turn depends on another variable x. • The most common scenario in ME451 is when the variable x is actually time, t • You have a function that depends on the generalized coordinates q, and in turn the generalized coordinates are functions of time (they change in time, since we are talking about kinematics/dynamics here…) • Case 1: scalar function that depends on an array of m time-dependent generalized coordinates: • Case 2: vector function (of dimension n) that depends on an array of m time-dependent generalized coordinates:

15. Chain RuleTime Derivatives • Question: what are the time derivatives of  and ? • Applying the chain rule of differentiation, the results in both cases can be written formally in the exact same way, except the dimension of the result will be different • Case 1: scalar function • Case 2: vector function

16. ExampleTime Derivatives

17. A Few More Useful Formulas

18. 2.6 Velocity and Acceleration of a Point Fixed in a Moving Frame

19. Velocity and Acceleration of a Point Fixed in a Moving Frame • A moving rigid body and a point P, fixed (rigidly attached) to the body • The position vector of point P, expressed in the GRF is:and changes in time because both (the body position) and (the body orientation) change. Questions: • What is the velocity of P?That is: what is ? • What is the acceleration of P?That is: what is ?

20. Some preliminaries • Orthogonal Rotation Matrix • Note that, when applied to a vector, this rotation matrix produces a new vector that is perpendicular to the original vector (counterclockwise rotation) • The matrix • The B matrix is always associated with a rotation matrix A. • Important relations (easy to check):

21. Velocity of a Point Fixed in a Moving Frame • Something to keep in mind: we’ll manipulate quantities that depend on the generalized coordinates, which in turn depend on time • Specifically, the orientation matrix A depends on the generalized coordinate  , which is itself a function of t • This is where the [time and partial] derivatives we discussed before come into play

22. Acceleration of a Point Fixed in a Moving Frame • Same idea as for velocity, except that you need two time derivatives to get accelerations

23. Example

24. Absolute (Cartesian) vs. Relative Generalized Coordinates

25. Generalized CoordinatesGeneral Comments • What are Generalized Coordinates (GC)? • A set of quantities (variables) that uniquely determine the state of the mechanism • the location of each body • the orientation of each body • (and from these, the position of any point on any body) • These quantities change in time since a mechanism allows motion • In other words, the generalized coordinates are functions of time • The rate at which the generalized coordinates change define the set of generalized velocities • Most often, obtained as the straight time derivative of the generalized coordinates • Sometimes this is not the case though Example: in 3D Kinematics, there is no generalized coordinate whose time derivative is the angular velocity • Important remark: there are multiple ways of choose the set of generalized coordinates that describe the state of your mechanism

26. [handout]Example (Relative GC) Use the array q of generalized coordinates to locate the point P in the GRF (Global Reference Frame)

27. [handout]Example (Absolute GC) Use the array q of generalized coordinates to locate the point P in the GRF (Global Reference Frame)

28. Relative vs. Absolute GCs (1) • A consequential question: • Where was it easier to come up with position of point P? • First Approach (Example RGC) – relies on relative coordinates: • Angle 1 uniquely specified both position and orientation of body 1 • Angle 12 uniquely specified the position and orientation of body 2 with respect to body 1 • To locate point P on body 2 w.r.t. the GRF, we need to first position body 1 w.r.t. the GRF (based on 1), then position body 2 w.r.t. to body 1 (based on 12) • Note that if there were 100 bodies, I would have to position body 1 w.r.t. to GRF, then body 2 w.r.t. body 1, then body 3 w.r.t. body 2, and so on, until we can position body 100 w.r.t. body 99

29. Relative vs. Absolute GCs (2) • Second Approach (Example AGC) – relies on absolute (Cartesian) generalized coordinates: • x1, y1, 1 define the position and orientation of body 1 w.r.t. the GRF • x2, y2, 2 define the position and orientation of body 2 w.r.t. the GRF • To express the location of P is then straightforward and uses only x2, y2, 2 and local information (local position of B in body 2): in other words, use only information associated with body 2. • For AGC, you handle many generalized coordinates • 3 for each body in the system (six for this example)

30. Relative vs. Absolute GCs:There is no such thing as a free lunch • 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…

31. Example 2.4.3Slider Crank • Based on information provided in figure (b), derive the position vector associated with point P (that is, find position of point P in the global reference frame OXY) O

32. What comes next… • Planar Cartesian Kinematics (Chapter 3) • Kinematics modeling: deriving the equations that describe motion of a mechanism, independent of the forces that produce the motion. • We will be using an Absolute (Cartesian) Coordinates formulation • Goals: • Develop a general library of constraints (the mathematical equations that model a certain physical constraint or joint) • Pose the Position, Velocity and Acceleration Analysis problems • Numerical Methods in Kinematics (Chapter 4) • Kinematics simulation: solving the equations that govern position, velocity and acceleration analysis