ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems Kinematics: Review October 9, 2013 Radu Serban University of Wisconsin, Madison

2. Vectorsand MatricesDifferential Calculus

3. Vectors in 2D • Geometric vectors → Reference Frames → Algebraic Vectors • We always use right-handed orthogonal RFs • Operations on geometric vectors ↔ Operations on algebraic vectors: • Scaling • Addition • Dot product • Calculating the angle between two vectors

4. Matrix Algebra • Operations on matrices • Scaling • Addition • Matrix-matrix multiplication • Matrix-vector multiplication • Transposition • Special matrices • Symmetric and skew-symmetric matrices • Singular matrices • Inverse matrix • Orthogonal matrix • Some important properties

5. Linear independence of vectorsMatrix rank • Row rank of a matrix • Largest number of rows of the matrix that are linearly independent • A matrix is said to have full row rank if the rank of the matrix is equal to the number of rows of that matrix • Column rank of a matrix • Largest number of columns of the matrix that are linearly independent

6. Matrix RankExample • What is the row rank of the following matrix? • What is the column rank of J?

7. Singular vs. Nonsingular Matrices • Let A be a square matrix of dimension n. The following are equivalent:

8. Transformation of Coordinates • Expressing a vector given in one reference frame (local) in adifferent reference frame (global):This is also called a change of base. • Since the rotation matrix is orthogonal, we have

9. Example

10. Kinematics of a Rigid Body • The position and orientation of a body(that is, position and orientation of the LRF)is completely defined by .The position of a pointP on the body is specified by: • in the LRF • in the GRF

11. Derivative, Partial Derivative,Total Derivative • The derivativeof a function (of a single variable) is a measure of how much the function changes due to a change in its argument. • A partial derivative of a function of several variables is the function derivative with respect to one of its variables when all other variables are held fixed. • The total derivative of a function of several variables is the derivative of the function when all variables are allowed to change.

12. Time Derivative of a Vector • Consider a vector whose components are functions of time:which is represented in a fixed (stationary) Cartesian RF. • In other words, the components of r change, but not the reference frame: the basis vectors and are constant. • Notation: • Then: 12

13. Time derivatives of a matrix

14. Partial Derivatives, General Case:Vector Function of Several Variables • You have a set of “m” functions each depending on a set of “n” variables: • Collect all “m” functions into an array F and collect all “n” variables into an array q: • So we can write:

15. Partial Derivatives, General Case:Vector Function of Several Variables • Then, in the most general case, we have • Example 2.5.2: The result is an m x n matrix!

16. 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:

17. Vector 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

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

19. Example

20. Time 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:

21. 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

22. 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 ?

23. Matrices of Interest • 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):

24. Velocity and Acceleration of a Point Fixed in a Moving Frame

25. Kinematics

26. What is Kinematics? • Study of the position, velocity, and acceleration of a system of interconnected bodies that make up a mechanism, independent of the forces that produce the motion • Flavors • Kinematic Analysis - Interested how components of a certain mechanism move when motion(s) are applied • Kinematic Synthesis – Interested in finding how to design a mechanism to perform a certain operation in a certain way

27. Nomenclature • Rigid body • Body-fixed Reference Frame (also called Local Reference Frame, LRF) • Absolute Reference frame (aka Global Reference Frame, GRF) • Generalized coordinates • Cartesian generalized coordinates • Constraints • Kinematic (scleronomic) • Driver (rheonomic) • Degrees of Freedom • KDOF • NDOF • Jacobian

28. Kinematic Analysis • We include as many actuators as kinematic degrees of freedom – that is, we impose KDOF driver constraints • We end up with NDOF = 0 – that is, we have as many constraints as generalized coordinates • We find the (generalized) positions, velocities, and accelerations by solving algebraic problems (both nonlinear and linear) • We do not care about forces, only that certain motions are imposed on the mechanism. We do not care about body shape nor inertia properties

29. Kinematic Analysis Stages • Position Analysis Stage • Challenging • Velocity Analysis Stage • Simple • Acceleration Analysis Stage • OK • To take care of all these stages, ONE step is critical: • Write down the constraint equations associated with the joints present in your mechanism • Once you have the constraints, the rest is boilerplate

30. Once you have the constraints… • Each of the three stages of Kinematics Analysis: position analysis, velocity analysis, and acceleration analysis, follow very similar recipes for finding the position, velocity and acceleration, respectively, of every body in the system. • All stages crucially rely on the Jacobianmatrix q • q – the partial derivative of the constraints w.r.t. the generalized coordinates • All stages require the solution of linear systems of equations of the form: • What is different between the three stages is the expression for the RHS b.

31. Position Analysis • As we pointed out, it all boils down to this: • Step 1: Write down the constraint equations associated with the model • Step 2: For each stage, construct q and the specific b, then solve for x • So how do you get the positionconfiguration of the mechanism? • Kinematic Analysis key observation: The number of constraints (kinematic and driving) should be equal to the number of generalized coordinates In other words, NDOF=0is a prerequisite for Kinematic Analysis IMPORTANT: This is a nonlinear systems with: • ncequations • and • ncunknowns • that must be solved for q

32. Velocity and Acceleration Analysis • Position analysis: The generalized coordinates (positions) are solution of the nonlinear system: • Take one time derivative of constraints (q,t) to obtain the velocity equation: • Take yet one more time derivative to obtain the acceleration equation:

33. Producing the RHS of theAcceleration Equation • The RHS of the acceleration equation was shown to be: • The terms in are pretty tedious to calculate by hand. • Note that the RHS contains (is made up of) everything that does not depend on the generalized accelerations • Implication: • When doing small examples in class, don’t bother to compute the RHS using expression above • You will do this in simEngine2D, where you aim for a uniform approach to all problems • Simply take two time derivatives of the (simple) constraints and move everything that does not depend on acceleration to the RHS

34. Kinematic Analysis Stages (1/2) • 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

35. Kinematic Analysis Stages (2/2) • 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

36. Implicit Function Theorem • 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.

37. Singular Configurations • Abnormal situations that should be avoided since they indicate either a malfunction of the mechanism (poor design), or a bad model associated with an otherwise well designed mechanism • Singular configurations come in two flavors: • Physical Singularities (PS): reflect bad design decisions • Modeling Singularities (MS): reflect bad modeling decisions • In a singular configuration, one of three things can happen: • PS1: Mechanism locks-up • PS2: Mechanism hits a bifurcation • MS1: Mechanism has redundant constraints • The important question:How can we characterize a singular configuration in a formal way such that we are able to diagnose it?

38. Identifying Singular Configurations

39. Newton-Raphson MethodGeometricInterpretation

40. Newton-RaphsonSystems of nonlinear equations • Let , with . Find such that , for a fixed value of • Algorithm becomesgiven an initial guess • Actual implementation • The Jacobian matrix is defined as

41. Newton-Raphson MethodPossible issues • Divergence • Division by zero close to stationary points • Convergence to a different root than the one desired (root jumping) • Cycle • Degradation of convergence rate if the Jacobian is close to singular at the root. • Nothing we can do to fix problem 5. This is a pathological (and rare) case. • All other issues can be resolved if the initial guess is close enough to the solution (in which case the method also has quadratic convergence). • Fortunately, when using N-R in Kinematic Analysis, we usually solve the position analysis problem on a time grid, . As such, we always have a good initial guess: the values of the generalized coordinates at the previous time step!(with one exception, the assembly problem at )

42. Systematic Derivation of Constraints • Kinematic (Geometric, Scleronomic) constraints • Time invariant • Absolute constraints • Location and/or orientation of a body w.r.t. GRF is constrained in a certain way • Examples: abs. x, abs. y, abs. angle, abs. dist • Using AGC, these constraints will only involve the GCs for a single body! • Relative constraints • Restrict relative motion of two bodies • Examples: rel. x, rel. y, rel. angle, rel. dist, rev., transl., gears, cam-followers • Using AGC, these constraints will only involve the GCs for two bodies! • Driver (Rheonemic) constraints • Explicit time dependency • Absolute drivers • Examples: abs. x, abs. y, abs. angle • Relative drivers • Examples: rel. x, rel. y, rel. angle, rel. dist, rev.-rotational, transl.-dist. • Recall:

43. Attributes of a Constraint • 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.)

44. Kinematics: Summary • 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

45. Putting it all together: Mechanism Analysis

46. Kinematics Modeling & Simulation • We are given a mechanism… • Describe how we would model this mechanism. How many bodies? How many GCs? • What kinematic constraints would we use? What is KDOF? • Write down the equations that model those constraints. How many equations do we end up with? • We are given a prescribed motion for this mechanism. How do we impose that? How many equations do we write down? • Can this mechanism reach a singular configuration? Why or why not? • Perform Kinematic Analysis at the initial time t=0. • Find the position, velocity, acceleration of some point on some body at t=0 • How would the N-R process start in order to solve for positions at the next point in the time grid (say, at t= 0.01)?

47. Mechanism Analysis

48. Mechanism Analysis