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

2. Vectors and Matrices Differential 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 Special 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 vectors Matrix 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 Rank Example 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 a different 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 𝐫 , 𝜙 . The position of a point P on the body is specified by:
𝐬′ 𝑃 in the LRF
𝐫 𝑃 in the GRF

11. Derivative, Partial Derivative, Total Derivative
The derivative of 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 𝐫 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 𝑖 and 𝑗 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 𝐱= 𝑥 1 , ⋯, 𝑥 𝑘 𝑇 :
A new function (x) is defined as:
Example: a force (which is a vector quantity), depends on the generalized positions and velocities

18. Chain Rule Vector Function of Vector Variables
Question: how do you compute 𝚽 𝐱 ?
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 Rule Time 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 𝐫 (the body position) and 𝐀 (the body orientation) change.
Questions:
What is the velocity of P? That is: what is 𝐫 𝑃 ?
What is the acceleration of P? That is: what is 𝐫 𝑃 ?

23. Matrices of Interest Important relations (easy to check):
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 Jacobian matrix 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 position configuration 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=0 is a prerequisite for Kinematic Analysis
IMPORTANT:
This is a nonlinear systems with:
nc equations
and
nc unknowns
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 the Acceleration Equation
The RHS of the acceleration equation was shown to be:
The terms in 𝛾 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 𝚽 𝐪,𝑡 =𝟎
Stage 3: Position Analysis Find the Generalized Coordinates as functions of time Needed: 𝚽 𝐪,𝑡 and 𝚽 𝐪
Stage 4: Velocity Analysis Find the Generalized Velocities as functions of time Needed: 𝚽 𝐪 and 𝛎(𝐪, 𝑡)
Stage 5: Acceleration Analysis Find the Generalized Accelerations as functions of time Needed: 𝚽 𝐪 and 𝛄(𝐪, 𝐪 ,𝑡)

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]:
Challenge: generating the RHS of acceleration equation, 𝛄(𝐪, 𝐪 ,𝑡)
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 a solution q(tk) of 𝚽 𝐪,𝑡 =𝟎.
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  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 𝑡∈ 𝑡 𝑘 −𝛿 , 𝑡 𝑘 +𝛿 . 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 Method Geometric Interpretation

40. Newton-Raphson Systems of nonlinear equations
Let 𝚽: ℝ 𝑛+1 → ℝ 𝑛 , with 𝚽=𝚽 𝐪,𝑡 . Find 𝐪 ∗ ∈ ℝ 𝑛 such that 𝚽 𝐪 ∗ ,𝑡 =0, for a fixed value of 𝑡
Algorithm becomes given an initial guess 𝐪 (0)
Actual implementation
The Jacobian matrix is defined as

41. Newton-Raphson Method Possible 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, 𝑡 𝑘 ;𝑘=0,1,2,… . 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 𝑡=0)

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 𝚽 𝐪,𝑡 =𝟎
Stage 3: Position Analysis – Find 𝐪 as functions of time
Stage 4: Velocity Analysis – Find 𝐪 as functions of time
Stage 5: Acceleration Analysis – Find 𝐪 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