ME451 Kinematics and Dynamics

of Machine Systems

Kinematics: ReviewOctober 9, 2013

Radu SerbanUniversity of Wisconsin, Madison

Vectors and MatricesDifferential Calculus

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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

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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

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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

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Matrix RankExample

What is the row rank of the following matrix?

What is the column rank of J?

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Singular vs. Nonsingular Matrices

Let A be a square matrix of dimension n. The following are equivalent:

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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

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Example

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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 point P on the body is specified by:• in the LRF• in the GRF

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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.

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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:

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Time derivatives of a matrix

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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:

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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!

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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:

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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

Vector Function of Vector Variables

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Question: how do you compute ?

Using our notation:

Chain Rule:

Chain RuleVector Function of Vector Variables

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Example

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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:

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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

Chain RuleTime Derivatives

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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 ?

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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):

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Velocity and Acceleration of a Point Fixed in a Moving Frame

Kinematics

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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

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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

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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

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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

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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.

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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

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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:

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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

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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

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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

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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 . 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 . That is, the constraint equations are well defined and the mechanism assumes a unique configuration at each time.

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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?

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Identifying Singular Configurations

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Newton-Raphson MethodGeometric Interpretation

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Newton-RaphsonSystems of nonlinear equations

Let , with . Find such that , for a fixed value of

Algorithm becomes

given an initial guess

Actual implementation

The Jacobian matrix is defined as

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Newton-Raphson MethodPossible issues

1. Divergence

2. Division by zero close to stationary points

3. Convergence to a different root than the one desired (root jumping)

4. Cycle

5. 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 )

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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:

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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.)

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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

Putting it all together: Mechanism Analysis

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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)?

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Mechanism Analysis

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Mechanism Analysis