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ME451 Kinematics and Dynamics of Machine Systems. Singular Configurations 3.7 October 07, 2013. Radu Serban University of Wisconsin-Madison. Before we get started…. Last Time: Numerical solution of systems of nonlinear equations Newton- Raphson method Today:

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me451 kinematics and dynamics of machine systems

ME451 Kinematics and Dynamics of Machine Systems

Singular Configurations

3.7

October 07, 2013

Radu Serban

University of Wisconsin-Madison

before we get started
Before we get started…
  • Last Time:
    • Numerical solution of systems of nonlinear equations
    • Newton-Raphsonmethod
  • Today:
    • Singular configurations (lock-up and bifurcations)
  • Assignments:
    • No book problems until midterm
    • Matlab 4 and Adams 2 – due October 9, [email protected] (11:59pm)
  • Midterm Exam
    • Friday, October 11 at 12:00pm in ME1143
    • Review session: Wednesday, October 9, 6:30pm in ME1152
kinematic analysis stages
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
position velocity acceleration analysis
Position, Velocity, Acceleration Analysis
  • The position analysis [Stage 3]:
    • The most difficult of the three as it requires the solution of a system of nonlinear equations.
    • Find generalized coordinates by solving the nonlinear equations:
  • The velocity analysis [Stage 4]:
    • Simple as it only requires the solution of a linear system of equations.
    • After completing position analysis, find generalized velocities from:
  • The acceleration analysis [Stage 5]:
    • Simple as it only requires the solution of a linear system of equations.
    • After completing velocity analysis, find generalized accelerations from:
ift implications for position analysis
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.
singular configurations1
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
  • Singular configurations do not represent the norm, but we must be aware of their existence
    • A PS is particularly bad and can lead to dangerous situations
singular configurations2
Singular Configurations
  • 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?
lock up ps1 example 3 7 5
Lock-up: PS1[Example 3.7.5]
  • The mechanism cannot proceed past this configuration
  • “No solution”
bifurcation ps2 example 3 7 5
Bifurcation: PS2[Example 3.7.5]
  • The mechanism cannot uniquely proceed from this configuration
  • “Multiple solution”
example 3 7 5 mechanism lock up 2
Example 3.7.5Mechanism Lock-Up (2)
  • Investigate rank of augmented Jacobian
  • Carry out position, velocity, and acceleration analysis ()

t = 1.90 it = 5 |Jac|= 1.493972e-01

q = [+1.028214e+00 +4.974188e-01 +3.034291e-01]

qd = [-8.597511e-01 +2.617994e-01 -1.540014e+00]

qdd = [-3.924469e+00 +0.000000e+00 -7.355873e+00]

t = 1.95 it = 5 |Jac|= 1.060626e-01

q = [+9.785586e-01 +5.105088e-01 +2.137492e-01]

qd = [-1.180227e+00 +2.617994e-01 -2.153623e+00]

qdd = [-1.083795e+01 +0.000000e+00 -2.105153e+01]

t = 2.00 it = 25 |Jac|= 8.718594e-09

q = [+8.660254e-01 +5.235988e-01 +1.743719e-08]

qd = [-1.300238e+07 +2.617994e-01 -2.600476e+07]

qdd = [-1.939095e+22 +0.000000e+00 -3.878190e+22]

t = 2.05 it = 100 |Jac|= 1.474421e-01

q = [+1.008677e+00 +5.366887e-01 -1.300261e+06]

qd = [-8.629133e-01 +2.617994e-01 -1.525969e+00]

qdd = [-3.893651e+00 +0.000000e+00 -7.307789e+00]

Jacobian ill conditioned

Start seeing convergence difficulties

Mechanism approaching speed of light

Failure to converge

Cannot solve for positions

(garbage)

example 3 7 5 mechanism bifurcation 2
Example 3.7.5Mechanism Bifurcation (2)
  • Use a time step-size
  • Carry out position, velocity, and acceleration analysis ()

t = 5.90 it = 3 |Jac| = 2.617695e-02

q = [+5.235390e-02 +1.544616e+00 +2.617994e-02]

qd = [-5.234194e-01 +2.617994e-01 -2.617994e-01]

qdd = [-3.588280e-03 +0.000000e+00 -1.998677e-13]

t = 5.95 it = 3 |Jac| = 1.308960e-02

q = [+2.617919e-02 +1.557706e+00 +1.308997e-02]

qd = [-5.235539e-01 +2.617994e-01 -2.617994e-01]

qdd = [-1.794293e-03 +0.000000e+00 -1.196983e-12]

t = 6.00 it = 3 |Jac| = -2.933417e-15

q = [-2.872185e-15 +1.570796e+00 -2.933417e-15]

qd = [-2.563346e-01 +2.617994e-01 +5.464817e-03]

qdd = [-2.335469e+13 +0.000000e+00 -2.335469e+13]

t = 6.05 it = 12 |Jac| = 1.308960e-02

q = [-9.066996e-16 +1.583886e+00 +1.308997e-02]

qd = [+1.815215e-14 +2.617994e-01 +2.617994e-01]

qdd = [-7.262474e-13 +0.000000e+00 -7.262474e-13]

t = 6.10 it = 2 |Jac| = 2.617695e-02

q = [-2.908636e-18 +1.596976e+00 +2.617994e-02]

qd = [-0.000000e+00 +2.617994e-01 +2.617994e-01]

qdd = [+2.168404e-19 +0.000000e+00 +0.000000e+00]

Jacobian is singular

Stepping over singularity and not knowing it

We ended up on one of the two possible branches

example 3 7 5 mechanism bifurcation 3
Example 3.7.5Mechanism Bifurcation (3)
  • Use a time step-size
  • Carry out position, velocity, and acceleration analysis ()

t = 5.88 it = 3 |Jac| = 3.141076e-02

q = [+6.282152e-02 +1.539380e+00 +3.141593e-02]

qd = [-5.233404e-01 +2.617994e-01 -2.617994e-01]

qdd = [-4.305719e-03 +0.000000e+00 +1.678902e-14]

t = 5.94 it = 3 |Jac| = 1.570732e-02

q = [+3.141463e-02 +1.555088e+00 +1.570796e-02]

qd = [-5.235342e-01 +2.617994e-01 -2.617994e-01]

qdd = [-2.153125e-03 +0.000000e+00 +9.807114e-14]

t = 6.00 it = 3 |Jac| = 4.163336e-16

q = [+4.775660e-16 +1.570796e+00 +4.163336e-16]

qd = [-3.003036e-01 +2.617994e-01 -3.850419e-02]

qdd = [+1.610640e+14 +0.000000e+00 +1.610640e+14]

t = 6.06 it = 9 |Jac| = -1.570732e-02

q = [-3.141463e-02 +1.586504e+00 -1.570796e-02]

qd = [-5.235342e-01 +2.617994e-01 -2.617994e-01]

qdd = [+2.153125e-03 +0.000000e+00 +1.112356e-12]

t = 6.12 it = 3 |Jac| = -3.141076e-02

q = [-6.282152e-02 +1.602212e+00 -3.141593e-02]

qd = [-5.233404e-01 +2.617994e-01 -2.617994e-01]

qdd = [+4.305719e-03 +0.000000e+00 -8.836328e-16]

Jacobian is singular

Stepping over singularity and not knowing it

We ended up on the other possible branch

singular configurations3
Singular Configurations
  • Remember that you seldom see singularities
  • Important: The only case when you run into problems is when the constraint Jacobian becomes singular:In this case, one of the following situations can occur:
    • You can be in a lock-up configuration (you won’t miss this, PS1)
    • You might face a bifurcation situation (very hard to spot, PS2)
    • You might have redundant constraints (MS1)
  • Otherwise, the Implicit Function Theorem (IFT) gives you the answer:If the constraint Jacobian is nonsingular, IFT says that you cannot be in a singular configuration.
summary of chapter 3
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
kinematics vs dynamics
Kinematics vs. Dynamics
  • Kinematics
    • 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
  • Dynamics
    • While we may impose some prescribed motions on the system, we assume that there are extra degrees of freedom – that is, NDOF > 0
    • The time evolution of the system is dictated by the applied external forces
    • The governing equations are differential or differential-algebraic equations
    • We very much care about applied forces and inertia properties of the bodies in the mechanism
dynamics m s
Dynamics M&S

Dynamics Modeling

  • Formulate the system of equations that govern the time evolution of a system of interconnected bodies undergoing planar motion under the action of applied (external) forces
    • These are differential-algebraic equations
    • Called Equations of Motion (EOM)
  • Understand how to handle various types of applied forces and properly include them in the EOM
  • Understand how to compute reaction forces in any joint connecting any two bodies in the mechanism

Dynamics Simulation

  • Understand under what conditions a solution to the EOM exists
  • Numerically solve the resulting (differential-algebraic) EOM
roadmap to deriving the eom
Roadmap to Deriving the EOM
  • Begin with deriving the variational EOM for a single rigid body
    • Principle of virtual work and D’Alembert’s principle
  • Consider the special case of centroidal reference frames
    • Centroid, polar moment of inertia, (Steiner’s) parallel axis theorem
  • Write the differential EOM for a single rigid body
    • Newton-Euler equations
  • Derive the variational EOM for constrained planar systems
    • Virtual work and generalized forces
  • Finally, write the mixed differential-algebraic EOM for constrained systems
    • Lagrange multiplier theorem

(This roadmap will take several lectures, with some side trips)

what are eom
What are EOM?
  • In classical mechanics, the EOM are equations that relate (generalized) accelerations to (generalized) forces
  • Why accelerations?
    • If we know the (generalized) accelerations as functions of time, they can be integrated once to obtain the (generalized) velocities and once more to obtain the (generalized) positions
    • Using absolute (Cartesian) coordinates, the acceleration of body i is the acceleration of the body’s LRF:
  • How do we relate accelerations and forces?
    • Newton’s laws of motion
    • In particular, Newton’s second law written as
newton s laws of motion
Newton’s Laws of Motion
  • 1st LawEvery body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.
  • 2nd LawA change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
  • 3rd LawTo any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction.
  • Newton’s laws
    • are applied to particles (idealized single point masses)
    • only hold in inertial frames
    • are valid only for non-relativistic speeds

Isaac Newton

(1642 – 1727)

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