The objective of this module is to show how constraint equations are

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Nov 14, 2013 (3 years and 11 months ago)

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Dynamic

Simulation
:

Constraint
Kinematics

Objective



The
objective of this module is to show how constraint equations are
used to compute the position, velocity, and acceleration of the
generalized coordinates
.


These equations
are kinematic in nature because they do not consider
the forces required to cause the
motion.


The
kinematic and motion constraints developed in
the previous
module (Module 3)
for the piston
-
crank mechanism are used to
demonstrate the mathematics.



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Notation


The total set of constraint equations
needed to define a mechanism
includes both kinematic constraints
and drive constraints.



There
are
15
generalized
coordinates
and 15 nonlinear constraint
equations for the piston
-
crank
assembly used in Module 3.



Since the piston
-
crank has a mobility
of one,
only one of the
fifteen
equations
will be a
motion constraint
that is an explicit function of time.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 2









0
,
,











t
q
q
t
q
d
k


q
k

i
s the set of kinematic
constraint equations



t
q
d
,

i
s the set of motion
constraint equations

q
i
s the set of generalized
coordinates

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Position


Solving
the constraint equations
will
yield the value of each
generalized
coordinate at a
specific instance of time.



The constraint equations are
non
-
linear and the Newton
-
Raphson

method is used as the
solution method.



The Newton
-
Raphson

method is
iterative and converges when
the constraint equations are
satisfied.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 3











0
,
,











t
q
q
t
q
d
k
Constraint Equations

Newton
-
Raphson

Equations











t
q
q
t
q
q
q
i
i
,
,
1
1














where












q
t
q
,
i
s the
Jacobian

matrix

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Velocity


The time derivative of the
constraint equations is used
to determine the velocities of
the generalized coordinates.



Since the generalized
coordinates are a function of
time and the constraint
equations are a function of
the generalized coordinates
and time, the chain rule for
partial differentiation must
be used.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 4











0
,
,











t
q
q
t
q
d
k
Constraint Equations

Time Derivative





0
,






































t
t
q
q
t
t
q
Velocities






























t
q
t
q
1
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Acceleration

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 5


The second time derivative of
the constraint equations is
used to determine the
accelerations of the
generalized coordinates.



Since the generalized
coordinates are a function of
time and the constraint
equations are a function of
the generalized coordinates
and time, the chain rule for
partial differentiation must
be used.

1
st

Time Derivative of Constraint Equations





0
,






































t
t
q
q
t
t
q
2
nd

Time Derivative of Constraint Equations









0
2
,
2
2
2
2
2
2
2



















































































t
t
q
q
t
q
t
q
q
q
q
q
t
t
q


Accelerations





























































































2
2
2
1
2
2
2
t
t
q
t
q
q
q
q
q
q
t
q


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Summary of Equations

Newton
-
Raphson

Equations

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 6











0
,
,











t
q
q
t
q
d
k
Constraint Equations











t
q
q
t
q
q
q
i
i
,
,
1
1














Used to determine the position
(values of the generalized
coordinates) at an instant in time.

Velocities of Generalized Coordinates






























t
q
t
q
1
Accelerations of Generalized Coordinates

























































































2
2
2
1
2
2
2
t
t
q
t
q
q
q
q
q
q
t
q


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Jacobian


The
Jacobian

and its inverse is
needed to determine the
position, velocity, and
acceleration of the generalized
coordinates.



Each
i,j

(
row,column
) term in
the
Jacobian

matrix is given by













q
J
Jacobian

matrix

j
i
j
i
q
J




,
i
th

constraint equation

j
th

generalized coordinate

Error messages indicating that
the
Jacobian

is singular are
sometimes encountered when
running multi
-
body dynamic
programs.


This occurs when there is not a
physically realizable solution.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 7

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Piston
-
Crank Constraint Equations

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 8


The fifteen constraint
equations developed
for the piston
-
crank
mechanism in Module
3 are given on the
right.



Note that only the
motion constraint is
an explicit function of
time.

0
)
6
0
)
5
0
)
4



E
cg
E
cg
E
cg
Y
X

0
)
3
0
)
2
0
8
.
156
)
1




A
cg
A
cg
A
cg
Y
X

0
cos
6
.
102
sin
28
)
10
0
sin
6
.
102
cos
28
)
9








C
C
cg
B
B
cg
C
C
cg
B
B
cg
Y
Y
X
X




0
cos
43
sin
3
.
41
)
8
0
sin
43
cos
3
.
41
)
7








D
D
cg
C
C
cg
D
D
cg
C
C
cg
Y
Y
X
X




0
)
12
0
)
11




E
cg
D
cg
E
cg
D
cg
Y
Y
X
X
0
0
1
cos
sin
sin
cos
cos
sin
sin
cos
0
1
1
0
0
1
)
14






































B
B
B
B
T
A
A
A
A
T








0
314
)
15



t
D

0
cos
sin
sin
cos
0
1
1
0
0
1
)
13






































A
CG
A
CG
B
CG
B
CG
T
A
A
A
A
T
Y
X
Y
X




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Piston
-
Crank
Jacobian

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 9




























































































































0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos
cos
sin
sin
0
0
sin
cos
sin
0
0
0
0
0
0
0
0
0
0
0
0
cos
sin
sin
cos
cos
sin
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
sin
6
.
102
1
0
cos
28
1
0
0
0
0
0
0
0
0
0
0
cos
6
.
102
0
1
sin
28
0
1
0
0
0
0
0
0
sin
43
1
0
cos
3
.
41
1
0
0
0
0
0
0
0
0
0
0
cos
43
0
1
sin
3
.
41
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
B
A
B
A
A
B
A
A
A
A
B
cg
A
cg
A
B
cg
A
cg
A
A
B
C
B
D
C
D
C
j
i
Y
Y
X
X
q
J





















The
Jacobian

of the constraint equations is given below. Although there are many
terms, there are a lot of zeros and the derivatives are easily computed.

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Velocities


The velocities of the generalized
coordinates are computed from
the equation





Since the
Jacobian

is known, this
equation can be solved if the array
containing the time derivatives of
the constraint equations is found.


Only the motion constraint,

(15),
is an explicit function of time.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page
10






























t
q
t
q
1





























































314
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
Motion Constraint

Required Array

0
314
)
15



t
D

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Acceleration

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 11

The accelerations can be computed if each term is found.

























































































2
2
2
1
2
2
2
t
t
q
t
q
q
q
q
q
q
t
q




The time
derivative of
the
Jacobian

is zero.

This term is
explained on
the next slide.

Inverse of the
Jacobian






























































314
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t




























































0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
t
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Acceleration Term





q
q
q
q



























This term is evaluated by breaking it down
into a series of operations that are easily
done on a computer.

Step 1) Multiply the
Jacobian

by the
velocities. This creates a
column array.



q
q










Step 2) Take the derivative of each row
with respect to each
generalized coordinate. This
operation is similar to finding
the
Jacobian

and results in a
matrix.




















q
q
q

Step 3) Multiply the matrix by the
velocities. This results in a
column array.





q
q
q
q



























Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 12

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Redundant Constraints


The
Jacobian

matrix is an important
quantity and enables the position,
velocity, and acceleration of the
generalized coordinates to be found.



Application of the methods contained
in this module requires that the
Jacobian

have an inverse.



This requires that the determinant of
the
Jacobian

be non
-
zero or that the
rank be equal to the number of
generalized coordinates.



The rank of a matrix is equal to the
number of independent rows or
columns.



Independent rows or columns can not
be written as a linear combination of
other rows or columns.



If rows or columns of the
Jacobian

are
not independent the
Jacobian

is
singular and the problem does not
have a solution.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 13

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Redundant Constraints: Detection


The Dynamic Simulation environment
within Autodesk Inventor software
assembles the
Jacobian

and
determines its rank as each constraint
is added.


The rank gives the number of
independent constraints.


The difference between the number of
generalized coordinates and the
number of independent constraints is
equal to the degree of mobility.


The difference between the number of
constraints and the number of
independent constraints is equal to
the degree of redundancy.

nq



number of generalized
coordinates

nc



number of constraints

nic



number of independent
constraints


Degree of Mobility

dom
=
nq



ni
c



Degree of Redundancy

dor

=
nc
-
nic

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 14

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Redundant Constraints: Reaction Forces


A redundant constraint occurs when the
motion associated with a DOF is enforced
by too many constraint specifications.


One or
more of the
constraint
specifications can be removed without
affecting the mobility of the system.


The joint reactions can not be
independently determined when
redundant constraints are present.


Although solutions can be obtained they
are based on assumptions by the program
as to which constraints to use.


Different assumptions will yield different
answers.

Joints having friction are
particularly effected by
redundant constraints.


Friction forces are based on
the joint normal forces.


Therefore, the friction forces
are incorrect if the joint
normal forces are incorrect
due to redundant
constraints.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 15

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Redundant Constraint: Example


A simple four bar mechanism will have
redundant constraints if revolute
joints are used at all joints.


The ground link shown in the figure is
fixed.


The revolute joint at 1 prevents the
drive link from rotating about its long
axis and moving normal to the joint
plane.


The revolute joint at 2 prevents the
coupler from rotating about its long
axis and moving normal to the joint
plane.


The revolute joint at 4 prevents the
rocker from rotating about its long axis
and moving normal to the joint plane.

Ground

Drive

Coupler

Rocker

1

2

3

4

A revolute
joint at 3
is redundant
because neither the rocker or
coupler can rotate about their long
axis or move normal to the joint
plane due to the other revolute
joints
. These degrees of freedom
are already restrained.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 16

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Redundant Constraints: Example


A point
-
line joint must be used at joint
3.



A point
-
line joint restricts a point
(the
center
point of the
hole at joint
3
on
the
coupler) to remain on a line
(the
centerline
of the
hole at joint
3
on
the
rocker
).



Redundant joints can be confusing and a
detailed analysis of what each joint is
doing is required to figure out how to
remove them.



An example of how to remove redundant
constraints is provided in the next
module: Module 5.


Ground

Drive

Coupler

Rocker

1

2

3

4

Revolute

Revolute

Revolute

Point
-

Line

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 17

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Module Summary


This module showed how the constraint equations can be used to
find the position, velocity, and acceleration of the generalized
coordinates.


Kinematic relationships were used during the derivation and no
mention of the forces required to impose the motion constraints was
made.


The constraint equations for the piston
-
crank introduced in the
previous module (Module 3) were used to demonstrate the
mathematical steps.


The Newton
-
Raphson

method is generally used to solve the
constraint equations.


The
Jacobian

is a key component of the overall solution process and
the rank of the
Jacobian

is used to detect redundant constraints.

Section 4


Dynamic Simulation

Module

4


Constraint Kinematics

Page 18