Review: Differential Kinematics

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13 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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Review:
Differential Kinematics

Find the relationship between the joint velocities
and the end
-
effector linear and angular velocities.

Linear velocity

Angular velocity

i
i
i
d
q

for a revolute joint

for a prismatic joint

Review:
Differential Kinematics

Approach 1

q
q
p
J
P

)
(

Review:
Differential Kinematics

Approach 2

Prismatic joint

Revolute joint

n
On
Pn
i
Oi
Pi
O
P
q
J
J
q
J
J
q
J
J
v

1
1
1
The contribution of single joint i to
the end
-
effector angular velocity

The contribution of single joint i to
the end
-
effector linear velocity

Review:
Differential Kinematics

Approach 3

Review:
Differential Kinematics

Approach 3

Kinematic Singularities

The Jacobian is, in general, a function of the
configuration q; those configurations at which J is
rank
-
deficient are termed
Kinematic singularities
.

Reasons to Find Singularities

Singularities represent configurations at which
mobility of the structure is reduced

Infinite solutions to the inverse kinematics problem
may exist

In the neighborhood of a singularity, small
velocities in the operational space may cause large
velocities in the joint space

Problems near Singular Positions

The robot is physically limited from unusually high joint
velocities by motor power constraints, etc. So the robot
will be unable to track this joint velocity trajectory
exactly, resulting in some perturbation to the
commanded cartesian velocity trajectory

The high accelerations that come from approaching too
close to a singularity have caused the destruction of
many robot gears and shafts over the years.

Classification of Singularities

Boundary singularities that occur when the
manipulator is either outstretched or retracted.

Not true drawback

Internal singularities that occur inside the
reachable workspace

Can cause serious problems

Example 3.2: Two
-

Consider only planar components of linear velocity

Consider determinant of J

Conditions for singularity

Example 3.2: Two
-

Conditions for sigularity

Jacobian when theta2=0

1
2
1
2
1
1
2
1
2
1
)
(
)
(
c
a
c
a
a
s
a
s
a
a
J

Computation of internal singularity via the
Jacobian determinant

Decoupling of singularity computation in the
case of spherical wrist

Wrist singularity

Arm singularity

Singularity Decoupling

Singularity Decoupling

Wrist Singularity

Z3, z4 and z5 are linearly dependent

orthogonal to z4 and z3

Singularity Decoupling

Elbow Singularity

Similar to two
-

The elbow is outstretched or retracted

Singularity Decoupling

Arm Singularity

The whole z0 axis describes a continuum
of singular configurations

0
0
0
23
3
2
2
y
x
p
p
c
a
c
a
Singularity Decoupling

Arm Singularity

A rotation of theta1 does not cause
any translation of the wrist position

The first column of J
P1
=0

Infinite solution

Cannot move along the z1 direction

The last two columns of J
P1
are
orthogonal to z1

Well identified in operational space;

Can be suitably avoided in the path
planning stage

Differential Kinematics Inversion

Inverse kinematics problem:

there is no general purpose technique

Multiple solutions may exist

Infinite solutions may exist

There might be no admissible solutions

Numerical solution technique

in general do not allow computation of all admissible
solutions

Differential Kinematics Inversion

Suppose that a motion trajectory is assigned to
the end effector in terms of v and the initial
conditions on position and orientations

The aim is to determine a feasible joint trajectory
(q(t), q’(t)) that reproduces the given trajectory

Should inverse kinematics problems be solved?

Differential Kinematics Inversion

Solution procedure:

If J is not square? (redundant)

If J is singular?

If J is near singularity?

Analytical Jacobian

The geometric Jacobian is computed by
following a geometric technique

Question: if the end effector position and
orientation are specified in terms of minimal
representation, is it possible to compute
Jacobian via differentiation of the direct
kinematics function?

Analytical Jacobian

Analytical technique

Analytical Jacobian

Analytical Jacobian

For the Euler angles ZYZ

Analytical Jacobian

From a physical viewpoint, the meaning of
ώ

is
more intuitive than that of
φ

On the other hand, while the integral of
φ
’ over
time gives
φ
, the integral of
ώ

clear physical interpretation

Example 3.3

Statics

Determine the relationship between the
generalized forces applied to the end
-
effector

and the
generalized forces applied to the
joints

-

forces for prismatic joints, torques for
revolute joints
-

with the manipulator at an
equilibrium configuration
.

X
0

Y
0

x
0

y
0

0

q
1

Y
1

X
1

0

x
2

a
1

q
2

R

a
2

y
2

f
x

f
y

Let
τ

denote the (n
×
1) vector of joint torques
and
γ
(r
×
1) vector of end effector forces
(exerted on the environment) where r is the
dimension of the operational space of interest

Statics

)
(
q
J
T

X
0

Y
0

x
0

y
0

0

q
1

Y
1

X
1

0

x
2

a
1

q
2

R

a
2

y
2

f
x

f
y

Manipulability Ellipsoids

Velocity manipulability ellipsoid

Capability of a manipulator to arbitrarily change the
end effector position and orientation

Manipulability Ellipsoids

Velocity manipulability ellipsoid

Manipulability measure: distance of the manipulator
from singular configurations

Example 3.6

Manipulability Ellipsoids

Force manipulability ellipsoid

Manipulability Ellipsoids

Manipulability ellipsoid can be used to analyze
compatibility of a structure to execute a task
assigned along a direction

Manipulability Ellipsoids

Fine control of the vertical force

Fine control of the horizontal velocity

Manipulability Ellipsoids