Review: Differential Kinematics

copygrouperMechanics

Nov 13, 2013 (3 years and 9 months ago)

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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
-
link Planar Arm


Consider only planar components of linear velocity





Consider determinant of J




Conditions for singularity


Example 3.2: Two
-
link Planar Arm


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







Cannot rotate about the axis


orthogonal to z4 and z3

Singularity Decoupling


Elbow Singularity


Similar to two
-
link planar arm







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

does not admit a
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


Actuation task of velocity (force)


Control task of velocity (force)

Manipulability Ellipsoids


Control task of velocity (force)


Fine control of the vertical force


Fine control of the horizontal velocity

Manipulability Ellipsoids


Actuation task of velocity (force)


Actuate a large vertical force (to
sustain the weight)


Actuate a large horizontal velocity