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
Comments 0
Log in to post a comment