Transcript Singularity Decoupling

Review: Differential Kinematics
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Find the relationship between the joint velocities
and the end-effector linear and angular velocities.
Linear velocity
Angular velocity
i for a revolute joint
qi   
 di for a prismatic joint
Review: Differential Kinematics
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Approach 1
p (q )
JP 
q

Review: Differential Kinematics
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Approach 2
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Prismatic joint
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Revolute joint
Review: Differential Kinematics
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Approach 3
The contribution of single joint i to
the end-effector linear velocity
 J P1 
 J Pi 
 J Pn 
v    q1      q i      q n
 J O1 
 J Oi 
 J On 
The contribution of single joint i to
the end-effector angular velocity
Review: Differential Kinematics
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Approach 3
Kinematic Singularities
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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
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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
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Boundary singularities that occur when the
manipulator is either outstretched or retracted.
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Not true drawback
Internal singularities that occur inside the
reachable workspace
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Can cause serious problems
Example 3.2: Two-link Planar Arm
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Consider only planar components of linear velocity
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Consider determinant of J
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Conditions for singularity
Example 3.2: Two-link Planar Arm
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Conditions for sigularity
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Jacobian when theta2=0
 (a1  a 2 ) s1
J 
 (a1  a 2 )c1
 a 2 s1 

a 2 c1 
Singularity Decoupling
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Computation of internal singularity via the
Jacobian determinant
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Decoupling of singularity computation in the
case of spherical wrist
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Wrist singularity
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Arm singularity
Singularity Decoupling
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Wrist Singularity
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Z3, z4 and z5 are linearly dependent
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Cannot rotate about the axis
orthogonal to z4 and z3
Singularity Decoupling
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Elbow Singularity
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Similar to two-link planar arm
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The elbow is outstretched or retracted
Singularity Decoupling
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Arm Singularity
 px  0
a2c2  a3c23   0  
 py  0
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The whole z0 axis describes a continuum
of singular configurations
Singularity Decoupling
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Arm Singularity
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A rotation of theta1 does not cause
any translation of the wrist position
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Cannot move along the z1 direction
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The first column of JP1=0
Infinite solution
The last two columns of JP1 are
orthogonal to z1
Well identified in operational space;
Can be suitably avoided in the path
planning stage
Differential Kinematics Inversion
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Inverse kinematics problem:
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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
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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
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Solution procedure:
If J is not square? (redundant)
If J is singular?
If J is near singularity?
Analytical Jacobian
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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
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Analytical technique
Analytical Jacobian
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Analytical Jacobian
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For the Euler angles ZYZ
Analytical Jacobian
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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
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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.
fy
Y0
R
y0
x2
y2
a2
Y1
fx
q2
X1
a1
q1
0
0
x0
X0
Statics
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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
  J (q)
T
fy
Y0
R
y0
x2
y2
a2
Y1
fx
q2
X1
a1
q1
0
0
x0
X0
Manipulability Ellipsoids
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Velocity manipulability ellipsoid
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Capability of a manipulator to arbitrarily change the
end effector position and orientation
Manipulability Ellipsoids
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Velocity manipulability ellipsoid
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Manipulability measure: distance of the manipulator
from singular configurations
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Example 3.6
Manipulability Ellipsoids
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Force manipulability ellipsoid
Manipulability Ellipsoids
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Manipulability ellipsoid can be used to analyze
compatibility of a structure to execute a task
assigned along a direction
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Actuation task of velocity (force)
Control task of velocity (force)
Manipulability Ellipsoids
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Control task of velocity (force)
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Fine control of the vertical force
Fine control of the horizontal velocity
Manipulability Ellipsoids
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Actuation task of velocity (force)
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Actuate a large vertical force (to
sustain the weight)
Actuate a large horizontal velocity