ROBOTICS
Singularities and
Inverse Velocity & Jacobian
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INVERSE VELOCITIES

As we know that if joint velocities are given then how linear velocities of end
effector can be found.
𝐉=

𝐽
𝐽
=
𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠
𝜁=
𝑉
𝜔
𝑉
𝑉
𝑉
=
𝜔
𝜔
𝜔
Similarly, if end effector velocities are given how we find joint velocities.
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1
INVERSE VELOCITIES



It is a need to calculate joint velocities continuously in order to ensure that the robot’s
hand maintains a desired velocity.
For this, there is a need to calculate the inverse of the Jacobian and use it in the
following equation:
This means that knowing the inverse of the Jacobian, we can calculate how fast each
joint must move, so that the robot’s hand will yield a desired differential motion or
velocity.
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INVERSE VELOCITIES





With the robot moving and its configuration changing, the actual magnitudes of all
elements of the Jacobian of a robot change continuously.
As a result, although the symbolic equations describing the Jacobian remain the
same, their numerical values change.
Consequently, it is necessary to calculate the Jacobian numerical values continuously.
This means, to calculate enough joint velocities per second to have accurate
velocities of hand, the process must be very efficient and quick; otherwise, the
motion will be inaccurate and useless.
Inverting the Jacobian is very difficult, computationally intensive, and time
consuming, considering that the Jacobian may be as large as 6 × 6.
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2
SINGULARITIES





If we try to control a manipulator in Cartesian space,
sometimes it runs into difficulties since the inverse mapping
from Cartesian space to joint space can sometimes become a
problem.
These problem positions of the robot are referred to as
singularities or degeneracies.
At a singularity, the mobility of a manipulator is reduced.
Usually, arbitrary motion of the manipulator in a Cartesian
direction is lost.
This is referred to as “Losing a DOF”.
A robot singularity is a configuration in which the robot endeffector becomes blocked in certain directions.
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SINGULARITIES



Boundary Singularities; (also known as workspace
singularities) are a common type of singularity.
They are usually caused by a full extension of a joint, and
asking the manipulator to move beyond where it can be
positioned.
Typically, this is trying to reach out of the workspace at the
extreme extent of the workspace.
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3
SINGULARITIES




Internal Singularities: (also known as joint space
singularities).
They are generally caused by an alignment of the robots
axes in space.
For example, if 2 axes become aligned in space, rotation of
one can be cancelled by counter rotation of the other,
leaving the actual joint location indeterminate.
Also, certain kinematic alignments specific to each
manipulator can cause these.
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SINGULARITIES
Identifying manipulator singularities is important for several reasons.
1. Singularities represent configurations from which certain directions of motion may be
unattainable.
2. At singularities, bounded end-effector velocities may correspond to unbounded joint
velocities.
3. At singularities, bounded end-effector forces and torques may correspond to unbounded
joint torques.
4. Singularities usually (but not always) correspond to points on the boundary of the
manipulator workspace, that is, to points of maximum reach of the manipulator.
5. Singularities correspond to points in the manipulator workspace that may be
unreachable under small perturbations of the link parameters, such as length, offset, etc.
6. Near singularities there will not exist a unique solution to the inverse kinematics problem.
In such cases there may be no solution or there may be infinitely many solutions.
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4
SINGULARITIES
A singularity occurs whenever the determinant of the Jacobian is 0.
If the determinant of Jacobian is 0 then we cannot reverse/ reverse, it.
The Jacobian is singular when its determinant is equal to 0.
The associated Jacobian matrix is said to be singular.
To find when this occurs, we set det (J) = 0
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SINGULARITIES
Problem:
Finding Singularities of the 2-Link Manipulator, where Joint 1 is a
revolute joint and joint 2 is a prismatic joint. Assume followings:
A singularity occurs when the joint velocity in joint space becomes
infinite to maintain Cartesian velocity.
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SINGULARITIES
Problem: Find Singularities and Velocities:
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SINGULARITIES
Problem:
Finding Singularities of the 2-Link Manipulator, where both Joints
are revolute joints. Assume followings:
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SINGULARITIES
Problem:
Finding Singularities of the 3-Link Manipulator, where two Joints
are revolute and one joint is prismatic. Assume followings:
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MANIPULABILITY
In kinematic analysis of robots, the manipulability index is a good measure for identifying manipulation in a
workspace.
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TORQUE (τ) and FORCE (F)
The Jacobian Matrix can also be used to relate forces F applied at the end effector frame to the induced joints
torques.
𝐹
𝐹
𝜏=𝐽 𝐹
𝐹=
𝐹
𝑛
𝜏
𝜏= 𝜏
𝜏
𝑛
𝑛
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BOOKS
1- Introduction to Robotics: Analysis, Control, Applications
By Saeed Benjamin Niku
Latest edition
2- Introduction to Robotics Mechanics and Control
By John J. Craig
3rd Edition
3- Theory of Applied Robotics
Kinematics, Dynamics and Control,
By Reza N. Jazar
2nd edition
4- Robotics, Vision and Control,
Fundamental Algorithms in Matlab
By Peter Corke
2nd Edition
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