Manipulator Jacobian
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Thanks Mark W. Spong et al for their great work of Robot Modeling and Control.
Thanks Zexiang Li et al for their great work of A Mathematical Introduction to Robotic Manipulation.
Thanks John J. Craig for his great work of Introduction to Robotics - Mechanics and Control - 3rd Edition.
We assume that all the frames mentioned below are right-handed.
- End-effector Velocity and Manipulator Jacobian
- Geometric Jacobian
- Angular Velocity
- Linear Velocity
- Translational Joints
- Rotational Joints
- Jacobian Matrix
- Changing a Jacobians Frame of Reference
- Analytical Jacobian
- Singularities
- Decoupling of Singularities
End-effector Velocity and Manipulator Jacobian
Let
It is also possible to define a body manipulator Jacobian,
Let
The relationship between joint velocity and end-effector velocity can be used to move a robot manipulator from one end-effector configuration to another without calculating the inverse kinematics for the manipulator.
Ex 1: SCARA robot
By inspection,
The next section shows an alternative expression of Manipulator Jacobian.
Geometric Jacobian
Consider a n-link manipulator whose joint variables are
Define
We would like to formulate the following equations:
where
Angular Velocity
We can determine the angular velocity of the end-effector relative to the base by expressing the angular velocity contributed by each joint in the orientation of the base frame and then summing these.
If joint
where
The angular velocity of the end-effector w.r.t. the base
When joint
All the
Linear Velocity
By the chain rule:
It is straightforward to see that the
Let us discuss two different types of joints separately.
Translational Joints
Rotational Joints
Jacobian Matrix
To calculate the Jacobian matrix, what we need is no more than the 3rd and 4th column of
Ex: Consider the two-link planar manipulator
Since both joints are revolute the Jacobian matrix, which in this case is
Changing a Jacobian’s Frame of Reference
Given a Jacobian written in frame
Analytical Jacobian
Analytical Jacobian,
Let
stand for the configuration of the end-effector.
We would like to find an equation of the following form:
If
where the
Then
i.e.
Singularities
A
For
It is straightforward to see that
It is important to identify the singularities, since:
- Singularities represent configurations from which certain directions of motion may be unattainable.
- At singularities, bounded end-effector velocities may correspond to unbounded joint velocities.
- At singularities, bounded end-effector forces and torques may correspond to unbounded joint torques.
- 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.
- Singularities correspond to points in the manipulator workspace that may be unreachable under small perturbations of the link parameters, such as length, offset, etc.
- 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.
All manipulators have singularities at the boundary of their workspace, and most have lots of singularities inside their workspace. We can class singularities into two categories:
- Workspace-boundary singularities occur when the manipulator is fully stretched out or folded back on itself in such a way that the end-effector is at or very near the boundary of the workspace.
- Workspace-inside singularities occur away from the boundary of workspace, and are usually generated from two or more collinear joint axes.
There are 3 common singularities for six degree of freedom manipulators:
- Two collinear revolute joints: This type of singularity is common in spherical wrist assemblies that are composed of three mutually orthogonal revolute joints whose axes intersect at a point. In this configuration, rotation about the axis normal to the plane defined by the first and second joints is not possible.
- Three parallel coplanar revolute joint axes: The Jacobian for a six degree of freedom manipulator is singular if there exist three revolute joints which satisfy the following conditions:(see also the figure below)
- The axes are parallel:
ωi=±ωj fori,j=1,2,3 . - The axes are coplanar : there exists a plane with unit normal
n such thatnTωi=0 andnT(qi−qj)=0,i,j=1,2,3 .
- The axes are parallel:
- Four intersecting revolute joint axes: The Jacobian for a six degree of freedom manipulator is singular if there exist four revolute joint axes that intersect at a point
q :ωi×(qi−1)=0,i=1,…,4 .
It is also possible for a manipulator to exhibit different types of singularities at a single configuration. In this case, depending on the number and type of the singularities, the manipulator may lose the ability to move in several different directions at once.
Decoupling of Singularities
We can decouple the determination of singular configurations, for those manipulators with spherical wrists, into two simpler problems. The first is to determine so-called arm singularities, that is, singularities resulting from motion of the arm, which consists of the first three or more links, while the second is to determine the wrist singularities resulting from motion of the spherical wrist.
Suppose that
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- Jacobian。。。
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