A robot configuration is either singular or
it's not.
But even if a configuration is nonsingular,
it still may be close to being singular.
The manipulability ellipsoid we saw in the
first video of this chapter is one way to
visualize how close a robot is to being singular.
For this 2_R robot, a circle of velocities
in the joint space maps through the Jacobian
to an ellipse of velocities at the tip of
the robot depending on the robot's configuration.
As the second joint angle approaches zero,
the ellipse squashes in the direction it is
difficult to move and stretches in the orthogonal
direction, until, at the singularity, the
ellipse collapses to a line segment.
As we will see shortly, we can assign a measure
of just how close the robot is to being singular
according to how close the ellipse is to collapsing.
To derive a general expression for the end-effector
velocity ellipsoid, let's begin by assuming
the end-effector's velocity is represented
as the m-dimensional vector v_tip.
The robot has n joints, so the Jacobian is
an m by n matrix.
A sphere of joint velocities, like the circle
shown here, is defined by the equation theta-dot
transpose times theta-dot equals 1.
If we assume the Jacobian is invertible, which
is not strictly necessary, then we can rewrite
the equation as shown here.
Rearranging, we get this, and rearranging
once more, we get this.
We summarize this equation as v_tip-transpose
times A-inverse times v_tip equals 1, where
A is the m-by-m matrix J times J-transpose.
The A matrix is both symmetric and positive
definite, and so is its inverse.
Now assume we take this same equation but
replace v_tip by a generic vector x.
The eigenvalues of the matrix A are called
lambda_1 to lambda_m, and the corresponding
eigenvectors are v_1 to v_m.
It is well known that the quadratic equation
x-transpose times A-inverse times x equals
1 defines an ellipsoid of x values that satisfy
the equation.
In general, this ellipsoid is an m-minus-1-dimensional
surface in the m-dimensional space of x, but
this figure shows the case where x is a 3-vector.
The principal axes of the ellipsoid are aligned
with the eigenvectors of A and the half-lengths
of the ellipsoid along the principal axes
are the square roots of the eigenvalues.
This geometric interpretation holds for any
symmetric positive definite matrix A, but
if we choose A equal to J times J-transpose,
then the x-vector can be interpreted as v_tip,
and the ellipsoid is called the manipulability
ellipsoid resulting from the unit sphere of
joint velocities.
If instead we set A equal to the inverse of
J times J-transpose, then the x-vector can
be interpreted as the end-effector forces
f_tip, and the ellipsoid is called the force
ellipsoid resulting from a unit sphere of
joint forces and torques.
This figure shows the manipulability ellipsoid
and the force ellipsoid for a 2R robot at
a particular configuration.
Since the matrix defining the manipulability
ellipsoid is just the inverse of the matrix
defining the force ellipsoid, the two ellipsoids
have the same principal axes, and the lengths
of the principal semi-axes are just the reciprocals
of each other.
In other words, only small forces can be applied
in directions where large velocities can be
attained, and only small velocities are possible
in directions where large forces can be applied.
Now that we can visualize the end-effector
motion capabilities as a manipulability ellipsoid,
we can assign a single number representing
how close the robot is to being singular.
These numbers are called manipulability measures.
The first manipulability measure is the ratio
of the longest axis to the shortest axis of
the ellipsoid.
This measure is lower-bounded by 1, and if
it is equal to 1, we say that the manipulability
ellipsoid is isotropic; it is equally easy
to move in any direction.
On the other hand, as the robot approaches
a singularity, this number grows large.
The second measure is just the square of the
first measure, often called the condition
number of the matrix A.
A final measure is the square root of the
product of the eigenvalues of A, which is
proportional to the volume of the manipulability
ellipsoid.
If the manipulability ellipsoid volume becomes
large, then the force ellipsoid volume becomes
small, and vice-versa.
Finally, consider the case that the Jacobian
corresponds to the body Jacobian derived in
this chapter.
The 6-by-n body Jacobian can be split into
the 3-by-n angular velocity Jacobian J-b-omega
and the 3-by-n linear velocity Jacobian J-b-v.
This separation into linear and angular components
is useful, because the units of the angular
velocity and linear velocity are different.
Then for any configuration of the robot, J_b-omega
can be used to create angular velocity manipulability
ellipsoids and angular moment ellipsoids and
J_bv can be used to create linear velocity
manipulability ellipsoids and linear force
ellipsoids.
So, this concludes Chapter 5.
You should now have a solid understanding
of how to derive and interpret the Jacobian,
a fundamental object in robotics that is heavily
used in many robot motion planners and controllers.
In Chapter 6, we study another key issue in
robot motion planning and control: inverse
kinematics, or, how to find joint positions
that achieve a desired end-effector configuration.
