We've seen two major uses of Jacobian matrices:
converting a set of joint velocities theta-dot
to an end-effector twist V and converting
an end-effector wrench F to a set of joint
forces and torques tau.
The twists and wrenches can be expressed in
the space frame {s} or the end-effector frame
{b}.
The Jacobian is a 6 by n matrix, where n is
the number of joints.
This means that the rank of the Jacobian can
be no greater than the minimum of 6 and n.
We say that the Jacobian is full rank at a
configuration theta if the rank is equal to
the minimum of 6 and n.
We say that the Jacobian is singular at a
configuration theta-star if the rank of the
Jacobian at theta-star is less than the maximum
rank the Jacobian can achieve at some configuration.
At a singular configuration, the robot loses
the ability to move in one or more directions.
We can also categorize Jacobians according
to the number of joints n.
If n is less than 6, the Jacobian is "tall,"
meaning it has more rows than columns.
The set of reachable configurations for the
end-effector is less than 6-dimensional, so
we call such robots kinematically deficient.
This does not mean the robot is not useful,
it just means it is not capable of general
motion at the end-effector.
An example robot is the 4-joint RRRP robot
shown here, which has a 6-by-4 Jacobian.
If n equals 6, the Jacobian is a 6-by-6 square
matrix, as for this 6R robot.
Such robots are often called general purpose
manipulators, because they are capable of
general 6-dimensional rigid-body motion at
their end-effectors.
If n is greater than 6, the Jacobian is "fat,"
meaning it has more columns than rows.
An example of such a robot is the 7R robot
pictured here, which has a 6-by-7 Jacobian.
Such robots are called redundant, because
they can achieve the same end-effector twist
with different joint velocities.
This capability can be useful in a number
of circumstances, allowing internal motion
of the arm that is not visible in motion at
the end-effector.
Your own arm has a redundancy like this: keeping
your hand stationary at a fixed configuration
in space, you can still move your arm internally.
It can be difficult to visualize 6-dimensional
motion of a robot, so to illustrate the shape
and rank properties of the Jacobian, we will
use a simple planar example.
In this example, the end-effector velocity
v_tip and force f_tip are 2-vectors, and the
Jacobian is 2 by n, where n is the number
of joints.
For the 3_R arm shown here, the number of
joints n is 3, the robot is redundant, and
its 2-by-3 Jacobian matrix is full rank, meaning
its rank is 2, at the configuration shown.
Since the Jacobian is rank 2, the robot can
generate any linear velocity at its end-effector,
and any force applied to the end-effector
must be actively resisted by at least one
of the joints.
Using the fact that v_tip equals J theta-dot,
we can always calculate v_tip given the joint
velocities theta-dot.
This figure shows the components of the endpoint
velocity caused by the individual joint velocities,
and we can sum them to get the end-effector
velocity v_tip.
Since the rank of J is 2, any v_tip can be
created by the joints.
You could imagine asking the inverse question,
given v_tip, what is theta-dot?
The answer to this question is not as straightforward,
however, because in general, as in this case,
the inverse of J does not exist, either because
J is not square or because it is singular.
Because this 3R robot is redundant, it turns
out that for any v_tip, there is a full one-dimensional
set of solutions of joint velocities that
achieves v_tip.
This inverse question will be addressed in
more detail in Chapter 6.
Moving on to forces, using the fact that tau
equals J-transpose times f_tip, we can always
find the joint forces and torques tau that
correspond to the end-effector force f_tip.
For the f_tip shown here, we can graphically
calculate tau_1, the torque about the first
joint, using the relationship tau_1 equals
minus r_1 times the magnitude of f_tip, where
r_1 is the vector perpendicular to f_tip from
the joint to the line of force.
Similarly, we can calculate the torques at
joints 2 and 3.
Each joint has to individually support the
endpoint force f_tip.
You could also imagine asking the inverse
question given tau, what is the endpoint force
f_tip, but this question is not as straightforward,
because the inverse of J-transpose may not
exist.
For the 3R arm, for most random choices of
joint torques, the arm will have internal
motion, and will not simply statically resist
an externally applied force minus f_tip.
Moving on, let's consider the redundant 3_R
arm when it is fully stretched out.
The rank of the 2-by-3 Jacobian drops to 1,
meaning the arm is at a singular configuration.
Rotation at joint 1, 2, and 3 produces only
vertical velocity at the end-effector; no
horizontal velocity can be achieved.
Also because of the singularity, a horizontal
force applied at the end-effector is resisted
by the mechanical structure of the robot;
no joint torques have to be applied.
This 2_R robot has a square Jacobian that
has rank equal to 2 at the configuration shown.
This means that any tip velocity is possible
and any force applied to the tip must be actively
resisted by the joints.
In this picture, the 2_R robot is at a singular
configuration, where only vertical velocities
are possible and horizontal forces can be
passively resisted by the mechanical structure
of the robot.
Finally, we have a 1_R robot.
The Jacobian is 2-by-1 and is full rank, meaning
the rank is equal to 1, at any configuration.
This robot is kinematically deficient for
the task of achieving arbitrary linear velocities
at the tip, as it can only achieve linear
velocities perpendicular to the link.
Any horizontal force is passively resisted
by the joint, while any vertical force must
be actively resisted by the joint torque.
In the next and final video of Chapter 5,
we will characterize how close a robot is
to being singular using the manipulability
ellipsoid touched on in the first video of
this chapter.
