In the previous video, we learned how to take
the joint screw axes S_1 to S_n, defined in
the space frame {s} when the robot is at the
zero configuration, and transform them to
the n columns of the space Jacobian at any
arbitrary joint configuration theta.
In this video, we construct the 6 by n body
Jacobian J_b from the screw axes B_1 to B_n,
expressed in the end-effector frame {b}.
The body Jacobian transforms joint velocities
to the body twist.
To derive the body Jacobian J_b, let's use
the 5R arm from the previous video as an example.
To derive J_b, we need to define the end-effector
frame {b}, but we don't need an {s} frame.
J_b has five columns, one for each joint,
and in this video we will focus on J_b3, the
third column, corresponding to the end-effector
twist when joint 3 moves with unit velocity.
First we set all joint angles equal to zero.
At this configuration, J_b3 is just B_3, the
screw axis of joint 3 expressed in the {b}
frame when the arm is at its zero configuration.
Now we rotate joint 1.
Notice that this rotation of joint 1 does
not change the relationship between joint
3 and the {b} frame, so J_b3 is still equal
to B_3.
Now we rotate joint 2.
Again, the relationship between joint 3 and
the {b} frame is unaffected by joint 2's motion,
so J_b3 is still equal to B_3.
Now we rotate joint 3.
As with joints 1 and 2, J_b3 is unaffected
by joint 3's motion.
Now we rotate joint 4 by theta_4.
This motion changes the configuration of joint
3 relative to the {b} frame, so J_b3 changes.
We define the frame {b-double-prime} to be
the {b} frame before joint 4 is rotated, and
the frame {b-prime} to be the {b} frame after
joint 4 is rotated.
The relationship between the two is given
by T_b-double-prime_b-prime equals e to the
bracket B_4 times theta_4.
We define the {b-double-prime} frame because
the screw axis for joint 3 is just B_3 in
this frame.
Finally, we rotate joint 5 by theta_5, giving
us the final end-effector frame {b}, obtained
by rotating the frame {b-prime} about the
joint 5 screw axis by theta_5.
To find the {b} frame relative to the {b-double-prime}
frame, we postmultiply T-b-double-prime-b-prime
by the body-frame transformation corresponding
to rotation about the body screw axis B_5,
giving us the equation shown here.
What we really want, though, is the configuration
of the {b-double-prime} frame relative to
the {b} frame, so we reverse the subscripts,
which is the same as taking the inverse of
the transformation matrix.
Making use of the fact that the inverse of
A times B, where A and B are invertible matrices,
is just B-inverse times A-inverse, we can
rewrite T_b_b-double-prime in this form.
Since the screw axis of joint 3 is just B_3
in the {b-double-prime} frame, to find J_b3
we just need to use our rule for changing
the frame of reference of a twist.
The final expression for the J_b3 column depends
on the screw axis for joint 3 as well as the
joint angles and screw axes for joints 4 and
5.
The same reasoning applies for any joint,
so we can generalize to this definition of
the body Jacobian J_b.
The last column of the body Jacobian is just
the screw axis B_n when the robot is at its
zero configuration.
It does not depend on the joint positions,
because no joint is between joint n and the
{b} frame.
Any other column i of the body Jacobian is
given by the screw axis B_i premultiplied
by the transformation that expresses the screw
axis in the {b} frame for arbitrary joint
positions.
You can see that J_b1 depends on the positions
of joints 2 through n, J_b2 depends on the
positions of joints 3 through n, etcetera.
You can also see that the body Jacobian is
independent of the choice of the space frame
{s}.
Since each column of a Jacobian is a twist,
we can use our rule for representing a twist
in a different frame to translate between
the space Jacobian J_s and the body Jacobian
J_b.
J_b is obtained from J_s by the matrix adjoint
of T_bs, and J_s is obtained from J_b by the
matrix adjoint of T_sb.
In the next video we will see that the Jacobian
is used not only to convert joint velocities
to end-effector twists, but also to understand
how end-effector wrenches are related to torques
and forces at the joints.
