A robot hand is holding this apple in gravity,
and the robot is equipped with a force-torque
sensor at its wrist.
It measures forces and torques in the frame
{f}.
If we know the mass of the apple, the direction
of gravity, and the location of the apple
in the hand, what are the forces and torques
measured by the sensor?
In this final video of Chapter 3 we will develop
the representations and transformations needed
to answer this question.
Here we see two frames, {s} and {b}.
A line of force f_b acts at the point r_b,
both represented in the {b} frame.
f_b is a 3-vector specifying the magnitude
of the force in 3 directions.
From physics we know that this force induces
a 3-vector torque, or moment, about the frame
{b} equal to r_b cross f_b.
We can package the moment and the force together
in a single 6-vector called the wrench, just
as we packaged the angular and linear velocity
of a rigid body into a twist.
Since we know the transform T_sb, we should
be able to represent this same wrench in the
{s} frame.
To derive the relationship between the wrenches
F_b and F_s, keep in mind this fact: the dot
product of a twist and a wrench is power.
Power does not depend on a coordinate frame,
and therefore the power must be the same whether
the wrench and twist are represented in the
{b} frame or in the {s} frame.
Using our rule to change the frame of representation
of a twist, we can express V_b in terms of
T_sb and V_s.
Since the transpose of the product of a matrix
and a vector is equal to the product of the
vector transposed and the matrix transposed,
we can rewrite the equation as shown here.
Finally, this equation holds for all twists
Vs, so it simplifies to the relationship we
are looking for, changing the coordinate frame
of the wrench from the {b} frame to the {s}
frame.
Returning to our apple example, we can define
a frame {a} at the center of mass of the apple.
In this frame, the force due to gravity is
mg in the minus y direction, and the moment
is zero, since the force vector passes through
the origin of the {a} frame.
To transform to the force sensor frame, we
use T_af, the configuration of the force sensor
frame relative to the apple frame, and we
see that the wrench F_f has a moment of negative
m_gL about the z-axis of the {f} frame.
So, this concludes Chapter 3.
The material in this chapter is fundamental
to representing motion and forces in three-dimensional
space, for robots and other types of mechanical
systems.
We're now equipped with the tools we need
to study the kinematics and statics of robots,
which begins in Chapter 4.
