Here you can see a 6R robot with a frame {b}
at the hand.
Imagine that the joints are moving according
to a trajectory theta of t.
The changing joint angles theta as a function
of time move the hand along the path shown
in yellow.
The hand is moving in free space, so it is
applying no forces to the environment.
In Chapter 8, when we study the inverse dynamics
of a robot, we will learn how the trajectory
theta of t can be turned into the torques
required to move the robot along the trajectory.
We call these torques tau-motion of t.
Now assume we choose a particular time instant
t, and let tau-motion be the joint torques
at this instant.
Now assume that someone applies a wrench to
the hand at this instant.
Perhaps someone grabbed the hand of the robot.
We will call this wrench minus F_b, consisting
of three angular moments and three linear
forces expressed in the {b} frame.
If we want the robot to continue to track
the planned trajectory, despite this disturbance
wrench, the robot's motors must create a wrench
F_b to balance the disturbance wrench.
Therefore, the joint torques should be tau-motion
plus tau, where we need to know how Fb relates
to tau.
To find this relationship, recall from physics
that force times velocity is power.
In the {b} frame, the wrench F_b created by
the motors multiplies the twist V_b to get
the mechanical power produced or consumed
at the hand.
This power must be coming from the motors,
and we know that the power produced or consumed
by the motors is the joint torques dotted
with the joint velocities.
If we plug in the identity J_b theta-dot equals
V_b, and recognize that the equality must
hold at all theta-dot, we get this equation,
and getting rid of the transposes we get the
relationship we were looking for, tau equals
J_b-transpose times F_b.
The exact same derivation holds for wrenches
and Jacobians expressed in the space frame
{s}, so we can generalize to the following
main result of this video:
To resist a wrench minus F applied to the
end-effector at a configuration theta, the
joint torques and forces tau must be J of
theta transposed times F.
This result holds no matter what frame the
Jacobian and wrench are expressed in.
This relationship can be useful in force control
of a robot: if we want the end-effector to
apply a wrench F to the environment, we use
this formula to calculate the joint forces
and torques tau.
In the next video we will consider the implications
of non-square and singular Jacobian matrices.
