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.