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.