#### 8.6. Dynamics in the Task Space

This video introduces task-space (or operational space) dynamics, where the joint-space robot dynamics are expressed in an equivalent form, but replacing the joint forces and torques, joint velocity and acceleration, and the joint-space mass matrix with the end-effector wrench, the end-effector twist and its time derivative, and the end-effector mass matrix, respectively.

Until now, we have been focusing on robot dynamics expressed in the space of joint motions and joint forces and torques. We could equivalently formulate the dynamics in the task space, that is, the space of end-effector motions and end-effector wrenches. We assume that the end-effector twist V equals the Jacobian times theta-dot, where the twist and Jacobian can either be in the space frame or the end-effector frame. If the Jacobian J is invertible, then V-dot equals J theta-double-dot plus J-dot theta-dot, and we can solve the equations for V and V-dot to find theta-dot and theta-double-dot. Plugging these into the joint-space dynamic equation, we get the task-space dynamics: the end-effector wrench is equal to Lambda of theta times V-dot plus eta of theta and V, where Lambda of theta is the robot's mass matrix expressed in the task space and eta of theta and V is the sum of the velocity-product and gravity terms expressed as an end-effector wrench. Each of Lambda and eta is expressed in terms of the joint positions theta, not the end-effector configuration X, since generally there could be more than one robot configuration for a given end-effector configuration.

If the end-effector applies a wrench F_tip, it is simply added to the total wrench.