Autoplay of the YouTube playlist for all videos in this chapter. This description box will not be updated with information about each video as the videos advance.

## Book Chapter: Chapter 8

Dynamics of Open Chains

This video introduces the Lagrangian approach to finding the dynamic equations of motion of robot and describes the structure of the dynamic equations, including the mass matrix, velocity-product terms (Coriolis and centripetal terms), and potential terms (e.g., gravity).

This video continues our study of the dynamic equations of motion of a robot, focusing on the velocity-product terms, namely, Coriolis terms and centripetal terms.

This video interprets the mass matrix of a robot in terms of how a sphere of joint torques maps to an ellipsoid of joint accelerations and vice-versa, and how a sphere of end-effector wrenches maps to an ellipsoid of end-effector accelerations and vice-versa.

This video introduces the center of mass of a rigid body; its 3×3 symmetric, positive-definite rotational inertia matrix; the principal axes and moments of inertia of an inertia matrix; and the equations governing the rotation of a rigid body.

This video introduces the 6×6 spatial inertia matrix of a rigid body, the Lie bracket of two twists, and the equation of motion governing the dynamics of a rotating and translating rigid body.

This video introduces the recursive Newton-Euler inverse dynamics for an open-chain robot. Forward iterations, from the base of the robot to the end-effector, calculate the configurations, twists, and accelerations of each link. Backward iterations then calculate the wrench applied to each link and the joint forces and torques needed to generate those wrenches.

This video shows how the recursive Newton-Euler dynamics can be used to solve for the forward dynamics of a robot (calculating the joint acceleration given the joint configuration, velocity, and forces/torques) and how the forward dynamics can be used to simulate the motion of a robot.

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.

This video describes the dynamics of robots when they are subject to constraints, such as loop-closure constraints or nonholonomic constraints. Lagrange multipliers, modeling forces against constraints, are introduced, as well as projection methods that eliminate explicit calculation of the Lagrange multipliers.

This video introduces the effect of gearing at the actuators to the Newton-Euler inverse dynamics and the concept of the apparent (or effective) inertia of a motor’s rotor when there is gearing.