## Book Chapter: Chapter 8

Dynamics of Open Chains

#### Chapter 8 Autoplay

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.

#### 8.1. Lagrangian Formulation of Dynamics (Part 1 of 2)

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).

#### 8.1. Lagrangian Formulation of Dynamics (Part 2 of 2)

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.

#### 8.1.3. Understanding the Mass Matrix

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.

#### 8.2. Dynamics of a Single Rigid Body (Part 1 of 2)

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.

#### 8.2. Dynamics of a Single Rigid Body (Part 2 of 2)

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.

#### 8.3. Newton-Euler Inverse Dynamics

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.

#### 8.5. Forward Dynamics of Open Chains

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.

#### 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.

#### 8.7. Constrained Dynamics

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.

#### 8.9. Actuation, Gearing, and Friction

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.