 ## Second Order Dynamic Systems

If your knowledge of dynamics is a bit rusty then let’s quickly revise the basics of second-order systems and the Laplace operator. Not rusty? Then go straight to the next section.

## Electric Motors

The most common type of actuator is a rotary electric motor so let’s look at the basic principles.

## Actuators

Actuators are the components that actually move the robot’s joint. So let’s look at a few different actuation technologies that are used in robots.

## Robot Joint Control Architecture

A robot joint is a mechatronic system comprising motors, sensors, electronics and embedded computing that implements a feedback control system.

## Introduction to Robot Joint Control

We will learn about how we make the the robot joints move to the angles or positions that are required in order to achieve the desired end-effector motion. This is the job of the robot’s joint controller and in this lecture we will learn how this works. This journey will take us in to the realms of control theory.

## Summary of Velocity Kinematics in 3D

This video gives summary of Velocity kinematics in 3D.

## Jacobian and Number of Robot Joints

A robot manipulator may have any number of joints. We look at how the shape of the Jacobian matrix changes depending on the number of joints of the robot.

## The Analytic Jacobian

Now we introduce a variant of the Jacobian matrix that can relate our angular velocity vector back to our rates of change of the roll, pitch and yaw angles.

## Mapping 3D Spatial Velocity Between Coordinate Frames

We previously learnt how to derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame. Now we extend that to the 3D case.

## Velocity Ellipsoid in 3D and Manipulability

The Jacobian matrix provides powerful diagnostics about how well the robot’s configuration is suited to the task. Wrist singularities can be easily detected and the concept of a velocity ellipse is extended to a 3-dimensional velocity ellipsoid.

## Inverting the Jacobian Matrix

As we did for the simple planar robots we can invert the Jacobian and perform resolved-rate motion control.

## Velocity of 6-Joint Robot Arm – Rotation

We resume our analysis of the 6-link robot Jacobian and focus on the rotational velocity part.

## Skew Symmetric Matrices

We have a quick revision of the skew-symmetric matrix. If you’re comfortable with this topic then go straight on to the next section.

## Velocity of 6-Joint Robot Arm – Translation

For a real 6-link robot our previous approach to computing the Jacobian becomes unwieldy so we will instead compute a numerical approximation to the forward kinematic function.

## Motion in 3D

A body moving in 3D space has a translational velocity and a rotational velocity. The combination is called spatial velocity and is described by a 6-element vector.

## Introduction to Velocity Kinematics in 3D

We will learn about the relationship, in 3D, between the velocity of the joints and the velocity of the end-effector — the velocity kinematics. This relationship is described by a Jacobian matrix which also provides information about how easily the end-effector can move in different Cartesian directions. To do this in 3D we need to learn about rate of change of orientation and the concept of angular velocity.

## Summary of Velocity Kinematics in 2D

This video gives summary of velocity Kinematics in 2D

## Mapping 2D Spatial Velocity Between Coordinate Frames

We can also derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame.

## Velocity of 3-Joint Planar Robot Arm

We extend what we have learnt to a 3-link planar robot where we can also consider the rotational velocity of the end-effector.

## Resolved Rate Motion Control in 2D

We will introduce resolved-rate motion control which is a classical Jacobian-based scheme for moving the end-effector at a specified velocity without having to compute inverse kinematics.