 ## 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.

## Velocity Ellipse in 2D

The end-effector is not able to move equally fast in all directions, and that in fact depends on the pose of the robot. We will introduce the velocity ellipse to illustrate this.

## Inverting the Jacobian Matrix

By inverting the Jacobian matrix we can find the joint velocities required to achieve a particular end-effector velocity, so long as the Jacobian is not singular.

## Velocity of 2-Joint Planar Robot Arm

For a simple 2-link planar robot we introduce and derive its Jacobian matrix, and also introduce the concept of spatial velocity.

## Rate of Change of Pose in 2D

We introduce the relationship between the velocity of the robot’s joints and the velocity of the end-effector in 3D space.

## Introduction to Velocity kinematics in 2D

We will learn about the relationship, in 2D, 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.

## Summary of Inverse Kinematics

This video gives summary of inverse kinematics.

## Cartesian interpolated Motion

An alternative for smooth motion between poses is Cartesian interpolated motion which leads to straight line motion in 3D space.

## Joint Interpolated Motion

To move a robot smoothly from one pose to another we need smooth and coordinated motion of all the joints. The simplest approach is called joint interpolated motion but it has some limitations.

## Redundant Robots

For a redundant robot the inverse kinematics can be easily solved using a numerical approach.