 ## Summary of paths and Trajectories

This video gives summary of paths and trajectories.

## Interpolating pose in 3D

We combine what we’ve learnt about smoothly varying position and orientation to create smoothly varying pose, often called Cartesian interpolation.

## Interpolating rotation in 3D

We learn how to create smoothly varying orientation in 3D by interpolating Euler angles and Quaternions.

## Multi-dimensional trajectory

We learn to compute a trajectory that involves simultaneous smooth motion of many robot joints.

## 1D trajectory with via points

Frequently we want a trajectory that moves smoothly through a series of points without stopping.

## 1D polynomial trajectory

The simplest smooth trajectory is a polynomial with boundary conditions on position, velocity and acceleration.

## Paths and Trajectories

Time varying coordinate frames are required to describe how the end-effector of a robot should move to grab an object, or to describe objects that are moving in the world. We make an important distinction between a path and a trajectory.

## Introduction to Paths and Trajectories

We will learn how to create coordinate frames that have smoothly changing position and orientation over time.

## Summary of 3D geometry and pose

This video gives summary of 3D geometry and pose.

## Describing rotation and translation in 3D

We learn how to describe the 3D pose of an object by a 4×4 homogeneous transformation matrix which has a special structure.

## Quaternions representation of rotation in 3D

The orientation of a body in 3D can also be described by a unit-Quaternion, an unusual but very useful mathematical object.

## Angle-axis representation of rotation in 3D

The orientation of a body in 3D can also be described by a single rotation about a particular axis in space.

## 2-vector representation of rotation in 3D

The orientation of a body in 3D can also be described by two vectors, often called the approach and orientation vectors.

## Singularity in 3D rotation angle sequences

A problem arises when using three-angle sequences and particular values of the middle angle leads to a condition called a singularity. This mathematical phenomena is related to a problem that occurs in the physical world with mechanical gimbal systems.

## Rotation angle sequences in 3D

The orientation of a body in 3D can be described by three angles, examples of which are Euler angles and roll-pitch-yaw angles.

## Rotations are non commutative in 3D

If we apply a sequence of 3D rotations to an objects we see that the order in which they are applied affects the final result.

## Describing rotation in 3D

We learn how to describe the orientation of an object by a 3×3 rotation matrix which has some special properties.

## Relative pose in 3D

We consider multiple objects each with their own 3D coordinate frame. Now we can describe the relationships between the frames and find a vector describing a point with respect to any of these frames. We extend our previous 2D algebraic notation to 3D and look again at pose graphs.

## Pose in 3D

We extend the idea of relative pose, introduced in the last lecture, to 3D. We learn another right-hand rule that indicates the direction of rotation about an axis, and we see how we can attach 3D coordinate frames to objects to determine their pose in 3D space.

## Build your own 3D coordinate frame

This is an exercise in which you can build a 3D coordinate frame by printing, cutting, folding and stapling.