## 1D polynomial trajectory

In Robotics | No commentThe simplest smooth trajectory is a polynomial with boundary conditions on position, velocity and acceleration.

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

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.

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

This video gives summary of 3D geometry and pose.

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

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

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

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

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.

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

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.

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

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.

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.

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

We discuss the structure of a right-handed 3D coordinate frame and the spatial relationship between its axes which is encoded in the right-hand rule.

We revisit the fundamentals of 3D geometry that you would have learned at school: coordinate frames, points and vectors.

We learn how to describe the position and orientation of objects in the 3-dimensional space that we live in. This builds on our understanding of describing position and orientation in two dimensions.

This video gives summary of 2D geometry and pose.

We learn how to describe the 2D pose of an object by a 3×3 homogeneous transformation matrix which has a special structure.