June, 23, 2021

Fundamentals of Robot Motions: Preliminaries

This post and the coming posts will be focused on the necessary tools needed to express the robot motions. In this post, you will become familiarized with the preliminaries and prerequisites needed before jumping into the main topics.

For the full comprehension of the Fundamentals of Robot Motions and the necessary tools to express motion in robotics, please also read all the lessons in the following link (some lessons will be added later):

https://www.mecharithm.com/category/fundamentals-of-robotics/fundamentals-of-robot-motions/

When we talk about robot motions, we will need fundamentals such as the representations of configurations, velocities, and forces that cause the motions. In this post and the coming posts, we will focus on these tools.

Free Vector vs. a Vector

A free vector is a geometric quantity and an arrow in n-dimensional flat space Rⁿ that is not rooted anywhere. It has a length and a direction. In robotics, a free vector is denoted by v.

A vector is a free vector expressed with its coordinates in a reference frame and length scale chosen for that space. In robotics, vectors are represented by an italic letter ν in Rⁿ.

A vector is dependent on the choice of the coordinate frame and length scale, whereas the underlying free vector is unchanged by choice of the coordinate frame or the length scale. In other words, v is coordinate-free.

free-vector-vs-a-vector — A free vector is coordinate-free, whereas a vector is dependent on the choice of the coordinate frames and length scale.

A vector can also represent a point P in the physical space. If we choose to give this physical space a reference frame and a length scale, the point is a vector from the origin of this reference frame to the point. It is then represented as an italic 𝑝 ∈ Rⁿ. The same point has a different representation by changing the length scale and the reference frame:

a-point-as-vector-in-two-coordinate-frames-1 — A physical point p can have different representations relative to different coordinate frames.

Frames in Robotics

In Robotics, frames are important. We use frames to represent the robot’s configurations, velocities, and forces causing the motion. Frames in robotics:

have an origin.
consist of orthogonal x, y, and z coordinate axes:

Right-handed-coordinate-frame — A right-handed coordinate frame consists of x, y, and z coordinate axes. The index finger is in the x-direction, the middle finger is in the y-direction, and the thumb is in the z-direction.

are right-handed, and this means that the cross product of the x and y axes is z, and so on:

\[\hat{x} \times \hat{y} = \hat{z}\\\hat{y} \times \hat{z} = \hat{x}\\\hat{z} \times \hat{x} = \hat{y}\]

are stationery! This is from Newton’s laws that the reference frames are always considered to be inertial.

A positive rotation about an axis follows the right-hand rule. If you align your thumb with the axis of rotation, then the positive rotation is in the direction that the fingers curl:

To conclude this lesson, let’s see a simple demonstration of the positive rotations around the coordinate axes attached to a robot joint:

The video version of the current lesson can be watched at the link below:

References:
Textbooks:
Modern Robotics: Mechanics, Planning, and Control by Frank Park and Kevin Lynch
A Mathematical Introduction to Robotic Manipulation by Murray, Lee, and Sastry