Homogeneous Transformation Matrices to Express Configurations in Robotics

Homogeneous Transformation Matrices to Express Configurations in Robotics

Up to this point, we have discussed orientations in robotics, and we have become familiarized with different representations to express orientations in robotics. In this lesson, we will start with configurations, and we will learn about homogeneous transformation matrices that are great tools to express configurations (both positions and orientations) in a compact matrix form. This lesson is part of the series of lessons on foundations necessary to express robot motions. For the full comprehension of the Fundamentals of Robot Motions and the necessary tools to represent the configurations, velocities, and forces causing the motion, please read the following lessons…
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Cayley-Rodrigues Parameters to Express Orientations in Robotics

Cayley-Rodrigues Parameters to Express Orientations in Robotics

In this lesson, we will become familiarized with another representation for orientations in robotics that is called Cayley-Rodrigues Parameters. Cayley-Rodrigues parameters provide local coordinates for SO(3). They are local coordinates because the representation is not singularity-free, and not all orientations can be expressed by them. However, they have properties that make them intriguing. This lesson is part of the series of lessons on foundations necessary to express robot motions. For the full comprehension of the Fundamentals of Robot Motions and the necessary tools to represent the configurations, velocities, and forces causing the motion, please read the following lessons (note that…
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Unit Quaternions to Express Orientations in Robotics

Unit Quaternions to Express Orientations in Robotics

In the lesson about the Exponential Coordinate Representation of the orientation, we saw that the logarithm could be numerically sensitive to small rotation angles ΞΈ because of the division by sinΞΈ. We also saw that all other three-parameter representations of SO(3), like Euler angles and roll-pitch-yaw angles, suffer from similar singularities in representation, and this means that the solution will not always exist for the inverse problem where we want to find a set of parameters for a given orientation. Therefore, an alternative representation of the orientation named Unit Quaternion is used that alleviates the singularity at the cost of…
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Explicit Representation of the Orientation: Exponential Coordinates

Explicit Representation of the Orientation: Exponential Coordinates

In the previous lesson, we became familiar with rotation matrices, and we saw that the nine-dimensional space of rotation matrices subject to six constraints (three unit norm constraints and three orthogonality constraints) could be used to implicitly represent the three-dimensional space of orientations. There are also methods to express the orientation with a minimum number of parameters (three in three-dimensional space). Exponential coordinates that define an axis of rotation and the angle rotated about that axis, the three-parameter Euler angles, the three-parameter roll-pitch-yaw angles, the Cayley-Rodrigues parameters, and the unit quaternions (use four variables subject to one constraint) are some of…
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Implicit Representation of the Orientation: a Rotation Matrix

Implicit Representation of the Orientation: a Rotation Matrix

In the previous lesson, we became familiar with the concept of the configuration for the robots, and we saw that the configuration of a robot could be expressed by the pair (R,p) in which R is the rotation matrix that implicitly represents the orientation of the body frame with respect to the reference frame and p is the position of the origin of the body frame relative to the space frame. In this lesson, we will focus on the orientation, and we will see that we can implicitly represent the orientation using powerful tools named rotation matrices, and we will…
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Fundamentals of Robot Motions: Configurations (Introduction)

Fundamentals of Robot Motions: Configurations (Introduction)

This post is part one of the series of lessons on the fundamentals necessary to represent the robot's configuration, and it gives an introduction to what we mean when we are talking about representing a robot's configuration. In previous lessons, we learned that the robot's configuration answers the question of where the robot is, and we saw that there are two ways to represent the robot's configuration: Implicit representation, where the configuration is represented by embedding the curved space in higher-dimensional Euclidean space subject to constraints and explicit representation where configuration is represented with a minimum number of coordinates. You…
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Configuration and Configuration Space (Topology and Representation) of a Robot

Configuration and Configuration Space (Topology and Representation) of a Robot

This is the second lesson on Fundamentals of Robotics, and it will be your guide to know everything you have ever needed to know about the configuration and configuration space (its topology and representation) of a robot. In this lesson, we will talk about the configuration of the robot. Then, the configuration space and topology of the configuration space will be discussed, and finally, we will talk about different ways to represent the configuration space. What is the Configuration of Something? The configuration of something answers the question, where is that thing? For example, to know where a door is,…
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