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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Other Explicit Representation for the Orientation in Robotics: Roll-Pitch-Yaw Angles

Other Explicit Representation for the Orientation in Robotics: Roll-Pitch-Yaw Angles

In the previous lesson, we learned about Euler Angles Representation which is one of the ways to explicitly represent an orientation. This lesson will continue with explicit ways to represent the orientation, and we will learn about Roll-Pitch-Yaw Angles. 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 more lessons will be added in the future): https://www.mecharithm.com/category/fundamentals-of-robotics/fundamentals-of-robot-motions/ Also, reading some lessons from the…
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Other Explicit Representation for the Orientation in Robotics: Euler Angles

Other Explicit Representation for the Orientation in Robotics: Euler Angles

In the lesson about the degrees of freedom of a robot, we learned that there are at least three independent parameters needed to express the orientation of a rigid body. In the previous lesson, we learned about the exponential coordinate representation for the orientation that is a three-parameter representation for a rotation matrix R and parameterizes the rotation matrix using a unit axis of rotation and the angle of rotation about this axis. There are also other explicit representations that are useful in different applications when dealing with orientations. In this lesson, we will talk about Euler Angles and will…
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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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