### Fundamentals of Robotics: Wrenches

In the previous lesson, we learned that screws are geometric interpretations of twists, and they can be used to express configurations in robotics. This lesson is about spatial forces or wrenches in robotics. We will talk about wrenches in this lesson, which are 6-vector representations of forces and moments in robotics. We will also become familiar with how to change the frame of representation of a wrench.  This lesson is part of the series of lessons on foundations necessary to express robot motions. For a complete comprehension of the Fundamentals of Robot Motions and the tools required to represent the…

### Screws: a Geometric Description of Twists in Robotics

In the previous lesson, we learned about velocities in robotics. We became familiar with angular and linear velocities and saw that stacking them together gives us the twist. We also saw how we could change the frame of reference for angular velocities and twists. This lesson is about screws as a geometric interpretation for twists and how they can be used to express configurations in robotics. This lesson is part of the series of lessons on foundations necessary to express robot motions. For the complete comprehension of the Fundamentals of Robot Motions and the tools required to represent the configurations,…

### Velocities in Robotics: Angular Velocities & Twists

In the previous lesson, we saw an introduction to screw theory and its applications in robotics. We saw that integrating a constant twist over time gives us the configuration. We also became familiar with the exponential coordinates of robot motions, and we saw that in order to define the screw axis, we need to understand angular and linear velocities. In this lesson, we will become familiar with the concept of velocities in robotics, and we will study angular velocities and twists. This lesson is part of the series of lessons on foundations necessary to express robot motions. For the complete…

### Screw Motion and Exponential Coordinates of Robot Motions

This lesson is part of the series of lessons on foundations necessary to express robot motions. For the complete comprehension of the Fundamentals of Robot Motions and the tools required 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 fromΒ the base lessons of the Fundamentals of Robotics courseΒ are deemed invaluable. In this lesson, we will see an introduction to screw motion in robotics, and we will also see how we can define exponential coordinates for robot motions. This is…

### 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…

### 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…

### 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…

### 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 fromΒ the…

### 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…

### 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…