Introducción a la robótica: Posición, orientación y tramas

Maestro Camacho - Mecatrónica & Manufactura

1 Feb 202208:27

Summary

TLDRThis video script delves into the fundamentals of robotics, focusing on the description of a point's position and orientation in space with respect to a coordinate system. The master explains the use of position vectors and rotation matrices to define a robot's arm orientation, emphasizing the importance of accurately representing both position and orientation. The concept of a 'transform', a set of four vectors encompassing position and orientation data, is introduced. The script also touches on the application of these principles in robotic arms, highlighting the kinematic chain's representation through coordinate systems and the practicality of using transforms for various robotic systems.

Takeaways

📍 The script introduces the concept of describing a point's position in space with reference to a coordinate system, specifically the A and Y coordinate system.
🔄 The point P can be represented by coordinates (x, y, z) in a right-handed coordinated system, with the superscript 'a' indicating reference to the A coordinate system.
🚦 The description of orientation is more complex than position and requires not one but three vectors to fully describe the orientation of a robotic arm or end effector.
📊 A 3x3 matrix, represented by 'R', is used to describe the orientation, which is essentially a rotation matrix from system A to system B.
🔢 The elements of the rotation matrix R are the projections of the principal vectors of system B onto the elements of system A, calculated using dot product and trigonometric functions.
⏬ The script explains that the rotation matrix R can also be represented as the transpose of matrix R, when analyzed by rows instead of columns.
🔍 By analyzing the rotation matrix by columns and rows, insights into the projections of the system B's vectors onto system A's vectors are gained.
💠 The term 'transform' (trama in Spanish) is introduced as a set of four vectors providing information about position and orientation, consisting of a 3x3 rotation matrix and a position vector.
🔄 The transform allows representing any coordinate system relative to a previous or 'base' coordinate system, which is crucial in robotics for representing the kinematic chains of an articulated robotic arm.
📈 The script concludes by emphasizing the practical application of transforms in robotics, particularly in understanding the relationship between different coordinate systems in the context of a robotic arm.

Q & A

What is the main topic of the video?
-The main topic of the video is the description of the position and orientation of a point in space with reference to a coordinate system, specifically focusing on robotics.
How is a point represented in a coordinate system?
-A point is represented by its coordinates with respect to a given coordinate system. In the video, the point P is represented by its coordinates (x, y, z) in the A coordinate system.
What are the three axes of a right-handed coordinate system mentioned in the video?
-The three axes of a right-handed coordinate system mentioned are the x, y, and z axes.
Why is it necessary to describe the orientation in robotics?
-Describing the orientation is necessary in robotics because a robotic arm or end effector can be rotated in any direction while keeping a point stationary, and this rotation needs to be accounted for in addition to the position.
How many vectors are required to fully describe the orientation of a system?
-Three vectors are required to fully describe the orientation of a system, which together form a 3x3 matrix.
What does the 3x3 matrix represent in the context of the video?
-The 3x3 matrix represents a rotation matrix, which describes the orientation of one coordinate system (B) with respect to another (A). It is a mathematical representation of how the basis vectors of one system are projected onto the other system.
How is the rotation matrix (R) related to the transpose of the matrix?
-The rotation matrix (R) is the transpose of the matrix because when analyzed by rows instead of columns, it represents the projection of the basis vectors of one system onto the other, thus giving the orientation of one system with respect to the other.
What is a transformation matrix (or 'trama' in Spanish) in robotics?
-A transformation matrix, or 'trama', is a set of four vectors that provide information about both the position and orientation of a coordinate system. It consists of a 3x3 rotation matrix and a position vector.
How many coordinate systems can be represented in relation to the initial system?
-Multiple coordinate systems can be represented in relation to the initial system. Each subsequent system can be described with respect to the previous one, forming a chain of transformations.
What does the base system (system U) represent in the context of a robotic arm?
-In the context of a robotic arm, the base system (system U) represents the fixed part of the arm where the coordinate system origin is placed, and it is the reference point for the rest of the moving segments in the robotic chain.
How are the transformation matrices used in the kinematic chain of a robotic arm?
-Transformation matrices are used to represent the position and orientation of each link in the kinematic chain of a robotic arm with respect to the previous link. They help in calculating the overall position and orientation of the end effector.

Outlines

00:00

🤖 Introduction to Coordinate Systems and Robotic Arm Orientation

This paragraph introduces the concept of describing a point's position in space with respect to a coordinate system, specifically the A coordinate system. It explains how a point P can be represented by coordinates in the X, Y, and Z axes within a right-handed coordinate system. The paragraph further delves into the complexity of describing orientation, differentiating it from position. It highlights the need for not just one vector but three vectors (xv, yv, zv) to fully describe orientation, resulting in a 3x3 matrix. The explanation includes understanding the projection of the coordinate system's basis vectors onto each other and introduces the rotation matrix (R) as a representation of orientation, emphasizing its significance in robotics.

05:00

📐 Understanding the Spatial Representation of Coordinate Systems and Dexterity in Robotics

This paragraph continues the discussion on coordinate systems by explaining how the orientation of one coordinate system (B) is represented with respect to another (A). It describes the process of obtaining the orientation matrix (R) through vector projections and how this matrix can be transposed to represent the inverse relationship. The concept of a 'frame', a set of four vectors providing information about position and orientation, is introduced. The paragraph clarifies that the position is given by a vector (pdv), and the orientation is given by a 3x3 rotation matrix. It concludes by illustrating how multiple frames can be used within a robotic arm to represent the kinematic chain, with each frame's origin and vectors representing the links of the arm.

Mindmap

Keywords

💡Robotics

Robotics is the branch of technology that deals with the design, construction, operation, and use of robots. In the context of the video, it is the main theme, as the speaker, 'Maestro Camacho', discusses various concepts related to robotic arms and their movements in space.

💡Coordinate System

A coordinate system is a geometrical framework that enables the precise determination of the position of points. In the video, the speaker introduces a coordinate system referred to as 'A' and explains how points, like the robotic arm's end effector, can be described in terms of their coordinates within this system.

💡Position

Position refers to the specific location of an object within a reference frame or coordinate system. In the video, the description of a point's position is crucial for understanding where the robotic arm or its end effector is located in space.

💡Orientation

Orientation is the alignment or posture of an object in space, which is as important as its position in robotics. The video discusses how the orientation of a robotic arm can vary, and it requires a more complex description than position, involving three vectors instead of one.

💡Transformation Matrix

A transformation matrix is a mathematical tool used in robotics to describe the position and orientation of a coordinate system relative to another. It encapsulates both the translation (position) and rotation (orientation) of a frame in space.

💡End Effector

The end effector is the part of a robotic arm that interacts with the environment, such as a gripper or tool. In the video, the end effector is used as an example to illustrate the concepts of position and orientation in space.

💡Chain of Motion

The chain of motion refers to the sequence of links or joints in an articulated robotic arm that allows it to move. In the video, the chain of motion is implied when discussing the representation of different coordinate systems along the arm.

💡Basis Vectors

Basis vectors are the fundamental unit vectors that define a coordinate system. They are used to express any other vector in the space spanned by the coordinate system. In the video, the basis vectors of one coordinate system are projected onto the basis vectors of another to describe orientation.

💡Vector

A vector is a mathematical object that represents both a direction and a magnitude. In the context of the video, vectors are used to describe positions and orientations in three-dimensional space.

💡Dot Product

The dot product is a binary operation that takes two vectors and returns a scalar value. It is used in the video to calculate the projections of the basis vectors onto each other when constructing the transformation matrix.

💡Transposition

Transposition is the process of converting a matrix from its original form to its transpose, which involves interchanging the rows and columns. In the video, the transposition of the transformation matrix 'R' is mentioned as being equivalent to 'R^T'.

Highlights

Introduction to robotics and coordinate systems, specifically the A and B coordinate systems.

Description of a point's position in space with respect to a coordinate system.

Representation of point P in terms of coordinates with respect to the reference system.

Explanation of the orientation description being more complex than position description.

Requirement of three vectors, not one, to fully describe orientation.

Introduction of the 3x3 matrix representing the orientation of system V with respect to system A.

Clarification that the elements of the matrix represent the projection of the main vectors of the coordinate system.

Explanation of the dot product and its role in vector projection.

The notation R for the rotation matrix and its representation of the orientation of system V with respect to system A.

The transposition of matrix R to obtain the orientation of system A with respect to system B.

Description of the transformation matrix (frame) and its components: position and orientation.

The position vector (p_A) and its representation of the coordinates of the frame's origin.

Application of frames in robotics, particularly in the kinematic chains of an arm.

Use of the transformation matrix equation to represent the system with respect to a previous coordinate system.

The concept of a base or universal system and how other systems are represented with respect to it.

The practical application of the transformation matrix in the kinematic chain of a robotic arm.

Conclusion of the video and a teaser for the next topic.