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Kinematics of Serial Robots: Position Analysis - Coggle Diagram
Kinematics of Serial Robots: Position Analysis
Introduction
Purpose
Study forward and inverse kinematics
Describe location/orientation with matrices
Kinematic Analysis
Forward: Known joint variables → end-effector location
Inverse: Desired location → joint variables
Robot Structure
Manipulator with changeable end effectors
Hand assumed to be a flat plate
Robots as Mechanisms
Multi-DOF Chains
Each joint must be known to locate the hand
Typical industrial robots have 6 DOF
Open vs. Closed Loop
Open-loop: prone to deflection; no feedback
Closed-loop: better accuracy but limited workspace
Deflection Handling
Use strong links
Add feedback sensors
Implement secondary links or lasers
Conventions
Symbols and Notation
Vectors: i, j, k; positions: P, x, y, z
Frames: Fxyz, Fnoa
Transformations: T, UTR
Trigonometric Notation
Sθ, Cθ used for sin and cos of angles
Matrix Representation
Points in Space
Position vector: P = axi + byj + czk
Matrix form with/without scale w
Vectors in Space
Start and end point define vector
Frames
Defined by origin and three orthogonal axes
includes directional cosines and origin
Constraints
Unit vectors must be perpendicular
Must form right-handed system
Homogeneous Transformation Matrices
Purpose
Maintain square matrix form (4×4)
Format
Combines rotation and translation
Used to multiply transformations efficiently
Representation of Transformations
Pure Translation
Only origin changes; direction vectors stay same
Pure Rotation
About x, y, or z axis
Rot x, Rot y, Rot z
Combined Transformations
non-commutative
Combine rotations and translations
Transformations Relative to Moving Frame
Post-multiplication when relative to current frame
Mixed Transformations
Pre- and post-multiplication combined
Used in complex robot motions
Inverse of Transformation Matrices
Purpose
Needed for solving kinematic chains
3x3 Rotation Matrix
Inverse is transpose
4x4 Homogeneous Matrix
Rotation: transpose
Position: negative dot product with rotation vectors
Forward and Inverse Kinematics
Forward Kinematics
Input: joint variables → Output: position/orientation
Inverse Kinematics
Input: desired position/orientation → Output: joint variables
Frames
Use attached frame to determine 6 DOF