The Denavit-Hartenberg (DH) convention simplifies robot kinematics by standardizing coordinate frame assignments. It uses four parameters to define each link, making it easier to derive forward kinematics equations for robotic manipulators.

Forward kinematics uses DH parameters to calculate end-effector position and orientation from joint angles. This process involves creating transformation matrices, multiplying them, and extracting position and orientation information from the final matrix.

Denavit-Hartenberg Convention and Forward Kinematics

Denavit-Hartenberg convention for robots

• DH convention standardizes robot kinematics description simplifying forward kinematics equations derivation
• Coordinate frame assignment rules align Z-axis with joint axis, X-axis perpendicular to current and previous Z-axes, Y-axis follows right-hand rule
• Link frame placement positions origin at intersection of common normal and joint axis with Frame i attached to link i
• Joint types consideration addresses revolute joints (Z-axis along rotation axis) and prismatic joints (Z-axis along translation direction)

DH parameters of robot manipulators

• Four DH parameters define each link: $a_i$ (link length along X-axis), $\alpha_i$ (link twist about X-axis), $d_i$ (link offset along Z-axis), $\theta_i$ (joint angle about Z-axis)
• Parameter identification process analyzes joint and link geometry, measures distances between joint axes, determines angular relationships between links
• Special cases include parallel joint axes, intersecting joint axes, prismatic joints requiring specific parameter considerations

Forward kinematics from DH parameters

• Homogeneous transformation matrices combine individual transformations for each parameter into composite transformation matrix $A_i$
• DH parameter matrix provides tabular representation of parameters for all links
• Chain of transformations multiplies individual link transformations $T_n = A_1 A_2 ... A_n$
• Symbolic representation uses variables for joint angles in revolute joints and joint displacements in prismatic joints

End-effector position from joint angles

• Process takes joint angles (or displacements for prismatic joints) as input
• Substitutes joint values into transformation matrices
• Performs matrix multiplication to obtain final transformation
• Extracts position information from last column of final transformation matrix
• Derives orientation information from rotation matrix in final transformation
• Represents orientation using Euler angles, Roll-Pitch-Yaw angles, or Quaternions
• Workspace analysis determines reachable positions of end-effector and identifies singularities affecting calculations