Understanding DH Parameters: A Simple Guide

Denavit-Hartenberg (DH) parameters is a standardized framework for modeling the geometry of robot arms.

Manipulator Components and Kinematics

A robot manipulator consists of:

Links: Rigid segments connecting joints
Joints: Points of motion, either
- Revolute (rotational) or
- Prismatic (linear)

Each joint contributes a degree of freedom (DOF). For full 3D control of a robot’s end-effector:

3 DOF for position (x, y, z)
3 DOF for orientation (roll, pitch, yaw)

Robots with fewer than six DOFs are underactuated; more DOFs provide redundancy but increase complexity.

Example: Most industrial arms, like the PUMA 560, have at least 6 DOFs to position and orient the end-effector anywhere in 3D space.

Denavit-Hartenberg (DH) Parameters

DH parameters provide a systematic way to describe the spatial relationship between consecutive links. Each joint is represented by four parameters:

Link length ( $a$ ) – distance along the common normal between joint axes
Link twist ( $\alpha$ ) – angle between joint axes along the common normal
Link offset ( $d$ ) – distance along the joint axis (variable for prismatic joints)
Joint angle ( $\theta$ ) – rotation about the joint axis (variable for revolute joints)

These parameters simplify forward and inverse kinematics.

T_i^{i-1} = \begin{bmatrix} \cos\theta_i & -\sin\theta_i \cos\alpha_i & \sin\theta_i \sin\alpha_i & a_i \cos\theta_i \\ \sin\theta_i & \cos\theta_i \cos\alpha_i & -\cos\theta_i \sin\alpha_i & a_i \sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}

This homogeneous transformation matrix maps frame $i$ to $i-1$ .

Attaching Coordinate Frames

Z-axis: aligned with the joint axis
X-axis: along the link between joints
Y-axis: determined by the right-hand rule

Consistent frame attachment ensures that the DH parameters accurately model the robot geometry.

Examples

PUMA 560 (6-DOF revolute arm):

Joint $i$	$a_i$ (m)	$\alpha_i$ (rad)	$d_i$ (m)	$\theta_i$ (rad)
1	0	π/2	0.675	θ1
2	0.26	0	0	θ2
3	0.68	0	0	θ3
…	…	…	…	…

Summary

DH parameters provide a compact, systematic way to model robotic arms, bridging physical construction (links and joints) and mathematical representation (transformation matrices). Mastering DH parameters is essential for: