Denavit-Hartenberg Parameters

Author: Rachid Manseur, ECE Dept. UWF

Robot manipulators are a sequence of links articulated at joints. To analyze the motion of robot manipulators, reference frames are attached to each link starting at frame Fo, attached to the fixed link, all the way to frame Fn, attached to the robot end-effector (assuming the robot has n joints). The process followed in assigning frames to links is based on the Denavit-Hartenberg parameters.

Given two consecutive link frames on a robot manipulator, frames Fi-1 and Fi, frame Fi will be uniquely determined from frame Fi-1 by use of the parameters di, ai, ai, and qi illustrated in Figure 1.


Figure1: DH-Parameters

Rules & Definitions

  • The z vector of any link frame is always on a joint axis.  The only exception to this rule is the end-effector (tool) of the robot which does not have a joint axis.

  • di is the algebraic distance along axis zi-1 to the point where the common perpendicular intersects axis zi-1.

  • ai is the length of the common perpendicular to axes zi-1 and zi.

  • qi is the angle, about zi-1, that the common perpendicular makes with vector xi-1.

  • ai is the angle, about xi, that vector zi makes with vector zi-1.




The rules and guidelines given here simplify the kinematic modeling and analysis of robot manipulators:

The reference vector z of a link-frame is always on a joint axis.

The parameter di is algebraic and may be negative. It is constant if joint i is revolute and variable when joint i is translational.

The parameter ai is always constant and positive.

a i is always chosen positive with the smallest possible magnitude.

q i is variable when joint i is revolute, and constant when joint i is translational. When joint i is translational, qi is constant and determined by the structure of the robot.

di is variable when joint i is translational, and constant when joint i is revolute.


Robot Manipulator Modeling.

A mathematical description for robot manipulators is usually given as a table of DH-parameters. The table contains one row of four parameters for each link frame. The Denavit-Hartenberg parameters allow one reference frame to be located exactly with respect to the preceding link frame. To understand how these four parameters can exactly locate a frame, consider, for example, that a frame B is determined from a Frame A by the four DH parameters d, a,
a, and q. This situation is illustrated in Figure 2.


Starting from frame A, frame B can be found by following the 4 steps outlined here:

1. From the origin of frame A, move a distance d on the zA axis. Note that d can be positive or negative.


2. Determine the direction of xB by rotating vector xA by an angle q about zA.


3. Move a distance a in the direction of vector xB. The position reached is the origin of Frame B. At this point vector xB is determined as well.


4. Rotate the vector zA about xB by an angle a to determine the unit vector zB.

Special cases:

If joint axes zi-1 and zi intersect, parameter ai is zero.
If the common perpendicular to zi-1 and zi intersects zi-1 at the origin of frame Fi-1, then di is zero.
If joint axes zi-1 and zi are parallel, angle
ai is zero.


The base frame, Fo, can always be located on joint axis zo at the intersection point with the common perpendicular to axis z1. Therefore, parameter d1 can always be chosen as zero.

The end-effector frame, Fn (for an n-DOF robot), is the only frame that does not have to be located on a joint axis. It is attached to the end-effector and can always be chosen such that parameters dn, an, and an are zero if joint n is revolute or parameters qn, an, and an are zero if joint n is translational.

Return to the Main Robot Modeling Page

This page was developed by :

Dr. Rachid Manseur, Associate Professor of Electrical Engineering
Robotics and Image Analysis
Electrical and Computer Engineering
University of West Florida

Last modified in  Aug.  2001