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 F_o, attached to the fixed link, all the way to frame F_n, 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 F_i-1 and F_i, frame F_i will be uniquely determined from frame F_i-1 by use of the parameters d_i, a_i, a_i, and q_i 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.

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Notes:

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

The parameter a_i 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, q_i is constant and determined by the structure of the robot.

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

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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.

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Special cases:

If joint axes z_i-1 and z_i intersect, parameter a_i is zero.
If the common perpendicular to z_i-1 and z_i intersects z_i-1 at the origin of frame F_i-1, then d_i is zero.
If joint axes z_i-1 and z_i are parallel, angle a_i is zero.

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End-Frames:

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

The end-effector frame, F_n (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 d_n, a_n, and a_n are zero if joint n is revolute or parameters q_n, a_n, and a_n are zero if joint n is translational.

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Return to the Main Robot Modeling Page

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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

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Last modified in  Aug.  2001