Denavit-Hartenberg Representation of Robots

The D-H model of representation is a simple way of modeling robot links and joints that can be used for any robot configuration. We will have to assign a z-axis and an x-axis for each link. The D-H representation does not use the y-axis at all. Let L_k be the frame associated with link k.

1. Define Z Axis from All Joints

From the axis of joint k+1, define the z^k axis for link k. If the joint is revolute, the z axis is in the direction of rotation as followed by the right hand rule for the rotations. If the joint is prismatic, the z-axis for the joint is along the direction of the linear movement.

2. Define Origins

The intersection of the z^k and z^k-1 axes is selected as the origin of L^k . If they do not intersect, use the intersection of z^k with a common normal between z^k and z^k-1. There is always one line mutually perpendicular to any two skew lines, called common normal, which is the shortest distance between them.

3. Define X-axis

Assign x^k in the direction of the common normal between z^k and z^k-1. If z-axes are intersecting, select x^k to be orthogonal to both z^k and z^k-1 (the direction of the cross-product of the two z-axes). If z^k and z^k-1 are parallel, point x^k away from z^k-1 colinear with the common normal of the previous joint.

4. Define Y-axis

Select y^k to form a right-handed orthonormal coordinate frame L_k.

5. Define the Four Kinematic Parameters

*θ_k is defined as the angle between x^k-1 and x^k axes about the z^k-1 axis. *d_k is the distance between x^k-1 and x^k axes along z^k-1 axis. *a_k is the distance between z^k-1 and z^k axes along x^k axis. *α_k is the angle between z^k-1 and z^k axes about x^k axis.

6. Transferring Frame k-1 to Frame k

*Rotation of L^k-1 about the z^k-1 axis by θ_k will make x^k-1 and x^k parallel to each other. This is true because the common normals a^k-1 and a^k are both perpendicular to z^k-1 axis. *Translation of L^k-1 along z^k-1 axis a distance of d_k will make x^k-1 and x^k colinear. *Translation of L^k-1 along x^k axis a distance of a_k will bring the origins of L^k-1 and L^k together. *Rotation of L^k-1 about x^k axis by α_k will make z^k-1 and z^k axes parallel. At this point, frames L^k-1 and L^k will be exactly the same.



For simulation, a nice free software called Robotassist can be downloaded at www.kinematics.com

Ref:
Introduction to Robotics -Analysis, Control, Applications; Second Edition, Saeed Benjamin Niku
Fundamental of Robotics -Analysis & Control, Robert J. Schilling