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Terminology

A robot is a collection of successive joints and links (or arms), each link has a start and an end joint, the end joint of a link being being identical to the start joint of the following link.
The Denavit-Hartenberg method associates, to each joint, a coordinate frame (B) - a system of frames, or coordinate axes (x,y,z) and, with each link, a system of parameters (a, α, d and θ).

• Link length (kinematic length) a_i is the distance between axes z_i-1 and z_i along their common perpedicular line, on which the x_i axis lies.
• Link twist α_i is the rotation of the z_i-1 axis about x_i to become parallel to the z_i axis. Note that, in general, alpha is equal to zero, 90, -90 or 180 degrees.
• Joint distance (link offset) d_i is the distance between the x_i-1 and x_i axes along the z_i-1 axis.
• Joint angle θ_i is the rotation of the x_i-1 axis about the z_i-1 axis to become parallel to the x_i axis.

• rotational, when the joint angle θ_i is variable, all other values being constant; such a link rotates around the joint axis z_i-1
• prismatic, when the joint distance d_i is variable, all other values being constant; such a link is moving along the z_i axis.

The end-point of the last link is the end-effector position.

Some technical details.

The Denavit-Hartenberg (DT) transformation of coordinates from frame i to frame i-1 is given by the matrix:

For a point with coordinates x_i, y_i, z_i in frame i, its coordinates in frame i-1 will be equal to

or

The trasformation, in detail:

x_i-1 = cosθ_i x_i - sinθ_i cosα_i y_i + sinθ_i sinα_i z_i + a_i cosθ_i
y_i-1 = sinθ_i x_i + cosθ_i cosα_i y_i - cosθ_i sinα_i z_i + a_i sinθ_i
z_i-1 = sinα_i y_i + cosα_i z_i + d_i

A point with coordinates (x_i, y_i, z_i) in the final frame, will have coordinates, in the original frame:

[ x₀ y₀ z₀ 1 ]^t = ⁰T_n [ x_n y_n z_n 1 ]^t

where

⁰T_n = ⁰T₁ ¹T₂ ... ^n-1T_n

is the composition of successive DT transformations. The coordinates, in the original frame, of end-effector position will be given by the formula abiove, where x_n = y_n = z_n = 0.
As mentioned above, each DT transformation is determined by some predefined, fixed parameters and one variable parameter (θ for rotational links, d for prismatic links); in this sense, the coordinates of the end-effector position become functions of n variables:

[ x₀ y₀ z₀ 1 ]^t = ⁰T₁(q₁) ¹T₂(q₂) ... ^n-1T(q_n) [ 0 0 0 1 ]^t

or

[ x₀ y₀ z₀ 1 ]^t = ⁰T_n(q₁, q₂, ... , q_n) [ 0 0 0 1 ]^t

where q_i = θ_i or d_i.

The task of inverse kinematics is to determine, for given initial values of parameters q_i and a given target point with coordinates (x_target, y_target, z_target) the necessary modifications of (q_i), 1 ≤ i ≤ n, such that the resulting end--effector position will become as close to target as possible. It is, in essence, an optimization process. Many procedures have been proposed, most of them for specific configurations. Here we use a method which is general, for any configuration and uses "classical" optimization methods, as in [3].

References

Credits

CGIC library
CGIC copyright 1996-2011 by Thomas Boutell and Boutell.Com.

Ogg-Theora-Vorbis library
Copyright (C) 2002-2009 Xiph.org Foundation.

WebM library
Copyright (c) 2010, Google Inc. All rights reserved

Apache Xerces for C++ XML Parser

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