Robot control

A simplistic model of a robot is to look at as a collection of links connected by joints. The tip of the robot is commonly referred to as the tool center point or TCP. As the joints rotate and the links contract and expand, the TCP will change position.

It is of great importance to know the position of the TCP in world coordinates. For example, for a robot to weld in a straight line, the actuators in the joints of the robot have to be controlled in complex manner.

Denavit-Hartenberg convention

To facilitate calculations, engineers use the Denavit-Hartenberg convention to help them describe the positions of links and joints unambiguously. Every link gets its own coordinate system. There are a few rules to consider in choosing the coordinate system:

1. the <math>z</math>-axis is in the direction of the joint axis
2. the <math>x</math>-axis is parallel to the common normal or if there is no common normal, <math>x_n = z_{n - 1} \times z_n</math>
3. the <math>y</math>-axis follows from the <math>x</math>- and <math>z</math>-axis by choosing it to be a right handed coordinate system.

Every link/joint pair can be described as a coordinate transformation from the previous coordinate system to the next coordinate system.

<math>{}^{n - 1}T_n

  = \operatorname{Rot}(z_{n - 1}, \theta_n) \cdot
    \operatorname{Trans}(z_{n - 1}, l_n) \cdot
    \operatorname{Trans}(x_n, d_n) \cdot
    \operatorname{Rot}(x_n, \alpha_n)</math>

The Rot(axis, angle) and Trans(axis, distance) are a shorthand for the transformation matrices.

<math>\operatorname{Trans}(z, d)

  = \begin{pmatrix}
    1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 \\
    0 & 0 & 1 & d \\
    0 & 0 & 0 & 1 \\
  \end{pmatrix}</math>

<math>\operatorname{Rot}(z, \theta)

  = \begin{pmatrix}
    \cos\theta & -\sin\theta & 0 & 0 \\
    \sin\theta &  \cos\theta & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1 \\
  \end{pmatrix}</math>

See also:

• robotic unicycle
• control theory
• ArtRobots