A-matrices

← Back: Denavit-Hartenberg parameters

Overview: Denavit-Hartenberg Convention

Next: Typical link examples →

The A-matrices describe the precise transformation between each two successive manipulator links. In the previous articles, it was described how the transformation can be described using local coordinate frames and the 4 Denavit-Hartenberg parameters. The parameters describe 2 translational and 2 rotational degrees of freedom, which correspond to 4 transformations, that are applied successively to transform coordinate frame $K_n$ with respect to frame $K_{n-1}$ like shown below.

The A-matrices now are used to combine the 4 successive transformations of the Denavit-Hartenberg parameters in one matrix. According to the figure above and following the rules for combinations of transformations, the combined A-matrix is determined as follows:

$\begin{align} A_n &= \mathbf{Trans}(0,0,d_n)\mathbf{Rot}(z,\theta_n)\mathbf{Trans}(l_n,0,0)\mathbf{Rot}(x, \alpha_n) \\ &= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d_n \\ 0 & 0 & 0 & 1 \end{array}\right] \left[\begin{array}{cccc} \cos{\theta_n} & -\sin{\theta_n} & 0 & 0 \\ \sin{\theta_n} & \cos{\theta_n} & 0 & 0 \\ 0 & 0 & 1 & d \\ 0 & 0 & 0 & 1 \end{array}\right] \left[\begin{array}{cccc} 1 & 0 & 0 & l_n \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \cos{\alpha_n} & -\sin{\alpha_n} & 0 \\ 0 & \sin{\alpha_n} & \cos{\alpha_n} & 0 \\ 0 & 0 & 1 & d \\ 0 & 0 & 0 & 1 \end{array}\right] \end{align}$