A-matrices

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Overview: Denavit-Hartenberg Convention

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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:

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

So the A-matrix for link $L_n$ can just be computed by setting in the parameters $\theta_n$, $d_n$, $l_n$ and $\alpha_n$ determined before. As shown in the figure on the right, $A_n$ then describes the transformation between the coordinate frames $K_{n-1}$ at the beginning and $K_n$ at the end of link $L_n$ including the static transformation as well as the translation or rotation caused by the joint $J_n$.