Difference between revisions of "A-matrices"

From Robotics
Jump to: navigation, search
Line 1: Line 1:
 
{{Navigation|before=[[Denavit-Hartenberg parameters]]|overview=[[Denavit-Hartenberg Convention]]|next=[[Typical link examples]]}}
 
{{Navigation|before=[[Denavit-Hartenberg parameters]]|overview=[[Denavit-Hartenberg Convention]]|next=[[Typical link examples]]}}
  
In the previous articles, it was described how the transformation between consecutive links of a manipulator can be described using [[Assigning coordinate frames|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 [[Assigning coordinate frames|coordinate frame]] <math>K_n</math> with respect to frame <math>K_{n-1}</math> like shown below.
+
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 [[Assigning coordinate frames|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 [[Assigning coordinate frames|coordinate frame]] <math>K_n</math> with respect to frame <math>K_{n-1}</math> like shown below.
  
 
[[File:dh-params-steps.png|center|950px]]
 
[[File:dh-params-steps.png|center|950px]]
  
Based on this, the A-matrices are used to describe the precise transformation of a link with respect to the previous link resulting of the 4 successive transformations described by the [[Denavit-Hartenberg parameters]].
 
  
 
[[File:amatrices.png|right|450px]]
 
[[File:amatrices.png|right|450px]]
 
+
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|combinations of transformations]], the combined A-matrix is determined as follows:
 +
:<math>
 +
\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}
 +
</math>
  
 
[[Category:Article]]
 
[[Category:Article]]
 
[[Category:Denavit-Hartenberg]]
 
[[Category:Denavit-Hartenberg]]

Revision as of 11:54, 17 November 2015

← 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.

Dh-params-steps.png


Amatrices.png

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}