Robot Mechanics and Control Robot Mechanics and Control

Example 3.3:: Determining an A-Matrix (Mathematica)

3R Spatial Manipulator

This example formulates the A-matrices for a 3R spatial manipulator.


Define a Generic A-Matrix Function

A[θ_, d_, a_, α_] := {{Cos[θ], -Cos[α] Sin[θ], Sin[α] Sin[θ], a Cos[θ]}, {Sin[θ], Cos[α] Cos[θ], -Cos[θ] Sin[α], a Sin[θ]}, {0, Sin[α], Cos[α], d}, {0, 0, 0, 1}};

DH Parameters

θ1=.; d1=-51; a1=178; α1=-90 Degree;
θ2=.; d2=0; a2=179; α2=-90 Degree;
θ3=.; d3=0; a3=153; α3=0 Degree;

A10: Transformation from the Link 1 Frame to the Link 0 Frame

A10 = A[θ1, d1, a1, α1];


A21: Transformation from the Link 2 Frame to the Link 1 Frame

A21 = A[θ2, d2, a2, α2];


A32: Transformation from the Link 3 Frame to the Link 2 Frame

A32 = A[θ3, d3, a3, α3];


This Mathematica file resides at For a complete description of this example, along with the requisite background, definitions, and notation, see the textbook Fundamentals of Robot Mechanics by G. L. Long, Quintus-Hyperion Press, 2015.