Chapter 2 : Homogeneous transformation matrices between
landmarks
2.1 Introduction
The synthesis of the control laws of manipulator robots requires the calculation of
certain mathematical models (such as: the direct geometric model (DGM)). The
calculation of this model is based on the use of homogeneous transformation matrices
between benchmarks.
2.2 Homogeneous coordinates
 Representation of a point
Let P be a point with Cartesian coordinates (Px, Py, Pz). We call homogeneous
coordinates of the point P the terms [w.Px, w.Py, w.Pz, w], where w is a scale factor. In
robotics, we take w=1. So, we represent the homogeneous coordinates of a point by the
vector:
 Px 
P 
P   y
 Pz 
 
1
 Representation of a direction
The representation of a direction (free vector) is also done by four components, but the
fourth is zero:
U x 
U 
U   y
U z 
 
1
 Representation of a plan
The plan α x + β y + γ z = δ is represented by a vector Q, where Q = [α β γ δ]. For any
point P belonging to the plan Q. the matrix product:
PQ  
 
 Px 
P 
  y   0
 Pz 
 
1
2.3 Homogeneous transformation
 transformations of landmarks
Let us undergo any transformation, translation and/or rotation, to the frame Ri, a
transformation which brings it to the frame Rj.
This transformation is defined by the matrix
where i s j , i n j , i a j respectively designate the unit vectors along the axes X j , Yj , Z j of
the frame Rj expressed in the frame Ri and where i p j is the vector expressing the origin
of the frame Rj in the frame Ri. It is also said that iT j defines the frame Rj in the frame
Ri and we also say, the matrix iT j represents the transformation making it possible to go
from the frame Ri to the frame Rj.
We also write:
 i Aj
Tj  
000
i
i
pj  isj

1  0
i
nj
0
i
aj
0
i
pj 

1 
where :
 the matrix A represents the rotation or orientation matrix of the frame Ri relative
to Rj.
 column p represents the translation of the frame Ri relative to the frame Rj
 in the case of a pure translation A=I3, such that I is the unit matrix.
Properties :
 the matrix A is orthogonal: A-1=AT
 jTi  iT j1
 rot(u,θ )-1 = rot(u,-θ) = rot(-u,θ )
 trans(u,θ )-1 = trans(u,-θ) = trans(-u,θ )

 sT p 
 T

A
 nT p 
 A p
1

 si T  
 then T  
 aT p 
000 1 


000
1


 translation of vectors
Let a vector i p j defining the point p1 in the frame Rj
We calculate the homogeneous coordinates of the point p1 in the frame Ri by the
following equation: 1 p j  iT j j p1
The matrix iTj therefore makes it possible to express in the frame Ri the coordinates of
a point in the frame Rj
Example :
We consider the transformation between two benchmarks presented by the following
figure:
-
From this figure, determine the matrix iTj and the matrix jTi by two methods
 pure transformation matrix
Let trans(a,b,c) be a transformation which denotes the translation a, b and c along the x,
y and z axes respectively. The transformation in this case is expressed by:
We subsequently use the notation trans(u,d) to designate a translation of a value d along
an axis u.
Property: trans(a,b,c)=trans(x,a)trans(y,b)trans(z,c).
 rotation matrices around the principal axes
- Let rot(x,θ) be a transformation which designates a rotation of θ with respect x
axis
Find the matrix iTj= rot(x,θ) ?
-
Let rot(y,θ) be a transformation which designates a rotation of θ with respect y
axis
Find the matrix iTj= rot(y,θ) ?
-
Let rot(z,θ) be a transformation which designates a rotation of θ with respect z
axis
Find the matrix iTj= rot(z,θ) ?
Properties :
-
composition of two matrices :
-
it is important to remember that the product of two transformation matrices is not
commutative T1T2 ≠ T2T1
-
if a frame has undergone k consecutive transformations and if each
transformation i (i=1,...k) is defined in relation to the current frame Ri-1, then the
transformation 0Tk can be deduced from the composition of the multiplications
to the right of these transformations: