.

and be the vector from the origin of frame o₀x₀y₀z₀ to the origin of frame o₁x₁y₁z₁. Suppose the point is rigidly attached to coordinate frame o₁x₁y₁z₁, with local coordinates . We can express the coordinates of with respect to frame o₀x₀y₀z₀ using

(2.58)

Now consider three coordinate frames o₀x₀y₀z₀, o₁x₁y₁z₁, and o₂x₂y₂z₂. Let d₁ be the vector from the origin of o₀x₀y₀z₀ to the origin of o₁x₁y₁z₁ and d₂ be the vector from the origin of o₁x₁y₁z₁ to the origin of o₂x₂y₂z₂. If the point p is attached to frame o₂x₂y₂z₂ with local coordinates , we can compute its coordinates relative to frame o₀x₀y₀z₀ using

(2.59)

      and

(2.60)

      The composition of these two equations defines a third rigid motion, which we can describe by substituting the expression for from Equation (2.59) into Equation (2.60)

(2.61)

      Since the relationship between and is also a rigid motion, we can equally describe it as

(2.62)

      Comparing Equations (2.61) and (2.62) we have the relationships

(2.63)

(2.64)

      Equation (2.63) shows that the orientation transformations can simply be multiplied together and Equation (2.64) shows that the vector from the origin o₀ to the origin o₂ has coordinates given by the sum of (the vector from o₀ to o₁ expressed with respect to o₀x₀y₀z₀) and (the vector from o₁ to o₂, expressed in the orientation of the coordinate frame o₀x₀y₀z₀).

      2.6.1 Homogeneous Transformations

      One can easily see that the calculation leading to Equation (2.61) would quickly become intractable if a long sequence of rigid motions were considered. In this section we show how rigid motions can be represented in matrix form so that composition of rigid motions can be reduced to matrix multiplication as was the case for composition of rotations.

      In fact, a comparison of Equations (2.63) and (2.64) with the matrix identity

      (2.65)

where 0 denotes the row vector (0, 0, 0), shows that the rigid motions can be represented by the set of matrices of the form

(2.66)

      Transformation matrices of the form given in Equation (2.66) are called homogeneous transformations. A homogeneous transformation is therefore nothing more than a matrix representation of a rigid motion and we will use SE(3) interchangeably to represent both the set of rigid motions and the set of all 4 × 4 matrices of the form given in Equation (2.66).

      Using the fact that is orthogonal it is an easy exercise to show that the inverse transformation is given by

(2.67)

      In order to represent the transformation given in Equation (2.58) by a matrix multiplication, we must augment the vectors and by the addition of a fourth component of 1 as follows,

      (2.68)

      (2.69)

      The vectors and are known as homogeneous representations of the vectors