Robot Modeling and Control. Mark W. Spong

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      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 and , respectively. It can now be seen directly that the transformation given in Equation (2.58) is equivalent to the (homogeneous) matrix equation

      (2.70)

      A set of basic homogeneous transformations generating SE(3) is given by

      (2.71)

      (2.72)

      (2.73)

for translation and rotation about the x, y, z-axes, respectively.

      The most general homogeneous transformation that we will consider may be written now as

      (2.74)

      In the above equation n = (nx, ny, nz) is a vector representing the direction of x₁ in the o₀x₀y₀z₀ frame, s = (sx, sy, sz) represents the direction of y₁, and a = (ax, ay, az) represents the direction of z₁. The vector d = (dx, dy, dz) represents the vector from the origin o₀ to the origin o₁ expressed in the frame o₀x₀y₀z₀. The rationale behind the choice of letters n, s, and a is explained in Chapter 3.

      The same interpretation regarding composition and ordering of transformations holds for 4 × 4 homogeneous transformations as for 3 × 3 rotations. Given a homogeneous transformation H⁰₁ relating two frames, if a second rigid motion, represented by H ∈ SE(3) is performed relative to the current frame, then

      whereas if the second rigid motion is performed relative to the fixed frame, then

       Example 2.10.

      The homogeneous transformation matrix that represents a rotation by angle α about the current x-axis followed by a translation of b units along the current x-axis, followed by a translation of d units along the current z-axis, followed by a rotation by angle θ about the current z-axis, is

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