Chapter2
                 Homogenoustransformation matrices
                 2.1 Translational transformation
                 In the introductory chapter we have seen that robots have either translational or
                 rotational joints. We therefore need a unified mathematical description of transla-
                 tional and rotational displacements. The translational displacement d,givenbythe
                 vector
                                                        d=ai+bj+ck,                                           (2.1)
                 can be described also by the following homogenoustransformation matrix H
                                                                      ⎡              ⎤
                                                                        100a
                                                                      ⎢              ⎥
                                                                        010b
                                             H=Trans(a,b,c)=⎢                        ⎥.                       (2.2)
                                                                      ⎣              ⎦
                                                                        001c
                                                                        0001
                 When using homogenous transformation matrices an arbitrary vector has the fol-
                 lowing 4×1form                           ⎡x⎤
                                                          ⎢y⎥              	
                                                     q=⎢ ⎥= xyz1 T.                                           (2.3)
                                                          ⎣z⎦
                                                            1
                     Atranslationaldisplacementofvectorq fora distance d is obtainedby multiply-
                 ing the vector q with the matrix H
                                                 ⎡100a⎤⎡x⎤ ⎡x+a⎤
                                                 ⎢010b⎥⎢y⎥ ⎢y+b⎥
                                            v=⎢                  ⎥⎢ ⎥=⎢               ⎥.                      (2.4)
                                                 ⎣001c⎦⎣z⎦ ⎣z+c⎦
                                                    0001 1                       1
                 Thetranslation, which is presented by multiplication with a homogenousmatrix, is
                 equivalent to the sum of vectors q and d
                                                                                              j+(z+c)k. (2.5)
                   v=q+d=(xi+yj+zk)+(ai+bj+ck)=(x+a)i+(y+b)
                 T. Bajd et al., Robotics, Intelligent Systems, Control and Automation: Science                   9
                 and Engineering 43, DOI 10.1007/978-90-481-3776-3_2,
                  c
                 Springer Science+Business Media B.V. 2010
                             10                                                                                              2 Homogenous transformation matrices
                                   In a simple example, the vector 2i+3j+2k is translationally displaced for the
                             distance 4iŠ3j+7k
                                                                                    ⎡                            ⎤⎡ ⎤ ⎡ ⎤
                                                                                        1004 2                                           6
                                                                                    ⎢010Š3⎥⎢3⎥ ⎢0⎥
                                                                           v=⎢                                   ⎥⎢ ⎥=⎢ ⎥.
                                                                                    ⎣                            ⎦⎣ ⎦ ⎣ ⎦
                                                                                        0017 2                                           9
                                                                                        0001 1                                           1
                                   Thesameresultis obtained by addingthe two vectors.
                             2.2 Rotational transformation
                             Rotational displacements will be described in a right-handedrectangularcoordinate
                             frame, where the rotations around the three axes, as shown in Figure 2.1, are con-
                             sideredaspositive.Positiverotationsaroundtheselectedaxisarecounter-clockwise
                             when looking from the positive end of the axis towards the origin of the frame O.
                             Thepositiverotationcanbedescribedalsobythesocalledrighthandrule,wherethe
                             thumbisdirectedalongtheaxistowardsitspositiveend,while the fingersshowthe
                             positivedirectionoftherotationaldisplacement.Thedirectionofrunningofathletes
                             onastadiumisalso an exampleof a positive rotation.
                                   Let us first take a closer look at the rotation around the x axis. The coordinate
                                            ′     ′     ′
                             frame x , y , z shown in Figure 2.2 was obtained by rotating the reference frame
                             x, y, z in the positive direction around the x axis for the angle α. The axes x and x′
                             are collinear.
                                   The rotational displacement is also described by a homogenous transformation
                             matrix. The first three rows of the transformationmatrix correspondto the x, y and z
                                                                                                                                                                            ′     ′            ′
                             axes of the reference frame, while the first three columns refer to the x , y and z
                                                                                                         z
                                                                                                                 Rot (z, γ )
                                                                                                      O
                                                                                                                                         Rot(y, β )          y
                                                               x          Rot(x, α )
                             Fig. 2.1 Right-hand rectangular frame with positive rotations
             2.2 Rotational transformation                                        11
                             z′        z
                                                                  y′
                                                    a
                                     x, x′                          y
             Fig. 2.2 Rotation around x axis
             axesoftherotatedframe.TheupperleftnineelementsofthematrixHrepresentthe
             3×3rotation matrix. The elements of the rotation matrix are cosines of the angles
             betweenthe axes given by the correspondingcolumn and row
                                       ′          ′          ′
                                   ⎡ x           y          z       ⎤
                                     cos0◦    cos90◦      cos90◦   0  x
                                   ⎢      ◦                  ◦      ⎥
                        Rot(x,α) =⎢cos90      cosα     cos(90 +α) 0⎥y
                                   ⎣      ◦      ◦                  ⎦
                                    cos90 cos(90 Šα)      cosα     0  z
                                       0001(2.6)
                                   ⎡                    ⎤
                                    10 00
                                   ⎢0cosα Šsinα 0⎥.
                                 =⎢                     ⎥
                                   ⎣0sinα      cosα   0⎦
                                    00 01
             The angle between the x′ and the x axes is 0◦, therefore we have cos0◦ in the
             intersection of the x′ column and the x row. The angle between the x′ and the y
             axesis 90◦, we put cos90◦ in the correspondingintersection.The angle betweenthe
             y′ and the y axes is α, the corresponding matrix element is cosα.
               To become more familiar with rotation matrices, we shall derive the matrix de-
             scribing a rotation around the y axis by using Figure 2.3. Now the collinear axes are
             y and y′
                                             y =y′.                             (2.7)
             By considering the similarity of triangles in Figure 2.3, it is not difficult to derive
             the following two equations
                                            ′        ′
                                        x=x cosβ+z sinβ
                                            ′       ′
                                      z =Šx sinβ +z cosβ.                       (2.8)
           12                                   2 Homogenous transformation matrices
                                                            z
                                              z′
                                                 z′
                                                       b
                                T
                                                            z
                        x       x                         y, y′
                                               b
                                           x′
                             x′
           Fig. 2.3 Rotation around y axis
              All three equations (2.7)and(2.8) can be rewritten in the matrix form
                                        ′     ′   ′
                                    ⎡ x      y   z     ⎤
                                      cosβ   0sinβ 0 x
                                    ⎢                  ⎥
                                        0100y
                           Rot(y,β)=⎢                  ⎥ .            (2.9)
                                    ⎣Šsinβ 0cosβ 0⎦z
                                        0001
              The rotation around the z axis is described by the following homogenous trans-
           formationmatrix           ⎡                 ⎤
                                     ⎢cosγ  Šsinγ  00
                                                       ⎥
                                       sinγ  cosγ  00
                            Rot(z,γ)=⎢                 ⎥.            (2.10)
                                     ⎣                 ⎦
                                        0010
                                        0001
              In a simple numerical example we wish to determine the vector w which is ob-
           tained by rotating the vector u = 7i+3j+0k for 90◦ in the counter clockwise i.e.
                                             ◦           ◦
           positive direction aroundthe z axis. As cos90 =0andsin90 =1,it is notdifficult
           to determine the matrix describing Rot(z,90◦) and multiplying it by the vector u