jagomart
digital resources
picture1_Matrix Pdf 173810 | Bajd Cap2 Matricitrasf


 143x       Filetype PDF       File size 0.37 MB       Source: didawiki.di.unipi.it


File: Matrix Pdf 173810 | Bajd Cap2 Matricitrasf
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 unied mathematical description of ...

icon picture PDF Filetype PDF | Posted on 27 Jan 2023 | 2 years ago
Partial capture of text on file.
                 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
The words contained in this file might help you see if this file matches what you are looking for:

...Chapter homogenoustransformation matrices translational transformation in the introductory we have seen that robots either or rotational joints therefore need a unied mathematical description of transla tional and displacements displacement d givenbythe vector ai bj ck can be described also by following matrix h b trans c when using homogenous an arbitrary has fol lowing form x y q xyz t z atranslationaldisplacementofvectorq fora distance is obtainedby multiply ing with ax v cz thetranslation which presented multiplication homogenousmatrix equivalent to sum vectors j k xi yj zk i bajd et al robotics intelligent systems control automation science engineering doi springer business media simple example translationally displaced for ij thesameresultis obtained addingthe two will right handedrectangularcoordinate frame where rotations around three axes as shown figure are con sideredaspositive positiverotationsaroundtheselectedaxisarecounter clockwise looking from positive end axis towards ...

no reviews yet
Please Login to review.