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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 4i3j+7k ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1004 2 6 ⎢0103⎥⎢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 α) cosα 0 z 0001(2.6) ⎡ ⎤ 10 00 ⎢0cosα sinα 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 =x 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) ⎣sinβ 0cosβ 0⎦z 0001 The rotation around the z axis is described by the following homogenous trans- formationmatrix ⎡ ⎤ ⎢cosγ sinγ 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
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