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Tham khảo tài liệu 'fundamentals_of_robotic_mechanical_systems part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 76 3. Fundamentals of Rigid-Body Mechanics coefficients a and I being determined from the condition that the product of At a by its inverse should be 1 which yields __d _ 3 3.16 2 1 cos ị 2 1 cos ự and hence ATA - J _1 1- T 3 17 2 1 cos ị 2 1 cos ị On the other hand ATb Q - lf Q - l a - dA 3.18 Upon solving eq. 3.13 for p. and substituting relations 3.17 and 3.18 into the expression thus resulting one finally obtains 2 1 cos p We have thus defined a line of the rigid body under study that is completely defined by its point p. of position vector p. and a unit vector e determining its direction. Moreover we have already defined the pitch of the associated motion eq. 3.8 . The line thus defined along with the pitch determines the screw of the motion under study. 3.2.2 The Placker Coordinates of a Line Alternatively the screw axis and any line for that matter can be defined more conveniently by its Pliicker coordinates. In motivating this concept we recall the equation of a line passing through two points p. and p. of position vectors p. and p. as shown in Fig. 3.1. If point p lies in then it must be collinear with p. and p. a property that is expressed as p. - p. X p - p. 0 FIGURE 3.1. A line passing through two points. 3.2 General Rigid-Body Motion and Its Associated Screw 77 or upon expansion p. - p. X p p. X p. - p. 0 3.20 If we now introduce the cross-product matrices p. and p. of vectors p. and p. in the above equation we have an alternative expression for the equation of the line namely P. - p. p p. X p. - p. 0 The above equation can be regarded as a linear equation in the homogeneous coordinates of point P namely P. -p. p. X p. -p. p 0 3.21 It is now apparent that the line is defined completely by two vectors the difference p. p. or its cross-product matrix for that matter and the cross product p. X p. p. . We will thus define a 6-dimensional array 7Z containing these two vectors namely 7z. p- p- 3.22 P X p- P whose six scalar entries are the Phicker .
