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Fundamentals_of_Robotic_Mechanical_Systems Part 7

Tham khảo tài liệu 'fundamentals_of_robotic_mechanical_systems part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 176 4. Kinetostatics of Simple Robotic Manipulators Notice that since the condition number is bounded from below the KCI is bounded from above by a value of 100 . Manipulators with a KCI of 100 are those identified above as isotropic because their Jacobians have at the configuration of minimum condition number all their singular values identical and different from zero. While the condition number of J defined in eq. 4.122 is conceptually simple for it derives from the polar-decomposition theorem it is by no means computationally simple. First it relies on the eigenvalues of JTJ which only in special cases can be found in symbolic form second even if eigenvalues are available symbolically their ordering from smallest to largest varies with the manipulator architecture and posture. An alternative definition of k J that is computationally simpler relies on the general definition of the concept namely Golub and van Loan 1989 J J J- 4.126a where II II stands for the matrix norm Golub and van Loan 1989 . While any norm can be used in the above definition the one that is most convenient for our purposes is the Frobenius norm II II J defined as J F 1 tr JJT n 4.126b where we have assumed that J is of n X n. Although a symbolic expression for J is not always possible this expression is more frequently available than one for the eigenvalues of JTJ. Moreover from the polar-decomposition theorem and Theorem 2.6.3 one can readily verify that J F Jr J J n 4.126c 4.9.1 Positioning Manipulators Here again we shall distinguish between planar and spatial manipulators. These are studied separately. Planar Manipulators If the manipulator of Fig. 4.24 is limited to positioning tasks we can dispense with its third axis the manipulator thus reducing to the one shown in Fig. 4.25 its Jacobian reduces correspondingly to J Es. Es. with Sj denoting the two-dimensional versions of vectors Tj of the Denavit-Hartenberg notation as introduced in Fig. 4.19. Now if we want to design this .