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Figure 9 – uploaded by Dr.Stanislav S.Makhanov

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Fig. 9. (a) The actual tool trajectory on the 1-1 machine before optimization. (b) The error. (c) The tool trajectory with an optimal setup rq,r,,T12. (d) The error. (e) The tool trajectory with an optimal setup r,,7),T12,T4. (f) The error. (g) The tool trajectory with an optimal setup 74,7p),T12,T34,T4. (h) The error.

Figure 9 (a) The actual tool trajectory on the 1-1 machine before optimization. (b) The error. (c) The tool trajectory with an optimal setup rq,r,,T12. (d) The error. (e) The tool trajectory with an optimal setup r,,7),T12,T4. (f) The error. (g) The tool trajectory with an optimal setup 74,7p),T12,T34,T4. (h) The error.

Related Figures (16)

Fig. 1. Changing an initial workpiece setup by rotating around the Z-axis and the Y-axis.
Fig. 1. Changing an initial workpiece setup by rotating around the Z-axis and the Y-axis.
Fig. 3. (a) A 2-0 machine. (b) The reference coordinate systems. The solution of this system establishes, for a particular machine and desired surface, not only the optimal position Five-axis machines are characterized by nonlinear kinematics due to the two additional degrees of freedom that control tool orientation. In this section, a closed-form
Fig. 3. (a) A 2-0 machine. (b) The reference coordinate systems. The solution of this system establishes, for a particular machine and desired surface, not only the optimal position Five-axis machines are characterized by nonlinear kinematics due to the two additional degrees of freedom that control tool orientation. In this section, a closed-form
Fig. 2. (a) The desired tool trajectory. (b) The actual tool trajectory before optimization. (c) The desired tool trajectory, (d) the actual tool trajectory for the new (optimal) setup (the workpiece is rotated by 99.23° around Z-axis and —41.28° around the Y,-axis and shifted by [21.47 49.33 33.67] mm). The desired and actual tool trajectories almost coincide in the optimal setup. The error has been reduced by 99.9%.
Fig. 2. (a) The desired tool trajectory. (b) The actual tool trajectory before optimization. (c) The desired tool trajectory, (d) the actual tool trajectory for the new (optimal) setup (the workpiece is rotated by 99.23° around Z-axis and —41.28° around the Y,-axis and shifted by [21.47 49.33 33.67] mm). The desired and actual tool trajectories almost coincide in the optimal setup. The error has been reduced by 99.9%.
Fig. 5. (a) 0-2 machine. (b) The reference coordinate systems. coordinate systems, respectively, and ,, the corresponding Now the machine coordinates M, and M,,, must be eliminated so that the resulting trajectory depends only on the workpiece coordinates and orientation of the tool. M,, M,+1 are eliminated by using the inverse transformation M,= RIM, R,, W,). Substituting M,,M,, +; into (4) yields
Fig. 5. (a) 0-2 machine. (b) The reference coordinate systems. coordinate systems, respectively, and ,, the corresponding Now the machine coordinates M, and M,,, must be eliminated so that the resulting trajectory depends only on the workpiece coordinates and orientation of the tool. M,, M,+1 are eliminated by using the inverse transformation M,= RIM, R,, W,). Substituting M,,M,, +; into (4) yields
Fig. 4. (a) A 1-1 machine. (b) The reference coordinate systems. representation of the trajectory of the tool tip is derived for general machine kinematics. This is followed by an analysis of the invariance of the tool tip trajectory for each of the three basic types of five-axis machines. Let us invoke the inverse kinematics to transform part- surface coordinates W, into machine coordinates M,=(x,, Yp:%p) as follows: M, = RTM, R,, W,). The rotation angles R = R(t) = (a(t), b(t)) and the machine coordinates of the tool tip M=M(t) are assumed to change linearly between the prescribed points, namely, M(t)=1M,,; +1—2M,, R(t) =fR,4, +(1—OR,, where ¢ is a fictitious time coordinate (O<t<1). Transforming machine coordinates M back to workpiece coordinates W for every t yields
Fig. 4. (a) A 1-1 machine. (b) The reference coordinate systems. representation of the trajectory of the tool tip is derived for general machine kinematics. This is followed by an analysis of the invariance of the tool tip trajectory for each of the three basic types of five-axis machines. Let us invoke the inverse kinematics to transform part- surface coordinates W, into machine coordinates M,=(x,, Yp:%p) as follows: M, = RTM, R,, W,). The rotation angles R = R(t) = (a(t), b(t)) and the machine coordinates of the tool tip M=M(t) are assumed to change linearly between the prescribed points, namely, M(t)=1M,,; +1—2M,, R(t) =fR,4, +(1—OR,, where ¢ is a fictitious time coordinate (O<t<1). Transforming machine coordinates M back to workpiece coordinates W for every t yields
Minimal sets of optimization parameters Table 1
Minimal sets of optimization parameters Table 1
Fig. 6. Dependent parameters T\7,+7>3,=c. The 2-0 machine.
Fig. 6. Dependent parameters T\7,+7>3,=c. The 2-0 machine.
Performance of the optimization method. The sweep surface Table 2 Table 3
Performance of the optimization method. The sweep surface Table 2 Table 3
Performance of the optimization method. The two-bell surface
Performance of the optimization method. The two-bell surface
Fig. 7. (a) Sweep surface. (b) Two-bell surface. W. Anotaipaiboon et al. / International Journal of Machine Tools & Manufacture 46 (2006) 964-977
Fig. 7. (a) Sweep surface. (b) Two-bell surface. W. Anotaipaiboon et al. / International Journal of Machine Tools & Manufacture 46 (2006) 964-977
Fig. 8. (a) CC points on the desired tool trajectory. (b) The extracted tool trajectory (sampling points). (c) The actual tool trajectory on the 2-0 machine before optimization. (d) The error. (e) The tool trajectory with an optimal setup r,, 7, 7:2. (f) The error. (g) The tool trajectory with an optimal setup rz, ry,712, T23. (h) The error.
Fig. 8. (a) CC points on the desired tool trajectory. (b) The extracted tool trajectory (sampling points). (c) The actual tool trajectory on the 2-0 machine before optimization. (d) The error. (e) The tool trajectory with an optimal setup r,, 7, 7:2. (f) The error. (g) The tool trajectory with an optimal setup rz, ry,712, T23. (h) The error.
Fig. 10. (a) The actual tool trajectory on the 0-2 machine before optimization. (b) The error. (c) The tool trajectory with an optimal setup r,,ry. (d) The error. (e) The tool trajectory with an optimal setup r,, 7», Ty. (f) The error. (g) The tool trajectory with an optimal setup rz, r,, T3, T34, Ty. (h) The error
Fig. 10. (a) The actual tool trajectory on the 0-2 machine before optimization. (b) The error. (c) The tool trajectory with an optimal setup r,,ry. (d) The error. (e) The tool trajectory with an optimal setup r,, 7», Ty. (f) The error. (g) The tool trajectory with an optimal setup rz, r,, T3, T34, Ty. (h) The error
Performance of the optimization measured by the required number of CC points 2-0 machine. The maximum allowed machining error=0.01 mm. Table 4
Performance of the optimization measured by the required number of CC points 2-0 machine. The maximum allowed machining error=0.01 mm. Table 4
Fig. 11. The sweep surface machined with different setups. The workpiece on the top (a), (b) has been cut with a conventional setup (r,=0, 7,=0). The workpiece at the bottom (c), (d) has been cut with r,= 1.6008 radian and r, =0.5236 radian (the optimal setup).
Fig. 11. The sweep surface machined with different setups. The workpiece on the top (a), (b) has been cut with a conventional setup (r,=0, 7,=0). The workpiece at the bottom (c), (d) has been cut with r,= 1.6008 radian and r, =0.5236 radian (the optimal setup).
Inverting with regard to a and b yields
Inverting with regard to a and b yields
Applying the same procedure to the for 0-2 machine (see Fig. 5) yields
Applying the same procedure to the for 0-2 machine (see Fig. 5) yields
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