# Numerical Analysis of the Kinematic Accuracy of the Hermetic Harmonic Drive in Space Vehicles
## 4 Results
Furthermore, in the case of this type of generator, it was noticed in the numerical simulation that there is a slight delay in the rotation of the circular spline relative to the  
Figure 11: Stress distribution in characteristics cooperation areas of harmonic drive major axis and minor axis. nominal value. This is the result of clearance occurring between the mating gears, the stiffness of the computational models, or the slide of the generator. At the beginning of cooperation, it also follows the \"tooth arrangement\" considered with a minimum deflection of the toothed wheel rim to match the flanks of the teeth of both gears. This phenomenon was not described earlier, but it was observed in the results from numerical calculations after applying a sufficiently large scale of deformation during presentation moves and selecting a small range of results (Figure 12). The graphs of the angle of rotation of the flexspline relative to the rotation angle of the generator help to indicate characteristic discrepancies relative to theoretical values. They allow one to determine whether there is a deceleration or acceleration relative to the nominal values of the angle of rotation. However, to accurately compare the angular value of the difference for different types of wave generators, additional calculations were made in the spreadsheet, and their result is shown in Figure 13. In this plot, the graphs  
Figure 12: The angle of the rotation of circular spline as the result of generator work: (**a**) two-roller; (**b**) four-roller; (**c**) double-disc; (**d**) cam. of the angle error of the flexspline relative to the nominal values were compiled, which corresponds to the axis of the abscissa (X) in the graph. All fragments of charts obtained in results of numerical calculations FEM for different generators above the axis means acceleration of circular spline relative nominal value, whereas negative values mean delay of the circular spline. The chart can read that the largest temporary angular error occurs for a four-roll generator and is equal to 0.0062247 radians, so 21 arc minutes and similar values are for the two-roll generator. The slightest error read for the cam generator was 0.003 radians, so about 4 arc minutes. The wave generator recommends that the angular error is kept in the range of 1 to 9 arc minutes [37]. The kinematic error was also lower for the double-disc generator than for the roller generator and was equal to 0.0041344 radians. The kinematic error for the roller generators was the highest and exceeded the values even recommended for standard harmonic drives. These two types were omitted in the next part of the analysis. Clearly, gears equipped with roller generators still have high kinematic accuracy and can be used successfully in valve mechanisms and regulators or machine tool setting systems. They definitely have a much simpler construction than cam generators, which in many cases is decided during the choice of these wave generators in the harmonic drive. Selected models of harmonic drives with double-disc and cam generators were further analyzed. The influence of the load of torque on the kinematic accuracy of the harmonic drive was tested. The same computational models as in the previous calculations were used, but the only difference was that the circular spline was loaded with an additional torque of about 200 Nm. Two variants of the calculation were assumed. In the first, the direction of the torque was consistent with the direction of rotation of the generator (positive) and with the direction of the circular spline. In the second step, the moment (negative) resisted the movement of the wheel because it was opposite to the direction of its rotation. In the graphs shown in Figures 14 and 15, the rotation angle without a load in relation to the rotation angle of the generator was marked by a black color (0). After the circular spline was loaded by torque in accordance with its rotation, there was not a clear difference in the view of the model without load.