Figure 13 shows an experimental X-2 motion platform. The spread-angle range can be adjusted from 21° to 69°. Fig. 14 shows the maximum pitch and roll angles with different spread-angles, revealing that the maximum pitch angle rises and the maximum roll angle falls as the spread-angle rises. Fig. 15 shows the workspaces for different spread-angle settings.
For instance, the maximum roll angle can reach about 21° but the maximum pitch angle falls to about 8° when the spread-angle φ = 21°. The maximum pitch angle remains to 21° while the maximum roll angle falls to 7.5°, when the spread-angle reaches φ = 69°. The different spread-angles result in a tradeoff between maximum roll and pitch angles.
Figure 16 shows the spread-angle that satisfies the workspace symmetry condition. The intersection point indicates that the spread-angle of a symmetrical workspace is 41.9°.
Figure 17 shows the variation of kinetic energy between different spread-angles, revealing that the kinetic energy rises as the spread-angle rises, and the slope becomes steeper for large spread-angles.
Figures 18 and 19 show the potential energy and the variation of gradient of potential energy between different spread-angles. Fig. 18 shows the potential energy field for different spread-angles, in which the arrows point toward the direction of rising potential energy. Fig.
20 shows the variation of the 2-norm of gradient of potential energy between different spread-angles. These figures indicate that the potential energy rises rapidly in two cases: when
both sliders go forward to their traversal limits, or when one slider goes forward and the other goes backward to the traversal limits. The first case presents a large pitch angle, and the second case exhibits a large roll angle. Thus, the slopes with maximum potential energy only occur at the maximum pitch or roll angle. Fig. 21 indicates that the maximum kinetic energy occurs at the largest spread-angle, and the minimum kinetic energy occurs at the smallest
spread-angle. Fig. 22 indicates that the minimum variation of potential energy occurs at about φ = 33°.
The individual performance indices are described qualitatively in the previous sequels.
The optimization was applied to cost function and fitness by the genetic algorithm for different purposes, and with different weights. Four sets of weights were adopted for different applications: (i) workspace symmetry, (ii) minimizing the infinity norm of the kinetic energy, (iii) minimizing the infinity norm of the gradient of potential energy, and (iv) multi-objective optimization.
Application (i) emphasized the workspace symmetry, with weights wb = 80, wT = 10, wV
= 10. The GA optimization yielded φ = 41.94° after 100 generations of searching.
Application (ii) emphasized minimizing the infinity norm of the kinetic energy, and adopted weights wb = 10, wT = 80, wV = 10. The GA optimization yielded φ = 21.22° after 100 generations of searching.
Application (iii) emphasized minimizing the infinity norm of the gradient of potential
energy, and weights wb = 10, wT = 10, wV = 80. The GA optimization yielded φ = 33.49°
after 100 generations of searching.
Application (iv) was subjected to a specific multi-objective optimization, and adopted weights of wb = 40, wT = 30, wV = 30. The GA optimization yielded φ = 36.53° after 100 generations of searching.
Figures 23, 24, 25 and 26 show analytical results for individual applications. Subplot (a) shows how the genetic algorithm generates the new best so far solution at each generation.
Subplot (b) shows the workspace. Subplot (c) shows the potential energy. Subplot (d) shows the gradient of potential energy. Subplot (e) shows the norm of the potential energy gradient.
The Subplot (f) shows the kinetic energy.
A simulation of Pseudo-Flight-Object (PFO) produced by IMON Corp. was applied in this study. The geocentric position and the body acceleration data of the aircraft produced from the equation of motion (EOM), were taken as inputs to the proposed motion cueing strategy in the PFO software. The outputs of the experiment were the motor position commands to the 3-rotational-DOF motion simulator, as shown in Fig. 27. The results were compared to the classical washout filter (CLWF). Figure 28 consists of four plots used to demonstrate the flight trajectory in 3-D view, front view, side view, and top view, respectively.
The flight data including longitudinal (pitch + x-acceleration), lateral (roll + y-acceleration), and yaw motions. Data of individual Euler angles (yaw, pitch and roll) were provided
simultaneously to the proposed ROMA algorithm to yield the motion cue of the pilot. Hence, one complex simulation was performed to test all motion cues simultaneously.
Various aspects of the experimental results are shown. The real-time optimal motion-cueing algorithm (ROMA) introduced in this paper is derived from the classical washout filter (CLWF), which is depicted in Fig. 8. ROMA should perform similar sustained motions, or low-frequency linear motions, to CLWF. The sustained motion activates only the attitude and the residual tilt control, for which the calculation is mainly derived from the CLWF method. There is merely difference when comparing the performances of sustained motion along the x and y axes between these two methods.
However, the CLWF is designed for the general 6-DOF motion simulator. The ROMA is designed for a 3-DOF flight simulator, specifically the rotational motion simulator with insufficient spatial DOF. The CLWF shows poor performance on the high-frequency linear motion when implemented on the rotational motion simulator.
Figures 29 and 30 show the comparison of the high-frequency (onset) linear motion cues along the x and y axes. In this case, the CLWF generates no output to the rotation motion. The proposed algorithm ROMA eventually converts the onset linear motion to a rotation command based on (3.14) and (3.20), and presents the onset linear motion on the motion simulator.
Figures 31 and 32 show the error between VR commands and actual linear acceleration by different motion cueing algorithms. The data indicate that the error rises rapidly as the
frequency of the linear motion increases when adopting the classical method.
Figures 33 and 34 show the results of a mixture of sustained and onset motion. The ROMA could optimize the motion cueing without violating the inequality equation, and remained within to the mechanical bounds of a motion simulator. The CLWF failed to do so;
therefore, the mechanical structure of the motion simulator can be damaged by CLWF.
Nevertheless, these figures indicate that the onset motion cue can be generated by ROMA rather than CLWF.
Figure 35 illustrates the effect of the yawing washout and ROMA. The washout motion continuously returns the cockpit to its home position when the indifference threshold is detected as in (3.54). The washout motion moving the cockpit back to its home position is performed with a velocity at the indifference threshold, as revealed in Fig. 36.