1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 7 Table 1.1 Summary of the transfer functions of the Stewart Platform’s legs i =1. . . 6 i =1 i =1 i =3 i =4 i =5 i =6 ˜GS,i 1011 s·(s+122.9) 1053.1 s·(s+55.5) 1123.5 s·(s+54.7) 780.9 s·(s+55.3) 988.1 s·(s+66.2) 1070.9 s·(s+59) have an influence up to 100 Hz. Therefore, we fitted a transfer function ˜GS of the following form to the measured data of the real plant GS: ˜GS,i(s) = k s · (Ts +1) = k · ωc s · (s +ωc) , with ωc = 1 T . (1.3) This dynamic behavior can also be interpreted as a mass with damping and no stiffness. This is quite illustrative, as the torques coming from the electric motors need to push the mass of the Stewart Platform’s legs and upper platform, which possess damping (friction, compliance, . . . ) but almost no stiffness. We identified the transfer functions listed in Table 1.1 for each leg. The cutoff frequencies, i.e., the frequency above which the amplitudes fall with a slope of −40 dB/decade is at frequencies of approximately 8–10 Hz. The amplitude falls below zero above 2–3 Hz, which implies that the input signal is mainly attenuated above and the dynamic limits of the Stewart Platform are in this range. The task of the feedforward controllers is to optimally use and even increase the dynamic range of the actuator to its mechanical limits such as velocity limit and maximum motor voltage. 1.3.3 Benchmark Problem To compare the tracking performance of the three different feedforward control schemes in combination with the existing P-PI-PI cascade, the following benchmark problem is used: A sine trajectory with frequency f and amplitude Aaround the initial positiont0 should be followed. Hence, the desired position trajectory td is of the form td(t) = ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎝ td,X td,Z td,Z td,φ td,θ td,ψ ⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎠ =t0 + ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎝ 0 0 A· sin(2πft) 0 0 0 ⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎠ , with t ∈ 0, 1 f (1.4) and the Stewart Platform is at its initial position t0 for t < 0 and t > 1 f . This trajectory command only prescribes the trajectory in Stewart Platform’s Z-direction, which is the vertical direction, and the other translations (XandY) and rotations (φ, θ, ψ) are set to 0. This trajectory commandtd is transformed by inverse kinematics, which gives the kinematic relationship between the upper platform and the leg lengths, to the desired leg lengths zi (compare to Fig. 1.2), for each leg i =1. . . 6. Between two successive sine trajectories there was a pause of 1 s at positiont0. 1.3.4 Parameters Setting for the Experiments To keep the results comparable, all experiments have been conducted several times with the same conditions, such as the same maximum actuator velocity of 40 mm s and the same constants in the P-PI-PI cascaded feedback controller. For MBDC, a scaling factor αMBDC was introduced to scale the feedforward signal. This is necessary because of the nonlinear transfer behavior of the Stewart Platform, i.e., the amplification from input voltage to leg displacement depends on the amplitude of the input voltage. In the tuning process, a continuous sine command with f = 2 Hz was used and the factor αMBDC was tuned for each leg to its optimum. The obtained values lie in the range of (1.8,3.8). Likewise, a scaling factor αMFIIC was used for MFIIC. Here, the scaling factor determines the convergence rate and the achievable minimum error with this scheme. If the scaling factor is chosen too small, the convergence is slow and the remaining error is large compared to the
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