i 1 i, : Om-1 Om_ i with m the system order G2GPID(s). With these values, (108) can be rewritten as
Additionally, combining (106) with (112) and the choice it yields
where the matrix (/ — kfJocT)A < 0 with only one eigenvalue zero, while (1 - ^(s)) is interpreted as a high-pass filter for the errors x and ec that are produced by fast changes of n in order to reach an effective tracking of/i«.
In order for (114) to give exponentially stable homogeneous solutions x(f), the first element of the initial condition vector x(0) must be set to null. Moreover, it is noticing that only high-frequency components of n can excite the state error dynamics and that these avoid vanishing errors. So, the price to be be paid for including the thruster dynamics in the control approach is the appearance of the vector error Af which is bounded and its magnitude depends just on the energy of the filtered n in the band of high frequencies. According to (46), the influence of n on Af is attenuated by small values of the axial velocity of the actuators va. In this way, the benefits in the control performance for including the thruster dynamics are significant larger than those of not to accomplish this, i.e., a vehicle model with dominant dynamics only.
Finally, the reference vector nr for the inputs of all the thrusters is calculated by means of (104) and (105) in vector form as
The estimation of nr closes the observer approach embedded in the extended structure of the adaptive control system described in Fig. 3.
To illustrate the performance of the adaptive guidance system presented in this Chapter, a case study is selected composed on one side of a real remotely teleoperated vehicle described in (Pinto, 1996, see also Fig. 1) and, on the other side, of a sampling mission application over the sea bottom with launch and return point from a mother ship. These results are obtained by numerical simulations.
The geometric reference path n for the mission is shown in Fig. 4. The on-board guidance system has to conduct the vehicle uniformly from the launch point down 10(m) and to rotate about the vertical line 3/4 n(rad) up to near the floor. Then, it has to advance straight 14(m) and to rotate again n/4 (rad) to the left before positioning correctly for a sampling operation. At this point, the vehicle performs the maneuver to approach 1(m) slant about n/4 (rad) to the bottom to take a mass of 0.5 (Kg), and it moves back till the previous position before the sampling maneuver. Afterwards it moves straight at a constant altitude, following the imperfections of the bottom (here supposed as a sine-curve profile). During this path the mass center G is perturbed periodically by the sloshing of the load. At this path end, the vehicle performs a new sampling maneuver taking again a mass of 0.5 (Kg). Finally the vehicle moves 1(m) sidewards to the left, rotates 3.535 (rad) to the left, slants up 0.289 (rad) and returns directly to the initial position of the mission. Moreover, the corners of the path are considered smoothed so that the high derivatives of n exist.
Here, the adaptive control system is applied according to the structure of the Fig. 3, i.e., with the vehicle dynamics in (24)-(25), the thruster dynamics in (46)-(50) and (52), the control law in (63), the adaptive laws in (72)-(78), and finally the estimation of the thruster shaft rate given in (115). The saturation values for the actuator thrust was set in ±30N.
ROV (at start position)
Moreover, the controller design gains are setup at large values according to theorem III in order to achieve a good all-round transient performance in the whole mission. These are
Besides, the design parameters for the observer are setup at values
The main design parameter kn was chosen roughly in such a way that a low perturbation norm | Af | „ in the path tracking and an acceptable rate in the vanishing of the error (nr -n) occur. The remainder observer parameters kn, k^ and k„2 were deduced from the thruster coefficients and kn according to (110), (111) and (113), respectively. Finally, the battery of filters g3(s) was selected with a structure like a second-order system.
Now we present simulation results of the evolutions of position and rate states in every mode. The vehicle starts from a position and orientation at rest at t0 = 0 that differs from the earth-fixed coordinate systems in
Moreover, the controller matrices Ui(0) are set to null, while no information of the system parameters was available for design aside from the thruster dynamics.
In Fig. 5 the evolutions of position and kinematics modes are illustrated (left and right, respectively). One sees that no appreciable tracking error occurs during the mission aside from moderate and short transients of about 5(s) of duration in the start phase above all in the velocities. During the phase of periodic parameter changes (160 (s) up to 340 (s)) and at the mass sampling points occurring at 130 (s) and 370.5 (s), no appreciable disturbance of the tracking errors was noticed. However in the kinematics, insignificant staggered changes were observed at these points and a rapid dissipation of the error energy took place. The sensibility of time-varying changes in the vehicle dynamics can be perceived above all in the thrust evolution. We reproduce in Fig. 6 the behavior of the eight thrusters of the ROV during the sampling mission; first the four vertical thrusts (2 and 3 in the bow, 1 and 4 in the stern) followed by the four horizontal ones (6 and 7 in the bow, 5 and 8 in the stern) (See Fig. 1). Both the elements of fideal and the ones of f are depicted together (see Fig. 6). It is noticing that almost all the time they are coincident and no saturation occurs in the whole mission time. Aside from the short transients of about 5(s) at the start phase, there is, however, very short periods of non coincidence between f and fideal. For instance, a transient at about 10(s) in the vertical thruster 3 occurs, where a separation in the form of an oscillation of (f - fideai) less than 4% of the full thrust range is observed (see f3 and n3 in Fig. 7, top). This is caused by jumps of the respective shaft rate by crossing discontinuity points around zero of the nonlinear characteristic.
Vertical Thrusters rJfAAAA/W\iv_
Horizontal Thrusters rJfAAAA/W\iv_
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