• the vectors Tl, Trof the motor moments at each waypoint;
• the vector δ of the steering angle at each waypoint.
The special cases with non-positive speeds discussed in Section 5.6.1 also result in additional tasks during the export of the results.
Zero speed Remind that for the waypoints where the aircraft is supposed to stop, a
small speed limit such as 0.1 m/s was set upon trajectory definition. An addi- tional piece of information is needed for the subsequent toolchain steps whether the aircraft should be stopped at those waypoints; also, the stop duration needs to be defined since the optimization results are a function of the path variable, hence they cannot contain any time-based information. For the former issue, a boolean flag is introduced for each waypoint assuming the value true if the air- craft needs to be stopped at that waypoint, and false elsewhere. For consistency, true values should only be assigned for waypoints subject to the small speed limit mentioned. All waypoint flags are summarized in a vector and exported. For the stopping times, data on the duration of each stop need to be made available, for example in form of a table.
Negative speed The trajectory portions where backward driving takes place were
mirrored upon trajectory definition and traveled on with positive speeds. This results in opposite signs for the motor torques. Consequently, the affected parts of the vectors Tl, Trneed to be changed in sign. Clearly, this also applies to the
affected parts of the speed vector vx. Note that the mirroring did not affect the
steering angle, thus no change is needed on the steering angle vector δ.
5.7 Summary
This chapter illustrated the generation of optimal path-following data based on a con- vex problem formulation. Starting from a three-wheeled vehicle model representing the aircraft on ground, the dynamic equations in the global reference system were derived (Section 5.1). The path-following optimization problem has then been formu- lated as a function of the path variable along the trajectory (Section 5.3). Elements of the problem such as the electric driving system and especially the APU have been modeled in convex form too and integrated into the optimization problem. A method for integrating thermal constraints has been presented in Section 5.3.5 by indirectly limiting the amount of heat produced by the motors. Two cost functions were pre- sented in Section 5.3.6: time minimization and fuel consumption minimization in the APU.
Having defined the problem completely, a discretization was performed (Section 5.4) and the problem was solved for a sample trajectory. The effect of different cost functions with and without thermal limits was shown. A significant result was the very small difference in travel time and fuel consumption when comparing the op-
timizations based on the different cost functions (Section 5.4.3). It is concluded that due to the high APU idle consumption and the relatively small amount of fuel used for power generation, a consumption minimizing driving strategy is only marginally different from a time minimizing one.
Constraining the time required to pass a waypoint was discussed in Section 5.5. The discussion focused in particular on minimum times that need to have elapsed from the start in order to clear the considered waypoints. A method to include minimum- time constraints into the convex problem in an approximate form was presented. An iterative algorithm was realized to find the related constraint parameters.
Finally, considerations of practical nature about using the convex path-following optimization were discussed in Section 5.6. In particular, Section 5.6.1 described how to discretize generic taxi trajectories and how to deal with limitations of the adopted problem formulation such as strictly positive speeds. The aircraft and system data needed for the problem definition were summarized in Section 5.6.2. The data from the optimization results that need to be treated and saved for the subsequent steps of the toolchain were listed in Section 5.6.4.
6 Ground Controller for Path
Following
This chapter discusses the development of a ground controller for the model-based path following of taxi trajectories. Within the general strategy of the work described in Chapter 1, model-based simulation is used to determine the performances of the electric taxi system. The aircraft commands and kinematic quantities that have been calculated through the convex optimization algorithm are used as inputs of the aircraft model during simulation. A number of reasons prevents the aircraft model from following the assigned path precisely, such as modeling and parameter uncertainties caused by the approximations and simplifications introduced in the model used in the convex optimization problem. For this reason, feedback control is necessary to eliminate the tracking error during the path following simulation. The aircraft ground motion will be governed by the sum of feed-forward commands calculated with the offline optimization and feedback commands dependent on the errors between the actual and the desired position. The control design task will be performed concretely on a narrow-body aircraft with characteristics shown in Table 6.1, which is consistent with the aircraft type used throughout the present work.
Beside stability and robustness, the ground controller should guarantee a similar tracking performance in the whole operating envelope to allow a sound comparison of different aircraft and system architectures.
In order to tune and analyze the behavior of the motion controller, the simplified nonlinear dynamic model of the aircraft on ground (5.1), (5.2), (5.3) already used in the definition of the convex optimization problem will be considered.
The longitudinal and lateral error will be controlled by two independent feedback loops, as illustrated in the following.
Table 6.1: Parameters of aircraft considered in the development of the ground controller
Parameter Symbol Value
Maximum Take-Off Weight (MTOW) m 73,500 kg
Operating Empty Weight (OEW) 41,100 kg
Inertia coeff. around z-axis Jz 3.78 · 106kg · m2
Main gear y-distance a 3.795 m
Main gear x-distance bh 2.51 m
Nose gear x-distance bf 10.19 m
Cornering stiffness
of nose gear cf 1.49 · 105N/rad
Cornering stiffness
of main gear cr 4.00 · 105N/rad
Coefficient of quadratic motion resistance kaero 1.297 N · s2/m2
Effective main gear wheel radius RR 0.5198 m
Reference speed for longitudinal control v¯x 5 m/s