4D Trajectory Generation for Transport Aircraft using Bezier
5.5 Load Factor and Curvature of the Trajectory
5.5 Load Factor and Curvature of the Trajectory
In Section 3.5, the expression relating the load factor that an aircraft would experi-ence if a curved horizontal path of radiusRwas flown, at a given speed, was stated by:
R= Va2 gp
n2z−1 (5.21)
Moreover, the equation relating the load factor and speed with the radiusR0 of the path in a pitch up motion was denoted by:
(nz−1)g Va
= Va
R0 (5.22)
Therefore, since the path has a curvature radius of1/κ, the curvatureκof the path is directly related to (5.21) and (5.22). In this manner, for independent lateral or vertical maneuvers, the path that an aircraft can follow without infringing load factor lim-its can be computed, taking into account the speed of the aircraft and the curvature of the path generated by the position of the control points. Then, path constraints regarding the maximum curvature can be established for different flyability require-ments based on load factor limits.
According to (5.21) and (5.22), maximum radius of curvature for circular motions can be computed for different speeds after defining load factor limits. For exam-ple, a2.5gload factor at 200m/s is generated by a circular trajectory of a radius of 1,779.55m, equivalent to a roll angle of66.5◦ using (3.75). This angle happens to be the maximum roll angle permitted for transport aircraft.
To better exemplify the relation of the load factor with respect to the velocity, con-sider a scenario of the trajectory of an aircraft changing airways by changing its heading (fly-by turn). Three control points located at the same altitude and sepa-rated by 50 nautical miles (nm) from each other are used (see Table 5.4). The control
TABLE5.4: Control points for an aircraft changing airways.
1nm=1852m X(m) Y(m) Z(m)
P1 0 92,600 10,000
P2 92,600 92,600 10,000
P3 92,600 0 10,000
points are shown in Figure 5.13, along with the computed Bezier curves ofG2 con-tinuity. Moreover, the arc-lengths of each curve and the points where the Bezier curves are joined (using their timestamps) are shown.
Now, consider three different speeds for an aircraft to go through these control points, the numerical values of the timestamps where the Bezier curves are joined change according to Table 5.5. Furthermore, the transition between legs will gen-erate different load factors depending on the speed, shown in Figure 5.14. On the other hand, independent from the speed, the curvature of the path remains constant (see Figure 5.15).
Once that the relation of the load-factor and velocity has been shown, we can ex-plore the curvature constraints due to load factor limits while keeping a constant velocity.
In the example above, the distance from the trajectory to the pointP2is7,161.66m.
Thus, consider a trajectory to allow an aircraft to turn closer toP2with a constant ve-locity of200m/s. The first guess is to use the path reshaping proposed in Section 5.4.
FIGURE5.13: Control points of example, Arc lengths, and Times.
TABLE5.5: Arc lengths and times at different velocities Time(s)
Arc length(m) 170m/s 200m/s 230m/s
t0 0 0 0
l1 46,300 t1 272.35 231.5 201.30 l2 84,571 t2 769.83 654.35 569 l3 46,300 t3 1,042.18 885.85 770.3
FIGURE5.14: Load factor at different velocities.
However, due to the design of the algorithm, even if the trajectory is to be designed with a maximum deviation of10m from P2, the algorithm modifies the trajectory such that the load factor remains within bounds. This scenario is depicted in Fig-ure 5.16, where a zoom toP2shows how the generated trajectory curves before and
5.5. Load Factor and Curvature of the Trajectory 107
FIGURE5.15: Curvature at different velocities.
after the control point to satisfy the maximum deviation constraint. This trajectory corresponds to an "optimized" fly-by turn. The load factor of the generated trajec-tory, shown in Figure 5.17(a), reaches a peek of≈1.425g. The curvature of the path, shown in Figure 5.17(b), reaches a value of≈2.5x10−4(m−1). The distance from this trajectory toP2is1.36m.
Consequently, in order to have a trajectory close to the control pointP2 while
in-FIGURE5.16: Zoom to the generated trajectory close to the control pointP2with a 10m maximum deviation.
creasing significantly the load factor, two auxiliary control points are added (see Table 5.6), such that the direction of the path is not curved, as in the case when the reshaping is done. This trajectory corresponds to a regular fly-by turn. A zoom in close to the control point P2 shows the generated trajectory, which passes at a distance of1,203m from P2 (see Figure 5.18). Under this conditions, the load fac-tor, shown in Figure 5.19(a) reaches its2.5g limit. The curvature of the path pikes
(a) Load factor. (b) Curvature.
FIGURE5.17: Load factor and Curvature of the trajectory with a 10m maximum deviation.
at≈5.6x10−4(m−1)(see Figure 5.19(b)), which is close enough to the1,779mradius limit for circular trajectories, stated before.
Thus, if an aircraft flying at 200m/s is commanded to pass through three control points distributed in a "L" shape, a single turn maneuver leaves the aircraft at a dis-tance of1,203mfrom the middle control point before infringing load factor limits.
On the other hand, the proposed reshaping algorithm generates a flyable trajectory at1.36mfrom the same control point, achieving an "optimal" fly-by turn. Numerical computations can be done for any other velocity and for the pitching motion.
TABLE5.6: Auxiliary control points to force a high load factor
X(m) Y(m) Z(m)
Paux1 77,043.2 92,600 10,000 Paux2 92,600 77,043.2 10,000
5.5. Load Factor and Curvature of the Trajectory 109
FIGURE5.18: Zoom to the generated trajectory close to the control point P2 forcing a maximum load factor compared with Reshaping
algorithm.
(a) Load factor. (b) Curvature.
FIGURE5.19: Load factor and Curvature of the trajectory close to the control pointP2with a maximum load factor.