IEEE - Aerospace and Electronic Systems - August 2022 - 27

Schwung and Lunze
Otherwise it assumes the data to be lost. If it does not
receive an ACK message it sends the new data SSðtc;k þ
2 ~tmaxðdðtkÞÞÞ at the time instant t ¼ tc;k þ 2 ~tmaxðdðtkÞÞ.
Furthermore, the event generator monitors the distance
between its trajectory wSðtÞ andadesiredtrajectory
wdðtÞ as
distðwSðtÞ;wwdðtÞÞ ¼ jjwSðtÞ wdðtÞjj
and changes its trajectory wSðtÞ by generating the event
eS1 : distðwSðtÞ;wwdðtÞÞ ¼eS
whenever it deviates from wdðtÞ and violates the threshold
eS. Following the desired trajectory leads to the collision
avoidance with static obstacles.
Due to the different conditions to invoke communication
with (11) and to trigger the change of the trajectory
with (13) or (14), a minimum time span between two consecutive
event time instants is ensured. Hence, the effect
of invoking communication infinitely (often called Zeno
behavior in the event-based literature) cannot occur.
TRAJECTORY PLANNING UNIT
PLANNING METHOD
The planning unit of both objects is split into the planning
task section and the planning algorithm section. In the first
section, the conditions on the trajectory are defined in a
way so that the resulting trajectory fulfills control aim
(A1).
In the planning algorithm section the trajectories are
determined using Bezier curves and the conditions on the
trajectory from the planning task section. As the dynamics
of quadrotors have the relative degree 4, smooth trajectories
of order m ¼ 9 need to be planned. A Bezier curve of
order m ¼ n 1 is a polynomial
wðtÞ¼ðwxðtÞ wyðtÞ wzðtÞT ¼
Xm
i¼0
bi Bm
i ðtÞ;t 2½ts;te
defined over an interval ½ts;te by n control points bi 2 R3.
Bm
i ðtÞ, ði ¼ 1; .. . ;mÞ represents the Bernstein polynomials
Bm
i
ðtÞ¼
1
ðte tsÞm
m
i
ðt tsÞiðte tÞmi;t 2½ts;te:
Different trajectories can be planned by only changing the
planning task while the planning method remains
unchanged. The planning method is stated in detail in [26].
In the following the ideas of the planning tasks for the
different trajectories of the give-way object are given. The
planning task of the reactive trajectory wG;rðtÞ is divided
AUGUST 2022
into three phases based on the communicated trajectory of
the stand-on object.
1) After the occurrence of the event eG1 a trajectory
wG;r1ðtÞ is planned to maintain the distances s þ
2eG ands 2eG to the stand-on object to satisfy
the control aim (A1). With this trajectory the giveway
object deviates from its initially planned
trajectory.
2) As long as the stand-on object moves in a way that
the give-way object cannot return to its initial trajectory,
a trajectory wG;r2ðtÞ is planned to keep the distances
s þ 2eG ands 2eG to the stand-on object.
3) If the return to the initial trajectory is possible, a trajectory
wG;r3ðtÞ is planned to lead the give-way
object back to this trajectory.
The avoidance trajectory wG;aðtÞ is planned in one
phase, because with this trajectory the give-way object
should just increase its distance to the stand-on object in
case of a packet loss. The trajectory wG;aðtÞ is planned so
that the give-way object increases the distance to the
stand-on object during the time span 2 ~tmaxðdð~tiÞÞ by
dS ¼jjvS;max 2 ~tmaxðdð~tiÞÞjj:
(16)
If a packet gets lost, the newly sent packet can be received
by the give-way object after the time span 2~tmaxðdð~tiÞÞ at
the latest, because it waits ~tmaxðdð~tiÞÞ for the lost packet
and the sending of the new packet lasts again ~tmaxðdð~tiÞÞ.
During this time span the stand-on object can cover at
most the distance (16) toward the give-way object. Hence,
in the worst-case scenario after the avoidance manoeuvre
the distance between the objects is maintained and no collision
can occur. In contrast, as the distance to the standon
object increases a violation of requirement (2) is possible.
The further trajectory is planned when new information
is received after a generation of the event eG1.
As the estimate ~tmaxðdð~tiÞÞ is used the give-way object
might evade the stand-on object unnecessarily if a packet is
not lost but it is received after the time span ~tmaxðdð~tiÞÞ.
Hence, the suitability of the control method highly depends
on the accuracy ofthe estimate ofthe delay estimator. In the
" Trajectory Planning Unit " section the estimation method is
analyzed and it is stated how often the give-way object plans
the trajectory wG;aðtÞ unnecessarily.
The trajectory of the stand-on object is planned in one
phase so as to avoid collision with static obstacles.
SUMMARY OF THE METHOD
The result of the control method is stated in the following
theorem. Requirement (1) can be guaranteed all the time,
while requirement (2) can be fulfilled only with a certain
probability due to the stochastic uncertainty of the
network.
IEEE A&E SYSTEMS MAGAZINE
27
IEEE - Aerospace and Electronic Systems - August 2022

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