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Optimal 3D trajectory generation in delivering missions under urban constraints for a flying robot

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Abstract

Interest in applying flying robots especially quadcopters for civil applications, in particular for delivering purposes, has dramatically grown in the recent years. In fact, since quadcopters are capable of vertical takeoff and landing, they can be widely employed for nearly any aerial task where a human presence is hazardous or response time is critical. In this regard, quadcopters come to be very beneficial in delivering packages; accordingly, generating an optimal flight trajectory plays a preponderant role for meeting this vision. This paper is concerned with generation of a time-optimal 3D path for a quadcopter under municipal restrictions in delivering tasks. To this end, the flying robot’s dynamics is first modeled through Newton–Euler method. Subsequently, the problem is formulated as a time-optimal control problem such that the urban constraints, which are safe-margins of high-rise buildings located throughout the course, are first modeled and then imposed to the trajectory optimization problem as inequality constraints. After discretizing the trajectory by means of Hermit–Simpson method, the optimal control problem is transformed into a nonlinear programming problem and finally is solved by the direct collocation technique. Extensive simulations demonstrate the efficacy of the proposed method and correspondingly verify the effectiveness of the suggested method in generation of optimum 3D routes while all constraints and mission requirements are satisfied.

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Abbreviations

b :

Constant thrust factor

d :

Constant drag factor

\(F_{sI}\) :

Force applied in the inertial frame

g :

Gravitational acceleration

\(h_{i}\) :

Time interval

\({\mathbf {I}}_\mathrm{CM}\) :

Body’s inertia tensor excluding rotors

\(\mathbf {I}_{{\varvec{X}}}\) :

X-axis inertia

\(I_{Y}\) :

Y-axis inertia

\(I_{Z}\) :

Z-axis inertia

\(I_{R}\) :

Single rotor inertia

J :

Objective function

l :

Distance from the center of the blade to the center of the body

\({\mathbf {L}}\) :

Angular momentum

m :

Mass of the quadcopter

R :

Rotation matrix

\(t_{{0}}\) :

Initial time

\(t_{f}\) :

Terminal time

u(t):

Control input

\(\hat{x}\) :

Standard unit vector

x :

Cartesian position

\(\dot{x}\) :

Velocity in the x-direction

\(\ddot{x}\) :

Acceleration in the x-direction

\(\hat{y}\) :

Standard unit vector

y :

Cartesian position

\(\dot{y}\) :

Velocity in the y-direction

\(\ddot{y}\) :

Acceleration in the y-direction

\(\hat{z}\) :

Standard unit vector

z :

Cartesian position

\(\dot{z}\) :

Velocity in the z-direction

\(\ddot{z}\) :

Acceleration in the z-direction

\(\theta \) :

Pitch angle around the Y-axis

\(\dot{\theta }\) :

Pitch rate

\(\ddot{\theta }\) :

Pitch acceleration

\({{\varvec{\tau }}}\) :

Applied torque

\({\emptyset }\) :

Roll angle around the X-axis

\(\dot{\emptyset }\) :

Roll rate

\(\ddot{\emptyset }\) :

Roll acceleration

\(\psi \) :

Yaw angle around the Z-axis

\(\dot{\psi }\) :

Yaw rate

\(\ddot{\psi }\) :

Yaw acceleration

\({{\varvec{\omega }} }\) :

Angular velocity of body

\({{\varvec{\Omega }} }\) :

Angular turn rate of rotor

\({\varOmega }_{R}\) :

Sum of the rotors’ angular velocities

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Acknowledgements

This research was partly funded by Iran National Science Foundation (INSF) under the Contract No. 93013017.

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Authors and Affiliations

  1. Faculty of New Sciences and Technologies, Flight Dynamics and Control, University of Tehran, Tehran, Iran

    Abolfazl Lavaei & M. A. Amiri Atashgah

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  1. Abolfazl Lavaei
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  2. M. A. Amiri Atashgah
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Correspondence to M. A. Amiri Atashgah.

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Lavaei, A., Atashgah, M.A.A. Optimal 3D trajectory generation in delivering missions under urban constraints for a flying robot. Intel Serv Robotics 10, 241–256 (2017). https://doi.org/10.1007/s11370-017-0225-x

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