an optimization problem for quadcopter reference flight

Research ArticleAn Optimization Problem for Quadcopter Reference FlightTrajectory Generation

Karam Eliker 1 Guoqing Zhang1 Said Grouni2 andWeidong Zhang 13

1Department of Automation Shanghai Jiao Tong University Shanghai 200240 China2Faculty of Sciences Engineering University of Boumerdes 35000 Boumerdes Algeria3School of Computer Engineering and Science Shanghai University Shanghai 200444 China

Correspondence should be addressed to Weidong Zhang wdzhangsjtueducn

Received 22 October 2017 Accepted 29 March 2018 Published 26 July 2018

Academic Editor Howard Li

Copyright copy 2018 Karam Eliker et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the reference flight trajectory generation and planning problems for quadcopter Unmanned Aerial Vehicle(UAV) The reference flight trajectory is defined as the composition of path and motion functions Both of them are generatedby using quintic B-spline functions Based on differential flatness approach the quadcopter dynamical constraints are satisfiedinstantaneously by computing the induced aerodynamical moments and lift force The optimal reference flight trajectory withrespect to the mission requirements and imposed constraints is reached by manipulating the control pointsrsquo vectors of B-splinefunctions via a nonlinear constrained optimization method The mission requirements are defined as a set of waypoints withtheir respective scheduled flight timetable A minimum-energy cost function is developed to minimize the consumed energy andinduced efforts by reference flight trajectory For the need of the optimal overfly-times schedule the overfly times with respect tothe defined constraints and performance criteria are calculated Numerical simulation results show the feasibility and effectivenessof the proposed optimization method

1 Introduction

With a vast proliferation of Manned Aerial Vehicles (MAVs)andUnmannedAerial Vehicles (UAVs) world air transporta-tion traffic has witnessed a continuous increase causing asaturation in the airspace The formal approach adopted byresearchers to design the UAVs is to separate the avion-ics systems into three distinct and independent systemswhich are Guidance Navigation and Control (GNC) Byusing this structure UAVs could perform a variety of plan-ning missions in either military or civilian fields Howeverthere are many incidents like a collision with civil avia-tion aircraft invasion of privacy and threats to securityreported by [1] Federal Aviation Administration (FAA)These incidents are because of the lack of transportationrules and regulations supporting UAVs and the weaknesscurrent structure UAVs Therefore the need to integratethe UAVs in National Airspace System (NAS) is highlyrecommended

To achieve this challenge an adequate flight trajectoryplanner that could be embedded on quadcopter UAV shouldbe developed The flight trajectory planner could reduceuncertainty and increase predictability for Air Traffic Man-agement (ATM) Planning missions in terms of waypointswith overfly times at which the UAV is scheduled to overflycould be a challenging problem for trajectory generationMost of the previous works have focused on 3D trajectoryplanning problem and consider only waypoints without anyoverfly-times constraints By contrast this paper proposesa new optimization method to generate the reference flighttrajectory that could fulfill some related requirements ofNASTheproposed flight trajectory planning should generatea trajectory that complies with the considered quadcopterdynamical constraints and meets the assigned missions Theassigned missions contain a set of waypoints with theirrespective scheduled flight timetable For the necessity ofoptimal overfly-times schedule the overfly times can becalculated in optimal way while considering the assigned

HindawiJournal of Advanced TransportationVolume 2018 Article ID 6574183 15 pageshttpsdoiorg10115520186574183

2 Journal of Advanced Transportation

constraints and performance criteria By generating suchreference flight trajectory different tasks such as deliveringand reconnaissance missions at acceptable overfly timescould be performed Moreover the traffic managements forUAV could be enhanced by managing the scheduled flighttimetable of waypoints

Several papers of 3D trajectory planning problems whileconsidering waypoints as physical points in the workspacehave been done On one hand the authors in [2ndash7] pro-posed an optimization approach to solve Time OptimalMotion Planning (TOMP) between two configuration pointsby using a Nonlinear Programming (NLP) method Onthe other hand the authors in [8ndash10] have proposed analgorithm based on Pontryaginrsquos minimum principle tosolve TOMP between two states In [11] the authors pre-sented an approach that could link separate trajectoriesto form smooth analytically defined paths while satisfy-ing a set of waypoint constraints Moreover in [12ndash14]a minimum snap cost function was designed to generatesmooth trajectory while satisfying a sequence of consecutivewaypoints In [15] a quasi-optimal trajectory generationwas developed to satisfy constraints concerning waypointsand their overfly times while minimizing the trajectorylength

Moreover some papers have dealt with the minimum-energy problem In [16] a quasi-optimal trajectory plannerwas proposed to generate trajectory between two pointsThe objective function was defined as a fuel consumptionproblem by considering the cost function as an averagevelocity The authors in [17] supposed that the consumedenergy was proportional to the rotors velocities In [18] theauthors designed a trajectory planning function for a small-scale helicopter in autorotation which could minimize thebattery power consumption and avoid bang-bang type solu-tions Nonlinear Programming problemwas used to computethe optimal polynomial coefficients of the trajectory whileminimizing the cost function written as the rate of all forcesand moments applied on helicopter In [19] a minimum-energy trajectory generation approach was proposed bydefining the cost function as a function of the voltageand current across the four rotors The similarity amongthese papers was the reference trajectory planning problemsdefined as longitudinal lateral and altitude displacementsas NLP problem between two configuration points while thedifferencewas related to the parameterization of the referencetrajectory

The objective of this paper is the reference trajectorygeneration and planning problems for quadrotor UAV Thereference flight trajectory is defined as a composition ofpath andmonotonically increasingmotion functions Both ofthemare generated by using quintic B-spline functions Basedon differential flatness approach the quadcopter dynamicalconstraints are checked instantaneously by computing theinduced aerodynamical roll and pitch headingmoments andlift force respectivelyThe optimal reference trajectory whichconsists of longitudinal lateral and altitude displacementsin addition to yaw angle is reached by manipulating thecontrol pointsrsquo vectors of B-spline functions via a nonlinearconstrained optimization method while taking into account

the mission requirements and defined constraints The mis-sion requirements are defined as a sequence ofwaypointswiththeir respective scheduled flight timetable and eachwaypointcould represent a particular posture A minimum-energycost function is designed to enhance the performance cri-teria regarding trajectory length induced efforts consumedenergy and trajectory smoothness For the optimal overfly-times schedule requirement the overfly times are calculatedby the proposed optimization method with regard to theinterested constraints and performance criteria Several con-figurations are studied and numerical simulation results showthe feasibility and effectiveness of the proposed optimizationmethod

2 Quadcopter Flight Dynamics

The quadcopter UAV is a vertical take-off and landing(VTOL) aircraft Its main characteristics are the ability ofhovering vertical take-off and landing in any terrain Thedisplacement and rotation motions are achieved by adjustingthe angular speed of each rotor To design the reference flighttrajectory a simplified quadcopter flight dynamics are usedto reduce the computation burden and to facilitate the opti-mization problem implementation By considering that thequadcopter UAV performs angular motions of low amplitudeand omitting the drag force aerodynamical frictions andgyroscopic effects moments the simplified dynamic modelof quadcopter UAV in contrast to full quadcopter dynamicspresented in [21 22] is given by

120601 = 1119868119909119880120601 (1a)120579 = 1119868119910119880120579 (1b)

= 1119868119911119880120595 (1c)

= 119880119911119898 (cos120601 sin 120579 cos120595 + sin120601 sin120595) (1d)

119910 = 119880119911119898 (cos120601 sin 120579 sin120595 minus sin120601 cos120595) (1e)

= minus119892 + 119880119911119898 (cos120601 cos 120579) (1f)

where 119909 119910 119911 are longitudinal lateral and altitude displace-ments and120601 120579 120595 are roll pitch and yaw angles in the Earth-fixed frame respectively 119892 is the gravity acceleration and 119898is the mass of quadcopter UAV 119868119909 119868119910 119868119911 are roll pitch andyaw inertia moments respectively The lift force 119880119911 and theaerodynamical roll 119880120601 pitch 119880120579 and heading 119880120595 momentsdeveloped by the quadcopter UAV are expressed with respectto the rotors velocities

Journal of Advanced Transportation 3

(119880119911119880120601119880120579119880120595)= ( 119862119871 119862119871 119862119871 1198621198710 119889119862119871 0 minus119889119862119871minus119889119862119871 0 119889119862119871 0minus119862119863 119862119863 minus119862119863 119862119863 )(12059621120596221205962312059624)

(2)

where 119862119871 and 119862119863 are lift and drag force coefficients respec-tively 119889 is the distance between the centermass of quadcopterand the axis of the propeller 120596119894 is the rotor 119894 velocity 119894 =1 2 3 4 From (1d) (1e) and (1f) roll (120601) and pitch (120579)angles are expressed as follows

120601 = arcsin( sin120595 minus 119910 cos120595radic2 + 1199102 + ( + 119892)2) (3a)

120579 = arctan( cos120595 + 119910 sin120595 + 119892 ) (3b)

Using (1a)ndash(1f) the aerodynamical roll pitch and head-ing moments and lift force which are induced by thereference flight trajectory are expressed as follows119880120601 = 119868119909 120601 (4a)119880120579 = 119868119910 120579 (4b)119880120595 = 119868119911 (4c)119880119911 = 119898radic2 + 1199102 + ( + 119892)2 (4d)

Based on (3a) and (3b) it can be concluded that theanalytical expressions of 120601 120579 120601 and 120579 are functions ofthe first second third and fourth derivatives of the vectorcomponents (119909 119910 119911 120595)3 Optimization Problem Formulation

The optimization problem is concerned with the referenceflight trajectory that satisfies waypoints constraints at specificschedule flight timetable Since the quadcopter UAV canperform six Degrees of Freedom (DOF) the reference flighttrajectory could contain six components

I (119905) = (119909119889 (119905) 119910119889 (119905) 119911119889 (119905) 120601119889 (119905) 120579119889 (119905) 120595119889 (119905)) (5)

However the quadcopter UAV is an underactuated sys-tem Only longitudinal lateral and altitude displacements(119909 119910 119911) and yaw angle (120595) can be directly controlled Roll (120601)and pitch (120579) angles are systemically computed with respectto quadcopter dynamics Differential flatness approach hasshown interesting performance regarding the trajectory plan-ning problem as demonstrated in [4 5 8 12ndash14 16ndash18] Using

the differential flatness method the state space variables andcontrol inputs can be defined in terms of flat output vectorFurthermore the computation load on optimization couldbe reduced considerably Equations (1a)ndash(1f) are differentiallyflat if there exists a flat output vector 120578 isin R119898 in the followingform [23]

120578 = 120577 (x u u u u(119903)) (6)

such that

x = 120572 (120578 (119902)) (7a)

u = 120573 (120578 (119902)) (7b)

where x and u are state space and control input vectorsrespectively From (1a)ndash(1f) (3a)-(3b) and (4a)ndash(4d) it canbe concluded that the vector (119909 119910 119911 120595) represents the flatoutput 120578(119905) vector Therefore the optimization problem canbe treated from the flat output 120578(119905) vector and expressed asNLP problem in the following form

min 119869 (120578 (119905)) for 119905 isin [0 119879] (8a)

such that ℎ (120578 (119905)) = 0 (8b)119892 (120578 (119905)) ⩽ 0 (8c)

where (8b) and (8c) denote a set of equality and inequalityconstraints respectively 119869(120578(119905)) is the objective function31 Mission Requirements and Constraints The consideredconstraints are classified as follows

(a) Mission Constraints The mission constraints are definedas a set of conditions as follows

(i) The quadcopter UAV posture

I (0) = Iinit (9a)

I (119905119894) = I119894 119894 = 1 119899waypoint (9b)0 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119899waypoint lt 119879 lt 119879aut (9c)

I (119879) = Ifin (9d)

where 119879aut and 119879 are autonomy flight and final timesrespectively 119899waypoint is the number of waypoints

(ii) The quadcopter UAV scheduled flight timetable119905lowast119894 minus 120575119905 ⩽ 119905119894 ⩽ 119905lowast119894 + 120575119905 (10)

where 119905lowast119894 is the predefined overfly time and 120575119905 is anallowed range of overfly time

(iii) The quadcopter UAV velocity

I (0) = 997888rarr0 (11a)

I (119879) = 997888rarr0 (11b)


(b) Workspace and Flight Envelope Constraints

(i) The workspace limits120601min ⩽ 120601119889 (119905) ⩽ 120601max (12a)120579min ⩽ 120579119889 (119905) ⩽ 120579max (12b)120595min ⩽ 120595119889 (119905) ⩽ 120595max (12c)119909min ⩽ 119909119889 (119905) ⩽ 119909max (12d)119910min ⩽ 119910119889 (119905) ⩽ 119910max (12e)119911min ⩽ 119911119889 (119905) ⩽ 119911max (12f)

(ii) The flight envelope limits the flight envelope is a setof flight conditions defined by the quadcopter heightand airspeed119911min ⩽ 119911119889 (119905) ⩽ 119911max (13a)min ⩽ 119889 (119905) ⩽ max (13b)0 ⩽ 119881119886 ⩽ 119881MO (13c)

where 119881MO is the maximum operating speed

(c) Induced Efforts Constraints From (2) the induced effortscan be expressed as follows0 ⩽ 119880119911 ⩽ 41198621198711205962max (14a)minus1198891198621198711205962max ⩽ 119880120601 ⩽ 1198891198621198711205962max (14b)minus1198891198621198711205962max ⩽ 119880120579 ⩽ 1198891198621198711205962max (14c)minus21198621198631205962max ⩽ 119880120595 ⩽ 21198621198631205962max (14d)

Since the quadcopter rotors are identical 120596max denotesthe maximum rotor velocity

32 Reference Flight Trajectory Parameterization The refer-ence trajectory can be defined as a composition [3 6 15 24]of path and motion functions Based on this definition thevelocity acceleration jerk and snap profiles of referenceflight trajectory could be adjusted to make them compliantwith the assigned constraints The reference flight trajectoryI(119905) = (119909119889(119905) 119910119889(119905) 119911119889(119905) 120595119889(119905)) is written as follows

I (119905) = 119875 (120582) ∘ 120582 (119905) = 119875 (120582 (119905)) (15a)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) (15b)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) + 2 (119905) 11988921198751198891205822 (120582 (119905)) (15c)

I (119905) = 120582 (119905) 119889119875119889120582 (120582 (119905)) + 3 (119905) (119905) 11988921198751198891205822 (120582 (119905))+ 3 (119905) 11988931198751198891205823 (120582 (119905)) (15d)

I(4) (119905) = 120582(4) (119905) 119889119875119889120582 (120582 (119905))+ 4 (119905) 120582 (119905) 11988921198751198891205822 (120582 (119905))+ 32 (119905) 11988921198751198891205822 (120582 (119905))+ 62 (119905) (119905) 11988931198751198891205823 (120582 (119905))+ 4 (119905) 11988941198751198891205824 (120582 (119905))

(15e)

where the path 119875(120582) is a set of time-independent configura-tions followed by the quadcopter UAV to fly from an initialconfiguration 119875init to a final one 119875fin The motion 120582(119905) is apositive scalar function that defines the way in which thepath is achieved with respect to the time 120582 isin [0 1] Inorder to construct the path 119875(120582) and motion 120582(119905) functionsa parametric function has to be selected For this purposeB-spline function is used for its advantage to support themanipulation of the reference trajectory as functions ofcontrol points and knot vectors Basically B-spline functionis demonstrated as follows [24]119878 (119906) = 119873sum

119894=1

119888119894119861119901119894 (119906) (16)

with 1199060 le sdot sdot sdot le 119906119895minus1 le 119906119895 le 119906119895+1 le sdot sdot sdot le 119906119899knot (17)

where 119888119894 is a control point and 119873 is a control points number119906 is a knot vector and 119899knot is the number of knots 119861119901119894 is abasis function of degree 119901 For this purpose the knot vectorsare selected as uniform sequences To construct the path andmotion functions the following conditions are defined

(a) Path Function 119875(120582)(i) Posture condition it satisfies the condition indicated

in (9a)ndash(9d)119875 (120582 = 0) = 119875init (18a)119875 (120582119894) = 119875119894 119894 = 1 119899waypoint (18b)119875 (120582 = 1) = 119875fin (18c)

(ii) Velocity condition it ensures the condition of(11a) and (11b) (120582 = 0) = 997888rarr0 (19a) (120582 = 1) = 997888rarr0 (19b)

(c) Motion Function 120582(119905)(i) The first derivative of 120582(119905)119889120582119889119905 ⩾ 0 forall119905 isin [0 119879] (20)


Motion control points(t)

I5

I4

I3

I2

I1

0 T

Time (s)

0

02

04

06

08

1M

otio

n fu

nctio

n(t)

(a)

Path control pointsP()

10

minus30

minus20

minus10

0

10

20

Path

func

tion

P(

)

(b)

Figure 1 An illustration example for (a) motion 120582(119905) and (b) path 119875(120582) functions respectively(ii) The motion must satisfy the following conditions120582 (0) = 0 (21a) (0) = 0 (21b) (0) = 0 (21c)

120582 (0) = 0 (21d)120582 (119879) = 1 (21e) (119879) = 0 (21f) (119879) = 0 (21g)120582 (119879) = 0 (21h)

(iii) The overfly-time conditions120582 (119905119894) = 120582119894 119894 = 1 119899waypoint (22)

From (3a)-(3b) and (15a)ndash(15e) the continuity of thesecond-order derivatives of the desired roll 120601119889(119905) and pitch120579119889(119905) angles depends on the continuity of the fourth derivativeof the path and motion functions Therefore the parametricfunction of 119875(120582) is modeled as a quintic B-spline functionusing (119873119901 + 2) control points number where 119873119901 ⩾ 5 The(119873119901) control points number represents a set of path controlpoints that will be manipulated by the proposed optimizationmethod The (2) control points number represents the fixedcontrol points that satisfy the condition expressed in (18a)and (18c) The parametric function of 120582(119905) is modeled asa quintic B-spline function using (119873119898 + 8) control pointsnumber The (119873119898) control points number is a set of motioncontrol points and the (8) control points number is a setof fixed control points that meet conditions defined by(11a)-(11b) and (21a)ndash(21h) In addition the motion control

pointsrsquo vector is uniformly distributed along the time scaleguarantying that each motion control point belongs to anincreasing bounded interval to satisfy the condition of (20)Figure 1 illustrates an example for path and monotonicallyincreasing motion functions respectively

4 Performance Criterion

For rotor-craft systems the electrical power is convertedto mechanical power in the form of thrust The developedmechanical power could be described as the product ofthe torque 120591119894 and angular velocity 120596119894 Therefore the powergenerated from each rotor 119894 could be computed as follows119875119894 = 120591119894120596119894 = (1198621198631205962119894 ) 120596119894 (23)

By using the first term 1198621198631205962119894 in (23) the proposed costfunction is expressed as a function of rotor velocities and yawdynamics

119869 = int1198790

[[( 4sum119894=1

1198621198631205962119894 )2 + 2]]119889119905 (24a)

= int1198790[1198622119863 (12059621 + 12059622 + 12059623 + 12059624)2 + 2] 119889119905 (24b)

= int1198790[120573 (2 + 1199102 + ( + 119892)2) + 2] 119889119905 (24c)

with 120573 = (119862119863 119898119862119871)2 (25)

where the term 2 is used to minimize the induced aerody-namical headingmoment and the second term120573(2+ 1199102+(+


Step 1

Step 2

Step 3

Definition of constraints

Suggestion of initial solution

Control points vectors

Path control points Motion control points

Uniform path knot sequences vector

Uniform motion knot sequences vector

Bounded interval of

each motion control point

Fix maximum tolerance and iteration

Initialization

Mission requirements

Workspace and flight envelope limits

Induced efforts constraints

Step 1

Step 2

Proposed optimization for reference trajectory

generation

Generating optimal reference trajectory using

B-spline functions

Optimization

Current solution

Generating reference trajectory using

B-spline functions

Induced efforts Computation

Cost function

Nonlinearconstraints

Equalityconstraints

Inequality constraints

Desired roll and pitch angles computation

Final time T

control pointsvectors

Minimize cost

function

Reach maximum tolerance or iteration

Update the current solution

Optimal Solution

Optimal overfly-times Optimal final Optimal control points vectors

No

Yes

time TＩＪＮＣＧＦ

ti_ＩＪＮＣＧＦi = 1 nＱＳＪＩＣＨＮＭ

(Nm + 8) vector(Np + 2) vector

i = 1 nＱＳＪＩＣＨＮＭ

Overfly-times ti

Figure 2 Proposed optimization method procedure

119892)2) is introduced in the proposed cost function to minimizethe induced efforts and consumed energy This yields119869 = int1

0[120573((119889211991111198891205822 )2 + (119889211991121198891205822 )2 + (119889211991131198891205822 + 119892)2)

+ (119889211991141198891205822 )2]119889120582 (26)

The synoptic procedure of optimizationmethod is shownin Figure 2 For fixed scheduled flight timetable problem onlythe positions of the control pointsrsquo vectors are manipulatedThe final solution will be the optimal control pointsrsquo vectorsFor this problem the overfly-time constraints defined shouldbe selected properly For the requirements of the optimaloverfly-times schedule the overfly times are calculated bythe proposed optimization problem with respect to theconstraints of interest and performance criteria

5 Numerical Simulations

51 Flight Trajectory 1 Let us consider the following con-figuration in Table 1 All simulations are done using 119873119898 =

Table 1 Waypoints configuration with their scheduled flight time-table

Waypoint 119905 (s) 119909 (m) 119910 (m) 119911 (m) 120595 (rad)Initial 0 5 minus5 0 01 10 10 0 20 minus017452 25 0 15708 20 034913 43 minus10 0 20 052364 68 0 minus15708 20 017455 80 10 0 20 0Final 95 5 5 0 012 and 119873119901 = 14 The constraints are listed in Table 3The real parameters identified in [20] are used to test theproposed optimization method (see Table 2) A numericalsolver in Matlab [25] via ldquofminconrdquo based upon SequentialQuadratic Programming (SQP) is used to solve the optimiza-tion method To demonstrate the effectiveness of proposedcost function in (26) different cost functions defined in[15 17] are used Figure 3 displays the 2D and 3D referenceflight trajectory with respect to the requirements of Table 1


0minus20

20

10

10Longitudinal x (m)

0

20

0

Alti

tude

z (m

)

Lateral y (m)minus10

30

20 minus20

40

Minimum-energyMinimum-length [15]Minimum-energy [17]

Initial configurationWaypointsFinal configuration

t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s

tＣＨＣＮ = 0 sT = 90 s

(a)

Longitudinal x (m)

Late

ral y

(m)

minus20 minus10 0 10 20minus20

minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)

Figure 3 Optimal 3D and 2D reference flight trajectory

Table 2 Quadcopter UAV parameters [20]

Parameter Value119868119909 00059 kgsdotm2119868119910 00059 kgsdotm2119868119911 00107 kgsdotm2119898 0694 kg119862119863 118 times 10minus7Nsdotmrads119862119871 38162 times 10minus6 Nrads119889 018mTable 3 Considered constraints

Constrainttypes Constraints Constraint ranges

Workspace

Longitudinaldisplacement (m) minus30 ⩽ 119909 ⩽ 30

Lateral displacement (m) minus40 ⩽ 119910 ⩽ 40Altitude (m) 0 ⩽ 119911 ⩽ 120

Roll angle (rad) minus05236 ⩽ 120601 ⩽ 05236Pitch angle (rad) minus06109 ⩽ 120579 ⩽ 06109Yaw angle (rad) minus07854 ⩽ 120595 ⩽ 07854

Flight envelopeHeight (m) 0 ⩽ 119911 ⩽ 120

Yaw rate (rads) minus17454 ⩽ ⩽ 17454Airspeed (ms) 0 ⩽ 119881119886 ⩽ 8

Actuators Maximum rotorvelocity (rads) 0 ⩽ 120596max ⩽ 1047

It could be noted that the proposed algorithm can dealwith different cost functions Moreover the smoothness ofthe generated reference flight trajectory depends on theconsidered cost functions Figure 4 shows the generated

Table 4 Comparison results of trajectory length and fitness value

Cost function type Trajectory length (m) Fitness value1 Minimum-energy 12916 11174 times 1032 Minimum-length [15] 11705 13609 times 1043 Minimum-energy [17] 13625 12241 times 104longitudinal lateral and altitude displacements in additionto yaw angle Figures 6 and 7 illustrate the induced effortsdeveloped by each cost function It could be seen from theproposed cost function in (26) that the mean values of theinduced efforts are considerably less than those related tothe cost functions in [15 17] Figure 5 displays the resultsgiven by each cost function in regards of trajectory lengthand consumed energy The fitness value developed by theproposed cost function which reflects consumed energyis less important than other cost functions The proposedcost function shows an interesting performance in termsof trajectory length induced efforts consumed energy andsmoothness as demonstrated in Tables 4 and 5

52 Flight Trajectory 2 Let us consider the configuration inTable 6 Simulations are done by considering three scenariosIn the first one the waypoints overfly times are fixed In thesecond one the waypoints overfly times are freely computedby the proposed optimization method However in the lastcase an allowed range of overfly time (120575119905 = 5 s) is used

Figure 8 displays the 3D and 2D generated referenceflight trajectory for the three scenarios Even if the imposedwaypoints overfly times defined in Table 6 are fixed withoutany prior knowledge on optimality aspect the proposed opti-mizationmethodwill show its ability to generate the referencetrajectory of desired configuration despite its weakness in


0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)

Figure 4 Desired (a) longitudinal (b) lateral and (c) altitude displacements and (d) heading angle respectively

2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue

2 31Cost function(b)

Figure 5 Performance criterion (a) trajectory length and (b) fitness value respectively

terms of induced efforts length trajectory and consumedenergy as demonstrated in Figures 9 and 11 Table 8 showsthe optimal overfly-times schedule computed in the secondand third scenarios

Figures 10 and 11 illustrate the induced lift force aerody-namical moments and their mean values computations withrespect to each scenario Based on the proposed optimizationproblem the overfly times are computed optimally andconsequently the induced efforts trajectory length andconsumed energy could be considerably reduced Tables 7and 9 show this aspect

53 Flight Trajectory 3 Let us consider the same configura-tion in Table 6 In this study different 120575119905 is used and 119879aut =

20min is fixed Figure 12 displays the 3D and 2D generatedreference flight trajectory and Table 10 shows the optimaloverfly-times schedule computed for each selected 120575119905 Figures13 14 and 15 display the trajectory length consumed energyand induced efforts developed by selecting 120575119905 Tables 11 and12 display the performance criteria assessment It can beseen that while 120575119905 increased the performance criteria showedenhancing results

6 Conclusion

An optimizationmethod for reference flight trajectory gener-ation has been proposed which allows the generation of feasi-ble and realistic flight plan for quadcopter UAV and manages

Journal of Advanced Transportation 9Li

ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)

Figure 6 Induced efforts (a) lift force and aerodynamical (b) roll and (c) pitch and (d) heading moments respectively

Lift

forc

e

0

2

4

6

8

Uz

(N)

2 31Cost function(a)

Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)

2 31Cost function(c)

Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)

2 31Cost function(d)

Figure 7 Mean values of (a) lift force and aerodynamical (b) roll and (c) pitch and (d) heading moments respectively


Table 5 Comparison results of induced efforts

Cost function type Lift force Roll moment Pitch moment Heading moment(N) (Nsdotm) (Nsdotm) (Nsdotm)

1 Minimum-energy 68092 50493 times 10minus6 74343 times 10minus6 23455 times 10minus52 Minimum-length [15] 6811 24896 times 10minus5 19401 times 10minus5 69616 times 10minus53 Minimum-energy [17] 68089 41299 times 10minus6 65995 times 10minus6 10246 times 10minus4

0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)

Scenario 1Scenario 2Scenario 3

(a)

minus30 minus15 0 15 30Longitudinal x (m)

minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)


adequately the scheduled flight timetable of waypoints In thispaper the flight trajectory generation problem was treatedas NLP problem and solved via a nonlinear constrainedoptimization technique The reference flight trajectory wasdefined as a composition of path and motion functions Bothof themwere generated by using B-spline functions Based ondifferential flatness approach the quadcopterUAVdynamicalconstraints were checked instantaneously by computing theinduced aerodynamical roll pitch and headingmoments andlift force respectively The optimal reference trajectory was

reached by manipulating the control pointsrsquo vectors of B-spline functions while taking into account the quadcoptermission requirements dynamical constraints flight envelopeandworkspace limitsThemission requirementswere definedas a series of waypoints with their respective scheduled flighttimetable A minimum-energy cost function has been pro-posed to minimize the consumed energy and induced effortsalong the path For requirement of optimal overfly-timesschedule the overfly times were computed in an optimal wayregarding the defined constraints and performance criteria


ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30

20minus100Longitudinal x (m)

0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)



Waypoint 119905 (s) 119909 (m) 119910 (m) 119911 (m) 120595 (rad)Initial 0 5 minus5 0 01 10 5 0 20 minus017452 25 0 7854 30 034913 43 minus5 0 40 052364 68 0 minus7854 50 017455 80 10 0 60 06 95 0 235619 70 017457 118 minus10 0 80 043638 130 0 minus235619 90 0Final 140 20 0 100 0Numerical simulations in this study have shown the feasibilityand effectiveness of the proposed optimization method to


Scenario Trajectory length (m) Fitness value1 24785 18187 times 1042 20377 47394 times 1033 20395 51734 times 103generate an optimal reference flight trajectory in terms oftrajectory length induced efforts consumed energy andsmoothness Moreover it shows its ability to manage thescheduled flight timetable of waypoints

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper


t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)



Table 8 Waypoints configuration with their optimal overfly-times schedule

Waypoint 119905scenario1 (s) 119905scenario2 (s) 119905scenario3 (s)Initial 0 0 01 10 1844 152 25 3455 28463 43 4726 45424 68 6024 63215 80 7167 75376 95 8944 94817 118 10539 1138 130 12076 125Final 140 140 140


Scenario Lift force Roll moment Pitch moment Heading moment(N) (Nsdotm) (Nsdotm) (Nsdotm)

1 68166 15055 times 10minus5 10145 times 10minus5 3111 times 10minus52 68092 25521 times 10minus6 13739 times 10minus6 24893 times 10minus53 68101 62292 times 10minus6 2631 times 10minus6 28259 times 10minus5


Waypoint 119905120575119905=06 (s) 119905120575119905=25 (s) 119905120575119905=4 (s) 119905120575119905=10 (s)Initial 0 0 0 01 106 125 14 17872 256 275 2739 33673 4322 455 47 45484 674 655 64 63465 794 7777 76 75486 956 9717 9561 92117 1174 1155 114 110178 1294 13045 13022 12727Final 14468 14622 14754 14946

Table 11 Comparison results of trajectory length and fitness value120575119905 (s) Trajectory length (m) Fitness value06 21629 81647 times 10325 20621 55334 times 1034 20397 48964 times 10310 20377 47394 times 103

Table 12 Comparison results of induced efforts120575119905 (s) Lift force Roll moment Pitch moment Heading moment(N) (Nsdotm) (Nsdotm) (Nsdotm)

06 68101 67975 times 10minus6 26577 times 10minus6 29807 times 10minus525 68094 41188 times 10minus6 17692 times 10minus6 25535 times 10minus54 68093 36939 times 10minus6 17541 times 10minus6 24895 times 10minus510 68089 25992 times 10minus6 15055 times 10minus6 22336 times 10minus5


Acknowledgments

Thispaper is partly supported by theNationalNatural ScienceFoundation of China (Grants nos 61473183 and U1509211)National Postdoctoral Program for Innovative Talents (noBX201600103) and the China Postdoctoral Science Founda-tion (no 2016M601600) The authors would like to thankBahij Eliker Zahia Oueldkherroubi and Nafissa Zouad fortheir support and helpful feedback for drafting the paper

References

[1] M Huerta Integration of civil Unmanned Aircraft Systems(UAS) in the National Airspace System (NAS) Roadmap FederalAviation Administration 2013

[2] L Lai C Yang and CWu ldquoTime-optimal control of a hoveringquad-rotor helicopterrdquo Journal of Intelligent amp Robotic Systemsvol 45 no 2 pp 115ndash135 2006

[3] Y BouktirMHaddad andT Chettibi ldquoTrajectory planning fora quadrotor helicopterrdquo in Proceedings of the 16thMediterraneanConference on Control and Automation pp 1258ndash1263 AjaccioFrance June 2008

[4] A Chamseddine Y Zhang and C-A Rabbath ldquoAdaptivetrajectory planning for a quad-rotor unmanned aerial vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 2ndash5 Toronto Canada August 2010

[5] A Chamseddine T Li Y Zhang C A Rabbath and DTheilliol ldquoFlatness-based trajectory planning for a quadrotorunmanned aerial vehicle test-bed considering actuator andsystem constraintsrdquo in Proceedings of the American ControlConference pp 920ndash925 Montreal Canada June 2012

[6] A P Schoellig F L Mueller and R DrsquoAndrea ldquoOptimization-based iterative learning for precise quadrocopter trajectorytrackingrdquoAutonomous Robots vol 33 no 1-2 pp 103ndash127 2012

[7] E Kahale P Castillo andY Bestaoui ldquoMinimum time referencetrajectory generation for an autonomous quadrotorrdquo in Pro-ceedings of the International Conference on Unmanned AircraftSystems pp 126ndash133 Orlando Fla USA March 2014

[8] MHehn andRDrsquoAndrea ldquoQuadrocopter trajectory generationand controlrdquo in Proceedings of the IFACWorld Congress vol 44pp 1485ndash1491 2011

[9] M Hehn R Ritz and R DrsquoAndrea ldquoPerformance bench-marking of quadrotor systems using time-optimal controlrdquoAutonomous Robots vol 33 no 1-2 pp 69ndash88 2012

[10] MHehn andRDandrea ldquoReal-TimeTrajectoryGeneration forQuadrocoptersrdquo IEEE Transactions on Robotics vol 31 no 4pp 877ndash892 2015

[11] J Jamieson and J Biggs ldquoPath planning using concate-nated analytically-defined trajectories for quadrotor UAVsrdquoAerospace vol 2 no 2 pp 155ndash170 2015

[12] D Mellinger and V Kumar ldquoMinimum snap trajectory gener-ation and control for quadrotorsrdquo in Proceedings of the IEEEInternational Conference on Robotics andAutomation pp 2520ndash2525 IEEE Shanghai China March 2011

[13] D Mellinger N Michael and V Kumar ldquoTrajectory generationand control for precise aggressive maneuvers with quadrotorsrdquoInternational Journal of Robotics Research vol 31 no 5 pp 664ndash674 2012

[14] J Yu Z Cai and Y Wang ldquoOptimal trajectory generation of aquadrotor based on the differential flatnessrdquo in Proceedings of

the 28th Chinese Control and Decision Conference pp 678ndash683Yinchuan China March 2016

[15] K Eliker H Bouadi and M Haddad ldquoFlight planning andguidance features for an UAV Flight Management Computerrdquoin Proceedings of the 21st IEEE International Conference onEmerging Technologies and Factory Automation 6 1 pagesBerlin Germany September 2016

[16] I D Cowling O A Yakimenko J F Whidborne and A KCooke ldquoDirect method based control system for an autono-mous quadrotorrdquo Journal of Intelligent amp Robotic Systems vol60 no 2 pp 285ndash316 2010

[17] A Chamseddine Y Zhang C A Rabbath C Join and DTheilliol ldquoFlatness-based trajectory planningreplanning fora quadrotor unmanned aerial vehiclerdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 2832ndash28482012

[18] S Taamallah X Bombois and P M J Van denHof ldquoTrajectoryplanning and trajectory tracking for a small-scale helicopter inautorotationrdquo Control Engineering Practice vol 58 pp 88ndash1062017

[19] F Morbidi R Cano and D Lara ldquoMinimum-energy pathgeneration for a quadrotorUAVrdquo in Proceedings of the 2016 IEEEInternational Conference on Robotics and Automation pp 1492ndash1498 Stockholm Sweden March 2016

[20] M De Lellis Costa de Oliveira Modeling identification andcontrol of a quadrotor aircraft [MS thesis] Faculty of ElectricalEngineering Prague Czech Republic 2011

[21] L Derafa T Madani and A Benallegue ldquoDynamic modellingand experimental identification of four rotors helicopter param-etersrdquo in Proceedings of the IEEE International Conference onIndustrial Technology (ICIT rsquo06) pp 1834ndash1839 Mumbai IndiaDecember 2006

[22] H Bouadi and F Mora-Camino ldquoModeling and adaptive flightcontrol for quadrotor trajectory trackingrdquo Journal of Aircraftvol 55 no 2 pp 666ndash681 2018

[23] G G RigatosNonlinear Control and Filtering Using DifferentialFlatness Approaches Applications to Electromechanical SystemsSpringer Cham Switzerland 2015

[24] L Biagiotti andCMelchiorriTrajectory Planning for AutomaticMachines and Robots Springer Berlin Germany 2009

[25] T ColemanM A Branch and A GraceOptimization Toolboxfor Use with Matlab MatlabThe Mathworks Inc 1999

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constraints and performance criteria By generating suchreference flight trajectory different tasks such as deliveringand reconnaissance missions at acceptable overfly timescould be performed Moreover the traffic managements forUAV could be enhanced by managing the scheduled flighttimetable of waypoints

Several papers of 3D trajectory planning problems whileconsidering waypoints as physical points in the workspacehave been done On one hand the authors in [2ndash7] pro-posed an optimization approach to solve Time OptimalMotion Planning (TOMP) between two configuration pointsby using a Nonlinear Programming (NLP) method Onthe other hand the authors in [8ndash10] have proposed analgorithm based on Pontryaginrsquos minimum principle tosolve TOMP between two states In [11] the authors pre-sented an approach that could link separate trajectoriesto form smooth analytically defined paths while satisfy-ing a set of waypoint constraints Moreover in [12ndash14]a minimum snap cost function was designed to generatesmooth trajectory while satisfying a sequence of consecutivewaypoints In [15] a quasi-optimal trajectory generationwas developed to satisfy constraints concerning waypointsand their overfly times while minimizing the trajectorylength

Moreover some papers have dealt with the minimum-energy problem In [16] a quasi-optimal trajectory plannerwas proposed to generate trajectory between two pointsThe objective function was defined as a fuel consumptionproblem by considering the cost function as an averagevelocity The authors in [17] supposed that the consumedenergy was proportional to the rotors velocities In [18] theauthors designed a trajectory planning function for a small-scale helicopter in autorotation which could minimize thebattery power consumption and avoid bang-bang type solu-tions Nonlinear Programming problemwas used to computethe optimal polynomial coefficients of the trajectory whileminimizing the cost function written as the rate of all forcesand moments applied on helicopter In [19] a minimum-energy trajectory generation approach was proposed bydefining the cost function as a function of the voltageand current across the four rotors The similarity amongthese papers was the reference trajectory planning problemsdefined as longitudinal lateral and altitude displacementsas NLP problem between two configuration points while thedifferencewas related to the parameterization of the referencetrajectory

The objective of this paper is the reference trajectorygeneration and planning problems for quadrotor UAV Thereference flight trajectory is defined as a composition ofpath andmonotonically increasingmotion functions Both ofthemare generated by using quintic B-spline functions Basedon differential flatness approach the quadcopter dynamicalconstraints are checked instantaneously by computing theinduced aerodynamical roll and pitch headingmoments andlift force respectivelyThe optimal reference trajectory whichconsists of longitudinal lateral and altitude displacementsin addition to yaw angle is reached by manipulating thecontrol pointsrsquo vectors of B-spline functions via a nonlinearconstrained optimization method while taking into account

the mission requirements and defined constraints The mis-sion requirements are defined as a sequence ofwaypointswiththeir respective scheduled flight timetable and eachwaypointcould represent a particular posture A minimum-energycost function is designed to enhance the performance cri-teria regarding trajectory length induced efforts consumedenergy and trajectory smoothness For the optimal overfly-times schedule requirement the overfly times are calculatedby the proposed optimization method with regard to theinterested constraints and performance criteria Several con-figurations are studied and numerical simulation results showthe feasibility and effectiveness of the proposed optimizationmethod

2 Quadcopter Flight Dynamics

The quadcopter UAV is a vertical take-off and landing(VTOL) aircraft Its main characteristics are the ability ofhovering vertical take-off and landing in any terrain Thedisplacement and rotation motions are achieved by adjustingthe angular speed of each rotor To design the reference flighttrajectory a simplified quadcopter flight dynamics are usedto reduce the computation burden and to facilitate the opti-mization problem implementation By considering that thequadcopter UAV performs angular motions of low amplitudeand omitting the drag force aerodynamical frictions andgyroscopic effects moments the simplified dynamic modelof quadcopter UAV in contrast to full quadcopter dynamicspresented in [21 22] is given by

120601 = 1119868119909119880120601 (1a)120579 = 1119868119910119880120579 (1b)

= 1119868119911119880120595 (1c)

= 119880119911119898 (cos120601 sin 120579 cos120595 + sin120601 sin120595) (1d)

119910 = 119880119911119898 (cos120601 sin 120579 sin120595 minus sin120601 cos120595) (1e)

= minus119892 + 119880119911119898 (cos120601 cos 120579) (1f)

where 119909 119910 119911 are longitudinal lateral and altitude displace-ments and120601 120579 120595 are roll pitch and yaw angles in the Earth-fixed frame respectively 119892 is the gravity acceleration and 119898is the mass of quadcopter UAV 119868119909 119868119910 119868119911 are roll pitch andyaw inertia moments respectively The lift force 119880119911 and theaerodynamical roll 119880120601 pitch 119880120579 and heading 119880120595 momentsdeveloped by the quadcopter UAV are expressed with respectto the rotors velocities



(2)



120579 = arctan( cos120595 + 119910 sin120595 + 119892 ) (3b)




I (119905) = (119909119889 (119905) 119910119889 (119905) 119911119889 (119905) 120601119889 (119905) 120579119889 (119905) 120595119889 (119905)) (5)



120578 = 120577 (x u u u u(119903)) (6)

such that

x = 120572 (120578 (119902)) (7a)

u = 120573 (120578 (119902)) (7b)


min 119869 (120578 (119905)) for 119905 isin [0 119879] (8a)

such that ℎ (120578 (119905)) = 0 (8b)119892 (120578 (119905)) ⩽ 0 (8c)




I (0) = Iinit (9a)


I (119879) = Ifin (9d)





I (0) = 997888rarr0 (11a)

I (119879) = 997888rarr0 (11b)









I (119905) = 119875 (120582) ∘ 120582 (119905) = 119875 (120582 (119905)) (15a)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) (15b)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) + 2 (119905) 11988921198751198891205822 (120582 (119905)) (15c)

I (119905) = 120582 (119905) 119889119875119889120582 (120582 (119905)) + 3 (119905) (119905) 11988921198751198891205822 (120582 (119905))+ 3 (119905) 11988931198751198891205823 (120582 (119905)) (15d)

I(4) (119905) = 120582(4) (119905) 119889119875119889120582 (120582 (119905))+ 4 (119905) 120582 (119905) 11988921198751198891205822 (120582 (119905))+ 32 (119905) 11988921198751198891205822 (120582 (119905))+ 62 (119905) (119905) 11988931198751198891205823 (120582 (119905))+ 4 (119905) 11988941198751198891205824 (120582 (119905))

(15e)


119894=1

119888119894119861119901119894 (119906) (16)









I5

I4

I3

I2

I1

0 T

Time (s)

0

02

04

06

08

1M

otio

n fu

nctio

n(t)

(a)


10

minus30

minus20

minus10

0

10

20

Path

func

tion

P(

)

(b)


120582 (0) = 0 (21d)120582 (119879) = 1 (21e) (119879) = 0 (21f) (119879) = 0 (21g)120582 (119879) = 0 (21h)







119869 = int1198790

[[( 4sum119894=1

1198621198631205962119894 )2 + 2]]119889119905 (24a)

= int1198790[1198622119863 (12059621 + 12059622 + 12059623 + 12059624)2 + 2] 119889119905 (24b)

= int1198790[120573 (2 + 1199102 + ( + 119892)2) + 2] 119889119905 (24c)

with 120573 = (119862119863 119898119862119871)2 (25)



Step 1

Step 2

Step 3







Bounded interval of



Initialization




Step 1

Step 2


generation


B-spline functions

Optimization

Current solution


B-spline functions


Cost function


Equalityconstraints



Final time T


Minimize cost

function



Optimal Solution


No

Yes





Overfly-times ti



0[120573((119889211991111198891205822 )2 + (119889211991121198891205822 )2 + (119889211991131198891205822 + 119892)2)

+ (119889211991141198891205822 )2]119889120582 (26)







0minus20

20

10


0

20

0

Alti

tude

z (m

)


30

20 minus20

40



t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s


(a)

Longitudinal x (m)

Late

ral y

(m)


minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)





Workspace













0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




(2)



120579 = arctan( cos120595 + 119910 sin120595 + 119892 ) (3b)




I (119905) = (119909119889 (119905) 119910119889 (119905) 119911119889 (119905) 120601119889 (119905) 120579119889 (119905) 120595119889 (119905)) (5)



120578 = 120577 (x u u u u(119903)) (6)

such that

x = 120572 (120578 (119902)) (7a)

u = 120573 (120578 (119902)) (7b)


min 119869 (120578 (119905)) for 119905 isin [0 119879] (8a)

such that ℎ (120578 (119905)) = 0 (8b)119892 (120578 (119905)) ⩽ 0 (8c)




I (0) = Iinit (9a)


I (119879) = Ifin (9d)





I (0) = 997888rarr0 (11a)

I (119879) = 997888rarr0 (11b)









I (119905) = 119875 (120582) ∘ 120582 (119905) = 119875 (120582 (119905)) (15a)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) (15b)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) + 2 (119905) 11988921198751198891205822 (120582 (119905)) (15c)

I (119905) = 120582 (119905) 119889119875119889120582 (120582 (119905)) + 3 (119905) (119905) 11988921198751198891205822 (120582 (119905))+ 3 (119905) 11988931198751198891205823 (120582 (119905)) (15d)

I(4) (119905) = 120582(4) (119905) 119889119875119889120582 (120582 (119905))+ 4 (119905) 120582 (119905) 11988921198751198891205822 (120582 (119905))+ 32 (119905) 11988921198751198891205822 (120582 (119905))+ 62 (119905) (119905) 11988931198751198891205823 (120582 (119905))+ 4 (119905) 11988941198751198891205824 (120582 (119905))

(15e)


119894=1

119888119894119861119901119894 (119906) (16)









I5

I4

I3

I2

I1

0 T

Time (s)

0

02

04

06

08

1M

otio

n fu

nctio

n(t)

(a)


10

minus30

minus20

minus10

0

10

20

Path

func

tion

P(

)

(b)


120582 (0) = 0 (21d)120582 (119879) = 1 (21e) (119879) = 0 (21f) (119879) = 0 (21g)120582 (119879) = 0 (21h)







119869 = int1198790

[[( 4sum119894=1

1198621198631205962119894 )2 + 2]]119889119905 (24a)

= int1198790[1198622119863 (12059621 + 12059622 + 12059623 + 12059624)2 + 2] 119889119905 (24b)

= int1198790[120573 (2 + 1199102 + ( + 119892)2) + 2] 119889119905 (24c)

with 120573 = (119862119863 119898119862119871)2 (25)



Step 1

Step 2

Step 3







Bounded interval of



Initialization




Step 1

Step 2


generation


B-spline functions

Optimization

Current solution


B-spline functions


Cost function


Equalityconstraints



Final time T


Minimize cost

function



Optimal Solution


No

Yes





Overfly-times ti



0[120573((119889211991111198891205822 )2 + (119889211991121198891205822 )2 + (119889211991131198891205822 + 119892)2)

+ (119889211991141198891205822 )2]119889120582 (26)







0minus20

20

10


0

20

0

Alti

tude

z (m

)


30

20 minus20

40



t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s


(a)

Longitudinal x (m)

Late

ral y

(m)


minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)





Workspace













0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










I (119905) = 119875 (120582) ∘ 120582 (119905) = 119875 (120582 (119905)) (15a)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) (15b)

I (119905) = (119905) 119889119875119889120582 (120582 (119905)) + 2 (119905) 11988921198751198891205822 (120582 (119905)) (15c)

I (119905) = 120582 (119905) 119889119875119889120582 (120582 (119905)) + 3 (119905) (119905) 11988921198751198891205822 (120582 (119905))+ 3 (119905) 11988931198751198891205823 (120582 (119905)) (15d)

I(4) (119905) = 120582(4) (119905) 119889119875119889120582 (120582 (119905))+ 4 (119905) 120582 (119905) 11988921198751198891205822 (120582 (119905))+ 32 (119905) 11988921198751198891205822 (120582 (119905))+ 62 (119905) (119905) 11988931198751198891205823 (120582 (119905))+ 4 (119905) 11988941198751198891205824 (120582 (119905))

(15e)


119894=1

119888119894119861119901119894 (119906) (16)









I5

I4

I3

I2

I1

0 T

Time (s)

0

02

04

06

08

1M

otio

n fu

nctio

n(t)

(a)


10

minus30

minus20

minus10

0

10

20

Path

func

tion

P(

)

(b)


120582 (0) = 0 (21d)120582 (119879) = 1 (21e) (119879) = 0 (21f) (119879) = 0 (21g)120582 (119879) = 0 (21h)







119869 = int1198790

[[( 4sum119894=1

1198621198631205962119894 )2 + 2]]119889119905 (24a)

= int1198790[1198622119863 (12059621 + 12059622 + 12059623 + 12059624)2 + 2] 119889119905 (24b)

= int1198790[120573 (2 + 1199102 + ( + 119892)2) + 2] 119889119905 (24c)

with 120573 = (119862119863 119898119862119871)2 (25)



Step 1

Step 2

Step 3







Bounded interval of



Initialization




Step 1

Step 2


generation


B-spline functions

Optimization

Current solution


B-spline functions


Cost function


Equalityconstraints



Final time T


Minimize cost

function



Optimal Solution


No

Yes





Overfly-times ti



0[120573((119889211991111198891205822 )2 + (119889211991121198891205822 )2 + (119889211991131198891205822 + 119892)2)

+ (119889211991141198891205822 )2]119889120582 (26)







0minus20

20

10


0

20

0

Alti

tude

z (m

)


30

20 minus20

40



t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s


(a)

Longitudinal x (m)

Late

ral y

(m)


minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)





Workspace













0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




I5

I4

I3

I2

I1

0 T

Time (s)

0

02

04

06

08

1M

otio

n fu

nctio

n(t)

(a)


10

minus30

minus20

minus10

0

10

20

Path

func

tion

P(

)

(b)


120582 (0) = 0 (21d)120582 (119879) = 1 (21e) (119879) = 0 (21f) (119879) = 0 (21g)120582 (119879) = 0 (21h)







119869 = int1198790

[[( 4sum119894=1

1198621198631205962119894 )2 + 2]]119889119905 (24a)

= int1198790[1198622119863 (12059621 + 12059622 + 12059623 + 12059624)2 + 2] 119889119905 (24b)

= int1198790[120573 (2 + 1199102 + ( + 119892)2) + 2] 119889119905 (24c)

with 120573 = (119862119863 119898119862119871)2 (25)



Step 1

Step 2

Step 3







Bounded interval of



Initialization




Step 1

Step 2


generation


B-spline functions

Optimization

Current solution


B-spline functions


Cost function


Equalityconstraints



Final time T


Minimize cost

function



Optimal Solution


No

Yes





Overfly-times ti



0[120573((119889211991111198891205822 )2 + (119889211991121198891205822 )2 + (119889211991131198891205822 + 119892)2)

+ (119889211991141198891205822 )2]119889120582 (26)







0minus20

20

10


0

20

0

Alti

tude

z (m

)


30

20 minus20

40



t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s


(a)

Longitudinal x (m)

Late

ral y

(m)


minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)





Workspace













0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Step 1

Step 2

Step 3







Bounded interval of



Initialization




Step 1

Step 2


generation


B-spline functions

Optimization

Current solution


B-spline functions


Cost function


Equalityconstraints



Final time T


Minimize cost

function



Optimal Solution


No

Yes





Overfly-times ti



0[120573((119889211991111198891205822 )2 + (119889211991121198891205822 )2 + (119889211991131198891205822 + 119892)2)

+ (119889211991141198891205822 )2]119889120582 (26)







0minus20

20

10


0

20

0

Alti

tude

z (m

)


30

20 minus20

40



t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s


(a)

Longitudinal x (m)

Late

ral y

(m)


minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)





Workspace













0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0minus20

20

10


0

20

0

Alti

tude

z (m

)


30

20 minus20

40



t3 = 43 st2 = 25 s

t1 = 10 st4 = 68 s t5 = 80 s


(a)

Longitudinal x (m)

Late

ral y

(m)


minus10

0

10

20

t3 = 43 s

t2 = 25 s

t1 = 10 s

t4 = 68 s

t5 = 80 s

tＣＨＣＮ = 0 s

T = 90 s



(b)





Workspace













0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

0 25 43 68 80 9510Time (s)

minus20

minus10

10

20di

spla

cmen

txd

(m)


Long

itudi

nal

(a)

Late

ral

minus20

minus10

0

10

20

disp

lacm

entyd

(m)


0 25 43 68 80 9510Time (s)

(b)


0

10

20

30

40

disp

lacm

entz d

(m)

0 25 43 68 80 9510Time (s)

Alti

tude

(c)

Yaw

minus1

0

1

angl

ed

(rad

)


0 25 43 68 80 9510Time (s)

(d)


2 31Cost function

0

50

100

150

Traj

ecto

ry le

ngth

(m)

(a)

0

5000

10000

15000

Fitn

ess v

alue







6 Conclusion



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ft fo

rce


6

8

Uz

(N)

20 40 60 80 950Time (s)

(a)

Roll

mom

ent

times10minus4

minus3minus2minus1

012

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(b)

Pitc

h m

omen

t

times10minus4

minus2

minus1

0

1

2

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus5

minus202

5

U

(Nmiddotm

)


20 40 60 80 950Time (s)

(d)


Lift

forc

e

0

2

4

6

8

Uz

(N)


Roll

mom

ent

times10minus5

2 31Cost function

0

1

2

3

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

times10minus5

0

1

2

3

U

(Nmiddotm

)


Hea

ding

mom

ent

times10minus4

0

05

1

15

U

(Nmiddotm

)







0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






0minus30 50

minus15

40

Lateral y (m)0 0

Alti

tude

z (m

)

15

80

30 minus50

120

Longitudinal x (m)


(a)


minus40

minus20

0

20

40

Late

ral y

(m)


(b)


2 31Scenario

0

100

200

300

Traj

ecto

ry le

ngth

(m)

(a)

2 31Scenario

0

05

1

15

2

Fitn

ess v

alue

times104

(b)





ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ft fo

rce


50 100 1400Time (s)

6

65

7

75

8

Uz

(N)

(a)

Roll

mom

ent

times10minus4

minus2

0

2

U

(Nmiddotm

)


50 100 1400Time (s)

(b)


Pitc

h m

omen

t

times10minus4

minus2

0

2

U

(Nmiddotm

)

50 100 1400Time (s)

(c)

Hea

ding

mom

ent

times10minus4

minus1

0

1

2

U

(Nmiddotm

)


50 100 1400Time (s)

(d)


Lift

forc

e

2 31Cost function

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

2 31Cost function

times10minus5

0

1

2

U

(Nmiddotm

)

(b)

Pitc

h m

omen

t

2 31Cost function

times10minus5

0

05

1

15

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

2 31Cost function

times10minus5

0

1

2

3

4

U

(Nmiddotm

)

(d)



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0minus20 30


0

40


Alti

tude

z (m

)

20 minus40

80

100

t = 10

t = 4t = 25t = 06

(a)


minus40

minus30

minus20

minus10

0

10

20

30

40

Late

ral y

(m)

t = 10

t = 4t = 25t = 06

(b)


0

100

200

250

Traj

ecto

ry le

ngth

(m)

25 4 1006t (s)

(a)

0

2000

4000

6000

8000

10000

Fitn

ess v

alue

25 4 1006t (s)

(b)









t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

Lift

forc

e

6

65

7

75

Uz

(N)

(a)

Roll

mom

ent

times10minus5

minus8

minus4

0

4

8

U

(Nmiddotm

)

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

(b)

Pitc

h m

omen

t

t = 10

t = 4

t = 25t = 06

times10minus5

minus3minus2minus1

0123

U

(Nmiddotm

)

50 100 149460Time (s)

(c)

Hea

ding

mom

ent

times10minus4

t = 10

t = 4

t = 25t = 06

50 100 149460Time (s)

minus2

minus1

0

1

2

3

U

(Nmiddotm

)

(d)


Lift

forc

e

25 4 1006t (s)

0

2

4

6

8

Uz

(N)

(a)

Roll

mom

ent

times10minus6

0

2

4

6

8

U

(Nmiddotm

)

25 4 1006t (s)

(b)

Pitc

h m

omen

t

times10minus6

25 4 1006t (s)

0

1

2

3

U

(Nmiddotm

)

(c)

Hea

ding

mom

ent

times10minus5

25 4 1006t (s)

0

1

2

3

4

U

(Nmiddotm

)

(d)














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia














Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


an optimization problem for quadcopter reference flight

Documents