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Autonomous Mars-Gravity Enabling Quadrotor
5th International Conference on Advances in Control and 5th International Conference on 5th International Conference on Advances Advances in in Control Control and and Optimization of Dynamical Systems 5th International Conference on Advances in Control Optimization of Systems Available onlineand at www.sciencedirect.com Optimization of Dynamical Dynamical Systems February 18-22, 2018. Hyderabad, India 5th International Conference on Advances in Control and Optimization of Dynamical Systems February 2018. India February 18-22, 18-22, 2018. Hyderabad, Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India
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IFAC PapersOnLine 51-1 (2018) 160–165
Autonomous Mars-Gravity Enabling Quadrotor Autonomous Mars-Gravity Enabling Quadrotor Autonomous Mars-Gravity Enabling Quadrotor Autonomous Mars-Gravity Enabling Quadrotor ∗ Autonomous Mars-Gravity Enabling Quadrotor Kedarisetty Siddhardha ∗ ∗
Kedarisetty Kedarisetty Siddhardha Siddhardha Kedarisetty Siddhardha ∗ ∗ Siddhardha is withKedarisetty Department Siddhardha Aerospace Engineering, IIT Madras, Siddhardha is with Aerospace Siddhardha is India with Department Department Aerospace Engineering, Engineering, IIT IIT Madras, Madras, Chennai, (e-mail: [email protected]). Siddhardha is India with Department Aerospace Engineering, IIT Madras, Chennai, (e-mail: [email protected]). Chennai, India (e-mail: [email protected]). Siddhardha is with Department Aerospace Engineering, IIT Chennai, India (e-mail: [email protected]). Madras, Chennai, India (e-mail: [email protected]). Abstract: Study of various physical phenomena in reduced gravity environments is of imporAbstract: Study of various physical phenomena in environments of Abstract: Study of various physical phenomena in reduced reduced gravity environments is of imporimportance to several branches of science. This paper describes thegravity trajectory design and is automation Abstract: Studybranches of various physical phenomena in reduced gravity environments is of importance to several of science. This paper describes the trajectory design and automation tance to several branches of science. This paper describes the trajectory design and automation control strategy for a quadrotor to maintain an acceleration such that a payload on-board Abstract: Studybranches of various phenomena inacceleration reduced environments of importance to strategy several ofphysical science. This paperan describes thegravity trajectory design and is automation control for to maintain such that aa with payload on-board control strategy for aa quadrotor quadrotor to This maintain an acceleration such that payload on-board experiences Mars-gravity forscience. a short timepaper period. A 1-D the vertical trajectory time varying tance to several branches of describes trajectory design and automation control strategy for a quadrotor to maintain an acceleration such that a payload on-board experiences Mars-gravity for aa short time period. A vertical trajectory with time varying experiences Mars-gravity forthis short time A period. A 1-D 1-D vertical trajectory with timeon-board varying acceleration is proposed for purpose. detailed kinematic analysis arrive control strategy for a quadrotor to maintain an acceleration such thatis apresented payload experiences Mars-gravity forthis a short time A period. A 1-D vertical trajectory with timeto varying acceleration is proposed for purpose. detailed kinematic analysis is presented to arrive acceleration is proposed for this purpose. A detailed kinematic analysis is presented to arrive at an acceleration schedule for the quadrotor to follow this trajectory. The analysis will also experiences Mars-gravity for a short time period. A 1-D vertical trajectory with time varying acceleration is proposed for for thisthe purpose. A detailed kinematic analysisThe is presented to arrive at an acceleration schedule quadrotor to follow this trajectory. analysis will also at an acceleration schedule for the quadrotor to follow this trajectory. The analysis will also enable a designer to choose an appropriate motor-propeller combination for a quadrotor to acceleration is proposed for this purpose. A detailed kinematic analysis is presented to arrive at an acceleration schedule for the quadrotor to follow this trajectory. The analysis will also enable a designer to choose an appropriate motor-propeller combination for a quadrotor to enable a designer to choose an appropriate motor-propeller combination for a quadrotor to follow this trajectory, given the constraints on peak altitude of the executed trajectory and at an acceleration schedule for quadrotormotor-propeller topeak follow this trajectory. Thefor analysis will also enable a designer to choose an the appropriate combination atrajectory quadrotor to follow this trajectory, given the constraints on altitude of the executed and follow this trajectory, given the constraints on peak altitude of the executed trajectory and duration for which reduced gravity needs to be maintained. The efficacy of the proposed enable a designer to choose an appropriate motor-propeller combination for a quadrotor to follow this trajectory, given the constraints on peak altitude of the executed trajectory and duration for which reduced gravity needs be maintained. The efficacy of the proposed duration for which reduced gravity needs to to be maintained. The efficacy of trajectory thesimulation. proposed approach and automation strategy is demonstrated using a detailed 6-DoF model follow this trajectory, given the constraints on peak altitude of the executed and duration for which reduced gravity needs to be maintained. The 6-DoF efficacymodel of thesimulation. proposed approach and automation strategy is using aa detailed approach and automation strategy is demonstrated demonstrated using detailed 6-DoF model simulation. Since the for proposed trajectory is highly unsteady, widely used state thrust model duration which reduced gravity needs to be the maintained. Thesteady efficacy of the proposed approach and automation strategy is demonstrated using a detailed 6-DoF model simulation. Since the proposed trajectory is highly unsteady, the widely used steady state thrust model Since the proposed trajectory is highly unsteady, the widely used steady state thrust model for quadrotors will not suffice. Therefore, a better thrust model is developed using blade approach automation strategy is demonstrated using a detailed 6-DoF model simulation. Since the and proposed trajectory is Therefore, highly unsteady, the widely used steady state thrust model for quadrotors will not suffice. a better thrust model is developed using blade for quadrotors will not suffice. Therefore, a better thrust model is developed using blade element theory where the model parameters are estimated using experiments conducted in Since the proposed trajectory is highly unsteady, the widely used steady state thrust model for quadrotors will notthe suffice. Therefore, a are better thrust model is developedconducted using blade element theory where model parameters estimated using experiments in element theory where the model parameters are estimated using experiments conducted in afor wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is quadrotors will not suffice. Therefore, a better thrust model is developed using blade element theory where the model parameters are estimated using experiments conducted in acapable wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is aelement wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is of replicating Mars gravity environment for duration of 4 seconds while executing theory where the model parameters are estimated using experiments conducted in acapable wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is of replicating Mars gravity environment for duration of 4 seconds while executing of replicating Mars gravity environment for duration of 4 seconds while executing acapable vertical trajectory with peak altitude less than 45 m, which is validated through simulation aacapable wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is of replicating Mars gravity environment for duration of 4 seconds while executing vertical trajectoryflight with peak altitude less which through simulation acapable vertical with peakgravity altitude less than than 45 45 m, m, duration which is is validated validated through and a preliminary test. oftrajectory replicating Mars environment of 4 seconds whilesimulation executing a vertical trajectory with peak altitude less than 45for m, which is validated through simulation and a preliminary flight test. and a preliminary aand vertical trajectoryflight withtest. peak altitude less than 45 m, which is validated through simulation a preliminary flight test. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and a preliminary test.quadrotor, autonomous control, system modelling. Keywords: Reducedflight gravity, Keywords: Keywords: Reduced Reduced gravity, gravity, quadrotor, quadrotor, autonomous autonomous control, control, system system modelling. modelling. Keywords: Reduced gravity, quadrotor, autonomous control, system modelling. Keywords: Reduced gravity, quadrotor, autonomous control, system 1. INTRODUCTION Moreover, owing tomodelling. the wind conditions at such altitudes, 1. INTRODUCTION INTRODUCTION Moreover, owing to the the wind wind conditions at such such altitudes, altitudes, 1. Moreover, owing to conditions at the g-quality obtained was very poor (disturbance band 1. INTRODUCTION Moreover, owing to the wind conditions at such altitudes, the g-quality obtained was very poor (disturbance band the g-quality obtained was fixed very poor band 0.4g magnitude). Unlike wing(disturbance UAVs, quadrotors INTRODUCTION Moreover, owing to the wind conditions at such altitudes, The study of fluid1.physics, flame and combustion char- of the g-quality obtained was very poor (disturbance band of 0.4g magnitude). Unlike fixed wing UAVs, quadrotors 0.4g magnitude). Unlike fixed wing UAVs, quadrotors The study of of fluid physics, physics, flame and combustionunder char- of can perform vertical maneouvres, thereby opening up the The study fluid flame and combustion charthe g-quality obtained was very poor (disturbance band acteristics, material science, and biotechnology of 0.4g magnitude). Unlike fixed wing UAVs, quadrotors can perform vertical maneouvres, thereby opening up the The study of fluid physics, flame and combustionunder char- can perform vertical maneouvres, thereby opening up the acteristics, material science, and biotechnology possibility of performing free-fall-like 1-D parabolic maacteristics, material science, and biotechnology under of 0.4g magnitude). Unlike fixed wing UAVs, quadrotors reduced-gravity (micro, Moon, andand Mars) conditions is a possibility can perform vertical maneouvres, thereby opening upmathe The study of fluid physics, flame combustion charof performing free-fall-like 1-D parabolic acteristics, material science, and biotechnology under possibility of performing free-fall-like 1-D parabolic mareduced-gravity (micro, Moon, and Mars) conditions is a toofcreate reduced gravitythereby environments (Afman reduced-gravity (micro, Moon,and and biotechnology Mars) conditions is a noeuvres can perform vertical maneouvres, upmathe major area of research (Ries-Kautt (1997); Alam (2016); possibility performing free-fall-like 1-Dopening parabolic acteristics, material science, under noeuvres to create reduced gravity environments (Afman reduced-gravity (micro, Moon, and Mars) conditions is a noeuvres create reduced gravity environments (Afman major area of research research (Ries-Kautt (1997); Alam Alam (2016); (2016)). Ato novel conceptual approach of using a variable major area of (Ries-Kautt (1997); (2016); possibility of performing free-fall-like 1-D parabolic maPletser (2012)). In order to simulate reduced gravity envinoeuvres to create reduced gravity environments (Afman reduced-gravity Moon, and reduced Mars) conditions is a (2016)). (2016)). A novel novelquadrotor conceptualtoapproach approach ofmicro-gravity using aa variable variable major area of research (Ries-Kautt (1997); Alam (2016); A conceptual using Pletser (2012)). In(micro, order to simulate gravity envipitch propeller achieve aof enPletser (2012)). In order to simulate reduced gravity envinoeuvres create reducedto gravity environments (Afman ronments onofEarth, researchers use drop towers, manned (2016)). Atonovel conceptual approach ofmicro-gravity using a variable major research (Ries-Kautt (1997); Alam (2016); pitch propeller quadrotor achieve a enPletserarea (2012)). In order to simulate reduced gravity envi- pitch propeller quadrotor to achieve a micro-gravity enronments on Earth, researchers use drop towers, manned vironment is proposed in Afman (2016). Their simulation ronments on Earth, researchers use drop towers, manned (2016)). A novel conceptual approach of using a variable fixed wing andtorockets (2016); Pletser pitch propeller quadrotor to achieve a micro-gravity enPletser (2012)). In order simulate reduced gravity envi- vironment vironment is proposed proposed in Afman Afman (2016). Their simulation ronments onaircraft, Earth, researchers use(Belser drop towers, manned is in (2016). Their simulation fixed wing aircraft, and rockets (Belser (2016); Pletser results show that it is possible to achieve a micro-gravity fixed wing aircraft, and (Belser (2016); Pletser results pitch propeller quadrotor to achieve a micro-gravity en(2008)). Despite theresearchers highrockets g-quality (lower the manned residual vironment is proposed in Afman (2016). Their simulation ronments on Earth, use drop towers, show that it is possible to achieve a micro-gravity fixed wing aircraft, and rockets (Belser (2016); Pletser show that it isApossible to achieve a micro-gravity (2008)). Despite the the highg-quality) g-qualityand (lower thecapability residual results for about 3is seconds. variable pitch propeller quadrotor (2008)). Despite the high g-quality (lower the residual vironment proposed in Afman (2016). Their simulation acceleration, higher their results show that it is possible to achieve a micro-gravity fixed wing aircraft, rockets (Belser (2016); Pletser for for aboutof 3 seconds. seconds. A variable variable pitch propeller propeller quadrotor (2008)). Despite the and highg-quality) g-quality (lower thecapability residual about 3 pitch quadrotor acceleration, higher the and their (instead conventional quadrotor) is required because acceleration, higher and their results show that it isA to achieve a micro-gravity to carry Despite heavy payloads, these methods that simulate for about 3 seconds. Apossible variable pitch propeller quadrotor (2008)). the the highg-quality) g-quality (lower thecapability residual (instead (instead of conventional quadrotor) is required because acceleration, higher the g-quality) and their capability of conventional quadrotor) is required because to carry heavy payloads, these methods that simulate both positive and negative thrust forces are necessary to carry heavy payloads, these methods that simulate for about 3 seconds. A variable pitch propeller quadrotor reduced-gravity environment aremethods costly, and proce- both (instead of conventional quadrotor) is required because acceleration, higher the g-quality) and their capability positive and negative negative thrust forces are necessary necessary to carry heavy payloads, these thatthe simulate both positive and thrust forces are reduced-gravity environment are costly, and the proceto maintain micro-gravity as well as to maintain attireduced-gravity environment costly, and the proce(instead of conventional quadrotor) is to required because dures areheavy involved limitingthese theare frequency at which they to both positive and negative thrust forces are necessary to carry payloads, methods that simulate maintain micro-gravity as well as maintain attireduced-gravity environment are costly, and the proceto maintain micro-gravity as well as to maintain attidures are involved limiting the frequency at which they tude control. This is, however, not possible to achieve dures are involved limiting theare frequency at which they tude both positivemicro-gravity and negative forces are tonecessary can conducted (Afman (2016)). to maintain asthrust well as to maintain attireduced-gravity environment costly, and the procecontrol. This is, however, not possible achieve dures are involved limiting the frequency at which they tude control. This is, however, not possible to achieve can conducted conducted (Afman (Afman (2016)). (2016)). by a conventional quadrotor. Awell conventional quadrotor can to maintain micro-gravity as as to maintain attitude control. This is, however, not possible to achieve dures are involved limiting the frequency at which they by conventional quadrotor. A conventional conventional quadrotor can conducted (2016)). aa conventional quadrotor. A quadrotor Meanwhile, the(Afman advances in Unmanned Aerial Vehicle by requires to switch of the propeller rotation to achieve tude control. This is, however, not possible to achieve by a conventional quadrotor. A conventional quadrotor Meanwhile, the(Afman advances inresearchers Unmanned to Aerial Vehicle can conducted (2016)). requires to switch of the the propeller rotation to achieve Meanwhile, the advances in Unmanned Aerial Vehicle to switch of rotation to achieve (UAV) technology, enabled conduct re- requires free thus leaving the propeller quadrotor uncontrollable to by a fall, conventional quadrotor. A conventional quadrotor Meanwhile, the advances inresearchers Unmanned to Aerial Vehicle requires to switch of the propeller rotation to achieve (UAV) technology, enabled conduct refree fall, thus leaving the quadrotor uncontrollable to (UAV) technology, enabled researchers to conduct refree fall, thus leaving the quadrotor uncontrollable to duced gravity experiments at low cost and more fredisturbances. Therefore, a conventional quadrotor canMeanwhile, the advances in Unmanned Aerial Vehicle requires switch of the propeller to achieve (UAV) gravity technology, enabled at researchers re- disturbances. free fall, to thus leaving the quadrotorrotation uncontrollable to duced experiments low cost cost to andconduct more payfreTherefore, a conventional quadrotor canduced gravity experiments at low and more fredisturbances. Therefore, a conventional quadrotor canquently in spite of the reduced g-quality and lower be used for simulating micro-gravity environments. (UAV) technology, enabled researchers re- not free fall, thus leaving the quadrotor uncontrollable to duced gravity experiments at low cost to andconduct more payfredisturbances. Therefore, a conventional quadrotor canquently in spite of the reduced g-quality and lower not be used for simulating micro-gravity environments. quently in spite of the reduced g-quality and lower paynot be used for simulating micro-gravity environments. load capacity. Inexperiments Higashino (2010), the authors proposed However, it is possible to produce a reduced-non-zeroduced gravity at low cost and more fredisturbances. Therefore, a conventional quadrotor canquently in spite of the reduced g-quality and lower paynot be used forpossible simulating micro-gravity environments. load capacity. In Higashino Higashino (2010), the authors authors proposed However, it is is to produce produce reduced-non-zeroload capacity. In (2010), the proposed it to aa reduced-non-zerothe use of spite a fixed wing UAV to create reduced gravgravity environments with amicro-gravity conventional quadrotor. quently in the reduced g-quality and lower pay- However, not be used forpossible simulating environments. load capacity. In of Higashino (2010), the authors proposed However, it is possible to produce a reduced-non-zerothe use of a fixed wing UAV to create reduced gravgravity environments with a conventional quadrotor. the use of a fixed wing UAV to create reduced gravenvironments with a conventional quadrotor. ity The simulation showproposed that an gravity load capacity. In Higashino (2010), the authors However, it is possible to produce a reduced-non-zerothe environment. use of a fixed wing UAV to results create reduced gravgravity environments with a conventional quadrotor. ity environment. The simulation results show that an major focus of this work is to design an autonomous ity simulation show for that environment than 0.15g be achieved 2 an to The the environment. use of a less fixedThe wing UAVcan to results create reduced gravThe major focus of this thiswith work is to to design design an an autonomous gravity environments a conventional quadrotor. ity environment. The simulation results show for that major focus of work is autonomous environment less than 0.15g can be achieved achieved 2 an to The conventional quadrotor towards gravity environment less than 0.15g can be for 2 to 3 seconds. Although the experimental results presented The major focus of this work is tosimulating design an Mars autonomous ity environment. The simulation results show that an conventional quadrotor towards simulating Mars gravity environment less than 0.15g can be achieved for 2 to conventional quadrotor towards simulating Mars gravity 3 seconds. Although the experimental results presented environment on Earth. A 1-D parabolic (in time) manoeu3 seconds. Although the experimental results presented The major focus of thisAwork is tosimulating design an autonomous were not as satisfactory as the simulation results, the conventional quadrotor towards Mars gravity environment less than 0.15g can be achieved for 2 to environment on Earth. 1-D parabolic (in time) manoeu3 seconds. Although the experimental results presented environment on Earth. A 1-D parabolic (in time) manoeuwere not as as satisfactory astothe the simulation results, the vre is proposed to achieve the required gravity levels. were not satisfactory as simulation results, the conventional quadrotor towards simulating Mars gravity feasibility of using UAVs produce reduced gravity environment on Earth. A 1-D parabolic (in time) manoeu3 seconds. Although the experimental results presented vre is proposed to achieve the required gravity levels. were not as satisfactory the simulation results, the vre is proposed tothis achieve the required gravity levels. feasibility of In using UAVsasto to(2011), produce reduced gravity The kinematics of flight trajectory are derived with feasibility of using UAVs produce reduced gravity environment on Earth. A 1-D parabolic (in time) manoeuwas proved. Hofmeister authors experimenvre is proposedoftothis achieve the required gravity levels. were not as satisfactory as the simulation results, the The kinematics flight trajectory are derived with feasibility of using UAVs to produce reduced gravity The kinematics ofto flight trajectory aregravity derived with was proved. In Hofmeister using (2011), authors experimenthe consideration ofthis flying constraints (maximum height, was proved. In Hofmeister (2011), authors experimenvre is proposed achieve the required levels. tally created micro-gravity a fixed wing UAV for The kinematics of this flight trajectory are derived with feasibility of using UAVs to produce reduced gravity the consideration of flying constraints (maximum height, was proved. In Hofmeister (2011), authors experimenthe consideration ofthis flying constraints (maximum height, tally created micro-gravity using fixed wing UAV for for reduced gravityofduration andtrajectory payload). The kinematic tally created micro-gravity using aa fixed wing UAV The kinematics flight are derived with about 10 seconds. However, to achieve such duration of the consideration of flying constraints (maximum height, was proved. In Hofmeister (2011), authors experimenreduced gravity duration and payload). The kinematic tally created micro-gravity using a fixed wing UAV for gravity duration and The kinematic about 10 seconds. seconds. However, tolength achieve such duration of reduced analysis also provides a base forpayload). the choice of the motorabout 10 However, to achieve such duration of the consideration of flying constraints (maximum height, micro-gravity, the flight path and the change in reduced gravity duration and payload). The kinematic tally created micro-gravity using a fixed wing UAV for analysis also provides a base for the choice of the motorabout 10 seconds. However, tolength achieveand such duration of analysis provides afor base forpayload). the choice ofautomation the motormicro-gravity, thewas flight path the change in in propeller combination this An micro-gravity, the flight path and the change reduced also gravity duration and The kinematic altitude 1 km and 400 meters respectively. analysis also provides afor base formission. the choice ofautomation the motorabout 10required seconds. However, tolength achieve such duration of propeller propeller combination this mission. An micro-gravity, thewas flight path length and the change in combination for this mission. An automation altitude required 1 km and 400 meters respectively. altitude required was 1 km and 400 meters respectively. analysis also provides a base for the choice of the motorpropeller combination for this mission. An automation micro-gravity, thewas flight path length and the change in altitude required 1 km and 400 meters respectively. propeller combination for this mission. An automation altitude required was 1 km and 400 meters respectively. Copyright © 2018 IFAC 176 ∗ K. ∗∗ K. ∗ K. K. ∗ K.
2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 176 Copyright ©under 2018 responsibility IFAC 176Control. Peer review© of International Federation of Automatic Copyright © 2018 IFAC 176 10.1016/j.ifacol.2018.05.027 Copyright © 2018 IFAC 176
5th International Conference on Advances in Control and Optimization of Dynamical Systems Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 February 18-22, 2018. Hyderabad, India
control strategy is proposed to enable the quadrotor to execute the desired trajectory autonomously. As the quadrotor has to perform ascent and descent flights to complete the proposed 1-D parabolic manoeuvre, a blade element theory based axial flight model is derived to calculate the thrust and counter torque of a propeller in axial flights. To validate the derived model in axial flight conditions and to estimate model parameters, wind-tunnel experiments are conducted. Finally, simulation and preliminary flight test results are presented to demonstrate the capability of conventional quadrotors to simulate reduced gravity environments.
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example, as per the Indian government regulations, the maximum altitude at which a micro UAV (weight less than 2 kg) can fly in an educational/ research institution without local authority permission is 200 ft (60 m approximately) (Air Transport Circular (2016)). • Reduced gravity time (trg ): It is the amount of time the quadrotor spends in a reduced gravity state. • Payload mass (mp ): It is the mass of the on-board equipment required to conduct experiments while in reduced gravity state. 2.3 Kinematics
2. TRAJECTORY DESIGN One way to create reduced gravity using a fixed wing aircraft is to perform a 2-D parabolic manoeuvre. Fixed wing aircraft require forward speed to be airborne, and therefore, to simulate a reduced gravity environment, they need to have a flight trajectory that spreads along both forward and vertical translational axes. However, owing to its capability to translate only along the vertical axis, a quadrotor can perform a 1-D parabolic (in time) manoeuvre to achieve reduced gravity. 2.1 The Trajectory The proposed trajectory a quadrotor need to follow to create Mars-gravity is shown in Fig. 1, and can be divided into five phases. In the first (take-off) phase, quadrotor takes off from ground and hovers at a certain altitude. In the second (climb) phase, it accelerates to reach a certain desired velocity. Once this velocity is achieved, the rotor angular speeds are varied such that the desired constant acceleration is maintained (third phase - reduced gravity phase). After completing the parabolic path, the quadrotor accelerates to break the fall in phase four (decent phase), and performs a controlled landing in phase five (landing phase). These five phases are indicated in Fig. 1. The circled number in the figure marks the beginning of the corresponding phase.
A detailed kinematic analysis is performed to compute the acceleration schedule required for a quadrotor to follow the proposed 1-D trajectory. First, the kinematic equations for phase-3 – the reduced gravity phase – is derived, as the required accelerations in the other phases, especially the desired acceleration in climb phase, will depend upon this. Phase-3 board is
The acceleration experienced by a payload on-
aexp = aE − ab (1) where, aE is gravitational acceleration of Earth, and ab is the acceleration of the body (quadrotor). We have ab = af + aE (2) where, af is the body acceleration due to thrust and drag forces acting on it. For the purpose of this paper, the required experienced acceleration aM – the acceleration due to gravity on Mars. From the equations (1) and (2), the desired acceleration due to thrust of the rotors (we neglect drag in this analysis) is af = −aM (3)
From the kinematic equations for one dimensional motion, we obtain arg trg 1 2 (4) hpeak = h3 − arg trg , v3 = −v5 = − 8 2 where, hi and vi are respectively the altitude and velocity at the beginning of phase-i, trg is the duration of phase-3, and arg = af + aE = −aM + aE is the body acceleration in phase-3. Phase-2
The kinematic equations for phase-2 gives us 2 a2rg trg arg trg (5) , t23 = − h3 = h2 + 8a23 2a23 where, t23 is the duration of phase-2, and a23 is body acceleration in phase-2. Note that in deriving the above relations, we used v3 from equation (4).
Fig. 1. Phases of 1-D parabolic manoeuvre 2.2 Flying Constraints There are three constraints that has to be considered while designing a quadrotor to conduct reduced gravity experiments. • Peak altitude (hpeak ): It is the maximum altitude a quadrotor achieves while executing the proposed trajectory. The peak altitude contraint usually arise from governmental regulations on UAV flying. For 177
From equations (4) and (5), we can obtain the acceleration requirement and its duration in phase 2, given the constraints on hpeak and trg as a23 =
2 a2rg trg 2 8(hpeak − h2 ) + arg trg
,
t23 = −
arg trg 2a23
(6)
Assuming a hover altitude of 5 m (h2 ) and using the above derived relations, we have computed the peak altitude
5th International Conference on Advances in Control and 162 Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
and the time duration of phase 2 (t23 ) as a function of acceleration requirement in phase 2 (a23 ) for various values of reduced gravity duration. These are plotted in Fig. 2. These charts will help a designer to estimate the maximum acceleration requirement, given the hpeak and trg constraints, to make an appropriate motor-propeller selection as described in the next section.
4. CONTROL AND AUTOMATION STRATEGY From the kinematic analysis presented in Section 2, the acceleration schedule – acceleration value and duration for each phase – is known, which when followed will result in the on-board payload experiencing reduced gravity (Mars-gravity in our case) for the required duration. To implement this acceleration schedule on a quadrotor with the motor and propeller appropriately chosen as in the previous section, we propose the following control and automation strategy. We propose to use a low-level attitude controller and higher level automation (altitudeacceleration controller for various flight phases) as described below. 4.1 Attitude Control
(a) hpeak vs a23
Fig. 3. Attitude controller structure A 2-loop feedback control structure is chosen for attitude controller as shown in Fig. 3. An outer loop with attitude feedback, and an inner loop with body rate feedback makes this control structure robust to disturbances such as model uncertainties, measurement noise, and environmental disturbances.
(b) t23 vs a23
Fig. 2. Design charts obtained from kinematic analysis 3. MOTOR-PROPELLER SELECTION Unlike the general quadrotor design for which the major design criteria are agility, range and endurance, the design criterion for a quadrotor to create reduced gravity is the ability to produce required acceleration in the ascent and descent phases. Thus the quadrotor should be designed such that maximum thrust produced by the motor-propeller combination meets this requirement. Fig. 2a provides the acceleration requirement (a23 ) in ascent and descent phases given the hpeak and trg constraints. Given this, the acceleration that the motorpropeller combination should produce is obtained from equation (2) as af = a23 − aE (7) Thus, the minimum thrust constraint can be written as T ≥ ks mq af (8) where, ks is the safety factor taken as 1.5 in practice. Here, the quadrotor mass mq is mq = 4ma + mk + mp
(9) where, mp is the payload mass, mk is the known mass, that is, the mass of the battery, frame, electronics, etc., and ma is the mass of motor-propeller module. Thus, any motor-propeller combination that satisfies the following constraint can be chosen of this mission T ≥ ks (4ma + mk + mp )af (10) Note that this is a ‘design relation’ wherein the quantities on both the right hand side (T ) and the left hand side (ma ) depends on the choice of motor-propeller combination. 178
4.2 Automation
Fig. 4. Automation strategy To enable a quadrotor to autonomously fly the five phases of the proposed reduced gravity manoeuvre, we suggest a state machine like automation as shown in Fig. 4. Depending upon the current phase, the desired acceleration is provided to a PI controller that uses an acceleration feedback to generate appropriate command to be given to the motor-propeller module. For take-off and landing phases (1 and 5), the desired acceleration is generated by an outer PID loop executing an altitude hold (see Fig. 4). In the climb, reduced gravity and decent phases (2, 3, and 4) required acceleration obtained from the kinematic analysis is used as desired acceleration input to the PI controller. In the case of communication loss or loss of stability, the phase state will be set as ‘fault’ (Phase f), and the desired acceleration is given as the acceleration due to gravity of Earth. This results in shutting down of all the motors and a resultant free-fall.
5th International Conference on Advances in Control and Optimization of Dynamical Systems Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 February 18-22, 2018. Hyderabad, India
5. SYSTEM MODELLING The quadrotor governing equations of motion, and the propeller thrust and counter torque model used in simulations to test the proposed automation strategy are described in this section. 5.1 Governing Equations We use the standard rigid body equations, specialized for quadrotors, given as d(mV b ) + Ω b × (mV b ) = F p + F g dt (11) d(IΩ b ) + Ω b × (IΩ b ) = τcg + τp + τgp dt where, V b and Ω b are the translation and angular velocities in directions of body reference frame axes, m is the mass of quadrotor, and I is the rotational inertia matrix of the quadrotor. The description of the forces and torques on the right hand side of equation (11) are as follows: F b - force due to propeller, F b - force due to gravity, τcg - torque due to center of gravity shift from geometric center, τp - torque due to propellers, and τgp - torque due to propeller gyroscopic effects. The widely used and popular thrust model for quadrotor assumes that the thrust produced is directly proportional to the square of RPM (Bouabdallah (2004)). This is true in hover and when the quadrotor is translating steady at very low speeds which usually is the case. However, our proposed trajectory involves varying accelerations and speeds as seen in Section 2, such a thrust model will not suffice. Therefore, we propose a new thrust model, that is more appropriate when axial speeds are non-zero, using blade element theory.
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Substituting in equation (12), we get 2 2 1 i 2 i dCT = clα σi r r θi r − r(λi + λa ) dr 2 i=0
(14)
i=0
By integrating along the blade length, the total thrust coefficient can be obtained as 1 (15) CT = clα k1 + k2 λi + λa 2 where, the constants k1 and k2 are functions of solidity ratio and pitch function coefficients σi , θi , i = 0, 1, 2. Thus, the total thrust produced by a propeller of radius R can be obtained by using the relation between thrust and non-dimensional thrust coefficient T = CT ρπR2 (ωR)2 as 1 k2 vi 1 4 πR ω 2 + clα k2 πR3 ωVa (16) T = clα k1 + 2 ωR 2 The relation between the inflow velocity and angular speed of the propeller is linear (vi = kv ω). This can be verified experimentally as shown in Fig. 6(b). Therefore, the thrust produced by the propeller in axial flight can be written as (17) T = bω 2 + ba ωVa k2 kv 1 1 4 3 where, b = 2 clα k1 + R πR and ba = 2 clα k2 πR are hover thrust and axial thrust coefficients respectively.
(a) Wind tunnel
5.2 Thrust model To develop a model that predicts the thrust and counter torque produced by a propeller in axial flights, we use the theory of blade element analysis (Leishman (2006)). Consider a propeller with solidity ratio σ, effective lift coefficient slope clα , blade pitch θ, non-dimensional inflow velocity λi , and non-dimensional axial velocity λa due to ascent or decent. Then from the blade element analysis, the elemental thrust coefficient can be written as (Leishman (2006)) 1 λ + λa (12) dCT = clα σr 2 θ − i 2 r where r is the non-dimensional radius of the propeller (r ∈ [0, 1]). The solidity ratio (σ) is constant for rectangular blades. However, for usual propellers σ varies along the length of the blade as the chord length is not constant. Similarly, the twist of the blade also varies along the blade length. For analysis, we assume second order polynomials to model the variation of σ and θ along the blade as σ=
2 i=0
σi r i ;
θ=
2
θi r i
(13)
i=0
179
(b) Motor-propeller setup
Fig. 5. Wind-tunnel experimental setup In order to experimentally validate the developed theoretical expressions, and to find the thrust coefficients, a sub-sonic wind tunnel is used to simulate the descent flight conditions for a propeller as shown in Fig. 5(a). The wind tunnel is capable of producing inflow velocity upto 12 m/s inside the chamber. The set-up inside the wind tunnel chamber as shown in Fig. 5(b) consists of motorpropeller combination, a six axis load cell to measure the thrust and counter torque, and an encoder to measure the angular speed of the propeller. In order to examine the wall-effects on motor-propeller inside the wind tunnel, the values of T , ω and vi are measured without any opposing inflow from wind tunnel (hover condition). The variation of these readings from the measurements obtained outside the wind tunnel are negligible. Thereby, in further wind-tunnel experiments, the wall-effect on motor-propeller module is considered to be negligible. The steady state measurements of T vs ω, and vi vs ω under hover condition are provided in
5th International Conference on Advances in Control and 164 Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
steady state (triggering criteria for phase 2), desired body acceleration of a23 (3.01 m/s2 ) is provided as command input. After achieving desired climb velocity (v3 (12.1 m/s), the flight transitions to reduced gravity phase. In the reduced gravity phase the commanded body ac(a) T vs ω at hover
(b) vi vs ω at hover
Fig. 6. Steady state thrust and inflow response of propeller at hover Fig. 6(a) and 6(b) respectively. From the Fig. 6(b), we can observe that the relation between inflow velocity and motor angular speed is linear (vi = kv ω).
(a) T vs ω at va = −6 m/s
(b) T vs ω at va = −7.3 m/s
Fig. 8. Simulation plots
(c) T vs ω at va = −8.5 m/s
(d) T vs ω at va = −9.8 m/s
Fig. 7. Steady state thrust measurements under descent condition To obtain thrust model coefficients (b and ba ), the measurements of T vs ω for various opposing airflow velocities are recorded as shown in Fig. 7. The combination of vehicle governing equations, motor dynamics, and thrust and counter torque steady state models completes the mathematical modelling of quadrotor in axial flights. This complex non-linear model is used for simulation purposes in the sections that follow. 6. RESULTS To demonstrate the efficacy of our proposed approach to generate reduced gravity using quadrotors, we perform a 6DoF simulation implementation of the suggested strategy. To validate the simulation results, preliminary (open-loop) experimental flight test results are also presented. 6.1 Simulation results The altitude, axial velocity and acceleration, and propeller angular speed simulation results for autonomous reduced gravity manoeuvre are shown in Fig. 8. In these plots, the transition in flight phase is indicated using a vertical dash line. Operation In the take-off phase, the quadrotor is commanded to hover at a height of 5 m. After achieving 180
celeration is arg (−6.099 m/s2 ), and the experienced acceleration on the quadrotor in this phase will mimic the Mars gravity. After trg (4 seconds) time period of reduced gravity phase (triggering criteria for phase 4), the descent phase is commenced by providing body acceleration command of a23 to break the fall. The descent phase is switched to landing phase once the velocity approaches near to zero. Analysis In order to maintain a constant acceleration in phases 2,3 and 4, the angular speed of the rotor is varied such that a constant thrust is achieved. In climb phase (refer rotor angular speed plot in Fig. 8), as the axial airflow increases, the angular speed of the propeller is increased such that a constant acceleration is produced. This is because in ascent the thrust produced by the propeller decreases when compared to the thrust produced in hover condition at same angular speeds. Similarly, during reduced gravity phase, as the quadrotor velocity decreases, the angular speed of the rotor also decreases to maintain constant thrust. Thus, the proposed autonomous flight strategy helps to produce a constant Mars gravity environment. 6.2 Flight Test Results Implementation In flight tests, a 3 phase open loop acceleration control is used to conduct preliminary reduced gravity experiments. The ascent, reduced gravity and descent are the three phases in the flight envelope. Based on the desired acceleration in each phase, BLDC motors are commanded appropriately (to produce desired thrust under hover conditions) for the calculated time interval using kinematics in Section 2. In the ascent and reduced gravity phases, the thrust command is autonomous, that is, for given time interval of each phase, desired actuation signals are provided to BLDC motors by the au-
5th International Conference on Advances in Control and Optimization of Dynamical Systems Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 February 18-22, 2018. Hyderabad, India
topilot code. Whereas, at the end of the reduced gravity phase, the quadrotor thrust control is switched to remote control. An experienced human pilot ensures the safe landing of the quadrotor by providing appropriate thrust commands.
165
acceleration towards the phase end, which is due to the thrust increase of the propeller caused by the downward axial velocity. These deviations in the acceleration can be easily controlled by implementing the developed autonomous strategy. Therefore, the preliminary flight test results indicate that the conventional quadrotors are feasible and capable to replicate reduced gravity environments with the exception of micro-gravity. 7. CONCLUSIONS In this paper, the design, modelling and automation of conventional quadrotor capable of replicating Mars gravity environment is proposed. An approximate thrust model is derived using blade element theory and validated through series of wind tunnel experiments. The simulations confirmed the feasibility of using the conventional quadrotor and the developed automation strategy to create Mars gravity for duration of 4 to 5 seconds. The practicality of this concept is also verified through preliminary flight test results.
Fig. 9. Flight test
REFERENCES
Fig. 10. Altitude, acceleration, and attitude plots Results and analysis The flight test snapshot collage of the quadrotor performing a 1-D parabolic like manoeuvre is shown in Fig. 9. The corresponding altitude, acceleration, and attitude plots are shown in Fig. 10. We can observe from Fig. 10(b) that a reduced gravity phase replicating the Mars gravity lasted approximately 4 seconds, and the peak altitude is within the prescribed limit. The designed attitude controller maintained the attitude of the vehicle within the range of ±3 degrees with an exception of a small disturbance oscillation due to sudden increase in the vehicles thrust. From the altitude and acceleration plots we can observe that the altitude does not decrease instantaneously with decrease in acceleration during the transition from acceleration to reduced gravity phase. This is due to the existing momentum of the body at the end of the acceleration phase, which enables the quadrotor to complete a 1-D parabolic like path. Since an open loop acceleration control strategy is chosen instead of closed loop control, the desired acceleration could not be maintained. From the analysis plot Fig. 12(b), in the ascent phase, the acceleration of the body decreases as the thrust produced by the propellers are decreased due to the upward axial flow. Similarly, in reduced gravity phase, we can observe increase in the 181
Ries-Kautt, M., et al. Crystallogenesis studies in microgravity with the Advanced Protein Crystallization Facility on SpaceHab-01. Journal of crystal growth, 181: 79–96, 1997. Alam, F. E., Frederick, L. D., & Tanvir, I. F. Effectiveness of xenon as a fire suppressant under microgravity combustion environment. Journal of Combustion Science and Technology, 188:145–165, 2016. Pletser, V., et al. The first joint European partial-g parabolic flight campaign at Moon and Mars gravity levels for science and exploration. Journal of Microgravity Science and Technology, 24:1-13, 2012. Belser, V., et al. Aerodynamic and engineering design of a 1.5 s high quality microgravity drop tower facility. Acta Astronautica, 129, pp. 335-344, 2016. Pletser, V., Anne, P., & Olivier, M. International heat and mass transfer experiments on the 48th ESA parabolic flight campaign of March 2008. Journal of Microgravity Science and Technology, 20, pp. 177-182, 2008. Higashino, S., & Shotaro, K. Automatic Microgravity Flight System and Flight Testing Using a Small Unmanned Aerial Vehicle. Journal of Japan Society Microgravity Application, 27, pp. 3-10, 2010. Hofmeister, P. G., & Jrgen, B. Parabolic flights@ home. Journal of Microgravity Science and Technology, 23, pp. 191-197, 2011. Afman, J. P., et al. On the Design and Optimization of an Autonomous Microgravity Enabling Aerial Robot. arXiv preprint, arXiv:1611.07650, 2016. Air Transport Circular. Office of the Director General of Civil Aviation, Government of India, 2016. Nelson, R. C. Flight stability and automatic control: Vol. 2. New York: WCB/McGraw Hill, 1998. Bouabdallah, S., Noth, A., & Siegwart, R. PID vs LQ control techniques applied to an indoor micro quadrotor. In Proc. of International Conference on Intelligent Robots and Systems, Sendai, Japan, pp. 2451-2456, 2004. Leishman, G. J. Principles of helicopter aerodynamics, Cambridge university press, 2006.
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IFAC PapersOnLine 51-1 (2018) 160–165
Autonomous Mars-Gravity Enabling Quadrotor Autonomous Mars-Gravity Enabling Quadrotor Autonomous Mars-Gravity Enabling Quadrotor Autonomous Mars-Gravity Enabling Quadrotor ∗ Autonomous Mars-Gravity Enabling Quadrotor Kedarisetty Siddhardha ∗ ∗
Kedarisetty Kedarisetty Siddhardha Siddhardha Kedarisetty Siddhardha ∗ ∗ Siddhardha is withKedarisetty Department Siddhardha Aerospace Engineering, IIT Madras, Siddhardha is with Aerospace Siddhardha is India with Department Department Aerospace Engineering, Engineering, IIT IIT Madras, Madras, Chennai, (e-mail: [email protected]). Siddhardha is India with Department Aerospace Engineering, IIT Madras, Chennai, (e-mail: [email protected]). Chennai, India (e-mail: [email protected]). Siddhardha is with Department Aerospace Engineering, IIT Chennai, India (e-mail: [email protected]). Madras, Chennai, India (e-mail: [email protected]). Abstract: Study of various physical phenomena in reduced gravity environments is of imporAbstract: Study of various physical phenomena in environments of Abstract: Study of various physical phenomena in reduced reduced gravity environments is of imporimportance to several branches of science. This paper describes thegravity trajectory design and is automation Abstract: Studybranches of various physical phenomena in reduced gravity environments is of importance to several of science. This paper describes the trajectory design and automation tance to several branches of science. This paper describes the trajectory design and automation control strategy for a quadrotor to maintain an acceleration such that a payload on-board Abstract: Studybranches of various phenomena inacceleration reduced environments of importance to strategy several ofphysical science. This paperan describes thegravity trajectory design and is automation control for to maintain such that aa with payload on-board control strategy for aa quadrotor quadrotor to This maintain an acceleration such that payload on-board experiences Mars-gravity forscience. a short timepaper period. A 1-D the vertical trajectory time varying tance to several branches of describes trajectory design and automation control strategy for a quadrotor to maintain an acceleration such that a payload on-board experiences Mars-gravity for aa short time period. A vertical trajectory with time varying experiences Mars-gravity forthis short time A period. A 1-D 1-D vertical trajectory with timeon-board varying acceleration is proposed for purpose. detailed kinematic analysis arrive control strategy for a quadrotor to maintain an acceleration such thatis apresented payload experiences Mars-gravity forthis a short time A period. A 1-D vertical trajectory with timeto varying acceleration is proposed for purpose. detailed kinematic analysis is presented to arrive acceleration is proposed for this purpose. A detailed kinematic analysis is presented to arrive at an acceleration schedule for the quadrotor to follow this trajectory. The analysis will also experiences Mars-gravity for a short time period. A 1-D vertical trajectory with time varying acceleration is proposed for for thisthe purpose. A detailed kinematic analysisThe is presented to arrive at an acceleration schedule quadrotor to follow this trajectory. analysis will also at an acceleration schedule for the quadrotor to follow this trajectory. The analysis will also enable a designer to choose an appropriate motor-propeller combination for a quadrotor to acceleration is proposed for this purpose. A detailed kinematic analysis is presented to arrive at an acceleration schedule for the quadrotor to follow this trajectory. The analysis will also enable a designer to choose an appropriate motor-propeller combination for a quadrotor to enable a designer to choose an appropriate motor-propeller combination for a quadrotor to follow this trajectory, given the constraints on peak altitude of the executed trajectory and at an acceleration schedule for quadrotormotor-propeller topeak follow this trajectory. Thefor analysis will also enable a designer to choose an the appropriate combination atrajectory quadrotor to follow this trajectory, given the constraints on altitude of the executed and follow this trajectory, given the constraints on peak altitude of the executed trajectory and duration for which reduced gravity needs to be maintained. The efficacy of the proposed enable a designer to choose an appropriate motor-propeller combination for a quadrotor to follow this trajectory, given the constraints on peak altitude of the executed trajectory and duration for which reduced gravity needs be maintained. The efficacy of the proposed duration for which reduced gravity needs to to be maintained. The efficacy of trajectory thesimulation. proposed approach and automation strategy is demonstrated using a detailed 6-DoF model follow this trajectory, given the constraints on peak altitude of the executed and duration for which reduced gravity needs to be maintained. The 6-DoF efficacymodel of thesimulation. proposed approach and automation strategy is using aa detailed approach and automation strategy is demonstrated demonstrated using detailed 6-DoF model simulation. Since the for proposed trajectory is highly unsteady, widely used state thrust model duration which reduced gravity needs to be the maintained. Thesteady efficacy of the proposed approach and automation strategy is demonstrated using a detailed 6-DoF model simulation. Since the proposed trajectory is highly unsteady, the widely used steady state thrust model Since the proposed trajectory is highly unsteady, the widely used steady state thrust model for quadrotors will not suffice. Therefore, a better thrust model is developed using blade approach automation strategy is demonstrated using a detailed 6-DoF model simulation. Since the and proposed trajectory is Therefore, highly unsteady, the widely used steady state thrust model for quadrotors will not suffice. a better thrust model is developed using blade for quadrotors will not suffice. Therefore, a better thrust model is developed using blade element theory where the model parameters are estimated using experiments conducted in Since the proposed trajectory is highly unsteady, the widely used steady state thrust model for quadrotors will notthe suffice. Therefore, a are better thrust model is developedconducted using blade element theory where model parameters estimated using experiments in element theory where the model parameters are estimated using experiments conducted in afor wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is quadrotors will not suffice. Therefore, a better thrust model is developed using blade element theory where the model parameters are estimated using experiments conducted in acapable wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is aelement wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is of replicating Mars gravity environment for duration of 4 seconds while executing theory where the model parameters are estimated using experiments conducted in acapable wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is of replicating Mars gravity environment for duration of 4 seconds while executing of replicating Mars gravity environment for duration of 4 seconds while executing acapable vertical trajectory with peak altitude less than 45 m, which is validated through simulation aacapable wind tunnel. Our kinematic analysis shows that an appropriately designed quadrotor is of replicating Mars gravity environment for duration of 4 seconds while executing vertical trajectoryflight with peak altitude less which through simulation acapable vertical with peakgravity altitude less than than 45 45 m, m, duration which is is validated validated through and a preliminary test. oftrajectory replicating Mars environment of 4 seconds whilesimulation executing a vertical trajectory with peak altitude less than 45for m, which is validated through simulation and a preliminary flight test. and a preliminary aand vertical trajectoryflight withtest. peak altitude less than 45 m, which is validated through simulation a preliminary flight test. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and a preliminary test.quadrotor, autonomous control, system modelling. Keywords: Reducedflight gravity, Keywords: Keywords: Reduced Reduced gravity, gravity, quadrotor, quadrotor, autonomous autonomous control, control, system system modelling. modelling. Keywords: Reduced gravity, quadrotor, autonomous control, system modelling. Keywords: Reduced gravity, quadrotor, autonomous control, system 1. INTRODUCTION Moreover, owing tomodelling. the wind conditions at such altitudes, 1. INTRODUCTION INTRODUCTION Moreover, owing to the the wind wind conditions at such such altitudes, altitudes, 1. Moreover, owing to conditions at the g-quality obtained was very poor (disturbance band 1. INTRODUCTION Moreover, owing to the wind conditions at such altitudes, the g-quality obtained was very poor (disturbance band the g-quality obtained was fixed very poor band 0.4g magnitude). Unlike wing(disturbance UAVs, quadrotors INTRODUCTION Moreover, owing to the wind conditions at such altitudes, The study of fluid1.physics, flame and combustion char- of the g-quality obtained was very poor (disturbance band of 0.4g magnitude). Unlike fixed wing UAVs, quadrotors 0.4g magnitude). Unlike fixed wing UAVs, quadrotors The study of of fluid physics, physics, flame and combustionunder char- of can perform vertical maneouvres, thereby opening up the The study fluid flame and combustion charthe g-quality obtained was very poor (disturbance band acteristics, material science, and biotechnology of 0.4g magnitude). Unlike fixed wing UAVs, quadrotors can perform vertical maneouvres, thereby opening up the The study of fluid physics, flame and combustionunder char- can perform vertical maneouvres, thereby opening up the acteristics, material science, and biotechnology possibility of performing free-fall-like 1-D parabolic maacteristics, material science, and biotechnology under of 0.4g magnitude). Unlike fixed wing UAVs, quadrotors reduced-gravity (micro, Moon, andand Mars) conditions is a possibility can perform vertical maneouvres, thereby opening upmathe The study of fluid physics, flame combustion charof performing free-fall-like 1-D parabolic acteristics, material science, and biotechnology under possibility of performing free-fall-like 1-D parabolic mareduced-gravity (micro, Moon, and Mars) conditions is a toofcreate reduced gravitythereby environments (Afman reduced-gravity (micro, Moon,and and biotechnology Mars) conditions is a noeuvres can perform vertical maneouvres, upmathe major area of research (Ries-Kautt (1997); Alam (2016); possibility performing free-fall-like 1-Dopening parabolic acteristics, material science, under noeuvres to create reduced gravity environments (Afman reduced-gravity (micro, Moon, and Mars) conditions is a noeuvres create reduced gravity environments (Afman major area of research research (Ries-Kautt (1997); Alam Alam (2016); (2016)). Ato novel conceptual approach of using a variable major area of (Ries-Kautt (1997); (2016); possibility of performing free-fall-like 1-D parabolic maPletser (2012)). In order to simulate reduced gravity envinoeuvres to create reduced gravity environments (Afman reduced-gravity Moon, and reduced Mars) conditions is a (2016)). (2016)). A novel novelquadrotor conceptualtoapproach approach ofmicro-gravity using aa variable variable major area of research (Ries-Kautt (1997); Alam (2016); A conceptual using Pletser (2012)). In(micro, order to simulate gravity envipitch propeller achieve aof enPletser (2012)). In order to simulate reduced gravity envinoeuvres create reducedto gravity environments (Afman ronments onofEarth, researchers use drop towers, manned (2016)). Atonovel conceptual approach ofmicro-gravity using a variable major research (Ries-Kautt (1997); Alam (2016); pitch propeller quadrotor achieve a enPletserarea (2012)). In order to simulate reduced gravity envi- pitch propeller quadrotor to achieve a micro-gravity enronments on Earth, researchers use drop towers, manned vironment is proposed in Afman (2016). Their simulation ronments on Earth, researchers use drop towers, manned (2016)). A novel conceptual approach of using a variable fixed wing andtorockets (2016); Pletser pitch propeller quadrotor to achieve a micro-gravity enPletser (2012)). In order simulate reduced gravity envi- vironment vironment is proposed proposed in Afman Afman (2016). Their simulation ronments onaircraft, Earth, researchers use(Belser drop towers, manned is in (2016). Their simulation fixed wing aircraft, and rockets (Belser (2016); Pletser results show that it is possible to achieve a micro-gravity fixed wing aircraft, and (Belser (2016); Pletser results pitch propeller quadrotor to achieve a micro-gravity en(2008)). Despite theresearchers highrockets g-quality (lower the manned residual vironment is proposed in Afman (2016). Their simulation ronments on Earth, use drop towers, show that it is possible to achieve a micro-gravity fixed wing aircraft, and rockets (Belser (2016); Pletser show that it isApossible to achieve a micro-gravity (2008)). Despite the the highg-quality) g-qualityand (lower thecapability residual results for about 3is seconds. variable pitch propeller quadrotor (2008)). Despite the high g-quality (lower the residual vironment proposed in Afman (2016). Their simulation acceleration, higher their results show that it is possible to achieve a micro-gravity fixed wing aircraft, rockets (Belser (2016); Pletser for for aboutof 3 seconds. seconds. A variable variable pitch propeller propeller quadrotor (2008)). Despite the and highg-quality) g-quality (lower thecapability residual about 3 pitch quadrotor acceleration, higher the and their (instead conventional quadrotor) is required because acceleration, higher and their results show that it isA to achieve a micro-gravity to carry Despite heavy payloads, these methods that simulate for about 3 seconds. Apossible variable pitch propeller quadrotor (2008)). the the highg-quality) g-quality (lower thecapability residual (instead (instead of conventional quadrotor) is required because acceleration, higher the g-quality) and their capability of conventional quadrotor) is required because to carry heavy payloads, these methods that simulate both positive and negative thrust forces are necessary to carry heavy payloads, these methods that simulate for about 3 seconds. A variable pitch propeller quadrotor reduced-gravity environment aremethods costly, and proce- both (instead of conventional quadrotor) is required because acceleration, higher the g-quality) and their capability positive and negative negative thrust forces are necessary necessary to carry heavy payloads, these thatthe simulate both positive and thrust forces are reduced-gravity environment are costly, and the proceto maintain micro-gravity as well as to maintain attireduced-gravity environment costly, and the proce(instead of conventional quadrotor) is to required because dures areheavy involved limitingthese theare frequency at which they to both positive and negative thrust forces are necessary to carry payloads, methods that simulate maintain micro-gravity as well as maintain attireduced-gravity environment are costly, and the proceto maintain micro-gravity as well as to maintain attidures are involved limiting the frequency at which they tude control. This is, however, not possible to achieve dures are involved limiting theare frequency at which they tude both positivemicro-gravity and negative forces are tonecessary can conducted (Afman (2016)). to maintain asthrust well as to maintain attireduced-gravity environment costly, and the procecontrol. This is, however, not possible achieve dures are involved limiting the frequency at which they tude control. This is, however, not possible to achieve can conducted conducted (Afman (Afman (2016)). (2016)). by a conventional quadrotor. Awell conventional quadrotor can to maintain micro-gravity as as to maintain attitude control. This is, however, not possible to achieve dures are involved limiting the frequency at which they by conventional quadrotor. A conventional conventional quadrotor can conducted (2016)). aa conventional quadrotor. A quadrotor Meanwhile, the(Afman advances in Unmanned Aerial Vehicle by requires to switch of the propeller rotation to achieve tude control. This is, however, not possible to achieve by a conventional quadrotor. A conventional quadrotor Meanwhile, the(Afman advances inresearchers Unmanned to Aerial Vehicle can conducted (2016)). requires to switch of the the propeller rotation to achieve Meanwhile, the advances in Unmanned Aerial Vehicle to switch of rotation to achieve (UAV) technology, enabled conduct re- requires free thus leaving the propeller quadrotor uncontrollable to by a fall, conventional quadrotor. A conventional quadrotor Meanwhile, the advances inresearchers Unmanned to Aerial Vehicle requires to switch of the propeller rotation to achieve (UAV) technology, enabled conduct refree fall, thus leaving the quadrotor uncontrollable to (UAV) technology, enabled researchers to conduct refree fall, thus leaving the quadrotor uncontrollable to duced gravity experiments at low cost and more fredisturbances. Therefore, a conventional quadrotor canMeanwhile, the advances in Unmanned Aerial Vehicle requires switch of the propeller to achieve (UAV) gravity technology, enabled at researchers re- disturbances. free fall, to thus leaving the quadrotorrotation uncontrollable to duced experiments low cost cost to andconduct more payfreTherefore, a conventional quadrotor canduced gravity experiments at low and more fredisturbances. Therefore, a conventional quadrotor canquently in spite of the reduced g-quality and lower be used for simulating micro-gravity environments. (UAV) technology, enabled researchers re- not free fall, thus leaving the quadrotor uncontrollable to duced gravity experiments at low cost to andconduct more payfredisturbances. Therefore, a conventional quadrotor canquently in spite of the reduced g-quality and lower not be used for simulating micro-gravity environments. quently in spite of the reduced g-quality and lower paynot be used for simulating micro-gravity environments. load capacity. Inexperiments Higashino (2010), the authors proposed However, it is possible to produce a reduced-non-zeroduced gravity at low cost and more fredisturbances. Therefore, a conventional quadrotor canquently in spite of the reduced g-quality and lower paynot be used forpossible simulating micro-gravity environments. load capacity. In Higashino Higashino (2010), the authors authors proposed However, it is is to produce produce reduced-non-zeroload capacity. In (2010), the proposed it to aa reduced-non-zerothe use of spite a fixed wing UAV to create reduced gravgravity environments with amicro-gravity conventional quadrotor. quently in the reduced g-quality and lower pay- However, not be used forpossible simulating environments. load capacity. In of Higashino (2010), the authors proposed However, it is possible to produce a reduced-non-zerothe use of a fixed wing UAV to create reduced gravgravity environments with a conventional quadrotor. the use of a fixed wing UAV to create reduced gravenvironments with a conventional quadrotor. ity The simulation showproposed that an gravity load capacity. In Higashino (2010), the authors However, it is possible to produce a reduced-non-zerothe environment. use of a fixed wing UAV to results create reduced gravgravity environments with a conventional quadrotor. ity environment. The simulation results show that an major focus of this work is to design an autonomous ity simulation show for that environment than 0.15g be achieved 2 an to The the environment. use of a less fixedThe wing UAVcan to results create reduced gravThe major focus of this thiswith work is to to design design an an autonomous gravity environments a conventional quadrotor. ity environment. The simulation results show for that major focus of work is autonomous environment less than 0.15g can be achieved achieved 2 an to The conventional quadrotor towards gravity environment less than 0.15g can be for 2 to 3 seconds. Although the experimental results presented The major focus of this work is tosimulating design an Mars autonomous ity environment. The simulation results show that an conventional quadrotor towards simulating Mars gravity environment less than 0.15g can be achieved for 2 to conventional quadrotor towards simulating Mars gravity 3 seconds. Although the experimental results presented environment on Earth. A 1-D parabolic (in time) manoeu3 seconds. Although the experimental results presented The major focus of thisAwork is tosimulating design an autonomous were not as satisfactory as the simulation results, the conventional quadrotor towards Mars gravity environment less than 0.15g can be achieved for 2 to environment on Earth. 1-D parabolic (in time) manoeu3 seconds. Although the experimental results presented environment on Earth. A 1-D parabolic (in time) manoeuwere not as as satisfactory astothe the simulation results, the vre is proposed to achieve the required gravity levels. were not satisfactory as simulation results, the conventional quadrotor towards simulating Mars gravity feasibility of using UAVs produce reduced gravity environment on Earth. A 1-D parabolic (in time) manoeu3 seconds. Although the experimental results presented vre is proposed to achieve the required gravity levels. were not as satisfactory the simulation results, the vre is proposed tothis achieve the required gravity levels. feasibility of In using UAVsasto to(2011), produce reduced gravity The kinematics of flight trajectory are derived with feasibility of using UAVs produce reduced gravity environment on Earth. A 1-D parabolic (in time) manoeuwas proved. Hofmeister authors experimenvre is proposedoftothis achieve the required gravity levels. were not as satisfactory as the simulation results, the The kinematics flight trajectory are derived with feasibility of using UAVs to produce reduced gravity The kinematics ofto flight trajectory aregravity derived with was proved. In Hofmeister using (2011), authors experimenthe consideration ofthis flying constraints (maximum height, was proved. In Hofmeister (2011), authors experimenvre is proposed achieve the required levels. tally created micro-gravity a fixed wing UAV for The kinematics of this flight trajectory are derived with feasibility of using UAVs to produce reduced gravity the consideration of flying constraints (maximum height, was proved. In Hofmeister (2011), authors experimenthe consideration ofthis flying constraints (maximum height, tally created micro-gravity using fixed wing UAV for for reduced gravityofduration andtrajectory payload). The kinematic tally created micro-gravity using aa fixed wing UAV The kinematics flight are derived with about 10 seconds. However, to achieve such duration of the consideration of flying constraints (maximum height, was proved. In Hofmeister (2011), authors experimenreduced gravity duration and payload). The kinematic tally created micro-gravity using a fixed wing UAV for gravity duration and The kinematic about 10 seconds. seconds. However, tolength achieve such duration of reduced analysis also provides a base forpayload). the choice of the motorabout 10 However, to achieve such duration of the consideration of flying constraints (maximum height, micro-gravity, the flight path and the change in reduced gravity duration and payload). The kinematic tally created micro-gravity using a fixed wing UAV for analysis also provides a base for the choice of the motorabout 10 seconds. However, tolength achieveand such duration of analysis provides afor base forpayload). the choice ofautomation the motormicro-gravity, thewas flight path the change in in propeller combination this An micro-gravity, the flight path and the change reduced also gravity duration and The kinematic altitude 1 km and 400 meters respectively. analysis also provides afor base formission. the choice ofautomation the motorabout 10required seconds. However, tolength achieve such duration of propeller propeller combination this mission. An micro-gravity, thewas flight path length and the change in combination for this mission. An automation altitude required 1 km and 400 meters respectively. altitude required was 1 km and 400 meters respectively. analysis also provides a base for the choice of the motorpropeller combination for this mission. An automation micro-gravity, thewas flight path length and the change in altitude required 1 km and 400 meters respectively. propeller combination for this mission. An automation altitude required was 1 km and 400 meters respectively. Copyright © 2018 IFAC 176 ∗ K. ∗∗ K. ∗ K. K. ∗ K.
2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 176 Copyright ©under 2018 responsibility IFAC 176Control. Peer review© of International Federation of Automatic Copyright © 2018 IFAC 176 10.1016/j.ifacol.2018.05.027 Copyright © 2018 IFAC 176
5th International Conference on Advances in Control and Optimization of Dynamical Systems Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 February 18-22, 2018. Hyderabad, India
control strategy is proposed to enable the quadrotor to execute the desired trajectory autonomously. As the quadrotor has to perform ascent and descent flights to complete the proposed 1-D parabolic manoeuvre, a blade element theory based axial flight model is derived to calculate the thrust and counter torque of a propeller in axial flights. To validate the derived model in axial flight conditions and to estimate model parameters, wind-tunnel experiments are conducted. Finally, simulation and preliminary flight test results are presented to demonstrate the capability of conventional quadrotors to simulate reduced gravity environments.
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example, as per the Indian government regulations, the maximum altitude at which a micro UAV (weight less than 2 kg) can fly in an educational/ research institution without local authority permission is 200 ft (60 m approximately) (Air Transport Circular (2016)). • Reduced gravity time (trg ): It is the amount of time the quadrotor spends in a reduced gravity state. • Payload mass (mp ): It is the mass of the on-board equipment required to conduct experiments while in reduced gravity state. 2.3 Kinematics
2. TRAJECTORY DESIGN One way to create reduced gravity using a fixed wing aircraft is to perform a 2-D parabolic manoeuvre. Fixed wing aircraft require forward speed to be airborne, and therefore, to simulate a reduced gravity environment, they need to have a flight trajectory that spreads along both forward and vertical translational axes. However, owing to its capability to translate only along the vertical axis, a quadrotor can perform a 1-D parabolic (in time) manoeuvre to achieve reduced gravity. 2.1 The Trajectory The proposed trajectory a quadrotor need to follow to create Mars-gravity is shown in Fig. 1, and can be divided into five phases. In the first (take-off) phase, quadrotor takes off from ground and hovers at a certain altitude. In the second (climb) phase, it accelerates to reach a certain desired velocity. Once this velocity is achieved, the rotor angular speeds are varied such that the desired constant acceleration is maintained (third phase - reduced gravity phase). After completing the parabolic path, the quadrotor accelerates to break the fall in phase four (decent phase), and performs a controlled landing in phase five (landing phase). These five phases are indicated in Fig. 1. The circled number in the figure marks the beginning of the corresponding phase.
A detailed kinematic analysis is performed to compute the acceleration schedule required for a quadrotor to follow the proposed 1-D trajectory. First, the kinematic equations for phase-3 – the reduced gravity phase – is derived, as the required accelerations in the other phases, especially the desired acceleration in climb phase, will depend upon this. Phase-3 board is
The acceleration experienced by a payload on-
aexp = aE − ab (1) where, aE is gravitational acceleration of Earth, and ab is the acceleration of the body (quadrotor). We have ab = af + aE (2) where, af is the body acceleration due to thrust and drag forces acting on it. For the purpose of this paper, the required experienced acceleration aM – the acceleration due to gravity on Mars. From the equations (1) and (2), the desired acceleration due to thrust of the rotors (we neglect drag in this analysis) is af = −aM (3)
From the kinematic equations for one dimensional motion, we obtain arg trg 1 2 (4) hpeak = h3 − arg trg , v3 = −v5 = − 8 2 where, hi and vi are respectively the altitude and velocity at the beginning of phase-i, trg is the duration of phase-3, and arg = af + aE = −aM + aE is the body acceleration in phase-3. Phase-2
The kinematic equations for phase-2 gives us 2 a2rg trg arg trg (5) , t23 = − h3 = h2 + 8a23 2a23 where, t23 is the duration of phase-2, and a23 is body acceleration in phase-2. Note that in deriving the above relations, we used v3 from equation (4).
Fig. 1. Phases of 1-D parabolic manoeuvre 2.2 Flying Constraints There are three constraints that has to be considered while designing a quadrotor to conduct reduced gravity experiments. • Peak altitude (hpeak ): It is the maximum altitude a quadrotor achieves while executing the proposed trajectory. The peak altitude contraint usually arise from governmental regulations on UAV flying. For 177
From equations (4) and (5), we can obtain the acceleration requirement and its duration in phase 2, given the constraints on hpeak and trg as a23 =
2 a2rg trg 2 8(hpeak − h2 ) + arg trg
,
t23 = −
arg trg 2a23
(6)
Assuming a hover altitude of 5 m (h2 ) and using the above derived relations, we have computed the peak altitude
5th International Conference on Advances in Control and 162 Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
and the time duration of phase 2 (t23 ) as a function of acceleration requirement in phase 2 (a23 ) for various values of reduced gravity duration. These are plotted in Fig. 2. These charts will help a designer to estimate the maximum acceleration requirement, given the hpeak and trg constraints, to make an appropriate motor-propeller selection as described in the next section.
4. CONTROL AND AUTOMATION STRATEGY From the kinematic analysis presented in Section 2, the acceleration schedule – acceleration value and duration for each phase – is known, which when followed will result in the on-board payload experiencing reduced gravity (Mars-gravity in our case) for the required duration. To implement this acceleration schedule on a quadrotor with the motor and propeller appropriately chosen as in the previous section, we propose the following control and automation strategy. We propose to use a low-level attitude controller and higher level automation (altitudeacceleration controller for various flight phases) as described below. 4.1 Attitude Control
(a) hpeak vs a23
Fig. 3. Attitude controller structure A 2-loop feedback control structure is chosen for attitude controller as shown in Fig. 3. An outer loop with attitude feedback, and an inner loop with body rate feedback makes this control structure robust to disturbances such as model uncertainties, measurement noise, and environmental disturbances.
(b) t23 vs a23
Fig. 2. Design charts obtained from kinematic analysis 3. MOTOR-PROPELLER SELECTION Unlike the general quadrotor design for which the major design criteria are agility, range and endurance, the design criterion for a quadrotor to create reduced gravity is the ability to produce required acceleration in the ascent and descent phases. Thus the quadrotor should be designed such that maximum thrust produced by the motor-propeller combination meets this requirement. Fig. 2a provides the acceleration requirement (a23 ) in ascent and descent phases given the hpeak and trg constraints. Given this, the acceleration that the motorpropeller combination should produce is obtained from equation (2) as af = a23 − aE (7) Thus, the minimum thrust constraint can be written as T ≥ ks mq af (8) where, ks is the safety factor taken as 1.5 in practice. Here, the quadrotor mass mq is mq = 4ma + mk + mp
(9) where, mp is the payload mass, mk is the known mass, that is, the mass of the battery, frame, electronics, etc., and ma is the mass of motor-propeller module. Thus, any motor-propeller combination that satisfies the following constraint can be chosen of this mission T ≥ ks (4ma + mk + mp )af (10) Note that this is a ‘design relation’ wherein the quantities on both the right hand side (T ) and the left hand side (ma ) depends on the choice of motor-propeller combination. 178
4.2 Automation
Fig. 4. Automation strategy To enable a quadrotor to autonomously fly the five phases of the proposed reduced gravity manoeuvre, we suggest a state machine like automation as shown in Fig. 4. Depending upon the current phase, the desired acceleration is provided to a PI controller that uses an acceleration feedback to generate appropriate command to be given to the motor-propeller module. For take-off and landing phases (1 and 5), the desired acceleration is generated by an outer PID loop executing an altitude hold (see Fig. 4). In the climb, reduced gravity and decent phases (2, 3, and 4) required acceleration obtained from the kinematic analysis is used as desired acceleration input to the PI controller. In the case of communication loss or loss of stability, the phase state will be set as ‘fault’ (Phase f), and the desired acceleration is given as the acceleration due to gravity of Earth. This results in shutting down of all the motors and a resultant free-fall.
5th International Conference on Advances in Control and Optimization of Dynamical Systems Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 February 18-22, 2018. Hyderabad, India
5. SYSTEM MODELLING The quadrotor governing equations of motion, and the propeller thrust and counter torque model used in simulations to test the proposed automation strategy are described in this section. 5.1 Governing Equations We use the standard rigid body equations, specialized for quadrotors, given as d(mV b ) + Ω b × (mV b ) = F p + F g dt (11) d(IΩ b ) + Ω b × (IΩ b ) = τcg + τp + τgp dt where, V b and Ω b are the translation and angular velocities in directions of body reference frame axes, m is the mass of quadrotor, and I is the rotational inertia matrix of the quadrotor. The description of the forces and torques on the right hand side of equation (11) are as follows: F b - force due to propeller, F b - force due to gravity, τcg - torque due to center of gravity shift from geometric center, τp - torque due to propellers, and τgp - torque due to propeller gyroscopic effects. The widely used and popular thrust model for quadrotor assumes that the thrust produced is directly proportional to the square of RPM (Bouabdallah (2004)). This is true in hover and when the quadrotor is translating steady at very low speeds which usually is the case. However, our proposed trajectory involves varying accelerations and speeds as seen in Section 2, such a thrust model will not suffice. Therefore, we propose a new thrust model, that is more appropriate when axial speeds are non-zero, using blade element theory.
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Substituting in equation (12), we get 2 2 1 i 2 i dCT = clα σi r r θi r − r(λi + λa ) dr 2 i=0
(14)
i=0
By integrating along the blade length, the total thrust coefficient can be obtained as 1 (15) CT = clα k1 + k2 λi + λa 2 where, the constants k1 and k2 are functions of solidity ratio and pitch function coefficients σi , θi , i = 0, 1, 2. Thus, the total thrust produced by a propeller of radius R can be obtained by using the relation between thrust and non-dimensional thrust coefficient T = CT ρπR2 (ωR)2 as 1 k2 vi 1 4 πR ω 2 + clα k2 πR3 ωVa (16) T = clα k1 + 2 ωR 2 The relation between the inflow velocity and angular speed of the propeller is linear (vi = kv ω). This can be verified experimentally as shown in Fig. 6(b). Therefore, the thrust produced by the propeller in axial flight can be written as (17) T = bω 2 + ba ωVa k2 kv 1 1 4 3 where, b = 2 clα k1 + R πR and ba = 2 clα k2 πR are hover thrust and axial thrust coefficients respectively.
(a) Wind tunnel
5.2 Thrust model To develop a model that predicts the thrust and counter torque produced by a propeller in axial flights, we use the theory of blade element analysis (Leishman (2006)). Consider a propeller with solidity ratio σ, effective lift coefficient slope clα , blade pitch θ, non-dimensional inflow velocity λi , and non-dimensional axial velocity λa due to ascent or decent. Then from the blade element analysis, the elemental thrust coefficient can be written as (Leishman (2006)) 1 λ + λa (12) dCT = clα σr 2 θ − i 2 r where r is the non-dimensional radius of the propeller (r ∈ [0, 1]). The solidity ratio (σ) is constant for rectangular blades. However, for usual propellers σ varies along the length of the blade as the chord length is not constant. Similarly, the twist of the blade also varies along the blade length. For analysis, we assume second order polynomials to model the variation of σ and θ along the blade as σ=
2 i=0
σi r i ;
θ=
2
θi r i
(13)
i=0
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(b) Motor-propeller setup
Fig. 5. Wind-tunnel experimental setup In order to experimentally validate the developed theoretical expressions, and to find the thrust coefficients, a sub-sonic wind tunnel is used to simulate the descent flight conditions for a propeller as shown in Fig. 5(a). The wind tunnel is capable of producing inflow velocity upto 12 m/s inside the chamber. The set-up inside the wind tunnel chamber as shown in Fig. 5(b) consists of motorpropeller combination, a six axis load cell to measure the thrust and counter torque, and an encoder to measure the angular speed of the propeller. In order to examine the wall-effects on motor-propeller inside the wind tunnel, the values of T , ω and vi are measured without any opposing inflow from wind tunnel (hover condition). The variation of these readings from the measurements obtained outside the wind tunnel are negligible. Thereby, in further wind-tunnel experiments, the wall-effect on motor-propeller module is considered to be negligible. The steady state measurements of T vs ω, and vi vs ω under hover condition are provided in
5th International Conference on Advances in Control and 164 Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
steady state (triggering criteria for phase 2), desired body acceleration of a23 (3.01 m/s2 ) is provided as command input. After achieving desired climb velocity (v3 (12.1 m/s), the flight transitions to reduced gravity phase. In the reduced gravity phase the commanded body ac(a) T vs ω at hover
(b) vi vs ω at hover
Fig. 6. Steady state thrust and inflow response of propeller at hover Fig. 6(a) and 6(b) respectively. From the Fig. 6(b), we can observe that the relation between inflow velocity and motor angular speed is linear (vi = kv ω).
(a) T vs ω at va = −6 m/s
(b) T vs ω at va = −7.3 m/s
Fig. 8. Simulation plots
(c) T vs ω at va = −8.5 m/s
(d) T vs ω at va = −9.8 m/s
Fig. 7. Steady state thrust measurements under descent condition To obtain thrust model coefficients (b and ba ), the measurements of T vs ω for various opposing airflow velocities are recorded as shown in Fig. 7. The combination of vehicle governing equations, motor dynamics, and thrust and counter torque steady state models completes the mathematical modelling of quadrotor in axial flights. This complex non-linear model is used for simulation purposes in the sections that follow. 6. RESULTS To demonstrate the efficacy of our proposed approach to generate reduced gravity using quadrotors, we perform a 6DoF simulation implementation of the suggested strategy. To validate the simulation results, preliminary (open-loop) experimental flight test results are also presented. 6.1 Simulation results The altitude, axial velocity and acceleration, and propeller angular speed simulation results for autonomous reduced gravity manoeuvre are shown in Fig. 8. In these plots, the transition in flight phase is indicated using a vertical dash line. Operation In the take-off phase, the quadrotor is commanded to hover at a height of 5 m. After achieving 180
celeration is arg (−6.099 m/s2 ), and the experienced acceleration on the quadrotor in this phase will mimic the Mars gravity. After trg (4 seconds) time period of reduced gravity phase (triggering criteria for phase 4), the descent phase is commenced by providing body acceleration command of a23 to break the fall. The descent phase is switched to landing phase once the velocity approaches near to zero. Analysis In order to maintain a constant acceleration in phases 2,3 and 4, the angular speed of the rotor is varied such that a constant thrust is achieved. In climb phase (refer rotor angular speed plot in Fig. 8), as the axial airflow increases, the angular speed of the propeller is increased such that a constant acceleration is produced. This is because in ascent the thrust produced by the propeller decreases when compared to the thrust produced in hover condition at same angular speeds. Similarly, during reduced gravity phase, as the quadrotor velocity decreases, the angular speed of the rotor also decreases to maintain constant thrust. Thus, the proposed autonomous flight strategy helps to produce a constant Mars gravity environment. 6.2 Flight Test Results Implementation In flight tests, a 3 phase open loop acceleration control is used to conduct preliminary reduced gravity experiments. The ascent, reduced gravity and descent are the three phases in the flight envelope. Based on the desired acceleration in each phase, BLDC motors are commanded appropriately (to produce desired thrust under hover conditions) for the calculated time interval using kinematics in Section 2. In the ascent and reduced gravity phases, the thrust command is autonomous, that is, for given time interval of each phase, desired actuation signals are provided to BLDC motors by the au-
5th International Conference on Advances in Control and Optimization of Dynamical Systems Kedarisetty Siddhardha / IFAC PapersOnLine 51-1 (2018) 160–165 February 18-22, 2018. Hyderabad, India
topilot code. Whereas, at the end of the reduced gravity phase, the quadrotor thrust control is switched to remote control. An experienced human pilot ensures the safe landing of the quadrotor by providing appropriate thrust commands.
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acceleration towards the phase end, which is due to the thrust increase of the propeller caused by the downward axial velocity. These deviations in the acceleration can be easily controlled by implementing the developed autonomous strategy. Therefore, the preliminary flight test results indicate that the conventional quadrotors are feasible and capable to replicate reduced gravity environments with the exception of micro-gravity. 7. CONCLUSIONS In this paper, the design, modelling and automation of conventional quadrotor capable of replicating Mars gravity environment is proposed. An approximate thrust model is derived using blade element theory and validated through series of wind tunnel experiments. The simulations confirmed the feasibility of using the conventional quadrotor and the developed automation strategy to create Mars gravity for duration of 4 to 5 seconds. The practicality of this concept is also verified through preliminary flight test results.
Fig. 9. Flight test
REFERENCES
Fig. 10. Altitude, acceleration, and attitude plots Results and analysis The flight test snapshot collage of the quadrotor performing a 1-D parabolic like manoeuvre is shown in Fig. 9. The corresponding altitude, acceleration, and attitude plots are shown in Fig. 10. We can observe from Fig. 10(b) that a reduced gravity phase replicating the Mars gravity lasted approximately 4 seconds, and the peak altitude is within the prescribed limit. The designed attitude controller maintained the attitude of the vehicle within the range of ±3 degrees with an exception of a small disturbance oscillation due to sudden increase in the vehicles thrust. From the altitude and acceleration plots we can observe that the altitude does not decrease instantaneously with decrease in acceleration during the transition from acceleration to reduced gravity phase. This is due to the existing momentum of the body at the end of the acceleration phase, which enables the quadrotor to complete a 1-D parabolic like path. Since an open loop acceleration control strategy is chosen instead of closed loop control, the desired acceleration could not be maintained. From the analysis plot Fig. 12(b), in the ascent phase, the acceleration of the body decreases as the thrust produced by the propellers are decreased due to the upward axial flow. Similarly, in reduced gravity phase, we can observe increase in the 181
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