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Maximum endurance for battery-powered rotary-wing aircraft
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Maximum endurance for battery-powered rotary-wing aircraft
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Mauro Gatti 1 , Fabrizio Giulietti ∗,2 , Matteo Turci 3
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Università di Bologna, Forlì, 47121, Italy
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Article history: Received 2 August 2014 Received in revised form 13 May 2015 Accepted 18 May 2015 Available online xxxx Keywords: Battery-powered aircraft Multi-rotor Endurance Preliminary design
The paper presents an analytical framework for addressing the hovering time prediction of rotary-wing aircraft, with a particular focus on multi-rotor platforms. By imposing the balance between required and available power, the endurance expression is derived as a function of airframe features, rotor parameters, and battery capacity. The best endurance condition is also obtained in terms of optimum capacity and hovering time, by means of two approximate closed-form solutions. The proposed methodology was validated by means of numerical simulation and flight testing. Results show the effectiveness of the proposed approach. © 2015 Published by Elsevier Masson SAS.
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1. Introduction
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The interest in electrically powered Remotely Piloted Aerial Systems (RPAS) among industry and academia has shown a steep increase along the last decade. This is mostly due to the potentiality of such platforms to play a leading role within a wide range of applications in the field of aerial photography and 3D reconstruction [1], search and rescue [2] and risk management, natural landscape monitoring, and disaster prevention [3]. In particular, among the available fixed and rotary-wing configurations, multi-rotor platforms gained a particular relevance, thanks to the simple configuration, simplicity of use (also in confined spaces), and hovering and vertical take-off and landing capability, which all make them an interesting alternative to fixed-wing aircraft in many practical applicative scenarios. Also, with respect to a conventional helicopter, a multi-rotor shows an increased manoeuvrability and a faster response to external disturbances together with a more compact size. This is achieved by spreading the total disc area into multiple rotor units, [4,5]. Thus, to provide the required thrust, smaller propellers rotating at higher speed are employed, at the cost of a loss in efficiency with respect to the conventional, single rotor configuration. This, in addition to the limited duration/weight ratio typical of electrically driven systems,
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*
Corresponding author. E-mail addresses: [email protected] (M. Gatti), [email protected] (F. Giulietti), [email protected] (M. Turci). 1 Ph.D. Student, Interdepartmental Center for Industrial Research (CIRI Aeronautica). 2 Associate Professor, Department of Industrial Engineering (DIN). 3 Research Fellow, Interdepartmental Center for Industrial Research (CIRI Aeronautica). http://dx.doi.org/10.1016/j.ast.2015.05.009 1270-9638/© 2015 Published by Elsevier Masson SAS.
makes the endurance in the hovering condition a critical, but challenging, issue in the framework of the multi-rotor platform design process. Studies addressing electric aircraft performance became available only in the last few years. Recent works present results related to fixed-wing aircraft, where the effects of Peukert’s law on the available battery capacity are taken into account. In Ref. [6] this approach was first applied to flight vehicles, later on in Ref. [7] the best range condition is also derived, whereas in Ref. [8] endurance estimates are validated by means of an experimental investigation. With respect to multi-rotor platforms, Ref. [9] presents a novel configuration where endurance is increased by using higher diameter propellers for lift generation and smaller propellers for attitude control, whereas in Ref. [10] the authors present an application of a statistically-based sizing methodology to multi-rotor RPAS, allowing for the estimation of the gross take-off weight as a function of hovering time and payload requirements, through the analysis of available data from platforms currently on the market. This paper presents the analytical study of battery-powered rotary-wing aircraft endurance in hovering flight condition, and its experimental validation carried out by using a multi-rotor platform. Starting from the balance equation of required and available power, and by taking into account airframe features, rotor figure of merit, and payload required power, two main results are obtained and discussed: 1) an analytical model for accurate estimate of hovering time as a function of on-board battery capacity, and 2) a set of approximate solutions for deriving the best endurance condition in terms of optimum capacity and maximum achievable hovering time for the cases when the figure of merit can be assumed to be constant, and the power required by the payload is small if compared to the hovering power. As a further contribu-
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tion, a simplified model not depending on rotor figure of merit is proposed, which can be used for estimating take-off weight during multi-rotor sizing at a preliminary/conceptual stage, [11]. The rest of the paper is structured as follows. In the next section the problem of hovering flight time is addressed and approximate analytical models describing the best endurance condition are derived. In the Results section numerical simulations and flight testing campaign data, validating the effectiveness of the proposed approach, are presented. A section of concluding remarks ends the paper.
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2. Problem statement and solution
For constant power applications, Li-Po cells show a linearly decreasing voltage as a function of residual capacity (with a negligible dependency on current draw) when discharged from the fully-charged voltage, V f , to the standard voltage, V0 , [14]. Let η express the fraction of the nominal capacity where the discharge process shows a linear behaviour. The available capacity becomes
C = ηC 0
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Let us consider a rotary-wing platform equipped with a payload for a specific mission. The total take-off weight can be expressed as
W to = W eo + W p + W b
where W eo is the empty-operative weight that includes a) the frame weight (structure and rigging), b) the driving system weight (motors, regulators, and propellers), and c) the avionics weight (autopilot and communication system), W p is the payload weight, and W b is the battery weight. For the purposes of the present analysis, the total power required for flight is
P r = P h + P ap
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3/2
Ph =
i=
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dt
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dC
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f
Pr
=
V
=
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V
3/2
W to
λf
+ P ap
(4)
(5)
3/2
W to + P ap λ f
C = C0
C0
k−1
C t= 0
it 0
Vλ f 3/2 W to
+ P ap λ f
dC
(7)
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t=
3/2 W to
Assumption 1. On the basis of validation data presented in Ref. [8], the battery voltage is supposed to be constant during the discharge.
C
+ P ap λ f
dC = 0
Ve λ f 3/2 W to
+ P ap λ f
ηC 0 η
C0 it 0
k−1
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(9)
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Assumption 2. Rotor figure of merit is supposed to be a slowlyvarying power function of the rotor thrust. The following model is proposed:
f = f0
W to
m
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(10)
T 0n
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where T 0 is the rotor thrust at a reference percentage of throttle position (i.e. at 60%), and n is the number of rotors. Model parameters, f 0 and m, are obtained by available data from the manufacturer and/or by means of an experimental characterisation.
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2.1. Maximum endurance condition
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Letting α indicate the battery weight/energy ratio, aircraft takeoff weight can be expressed as
(11)
where W 0 = W eo + W p is the zero-capacity weight, representing the weight of the aircraft without the battery system. By substituting Eqs. (4) and (10) into Eq. (7), and by taking into account Eq. (11), endurance in hovering flight is given by
t = t0
Ve η f 0 λ t 0 ( T 0 n)m
(C 0 )
(C 0 ) = C 0k ( W 0 + C 0 Ve α )m−3/2 +
(12)
f 0λ
( T 0n)m
P ap
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(13)
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( W 0 + C 0 Ve α )3/2 ( T 0n)m + P ap f 0 λ ( W 0 + C 0 Ve α )m C0
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−k
Ve α (2m − 3) ( W 0 + C 0 Ve α )1/2 ( T 0 n)m 2
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depends on the nominal battery capacity. Provided that all the parameters in the bracketed coefficient of Eq. (12) are constant, the best endurance condition is obtained by taking the derivative of with respect to C 0 and imposing d(C 0 )/dC 0 = 0. The nominal capacity allowing for the maximum endurance is thus obtained by solving the equation
+
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k
where the function
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W to = W 0 + W b = W 0 + α Ve C 0
(6)
where t 0 is the rated discharge time, and k is Peukert’s coefficient which can be estimated by using the method described in Ref. [6]. By considering the complete discharge of the available capacity, flight endurance is given by the following integral:
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For a nominal battery capacity C 0 , the actual available capacity C at the discharge rate i is provided by the Peukert equation, [13],
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(3)
2ρ A t
Vλ f
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√
where V is the battery voltage that, in general, is a function of both current draw and capacity. From the definition of the discharge ratio, dC /dt = i, it is straightforward to obtain the specific endurance
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W to
where f is the figure of merit of the rotor, √ ρ is air density, and A t is the total disc area. After naming λ = 2ρ A t , by imposing the balance between the required and available power from the battery, the current draw, i, for the hovering condition is given by
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(2)
where P h is the power required for the hovering condition and P ap is the power required for avionics and payload. For rotary-wing aircraft, the power required in hovering condition is given by [12]
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(1)
it 0
Ve λ f
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(8)
By imposing in Eq. (7) the discharge at the equivalent constant voltage, Ve = (V f + V0 )/2, the endurance turns to
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ηC 0
k−1
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=0
(14)
To the best of the authors’ knowledge, an analytical solution to the problem cannot be found when (C 0 ) assumes the form expressed in Eq. (13). Nonetheless approximate closed-form solutions can be obtained on the basis of two different simplifying assumptions.
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Case 1. The rotor figure of merit is assumed to be constant, f ≈ f 0 . This assumption holds when f shows negligible variations within
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Table 1 Relevant parameters of a selection of existing platforms.
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Platform
W to [N]
Wp [N]
W eo / W to
W b /W 0
P ap / P h
C0 [Ah]
t [min]
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DJI S1000 Altura ATX8 DJI S900 DJI S800 evo Altura ATX4 DJI F550 HighOne Quad tarot t810 E1100-V2 Aibot-X6
107.91 61.80 80.44 68.67 51.99 22.56 76.51 66.70 83.38 65.23
18.63 24.52 18.63 18.63 9.81 2.25 12.75 14.71 13.73 19.62
0.56 0.51 0.55 0.52 0.41 0.51 0.58 0.51 0.65 0.51
0.26 0.37 0.23 0.21 0.66 0.35 0.34 0.36 0.22 0.23
< 0.01 < 0.01 < 0.01 < 0.01 < 0.01 0.023 < 0.01 < 0.01 < 0.01 < 0.01
15 16.6 12 15 16.6 5 .4 16 11.6 12 10
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2
21
C0
−
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3Ve α ( W 0 + C 0 Ve α )1/2
( W 0 + C 0 Ve α )3/2 + P ap f 0 λ
1
(1 )
C 0 |me =
(15)
Ve α
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0 =
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(16)
2
2/3 2 f 02 λ2 P ap − W 03 + P ap f 0 λ
+ W0 2/3
2 f 02 λ2 P ap
−
W 03
(17)
+ P ap f 0 λ
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The maximum value for hovering time is obtained by substituting Eq. (16) into Eq. (13), providing the following expression:
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⎤k
⎡
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t |me = t 0 ⎣ (1 )
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f 0 ηλ (0 − W 0 )
3/2
t 0 α 0
+ P ap f 0 λ
⎦
(18)
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Results obtained under the hypothesis of constant figure of merit are effective in all applications where the aircraft take-off weight , because of mission and/or is limited to a maximum value, W to regulation constraints. In this case, for a given take-off weight, the platform figure of merit is fixed, making Eqs. (16) and (18) provide an accurate estimation of battery capacity and endurance. By taking into account Eq. (11), it is straightforward to obtain from Eq. (15) the following expression,
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3/2
C 0 | W to =
2 W to 3
+ P ap f 0 λ
α Ve W to 1/2
(19)
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which represents the nominal capacity for the maximum endurance in the presence of a constraint on the maximum take-off weight.
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2
+
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C0
Case 2. The power required for systems and payload is neglected, P r ≈ P h . This assumption is usually valid in the following three scenarios: 1) there is no power-consuming payload on board and aircraft systems only consist of autopilot and communication system whose power consumption usually has a negligible effect on endurance, 2) the payload uses a dedicated battery system and the weight of such battery is already included in W 0 , 3) the power required for avionics and payload is small if compared to the power
(20)
1
87
(21)
α Ve 1 − 2m
t |me = t 0
90
m−1/2
W0
2 f 0 ηλ
t 0 α (1 − 2m) ( T 0
n)m
2m − 3 2m − 1
m−3/2
k
91 92
(22)
C 0 |me = and
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1
α Ve
(2W 0 )
(23)
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2 f 0 ηλ
( 0)
t |me = t 0
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Optimal capacity and endurance estimation carried out with variable figure of merit represent a suitable solution for addressing the performance analysis with respect to battery capacity of already existing platforms. In particular, the endurance prediction shows an increased accuracy for a wide range of commercial multirotor platforms equipped with a stabilised photo/video payload for surveillance, aerial photography, and low-cost photogrammetry applications, where the ratio P ap / P h has the same order of magnitude of 10−2 . Table 1 shows P ap / P h and other design parameters values for a selection of platforms available on the market. (1) (2) Although C 0 |me and C 0 |me represent acceptable approximations for a class of problems involving hovering flight endurance, there are cases where a further simplified result would be useful. As an example, let consider the preliminary phase of the design process, where the selection and/or characterisation of the driving system is still an occurring problem. Thus, a simpler solution, describing the best endurance condition without the need of detailed information on rotor features, may represent a valuable starting guess for the aircraft take-off weight estimation. By imposing m = 0 into Eq. (21) one gets ( 0)
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By substituting this latter result into Eq. (13) one has (2 )
84 86
2W 0
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85
which solution provides
C 0 |me =
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Ve α (2m − 3) = 0, W 0 + C 0 Ve α
(2 )
(0 − W 0 )
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36
required for the hovering condition, P ap / P h 1. After imposing P ap = 0 into Eq. (14), the problem is thus reduced to
provided
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32
=0
By solving with respect C 0 , the nominal capacity for the maximum endurance becomes
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72
80
the operational range of motor throttle (i.e |m| 1), or when the aircraft is sized so that the driving system operates in the neighbourhood of an assigned throttle position. By imposing m = 0 in Eq. (14) the problem reduces to:
19
23
70
√
3 3αt 0
−1/2
k
W0
119
(24)
121
By recalling that W b = α Ve C 0 , from Eq. (21) the weight ratio for maximum endurance is
(0) =2 W 0 me Wb
(25)
Note that the last result does not depend on rotor figure of merit. Thus, it can be used for addressing preliminary sizing of the platform according to mission requirements without the need for any detail on the rotor system. To this aim, Eq. (1) is reshaped as
W to = keo W to + kb W to + W p
120
(26)
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Table 2 Multi-rotor platform parameters.
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Parameter
Symbol
Value
Airframe Number of rotors Airframe size Empty weight Payload weight Avionics and payload power
n dA W eo Wp P ap
6 750 19.62 2.32 18
KV de dp
700 31 10 4.5 0.3814 0.1617 0.4068
V rpm/min mm in in
16.85/14.82 10–40 1 0.0509 0.71 1.051
V Ah h N/Wh
Rotor Motor parameter Motor size Propeller diameter Propeller pitch Figure of merit constant parameter Figure of merit power parameter Mean value 65–100% Battery Voltage (fully charged/standard) Nominal capacity Rated discharge time Weight/energy ratio Discharge fraction Peukert coefficient
Units
70 71
mm N N W
72 73 74 75 76
β f0 m f 65
77 78 79 80 81 82 83
V f , V0 C0 t0
α η k
84 85
Fig. 1. Measured rotor thrust and power as a function of the throttle.
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where keo = W eo / W to and kb = W b / W to . Since payload weight, W p , is usually defined as a requirement, the initial guess value for take-off weight is then given by
Wp
( 0)
91 92 93
(27)
94
where keo is obtained from a statistical analysis of existing platforms, and kb is chosen to be 2/3 according to Eqs. (11) and (25).
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3. Results
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W to =
1 − keo − kb
95 97 98
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87 88
23 24
86
100
The numerical model for hovering time prediction and approximate solutions describing the best endurance condition are applied to a 750 mm size six-rotor platform equipped with a 2-axis stabilised camera. Aircraft data and parameters relevant for the proposed applications are reported in Table 2 whereas Fig. 5 shows the test platform. Figure of merit model described by Eq. (10) is first validated by means of a test rig for measuring the thrust delivered by the rotor and the power required for driving it. Fig. 1 shows the data acquired for 13 different throttle settings between 40% and 100%. Model parameters, f 0 and m, are then obtained by a least-square estimate, and Fig. 2 presents measured and approximated figure of merit as a function of the throttle. The proposed model provides an accurate fit up to 75% of the throttle setting, whereas an increasing percentage error is found at higher values (5 @ 80%, 8 @ 90%, 10 @ 100%). Nonetheless, for many multi-rotor platforms, the driving system is sized so that the power margin, intended as the ratio between the power available on-board and the power required at hovering, lies between 1.2 and 2.2 [9]. These values for the power margin allow for an increased manoeuvrability and, in case of a 6-rotor configuration, they allow for keeping the hovering condition with a residual manoeuvre capability even in case of engine failure. This makes the 50–70% range suitable for operating the hovering flight condition. On the basis of the model for figure of merit, multi-rotor endurance is computed as a function of battery ratio, W b / W 0 . Fig. 3 compares numerical prediction expressed by Eq. (12) and approximate solutions derived in Section 2.1 with endurance calculated by using the measured values of figure of merit. Results show how in the range where the figure of merit model fits the measured data, the hovering time prediction is accurate as well. At larger values of the throttle position, where the figure of merit model is
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Fig. 2. Rotor figure of merit as a function of the throttle.
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Fig. 3. Hovering time as a function of the battery ratio.
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Fig. 4. Endurance error of approximate models in the range of 50–70% of the motor throttle.
Fig. 6. Experimental validation.
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Table 3 Experimental results.
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Capacity [Ah]
Take-off weight [N]
Endurance [min]
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26
C0
W to
Eq. (12)
Experimental
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t
t¯
σt
93
15.12 22.90 27.42 29.90
15.57 22.86 27.28 29.24
0.2902 0.1630 0.056 0.1522
27 28
10 20 30 40
29 30 31
29.20 36.84 44.49 52.13
32 34 35 36 37 38 39 40 41
Fig. 5. Multi-rotor platform used for validation.
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affected by an increasing error, the estimation provided Eq. (12) looses accuracy. Nonetheless, since the measured figure of merit exhibits small variations beyond the 65% throttle position, the hovering time can be estimated according to Eq. (18) by using f = f 65 , that is the mean value of figure of merit in the range of 65 to 100%. This produces suitable estimates, especially for the best endurance battery ratio which is overestimated when Eq. (12) is used. Accuracy of approximate solutions in the range of 50–70% is highlighted in Fig. 4 in terms of the percentage error. The model derived under the hypothesis of small payload required power provides, as expected, an overestimation of the hovering time in the whole range of the throttle, with a maximum error of 6%. On the other hand, under the hypothesis of constant figure of merit, the corresponding model shows a larger error, as high as 8% far from the propeller regime where f = f 0 , but it provides the exact estimation when the actual figure of merit lies in the neighbourhood of f 0 . Also, it is worth noting that the simplest approximate model, given by m = 0 and P ap = 0, still provides an acceptable estimation of the hovering time, with a percentage error of about 15% in the absence of detailed information on rotor and payload characterisation.
To validate the proposed methodology, an experimental test campaign is performed. A total of twenty hovering flights are completed by using batteries with identical weight/energy ratio, α , along four different nominal capacity values. Fig. 6 presents the experimental hovering time plotted vs. the nominal capacity. Experimental data are compared to the analytical prediction of Eq. (12) and constant figure of merit approximation of Eq. (18). Statistics of the test campaign are provided in Table 3 in terms of mean value, t¯, and standard deviation, σt . Results show how the proposed approach provides a satisfactory estimation of the hovering time, making the model a suitable analytical tool for the preliminary design phase. In this respect, it is important to note that following an increase of battery capacity, the corresponding increase in hovering time shows a negative trend. By increasing the capacity from 10 to 20 Ah, one gains about 10 min of hovering time, for the considered platform, which largely increases the initial endurance. A further addition of 10 Ah increases endurance by less than 5 min. A further increase from 30 to 40 Ah provides only 2 min of additional hovering capability. These data represents a further proof for the importance of correctly addressing the issue of preliminary sizing the batterypowered aircraft, especially for multi-rotor platforms, on the basis of a physically sound methodology. As a matter of fact, trying to increase the endurance of an already existing platform by simply adding batteries may not be an effective solution, as propeller operates at higher motor regimes, where the decreasing slope of the endurance vs. capacity plot provides only a marginal improvement.
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4. Conclusions
128 129
A rigorous analytical framework has been derived for addressing the study of the hovering flight condition of electrically powered multi-rotor platforms equipped with a payload. Under the
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assumptions of constant voltage battery discharge and small variations of rotor figure of merit, a mathematical model providing an accurate estimation of the hovering time has been found. In addition, closed-form analytical solutions for the best endurance battery capacity have been found for two cases, when the figure of merit is constant, and when the power required by the payload is small if compared to the power required for the hovering condition. As a further result, a simpler approximate expression of the best endurance capacity not depending on rotor features and payload has been derived, showing that, along the preliminary sizing process, a first valuable guess for the battery weight is obtained by doubling the zero-capacity weight of the platform. The consistency of the proposed theoretical approach has been demonstrated by numerical simulations and an experimental flight test campaign by using a six-rotor platform.
16 17
Conflict of interest statement
18 19
None declared.
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References [1] S. Harwin, A. Lucieer, Assessing the accuracy of georeferenced point clouds produced via multi-view stereopsis from unmanned aerial vehicle (UAV) imagery, Int. J. Remote Sens. 4 (6) (2012). [2] P. Rudol, P. Doherty, Human body detection and geolocalization for UAV search and rescue mission using color and thermal imagery, in: 2008 IEEE Aerospace Conference, Big Sky, MT, 1–8 March 2008.
[3] D.W. Casber, R.W. Beard, T.W. McLain, Forest fire monitoring with multiple small UAVs, in: 2005 American Control Conference, June 8–10, 2005, Portland, OR, USA. [4] Markus Ryll, Heinrich H. Bülthoff, Paolo Robuffo Giordano, First flight tests for a quadrotor UAV with tilting propellers, in: IEEE International Conference on Robotics and Automation, ICRA 2012, IEEE, Piscataway, NJ, USA, 2012, pp. 4606–4613. [5] Fatih Senkul, Erdinç Altuˇg, Modeling and control of a novel tilt roll rotor quadrotor UAV, in: 2013 International Conference on Unmanned Aircraft Systems (ICUAS), May 28–31, 2013, Grand Hyatt Atlanta, Atlanta, GA. [6] L.W. Traub, Range and endurance estimates for battery-powered aircraft, J. Aircr. 48 (2) (March–April 2011). [7] G. Avanzini, F. Giulietti, Maximum range for battery-powered aircraft, J. Aircr. 50 (1) (March–April 2013). [8] L.W. Traub, Validation of endurance estimates for battery powered UAVs, Aeronaut. J. 117 (1197) (November 2013). [9] J. Verbeke, D. Hulens, H. Ramon, T. Goedemé, J. De Schutter, The design and construction of a high endurance hexacopter suited for narrow corridors, in: ICUAS, 27–30 May 2014, Orlando, FL, USA. [10] M. Gatti, F. Giulietti, Preliminary design analysis methodology for electric multirotor, in: RED-UAS 2013, 2nd IFAC Workshop on Research, Education and Development of Unmanned Aerial Systems, November 20–22, 2013, Compiegne, France. [11] D.P. Raymer, Aircraft Design: A Conceptual Approach, 4th ed., AIAA, Reston (VA), 2006, Chap. 17. [12] J. Gordon Leishman, Principles of Helicopter Aerodynamics, 2nd ed., Cambridge Aerospace Series, Cambridge University Press, Cambridge, New York, 2000. [13] D. Doerffel, S.A. Sharkh, A critical review of using Peukert equation for determining the remaining capacity of lead-acid and lithium-ion batteries, J. Power Sources 155 (2006) 395–400. [14] O. Tremblay, L.A. Dessaint, Experimental validation of a battery dynamic model for EV application, World Electr. Veh. J. (ISSN 2032-6653) 3 (2009).
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Maximum endurance for battery-powered rotary-wing aircraft
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Mauro Gatti 1 , Fabrizio Giulietti ∗,2 , Matteo Turci 3
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Università di Bologna, Forlì, 47121, Italy
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Article history: Received 2 August 2014 Received in revised form 13 May 2015 Accepted 18 May 2015 Available online xxxx Keywords: Battery-powered aircraft Multi-rotor Endurance Preliminary design
The paper presents an analytical framework for addressing the hovering time prediction of rotary-wing aircraft, with a particular focus on multi-rotor platforms. By imposing the balance between required and available power, the endurance expression is derived as a function of airframe features, rotor parameters, and battery capacity. The best endurance condition is also obtained in terms of optimum capacity and hovering time, by means of two approximate closed-form solutions. The proposed methodology was validated by means of numerical simulation and flight testing. Results show the effectiveness of the proposed approach. © 2015 Published by Elsevier Masson SAS.
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1. Introduction
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
The interest in electrically powered Remotely Piloted Aerial Systems (RPAS) among industry and academia has shown a steep increase along the last decade. This is mostly due to the potentiality of such platforms to play a leading role within a wide range of applications in the field of aerial photography and 3D reconstruction [1], search and rescue [2] and risk management, natural landscape monitoring, and disaster prevention [3]. In particular, among the available fixed and rotary-wing configurations, multi-rotor platforms gained a particular relevance, thanks to the simple configuration, simplicity of use (also in confined spaces), and hovering and vertical take-off and landing capability, which all make them an interesting alternative to fixed-wing aircraft in many practical applicative scenarios. Also, with respect to a conventional helicopter, a multi-rotor shows an increased manoeuvrability and a faster response to external disturbances together with a more compact size. This is achieved by spreading the total disc area into multiple rotor units, [4,5]. Thus, to provide the required thrust, smaller propellers rotating at higher speed are employed, at the cost of a loss in efficiency with respect to the conventional, single rotor configuration. This, in addition to the limited duration/weight ratio typical of electrically driven systems,
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*
Corresponding author. E-mail addresses: [email protected] (M. Gatti), [email protected] (F. Giulietti), [email protected] (M. Turci). 1 Ph.D. Student, Interdepartmental Center for Industrial Research (CIRI Aeronautica). 2 Associate Professor, Department of Industrial Engineering (DIN). 3 Research Fellow, Interdepartmental Center for Industrial Research (CIRI Aeronautica). http://dx.doi.org/10.1016/j.ast.2015.05.009 1270-9638/© 2015 Published by Elsevier Masson SAS.
makes the endurance in the hovering condition a critical, but challenging, issue in the framework of the multi-rotor platform design process. Studies addressing electric aircraft performance became available only in the last few years. Recent works present results related to fixed-wing aircraft, where the effects of Peukert’s law on the available battery capacity are taken into account. In Ref. [6] this approach was first applied to flight vehicles, later on in Ref. [7] the best range condition is also derived, whereas in Ref. [8] endurance estimates are validated by means of an experimental investigation. With respect to multi-rotor platforms, Ref. [9] presents a novel configuration where endurance is increased by using higher diameter propellers for lift generation and smaller propellers for attitude control, whereas in Ref. [10] the authors present an application of a statistically-based sizing methodology to multi-rotor RPAS, allowing for the estimation of the gross take-off weight as a function of hovering time and payload requirements, through the analysis of available data from platforms currently on the market. This paper presents the analytical study of battery-powered rotary-wing aircraft endurance in hovering flight condition, and its experimental validation carried out by using a multi-rotor platform. Starting from the balance equation of required and available power, and by taking into account airframe features, rotor figure of merit, and payload required power, two main results are obtained and discussed: 1) an analytical model for accurate estimate of hovering time as a function of on-board battery capacity, and 2) a set of approximate solutions for deriving the best endurance condition in terms of optimum capacity and maximum achievable hovering time for the cases when the figure of merit can be assumed to be constant, and the power required by the payload is small if compared to the hovering power. As a further contribu-
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tion, a simplified model not depending on rotor figure of merit is proposed, which can be used for estimating take-off weight during multi-rotor sizing at a preliminary/conceptual stage, [11]. The rest of the paper is structured as follows. In the next section the problem of hovering flight time is addressed and approximate analytical models describing the best endurance condition are derived. In the Results section numerical simulations and flight testing campaign data, validating the effectiveness of the proposed approach, are presented. A section of concluding remarks ends the paper.
11 12
2. Problem statement and solution
For constant power applications, Li-Po cells show a linearly decreasing voltage as a function of residual capacity (with a negligible dependency on current draw) when discharged from the fully-charged voltage, V f , to the standard voltage, V0 , [14]. Let η express the fraction of the nominal capacity where the discharge process shows a linear behaviour. The available capacity becomes
C = ηC 0
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Let us consider a rotary-wing platform equipped with a payload for a specific mission. The total take-off weight can be expressed as
W to = W eo + W p + W b
where W eo is the empty-operative weight that includes a) the frame weight (structure and rigging), b) the driving system weight (motors, regulators, and propellers), and c) the avionics weight (autopilot and communication system), W p is the payload weight, and W b is the battery weight. For the purposes of the present analysis, the total power required for flight is
P r = P h + P ap
33 34 35 36 37 38
3/2
Ph =
i=
41 42 43 44 45 46
dt
48
dC
50 51
f
Pr
=
V
=
56 57 58
61 62
V
3/2
W to
λf
+ P ap
(4)
(5)
3/2
W to + P ap λ f
C = C0
C0
k−1
C t= 0
it 0
Vλ f 3/2 W to
+ P ap λ f
dC
(7)
63 64 65 66
t=
3/2 W to
Assumption 1. On the basis of validation data presented in Ref. [8], the battery voltage is supposed to be constant during the discharge.
C
+ P ap λ f
dC = 0
Ve λ f 3/2 W to
+ P ap λ f
ηC 0 η
C0 it 0
k−1
69 70 71 72 74 75 76 77 78 79
(9)
80 81 82 83
Assumption 2. Rotor figure of merit is supposed to be a slowlyvarying power function of the rotor thrust. The following model is proposed:
f = f0
W to
m
84 85 86 87
(10)
T 0n
88 89
where T 0 is the rotor thrust at a reference percentage of throttle position (i.e. at 60%), and n is the number of rotors. Model parameters, f 0 and m, are obtained by available data from the manufacturer and/or by means of an experimental characterisation.
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2.1. Maximum endurance condition
96
Letting α indicate the battery weight/energy ratio, aircraft takeoff weight can be expressed as
(11)
where W 0 = W eo + W p is the zero-capacity weight, representing the weight of the aircraft without the battery system. By substituting Eqs. (4) and (10) into Eq. (7), and by taking into account Eq. (11), endurance in hovering flight is given by
t = t0
Ve η f 0 λ t 0 ( T 0 n)m
(C 0 )
(C 0 ) = C 0k ( W 0 + C 0 Ve α )m−3/2 +
(12)
f 0λ
( T 0n)m
P ap
102 103 104 106 107
110
(13)
111 112 113 114 115 116 117 118 119 120
( W 0 + C 0 Ve α )3/2 ( T 0n)m + P ap f 0 λ ( W 0 + C 0 Ve α )m C0
101
109
−k
Ve α (2m − 3) ( W 0 + C 0 Ve α )1/2 ( T 0 n)m 2
100
108
depends on the nominal battery capacity. Provided that all the parameters in the bracketed coefficient of Eq. (12) are constant, the best endurance condition is obtained by taking the derivative of with respect to C 0 and imposing d(C 0 )/dC 0 = 0. The nominal capacity allowing for the maximum endurance is thus obtained by solving the equation
+
98
105
k
where the function
97 99
W to = W 0 + W b = W 0 + α Ve C 0
(6)
where t 0 is the rated discharge time, and k is Peukert’s coefficient which can be estimated by using the method described in Ref. [6]. By considering the complete discharge of the available capacity, flight endurance is given by the following integral:
59 60
1
For a nominal battery capacity C 0 , the actual available capacity C at the discharge rate i is provided by the Peukert equation, [13],
54 55
(3)
2ρ A t
Vλ f
52 53
√
where V is the battery voltage that, in general, is a function of both current draw and capacity. From the definition of the discharge ratio, dC /dt = i, it is straightforward to obtain the specific endurance
47 49
W to
where f is the figure of merit of the rotor, √ ρ is air density, and A t is the total disc area. After naming λ = 2ρ A t , by imposing the balance between the required and available power from the battery, the current draw, i, for the hovering condition is given by
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(2)
where P h is the power required for the hovering condition and P ap is the power required for avionics and payload. For rotary-wing aircraft, the power required in hovering condition is given by [12]
31 32
(1)
it 0
Ve λ f
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(8)
By imposing in Eq. (7) the discharge at the equivalent constant voltage, Ve = (V f + V0 )/2, the endurance turns to
13 14
ηC 0
k−1
67
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=0
(14)
To the best of the authors’ knowledge, an analytical solution to the problem cannot be found when (C 0 ) assumes the form expressed in Eq. (13). Nonetheless approximate closed-form solutions can be obtained on the basis of two different simplifying assumptions.
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Case 1. The rotor figure of merit is assumed to be constant, f ≈ f 0 . This assumption holds when f shows negligible variations within
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Table 1 Relevant parameters of a selection of existing platforms.
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Platform
W to [N]
Wp [N]
W eo / W to
W b /W 0
P ap / P h
C0 [Ah]
t [min]
69
DJI S1000 Altura ATX8 DJI S900 DJI S800 evo Altura ATX4 DJI F550 HighOne Quad tarot t810 E1100-V2 Aibot-X6
107.91 61.80 80.44 68.67 51.99 22.56 76.51 66.70 83.38 65.23
18.63 24.52 18.63 18.63 9.81 2.25 12.75 14.71 13.73 19.62
0.56 0.51 0.55 0.52 0.41 0.51 0.58 0.51 0.65 0.51
0.26 0.37 0.23 0.21 0.66 0.35 0.34 0.36 0.22 0.23
< 0.01 < 0.01 < 0.01 < 0.01 < 0.01 0.023 < 0.01 < 0.01 < 0.01 < 0.01
15 16.6 12 15 16.6 5 .4 16 11.6 12 10
15 19 18 20 35 13 16 14 11 12
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4 5
3
14 15 16 17 18 20
2
21
C0
−
22 24 25 26
3Ve α ( W 0 + C 0 Ve α )1/2
( W 0 + C 0 Ve α )3/2 + P ap f 0 λ
1
(1 )
C 0 |me =
(15)
Ve α
30
31
0 =
34
(16)
2
2/3 2 f 02 λ2 P ap − W 03 + P ap f 0 λ
+ W0 2/3
2 f 02 λ2 P ap
−
W 03
(17)
+ P ap f 0 λ
35 37
The maximum value for hovering time is obtained by substituting Eq. (16) into Eq. (13), providing the following expression:
40
⎤k
⎡
38 39
t |me = t 0 ⎣ (1 )
41
f 0 ηλ (0 − W 0 )
3/2
t 0 α 0
+ P ap f 0 λ
⎦
(18)
42 43 44 45 46 47 48 49 50
Results obtained under the hypothesis of constant figure of merit are effective in all applications where the aircraft take-off weight , because of mission and/or is limited to a maximum value, W to regulation constraints. In this case, for a given take-off weight, the platform figure of merit is fixed, making Eqs. (16) and (18) provide an accurate estimation of battery capacity and endurance. By taking into account Eq. (11), it is straightforward to obtain from Eq. (15) the following expression,
51 52 53
3/2
C 0 | W to =
2 W to 3
+ P ap f 0 λ
α Ve W to 1/2
(19)
54 55 56 57
which represents the nominal capacity for the maximum endurance in the presence of a constraint on the maximum take-off weight.
58 59 60 61 62 63 64 65 66
2
+
73 74 75 76 77 78 79
C0
Case 2. The power required for systems and payload is neglected, P r ≈ P h . This assumption is usually valid in the following three scenarios: 1) there is no power-consuming payload on board and aircraft systems only consist of autopilot and communication system whose power consumption usually has a negligible effect on endurance, 2) the payload uses a dedicated battery system and the weight of such battery is already included in W 0 , 3) the power required for avionics and payload is small if compared to the power
(20)
1
87
(21)
α Ve 1 − 2m
t |me = t 0
90
m−1/2
W0
2 f 0 ηλ
t 0 α (1 − 2m) ( T 0
n)m
2m − 3 2m − 1
m−3/2
k
91 92
(22)
C 0 |me = and
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1
α Ve
(2W 0 )
(23)
116 117 118
2 f 0 ηλ
( 0)
t |me = t 0
93 94
Optimal capacity and endurance estimation carried out with variable figure of merit represent a suitable solution for addressing the performance analysis with respect to battery capacity of already existing platforms. In particular, the endurance prediction shows an increased accuracy for a wide range of commercial multirotor platforms equipped with a stabilised photo/video payload for surveillance, aerial photography, and low-cost photogrammetry applications, where the ratio P ap / P h has the same order of magnitude of 10−2 . Table 1 shows P ap / P h and other design parameters values for a selection of platforms available on the market. (1) (2) Although C 0 |me and C 0 |me represent acceptable approximations for a class of problems involving hovering flight endurance, there are cases where a further simplified result would be useful. As an example, let consider the preliminary phase of the design process, where the selection and/or characterisation of the driving system is still an occurring problem. Thus, a simpler solution, describing the best endurance condition without the need of detailed information on rotor features, may represent a valuable starting guess for the aircraft take-off weight estimation. By imposing m = 0 into Eq. (21) one gets ( 0)
88 89
By substituting this latter result into Eq. (13) one has (2 )
84 86
2W 0
82
85
which solution provides
C 0 |me =
81 83
Ve α (2m − 3) = 0, W 0 + C 0 Ve α
(2 )
(0 − W 0 )
33
36
required for the hovering condition, P ap / P h 1. After imposing P ap = 0 into Eq. (14), the problem is thus reduced to
provided
29
32
=0
By solving with respect C 0 , the nominal capacity for the maximum endurance becomes
27 28
72
80
the operational range of motor throttle (i.e |m| 1), or when the aircraft is sized so that the driving system operates in the neighbourhood of an assigned throttle position. By imposing m = 0 in Eq. (14) the problem reduces to:
19
23
70
√
3 3αt 0
−1/2
k
W0
119
(24)
121
By recalling that W b = α Ve C 0 , from Eq. (21) the weight ratio for maximum endurance is
(0) =2 W 0 me Wb
(25)
Note that the last result does not depend on rotor figure of merit. Thus, it can be used for addressing preliminary sizing of the platform according to mission requirements without the need for any detail on the rotor system. To this aim, Eq. (1) is reshaped as
W to = keo W to + kb W to + W p
120
(26)
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Table 2 Multi-rotor platform parameters.
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Parameter
Symbol
Value
Airframe Number of rotors Airframe size Empty weight Payload weight Avionics and payload power
n dA W eo Wp P ap
6 750 19.62 2.32 18
KV de dp
700 31 10 4.5 0.3814 0.1617 0.4068
V rpm/min mm in in
16.85/14.82 10–40 1 0.0509 0.71 1.051
V Ah h N/Wh
Rotor Motor parameter Motor size Propeller diameter Propeller pitch Figure of merit constant parameter Figure of merit power parameter Mean value 65–100% Battery Voltage (fully charged/standard) Nominal capacity Rated discharge time Weight/energy ratio Discharge fraction Peukert coefficient
Units
70 71
mm N N W
72 73 74 75 76
β f0 m f 65
77 78 79 80 81 82 83
V f , V0 C0 t0
α η k
84 85
Fig. 1. Measured rotor thrust and power as a function of the throttle.
25 26 27 28 29 30 31 32 33
89 90
where keo = W eo / W to and kb = W b / W to . Since payload weight, W p , is usually defined as a requirement, the initial guess value for take-off weight is then given by
Wp
( 0)
91 92 93
(27)
94
where keo is obtained from a statistical analysis of existing platforms, and kb is chosen to be 2/3 according to Eqs. (11) and (25).
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3. Results
99
W to =
1 − keo − kb
95 97 98
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87 88
23 24
86
100
The numerical model for hovering time prediction and approximate solutions describing the best endurance condition are applied to a 750 mm size six-rotor platform equipped with a 2-axis stabilised camera. Aircraft data and parameters relevant for the proposed applications are reported in Table 2 whereas Fig. 5 shows the test platform. Figure of merit model described by Eq. (10) is first validated by means of a test rig for measuring the thrust delivered by the rotor and the power required for driving it. Fig. 1 shows the data acquired for 13 different throttle settings between 40% and 100%. Model parameters, f 0 and m, are then obtained by a least-square estimate, and Fig. 2 presents measured and approximated figure of merit as a function of the throttle. The proposed model provides an accurate fit up to 75% of the throttle setting, whereas an increasing percentage error is found at higher values (5 @ 80%, 8 @ 90%, 10 @ 100%). Nonetheless, for many multi-rotor platforms, the driving system is sized so that the power margin, intended as the ratio between the power available on-board and the power required at hovering, lies between 1.2 and 2.2 [9]. These values for the power margin allow for an increased manoeuvrability and, in case of a 6-rotor configuration, they allow for keeping the hovering condition with a residual manoeuvre capability even in case of engine failure. This makes the 50–70% range suitable for operating the hovering flight condition. On the basis of the model for figure of merit, multi-rotor endurance is computed as a function of battery ratio, W b / W 0 . Fig. 3 compares numerical prediction expressed by Eq. (12) and approximate solutions derived in Section 2.1 with endurance calculated by using the measured values of figure of merit. Results show how in the range where the figure of merit model fits the measured data, the hovering time prediction is accurate as well. At larger values of the throttle position, where the figure of merit model is
101 102 103 104 105 106 107 108
Fig. 2. Rotor figure of merit as a function of the throttle.
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Fig. 3. Hovering time as a function of the battery ratio.
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Fig. 4. Endurance error of approximate models in the range of 50–70% of the motor throttle.
Fig. 6. Experimental validation.
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Table 3 Experimental results.
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25
Capacity [Ah]
Take-off weight [N]
Endurance [min]
91
26
C0
W to
Eq. (12)
Experimental
92
t
t¯
σt
93
15.12 22.90 27.42 29.90
15.57 22.86 27.28 29.24
0.2902 0.1630 0.056 0.1522
27 28
10 20 30 40
29 30 31
29.20 36.84 44.49 52.13
32 34 35 36 37 38 39 40 41
Fig. 5. Multi-rotor platform used for validation.
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95 96 97 98
33
42
94
affected by an increasing error, the estimation provided Eq. (12) looses accuracy. Nonetheless, since the measured figure of merit exhibits small variations beyond the 65% throttle position, the hovering time can be estimated according to Eq. (18) by using f = f 65 , that is the mean value of figure of merit in the range of 65 to 100%. This produces suitable estimates, especially for the best endurance battery ratio which is overestimated when Eq. (12) is used. Accuracy of approximate solutions in the range of 50–70% is highlighted in Fig. 4 in terms of the percentage error. The model derived under the hypothesis of small payload required power provides, as expected, an overestimation of the hovering time in the whole range of the throttle, with a maximum error of 6%. On the other hand, under the hypothesis of constant figure of merit, the corresponding model shows a larger error, as high as 8% far from the propeller regime where f = f 0 , but it provides the exact estimation when the actual figure of merit lies in the neighbourhood of f 0 . Also, it is worth noting that the simplest approximate model, given by m = 0 and P ap = 0, still provides an acceptable estimation of the hovering time, with a percentage error of about 15% in the absence of detailed information on rotor and payload characterisation.
To validate the proposed methodology, an experimental test campaign is performed. A total of twenty hovering flights are completed by using batteries with identical weight/energy ratio, α , along four different nominal capacity values. Fig. 6 presents the experimental hovering time plotted vs. the nominal capacity. Experimental data are compared to the analytical prediction of Eq. (12) and constant figure of merit approximation of Eq. (18). Statistics of the test campaign are provided in Table 3 in terms of mean value, t¯, and standard deviation, σt . Results show how the proposed approach provides a satisfactory estimation of the hovering time, making the model a suitable analytical tool for the preliminary design phase. In this respect, it is important to note that following an increase of battery capacity, the corresponding increase in hovering time shows a negative trend. By increasing the capacity from 10 to 20 Ah, one gains about 10 min of hovering time, for the considered platform, which largely increases the initial endurance. A further addition of 10 Ah increases endurance by less than 5 min. A further increase from 30 to 40 Ah provides only 2 min of additional hovering capability. These data represents a further proof for the importance of correctly addressing the issue of preliminary sizing the batterypowered aircraft, especially for multi-rotor platforms, on the basis of a physically sound methodology. As a matter of fact, trying to increase the endurance of an already existing platform by simply adding batteries may not be an effective solution, as propeller operates at higher motor regimes, where the decreasing slope of the endurance vs. capacity plot provides only a marginal improvement.
99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127
4. Conclusions
128 129
A rigorous analytical framework has been derived for addressing the study of the hovering flight condition of electrically powered multi-rotor platforms equipped with a payload. Under the
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assumptions of constant voltage battery discharge and small variations of rotor figure of merit, a mathematical model providing an accurate estimation of the hovering time has been found. In addition, closed-form analytical solutions for the best endurance battery capacity have been found for two cases, when the figure of merit is constant, and when the power required by the payload is small if compared to the power required for the hovering condition. As a further result, a simpler approximate expression of the best endurance capacity not depending on rotor features and payload has been derived, showing that, along the preliminary sizing process, a first valuable guess for the battery weight is obtained by doubling the zero-capacity weight of the platform. The consistency of the proposed theoretical approach has been demonstrated by numerical simulations and an experimental flight test campaign by using a six-rotor platform.
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Conflict of interest statement
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None declared.
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