Flight test methods for assessing the performance of manned airplanes powered by gas turbine and reciprocating engines have been well established for many years. However, traditional flight-testing methods may need to be improvised for it to be directly applicable for small UAVs.

This article discusses a straight forward Flight test method for testing and establishing the performance (more precisely, ‘Static performance’) for small unmanned aerial vehicles powered by electric motors with fixed pitch-propellers.

In a straight and level flight, the UAV’s battery pack is required to deliver electrical power to counter the aerodynamic drag, accounting for losses due to the propeller, motor, and motor controller. Since the same battery pack also supplies energy to payload and avionic system, a further share of electrical power needs to be considered in the energy balance equation.

The first part wherein aerodynamic efficiency requires treatment to arrive at performance metrics remains the same for any aircraft, although significant differences are that for such aircraft, in-flight engine power cannot be determined by measuring fuel flow, as it is done for engines that burn fuel. Secondly, electric powered UAVs do not decrease in weight over time as fuel burning planes do. Establishing Performance metrics for aircraft includes finding the maximum speed, minimum speed, maximum range speed, maximum endurance speed, maximum rate of climb, maximum climb angle, and turn performance.

The second part comprising prediction modeling of the battery behavior is a challenging task since effective battery capacity depends on temperature, aging, cycling, and current drawn.

Basic Theory

Range & Endurance

We know that for any propeller driven aircraft,

Range is maximum when Drag is minimum. This corresponds to a speed at which $({{C_L} \over C_D})$ is maximum. This speed is called the Range speed.

Endurance is maximized when ‘power required‘ is minimized, which corresponds to a speed at which $({{C_L^{3/2}} \over C_D})$ is maximum. This speed is called the endurance speed.

During steady level flight , Drag = Thrust Required (for small values of angle of attack).

Also, Power Required= Thrust Required x Velocity

$V_{\alpha} $ for a given $T_r$ depends on

  1. Thrust-to-weight ratio ${T_r \over W}$
  2. Wing loading ${W \over S}$
  3. The drag polar, that is, $ C_{D0}$ and $K$

Thrust-to-weight ratio and the wing loading are therefore fundamental parameters for airplane performance apart from the drag polar coefficients.

Unlike a gas turbine or ICT engine powered aircraft, for a battery powered UAV it is important to note that

  • Available thrust may be significantly affected by battery state of charge, current drawn and voltage. Combination of motor efficiency (including motor controller) and propeller efficiency would determine the actual propulsive thrust available to the aircraft.
  • W is constant throughout the flight. Hence wing loading is a constant.

The Drag Polar, Coefficients

The drag polar can be represented as,

$C_D=C_{D0}+K_1 C_L+K_2 C_L^2$

Or, in its simplified form, drag polar can be represented as

$C_D=C_{D0}+K C_L^2$

The parameters in the drag polar that are not known are $C_{D0}$ and either $K_2$ and $K_1$, (or $K$ for a simplified Drag polar). $C_{D0}$, $K_2$, and $K_1$ can be found graphically by plotting $C_D$ verses $C_L$ and then fitting the data with a second order polynomial in order to find the constants. In the case of the simplified drag polar, if $C_D$ is plotted verses $C_L^2$, the slope of this plot is equal to $K$ and the y-intercept is equal to $C_{D0}$.

Battery Modelling and Net Performance

The rated capacity of a battery is quoted in ampere hours or milli-ampere hours (mAH), typically based upon rated current for a 1 hour discharge. However, the battery typically would not be capable of supplying double the rated current for a half hour discharge, a consequence of Peukert effect. Similarly, if
the current draw was double the rated, the battery might show an increase in effective capacity. Hence the higher the current draw, the less the effective battery capacity. Hence we need a power relationship between discharge current and delivered capacity over the specified range of discharge currents as applicable during the flight profile. Using Peukert law, battery discharge time is modeled as a function of the current drawn, which in turn is arrived from the Power required. Endurance and range are now found taking this into consideration.

Peukert’s equation, $t={{R_t} \over {i^n}} ({C \over {Rt}})^n$,

where,

  • $t$ is the time in hours,
  • $i$ is the discharge current (amperes),
  • $C$ is the battery capacity in ampere hours,
  • ${R_t}$ is the battery hour rating (in hours): i.e., the discharge time over which the capacity was determined (typically 1 h for small rechargeable battery packs).
  • $n$ is a ‘discharge parameter’ dependent on the battery type and temperature.

$n$ typically changes for a given battery as it ages and cycles such that capacity usually diminishes.

Preparatory Ground Tests

Before the drag polar can be found from flight tests, the efficiency of the motor-propeller combination must be determined over the range of possible throttle settings and vehicle velocities.

Wind Tunnel Testing of motor-propeller

The motor-propeller efficiency can be found from wind tunnel tests, where an electric motor and propeller are mounted on a force balance and the thrust produced, $T$, is measured for various air speeds.

The thrust multiplied by the velocity ($V$) of the air moving through the wind tunnel is the mechanical power out.

Output Power, $P_o= T V$

Electrical power ($P_e$) supplied to the motor is the product of the voltage and the current into the electric Motor.

Input Power, $P_e= \nu I$ , where $\nu$ is the voltage and $I$ is the current

The ratio of the mechanical power ($P_o$) to the electrical power ($P_e$) supplied to the motor is the efficiency ($η$).

Motor-Propeller Efficiency, $η= {P_o \over P_e } $

Once the motor-propeller efficiency is measured over a range of power settings and wind tunnel air speeds, a two-dimensional look-up table can then be created with $P_e$, $V$ as the inputs and $η$ as the output. This look-up table is used to determine the operating efficiency of the motor-propeller combination.  

During wind tunnel testing the maximum thrust is also recorded for each airspeed. These values are used to construct the thrust available ($T_a$)and power available ($P_a$) curves, which can then be used in the calculation of other performance parameters.

Other vital data to gather are the aircraft’s total weight ($W$, in Newtons), wing area ($S$), the air temperature, and the barometric pressure. Air temperature and barometric pressure are used to calculate air density ($ρ$).

Flight Test

The fundamental approach, as applicable for any aircraft, is to first determine the drag polar from flight tests and then calculate other performance parameters from this drag polar. When possible, the parameters found from calculations are validated with further flight tests.

The test aircraft is flown straight and level at a constant air speed on a straight level course with constant throttle setting.  The runs are repeated for a range of throttle settings for different airspeeds. During each constant-throttle run airspeed, current, and voltage are measured and recorded multiple times. These data points are then averaged to eliminate precision error.

For steady level flight,

$C_L= {2 W \over {ρ V^2 S}} $ ——————(1)

and,

$C_D= {2 P_r \over {ρ V^3 S}} $ ——————(2)

where, $P_r=P_o=η P_e=η \nu I$

Therefore,

$C_D={{2 η \nu I} \over {ρ V^3 S}} $ ——————(3)

Since all the values on the right side are known, $C_L$ and $C_D$ can be calculated for each speed.

Based on the calculated values of $C_L$ and $C_D$, the drag polar can be plotted.


Performance metrics

$({L \over D})_{max}= \sqrt {1 \over {4 C_{D0} K_2}}$

Endurance Speed, $V_{({{C_L^{3/2}} \over C_D})_{max}} = \sqrt {({{2 \over ρ} \sqrt {K_2 \over {3 C_{D0}}} ({W \over S})})}$

Range Speed, $V_{({{C_L} \over C_D})_{max}} = \sqrt {({{2 \over ρ} \sqrt {K_2 \over { C_{D0}}} ({W \over S})})}$

i.e. ${Range Speed} = \sqrt {\sqrt{3}} . {Endurance Speed}$

It can be observed that the Wing loading, ${W \over S}$ directly influences the performance speeds.


Rate of Climb,${ROC}={{P_a-P_r} \over W}$

Angle of Climb,${\gamma} = Sin^{-1} ({{ROC} \over V})$

For a range of velocities the above parameters can be plotted to find the maximum ROC and Max angle of climb.


Level turn load factor (Speed limited), $n_{max}={{ρ V^2 S C_{Lmax}} \over {2 W}}$

Level turn load factor (Thrust limited), $n={{{- K_1 W}+ \sqrt {({K_1 W})^2 -4 K_2 W^2 ({C_{D0}+{2 T \over {ρ V^2 S}}}) }} \over {{4 K_2 W^2} \over {ρ V^2 S}}} $


Maximum Endurance in hours is given by:

$E_{max}=R_t^{1-n} [{{η \nu C} \over {({2 \over \sqrt {ρS}} C_{D0}^{1 \over 4}{2W {\sqrt {k \over 3}}})^{3 \over 2}}}]^n$ , where $\nu$ is the voltage and $\eta$ is the combined efficiency of motor,controller and propeller

Max Range is given by,

$R_{max}=R_t^{1-n} [{{η \nu C} \over {({1 \over \sqrt {ρS}} C_{D0}^{1 \over 4}{2W {\sqrt {k}}})^{3 \over 2}}}]^n \times \sqrt {{{2W} \over {ρS}} {\sqrt {K \over C_{D0}}}} \times 3.6$

Note: For most batteries used in UAV applications, voltage drop during discharge is moderate. However, if potentially greater accuracy is required, voltage drop effects needs to be incorporated.


References:

  • Aircraft Performance & Design” by John D Anderson, Jr. , McGraw Hill Pub.
  • Performance Characterization of a Lithium–Ion Gel Polymer Battery Power Supply for an Unmanned Aerial Vehicle,” Reid, C., and Manzo, M., NASA TM 2004-213401

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