Quadrotor 6-DOF HIL Simulation and Verification Using a 6-axis Load ...

Quadrotor 6-DOF HIL Simulation and Verification Using a 6-axis Load Cell Travis D. Fields,∗ Logan M. Ellis† and Gregory W. King‡ University of Missouri-Kansas City, MO, 64110, USA

Over the past decade, multirotor unmanned aircraft systems have seen tremendous growth in research and commercial applications. Much research has been done on developing complex control systems; however, due to current Federal Aviation Administration limitations, many systems can not be flown legally outdoors in the National Airspace. As an alternative approach to vehicle testing, this paper discusses the use of a six-axis load cell coupled with a physical quadrotor aircraft. The load cell measures the quadrotor-generated forces and moments, which are then integrated with the use of a forward dynamic model to provide real-time estimates of the system states. State information is telemetered from the simulation program back to the vehicle, overwriting onboard attitude/location estimates. To validate the use of a hardware-in-the-loop (HIL) simulation, load cell experiments were performed and compared with free-flight responses within an indoor motion capture facility. Step responses were performed in both configurations. Results show good agreement between the free-flight system and the rotational HIL simulation. Rotational response tests exhibited good agreement between simulated and experimental trials. Translational simulation step response data also demonstrated strong correlation with experimentally collected responses (both X and Z translations). In each of the tests, the added drag coefficient was relatively insensitive to the simulated response, providing additional confidence in HIL simulation as a path-planning testing tool with no/minimal experimental data collection.

Nomenclature {ˆbx , ˆby , ˆbz } B B cB x , cy , cz FBcm D ~ Fg fmed FC R ~B/C I ~I B/Bcm xx B/Bcm Iyy B/Bcm Izz kxB , kyB , kzB mB B/C MR ~ MB D ~

= = = = = = = = = = = = = =

Orthogonal unitary basis fixed to quadrotor frame Moment (rotational) drag coefficients Drag force vector with components x, y, and z Force of gravity Median power frequency of oscillation Reaction force vector (components x, y, and z) measured by load cell Quadrotor inertia dyadic about load cell attachment point C Quadrotor moment of inertia about center of mass along ˆbx axis Quadrotor moment of inertia about center of mass along ˆby axis Quadrotor moment of inertia about center of mass along ˆbz axis Force (translational) drag coefficients Mass of quadrotor Reaction moment vector on quadrotor B about load cell attachment point C Drag moment vector

∗ Assistant Professor, Civil and Mechanical Engineering, 5110 Rockhill Rd, Kansas City, MO, [email protected], AIAA Member. † Undergraduate Research Assistant, Civil and Mechanical Engineering, 5110 Rockhill Rd, Kansas City, MO, AIAA Student Member. ‡ Associate Professor, Civil and Mechanical Engineering, 5110 Rockhill Rd, Kansas City, MO, [email protected]

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O {ˆ nx , n ˆy , n ˆz } P W M1 − P W M4 qR,d ~q˙ R ~q ¨R ~q T,d ~q˙ T ~q ¨T ~R RM SE rBcm/C ~ UY , UP , UR , UZ

= = = = = = = = = = = = =

Maximum percent overshoot Orthogonal unitary basis fixed in Newtonian frame (North-East-Down convention) Output motor commands in pulse-width-modulation form Desired rotational position vector Quadrotor angular velocity vector in Newtonian frame N Quadrotor angular acceleration vector in Newtonian frame N Desired translational position vector Velocity vector of the quadrotor center of mass in the Newtonian frame Acceleration vector of the quadrotor center of mass in the Newtonian frame Pearson correlation coefficient Root mean squared error Position vector from point C to quadrotor mass center, Bcm Controller output gains from yaw, pitch, roll, and altitude controllers

I.

Introduction

Multi-rotor helicopter platforms have seen tremendous growth due to their ability to hover and the relative ease with which hardware and software can be developed for specific needs. A wide variety of control approaches have been developed to stabilize the inherently unstable multi-rotor aircraft, from the relatively simple PID controller1, 2, 3 to the more complex control designs such as Model Predictive Control,4, 5 Sliding-mode Control,6 and state estimators such as extended Kalman Filtering.7 All of the identified flight controllers provide adequate vehicle stabilization; however, physical aircraft testing is often overlooked due to the difficulties and limitations in operating such aircraft. Outdoor testing requires prior authorization from the Federal Aviation Administration (which can be impractical for many applications), and indoor testing incurs obvious airspace limitations for the aircraft. As an alternative, test stands have been used for development and testing of particular aspects of the vehicle rotational controller (pitch, roll, and yaw). Tayebi and McGilvray developed a simple ball mount base which allows for unrestricted yaw and ±30◦ of pitch and roll while completely restricting translational motion.8 Hoffman et. al. developed a gimbaled mechanism into the structure of the quadrotor, producing a robust benchmark platform; however the test bench vehicle is not capable of free flight as it is built into the test bench directly.9 These test benches (along with many other 3 DOF test stands) provide the ability to test and tune the rotational control system with physical hardware, rather than relying solely on simulations. To develop a fully capable control system/autopilot, both the rotational and translational controllers must be tested and tuned on the actual flight vehicle. The primary difficulty in testing the translational controller is the need for free flight testing, ideally performed outdoors, and the heavy restrictions imposed on researchers (or moratorium on flight testing for commercial applications) imposed by the Federal Aviation Administration. Even with the eventual integration of civil unmanned aircraft systems into the national airspace, outdoor testing introduces several added difficulties and risks including: unexpected atmospheric disturbances, unretrievable aircraft in the presence of software errors, and risk to people or property. These risk factors are particularly prevalent during preliminary/initial controller tuning as the vehicle is likely still in a prototype state. Another approach to tuning the translational controller is to use an indoor motion capture facility such as the one used in advanced robot cooperation development by How et. al.;10 however, these facilities can easily exceed $100,000 dollars to acquire and operate. Even with an expensive motion capture facility, testing and tuning is limited to the translational and rotational controller, severely limiting any path-planning/trajectory generation algorithm testing (due to space limitations). A lower cost alternative is a rotational and translational controller testbed that utilizes hardware-in-theloop (HIL) simulation in conjunction with a six-axis load cell to perform a forward dynamics simulation. The multi-axis load cell is capable of measuring the reaction forces and torques, which can then be integrated to compute the system states. In addition to providing a safe, cost-effective means of tuning a multirotor flight controller, the HIL simulation environment opens up the possibility of adding in specific atmospheric disturbances and/or measurement uncertainties, providing the ability to test path-planning and trajectory generation algorithms in addition to the lower-level stability controllers. Recent work has introduced the HIL concept, with published simulation results; however, verification that the simulation matches the free-flight

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system has yet to be performed.11 In order to fully realize the benefits of such a HIL simulation approach, the simulation must be compared with free flight testing results. This paper is focused on validating HIL platform experiments by comparing with free flight tests obtained in an indoor motion capture facility. Rotational and translational controllers are compared individually in an effort to fully quantify the capabilities of the HIL test stand.

II.

Methodology

The HIL simulation can be broken down into two subsystems: the LabVIEW simulation and the quadrotor hardware with onboard autopilot. A block diagram illustrating the experimental setup is shown in Figure 1 for a fully integrated autopilot (only input from operator is desired global X, Y , Z location. As the subsystems only interact by sending data and measuring forces/moments (from the load cell), they can be examined independently.

Multirotor Hardware with Onboard Autopilot qT,d

qR,d

Translational PID Controller

+-

qT , qR

∫

q&T , q& R

∫

Rotational PID Controller

q&&T , q&&R

Multirotor Dynamics

Motor Out

FRC , M RB/C

Physical Multirotor

Load Cell

LabVIEW Simulation Figure 1: Conceptual flowchart of HIL simulation structure and control structure. A.

Simulation Development

The LabVIEW simulation requires data from a six-axis load cell (Bertec Corporation, Columbus, OH, USA) and previous estimates of the rotational and translational positions and velocities in order to estimate the current attitude and location. For simplicity the following assumptions have been made in developing the simulation model: 1. Multirotor and all associated components are considered to be a single rigid body. 2. Rotor gyroscopic effects have negligible influence on quadrotor motion. Once the attitude and location estimates have been calculated, the data is telemetered to the flight controller onboard the quadrotor via 2.4 GHz RF modems (XBee). The multirotor dynamics equations can be readily developed from rigid body dynamics. Using Newton’s second law of motion, the translational equations of motion can be derived as shown in Eq. 1. FC ˆ z − FBcm = N aBcm mB (1) R + Fg n D ~ ~ ~ Where the only external forces are the measured reaction forces from the load cell, gravity, and drag. Aerodynamic drag is assumed to be linearly proportional to the translational velocity, and serves to stabilize the forward dynamic simulation during integration of the kinematics and kinetics equations.

FBcm D ~

  kxB  B = ky  q˙ T kzB ~

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(2)

The rotational equations of motion can be derived from Euler’s equations for a three-dimensional rigid body.   B/Bcm ˙ B + N ω B × IB/Bcm · N ω B (3) MR − MB = IB/BcmN ω D ~ ~ ~ ~ ~ ~ ~ Where the only external moments acting on the quadrotor are the reaction moments measured by the load cell and the applied drag moments. Similar to the translational aerodynamic drag, a drag moment has been incorporated into the model primarily for simulation stabilization (and is also linearly proportional to the rotational velocities).

MB D ~

  cB x  B N B = cy  ω ~ cB z

(4)

The scalar equations of motion are created using Kane’s method, which utilizes generalized speeds (rather than generalized coordinates used in Lagrange’s equations). The set of generalized speeds were selected such that the kinematical equations are:     cos qrz U1 −sin qrz U2 q˙rx cos qry      q˙ry   sin qrz U1 + cos qrz U2     U3 + tan qry (sin qrz U2 − cos qrz U1 ) q˙rz    =   (5)   q˙   U4   tx         q˙ty   U5 q˙tz U6 This selection of generalized speeds greatly reduces the size of the equations of motion (compared to the trivial q˙rx = U1 form of generalized speeds). Kane’s equations also require partial velocities and partial angular velocities of the form:

vrBcm ~

=

∂ N vBcm ~ ∂Ur

(6)

and ∂ N ωB ~ ∂Ur

(7)

= U4 n ˆ x + U5 n ˆ y + U6 n ˆz

(8)

ωB r ~

=

where r = 1, 2, ...6 and, N

vBcm ~

and ω B = U1ˆbx + U2ˆby + U3ˆbz ~ Finally, to get the six equations of motion (r = 1, ..., 6): N

(9)


Bcm B N Bcm FB a · vr + resultant − m h~  i ~ ~ B/Bcm Mresultant − IB/Bcm · N ω ˙ B + N ω B × IB/Bcm · N ω B · ω B (10) r ~ ~ ~ ~ ~ ~ ~ To compute the dot and cross products, the following rotation matrix is required between the body fixed and Newtonian frames. The rotation matrix B RN follows the standard trigonometry naming simplification c2 = cos qR2 and s2 = sin qR2 . 0

=

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 B

RN

=

c2 c3  −s3 c2 s2

s3 c1 + s1 s2 c3 c1 c3 − s1 s2 s3 −s1 c2

 s1 s3 − s2 c1 c3  s1 c3 + s2 s3 c1  c1 c2

(11)

The resulting six equations (omitted for clarity) then provide an estimate of the accelerations based on the measured load cell data as well as previous state information. In the LabVIEW environment, the Control and Simulation Loop is used to perform the dynamic simulation. This loop performs in much the same way as the widely-used Matlab Simulink toolbox. A double integrating Control and Simulation Loop used to estimate the new quadrotor states (double integrates one time step forward) is shown in Figure 2. The algebraic acceleration equations are incorporated into a SubVI (Dynamic Solver in Figure 2) in a Mathscript node. This allows for easy modifications to a Matlab-like script, while still maintaining the benefits and ease-of-use of the LabVIEW programming environment.

Figure 2: LabVIEW simulation loop used to estimate quadrotor states one time step forward. The complete LabVIEW graphical user interface (GUI) is shown in Figure 3. The LabVIEW GUI shows the user the current state estimates, as well as a three-dimensional representation of the multirotor (represented as a flat plate). Additionally, the GUI shows the user the data read from the multirotor (including current and desired rotational states) and the current state information to the multirotor. When desired, the measured loads and state estimates can be stored to a spreadsheet for post-processing by simply clicking on the appropriate boolean switch. Parameters used in performing the simulations are shown in Table 1. The drag coefficients were varied to find the most realistic simulation as discussed in the results section. Parameter B/Bcm Ixx B/Bcm Iyy B/Bcm Izz rBcm/C mb

Value 0.0112 0.0112 0.0205 -0.06 ˆbz 1.20

Units N m2 N m2 N m2 m kg

Table 1: Estimated physical parameters used in rotational simulation verification testing. B.

Hardware and Controller Development

A custom quadrotor was constructed from various commercial products. The constructed quadrotor has a frame diameter of 0.45 m (1.5 ft) and a total takeoff weight of 11.8 N (2.64 lb) including the battery. Maximum total thrust with a three-cell and four-cell Lithium-Polymer battery is 120 N and 140 N, respectively. The flight controller is an inexpensive Arduino-based flight controller (MultiWii) containing a barometric pressure sensor, thee-axis magnetometer, gyroscope, and accelerometer. The processor is an ATmega 2560 chip with 256 kb flash memory, 8kb RAM, 4 kb EEPROM, 86 GPIOs, 4 USARTs, and a 16-channel 10-bit

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Figure 3: LabVIEW developed HIL simulation graphical user interface.

A/D converter. The MultiWii board has seen extensive use in the low-cost hobby market, with an elaborate autopilot and associated ground station freely distributed. For this study, the MultiWii code was used as a framework to gather and interpret sensor data, remotely control inputs, and send pulse width modulation commands to the electronic speed controllers. All controller routines were developed in-house, enabling an easily tunable control interface for the various experiments required for this study. Figure 4 shows the constructed quadrotor mounted to the six-axis load cell and test stand.

Figure 4: Quadrotor mounted on test stand with 6-axis load cell. The attitude and translational control structure developed is a rudimentary nested Proportional-IntegralDerivative (PID) controller as shown in Figure 5. Depending on the flight mode, the three desired translational positions and yaw angle can be defined either by preprogrammed trajectory, or with a standard hobby R/C controller. Using the desired and actual horizontal positions and yaw angle, the lateral and longitudinal controllers calculate the estimated pitch and roll necessary to reach the desired location. The lower-level attitude controllers (yaw, pitch, and roll) and the altitude controller each provide an output gain. These output gains are used to manipulate the power going to each of the four motors according to Eq. 12.

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qR3,d

Yaw Controller

UY

+

Pitch Controller

UP

+

Roll Controller

UR

qT1,d qT2,d qT3,d -

+

Lateral Controller

qR2,d

Longitudinal Controller

qR1,d

qR3

- qR2

qT1

+

+

- qR1

qT2

UZ

Altitude Controller

+ qT3

Figure 5: Multirotor nested PID control structure.

  P W M1   P W M2    P W M3  P W M4

 −1  +1 =  +1 −1

  −1 0 +1 UY   0 −1 +1 UP     + P W Mmin 0 +1 +1 UR  UZ +1 0 +1

(12)

The minimum PWM command, P W Mmin , is derived from the minimum pulse width (in microseconds) that the electronic speed controllers (ESC) can interpret. This value is typically around 900-1000 µs, depending on the specific ESC. The resulting motor output commands, P W M1 − P W M4 , are sent directly to the ESCs. Figure 6 shows the motor/ESC numbering scheme used for this study.

bˆx PWM3

bˆy PWM4

PWM2

PWM1 Figure 6: Quadrotor sketch with motors/ESCs numbering scheme. C.

Verification Testing Methods

Experimental and HIL-based simulation tests were conducted for the rotational and translational controllers. Step responses were performed in both a motion capture laboratory and on the HIL test stand for each degree of freedom (pitch, roll, yaw, x, y, and z). All state estimates (whether from onboard or simulated sources) were performed and telemetered at 125 Hz in an effort to reduce controller instabilities with low frequency state estimations. For simplicity, step responses were conducted about a single axis and stored prior to testing the next axis. HIL simulation testing was conducted with the six-axis load cell, LabVIEW simulation, and physical quadrotor with onboard autopilot. Pitch/roll step responses were performed with a 1-DOF test stand (isolates only pitch or roll rotation), and translational and yaw step responses were performed in a motion capture facility. HIL simulations for both the rotational and translational step responses were conducted with the quadrotor mounted on the six-axis load cell. Motion was completely restricted on the load cell, as all three forces

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and moments could be measured directly. Data was acquired by LabVIEW and then integrated to provide an estimate of the quadrotor states as previously discussed. State information was then telemetered to the quadrotor, overwriting internally estimated states. Based on the error between current and desired angle/position, adjustments to the output motors were made, thereby changing the measured forces and moments of the load cell. It is important to note that simulation equations of motion were developed to include the inevitable separation distance between the load cell mounting plate and the quadrotor’s center of mass. Rotational experiments were performed with the quadrotor mounted on either a single-axis test apparatus (pitch and roll) or with the quadrotor flown in a motion capture facility (yaw). To reduce coupling effects present during multi-axis motion, testing was conducted on one axis at a time. For pitch and roll, the initial desired angle was set to zero degrees (level). The amplitude of the step response was set to 15◦ . Due to the aircraft symmetry, only pitch step responses were conducted as preliminary data showed similar results between pitch and roll. For yaw the step response amplitude was 30◦ from the initial angle. As discussed above, drag terms were added to create more realistic simulations. Testing was conducted with many different drag values in order to find a realistic model as well as quantify the sensitivity of the drag forces and moments on the vehicle motion. Collected data from experimental and simulation trials were compared to quantify correlation between the simulation and the physical system motion. Resulting datasets were truncated to begin at the instant the aircraft began moving in the desired direction (after the input step response was transmitted to the flight controller), and end a time selected to fully capture transient effects (5 seconds for both pitch and yaw). Step response magnitudes of 30◦ and 15◦ were used for yaw and pitch responses, respectively. All flight data was resampled via linear interpolation such that data points were aligned to discrete time values corresponding with a sampling frequency of 100 Hz. RMS errors (RM SE) and Pearson correlation coefficients (R) were calculated between each simulated profile and the single measured profile to evaluate how well each simulated drag coefficient approximated actual conditions. Two additional variables, percent overshoot (O) and median power frequency of oscillation (fmed ), were calculated for each profile. Percent overshoot was defined as the maximal angle reached immediately following transition of the target signal from low to high, expressed as a percentage of target level. Median frequency was calculated by first performing a Fourier series analysis to derive a power spectrum, then extracting the frequency at which median cumulative power occurs (i.e. the frequency dividing the area under the power spectrum into two equal halves). While the latter two variables may be used to quantify the absolute performance of each condition (simulated and actual) in meeting the target angle, this is beyond the scope of the current work as the main objective was to find which simulated trial(s) most closely approximated the actual testing conditions. Translational model verification was performed in UMKC’s Human Motion Laboratory (HML) (Figure 7). Three retroreflective motion capture markers were placed on the quadrotor tracked in 3D space by the HML’s 7-camera motion capture system (Vicon, Inc., Los Angeles, CA, USA). A LabVIEW Virtual Instrument (VI) was created, using the Vicon DataStream Software Development Kit, to retrieve marker X,Y, and Z position data for the three markers from the Vicon Nexus software. Yaw attitude estimates were calculated using the relative position of the three markers and was telemetered with the X,Y, and Z position information to the flight controller onboard the multi-rotor aircraft. Calculated yaw data was incorporated into the data stream because of the magnetic interference in the HML (causing inevitable instabilities in the yaw estimate using onboard sensors); however, onboard pitch and roll estimates were unaffected and therefore used for aircraft control. A series of flight tests were conducted in which the quad rotor was flown in a straight line distance of 0.75 m and 1.25 m along the lab’s X and Z axes, respectively, from an initial hovering position approximately centered on the lab’s Newtonian global origin. Nine trials were performed for Z testing, and six trials were conducted in the X translational testing. During each trial, the Vicon system recorded the quadrotor’s 3D coordinates at a sampling rate of 125 Hz. This data was streamed to the quadrotor’s onboard controller to provide real-time position feedback, and also stored on an onboard SD card along with HIL-simulated trajectory data for offline processing. Simulated and actual translation data were synchronized and time scaled between the time at which the simulator control was initiated and a time chosen to fully capture transient effects (14 and 10 seconds for Z and X responses, respectively). RM SE and R values were again calculated from the resulting pairs of datasets to quantify the deviation between simulated and actual trajectories for the various friction coefficients tested on each axis. Additionally, for Z step response testing a second experimental test was performed in order to provide a RM SE and R thereby quantifying the repeatability of the experimental setup.

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Figure 7: UMKC’s Human Motion Laboratory (HML).

III. A.

Results

Yaw Step Response

Comparisons between simulated trials and experimental results are depicted in Table 2, and a representative plot of experimental and simulated results is included in Figure 8. RM SE and R represent comparisons between simulated and measured trials; O and fmed represent characteristics of individual configurations. There was no one drag condition that clearly exhibited the best agreement with the experimental condiB tions. Drag values of cB z = 0.005 and cz = 0.006 yielded minimal RM SE and maximal R, respectively, in comparison to experimental trials; however these simulations yielded overshoot levels much higher (70 137%) than in experimental conditions. The drag value of cB z = 0.003 yielded overshoot levels more closely matching experimental trials; however this condition yielded poorer agreement (RM SE = 7.14◦ , R = 0.56). While it was not clear which drag coefficient yielded ideal comparison with experimental results, there was a modest linear trend (R2 = 0.63) indicating better correlation with experimental conditions with increasing drag coefficient. It is important to note that yaw step response testing was performed in the motion capture facility, which allows for pitch and roll motion. Trial HIL Sim #1 HIL Sim #2 HIL Sim #3 HIL Sim #4 HIL Sim #5 HIL Sim #6 Experimental

cB z 0.00006 0.003 0.0045 0.005 0.006 0.01 -

RMSE (deg) 7.71◦ 7.14◦ 10.25◦ 5.57◦ 7.93◦ 7.21◦ -

R 0.52 0.56 0.65 0.82 0.90 0.84 -

O(%) 68.60 51.66 55.98 74.31 103.64 68.00 43.67

fmedian (Hz) 0.57 0.43 0.43 0.43 0.43 0.43 0.43

Table 2: Results for simulated and experimental yaw trials. B.

Pitch Step Response

For pitch simulations, an analysis of varied drag coefficients was carried out similar to the yaw analysis presented above. Comparisons between simulated trials and experimental results are depicted in Table 3, and a representative plot of experimental and simulated results is included in Figure 9. Although varying the drag coefficient had little effect on simulations, a pitch drag coefficient of 0.165 was selected as the one yielding minimal error (RM SE = 1.96◦ , R = 0.84) between simulated and measured profiles (Figure 9). Overshoot and median frequency for simulated and experimental pitch trials are also depicted in Table 3. 9 of 14 American Institute of Aeronautics and Astronautics

60 Experimental Trajectory Simulated Trajectory Desired Trajectory

50

Heading (Degrees)

40 30 20 10 0 −10 0

1

2

3

4

5

Time (s) Figure 8: Experimental yaw response compared to a simulated yaw response (cB z = 0.005). Trial cB RMSE (deg) R O(%) f (Hz) median y HIL Sim #1 0.10 3.17◦ 0.52 2.00 1.60 ◦ HIL Sim #2 0.13 3.38 0.63 15.02 2.00 HIL Sim #3 0.14 2.70◦ 0.65 7.96 2.00 ◦ HIL Sim #4 0.15 2.96 0.63 8.03 1.66 HIL Sim #5 0.165 1.96◦ 0.84 3.28 1.60 Experimental 23.86 1.80 Table 3: Results for simulated and experimental pitch trials. 20 Experimental Trajectory Simlated Trajectory Desired Trajectory

Heading (Degrees)

15

10

5

0

−5 0

1

2

3

4

5

Time (s) Figure 9: Experimental pitch response compared to a simulated pitch response (cB y = 0.165). C.

Vertical Translation (Z) Step Response

Comparisons between simulated vertical translation trials and experimental results are depicted in Table 4, and a representative plot of experimental and simulated results is included in Figure 10. Although nearly all simulated trials exhibited good agreement with experimental results, simulations with kzB = 0.10 exhibited the best agreement (RM SE = 0.07m, R = 0.95). Modest positive and negative linear trends were observed for RM SE (R2 = 0.27) and R (R2 = 0.38) with increasing simulated drag coefficients.

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kzB 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 -

Trial HIL Sim #1 HIL Sim #2 HIL Sim #3 HIL Sim #4 HIL Sim #5 HIL Sim #6 HIL Sim #7 HIL Sim #8 HIL Sim #9 Experimental

RMSE (m) 0.07 0.14 0.11 0.12 0.08 0.13 0.13 0.16 0.13 -

R 0.95 0.92 0.92 0.87 0.94 0.87 0.87 0.82 0.88 -

O(%) 43.33 70.59 59.33 18.00 26.07 55.99 58.82 56.76 49.33 29.96

fmedian (Hz) 0.29 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21

Table 4: Results for simulated and experimental vertical translation trials. 1.4 Experimental Trajectory Simulated Trajectory Desired Trajectory

1.2

Position (m)

1 0.8 0.6 0.4 0.2 0 0

2

4

6

8

10

12

14

Time (s) Figure 10: Experimental vertical translation response compared to a simulated vertical translation response (kzB = 0.10). D.

X Step Response

Comparisons between simulated global X translation trials and experimental results are depicted in Table 5, and a representative plot of experimental and simulated results is included in Figure 11. Although nearly all simulated trials exhibited good agreement with experimental results, simulations with kxB = 1.25 exhibited the best agreement (RM SE = 0.18m, R = 0.96). Positive and negative linear trends were observed for RM SE (R2 = 0.70) and R (R2 = 0.90) with increasing simulated drag coefficients. Note for all experiments the body axis drag coefficient kxB was applied to both the X and Y direction.

IV. A.

Discussion

Rotational Simulation and Verification

RMS errors calculated between simulated yaw profiles with different drag coefficients and experimentally collected yaw step response data were similar with increasing drag coefficients resulting in generally better agreement with the experimental results. Fortunately, the results suggest that varying the simulated yaw drag coefficient cB z does not have a significant effect on the HIL simulation even with the large range of coefficients tested. Based on the data collected, the best overall yaw drag coefficient is cB z = 0.005 as the results indicate the lowest RMSE and strong correlation coefficient at the cost of larger than experimentally observed overshoot. Pitch step response results also showed generally good agreement for the range of pitch drag coefficients

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Trial HIL Sim #1 HIL Sim #2 HIL Sim #3 HIL Sim #4 HIL Sim #5 HIL Sim #6 Experimental

kxB 0.65 1.00 1.25 1.50 2.00 5.00 -

RMSE (m) 0.43 0.20 0.18 0.39 0.36 0.73 -

R 0.87 0.95 0.96 0.75 0.76 0.38 -

O(%) 90.00 66.06 60.38 33.60 30.39 14.42 36.94

fmedian (Hz) 0.20 0.20 0.20 0.20 0.20 0.20 0.20

Table 5: Results for simulated and experimental forward translation trials. 2.5 Experimental Trajectory Simulated Trajectory Desired Trajectory

Position (m)

2

1.5

1

0.5

0

−0.5 0

2

4

6

8

10

Time (s) Figure 11: Experimental forward translation response compared to a simulated forward translation response (kxB = 1.25). tested. Although a smaller range of coefficients was tested in comparison to the range of yaw drag coefficients, the results demonstrate the pitch drag coefficient has little effect on the simulation results. For all translational simulations, a pitch drag coefficient cB y of 0.165 was used as it produced the dataset with the lowest RMSE and best Pearson correlation coefficient. For cB y = 0.165, percent overshoot did not match the experimentally collected data; however, upon inspection of Figure 9 the simulation (shown as the dotted line) does match the lower oscillation peaks in magnitude. Due to the airframe symmetry, the pitch drag coefficient value was also applied directly as the roll drag coefficient cB x. The yaw and pitch step response results suggest that varying the simulated drag coefficient, at least within the range studied, does not have a significant effect on the simulated angle response. These results are encouraging for developing a HIL simulation platform, because prior to testing the vehicle, an estimated drag coefficient must be used. The goal of the HIL simulation is to eliminate the need for free-flight testing until a mature prototype has been developed, making experimental measurements of the rotational drag coefficients impossible. The results show that the rotational drag coefficients are relatively insensitive, therefore providing confidence in using a general drag coefficient term for a range of vehicle sizes/shapes. It is important to note that the applicability of the drag coefficients on different airframes is beyond the scope of this work, but will be investigated in future work. B.

Translational Simulation and Verification

The vertical drag coefficient has a relatively small impact on the simulated response data. A vertical drag coefficient kzB of 0.10 produced the best agreement between the simulated and actual flight tests, although all tests produced adequate results. Two experimental step responses were conducted to provide repeatability estimates of the RMSE. Between two identical experimental step responses, the RSME was found to be 0.04 m with a correlation coefficient of 0.98. This repeatability comparison provides the needed assurance that

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the best simulated trial is close to the best expected RSME/correlation of repeated free flight experiments with a RSME of 0.07m and correlation coefficient of 0.95. As shown in Figure 10, the simulated trajectory closely follows the experimental trajectory for a coefficient of kzB = 0.10; however, the simulated trajectory does not dampen out as quickly as in the experimental results. This is likely due to the fact that the quadrotor is free to pitch, roll, and yaw during translational step response testing. As discussed previously, the rotational stability does have good agreement with the experimental results, but the errors from each degree of freedom can compound when allowing for full six-DOF motion. Additional variations between experimental and simulated results could be due to different battery charge states. During testing it was noted that the lower battery voltages produced slower responses. It is important to note that during HIL simulation experiments, the X and Y translations were constrained to zero. During testing in the HML, minor pitch and roll inputs were provided by the operator to maintain flight in the Vicon camera volume. X translation step response testing represents the complete six-DOF control/simulation system. As a reminder, X is a global translation with respect to the HML’s defined coordinate system. The desired pitch and roll angle is determined from the current yaw angle and distance from the desired global coordinate. For example, if yaw = 0.0◦ and the desired X coordinate is positive, the vehicle will have a negative desired pitch angle (pitch down). Based on the results provided in Section D, the best X drag coefficient kxB is 1.25. The HIL simulation corresponding to kxB = 1.25 is shown in Figure 11. Similar to the Z step response data, the simulation has a more idealized step response compared to the experimental response; however the overall response matches quite well considering the pitch, roll, yaw, and Z are also controlled during X testing. Due to the symmetry of the aircraft, the X drag coefficient is also used for the Y drag coefficient kyB . Although not shown in the results, the Y displacement during X step response testing shows similar response between the simulated and experimental testing in much the same way as the X displacement data.

V.

Conclusion

The objective of this study was to establish the feasibility of using a load cell test stand hardware-in-theloop simulation as a method for testing multirotor vehicles. HIL simulations were conducted and compared against unconstrained motion of a quadrotor unmanned aircraft system. HIL simulations were performed with a quadrotor mounted on a six-axis load cell, coupled with the LabVIEW programming environment. Yaw testing was performed in a motion capture facility, and pitch response testing was performed by constraining all but the pitch axis. Translational experimental testing was conducted in a motion capture facility, in which the motion capture equipment estimated the location of the quadrotor and telemetered the current position to the vehicle. Rotational and translational step responses were performed with both the HIL simulation and the unconstrained vehicle. Data was collected from one axis at a time as a means of reducing post-processing analysis. In the HIL simulation, friction/damping terms were added to increase simulation realism. Tests were conducted to verify simulation realism, as well as sensitivity of the added friction parameters. Rotational results show good agreement between the HIL simulation and the experimental testing. Yaw testing demonstrated only a weak sensitivity to the yaw drag coefficient, providing encouraging results for estimating adequate drag coefficients without the need for experimental testing. The best correlation between the actual and simulated yaw motion was found with yaw drag coefficient cB z = 0.005. Pitch testing shows similar correlation, although a smaller range of drag coefficients were tested. The best pitch (and corresponding roll) drag coefficient was found to be 0.10. Translational results show good agreement for a forward dynamics simulation. The Z step response testing demonstrated that the simulated response was relatively insensitive to the linear drag coefficient. Even in the presence of unconstrained flight testing (with minor operator roll/pitch corrections), the HIL simulation with kzB = 0.10 matches the experimental test quite well. Repeated experimental step responses were conducted to provide a baseline RMSE, with the kzB = 0.10 simulation only 0.03m larger than the RMSE between the two repeated experimental tests. X translational testing again showed the same general insensitivity to drag coefficient, and the HIL simulation also produced excellent results. The best X (and corresponding Y ) drag coefficient was found to be kxB = 1.25.

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VI.

Acknowledgements

The authors would like to thank Burton Smith for his assistance in tuning and testing the simulation environment, and performing rotational verification testing on an early prototype airframe. Student support for this research was funded from the UMKC Summer Undergraduate Research Opportunity Program (SUROP) and Students Engaged in Artistic and Academic Research (SEARCH) programs.

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