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System Identification of an Octocopter in Hover using Full-Harmonic Orthogonal Multisine Inputs

Justin J. Matt, George V. Altamirano

arXiv:2609.29832v1eess.SYcs.RO

TL;DR

Multirotor system identification faces challenges from constrained flight maneuvers, feedback, and the lack of multi-input, multi-axis methods addressing individual-rotor aerodynamics. This paper develops full-harmonic orthogonal multisines and input allocation, then demonstrates them in near-hover octocopter flight tests. The allocation increases yaw-rate response, and the identified models predict vehicle and motor responses.

  • Problem

    Dynamic flight testing has generally not been used to characterize multirotor aerodynamics at the individual-rotor level, and prior octocopter allocation work did not address this within multi-input, multi-axis maneuvers.

    Source: §I · p. 3

  • Method

    The method injects full-harmonic orthogonal multisines with shared frequency content and uses input allocation to adjust response magnitude about a selected degree of freedom.

    Source: §I · p. 3

  • Results

    120% average increase in yaw-rate response RMS with allocation; identified models accurately predicted power consumption, motor speed, and rigid-body forces and moments in near-hover conditions.

    Source: §V · p. 27, §VI · p. 28

  • Takeaways & Limitations

    Models parameterized primarily by individual rotor thrust and torque coefficients are suitable for analyzing aircraft flight dynamics and individual-rotor aerodynamics.

    Source: §I · p. 3, §V · p. 27

    The flight-test and system-identification results are demonstrated for GAMMA in near-hover conditions.

    Source: §I · p. 3

Abstract

from arXiv · show

A new method for multi-input flight maneuver design for system identification is presented. The method consists of injecting "full-harmonic" orthogonal multisine signals into the flight control system. Orthogonality is achieved by repeating maneuvers with changing multisine polarities. The multisines can contain the same frequency content, which can simplify frequency response estimation and allow for long flight maneuvers to be split into several shorter maneuvers while maintaining the same frequency resolution and minimum frequency. An input allocation scheme is presented that augments the multisines to size the vehicle response amplitude about a specific degree of freedom. The developed approach was demonstrated through flight testing of a small octocopter in near-hover conditions. The input allocation scheme was utilized successfully to increase excitation about the yaw axis. Electrical power, motor speed, and rigid-body dynamic models were identified and are shown to predict the vehicle and motor responses accurately. The models are parameterized primarily by rotor thrust and torque coefficients, making them suitable for analysis of aircraft flight dynamics and individual rotor aerodynamics. The results demonstrate that the near-hover flight dynamics can be modeled accurately by neglecting rotor hub moments, variations in rotor coefficients, gyroscopic moments in roll and pitch, and aerodynamic interaction effects.

Nomenclature

The paper addresses gaps in multirotor system identification and presents a multi-input maneuver design demonstrated on the GAMMA octocopter. Its introduction connects flight-test data to models for aircraft dynamics and individual-rotor analysis.

Nomenclature

  • The nomenclature defines vehicle states, body-axis forces and moments, rotor thrust and torque coefficients, and motor power and rotational speed.It also defines symbols for vehicle and rotor-system inertia, multisine period and harmonic number, and normalized residual.Source: Nomenclature · p. 1

I. Introduction

  • Constrained maneuvers and feedback control complicate multirotor identification, while prior work had not combined multi-input, multi-axis maneuvers with allocation or broadly used dynamic testing to characterize individual-rotor aerodynamics.Single-axis tests can suppress off-axis responses and miss coupling; the paper also notes that aerodynamic-characterization studies generally had not used dynamic flight testing and system identification.Source: §I · pp. 2–3
  • The proposed method uses full-harmonic orthogonal multisines with shared frequency content, making frequency-response estimation straightforward and maneuver duration independent of the number of inputs.Orthogonality comes from repeating maneuvers with changed input polarities; a PTI allocation procedure can augment inputs to adjust response magnitude about a selected degree of freedom.Source: §I · p. 3
  • The GAMMA flight tests support models based on individual rotor speeds and thrust and torque coefficients, intended for flight-dynamics and individual-rotor aerodynamic analysis.The study also models motor-speed and electrical-power dynamics from the same flight data; its results are presented for near-hover conditions.Source: §I · pp. 2–3

II. Aircraft Platform

  • GAMMA is a small octocopter with eight canted arms, brushless motors, and rotors; its ESCs record motor speed, voltage, and current.The aircraft's inertia estimates use an empty-airframe trifilar pendulum measurement and the parallel-axis theorem for the equipped vehicle.Source: §II · p. 4

III. Experiment Design

The experiment design injects test signals downstream of the flight controller to independently excite actuators. The section reviews multisine construction and practical frequency and phase choices for flight maneuvers.

III. Experiment Design

  • Test signals are summed with the PX4 control-system output so each actuator can receive an independently excited, decorrelated input downstream of feedback control.This placement addresses the tendency of feedback to suppress excitation and transient motion relevant to modeling.Source: §III · p. 5, Figure 3 · p. 5
  • Multirotor tests face weak yaw authority and control-allocation constraints, while alternating frequency assignments can make maneuver duration grow with input count.The section frames the developed design as addressing these issues when identifying dynamics as functions of individual motor speeds.Source: §III · p. 5

A. Multisine Input Design

  • A multisine sums sinusoids at harmonically related frequencies; its frequency resolution is 1/T, and flight tests often choose a minimum frequency of 2/T to measure at least two cycles.The minimum frequency and desired frequency resolution constrain maneuver duration, while practical limits can cap maneuver time.Source: A. Multisine Input Design · pp. 5–6
  • Phase optimization can reduce a multisine's peak factor, producing a lower peak-to-peak vehicle response suited to aircraft system identification.The described algorithm iteratively clips time-domain peaks while enforcing the designed amplitude spectrum in the frequency domain.Source: A. Multisine Input Design · p. 6

1. Alternating-Harmonic Orthogonal Multisines

Alternating-harmonic multisines assign different harmonics to each input, while full-harmonic multisines achieve orthogonality by changing signal polarities across repeated maneuvers. The full-harmonic approach allows shared frequency content and shorter individual maneuvers, with additional transient intervals as a trade-off.

1. Alternating-Harmonic Orthogonal Multisines

  • Alternating-harmonic multisines assign unique harmonic frequencies to each input, which simplifies data analysis but can require longer maneuvers to achieve a shared minimum frequency.For GAMMA's eight inputs, a 0.5 rad/s minimum frequency was estimated to require a 113 s maneuver.Source: §1 · p. 6

2. Full-Harmonic Orthogonal Multisines

  • Full-harmonic multisines achieve orthogonality across repeated maneuvers by selectively changing input polarities, allowing all inputs to share frequency content.The same multisine set is reused across maneuvers, with the orthogonalization matrix determining signal scaling and polarity.Source: §2 · p. 7
  • For the same minimum frequency, a single alternating-harmonic maneuver is (N_u + 1)/2 times longer; matching frequency resolution requires N_u times longer maneuvers, though total experiment duration is equal.Full-harmonic inputs can share the same minimum frequency and have frequency resolution 1/T, versus N_u/T for alternating-harmonic inputs.Source: §2 · p. 8
  • Full-harmonic designs have N_u transient intervals rather than one, so transient-response spectral degradation may be greater.The multiple maneuvers can be concatenated when duration constraints permit, addressing this difference.Source: §2 · p. 9
  • In the two-input example, alternating-harmonic inputs used one 15 s maneuver, while full-harmonic inputs used two 10 s maneuvers with Input 2 reversed between runs.The example's full-harmonic inputs shared frequency content and had a 0.6 rad/s resolution, compared with 0.8 rad/s for alternating-harmonic inputs.Source: §2 · p. 9, Figure 4 · p. 9, Figure 5 · p. 10

3. PTI Allocation

PTI allocation scales virtual roll, pitch, yaw, and heave inputs to shape vehicle response despite redundant multirotor controls. An orthogonal allocation matrix can reduce the correlation introduced by scaling, while larger scaling can degrade signal properties.

3. PTI Allocation

  • The PTI allocation scheme scales virtual roll, pitch, yaw, and heave inputs to shape response despite redundant multirotor controls.Redundant controls can make desirable axis-specific response magnitudes difficult to achieve, particularly for yaw, which may have less control power.Source: §3 · p. 10
  • The scheme transforms designed signals into virtual commands with C^-1, scales them with K, and maps them back to motor commands with C.C maps eight virtual inputs to eight motor speed commands, while K is diagonal and scales virtual-input magnitudes.Source: §3 · p. 10
  • For GAMMA, the second column of the 8th-order Hadamard allocation matrix produces a pure yaw response without other columns affecting yaw.The columns' effects are analyzed using the traditional four-axis control allocation matrix.Source: §3 · pp. 10–11
  • Scaling with K can reduce power at some inputs and frequencies, increase correlation, and make peak factors non-optimal; correlation typically rises with K's condition number.Choosing C orthogonal can minimize the correlation introduced by K.Source: §3 · pp. 10–11

B. Flight Test Procedure

The flight tests used full-harmonic multisine inputs on GAMMA in indoor hover, with yaw gain increased to obtain a sufficient yaw response. The identification methodology combines physics-based force, moment, motor-speed, and electrical-power models with measured and reconstructed flight data.

B. Flight Test Procedure

  • GAMMA's indoor hover tests used multisines with an 18.9 s period and 12% throttle amplitude, with yaw gain 2.3; two sets yielded 16 maneuvers.The tests were flown near the center of ALIFT to avoid ground effect and mitigate recirculation; roll and pitch response constrained maneuver sizing.Source: B. Flight Test Procedure · p. 11
  • Four additional yaw-only maneuvers, spanning 0.6–50 rad/s, supplied higher-frequency data for electrical-power modeling.The maximum frequency was increased gradually and motor temperature checked as a safety measure.Source: B. Flight Test Procedure · pp. 11–12

IV. System Identification Methodology

  • The physics-based model outputs rigid-body forces and moments, motor speeds, and electrical power; forces and moments are reconstructed from vehicle-state measurements and mass and inertia properties.Motor speed, voltage, and current were recorded by the ESCs, with power computed as voltage times current.Source: §IV · p. 12
  • Rigid-body forces and moments use constant rotor coefficients, with separate body-frame x- and y-force coefficients because thrust projection alone did not model those forces sufficiently.The model neglects hub moments and aerodynamic interactions; rotor thrust is treated as parallel to the body z-axis for the force and moment equations.Source: §IV · pp. 12–13
  • Motor angular acceleration is modeled from net shaft and aerodynamic torque, with shaft torque computed as mechanical power divided by motor speed.Electrical power is modeled using a linear transfer function driven by motor-speed command; insufficient data prevented a physics-based motor and ESC model.Source: §IV · p. 14

A. Equation-Error Parameter Estimation in the Frequency Domain

Frequency-domain equation-error estimation fits the rigid-body and motor-dynamics models at excitation frequencies, concatenating data across maneuvers and updating shared parameters using a Bayesian-like formulation.

A. Equation-Error Parameter Estimation in the Frequency Domain

  • Frequency-domain equation-error estimation fits rigid-body and motor-dynamics models at excitation frequencies, updating parameters shared across equations with a Bayesian-like formulation.Response and regressor data are detrended, transformed using SIDPAC's chirp z-transform with cubic interpolation and endpoint corrections, then concatenated across maneuvers.Source: A. Equation-Error Parameter Estimation… · p. 14

1. Ordinary Least Squares Regression

Ordinary least squares estimates unknown model parameters by minimizing the squared discrepancy between measured and predicted outputs. Bias terms are fitted separately in the time domain because detrending prevents their identification in the frequency domain.

1. Ordinary Least Squares Regression

  • OLS estimates unknown parameters by minimizing the sum of squared errors between measured outputs and model predictions.The regression uses measured outputs, model-predicted outputs, regressors, and equation errors.Source: §1 · p. 14
  • The parameter-estimate covariance matrix provides standard errors as the square roots of its diagonal elements.Source: §1 · p. 15
  • Bias terms cannot be identified after frequency-domain detrending, so they are fitted to time-domain model errors after parameter estimation.Source: §1 · p. 15

2. Least Squares Using Prior Information

A modified least-squares framework produces consistent shared parameters across model equations by updating estimates from prior values. Its use assumes zero-mean, uncorrelated noise with equal constant variance across equations.

2. Least Squares Using Prior Information

  • Shared parameters are estimated consistently across model equations by concatenating regressions or updating prior estimates in a Bayesian-like framework.Both approaches assume equation noise is zero-mean, uncorrelated, and has the same constant variance.Source: §2 · p. 15
  • Thrust coefficients are updated across vertical-force, rolling-moment, and pitching-moment models, while torque coefficients are shared across yawing-moment and motor-speed models.Source: §2 · p. 15
  • Rotor-system inertia is estimated from the yawing-moment model and then held constant in the motor-speed model to preserve linearity in the parameters.Parameters without prior information can be assigned infinite prior variance, equivalent to ignoring that information.Source: §2 · p. 15

3. Model Structure Determination and Evaluation of Goodness of Fit

Model structure is built primarily from first principles, with candidate regressors added or retained according to predictive fit criteria. The section also describes estimating motor frequency-response models from flight data.

3. Model Structure Determination and Evaluation of Goodness of Fit

  • Regressors are added sequentially by correlation with the output and removed if they do not reduce predicted squared error (PSE).PSE combines mean squared fit error (MSFE) with a penalty that grows with the number of model terms.Source: §3 · pp. 15–16
  • Normalized RMSE uses a normalization applicable to real and complex data that is insensitive to large means but sensitive to spikes.The analyzed data did not exhibit the spike issue noted for this normalization.Source: §3 · p. 16
  • With prior information, R^2 is a normalized squared-fit-error measure rather than the proportion of output variance explained, so it was not used to stop model-structure selection.It was still computed to provide insight into fit quality.Source: §3 · p. 16

B. Frequency Response Estimation and Output-Error Parameter Estimation

  • The power-to-motor-speed-command frequency response is estimated from flight data and fitted to a transfer function using an output-error framework.The approach was chosen because the measured relationship appeared highly linear and allows SISO model-structure inspection from the frequency response.Source: B. Frequency Response Estimation and… · p. 17

1. Frequency Response Estimation

Full-harmonic orthogonal multisine inputs support direct frequency-response matrix estimation from periodic flight data. Estimator quality depends on input conditioning, and closed-loop data can require a joint input-output method to address input-correlation bias.

1. Frequency Response Estimation

  • For full-harmonic MIMO excitation, the frequency-response matrix is estimated from output and input DFT spectra across the maneuver set.Periodic multisines have known discrete frequency content and avoid spectral leakage when an integer number of steady-state periods is measured.Source: §1 · p. 17
  • Full-harmonic orthogonal multisine design yields a low input-matrix condition number, supporting accurate inversion and lower FRM-estimate covariance.Source: §1 · p. 17
  • Closed-loop frequency-response estimates can use the joint input-output method to resolve bias from correlated inputs, but summed pilot inputs are suppressed in the estimate.Unique excitation phases across maneuver sets yield distinct nonlinear responses that are averaged, producing best linear approximations.Source: §1 · p. 17
  • FRM variance depends strongly on the input-spectrum condition number, while covariance reflects both nonlinearities and noise.The covariance can support coherence estimation; separating nonlinearity and noise requires specific experiment-design and data-analysis approaches.Source: §1 · p. 18
  • Closed-loop estimator conditioning can be ill-conditioned by the system itself, requiring experiment-design techniques to produce a well-conditioned matrix.Source: §1 · p. 18

2. Output-Error Parameter Estimation from Frequency Response Data

Measured SISO power-to-motor-speed command frequency responses are used in an output-error parameter estimation framework. Because it solves state equations in the frequency domain, the framework is limited to linear time-invariant systems.

2. Output-Error Parameter Estimation from Frequency Response Data

  • The method formulates output-error parameter estimation from measured SISO power-to-motor-speed command frequency responses.Its cost function is derived from the negative log-likelihood function.Source: §2 · p. 18
  • The framework is limited to linear time-invariant systems because it solves the state equations in the frequency domain.Source: §2 · p. 18
  • The residual is the difference between measured and modeled frequency responses; with no input noise, the cost function is the maximum-likelihood estimator.The residual spectral density is assumed constant across analysis frequencies.Source: §2 · p. 18
  • Modified Newton–Raphson minimization estimates transfer-function parameters and parameter variances or Cramér–Rao bounds using a numerically perturbed Fisher information matrix.Source: §2 · p. 18
  • Inspection of the measured frequency response and model comparison selected a second-order transfer function with one real numerator zero.Source: §2 · p. 18

V. Results

The identified models predict near-hover power consumption, motor speed, and rigid-body forces and moments, while relying primarily on rotor thrust and torque coefficients. Validation also supports neglecting several effects, although body x- and y-axis forces are less accurately captured.

V. Results

  • A yaw-rate allocation gain of 2.3 yielded 120% higher yaw-rate RMS than maneuvers flown without PTI allocation.Other accelerometer and angular-rate RMS values increased by 15%–30%, likely due to coupling.Source: §V · p. 19, Figure 8 · p. 19
  • Frequency-domain rigid-body fits closely match moments and vertical force, while horizontal forces show more error and normalized residuals appear complex-normally distributed.The residual distribution indicates that the model captures deterministic trends in the data.Source: §V · p. 19, Figure 9 · p. 20, Figure 10 · p. 20
  • Motor-speed models fit all eight motors similarly, and the results suggest they capture dynamics across the tested voltage levels.The physics-based models predict motor acceleration in response to electrical power.Source: Figure 11 · p. 21, Figure 12 · p. 21, §V · p. 22
  • Rotor thrust coefficients were about 9% below isolated-rotor wind-tunnel values, and average torque coefficients were 18% lower.The authors suggest close rotor spacing or greater wake interference from vehicle arms could explain the thrust difference; electrical efficiency was around 90% per motor.Source: §V · p. 22, Figure 13 · p. 22
  • Added translational damping and coupling derivatives have limited excitation support in lateral motion, yet their inclusion yields an unstable complex pole in linearized roll and pitch rate dynamics.The authors identify these terms as critical for control-design applications, even if their effect on predictive accuracy may be small at low hover translational velocities.Source: §V · pp. 22–23
  • Power-model frequency responses match measurements, but the power-dynamics bandwidth was not captured despite higher-frequency testing.The reported parameter standard errors were all below 11%.Source: §V · pp. 24–25, Figure 14 · p. 24, Figure 15 · p. 24
  • Held-out multisine data show close agreement between measured and predicted power consumption, motor speed, and forces and moments.The force and moment predictions have somewhat larger errors for body x- and y-axis forces than for the other outputs.Source: §V · pp. 25–27, Figure 16 · p. 25, Figure 17 · p. 26
  • The models predict power, motor speed, and rigid-body forces and moments in near-hover conditions across the operational battery-voltage range, with rotor hub moments and other effects neglected.The models primarily use individual rotor thrust and torque coefficients; the reported negligible effects include coefficient variations, aerodynamic interactions, and gyroscopic moments in roll and pitch.Source: §V · p. 27

VI. Conclusion

The full-harmonic multisine method enables simultaneous multi-input excitation and can split maneuvers without losing frequency resolution or minimum frequency. Flight tests show increased yaw response and accurate near-hover predictions from models parameterized mainly by rotor coefficients.

VI. Conclusion

  • Full-harmonic orthogonal multisines excite all degrees of freedom simultaneously; changing polarity across repeated maneuvers establishes orthogonality.Shared frequency content supports matrix-based frequency-response estimation without interpolation and allows long maneuvers to be split while preserving frequency resolution and minimum frequency.Source: §VI · p. 28
  • Yaw-rate RMS increased by an average of 120% with input allocation, while other measurements increased only slightly.The identified models accurately predicted power consumption, motor speed, and rigid-body forces and moments in near-hover conditions and at various voltage levels.Source: §VI · p. 28
  • Near-hover dynamics were accurately modeled primarily with constant individual rotor force and moment coefficients, with several rotor and aerodynamic effects negligible in the tested conditions.The study identifies rotor hub moments, coefficient variations, roll- and pitch-axis gyroscopic moments, aerodynamic interactions, and airframe effects as negligible or minimal.Source: §VI · p. 28
  • The model's physics-based structure supports comparison with models from other sources, and static test data could potentially suffice for identifying its structure.The authors propose testing and identifying models in cruise, climb, and descent as future work, where rotor interactions and non-uniform inflow make aerodynamics more complex.Source: §VI · p. 28
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