Non-Parametric UAV System Identification with Dependent Gaussian Processes

Transcription

1 Non-Parametric UAV System Identification with Dependent Gaussian Processes Prasad Hemakumara and Salah Sukkarieh Abstract A mathematical model for a complex system such as an Unmanned Aerial Vehicle (UAV) requires estimation of aerodynamic, inertial and structural properties of the many elements of the platform. This physical modeling approach is labor intensive and requires coarse approximations to be made in calculations. Similarly, models constructed through flight tests are only applicable to a narrow flight envelope and classical system identification approaches require prior knowledge of the model structure, which in some instance may only be partially known. To tackle these problems, we introduce a novel aircraft system identification method based on dependent Gaussian processes. The approach allows high fidelity nonlinear flight dynamic models to be constructed through flight testing. The proposed algorithm learns the system parameters as well as captures any dependencies between them. The method is demonstrated by generating a model of the force and moment coefficients for the Brumby MkIII UAV from real flight data. The learnt dynamic model identifies coupling between flight modes, provides an estimate of uncertainty, and is applicable to a broader range of the flight envelope. I. INTRODUCTION Unmanned Aerial Vehicle (UAV) system identification involves flight testing where flight data is collected to accurately characterize the dynamic response of the platform. The models constructed can be used for modern control designs, which rely on feedback of the state vector. Hence, required flight models are differentiable with respect to the state and control inputs. These can be constructed from aerodynamic, inertial and structural characterizations of the aircraft s individual component elements such as the fuselage, wing and empennage. Aerodynamic models are based on first principles, such as the finite wing theory, or by using empirical data of a similar platform. More complex models may involve wind-tunnel tests and analytical methods such as computational fluid dynamics. Estimates of the mass and inertia typically involve coarse approximations. Thus, constructing a mathematical model for a flight is a modular systematic process that is prone to error. In system identification, test maneuvers are designed to excite the dynamic response modes of the aircraft so a dynamic model can be extracted from the logged flight data. To construct a complete model, parameters are identified for different speeds and different dynamic pressures. The This work is supported in part by the ARC Centre of Excellence programme, funded by the Australian Research Council (ARC) and the New South Wales State Government/ Australian Centre for Field Robotics, The University of Sydney. The authors are with the Australian Center for Field Robotics, The University of Sydney, Sydney, NSW 26, Australia {prasad,salah}@acfr.usyd.edu.au Fig.. The Brumby MkIII UAV. estimated coefficients are then regressed to a hypothesized model []. There has been work done on developing models using both parametric and non-parametric approaches. Parametric approaches involve techniques such as ordinary least squares [], maximum likelihood [2] and frequency domain methods [3]. However, these models are limited by requiring some a priori information about the model structure. Also, the regressed flight models are only applicable to a narrow band of the tested flight envelope. Non-parametric system identification has been developed using techniques such as neural networks [4] and apprenticeship learning [5], [6]. Nevertheless, the constructed models were confined to operate within a trained flight path or have limited robustness to uncertainty. Gaussian Processes (GP) is a probabilistic framework approach [7] that presents a generalization of probability distribution to functions. GPs have been applied to the problem of system identification from training data in [8], [9], [], []. The work in [8] demonstrated GPs and reinforcement learning could be used to identify and control a blimp. Here a GP model was used in conjunction with a non-linear model to learn the errors. The approach is limited in identifying single independent outputs and hence can not capture any dependencies between the identified parameters. The experiments were also conducted in a lab environment without any wind disturbance. These limitations can be tackled by learning multiple outputs, while capturing any dependencies. However, due to the difficulty of maintaining positive definiteness, the parameterization of the covariance function makes it difficult to deal with multiple outputs. This has previously been addressed in geostatistical literature under the name

2 co-kriging [2]. The method generally involves using an independent model for each output. An alternative approach, Dependent Gaussian Processes (DGP) was proposed in [3], [4] where Gaussian processes are treated as white noise sources convolved with smoothing kernels. This allows the Gaussian process to handle multiple and coupled outputs. The sole dependency between different tasks comes from sharing parameters of the underlying covariance function. In this paper, we show how to identify system parameters of aircraft using DGP. The method is demonstrated by generating a model of the force and moment coefficients of the Brumby MkIII UAV (see Fig. ) from real flight data. The non-parametric nature of the model enables it to capture a wide range of the dynamics. Hence, prior knowledge of the model structure is not required. The ability to capture dependencies enables the models to be applicable to a broader flight envelope than existing methods as a result saving significant costs on flight testing. It enables the model to identify any coupling between longitudinal and lateral parameters. Finally, the predictions from the model also come with uncertainty estimates which can be used in maneuver design for system identification and for flight controller design. The paper is organized as follows: Section II outlines the formulation of the force and moment coefficients; Section III describes the background theory of using dependent Gaussian processes for learning the coefficients; Section IV details the implementation procedure. We present the results of our ongoing work in Section V and Section VI concludes the paper with a discussion of future work. II. PROBLEM FORMULATION UAVs are nonlinear systems. In order for linear approximations to remain valid the system must operate over a restricted range of conditions, such as at a fixed altitude and dynamic pressure. The continuous state and output equations can be expressed as, ẋ(s) = f[x(s),u(s),w(s),s] () y(s) = h[x(s),u(s),s] (2) Here the state vector is x R n s, control input is u R n i, system output is y R n o and w denotes zero mean system noise. The entire process varies from time s to time s s The global representation of the flight model is contained in f( ) and h( ). A. Dynamics of the UAV Using Newton s second law of motion in translation and rotational forms, forces and moments can be obtained by, F G + F T + F A = m V b + ω b mv b (3) M T + M A = I ω b + ω b Iω b (4) where V b = [u,v,w] are the aircraft body axis translational velocity components, ω b = [p,q,r] are the aircraft body axis rotation rates and m m TO is the platform mass. Moments are slightly more involved based upon the aircraft s inertia Fig. 2. Brumby MKIII frames of reference, in body axes (x b,y b,z b ) and earth axes (x e,y e,z e ). matrix I, the angular rates ω b and the rate of change of the angular rates ω b. The applied forces and moments on the left hand side of equations (3) - (4) arise from the aerodynamics of the platform, gravity and propulsion. Since, the gravitational component acts through the center of gravity (C.G.) and is assumed to be uniform there is no gravity moment on the platform. Applied forces in these equations come from gravity (F G ), thrust (F T ), and aerodynamics (F A ). Applied moments are a resultant of thrust (M T ) and aerodynamics (M A ). The contribution of gravity in vector form is: F G = mgsin(θ) mgsin(φ)cos(θ) (5) mgcos(φ)cos(θ) where, Ψ b = [φ,θ,ψ] are the Euler angles. This takes into account the dependency of gravity components to aircraft orientation relative to the earth axis. The applied forces and moments due to thrust are modeled using the geometry of the installation and engine tests done on the ground. Engine thrust forces are described as, Γ rx F T = T Γ ry (6) Γ rz where Γ r( ) are the thrust-line offset rotation constants and engine thrust is described by T = f (Ω,V ), which is a function of propeller rotational rate (Ω) and true velocity (V ). Engine thrust moments are given by, F Ty Γ cgz + F Tz Γ cgy M T = F Tx Γ cgz + F Tz Γ cgx (7) F Tx Γ cgy + F Ty Γ cgx where Γ cg( ) are the thrust line offsets relative to aircraft C.G. The system identification problem then reduces to determining the aerodynamic forces (F A ) and moments (M A ). These can be expressed in terms of non-dimensional coefficients, F A = qs C X C Y, M A = qs bc l cc m (8) C Z bc n

3 where S is the wing reference area, b is the wing span, c is the mean aerodynamic chord of the wing and q = (/2)ρV 2 is the dynamic pressure, ρ is the air density and V is the true velocity. The non-dimensionalized force (C x,c y,c z ) and moment (C l,c m,c n ) coefficients depend nonlinearly on the UAV translational and angular velocity components, control inputs and possibly their time derivatives. This dependence is usually characterized mathematically using parametric system identification methods as in [], [5]. In this paper, we propose to learn these parameters via modeling them as dependent Gaussian processes. III. SYSTEM MODELING WITH DEPENDENT GAUSSIAN PROCESSES A Gaussian process (GP) is a collection of variables (y (s)...y N (s)) which have a joint distribution p(y C,{X M }) for any input {X M } where C i, j = C(s i,s j ;Θ) is a parameterized covariance function with hyperparameters Θ. The output of the GP model is a normal distribution, expressed with a mean and the variance. The mean value represents the most likely outcome and variance can be interpreted as the confidence level of the outcome. The key advantages of this approach is its ability to provide uncertainty estimates, model flexibility, and to learn noise and smoothness estimates from test data [7]. In addition, with the extension of dependent Gaussian processes [3], [4], the model can capture dependency between outputs y R N of N dimension. The following subsections detail the background theory that forms the theoretical basis of this work. A. Assumptions In formulating the problem certain assumptions about the underlying process are made, this is to satisfy the condition for the kernel function. System inputs are independent and stationary. Hence, the value of the covariance function C i, j = C(s i,s j ) between inputs s i and s j are only dependent on their distance and does not change within the input space. The underlying process has zero mean and the training data is drawn from a Gaussian white noise process. B. Modeling with Dependent Gaussian Processes The objective of dependent Gaussian processes is to infer multiple outputs jointly while capturing any dependencies. By learning outputs in parallel the performance of the GP model is improved in comparison to learning independent outputs because the outputs are derived from a common set of input sources. Consider M-input processes, x (s)...x M (s),s R D, and N-output processes y (s)...y N (s). The training set is assumed to be drawn from the noisy process represented by, y n (s) = u n (s) + w n (s) (9) where w n (s) is a stationary Gaussian white noise drawn from N (,σ 2 n ) with variance σ 2 n. u n (s) is defined by, M u n (s) = h mn (s) x m (s) () m= M = h mn(α)x m (s α)d D α () m= R D The above equation calculates the sum of convolutions of the h mn kernel connecting the input m to output n. The marginal likelihood of y given inputs X is of the form, p(y X) = N (,K(X,X) + σ 2 n I) (2) where K is the kernel matrix, which defines the connection from input m to output n. K can be described as a function cov y i j (s a,s b ) which defines the auto (i = j) and cross covariance (i j) between y i (s a ) and y j (s b ). cov y i j (s a,s b ) = cov u i j(s a,s b ) + σ 2 i (3) Given the assumption that the inputs are stationary, the kernels are likewise stationary. Thus, a separation vector can be defined as d s = s a s b. Next, cov u i j (s a,s b ) is maximized for E{u i (s a )u j (s b )}. Solving this integral results in, cov y M i j (d s) = h m= R D m j(β)h mi (β + d s )d D β (4) The kernels are set to be parameterized Gaussians, h mn (s) = v mn exp( 2 (s µ mn) T A mn (s µ mn )) (5) where the hyperparameters are defined by v mn R, A mn and µ mn R. Substituting kernel function (5) into (4) and solving the integral results in, cov y i j (d s) = where, M m= (2π) D 2 v mi v m j Am j + A mi exp( 2 (d s [µ mi µ m j ]) T Σ(d s [µ mi µ m j ])) Σ = A mi (A mi + A m j ) A m j (6) Together these define the positive definite covariance matrix K for the combined output processes N, K, K,N K =..... K N, K N,N Now, given a set of test inputs X, the predictive output f can be obtained by, ( [ ]) K(X,X p( f,y X,X) = N, ) K(X,X) K(X,X ) K(X,X) + σn 2 I (7) Since, the training output y is known, the Gaussian distribution can be conditioned on y to obtain the predictive mean and variance for X. p( f X,y,X) = N (µ,σ) (8)

4 Where, µ = K(X,X )[K(X,X) + σ 2 n I] y Σ = K(X,X ) K(X,X)[K(X,X) + σ 2 n I] K(X,X ) The formulation above assumes the underlying processes have zero mean. This is not a significant limitation, since the mean of the posterior process is not confined to be zero. The outputs were centered to have zero mean on the training set. The subtracted values were then added to the predicted outputs f. C. Hyperparameter Optimization This involves learning the appropriate hyperparameters and noise variance given the observations. Hyperparameters are free parameters of the covariance function. In this case the parameters of the kernel function are Θ = [v m,a m, µ m ]. These parameters are learned by maximizing the log marginal likelihood of the training outputs given the inputs, where, [Θ,σn 2 ] = argmax{log(p(y X,Θ,σn 2 ))} (9) Θ,σn 2 log(p(y X)) = 2 yt (K(X,X) + σ 2 n I) 2 log K(X,X) + σ 2 n I N i= n i 2 log2π (2) In GP model training the hyperparameters Θ and noise variance σn 2 are optimized. In the model hyperparameters v m and A m express the relative significance of the associated regressors, µ m expresses the relative dependence between the outputs and hyperparameter σn 2 accounts for the influence of noise. Once these are found, predictions can be made on the posterior outputs f by substituting Θ into (6) and σn 2 into (3). IV. IMPLEMENTATION This section presents the development of the model, the UAV used and the data collection procedure. A. Overview The implementation framework can be broken down into three main sections, flight testing, model training and predictions through inference as shown in Fig. 3. Flight testing involves designing maneuvers to maximize the information content and collect flight data. In the next phase the system inputs and outputs are used to learn the hyperparameters for the flight model. Lastly, the chosen DGP models are passed on to estimate the aerodynamic coefficients given the system inputs. The resultant aerodynamic forces and moments can be calculated using (8). This is then used along with the gravity vector and the thrust model to calculate the total applied forces and moments on the platform. With the new {F ( ),M ( ) } estimates and the current state vector of the UAV, state derivatives can be determined. The output can now be used for flight controller design, simulator development and to understand the UAV handling qualities. Fig. 3. Block diagram of the dependent Gaussian Processes aircraft system identification procedure from flight testing, training to model inference. B. Brumby-MkIII The Brumby MkIII is a delta winged, pusher UAV (see Fig. ). The platform has a weight of approximately 4 kg, a wing span of 2.8 m, a payload capacity of 3.5 kg and is capable of flying at 55 kts. It is controlled through rudders and elevons in each side of the delta wing. This flight vehicle is mainly used as a research testbed to demonstrate real-time algorithms for decentralized data fusion [6] and cooperative control strategies involving multiple vehicles [7]. Flight sensors include an IMU, GPS and a pitot tube. The on-board navigation system uses differentially corrected GPS to aid the onboard IMU. The data from the IMU are logged at 4 Hz and the filtered navigation solution is logged at 2 Hz. A PC4 computer setup onboard is used to log this flight data. C. Measurands System identification and DGP regression requires measurements of the aerodynamic forces and moments that the aircraft is experiencing, as well as measurements of the regressors. These forces and moments derived in (8) are modeled as the output states to be learned: y = [ ] C X C Y C Z C l C m C n (2) These output states are assumed to be zero mean processes. Forces are recorded in the body centered frame of reference using the internal IMU. Moments are calculated using the logged rotational rates ω, change in rotational rates ω and inertia of the platform. Having determined the coefficients to perform parameter estimation on, the regressors can be measured. These are: X = [ a ω ω α β ] δ (22) The state is composed of the body axis accelerations a [a x,a y,a z ], rotation rates ω [ p, q, r], change in rotation rates ω [ ṗ, q, ṙ], change in angle of attack α, change in side slip angle β, and control inputs δ [δt,δ e,δ a,δ r ]. Angular

5 δ T () δ e (deg) δ a (deg) δ r (deg) Fig. 4. Control inputs used on the training flight. rates, angle of attack and side slip parameters were nondimensionalised with respect to true velocity, wing mean geometric chord and wing span. The control deflections are throttle δ T, elevator δ e, aileron δ a and rudder δ r. The inputs from the elevon were separated into aileron and elevator commands based on parity. Filtered outputs of these regressors were used for training. It is assumed that these inputs are independent and stationary to satisfy the condition for the kernel function. D. Flight Testing and Model Training The purpose of flight tests is to design maneuvers to maximize information content in the training data. To achieve this, the system modes must be excited such that the sensitivities of the model outputs to the parameters are high. For this initial test, maneuvers were designed to excite only the lateral dynamics of the system. This involved the UAV performing orbit type maneuvers with aileron, elevator and rudder inputs. A limited control space was explored due to high responsiveness of the platform. The inputs used for training are shown in Fig. 4. The training flight was performed at an altitude of m above ground level (AGL). The platform had an average mass of kg, C.G. of.735 m and maintained an average air speed of 78.8 knots. Training data for the flight was sampled at Hz, which was determined to be sufficient to capture the dynamics of the platform. Of these experimental results, 755 points were chosen to learn the hyperparameters for the DGP model by maximizing the log marginal likelihood in (9). The model learnt was then tested against a different flight. E. Model Inference and Testing Maneuvers were designed to test the learnt lateral dynamics of the system. This involved testing over a complex flight path with lateral orbit type maneuvers as shown in Fig. 5. Both the training and test flights were performed in the Marulan flight test facility. The test flight had a total flight time of 2 minutes, an average mass of kg, C.G. of.735 m and maintained an average air speed of X Northing MGA (m) Fig. 5. x Y Easting MGA (m) x 5 Test flight path over the Marulan flight test facility kts. The flight also spanned across two different altitude regimes, m and 2 m AGL respectively. Aerodynamic coefficients were estimated for the whole flight using the trained DGP model. The estimates and the results from this test are presented in the following section. V. EXPERIMENTAL RESULTS The non-dimensional aerodynamic coefficients expressed in (8) can be divided into longitudinal and lateral modes. Longitudinal modes being forward force (C x ), downward force (C z ) and pitching moment (C m ). Lateral modes being sideways force (C Y ), roll moment (C l ) and yaw moment (C n ). The primary goal of this flight test was to learn the lateral dynamics and the longitudinal dynamics, as a secondary effect. The estimated coefficients were then compared against those measured coefficients for the entire test flight (Fig. 5). These measured parameters were calculated using equations (3), (4) and the navigation solution which was used by the Brumby MkIII flight controller [8]. The aforementioned navigation solution has been used by the flight control system to perform over 3 autonomous flights [9], including cooperative missions [7]. Hence, it provided sufficient experimental evidence to be used as the ground truth. The following subsections detail the results obtained. A. Estimated Aerodynamic Coefficients The estimated coefficients and the prediction uncertainties for a segment of the test flight are shown in Fig. 6. For the time segment between 7 s to 85 s the maneuvers were performed at m AGL. Between the segment 85 s - 9 s, the UAV climbs to its new altitude regime of 2 m AGL for rest of the test flight. For the complete duration of the flight the lateral parameters estimated are nearly identical to the measured. Longitudinal estimates are inferred through learning any coupling between the lateral modes. Even in

6 C x C y C z Uncertainty Measured Estimated x 3 C l C m C n x Fig. 6. Estimated and measured non-dimensional aerodynamic coefficients with prediction uncertainties on the test flight. regions of relatively high uncertainty it is able to infer a solution closer to the measured. This is only possible due to the µ mn hyperparameter in (6), which describes the outputs that are coupled and translated relative to each other. Given this, longitudinal parameters were estimated with nearly identical results with slight discrepancies in the pitch moment and the downward force coefficients. This result was expected, since test maneuvers were designed to extract information from the lateral modes. Apart from identifying accurate parameters, the next most useful outcome of the proposed approach is the estimates of uncertainty of the predictions. These can be used as a verification measure to validate the model s usefulness. Lack of confidence in model predictions can serve as the grounds to learn from new information. Nevertheless, it is observable in Fig. 6 that even in the regions of high uncertainty the estimates are fairly close to the measured. This again is due to the dependency learnt between parameters, thus making the model valid for a broader flight envelope. The uncertainty estimates can also be used in conjunction with a controller to provide confidence on the model. The relative errors on the estimated parameters were calculated by comparing against the measured. It was found that the median error on the lateral coefficients is ±8.88% and on the longitudinal coefficients is ±23.56% of the measured. This is given that the UAV was experiencing wind gusts of up to 4 kts, and there are propagated errors from the sensor noise. Also, the coarsely known mass and inertia properties of the platform and any unmodeled characteristics are captured in the response. In summary, the model parameters were identified at an altitude of m AGL and were demonstrated to be valid

7 up to 2 m AGL. Next, these coefficient estimates were used to predict the system states. Only the lateral directional dynamics were analyzed since the training maneuvers were designed to extract these. B. Estimated Lateral States The lateral states considered here are dependent on the aerodynamic coefficients. These include the side velocity (v), roll rate (p) and yaw rate (r), which are dependent on C y, C l and C n respectively. The estimated lateral aerodynamic parameters along with the thrust model and the gravity vector were used to find the total applied forces and moments (see Section IV-A). These were then used to calculate the current state derivatives using the equations from Newton s second law of motion in translation and rotational forms (see Section II-A). In order to calculate the states, the derivatives were integrated using the 4th-order Runge-Kutta method. V E (m/s) p (deg/s) r (deg/s) Uncertainty Measured Estimated Fig. 7. Estimated and measured lateral states. The estimated outputs were compared against the measured navigation solution. The results are shown in Fig. 7. It can be seen that the propagated error from the coefficients to the system states are negligible. The median error for the estimated East velocity was calculated to be.47 m/s, for roll rate it was.89 deg/s and.62 deg/s for yaw rate. Hence, a very promising result. The relatively high error in yaw rate could be a resultant due to poor observability. This could be taken into account in future flight test maneuver design. VI. CONCLUSION AND FUTURE WORK In this paper, we introduced a novel system identification approach for UAVs based on dependent Gaussian processes. The proposed model is non-parametric and alleviates the labor intensive mathematical modeling process. The model was demonstrated through training and testing with real flight data from the Brumby MkIII UAV. 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