A new modeling scheme for powered parafoil unmanned aerial vehicle platforms: Theory and experiments

Bingbing LI, Yuqing HE, Jianda HAN, Jizhong XIAO

a State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China

b University of Chinese Academy of Sciences, Beijing 100049, China

c Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110016, China

d College of Artificial Intelligence, Nankai University, Tianjin 300350, China

e The City College, City University of New York, New York 10031, USA

KEYWORDS

Abstract A novel framework is established for accurate modeling of Powered Parafoil Unmanned Aerial Vehicle (PPUAV). The model is developed in the following three steps: obtaining a linear dynamic model,simplifying the model structure,and estimating the model mismatch due to model variance and external disturbance factors.First,a six degree-of-freedom linear model,or the structured model,is obtained through dynamic establishment and linearization.Second,the data correlation analysis is adopted to determine the criterion for proper model complexity and to simplify the structured model. Next, an active model is established, combining the simplified model with the model mismatch estimator. An adapted Kalman filter is utilized for the real-time estimation of states and model mismatch. We finally derive a linear system model while taking into account of model variance and external disturbance. Actual flight tests verify the effectiveness of our active model in different flight scenarios.

1. Introduction

Powered Parafoil Unmanned Aerial Vehicles (PPUAVs) are widely accepted as a suitable platform for long-distance delivery and heavy payload transportation due to their compact structure, low costs, marvelous adaptability to various environments,and inherent stability.1-3However,system modeling for PPUAVs has been a fundamental challenge because of the following four special characteristics4:(A)Separation between the Center of Mass (CM) of the parafoil canopy and the payload produces a swing motion;(B)Changing thrust induces a considerable pitching motion; (C) Flexible connection between the canopy and the payload produces relative pitching and yawing motions; (D) Additional mass of the parafoil canopy exists because the canopy is full of air and thus any change of parafoil motion simultaneously generates changes of fluid flow.5

Over the past few decades,a series of studies have been conducted regarding the modeling of PPUAVs, and different kinds of models have been proposed.In special circumstances,the 3 Degree-of-Freedom (DOF) model is usable to describe the most important vehicle characteristics and to test the functionality of Guidance,Navigation,and Control(GN&C)algorithms.6However, this low DOF model could overlook many influential factors, for example, the abrupt change of the roll angle of the system, which is an important factor for turning motion modeling,and horizontal and vertical velocities,which are the model constants but not correct in many cases. Therefore,a 4-DOF model7is constructed to simulate the increasing sink rate during turning and the effects of symmetric deflection on velocity and lift/drag variations during steady flights. In contrast to the 3-DOF model, the 4-DOF model better presents the status of forward velocity during turning. However,in this model, the CM of the whole system is set at the point of the aerodynamic center, which reduces the precision of the model during high maneuver motions. In most references,the 6-DOF model, describing both three inertial positions and three Euler orientation angles,is utilized as the basic structure and a sufficient tool when the autonomous GN&C algorithm uses only position and velocity feedback.1Based on the 6-DOF model, a 7-DOF model is established by taking into account of the roll movement of the payload with respect to the parafoil.8Recently,much higher DOF models are developed to measure more detailed movements of the vehicles,including 8-DOF,99-DOF,10,1110-DOF,12and 12-DOF models.13As shown in Table 1, different models have been proposed while a wide array of GN&C algorithms have been developed for application.14Simpler models are usually preferred for controller design, but they may have inherent flaws in presenting practical dynamics, especially when external disturbance exists. The simplest but highly efficient models are expected, which are however unavailable because of the lack of proper criteria for model complexity selection.

Most established models in the previous studies are structured models composed of fixed structures and some parameters obtained through offline identification.20A structured model is usually derived based on proper model simplification assumptions. These assumptions, however, cannot always satisfy real-world scenarios, especially in which vehicles perform different maneuvers,thus making parameter identification difficult,if not impossible.21-23In some cases with external disturbance,it is even more difficult to utilize practical flight data to derive structured models. Furthermore, even if a structured model is obtained, it may not be accurate enough to describe all dynamic characteristics of the PPUAV system. An additional problem of this model structure is that it does not consider many environmental factors in real scenarios, which usually leads to control failure.9,24

Several methods have been proposed to deal with the unmodeled factors in real-world scenarios.For example,some researchers have adopted data-driven methods to construct a mathematical description of a dynamical system. Neural networks, fuzzy logic, and expert system methods have proven to be useful in some cases, especially the cases with process control problems.25-27However,these revised methods are difficult to be utilized in motion control of UAV systems since system stability based on the sole application of these methods cannot be guaranteed. Some data-driven methods are thus combined with model-based schemes to achieve accurate modeling. Online mode disturbance observer,28active disturbance rejection control29and active models30for unmanned vehicles have been developed to estimate unknown external disturbance and possible model mismatch. An active model is composed of a structured model with unstructured factors, which can be estimated in real cases and real time, and has been shown to be effective in many systems, such as the rotorcraft flying robot,31surface robot,32and even pneumatic artificial muscle.33This new modeling scheme is advantageous in two aspects. First, compared with normal linear and nonlinear structured models, the active model presents a more accurate description of system characteristics due to the introduction of online model error estimation technique which is used to handle external disturbance and model mismatch. Second,state mismatch, which is the difference between actual states and model outputs, can be selected as the description of the model error, so a more realistic linear active model could be derived,which makes future controller design easier and more realizable.

In this paper,an active model is developed through theoretical and experimental analysis to address the problems of model complexity selection and accurate modeling. In Section 2, a comprehensive theoretical analysis is presented,including theories of nonlinear dynamic model establishment,linearization, model simplification based on correlation analysis, and active model establishment. The linearized model is referred to as the structured model, from which a simplified model is derived by setting several parameters as zero according to the criterion for proper model complexity.The criterion is based on the correlation analysis, as described in detail in Section 4. Then an active model is obtained by combining the simplified model with the real-time model error estimator.Section 3 introduces the experimental platform used in this research,and Section 4 describes three experiments conducted.The first experiment concerns correlation analysis and linear model simplification; the second one aims at identifying the structured and simplified models; the third one validates the active model by operating the system in different flights. The experimental results indicate significant improvement of the structured model, thus proving the effectiveness of the active model developed.

Table 1 Studies on system modeling.

2. Theoretical analysis of PPUAV

2.1. Dynamic model

A PPUAV consists of a parafoil canopy,payload,GN&C system,and suspension lines, and its structure is shown in Fig.1.Deployment of the right (or left) brake causes a significant drag rise and a small lift rise on the right (or left) side of the parafoil canopy, so that the PPUAV turns right (or left) with slight roll movement. With an engine installed on the back of the payload,the PPUAV can adjust its forward and vertical velocities.34-36

Fig.1 Structure of PPUAV.

Fig.2 Coordinate systems of PPUAV.

The rotation matrix from the body frame to the inertial frame is

where φ, θ, and ψ are Euler angles.

The dynamic model of the PPUAV expressed in the body frame is

and

The aerodynamic moment on the canopy is

whereClp, Clφ, Cmp, Cm, and Cnrare aerodynamic coefficients;b and c are the canopy span and chord, respectively.

Toglia37proposes the following four assumptions of the theoretical model of PPUAVs:

(A) Apparent mass and inertia effects are negligible in near steady-state conditions;

(B) Moments of aerodynamic forces on the CM are much smaller than those of the aerodynamic rolling, pitching and yawing;

(C) Payload drag is proportional to the surface area and therefore much smaller than the parafoil drag;

(D) The contribution of the parafoil to the velocity due to different locations of the center of parafoil pressure and CM is negligible (i.e., the velocity of the parafoil is considered the same as that of the CM).

Then the nonlinear theoretical model is derived as follows:

2.2. Structured model

The structured model is derived by calculating the Jacobian linearization of Eqs.(9)and(10)around the equilibrium point

where

C=I6×6

Im×nis an m×n unit matrix; W t（ ） is the process noise;Astructuredand Bstructuredare the functions of the stable states,stable inputs, aerodynamic coefficients, and other system parameters.

2.3. Simplified model based on correlation analysis

For different PPUAV systems, their dynamic characteristics are slightly different. If dynamics is presented in the form of linear models, the differences are reflected in the values of parameters, which represent the linear correlation of state derivatives with states and inputs (hereafter the correlation).Therefore, the data-driven analysis—correlation analysis—is performed to find a proper linear model structure for our PPUAV platform.

To ensure the consistency of findings, we choose fourteen groups of experiment data in different conditions.The correlation analysis is implemented in the following steps.

Step 2. The mean and median of each line of the matrix in Eq. (16) are given as follows:

Step 3.Two bases for correlation judgment are given below:

where α is the parameter that shows the extent of separation between the omitted and retained states, and can be selected manually.

Step 4. Then Cor_matrix（i ,j） is set zero if and only if

which yields consistent results of correlation analysis of all flight data.

Finally, the correlation between x_dot（i） and x_comb（j） is determined by the value of Cor_matrix（i ,j） and is omitted if the value of Cor_matrix（i ,j） equals zero, which provides the basis for model structure simplification.

The derived model structure and the simplified model are described in Section 4.

2.4. Active model of PPUAV

The problem is solved through the introduction of active modeling. An active model consists of the simplified model and the real-time model error, which is the difference between the actual dynamics and the output of the simplified model,as shown in Fig.3. The simplified model is derived through offline parameter identification,while the model error is obtained through online estimation. The active model considers model mismatch caused by model variation and external disturbance.Furthermore, due to the linear structure of the active model,linear controllers can be applied with a regulator introduced to minimize the difference between the desired and the actual states31.

All the model drifts caused by linearization, disturbance and unmodeled dynamics can be considered as additive process noise.According to Song et al.31,when U0=0,the model error can be formulated as

The actual model of the PPUAV can be expressed as

Fig.3 Procedure of active modeling and controller design.

The state X t（ ） and the model error f t（ ） can be estimated simultaneously by ‘‘joint estimation” in which we define the extended state as

Due to the high sampling frequency (50 Hz), the model error f t（ ） is treated as a slowly time-varying vector and can be formulated as

fk+1=fk+hk

Thus, the discrete model is derived as

where

Fig.4 PPUAV structure and GN&C system.

Thus, the active model is established as

which is a combination of a simplified model and the estimated model error.

3. Experimental platform

The GN&C system of a PPUAV consists of winches,a Global Position System (GPS), magnetic compass, Inertial Measurement Unit (IMU), pitot tube, flight computer, and datatransmit module to uplink commands and to downlink data.The structure of the PPUAV is shown in Figs. 4 and 5, and the modules are described in Table 2. The parameters of the platform are shown in Table 3.

4. Experiments and result analysis

Three experiments are conducted to (A) acquire the correlation, (B) execute parameters identifications of the structured model and the simplified model, and (C) verify the derived active model. Fourteen groups of data are saved, in which a wide range of maneuvers are performed.

Fig.5 Experimental platform of PPUAV.

4.1. Experiment 1: Correlation analysis

The values of coefficient α in Eqs.((20)and(21))determine the correlation, used for system simplification. A larger α usually leads to more system simplifications and a further simpler linear model. We try different values of α for simplification and finally choose a proper one to determine the structure of the derived simplified model.

4.1.1. α=0.5

Given that α=0.5, we obtain the isolated results of one given group of data and then consistent results of fourteen groups of data.

Step 1. Correlations are obtained for one given group of data.

For one group of data,correlation analysis is carried out as shown in Figs. 6 and 7 and the correlation matrices are obtained as follows:

Table 3 System parameters.

Fig.6 Flight trajectory for correlation analysis.

Table 2 PPUAV structure.

Findings can be obtained for this group of data based on correlation judgement:

Then we calculate all correlation matrices of fourteen groups of data and get consistent results

Step 2. Consistent correlations are found of fourteen groups of data.

According to the value of Cor_matrix, four consistent correlations are shown:

4.1.2. α=0.2

After correlation analysis of fourteen groups of data,only one consistent correlation is found:

4.1.3. α=0.8

After correlation analysis of fourteen groups of data, six consistent correlations are found:

Fig.7 Flight data for correlation analysis.

In summary, the value of α determines the number of consistent correlations.It seems better that more consistent correlations can be found if we choose a larger α but not that wrong findings may be gotten if the parameter is not proper. As shown in Section 4.1.3, if we choose α=0.8, then the thrust T has no effect on the system, which is contrary to the fact and will result in a wrong system structure and poor system identification.

After several adjustments of α,we choose α=0.5.Then the four consistent correlations are utilized to simplify the structured model in Eq.(11),and the simplified model is derived as

where

Fig.8 Flight trajectory for system identification.

Fig.9 System identification results of structured model.

X（t ）= ［u（t ）-u0,v（t ）-v0,w（ t）-w0,p（t ）-p0,q（t ）-q0,r（t ）-r0］T

U（t ）= ［δa（t ）-δa0,δs（t ）-δs0,T（t ）-T0］T

C=I6×6

Aijand Bijare the functions of the stable states, stable inputs, aerodynamic coefficients, and system parameters.

4.2. Experiment 2: Model identification

Fig.10 System identification results of simplified model.

Fig.11 Flight trajectory of turning movement for active model evaluation.

The identification results of the structured and simplified models are shown in Figs. 9 and 10, respectively. The means and variances of the errors are listed in Table 4. The results demonstrate that the structured and simplified models are consistent with the actual flight data well.The errors of the simplified model are slightly larger than those of the structured model, but the results are still acceptable.

Fig.12 Estimation results of model errors.

4.3. Experiment 3: Active modeling

The active model consists of the simplified model and model error estimation, as shown in Fig.3. By choosing another group of flight data (Fig.11), we obtain the model error estimation and then validate the performance of the active model.First,the difference between the real flight data and the output of the simplified model is selected as the model error.Then the Kalman filter in Eqs. ((26) and (27)) is utilized to verify the estimator. Fig.12 shows the performance of online model error estimation. The means and variances of the differences between actual model errors and estimated results are listed in Table 5. The estimation results are stable and remarkably consistent with the actual data.

The performances of the structured model and the active model are shown in Fig.13;the means and variances of the prediction errors are listed in Table 6.The mean prediction errors of linear velocities of the active model are less than 0.003 m/s and the mean errors of angular velocities are less than 0.001 rad/s. The error variances of six states obtained from the active model are less than1/6, 1/18, 1/2, 1/3, 1/4, and 1/4 of those from the structured model. The results demonstrate that the active model is consistent with the actual and achieves much better performance than the structured model.

Table 5 Means and variances of estimation errors.

Fig.13 Performances of structured model and active model.

Table 6 Means and variances of prediction errors while performing turning maneuver.

Fig.14 Flight trajectory of stable movement for active model evaluation.

Another flight test with a maneuver of relatively stable movement is conducted to validate the performance of the active model, as shown in Fig.14. The performances of the structured model and the active model based on the test data are shown in Fig.15, and the means and variances of the prediction errors are listed in Table 7.The mean prediction errors of linear velocities of the active model are less than 0.001 m/s and the mean errors of angular velocities are less than 0.0004 rad/s. The error variances of the six states obtained from the active model are less than 1/1161, 1/1452, 1/2740,1/7, 1/18, and 1/10 of those from the structured model. These results demonstrate that the active model is consistent with the actual data and achieves much better performance than the structured model.

Fig.15 Prediction result of structured model and active model in a different flight status.

Table 7 Means and variances of prediction errors while flying in a relatively stable state.

5. Conclusions

(1) A novel linear model, or active model, of PPUAVs is proposed in this paper based on Newton-Euler equations, linearization, correlation analysis, and real-time model error estimation. Compared with the structured model, the active model has a simpler structure and is more accurate. This study provides insight into establishment of a linear model with model variation and external disturbance considered.

(2) Correlation analysis has been introduced as a datadriven method to determine the criterion for choosing proper model complexity,in order to obtain the simplest possible linear model which is equally effective.

(3) Real-time model error estimation is utilized in the active model and is achieved by means of an adapted Kalman filter.

(4) Fourteen flight tests have been conducted to obtain correlations and proved the effectiveness of the simplified and active model. A simplified model is derived according to the correlations and is proved to be effective in the real flight test. The results of different flight tests indicate that significant improvement of the structured model has been achieved,thus verifying the effectiveness of the active model.

(5) The method proposed is not limited to the application of PPUAVs, but can be utilized for other underactuated systems.

(6) Linear controllers can be applied to the active model,introducing a model error regulator.

Acknowledgements

This study was co-supported by the National Nature Sciences Foundation of China(Nos.61503369 and 61528303),the State Key Laboratory of Robotics, and the Chinese National Key Technology R&D Program (No. Y4A12081010).