Nonlinear Control Approaches for an Autonomous ...

7th AIAA Aviation Technology, Integration and Operations Conference (ATIO) 18 - 20 September 2007, Belfast, Northern Ireland

AIAA 2007-7782

Nonlinear Control Approaches for an Autonomous Unmanned Robotic Airship Ely Carneiro de Paiva∗, Fábio Benjovengo+, Samuel Siqueira Bueno± Research Center Renato Archer Rod. D. Pedro I, km 143,6 – 13082-120 – Campinas - Brazil [email protected] José Raul Azinheira#, Alexandra MoutinhoΘ Instituto Superior Técnico - IDMEC / IST Av. Rovisco Pais 1049–001 Lisboa - Portugal [email protected] [Abstract] This paper presents the research developments for the global nonlinear control of an autonomous airship, covering the full flight envelope from hovering to aerodynamic flight. The preliminary reports for the three nonlinear control solutions under investigation were presented, that are Dynamic Inversion, Backstepping, and the Sliding Mode Control, along with some representative simulation results. A complete airship mission, with vertical take-off, path tracking, hovering, and vertical landing was successfully simulated using the Backstepping Aproach.

B

I. Introduction

ESIDES their use as military surveillance platforms, Unmanned Aerial Vehicles (UAVs) have a wide spectrum of potential civilian applications as observation and data acquisition platforms. They can be utilized in several environmental monitoring applications related to biodiversity, ecological and climate research and monitoring. Inspection oriented applications cover different areas such as mineral and archaeological prospecting, agricultural and livestock studies, crop field prediction, land use surveys in rural and urban regions, and also inspection of manmade structures such as pipelines, power transmission lines, dams and roads. UAV gathered data can also be used in a complementary way concerning information obtained by satellites, balloons, manned aircraft or on ground. Most of the applications before cited have profiles that require maneuverable low altitude, low speed airborne data gathering platforms. The vehicle should ideally be able to hover above an area, present extended airborne capabilities for long duration studies, take-off and land vertically without the need of runway infrastructures, have a large payload to weight ratio, among other requisites. For this scenario, lighter-than-air (LTA) vehicles are better suited than airplanes and helicopters1 mainly because: they derive the largest part of their lift from aerostatic, rather than aerodynamic forces; they are safer and, in case of failure, present a graceful degradation; they are intrinsically of higher stability than other platforms. In this context, Project AURORA - Autonomous Unmanned Remote mOnitoring Robotic Airship – was proposed2. AURORA focuses on the establishment of the technologies required to substantiate autonomous operation of unmanned robotic airships for environmental monitoring and aerial inspection missions. This includes sensing and processing infrastructures, control and guidance capabilities, and the ability to perform mission, navigation, and sensor deployment planning and execution. Aiming the autonomous airship goal, aerial platform positioning and path tracking should be assured by a control and navigation system. Such a system needs to cope with the highly nonlinear, flight-dependent and underactuated airship dynamics, ranging from the hovering flight (HF) to the aerodynamic flight (AF) or cruise flight. Hovering flight is defined here as a flight at low airspeed condition. ∗

Research Engineer, DRVC/CenPRA, Rod. D. Pedro I, km 143,6 – 13082-120 – Campinas – Brazil. Graduate Student, DRVC/CenPRA, Rod. D. Pedro I, km 143,6 – 13082-120 – Campinas – Brazil. ± Research Engineer, DRVC/CenPRA, Rod. D. Pedro I, km 143,6 – 13082-120 – Campinas – Brazil. # Professor IDMEC / IST Av. Rovisco Pais 1049–001 Lisboa - Portugal Θ Professor IDMEC / IST Av. Rovisco Pais 1049–001 Lisboa - Portugal +

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Basically, two main approaches can be considered for the automatic control and navigation system of an airship. The first one relies on the linear control theory to design individual compensators to satisfy closed-loop specifications, based on linearized models of the airship dynamics. One important result of the linearization approach is the separation of two independent (decoupled) motions: the motion in the vertical plane, named longitudinal, and the motion in the horizontal plane, named lateral. The linear based controllers may be a sufficient solution for some kind of specific applications, especially in the aerodynamic flight1,3. We can cite the experimental results obtained for the AURORA airship for path following in AF (and with constant airspeed) through a set of predefined points in latitude/longitude, along with an automatic altitude control reported in Ramos et al.4 and de Paiva et al.5 However, for applications where the airspeed (and thus the dynamics) has a large range of variation, the linearized controllers may not yield a good performance. The second approach for the airship automatic control system consists on the search for a single global control scheme covering the full flight envelope from HF to AF. For security reasons, simplicity and flexibility, a global nonlinear control is more interesting than a linearized one, and may lead to better performance results. The main challenges for the control system are the nonlinear and abrupt behavior change of the dynamics between the HF and AF zones, the very different dynamics in low and high airspeeds, and the modeling uncertainties. Three nonlinear control solutions are under investigation for the AURORA airship at present, namely Dynamic Inversion11,12,13, Backstepping6,19, and the Sliding Mode Control7. The preliminary reports for each of these methodologies are presented in the sequel along with some simulation results. In a near future, the one presenting the best results will be implemented in the airship onboard system for flight tests and validation.

II. Dynamic Inversion A Dynamic Inversion is a methodology to design closed-loop control laws for nonlinear systems8,9, searching for a global controller from a nonlinear model of the plant. Its application in flight control10 is justified since it can explicitly address the nonlinearities in the aircraft dynamics and provides a control law that is valid over the entire flight envelope. By using dynamic inversion, the set of existing deficient or undesirable dynamics are cancelled out and replaced by a designer selected set of favoured dynamics. This is accomplished by careful algebraic selection of a feedback function, reason for which the dynamic inversion methodology is also called Feedback Linearization. The developments of the Dynamic Inversion approach for the AS800 AURORA airship are being carried out by the researchers of the Instituto Superior Técnico - IDMEC / IST of Lisbon, Portugal. The preliminary investigation of dynamic inversion considered the airship equations of motion compounded by linearized dynamics and nonlinear kinematics. This lead to an airship control design still divided into longitudinal and lateral motions11,12. Yet, a nondecoupled approach that considers the airship nonlinear dynamics is possible and is currently being developed, having obtained so far good results for the AURORA airship stabilization problem13. A fundamental assumption in the dynamic inversion methodology is that the plant dynamics is perfectly modeled and may be cancelled exactly. In practice this assumption is obviously not realistic, and the robustness of the closedloop dynamics must be secured in order to suppress any undesired behavior due to plant uncertainties. For this reason, the stability of the dynamic inversion control system of the AURORA airship is analyzed applying Lyapunov’s stability tools by Moutinho and Azinheira12. In the same work, the closed-loop robustness to wind disturbances, as well as to uncertainties in the model parameters, is verified. After some simulation tests where either different models were used for the dynamic inversion controller synthesis or wind disturbance was present, good results were still obtained. Moreover, the more significant model parameters are pointed out, for which a more careful identification is necessary.

III. Backstepping Another nonlinear control approach under development for the AURORA robotic airship is the Backstepping technique14,15, which is a Lyapunov-based nonlinear control design approach that presents important robustness properties against unmatched uncertainties, like unmodelled dynamics. By formulating a scalar positive function of the system states and then choosing a control law to make this function decrease, the stability of the resulting nonlinear control system is assured. Backstepping has recently been considered in different flight control applications. Mahouny et al.16 have applied it to helicopters control, while Beji and Abichou17 and Hygounenc and Souères18 have simulated it with airships.

The developments of the Backstepping approach for the AS800 AURORA airship are being carried out in partnership with Instituto Superior Técnico - IDMEC / IST of Lisbon, Portugal. The first simulation results were reported in the AIAA - Journal of Guidance, Control, and Dynamics6, and further improvements will appear soon in the Journal of Robust and Nonlinear Control (IJRNL)19. In a first attempt, a backstepping nonlinear control law was developed just for the stabilization of the AURORA robotic airship6. The controller is based on a nonlinear dynamic model valid for the hover flight over a ground target. Prior to the application of the Backstepping formulation, a new and synthetic modelling of the airship dynamics was introduced, using a modified quaternion formulation of the kinematics equations. The saturation of the control signals is considered in the design, and a guidance strategy is proposed to deal with the airship lateral underactuation in face of wind disturbances. A brief description of this first Backstepping aproach (for stabilization, or hovering control) is given in the sequel. Initialy, a new analytical and compact model for the airship dynamics and kinematics was developed using the quaternions, and is given as:

x = Kx + M −1 (ESg + F + f ) η = DCx S = −Ω S

(1)

3

 = DΩ D 7 where x = [ v , ω ] is the state vector composed by the linear and angular airship inertial velocities represented T

T T

in the local framel, η = [p , q ] is the airship position error described in cartesian coordinates of the earth frame (p), together with the airship attitude represented by the quaternions (q), M is the Inertia matrix, S is the transformation matrix from earth to local reference, g is the gravity vector, F represents the aerodynamic forces, T

T T

f = [FuT , TuT ]T is the vector of forces and torques generated by the actuators (tail surfaces and motors), and Ω 3 = ω × and K = −M −1Ω 6 M (For further details the reader is referred to Azinheira et al6.) The required stabilization corresponds to the control objective where both the velocity x as well as the position error η are regulated to zero (hovering condition). Assume the intermediate variables y1 and y2 as:

y 1 = aη + bTx  y 2 = ∆x

(2)

where a, b are positive scalars, ∆ is a diagonal positive definite matrix, and T is defined as T = DC . Let us consider the following candidate Lyapunov function:

W =

1 T 1 y 1 y 1 + y T2 y 2 2 2

(3)

If the control law in (1) is assumed as:

f = −MT + ( ρ 2 Λη + Λ 12 Tx) − M (CT Ω 7 C + K )x − ESg − F where

ρ 2 = a / b , T+

is the pseudoinverse of T , and Λ 1 is defined as Λ 1 = 2

(4)

ρ 2 I 7 + (1 +

1 ) Λ , then it b2

can be shown6 that the derivative of the Lyapunov function considered is negative definite, and thus the system is assimptotically stable. Some simulation results for the ground hovering flight with Backstepping control are shown in the sequel. We use the fully nonlinear platform developed in the AURORA project1,4 to simulate the motion of the AURORA

airship prototype weighting 33kg and with a volume of 30m3. The case described here corresponds to the stabilization of the airship at (Nr=0m, Er=0m) and constant altitude. The airship starts from an initial horizontal error (Ni=−25m, Ei=5m), where N and E stand for the North and East position. For illustrative purposes, the attitude of the airship is also set out of equilibrium, with 10º in each one of the Euler angles. We shall examine the airship behavior when in the presence of a constant wind of 3m/s blowing from North, plus a turbulent gust of 3m/s. This is an intermediate condition with a relatively high turbulence, with wind heading changes reaching 40º, but one may consider the airship should achieve stabilize in these conditions, for ground or target observation.

Figure 1: Simulation results of backstepping stabilization of the airship in face of wind disturbances: North x East plot, Euler angles, aerodynamic variables, and control inputs. The results of this simulation are presented in Fig. 1. The horizontal North−East path is represented on the left upper corner. We can observe that the reference shaping leads to a priority in the alignment of the airship against the wind. When the airship reaches the target point, the wind input results in a mostly lateral oscillation around 2m wide. The influence of the turbulent disturbance is particularly visible for the aerodynamic variables (airspeed Vt, angle of attack α and sideslip angle β) depicted below the N−E path. With airspeed oscillations above 1m/s, the aerodynamic angles go up to almost 15º. The longitudinal control is still quite good, suffering very little with from the disturbance. The lateral control is more difficult, which is simply explained by the high aerodynamic lateral forces due to the wind inputs, being the airship clearly underactuated in this axis. The overall stabilization objective is still achieved, and namely the roll φ and pitch θ angles are very well regulated as seen on the right of the N−E path. As an evolution to this previous work on Backstepping control for the hovering flight, a further development has been made19. The new approach allows to describe both the airship stabilization and the path-tracking in a single and global design frame. The knowledge of wind disturbances is included based on a linear wind estimator assuming constant wind velocity, while the airship dynamics is expressed as function of the air velocity. Based on the desired ground velocity and estimated wind, a reference air velocity and attitude is obtained, which is valid both for

hovering and aerodynamic flights. In order to evaluate the new approach, representative simulation tests were performed using a complete airship mission, consisting of vertical take-off, path tracking under turbulent wind, hovering over a target, and finally, vertical landing. The results confirming the strong and robust tool for the nonlinear airship control will appear soon in the Journal of Robust and Nonlinear Control (IJRNL) 19.

IV. Sliding Modes The Sliding Mode Control (SMC) is a class of nonlinear, variable structure method, presenting one or more discontinuities in the space of states of a system20. Therefore, the structure of the system is changed or switched each time the state crosses this discontinuous surface, called sliding surface. A possible drawback is that the control signal tends to switch around the zero error region giving a high frequency input to the control actuator, called chattering. However this can be avoided by the use of a soft (or pseudo) switching structure. It is a powerful control technique yielding high robustness to model uncertainty and external disturbances. SMC is commonly used in the control of ROVs21, and also in the aeronautics field22,23. In the sequel the derivation of the sliding mode control law for the longitudinal and lateral modes of the AURORA airship for path tracking purposes7 is presented. The MIMO SMC developed in this first phase, for path tracking purposes, is adapted from the approach presented in Healey and Lienard21. Two controllers are derived (for the longitudinal and the lateral modes) based on linearized models of the airship. Although the design of the SMC controllers relies in the model linearization for a given airspeed, the results are however applied to a wide range of airspeeds due to the high robustness of the nonlinear switching control. In addition, the lateral controller uses the trajectory error generator introduced in Azinheira et al.3 The path tracking strategy minimizes the angular and distance tracking errors with respect to a given reference trajectory. A methodology is presented for the derivation of the sliding surface, and the bounded stability is proved. In order to evaluate the approach, a representative simulation test was performed, using the 6 DOF airship simulation environment, where the airship is submitted to a large variation in airspeed. First, let us rewrite the airship dynamics (either longitudinal or lateral), and the kinematics by the following equations:

 v = f (v,p) + g (v,p)u  p = h(v,p)   y = v p    

(5)

where v and p are the velocity and position state vectors, with f,g,h as nonlinear functions, and u is the control input vector.

~ T ]T The output y is required to follow a given reference y com , such that the error e = y − y com = [ ~v T p tends to zero with global stability. The objective of the design of the SMC is to define the sliding hyperplane S, and a discontinuous control law ensuring that the error state attains a sliding mode. The hyperplane in the error space is given by: ~ ~) = Se = [S S ] v  σ(~ v, p 1 2 ~  p  Let us assume the linearized error dynamics/kinematics for a given trimmed condition as ~ v   A 1 ~  =  p   A 3

A 2  ~ v  B  ~ + u + δf   ~ A 4  p   0 

(6)

where δf denotes the differences between the linear and nonlinear models. And let us propose the following control law for the system as

~(t ) = −[S B ]+ [S u 1 1

 A S 2 ] 1  A 3

A 2  ~ v      ~ A 4  p  

(7)

− [S 1 B ]+ F(σ ) = u equivalent + u switching

~ (t ) is the control effort comprised of two parts, a linear where the superscript “+” denotes the pseudoinverse, and u equivalent control uequivalent, and a nonlinear switching term uswitching in F(σ ) , whose components are defined as ~ )) Fi (σ ) = η i sgn(σ i (~ v, p

(8)

where ηi are positive constant parameters. Supposing that matrix B is full rank, we can select matrix S1 as identity, and the substitution of the control law (7) in (6) leads to ~ v  − S 2 A 3 ~  =  p   A 3

− S 2 A 4  ~ v  F(σ ) − + δf   ~ A 4  p   0 

(9)

such that ~) = [S σ (~ v, p 1

~v  S 2 ]~  = − F(σ ) + δf p 

(10)

The global stability of σ will now be proved. Let us consider the Lyapunov function V(σ ) =

1 T ~ ~ ~ ~ σ ( v, p)σ ( v, p ) 2

where V is positive definite. Then, from (8) and (10), and considering sufficiently high gains ηi, the global assimptotic stability of the system is assured as d V(σ ) = σ T σ < 0 dt

However, to avoid the usual chattering problems, associated to the switching control, it is necessary to redefine F(σ ) in (8) as ~) / φ ) Fi (σ ) = η i tanh(σ i (~v, p i

(11)

where the parameters φi are used to adjust the chattering smoothing. And in this case, the assimptotic stability turns into a bounded stability. Now, let us derive the system dynamics in the sliding surface, when we have σ = 0 . In this case, the substitution of F(σ ) = 0 in (9), and supposing matched dynamics (δf = 0) leads to ~ v  − S A e = ~  =  2 3 p   A 3

− S 2 A 4  ~ v   ~ A 4  p 

Such that in the sliding condition, not only σ , but also σ will vanish, as σ = [S 1

S 2 ]e .

Therefore, the dynamics of the system in the sliding surface is given by the choice of matrix S2. As we have σ = 0 ,  = −S 2 p , such that the dynamics of p can be specified by the closed as well σ = 0 and S1=I, we can consider v loop matrix A c or

p = A cp

where A c = ( A 4 − A 3 S 2 )

Assuming that the pair (A4,A3) is controllable, then S2 can be determined through pole placement or LQR solution. Therefore, the dynamics of the vehicle in the sliding condition has the same number of integrators as the number of control actuators.

Thus, with S1 and S2 determined, the control law in (7) is defined. The choice of parameters ηi and φi depends on the level of activity desired for the respective actuators input, and the level of the reducing (smoothing) in chattering. Therefore, the closed loop system dynamics is represented by a sliding manifold, that is a hyperplane in the error space, which the controller strives to converge the system toward. The switching term is effective when the system diverges from this zero sliding surface, forcing it to come back (despite the unmodeled dynamics and nonlinearities), and the equivalent controller is effective when upon the sliding manifold. In order to validate the approach, a simulation test using the Sliding Modes control approach was performed, using a 6 DOF nonlinear model based simulation environment (de Paiva et al., 1999). The path to be followed is a rectangle with sides of 200m and 300m (Figure 4). There is a constant wind of 3 m/s speed blowing from North. The groundspeed should be regulated to 5 m/s and the airship height to 100m. The condition is somehow aggressive as the airship airspeed varies from 2 m/s to 8 m/s, with a large variation in the airship dynamics (from HF to AF). Previous results with PID controllers were obtained for constant airspeed in AF only (de Paiva et al., 2006). The longitudinal SMC controller should regulate the groundspeed and height, while the lateral SMC controller is set to minimize the path tracking errors. The parameters and matrices (linearized for an airspeed of 5 m/s) of the longitudinal control law according to (7) are: v=[u w q]T

p=[θ h]T

η=[2 2 20]T

u=[δ e XΤ δ v ]T

φ=[0.25 0.25 0.25]T

- 0.061 2.208 - 0.065 0.033  - 0.029 - 0.436 5.740 0.004  A = − 0.016 0.132 − 3.448 − 1.200  0 1 0  0  0 −1 0 5 

0 0 0  0 0

0.004 − 0.592  0.328 0  1 0 0 0  B = − 0.565 − 0.015 − 0.226 S = 0 1 0 − 5 − 0.8   − 0.528 0.001 − 0.111 0  0 0 1 1

The parameters and matrices (linearized for an airspeed of 2 m/s) of the lateral control law according to (7) are: v=[r]

p=[ε δ]T η=[0.8]

− 0.47 0 0 A =  − 1 0 0  0 20 0

u=[δ r TY ]T φ=[0.15]

B = [− 0.029

− 0.020 ]

S = [1 − 1.8 − 0.04]

The path tracking results are shown in Figure 4, together with the main states and control inputs. The airship flies in a clockwise direction beginning from (0,0). The path is followed with reasonable accuracy except for the corners numbered as 3 and 4, where the wind has a backward incidence on the airship reducing the tail authority. The rudder input (δr) is shown to saturate in the low airspeed region of the path, when the tail thrust (TY) becomes fundamental for the control. It also exhibits some oscillatory behavior in other regions, mainly due to the abrupt airspeed changes (that could be diminished by the use of smooth transition paths, like arcs of circles). The groundspeed (u) and the height (h) are well regulated except for the 4th corner where the abrupt change of direction with low airspeed generates a great sideslip angle with some loss of velocity that is quickly recovered. The airspeed curve (Vt) confirms the large range of values assumed. Note that although the design of the SMC controllers are based upon linearized models for a given airspeed, the results are however applied to a wide range of airspeeds due to the high robustness of the sliding surface.

2

3

1

4 2

3

4

1

Figure 4. At left: Path tracking results of simulation. At right: Main Signals: groundspeed (u), height (h), airspeed (Vt), rudder deflection (δr), tail thrust (TY).

V. Conclusion In this paper the authors presented the research developments for the global nonlinear control of an autonomous airship, covering the full flight envelope from hovering to aerodynamic flight. Three nonlinear control solutions are under investigation for the AURORA airship at present, namely Dynamic Inversion, Backstepping, and the Sliding Mode Control. The preliminary reports for each of these methodologies were presented along with some simulation results. For the Backstepping approach it was possible to simulate a representative flight test using a complete airship mission, consisting of vertical take-off, path tracking under turbulent wind, hovering over a target, and finally, vertical landing. The results confirmed the strong and robust tool for the nonlinear airship control. In a near future, the nonlinear control approach presenting the best results will be implemented in the airship onboard system for flight tests and validation.

Acknowledgments The present work was sponsored by the Brazilian agencies Fapesp under grant n. 04/13467-5, and CNPq under grants n. 150181/2003-5, and 382739/2005-1. It is also in the scope of the cooperative agreement CNPq/GRICES, with the SISROB Project n. 490769/2006-3. The work was also partly developed in the frame of project DIVAPOSI/SRI/45040/2002, approved by the Portuguese Ministry of Science FCT and POSI program, with support from FEDER European Fund.

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