Adaptive Guidance and Control for Autonomous Formation Flight Jongki Moon∗, Ramachandra Sattigeri†, J.V.R. Prasad ‡, Anthony J. Calise §
[email protected],
[email protected] {jvr.prasad, anthony.calise} @ae.gatech.edu Georgia Institute of Technology, Atlanta, GA 30332 Abstract Autonomous formation flight is a mechanism for achieving a pre-specified formation between a group of unmanned aerial vehicles. This paper presents an approach to autonomous formation flight in a leader-follower configuration using an adaptive output feedback control technique. Using measurements of the line-of-sight range and angles, an adaptive guidance law is formulated for the follower that generates velocity commands so that the follower maintains a prescribed range from the leader in the presence of leader maneuvers. A method to integrate the guidance system with the adaptive trajectory following autopilot controller of the Georgia Tech helicopter UAV is also studied. The overall architecture for autonomous formation flight is evaluated using software-in-theloop simulations. Simulation results show that the proposed adaptive formation controller for helicopter UAVs maintains good range tracking performance in the presence of leader maneuvers.
I. Introduction Formation control of multiple unmanned aerial vehicles (UAVs) has attracted significant attention from the UAV research community. The objective of formation control is to obtain a group of autonomous agents to move together in a desired formation and accomplish desired tasks such as reconnaissance, surveillance and precision strike. To complete such tasks without human intervention and in the presence of large external disturbances or flight critical failure, one of the problems of particular interest to researchers has been the automatic control of a group of UAVs flying in close formation. Most of the research done in the recent past has focused on the coordination and station-keeping of multiple UAVs so as to maintain the relative separation and orientations between the UAVs in the formation and to track desired flight trajectories. In most autonomous formation flight (AFF) designs, communications between UAVs in formation is required. In general, information about a vehicle in formation is broadcast to the entire group or only to the adjacent vehicles in close proximity. Recently, many different approaches to communication for formation control are introduced. If an active communication ∗ Graduate
Student, School of Aerospace Engineering Student, School of Aerospace Engineering ‡ Professor, School of Aerospace Engineering § Professor, School of Aerospace Engineering † Graduate
Presented at the American Helicopter Society 63rd Annual Foc rum, Virginia Beach, VA, May 1-3, 2007. Copyright 2007 by the American Helicopter Society International, Inc. All rights reserved.
is available, formation flight controller shows a desirable performance [1 and 2]. Ref.[1] shows that wireless communication network can be used in order to get position and velocity data of each vehicle. Globally stable AFF derived in Ref.[2] uses position, velocity, heading, and leader’s input data. Even though an active communication link between the vehicles can increase the performance of the AFF control, it requires the receiver and transmitter combination to be healthy for mission completion. Since a failure of any part of the communication equipment results in a failure of entire task, passive detection methods would be much preferred. One of the methods for passive detection is to use the wake of the leading aircraft, which can characterize the relative position from the leader. In Ref.[3], a neural network (NN) is used to find the relationship between wake and position from the leader. Another passive method for detecting nearby vehicles is to use a vision sensor [4 and 5]. An omnidirectional camera in conjunction with image processing algorithms and nonlinear filters, such as an Extended Kalman Filter (EKF), can be used to estimate relative position, velocity and attitude with respect to a nearby vehicle. Refs.[6 and 7] present an approach for formation flight controller design for fixed-wing aircraft using information on relative positions obtained via the image processing. It is assumed that every vehicle in formation knows its own speed and heading and can measure line-of-sight (LOS) range and angle to an adjacent aircraft in formation. Formation guid-
ance commands are given in the form of velocities[6] or accelerations[7]. In Ref.[8], a guidance law using only LOS angle and angular rate was proposed. The developed formation control scheme divided the formation flight procedure into an approaching mode and a guidance mode. A closed form guidance method for formation flight and its stability analysis are presented in Ref.[9]. This guidance law is only valid under the assumption that all information about the speed and the path angle of the nearest vehicle is available. In this paper, assuming that the only information available is measurements of LOS range and angle to a leader vehicle and own-aircraft navigation data, an adaptive guidance law is developed for the formation flight of helicopter UAVs. The objective is for a follower aircraft to maintain a desired range from a maneuvering leader aircraft. Since a helicopter can perform its unique maneuvers, including hover, transition to cruise, and back to hover, the guidance law shown here should be suitable for these conditions. This research investigates a velocity-command guidance for formation flight and compares the results to that of approach of Ref.[10], an acceleration-command guidance. The adaptive nature of the follower’s guidance law prevents significant degradation of the tracking performance in the presence of the leader’s maneuvers. We also integrate the guidance system with the adaptive trajectory autopilot of the Georgia Tech UAV simulation tool (GUST) [17]. Software-in-the-loop (SITL) simulations are presented to evaluate the performance of the adaptive formation controller. The paper is organized as follows. In Section II, we present problem formulation for a leader-follower formation flight configuration. The details of adaptive velocity guidance are presented in Section III. Section IV deals with the integration of the adaptive guidance and the adaptive autopilot systems for a nonlinear model of a helicopter UAV. Simulation results showing the validity of the overall approach are presented and discussed in Section V. Conclusions and directions for future research are given in Section VI.
II. Problem Formulation Figure 1 shows the basic geometry associated with formation flight. It shows a formation consisting of a leader(L) and a follower(F ). The terms, VL and ψL , represent the speed and the heading angle of the leader, respectively. Likewise, VF and ψF is the follower’s speed and heading angle. The measured joint performance variables are the relative distance R and the LOS angle λA which are defined as follows:
VL
North VF Follower
λA
ψL
R
ψF
Leader
East Fig. 1
Configuration of formation flight
p (xL − xF )2 + (yL − yF )2 , yL − yF ) λA = arctan( xL − xF
R=
(1)
where x and y represent the position along North and East axes, respectively. The follower also knows its own position and velocity. The main objective of the AFF control system is to maintain relative distance as commanded. In general, there are two approaches to the problem of trajectory tracking. One way is based on the fact that the control law design for the formation flight problem exhibits a two time-scale feature like other conventional flight trajectory control problems [10 and 11]. This is because the trajectory dynamics (relative position and velocity) are relatively slower than the attitude dynamics (angular velocity and orientation). Ref.[11] shows experimental results using this concept of controller design. The alternative method uses an integrated approach wherein the guidance loop and the autopilot controller are designed simultaneously [12 and 13]. In most actual flight applications, the separate inner and outer loop design is more commonly taken because it is usually simpler and well-designed autopilot controllers are available. In this paper, we will approach the formation flight control problem using time-scale separation assumption. Hence, the overall architecture of the AFF control system consists of a guidance system and an autopilot system. Figure 2 shows the complete closed-loop system of UAV [10], which is called a two-degree of freedom design. In this research, only two-dimensional formation guidance is considered. However, the guidance law proposed here can be extended to three-dimensional formation with simple modification.
III. Guidance Law Design We assume that inputs into the guidance system are the commanded relative positions and outputs are the
Mission Objective
Guidance Law
Autopilot Controller
where Rc is the filtered range command. Then, using Eq.(5), the error dynamics can be expressed as
UAV Dynamics Measure& Estimation
Fig. 2
Overall controller architecture
velocity commands. A leader is chosen to direct the formation and can be either a piloted vehicle or another UAV. Thus the objective of the AFF controller will be to maintain a prescribed relative range with respect to the leader in the presence of leader maneuvers. The only information about the leader is the LOS data assumed to be available by means of a passive sensor system. In order to determine the velocity commands from the measured joint performance variables, we will mainly use the theory for multi-input multioutput (MIMO) adaptive output feedback, which is detailed in Refs.[14 and 15]. Relative distance control
The heading command of the follower is set equal to the LOS angle, as in a pure following situation, ψF,com = λA .
(2)
Starting with Eq.(1), the magnitude of velocity vector will be determined. The time derivative of the relative distance is given by 1 R˙ = {(xL −xF )(x˙ L − x˙ F )+(yL −yF )(y˙ L − y˙ F )}. (3) R Using Eq.(1), we can rewrite Eq.(3) as follows: R˙ = cos(λA )(x˙ L − x˙ F ) + sin(λA )(y˙ L − y˙ F ).
(4)
Since x˙ = V cos ψ and y˙ = V sin ψ for both the leader and the follower, Eq.(4) can be written as: R˙ = cos(λA )(VL cos ψL − VF cos ψF ) + sin(λA )(VL sin ψL − VF sin ψF ) = −VF cos(ψF − λA ) + VL cos(ψL − λA )
= −VF cos(ψF − λA ) + VL cos(ψL − λA ) − R˙ c .
e˙ = −αe − Vad + VL cos(ψL − λA ).
(9)
As shown in the above equation, the tracking error converges to zero as long as the NN output effectively cancels the leader velocity out. The way to design the NN will be discussed later. Remark 1. It is noticed that the speed command cannot be determined uniquely if cos(ψF − λA ) is equal to zero. This implies the range cannot be controlled using VF , and the follower should simply change the heading to avoid the singularity without changing speed. Adaptive neural network design
A single-hidden-layer (SHL) neural network is used to approximate VL cos(ψL − λA ) in Eq.(9). The result in Ref.[15] establishes a universal approximation for unknown continuous function ∆(x, u) of states and control in a bounded, observable process using a memory unit of sampled input/output pairs. Figure 3 shows the generic structure of a SHL neural network. b2 V
σ(·)
b1 σ(·)
2
μ2
· · · · σ(·)
Fig. 3
Vad = W T σ (V T μ )
σ(·) · · · ·
μn
W
1
(5)
(6)
(7)
Hence, the follower speed command can be obtained as −R˙ c + αe + Vad , (8) VF,com = cos(ψF − λA ) where α > 0 is a design parameter, and Vad is the output of an adaptive neural network (NN), designed to cancel out the leader velocity along the LOS. If Eq.(8) is used for the follower speed command, then the error dynamics in Eq.(7) reduces to:
μ1
In Eq.(5), the first term in the right-hand side is the follower velocity along the LOS, which has to be determined, and the second term is the leader velocity along the LOS, which is unknown to the follower. Furthermore, the direction of desired follower velocity is determined as Eq.(2), and the only thing left to be determined is its magnitude. In order to do this, let us define a tracking error as follows: e = R − Rc ,
e˙ = R˙ − R˙ c
3
Basis function: 1 σ ( x) = 1 + e − a ( x −c ) m
Generic structure of a SHL neural network
For arbitrary ∗ > 0, there exist bounded unknown constant weights, W and V , such that: ∆(x, u) = W T σ(V T µ) + ε(µ), |ε(µ)| ≤ ∗ ,
(10)
where ε(µ) is the NN reconstruction error, and µ is the NN input vector ¯ Td (t) y ¯ Td (t) ]T µ(t) = [ 1 u ¯ Td (t) = [u(t) u(t − d) u
T
· · · u(t − (n1 − r − 1)d)]
¯ Td (t) = [y(t) y(t − d) · · · y(t − (n1 − 1)d)]T y (11) in which n1 is the length of the time window and is generally required to be greater than or equal to the system dimension, d > 0 is a time-delay, r is the relative degree of the output, σ is a vector of squashing functions, σ(·), whose ith element is defined as [σ(µ)]i = σ [(µ)i ]. Since the guidance system has a relative degree of one here, unlike Ref.[10], the tracking error is directly available from measurements. Thus, we don’t need an observer for the tracking error dynamics. The SHL NN weights are updated on-line using the adaptive laws given in Ref.[15] ˙ = −ΓW [2(σ − σ 0 V T )eP B + kW W ] W 0 V˙ = −ΓV [2µeP BW T σ + kV V ]
where h(·) represents the dynamics of the command filter. Since the relative degree of relative distance is one, a first order command filter is used here.
IV. Integrated autonomous controller for Helicopter UAV The guidance law shown in section III is applied to the GUST as a follower. The flight architecture of the GUST has been developed by the Georgia Tech UAV program and facilitates smooth transition from software-in-the-loop (SILT) to hardware-in-the-loop (HILT) simulation, followed by flight testing. Detailed description of the GTMax configuration can be found in Ref.[16]. Figure 4 is a picture of the Georgia Tech Yamaha R-Max helicopter (GTMax).
(12)
in which ΓW > 0 and ΓV > 0 are adaptation gains for the output layer and the hidden layer, respectively. 0 In addition, σ denotes the Jacobian matrix, P is the solution of the Lyapunov equation αP + P α = Q
(13)
for some Q > 0, and kW > 0 and kV > 0 are the σ-modification gains. Pseudo-control hedging (PCH) is introduced to prevent the adaptive element of an adaptive control system from attempting to adapt to selected plant input characteristics [16]. These characteristics include the position limits, rate limits, actuator dynamics, etc. The main idea of the PCH methodology is to modify the reference command in order to prevent the adaptive law from ”seeing” these system characteristics as reference model tracking error. The reference model is ’hedged’ by an amount of the difference between the commanded pseudo-control and the achieved pseudocontrol. Note that since we are commanding velocity, the velocity dynamics are treated as actuator dynamics and hedged. The achieved pseudo-control can be expressed by −VF cos(ψF − λA ), and the hedging signal, Vh , is obtained as follows: Vh = (−VF,com + VF ) cos(ψF − λA ).
(14)
Then, the dynamics of the reference model is modified as shown below R˙ c = h(Rc , Rcom ) − Vh
(15)
Fig. 4
GTMax Helicopter UAV
Figure 5. The GTMax Helicopter.
GUST controller
An autopilot controller for adaptive trajectory tracking was presented in Ref. [17] along with flight test results. This adaptive NN-based controller has 18 inputs for the NN, 5 hidden-layer neurons, and 7 outputs for 6 rigid-body degrees of freedom and a degree of freedom for rotor RPM. It consists of an outer-loop part and an inner-loop part. While the outer-loop deals with translational (force) dynamics, the inner-loop deals with attitude (moment) dynamics. An approximate model for the attitude dynamics of the helicopter was generated by the linearizing the nonlinear model around hover and neglecting coupling between the attitude and 37the of 49translational dynamics as well as theTrajectory stabilizer rotor. InHelicopters, addition, the Adaptive Control for Autonomous JOHNSON and navigation KANNAN system is a 17 state extended Kalman filter that fuses information from sensors such as GPS, IMU, sonar, radar, and magnetometer to provide estimates of vehicle position, velocity, attitude, accelerometer biases, and terrian height.
One may address model error and stabilize the linearized system by designing the pseudo-controls as ades = acr + apd − aad αdes = αcr + αpd − αad ,
(16)
where acr and αcr are the outputs of reference models for translational and attitude dynamics, respectively. apd and αpd are outputs of PD compensators, and aad and αad are outputs of an adaptive NN in the autopilot controller. In order to obtain an approximate model for pseudo-controls ades and αdes , Ref.[17] uses the following procedure. The translational dynamics are modeled as a point mass with a thrust vector that may be oriented in a given direction as shown in Figure 5. The desired specific force along the body z axis is computed as fsf = (ades − Lbv g)3 .
(17)
The desired collective input may be evaluated as δcdes =
fsf + δctrim . Zδc
(18)
The attitude augmentation required in order to orient the thrust vector to obtain the desired translational acceleration are given by the following small angle corrections from the current reference body attitude ∆φ =
ades2 ades1 , ∆θ = − , ∆ψ = 0. fsf fsf
(19)
where q(·) is a function that express an Euler-anglesbased rotation as a quaternion [18]. The attitude dynamics are modeled as the following approximate model b B +B b (δmdes − δmtrim ) , αdes = Aω
(21)
b represents the attitude dynamics, ωB reprewhere A sents the angular velocity of the body with respect to b such that it the earth. Choosing the control matrix B is invertible, the moment controls may be evaluated as b −1 (αdes − Aω b B ) + δmtrim δmdes = B
(22)
Integration of guidance system and autopilot
The autopilot controller designed in Ref.[17] is basically a trajectory tracking controller. Thus, it requires position commands as well as velocity commands.Integrating velocity commands gives us reasonable position commands. Since we have all required commands for the outer-loop, we now determine commands for the inner-loop. Note that we use a point mass model when we design a guidance law, which means that the direction of the velocity vector represents the direction of the body X-axis. Hence, only the yaw attitude command is fed into the inner-loop as ψcom = λA . Eq.(23) summarizes translational and rotational command inputs to the autopilot controller Vcom , Z
For this simplified helicopter model, heading change
f sp
∆φ
(23) ψcom = λA ,
Figure 6 shows the overall autonomous formation flight architecture for the GUST.
yB initial
V. Numerical Evaluation
yB final
ZB final ZB initial Fig. 5
Vcom ,
θcom = 0 φcom = 0 ωcom = 0.
a des,2
g initial
Pcom =
Point mass model for Outer loop Inversion
has no effect on acceleration and hence ∆ψ = 0. These three correction angles can be used to generate the attitude quaternion correction by the outer loop. Thus, qdes = q(∆φ, ∆θ, ∆ψ)
(20)
The AFF system for the GUST is evaluated on a desktop computer using SITL simulation. The SITL simulation configuration refers to the combined simulation of the ground control station (GCS), onboard routines, and simulated sensors and vehicle dynamics on any desktop computer.In this configuration, all hardware is simulated to the level of its digital communication with other components. This configuration is useful for rigorous software testing without requiring any actual flight hardware. In the test architecture, shown in figure 7, a communication link is established between the primary and secondary flight computers using routines contained within the data communications software. When this datalink
Guidance loop Hedge
Rcom
Proportional Compensator
Command Filter
+
Vc
Dynamic Inversion
+
+ -
R, p,Vc
NN
αdes
ades
Fig. 6
Guidance Loop
V
Simulation on a single desktop computer
Integrated guidance law with autopilot of GUST.
is enabled the following information is sent from the primary to the secondary flight computer: • time onboard the primary computer • vehicle position vector expressed in inertial frame • vehicle velocity expressed in body frame • vehicle angular rate commands expressed in body frame.
Desktop Computer Sensor Emulation
State
Control
Vehicle Model
(w/ Error Model)
Sensor Raw Data
Sensor Drivers
Sensor Data
Actuator Raw Data
Other Systems
Navigation Filter
Actuator Model
State Estimate
Flight Controller
Control
Actuator Driver
Command Vector
A leader is chosen to be a ground vehicle, and it moves in the horizontal plane. Its trajectory is also assumed to be pre-determined. The GTMax is the follower in simulation. For the guidance law design, the proportional gain, α, in Eq.(8) is set to α = 0.25. The basis functions of the SHL NN are sigmoidal activation functions given by: σi (x) =
1 , 1 + exp(−ai x)
i = 1, · · ·, 8,
(24)
where ai is an activation potential. The design parameters of the adaptive NN compensator in Eq.(12) are
GCS
Fig. 7 ture
Guidance System
Software-in-the-loop evaluation architec22
chosen as follows: ΓV = ΓW = 0.5, kV = 0.1, kW = 0.2.
(25)
A first-order command filter for the range command
in Eq.(15) is implemented given by
220
1 τs + 1
200
(26)
180 Rcom Range (ft)
with τ = 3.
R
120
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80 20
600
Rc
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350 LOS angle and Heading Angles (deg)
North (ft)
160
200 0 -200 Leader -400 -600
300 250 200 150 100 50 0 -50 20
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1400 1600
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1800
Trajectory of a ground leader
The formation flight consists of a leader and a follower. Since the follower tries to align its velocity vector with the LOS, this formation flight corresponds to a pure pursuit. The leader starts from rest and accelerates upto 30 f t/sec. Then it makes three successive turns and returns to rest. The GTMax follower also starts from hover. The initial range is R0 =200f t and the range command is set equal to 100 f t. As shown in figure 8, the leader moves along a boxedshape trajectory. Figure 9 shows the simulation results without an adaptive NN compensator in the guidance system. It can be noticed that there is an offset error in range tracking. This is because the speed command is simply proportional to the range tracking error. In contrast, as shown in figure 10, if an adaptive NN is used in the guidance system, then the range response shows a good tracking performance. We can see in both figure 9 and 10 that a large initial offset in relative distance results in rapid oscillations of Euler angles during the initial phase. Figure 11 explains the reason why the NN-based adaptive guidance law works better than the proportional error based linear guidance law. The output of the NN effectively cancels out the leader velocity along the LOS.
Bank Angle (deg)
Fig. 8
200
Pitch Angle (deg)
0
200
220
10
0 Bank Angle -10 20
200
220
20 Pitch Angle 0 -20 20
Yaw angle (deg)
-800 -200
40
60
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0
-200 20
Yaw Angle
40
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Fig. 9 Formation flight results w/o NN compensator in the guidance system
Simulation results using the AFF controller based on the acceleration command guidance system are also shown in figure 12. As shown in Ref.[10], we can see that the acceleration command guidance system results in the noticeable tracking error during the turning maneuvers. This result is not surprising since acceleration commands from the guidance system are integrated twice to generate the needed position commands to the GTMax trajectory controller. In contrast, in a velocity command guidance, the velocity command from the guidance system needs to
35
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25 Rcom
160
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Range (ft)
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Rc R
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20 15 10
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5 80 20
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LOS angle and Heading Angles (deg)
350 300
Leader LOS Velocity NN out Follower Velocity
0
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Fig. 11
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250 200 150 100 50 0
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Fig. 10 Formation flight results with NN compensator in the guidance system
be integrated only once to get the required position command.
VI. Conclusions An adaptive approach to designing AFF controller for helicopter UAVs is studied. After designing a guidance law using an adaptive NN compensator, the integration of the guidance system with the adaptive autopilot of the Georgia Tech helicopter UAV has been
Fig. 12 Formation flight results using the acceleration-command guidance system
proposed. Since the only measured information related to the leader aircraft are the LOS range and angle, this technique can be applied to the AFF control system that uses a passive sensing method. The adaptive AFF controller is successfully evaluated using the Georgia Tech UAV simulation tool (GUST). Software-in-theloop (SITL) simulation results show that the adaptive
compensator in the guidance loop compensates for the unknown component of the leader’s velocity along the LOS. It is also noticed that the velocity command guidance system shows good range tracking performance in the presence of leader maneuvers. We can extend this approach to three-dimensional formation flight by modifying the joint performance variables.
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