- Email: [email protected]
Nonlinear Decentralized Model Predictive Control Strategy for a Formation of Unmanned Aerial Vehicles
2nd IFAC Workshop on Multivehicle Systems The International Federation of Automatic Control Espoo, Finland. October 3-4, 2012
Nonlinear Decentralized Model Predictive Control Strategy for a Formation of Unmanned Aerial Vehicles A. Freddi ∗ S. Longhi ∗ A. Monteriù ∗ ∗
Dipartimento di Ingegneria dell’Informazione – Università Politecnica delle Marche, Ancona, Italy (email: [email protected], [email protected], [email protected]) Abstract: Aerial vehicles flying in formation may perform more complex tasks than vehicles flying independently. Formation control, however, is a difficult problem to cope with for these kind of vehicles, since they are highly nonlinear, underactuated and the feasible control actions are constrained. Moreover the computational capabilities which can be mounted on aerial vehicles are usually limited, thus centralized solutions should be avoided. To solve these problems, a Nonlinear Decentralized Model Predictive Control algorithm is presented in this paper, taking into account both physical and actuation constraints. Each vehicle is equipped with sensors providing inertial measurements and communicates with its neighbours using a Wireless Local Area Network. Simulation results applied to a formation of quadrotor vehicles show that the proposed technique is a valid way to solve the control problem for a generic formation, granting at the same time the possibility to deal with constraints. Keywords: Predictive Control, Co-operative Control, Autonomous Vehicles, Decentralized Systems. 1. INTRODUCTION
Coordinated formations of Unmanned Aerial Vehicles (UAVs) can provide significant benefit with respect to single vehicle in a number of applications, including photogrammetry, sampling and surveillance. The formation control problem is challenging for these kind of vehicles since they are highly nonlinear, underactuated and the feasible control actions are constrained. In order to minimize control complexity and allow easy reconfiguration of formation, decentralized solution are usually preferred to the centralized ones. Different decentralized solutions to the formation control of multiple autonomous vehicles have been developed (Keviczky et al. (2006); Dunbar and Murray (2006); Fang and Antsaklis (2006)). A decentralized Model Predictive Control algorithm for formation keeping have been developed, analyzed and compared with the centralized solution by Vaccarini and Longhi (2007a,b) for motion on the horizontal plane, and applied to the control of a fleet of unmanned underwater gliders in Fonti et al. (2011).
In the last decades unmanned aerial, underwater and ground vehicles have generated considerable attraction due to their strong autonomy and ability to perform relatively difficult tasks in remote, uncertain or hazardous environments where human beings are unable to go. The purposes of such vehicles are extremely various, ranging from scientific exploration, data collection and remote sensing, to provision of commercial services, military reconnaissance and intelligence gathering. A number of unmanned systems have become available and research is ongoing in a number of areas that will significantly advance the state of the art in unmanned vehicles technology. Moreover designers have more freedom in the development of such vehicles, not having to account for the presence of a pilot and the associated life-support systems. This potentially results in cost and size savings, as well as increased operational capabilities (Lyon (2004); Valavanis (2007); Mohr and Fitzpatrick (2008); Bethke et al. (2008)).
In this paper the formation control of a system of multiple unmanned aerial vehicles moving according a trajectory is addressed. The approach is based on a leader-follower architecture: each vehicle has to keep a reference distance from its leader, where the main leader follows the desired trajectory. The algorithm implements a Nonlinear Decentralized Model Predictive Control (ND-MPC) technique which takes into account physical and predictive model constraints. The single control agents communicate using a Wireless Local Area Network (WLAN) in which each neighbour exchanges information about its current state and predicted control efforts.
From a formal point of view, Unmanned Vehicles (UVs) can be defined as vehicles that can accomplish a task without the aid of a human guide. To do that, they rely on sensors, that provide information about the external environment or the internal system states; actuators, that physically realize the desired motion; controller, that drives the actuators according to the measurements and the task that need to be accomplished. With an increasing availability of modern sensors and effective communication channels, coordination of unmanned vehicles has become feasible and attractive at the same time. 978-3-902823-15-1/12/$20.00 © 2012 IFAC
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2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
The proposed approach extends in three dimensions previous results obtained for the motion on the horizontal plane only (Vaccarini and Longhi (2007a,b)), and is applied for formation keeping and trajectory planning of unmanned aerial vehicles.
N + 1 frames are used to study the formation motion (see Fig. 1): a frame integral with the earth {R} (O, x, y, z), which to be inertial, and N body-fixed frames i is supposed i i i i RB (OB , xiB , yB , zB ), where iO B is fixed to the center of mass of the i-th vehicle. RB is related to {R} by a T position vector ξ i = xi y i z i , describing the position i of the center of gravity of vehicle V i (i.e. OB ) with respect to {R} and by a vector of three independent angles η i = i i i T φ θ ψ , which represent the orientation of the vehicle i (i.e. the orientation of the body-fixed frame RB ), with respect to the earth frame {R}, using the so-called yaw, pitch and roll notation (referred to as Euler angles, Fossen (2011)).
The paper is organized as follows. Section 2 describes the physical requirements which the aerial vehicles must posses in order to apply the algorithm. Section 3 defines the kinematic model, the formation vector and how to use them for the proposed control algorithm. Section 4 provides the results of a simulated scenario in which a formation of unmanned quadrotor vehicles is adopted. Conclusions and future works are finally provided in Section 5. 2. UAV REQUIREMENTS
j
yB
The control algorithm presented in this paper can be applied to a wide class of unmanned aerial vehicles as long as they possess a minimum set of requirements.
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Sensors: each vehicle must be equipped with a heading sensor, an altitude sensor and either a Global Positioning System (GPS) or a vision system for outdoor/indoor absolute localization (i.e. linear displacements referred to an inertial reference system). Communication: vehicles must be connected to a WLAN to exchange sensor information and predicted control efforts; the required bandwidth is low since the algorithm requires only a limited amount of information (i.e. measurements and control effort predictions among neighbours) to be transmitted. Low-level controller: each vehicle must posses an inner controller for the internal dynamics which can track a reference velocity vector, as described in Section 3. When these requirements are satisfied, then it is possible to apply a ND-MPC control algorithm for formation control and trajectory planning of UAVs as described in the following section.
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Fig. 1. The reference frames adopted: each agent has a solidal BF whose origin is in the center of gravity. T T The vectors ξ i = xi y i z i and η i = φi θi ψ i fully describe, respectively, the translational and the rotational movement of the i-th vehicle with respect to the earth frame. Let define the rotation matrix which maps the linear velocity of V i from the i-th body frame into the earth frame as # " Cθi Cψi Cψi Sθi Sφi − Cφi Sψi Cφi Cψi Sθi + Sφi Sψi R , Cθi Sψi Sθi Sφi Sψi + Cφi Cψi Cφi Sθi Sψi − Cψi Sφi −Sθi Cθi Sφi Cθi Cφi (1) where S(.) and C(.) represent sin (.) and cos (.), respectively.
3. PROBLEM STATEMENT The control strategy presented in this paper aims to solve the problem of formation keeping for unmanned aerial vehicles flying in formation. The problem is addressed using a ND-MPC algorithm for agents flying in leader-follower formation: each vehicle flies at a fixed and predefined distance from its leader, where the main leader follows a desired trajectory.
Define the matrix which maps the angular velocity of V i , expressed in the coordinates of the i-th body frame, into the rate of change of the Euler angles as " # 1 Sφi Tθi Cφi Tθi −Sφi W , 0 Cφi (2) 0 Sφi /Cφi Cφi /Cθi where T(.) represents tan (.).
The formation is modelled in order to predict the vehicles motion within a predictive horizon. These predictions allow to formulate a nonlinear optimization problem, whose solutions are the desired values of linear and angular velocities which must be tracked by each agent to maintain the desired formation and follow the desired trajectory. The algorithm can be successfully applied to a wide class of unmanned aerial vehicles as long as they satisfy the physical requirements of Section 2.
Denote the linear velocity vector of the i-th vehicle along T the axes of the i-th body frame as v i = vxi vyi vzi , and the angular velocity vector of the i-th vehicle around T the axes of the i-th body frame as ω i = ωpi ωqi ωri , as described in Fig. 1.
3.1 Kinematic Model Let consider a set of N aerial vehicles V i i = 1, . . . , N , which fly at a fixed distance from one another, following a leader vehicle whose trajectory can be arbitrarily chosen.
The kinematic model for V i is described by 50
2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
i ξ˙ = R(φi , θi , ψ i )v iB
(3a)
η˙ i = W (φi , θi )ω i
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where v (x,y,z) and v¯(x,y,z) are the constant linear velocity bounds, ω r and ω r are the constant angular velocity bounds, while linear and angular velocities variations are limited by ∆v (x,y,z) and ∆ω r respectively.
Define the absolute configuration vector (i.e. referred to T the earth frame) for the i-th vehicle as q i , xi y i z i ψ i and assume that the roll (φi ) and pitch (θi ) angles of vehicle V i are both close to zero (i.e. stabilized) for the entire flight duration. Then the time-continuous kinematic model for the i-th vehicle can be derived from (3) as q˙ i (t) = T −1 (ψ i )ui (t) (4a) Cψi −Sψi 0 0 Sψi Cψi 0 0 −1 i (4b) T (ψ ) , 0 0 1 0 0 0 01 i i i i T i (4c) u (t) , vx vy vz ωr
The time continuous eq. (4) can now be discretized into q ik+1 = q ik + T −1 (ψki )uik
Defining the displacement of vehicle V j referred to the frame fixed to vehicle V i as h iT ji ji ji ji dji (11) k , dxk dyk dzk dψk where dji (·)k represent the linear and angular distances among vehicles, the absolute configuration vector can be expressed as j i i dji (12) k = T(ψk )(qk − qk ) which leads to the following discrete-time formation vector model ji j i ji i i dji (13) k+1 = Ak dk + Bk uk + Ek uk where
From a physical point of view the constraints on roll and pitch angles imply that the linear dynamics, together with the rotational dynamics around the vertical axis, can be decoupled from the dynamics of roll and pitch. This is usually valid for several aerial vehicles in conservative flight conditions (Castillo et al. (2005)). 3.2 Formation vector Let assume to sample the continuous-time variables with sampling interval Ts and define the sampled input vector as i T i i i vy,k vz,k ωr,k uik , vx,k (5) where i vx,k , Ts vxi (kTs ) (6a) i vy,k , Ts vyi (kTs )
(6b)
i vz,k i ωr,k
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, ,
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−1
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i (ωr,k )
i , T(ωr,k )T−1 (ψkji )
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(a) the reference trajectory T ∗ is generated by a virtual reference vehicle V 0 which moves according to the considered model; (b) each vehicle V i follows one and only one leader V j , j 6= i; V 1 follows virtual vehicle V 0 which exactly tracks the reference trajectory T ∗ ; (c) each vehicle V i should keep the reference formation h iT ¯ ji = d¯ji d¯ji d¯ji d¯ji , from its leader V j . pattern d x
y
z
ψ
In the proposed control algorithm, each vehicle V i is equipped with an independent control agent Ai whose tasks are to collect both local and remote information and iteratively performs a nonlinear optimization for computing the local control action. As previously stated, each vehicle V i tracks a leader V j with a defined displacement. The set of all displacements defines the formation. In this paper each vehicle is supposed to know its absolute configuration vector (i.e. absolute position and heading) which is the common case for aerial vehicles flying in formation. In the implementation of the proposed formation control law, the following set of assumptions is made as well. Assumptions 2.
(8d)
Due to physical limits, the velocities of each vehicle are constrained and their limits depend on the lower level controller and on the dynamic behavior of each vehicle. Fixed constraints are assumed in the following without loss of generalization: i i v x ≤ vx,k ≤ vx , |∆vx,k | ≤ ∆v x (9a) i v y ≤ vy,k ≤ vy ,
(14a)
Bik Eji k
The control system for the considered formation is based on a three dimensional extension of the cascaded leaderfollower approach presented in Vaccarini and Longhi (2007a,b). In order to apply the algorithm, two different sets of assumptions are requested. Assumptions 1.
Define the discretized absolute configuration vector for the i-th vehicle as i T i i i qy,k qz,k qψ,k q ik , qx,k (7) where i qx,k , qxi (kTs ) (8a) i qz,k i qψ,k
i Aik , T−1 (ωr,k )
3.3 Formation Control
, (6d) Eq. (6) represents finite movements within each sampling interval Ts . These movements can also be seen as velocities normalized w.r.t. the sampling interval Ts and, in the following, they will be referred to as velocities.
i qy,k , qyi (kTs )
(10)
(9b)
(a) Each control agent Ai communicates with its neighboring agents using a WLAN only once within a sampling interval. (b) The communication network introduces a delay τ = 1.
(9c) (9d) 51
2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
minimization algorithms and allow to compute the control efforts uik for each V i of the formation.
(c) The agents are synchronous. (d) Each control agent knows the absolute configurations of the neighbours.
In order to avoid possible contacts between vehicles, a collision-free constraint is finally added to the set of velocities constraints (9) √ i i ||M Aik dli 2¯ v k + Bk uk || ≥ d + (17) (l 6= i, l = 1, . . . , N )
Note that condition (d) is not strictly necessary and can be relaxed when one or more absolute measurements are missing, by using proper sensor fusion algorithms (see Freddi et al. (2012)). The formation control problem is then decomposed into an inner-loop dynamic task, which consists of making the vehicle’s velocities track a set of references, and an outerloop kinematic task, which assigns the reference velocities to be tracked for the desired trajectory.
where d is the safe constant distance, v¯ = max(|v x |, |v x |)+ 1 0 0 0 max(|v y |, |v y |) and M = . 0 1 0 0 4. SIMULATION RESULTS
Inner-loop dynamics controller For each vehicle V i , the low level controller is assumed to drive the actuators in order to track velocities vxi , vyi , vzi , ωri . This is possible if the inner control loop acts much faster than the outer control loop. With these assumptions, the considered high level control problem becomes a path planning problem for the low level controller. The high level controller should define the optimal speeds vxi , vyi , vzi , ωri that allow to keep the desired formation with the minimum possible efforts. The formation input vector uik = i T i i i vx,k vy,k vz,k ωr,k is the reference vector for the low level controller. No other communication is needed between the two control loops, since the MPC requires position feedback only, as described below.
The developed strategy has been tested in a simulated R scenario, created using MATLAB to control a formation composed by N = 5 quadrotor vehicles. A quadrotor consists in four DC motors on which propellers are fixed. These motors are arranged to the extremities of a Xshaped frame, where all the arms make an angle of 90 degrees with one another. The speed of rotation of the motors (i.e. the lift force associated to the propeller attached to that motor) can be individually changed, thus modifying the attitude of the vehicle allowing the quadrotor to translate into the space. Quadrotor vehicles have been chosen since they are often used in literature as UAVs, due to their high manoeuvrability, simple modelling and low maintenance costs (Bouabdallah et al. (2007)). The quadrotors adopted in simulation have been modelled according to Freddi et al. (2011), including nonlinear dynamics, propeller drag and air friction.
Outer-loop formation controller The following scalar is considered here as a measure of the performance for control agent Ai : ji ji ¯ ji 2 ¯ji 2 ¯ji 2 hdji k − d i , ρx (dxk − dx ) + ρy (dyk − dy ) + ! ji ¯ji d − d ψ ψ 2 k ¯ji 2 + ρz (dji (15) zk − dz ) + ρψ sin 2
The starting and objective formations are described in Fig. 2. V 1 | h1 = 0 m
¯ ji is the constant desired displacement and ρx , where d ρy , ρz , ρψ are arbitrary weights. The cost function to be minimized is p X ˆ ji ¯ ji 2 Jki = hd uik+h−1|k |2 + k+h|k − d i + µ|ˆ
V0 V 2 | h2 = 0 m
+λ
p−1 X h=1
ˆ ik+h−1|k−1 |2 |ˆ uik+h−1|k − u
V3 | h3 = 3 m
V3 | h 3 = 0 m
h=1
σ|∆ˆ uik+h−1|k |2
V 1 | h1 = 3 m
2m
(16)
V4 | h4 = 0 m
V 5 | h5 = 3 m
V2 | h2 = 3 m 2m
where V 5 | h5 = 0 m
• µ is the weight which penalizes the control efforts at time k; • σ is the weight which penalizes large variation of the control efforts; • λ is the weight which penalizes the variation of the control efforts between two successive predictions; • p is the prediction horizon; ˆ jk|h is the j-th control agent predicted control effort • u at time h for time k.
V 4 | h4 = 3 m
2m
Fig. 2. The considered leader-follower formation: the NDMPC has the task to maintain a V formation of five vehicles, flying on the same plane at a fixed distance, while following the main leader. The physical constraints used in simulation are:
The parameters are chosen as trade-off between MPC convergence speed, control efforts magnitude and computational complexity. The above nonlinear constrained optimization problem is iteratively set-up and solved at each sample time by proper
i −2.5 m s−1 ≤ vx,k ≤ 2.5 m s−1 ,
|∆vx,k | ≤ 0.25 m s−2 ,
i −2.5 m s−1 ≤ vy,k ≤ 2.5 m s−1 ,
|∆vy,k | ≤ 0.25 m s−2 ,
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,
≤ 0.075 rad s
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,
|∆ωr,k | ≤ 0.005 rad s−2 .
for i = 1 . . . 5. The parameters used in simulation are: 52
2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
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Fig. 3. Trajectories followed by the five quadrotors in the horizontal plane (a) and in the vertical plane (b). Each vehicle is identified by a colour and a marker. The simulation is frozen at sample times k = 0, k = 75 and k = 150. Figure (b) is limited to the first 27 samples, since height is almost constant later on. • cost function weights: ρx = 10, ρy = 10, ρz = 10, ρψ = 200, µ = 0.5, σ = 1 and λ = 0.4, • prediction horizon p = 3, • safe distance d equal to 0.75 m.
and angular relative distances between vehicle V j and vehicle V i at time k, namely the components of the displacement vector dji k . The dotted black lines represent the desired displacement values, while the coloured solid lines represent the real displacement values.
The total number of simulation steps is K = 150 and the chosen trajectory has a curvilinear behaviour.
5. CONCLUSIONS
As it can be seen in Fig. 3 the main leader follows the virtual leader while the followers quickly assume the desired formation, reaching the same height of the main leader and keeping the desired distances.
Unmanned vehicles flying in formation may perform tasks better than single vehicles flying independently. The formation control problem, however, is difficult to solve for this kind of vehicles since they are nonlinear, underactuated and the feasible control actions are constrained. Moreover decentralized solutions are usually preferred to the centralized ones because of possible computational problems.
The transient phase for this kind of configuration is short. A similar result can be achieved changing the reference trajectory and the formation pattern. During this transient time the vehicle positions differ from the desired position, however at steady state all the distances between leaders and followers converge to the desired values, as it can ji ji be seen in Fig. 4 which shows the dji xk , dyk , dzk and ji dψk variables representing lateral, longitudinal, vertical
To solve the formation control problem a ND-MPC algorithm is developed in this paper, according to a leaderfollower approach: each vehicle flies at a desired distance 53
2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
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Fig. 4. Components of the displacement vectors. The ideal values are reported with a black dashed line, while the real values are reported with different colours. The i-th column contains the relative distance between vehicle i and vehicle i − 1, read into the frame fixed to V i . The main leader distances (column 1) are referred instead to virtual leader V 0 (i.e.: point moving along the planar reference trajectory). from its leader, where the main leader follows a desired trajectory. A nonlinear optimization problem is formulated within a certain prediction horizon, whose solutions are the desired values of linear and angular velocities which must be tracked by each vehicle to maintain the desired formation and follow the desired trajectory. In this way whenever the desired velocities can be correctly set by a low-level controller, the high-level formation control is achievable taking into account physical constraints, actuation constraints and collision-free constraints.
ics. In 14th Mediterranean Conference on Control and Automation (MED). Ancona (Italy). Pages 1-6. Fonti, A., Freddi, A., Longhi, S., and Monteriù, A. (2011). Cooperative and decentralized navigation of autonomous underwater gliders using predictive control. In 18th IFAC World Congress. Milan (Italy). Vol. 18, no. 1, pp. 12813-12818. Fossen, T. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons Ltd. Freddi, A., Lanzon, A., and Longhi, S. (2011). A Feedback Linearization Approach to Fault Tolerance in Unmanned Quadrotor Vehicles. In 18th IFAC World Congress. Milan (Italy). Vol. 8, no. 1, pp. 5413-5418. Freddi, A., Longhi, S., and Monteriù, A. (2012). A coordination architecture for UUV fleets. Journal of Intelligent Service Robotics. Vol. 5, no. 2, pp. 133-146. Keviczky, T., Borrelli, F., and Balas, G.J. (2006). Decentralized receding horizon control for large scale dynamically decoupled systems. Automatica. Vol. 42, no. 12, pp. 2105-2115. Lyon, D. (2004). A military perspective on small unmanned aerial vehicles. IEEE Instrumentation Measurement Magazine. Vol. 7, no. 3, pp. 27-31. Mohr, B. and Fitzpatrick, D. (2008). Micro air vehicle navigation system. IEEE Aerospace and Electronic Systems Magazine. Vol. 23, no. 4, pp. 19-24. Vaccarini, M. and Longhi, S. (2007a). Networked decentralized MPC for formation control of underwater glider fleets. In Proc. of Control Applications in Marine Systems. Bol (Croatia). Vaccarini, M. and Longhi, S. (2007b). Networked decentralized MPC for unicycle vehicles formation. In 7th IFAC Symposium on Nonlinear Control Systems. Pretoria (South Africa). Valavanis, K. (2007). Advances in unmanned aerial vehicles: state of the art and the road to autonomy. Springer Verlag.
The proposed solution proves to be effective in simulated scenarios with reasonable computational efforts. Its decentralized architecture allows it to be extended to deal with faults on the communication channels and/or the vehicle sensors, simply formulating a new optimization problem for each single agent, similarly to what have been proposed in Freddi et al. (2012). This aspect, together with a practical implementation of the proposed algorithm on real quadrotors, is actually under investigation.
REFERENCES Bethke, B., Valenti, M., and How, J. (2008). Uav task assignment. IEEE Robotics Automation Magazine. Vol. 15, no. 1, pp. 39-44. Bouabdallah, S., Becker, M., and Siegwart, R. (2007). Autonomous miniature flying robots: coming soon! research, development, and results. IEEE Robotics Automation Magazine. Vol. 14, no. 3, pp. 88-98. Castillo, P., Lozano, R., and Dzul, A. (2005). Modelling and control of mini-flying machines. Springer-Verlag New York Inc., 1st edition. Dunbar, W.B. and Murray, R.M. (2006). Distributed receding horizon control for multi-vehicle formation stabilization. Automatica. Vol. 42, no. 4, pp. 549-558. Fang, L. and Antsaklis, P. (2006). Decentralized formation tracking of multi-vehicle systems with nonlinear dynam54
Nonlinear Decentralized Model Predictive Control Strategy for a Formation of Unmanned Aerial Vehicles A. Freddi ∗ S. Longhi ∗ A. Monteriù ∗ ∗
Dipartimento di Ingegneria dell’Informazione – Università Politecnica delle Marche, Ancona, Italy (email: [email protected], [email protected], [email protected]) Abstract: Aerial vehicles flying in formation may perform more complex tasks than vehicles flying independently. Formation control, however, is a difficult problem to cope with for these kind of vehicles, since they are highly nonlinear, underactuated and the feasible control actions are constrained. Moreover the computational capabilities which can be mounted on aerial vehicles are usually limited, thus centralized solutions should be avoided. To solve these problems, a Nonlinear Decentralized Model Predictive Control algorithm is presented in this paper, taking into account both physical and actuation constraints. Each vehicle is equipped with sensors providing inertial measurements and communicates with its neighbours using a Wireless Local Area Network. Simulation results applied to a formation of quadrotor vehicles show that the proposed technique is a valid way to solve the control problem for a generic formation, granting at the same time the possibility to deal with constraints. Keywords: Predictive Control, Co-operative Control, Autonomous Vehicles, Decentralized Systems. 1. INTRODUCTION
Coordinated formations of Unmanned Aerial Vehicles (UAVs) can provide significant benefit with respect to single vehicle in a number of applications, including photogrammetry, sampling and surveillance. The formation control problem is challenging for these kind of vehicles since they are highly nonlinear, underactuated and the feasible control actions are constrained. In order to minimize control complexity and allow easy reconfiguration of formation, decentralized solution are usually preferred to the centralized ones. Different decentralized solutions to the formation control of multiple autonomous vehicles have been developed (Keviczky et al. (2006); Dunbar and Murray (2006); Fang and Antsaklis (2006)). A decentralized Model Predictive Control algorithm for formation keeping have been developed, analyzed and compared with the centralized solution by Vaccarini and Longhi (2007a,b) for motion on the horizontal plane, and applied to the control of a fleet of unmanned underwater gliders in Fonti et al. (2011).
In the last decades unmanned aerial, underwater and ground vehicles have generated considerable attraction due to their strong autonomy and ability to perform relatively difficult tasks in remote, uncertain or hazardous environments where human beings are unable to go. The purposes of such vehicles are extremely various, ranging from scientific exploration, data collection and remote sensing, to provision of commercial services, military reconnaissance and intelligence gathering. A number of unmanned systems have become available and research is ongoing in a number of areas that will significantly advance the state of the art in unmanned vehicles technology. Moreover designers have more freedom in the development of such vehicles, not having to account for the presence of a pilot and the associated life-support systems. This potentially results in cost and size savings, as well as increased operational capabilities (Lyon (2004); Valavanis (2007); Mohr and Fitzpatrick (2008); Bethke et al. (2008)).
In this paper the formation control of a system of multiple unmanned aerial vehicles moving according a trajectory is addressed. The approach is based on a leader-follower architecture: each vehicle has to keep a reference distance from its leader, where the main leader follows the desired trajectory. The algorithm implements a Nonlinear Decentralized Model Predictive Control (ND-MPC) technique which takes into account physical and predictive model constraints. The single control agents communicate using a Wireless Local Area Network (WLAN) in which each neighbour exchanges information about its current state and predicted control efforts.
From a formal point of view, Unmanned Vehicles (UVs) can be defined as vehicles that can accomplish a task without the aid of a human guide. To do that, they rely on sensors, that provide information about the external environment or the internal system states; actuators, that physically realize the desired motion; controller, that drives the actuators according to the measurements and the task that need to be accomplished. With an increasing availability of modern sensors and effective communication channels, coordination of unmanned vehicles has become feasible and attractive at the same time. 978-3-902823-15-1/12/$20.00 © 2012 IFAC
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The proposed approach extends in three dimensions previous results obtained for the motion on the horizontal plane only (Vaccarini and Longhi (2007a,b)), and is applied for formation keeping and trajectory planning of unmanned aerial vehicles.
N + 1 frames are used to study the formation motion (see Fig. 1): a frame integral with the earth {R} (O, x, y, z), which to be inertial, and N body-fixed frames i is supposed i i i i RB (OB , xiB , yB , zB ), where iO B is fixed to the center of mass of the i-th vehicle. RB is related to {R} by a T position vector ξ i = xi y i z i , describing the position i of the center of gravity of vehicle V i (i.e. OB ) with respect to {R} and by a vector of three independent angles η i = i i i T φ θ ψ , which represent the orientation of the vehicle i (i.e. the orientation of the body-fixed frame RB ), with respect to the earth frame {R}, using the so-called yaw, pitch and roll notation (referred to as Euler angles, Fossen (2011)).
The paper is organized as follows. Section 2 describes the physical requirements which the aerial vehicles must posses in order to apply the algorithm. Section 3 defines the kinematic model, the formation vector and how to use them for the proposed control algorithm. Section 4 provides the results of a simulated scenario in which a formation of unmanned quadrotor vehicles is adopted. Conclusions and future works are finally provided in Section 5. 2. UAV REQUIREMENTS
j
yB
The control algorithm presented in this paper can be applied to a wide class of unmanned aerial vehicles as long as they possess a minimum set of requirements.
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Sensors: each vehicle must be equipped with a heading sensor, an altitude sensor and either a Global Positioning System (GPS) or a vision system for outdoor/indoor absolute localization (i.e. linear displacements referred to an inertial reference system). Communication: vehicles must be connected to a WLAN to exchange sensor information and predicted control efforts; the required bandwidth is low since the algorithm requires only a limited amount of information (i.e. measurements and control effort predictions among neighbours) to be transmitted. Low-level controller: each vehicle must posses an inner controller for the internal dynamics which can track a reference velocity vector, as described in Section 3. When these requirements are satisfied, then it is possible to apply a ND-MPC control algorithm for formation control and trajectory planning of UAVs as described in the following section.
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Fig. 1. The reference frames adopted: each agent has a solidal BF whose origin is in the center of gravity. T T The vectors ξ i = xi y i z i and η i = φi θi ψ i fully describe, respectively, the translational and the rotational movement of the i-th vehicle with respect to the earth frame. Let define the rotation matrix which maps the linear velocity of V i from the i-th body frame into the earth frame as # " Cθi Cψi Cψi Sθi Sφi − Cφi Sψi Cφi Cψi Sθi + Sφi Sψi R , Cθi Sψi Sθi Sφi Sψi + Cφi Cψi Cφi Sθi Sψi − Cψi Sφi −Sθi Cθi Sφi Cθi Cφi (1) where S(.) and C(.) represent sin (.) and cos (.), respectively.
3. PROBLEM STATEMENT The control strategy presented in this paper aims to solve the problem of formation keeping for unmanned aerial vehicles flying in formation. The problem is addressed using a ND-MPC algorithm for agents flying in leader-follower formation: each vehicle flies at a fixed and predefined distance from its leader, where the main leader follows a desired trajectory.
Define the matrix which maps the angular velocity of V i , expressed in the coordinates of the i-th body frame, into the rate of change of the Euler angles as " # 1 Sφi Tθi Cφi Tθi −Sφi W , 0 Cφi (2) 0 Sφi /Cφi Cφi /Cθi where T(.) represents tan (.).
The formation is modelled in order to predict the vehicles motion within a predictive horizon. These predictions allow to formulate a nonlinear optimization problem, whose solutions are the desired values of linear and angular velocities which must be tracked by each agent to maintain the desired formation and follow the desired trajectory. The algorithm can be successfully applied to a wide class of unmanned aerial vehicles as long as they satisfy the physical requirements of Section 2.
Denote the linear velocity vector of the i-th vehicle along T the axes of the i-th body frame as v i = vxi vyi vzi , and the angular velocity vector of the i-th vehicle around T the axes of the i-th body frame as ω i = ωpi ωqi ωri , as described in Fig. 1.
3.1 Kinematic Model Let consider a set of N aerial vehicles V i i = 1, . . . , N , which fly at a fixed distance from one another, following a leader vehicle whose trajectory can be arbitrarily chosen.
The kinematic model for V i is described by 50
2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
i ξ˙ = R(φi , θi , ψ i )v iB
(3a)
η˙ i = W (φi , θi )ω i
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where v (x,y,z) and v¯(x,y,z) are the constant linear velocity bounds, ω r and ω r are the constant angular velocity bounds, while linear and angular velocities variations are limited by ∆v (x,y,z) and ∆ω r respectively.
Define the absolute configuration vector (i.e. referred to T the earth frame) for the i-th vehicle as q i , xi y i z i ψ i and assume that the roll (φi ) and pitch (θi ) angles of vehicle V i are both close to zero (i.e. stabilized) for the entire flight duration. Then the time-continuous kinematic model for the i-th vehicle can be derived from (3) as q˙ i (t) = T −1 (ψ i )ui (t) (4a) Cψi −Sψi 0 0 Sψi Cψi 0 0 −1 i (4b) T (ψ ) , 0 0 1 0 0 0 01 i i i i T i (4c) u (t) , vx vy vz ωr
The time continuous eq. (4) can now be discretized into q ik+1 = q ik + T −1 (ψki )uik
Defining the displacement of vehicle V j referred to the frame fixed to vehicle V i as h iT ji ji ji ji dji (11) k , dxk dyk dzk dψk where dji (·)k represent the linear and angular distances among vehicles, the absolute configuration vector can be expressed as j i i dji (12) k = T(ψk )(qk − qk ) which leads to the following discrete-time formation vector model ji j i ji i i dji (13) k+1 = Ak dk + Bk uk + Ek uk where
From a physical point of view the constraints on roll and pitch angles imply that the linear dynamics, together with the rotational dynamics around the vertical axis, can be decoupled from the dynamics of roll and pitch. This is usually valid for several aerial vehicles in conservative flight conditions (Castillo et al. (2005)). 3.2 Formation vector Let assume to sample the continuous-time variables with sampling interval Ts and define the sampled input vector as i T i i i vy,k vz,k ωr,k uik , vx,k (5) where i vx,k , Ts vxi (kTs ) (6a) i vy,k , Ts vyi (kTs )
(6b)
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i , T(ωr,k )T−1 (ψkji )
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(a) the reference trajectory T ∗ is generated by a virtual reference vehicle V 0 which moves according to the considered model; (b) each vehicle V i follows one and only one leader V j , j 6= i; V 1 follows virtual vehicle V 0 which exactly tracks the reference trajectory T ∗ ; (c) each vehicle V i should keep the reference formation h iT ¯ ji = d¯ji d¯ji d¯ji d¯ji , from its leader V j . pattern d x
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In the proposed control algorithm, each vehicle V i is equipped with an independent control agent Ai whose tasks are to collect both local and remote information and iteratively performs a nonlinear optimization for computing the local control action. As previously stated, each vehicle V i tracks a leader V j with a defined displacement. The set of all displacements defines the formation. In this paper each vehicle is supposed to know its absolute configuration vector (i.e. absolute position and heading) which is the common case for aerial vehicles flying in formation. In the implementation of the proposed formation control law, the following set of assumptions is made as well. Assumptions 2.
(8d)
Due to physical limits, the velocities of each vehicle are constrained and their limits depend on the lower level controller and on the dynamic behavior of each vehicle. Fixed constraints are assumed in the following without loss of generalization: i i v x ≤ vx,k ≤ vx , |∆vx,k | ≤ ∆v x (9a) i v y ≤ vy,k ≤ vy ,
(14a)
Bik Eji k
The control system for the considered formation is based on a three dimensional extension of the cascaded leaderfollower approach presented in Vaccarini and Longhi (2007a,b). In order to apply the algorithm, two different sets of assumptions are requested. Assumptions 1.
Define the discretized absolute configuration vector for the i-th vehicle as i T i i i qy,k qz,k qψ,k q ik , qx,k (7) where i qx,k , qxi (kTs ) (8a) i qz,k i qψ,k
i Aik , T−1 (ωr,k )
3.3 Formation Control
, (6d) Eq. (6) represents finite movements within each sampling interval Ts . These movements can also be seen as velocities normalized w.r.t. the sampling interval Ts and, in the following, they will be referred to as velocities.
i qy,k , qyi (kTs )
(10)
(9b)
(a) Each control agent Ai communicates with its neighboring agents using a WLAN only once within a sampling interval. (b) The communication network introduces a delay τ = 1.
(9c) (9d) 51
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minimization algorithms and allow to compute the control efforts uik for each V i of the formation.
(c) The agents are synchronous. (d) Each control agent knows the absolute configurations of the neighbours.
In order to avoid possible contacts between vehicles, a collision-free constraint is finally added to the set of velocities constraints (9) √ i i ||M Aik dli 2¯ v k + Bk uk || ≥ d + (17) (l 6= i, l = 1, . . . , N )
Note that condition (d) is not strictly necessary and can be relaxed when one or more absolute measurements are missing, by using proper sensor fusion algorithms (see Freddi et al. (2012)). The formation control problem is then decomposed into an inner-loop dynamic task, which consists of making the vehicle’s velocities track a set of references, and an outerloop kinematic task, which assigns the reference velocities to be tracked for the desired trajectory.
where d is the safe constant distance, v¯ = max(|v x |, |v x |)+ 1 0 0 0 max(|v y |, |v y |) and M = . 0 1 0 0 4. SIMULATION RESULTS
Inner-loop dynamics controller For each vehicle V i , the low level controller is assumed to drive the actuators in order to track velocities vxi , vyi , vzi , ωri . This is possible if the inner control loop acts much faster than the outer control loop. With these assumptions, the considered high level control problem becomes a path planning problem for the low level controller. The high level controller should define the optimal speeds vxi , vyi , vzi , ωri that allow to keep the desired formation with the minimum possible efforts. The formation input vector uik = i T i i i vx,k vy,k vz,k ωr,k is the reference vector for the low level controller. No other communication is needed between the two control loops, since the MPC requires position feedback only, as described below.
The developed strategy has been tested in a simulated R scenario, created using MATLAB to control a formation composed by N = 5 quadrotor vehicles. A quadrotor consists in four DC motors on which propellers are fixed. These motors are arranged to the extremities of a Xshaped frame, where all the arms make an angle of 90 degrees with one another. The speed of rotation of the motors (i.e. the lift force associated to the propeller attached to that motor) can be individually changed, thus modifying the attitude of the vehicle allowing the quadrotor to translate into the space. Quadrotor vehicles have been chosen since they are often used in literature as UAVs, due to their high manoeuvrability, simple modelling and low maintenance costs (Bouabdallah et al. (2007)). The quadrotors adopted in simulation have been modelled according to Freddi et al. (2011), including nonlinear dynamics, propeller drag and air friction.
Outer-loop formation controller The following scalar is considered here as a measure of the performance for control agent Ai : ji ji ¯ ji 2 ¯ji 2 ¯ji 2 hdji k − d i , ρx (dxk − dx ) + ρy (dyk − dy ) + ! ji ¯ji d − d ψ ψ 2 k ¯ji 2 + ρz (dji (15) zk − dz ) + ρψ sin 2
The starting and objective formations are described in Fig. 2. V 1 | h1 = 0 m
¯ ji is the constant desired displacement and ρx , where d ρy , ρz , ρψ are arbitrary weights. The cost function to be minimized is p X ˆ ji ¯ ji 2 Jki = hd uik+h−1|k |2 + k+h|k − d i + µ|ˆ
V0 V 2 | h2 = 0 m
+λ
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V3 | h3 = 3 m
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V4 | h4 = 0 m
V 5 | h5 = 3 m
V2 | h2 = 3 m 2m
where V 5 | h5 = 0 m
• µ is the weight which penalizes the control efforts at time k; • σ is the weight which penalizes large variation of the control efforts; • λ is the weight which penalizes the variation of the control efforts between two successive predictions; • p is the prediction horizon; ˆ jk|h is the j-th control agent predicted control effort • u at time h for time k.
V 4 | h4 = 3 m
2m
Fig. 2. The considered leader-follower formation: the NDMPC has the task to maintain a V formation of five vehicles, flying on the same plane at a fixed distance, while following the main leader. The physical constraints used in simulation are:
The parameters are chosen as trade-off between MPC convergence speed, control efforts magnitude and computational complexity. The above nonlinear constrained optimization problem is iteratively set-up and solved at each sample time by proper
i −2.5 m s−1 ≤ vx,k ≤ 2.5 m s−1 ,
|∆vx,k | ≤ 0.25 m s−2 ,
i −2.5 m s−1 ≤ vy,k ≤ 2.5 m s−1 ,
|∆vy,k | ≤ 0.25 m s−2 ,
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for i = 1 . . . 5. The parameters used in simulation are: 52
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Fig. 3. Trajectories followed by the five quadrotors in the horizontal plane (a) and in the vertical plane (b). Each vehicle is identified by a colour and a marker. The simulation is frozen at sample times k = 0, k = 75 and k = 150. Figure (b) is limited to the first 27 samples, since height is almost constant later on. • cost function weights: ρx = 10, ρy = 10, ρz = 10, ρψ = 200, µ = 0.5, σ = 1 and λ = 0.4, • prediction horizon p = 3, • safe distance d equal to 0.75 m.
and angular relative distances between vehicle V j and vehicle V i at time k, namely the components of the displacement vector dji k . The dotted black lines represent the desired displacement values, while the coloured solid lines represent the real displacement values.
The total number of simulation steps is K = 150 and the chosen trajectory has a curvilinear behaviour.
5. CONCLUSIONS
As it can be seen in Fig. 3 the main leader follows the virtual leader while the followers quickly assume the desired formation, reaching the same height of the main leader and keeping the desired distances.
Unmanned vehicles flying in formation may perform tasks better than single vehicles flying independently. The formation control problem, however, is difficult to solve for this kind of vehicles since they are nonlinear, underactuated and the feasible control actions are constrained. Moreover decentralized solutions are usually preferred to the centralized ones because of possible computational problems.
The transient phase for this kind of configuration is short. A similar result can be achieved changing the reference trajectory and the formation pattern. During this transient time the vehicle positions differ from the desired position, however at steady state all the distances between leaders and followers converge to the desired values, as it can ji ji be seen in Fig. 4 which shows the dji xk , dyk , dzk and ji dψk variables representing lateral, longitudinal, vertical
To solve the formation control problem a ND-MPC algorithm is developed in this paper, according to a leaderfollower approach: each vehicle flies at a desired distance 53
2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012
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Fig. 4. Components of the displacement vectors. The ideal values are reported with a black dashed line, while the real values are reported with different colours. The i-th column contains the relative distance between vehicle i and vehicle i − 1, read into the frame fixed to V i . The main leader distances (column 1) are referred instead to virtual leader V 0 (i.e.: point moving along the planar reference trajectory). from its leader, where the main leader follows a desired trajectory. A nonlinear optimization problem is formulated within a certain prediction horizon, whose solutions are the desired values of linear and angular velocities which must be tracked by each vehicle to maintain the desired formation and follow the desired trajectory. In this way whenever the desired velocities can be correctly set by a low-level controller, the high-level formation control is achievable taking into account physical constraints, actuation constraints and collision-free constraints.
ics. In 14th Mediterranean Conference on Control and Automation (MED). Ancona (Italy). Pages 1-6. Fonti, A., Freddi, A., Longhi, S., and Monteriù, A. (2011). Cooperative and decentralized navigation of autonomous underwater gliders using predictive control. In 18th IFAC World Congress. Milan (Italy). Vol. 18, no. 1, pp. 12813-12818. Fossen, T. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons Ltd. Freddi, A., Lanzon, A., and Longhi, S. (2011). A Feedback Linearization Approach to Fault Tolerance in Unmanned Quadrotor Vehicles. In 18th IFAC World Congress. Milan (Italy). Vol. 8, no. 1, pp. 5413-5418. Freddi, A., Longhi, S., and Monteriù, A. (2012). A coordination architecture for UUV fleets. Journal of Intelligent Service Robotics. Vol. 5, no. 2, pp. 133-146. Keviczky, T., Borrelli, F., and Balas, G.J. (2006). Decentralized receding horizon control for large scale dynamically decoupled systems. Automatica. Vol. 42, no. 12, pp. 2105-2115. Lyon, D. (2004). A military perspective on small unmanned aerial vehicles. IEEE Instrumentation Measurement Magazine. Vol. 7, no. 3, pp. 27-31. Mohr, B. and Fitzpatrick, D. (2008). Micro air vehicle navigation system. IEEE Aerospace and Electronic Systems Magazine. Vol. 23, no. 4, pp. 19-24. Vaccarini, M. and Longhi, S. (2007a). Networked decentralized MPC for formation control of underwater glider fleets. In Proc. of Control Applications in Marine Systems. Bol (Croatia). Vaccarini, M. and Longhi, S. (2007b). Networked decentralized MPC for unicycle vehicles formation. In 7th IFAC Symposium on Nonlinear Control Systems. Pretoria (South Africa). Valavanis, K. (2007). Advances in unmanned aerial vehicles: state of the art and the road to autonomy. Springer Verlag.
The proposed solution proves to be effective in simulated scenarios with reasonable computational efforts. Its decentralized architecture allows it to be extended to deal with faults on the communication channels and/or the vehicle sensors, simply formulating a new optimization problem for each single agent, similarly to what have been proposed in Freddi et al. (2012). This aspect, together with a practical implementation of the proposed algorithm on real quadrotors, is actually under investigation.
REFERENCES Bethke, B., Valenti, M., and How, J. (2008). Uav task assignment. IEEE Robotics Automation Magazine. Vol. 15, no. 1, pp. 39-44. Bouabdallah, S., Becker, M., and Siegwart, R. (2007). Autonomous miniature flying robots: coming soon! research, development, and results. IEEE Robotics Automation Magazine. Vol. 14, no. 3, pp. 88-98. Castillo, P., Lozano, R., and Dzul, A. (2005). Modelling and control of mini-flying machines. Springer-Verlag New York Inc., 1st edition. Dunbar, W.B. and Murray, R.M. (2006). Distributed receding horizon control for multi-vehicle formation stabilization. Automatica. Vol. 42, no. 4, pp. 549-558. Fang, L. and Antsaklis, P. (2006). Decentralized formation tracking of multi-vehicle systems with nonlinear dynam54