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Research Article | Open Access
Jawhar Ghommam, Luis F. Luque-Vega, Maarouf Saad, "Distance-Based Formation Control for Quadrotors with Collision Avoidance via Lyapunov Barrier Functions", International Journal of Aerospace Engineering, vol. 2020, Article ID 2069631, 17 pages, 2020. https://doi.org/10.1155/2020/2069631
Distance-Based Formation Control for Quadrotors with Collision Avoidance via Lyapunov Barrier Functions
In this paper, group formation control with collision avoidance is investigated for heterogeneous multiquadrotor vehicles. Specifically, the distance-based formation and tracking control problem are addressed in the framework of leader-follower architecture. In this scheme, the leader is assigned the task of intercepting a target whose velocity is unknown, while the follower quadrotors are arranged to set up a predefined rigid formation pattern, ensuring simultaneously interagent collision avoidance and relative localization. The adopted strategy for the control design consists in decoupling the quadrotor dynamics in a cascaded structure to handle its underactuated property. Furthermore, by imposing constraints on the orientation angles, the follower will never be overturned. Rigorous stability analysis is presented to prove the stability of the entire closed-loop system. Numerical simulation results are presented to validate the proposed control strategy.
1.1. Background and Related Works
Quadrotors are agile mechanical systems which have drawn the attention of researchers for the last decades [1–3], mainly because the use of these types of aerial vehicles is appealing for several real-world applications. Due to their excellent performances, quadrotors are likely to be adopted in various civil potential applications such as inspection of power lines, oil platforms, search and rescue operations, and surveillance [4, 5]. On the other hand, it is worth pointing out that a team of quadrotors has many benefits over using a single quadrotor in several applications where it results in faster and more efficient operations. The coordination of multiple quadrotors to perform cooperative tasks has received considerable attention due to the challenges arising in their control design. Coordinated movement of quadrotors comes with several methodologies and approaches proposed over the last few years [6–10]. From the control structure perspective, the approaches used for coordinated control can be categorized into centralized and decentralized methods. In the centralized method, the control signals for all the agents are generated in the leader agent or in a central station [7, 11]. Whereas, in the decentralized method also known as the distributed method, each member of the team uses its own local controller enhanced with local information from its close neighbors [6, 8, 10].
From the control mechanism perspective, there are roughly five approaches to cooperative control of multiple aerial vehicles: the behavioral-based approach [12, 13] where each agent locally reacts to actions of its neighbors. This method is categorized under the decentralized approach and is suitable for a large number of agents in the group. The disadvantage of this approach, however, is that the complexity of the dynamics of the group of robots cannot be amenable to mathematical stability analysis. Another important approach is the virtual structure approach [14–17]. This approach considers all the agents in the group as a single entity, thus limiting the formation scenarios that can be anticipated to merely static formation patterns. Moreover, the priority of agents in the group team is to respect one of the two rules: following their individual trajectories  or maintaining the group formation. The artificial potential function-based approach [6, 19, 20] is a powerful tool as it allows the agents to evolve in a cohesive motion while avoiding obstacles. The disadvantage of this approach is that local minima may appear where the vector field vanishes, therefore forcing the agents to get trapped into undesired equilibrium points. The leader-following approach [11, 21, 22], on the other hand, seems to be particularly attractive and preferred over the aforementioned approaches due to its simplicity and scalability. In this approach, some agents are designated as leaders aimed at tracking predefined trajectories, while others are treated as followers whose actions are merely to follow the leader(s) in order to keep the formation. The followers can in turn be designated as leaders for other vehicles in the group resulting in a scalability of the formation. The advantage of this approach is that from the definition of the motion of one entity, the overall group behavior can subsequently be defined. Also, it should be pointed out that the internal formation stability is induced by the control laws of individual vehicles. A major drawback of this approach, however, is that there exists no feedback from the followers to the leaders. Consequently, if the followers are perturbed, the coordination shape of the group team cannot be maintained.
A leader-follower formation control algorithm for multiquadrotors was proposed in [22, 23] that involves a linear inner-loop and nonlinear outer-loop control structure, where a combination of sliding mode control and LQR/PD technique was deployed for the trajectory tracking control of a follower to track a leader quadrotor. The formation shape is formed then maintained by keeping a constant distance and a desired angle between each follower and the leader. The authors in  proposed a control strategy for a string-like formation of multiple autonomous quadrotor vehicles. In this approach, the objective is to design a maneuvering control for the quadrotor group along a geometric path while respecting an assigned timing law that determines the formation rate of advancement. A formation control strategy has been proposed  using the leader-follower approach. The suggested approach differs from common strategies employed for leader-follower schemes and consists in stabilizing, at a predefined distance vector, the position of the leader in the body frame of a virtual follower which in turn generates a trajectory which is used as a reference trajectory to be tracked by the real follower vehicle. The proposed strategy allows for generating curvature-rich formation trajectories. The authors in  proposed a 3D leader-follower formation control strategy for a quadrotor with completely unknown dynamics. Their presented strategy relies principally on the use of the spherical coordinate method where the desired position of the follower quadrotor is specified using a desired separation distance along with a desired angle of incidence and bearing. The model unknown nonlinearities are compensated using the neural network techniques. However, a common drawback of many previous leader-follower formation controllers is that they ignore the physical quadrotor flight limitation. In fact, for a safe flight, it is required that the quadrotor avoid singularities in the rotation matrix represented by the Euler angles. Essentially, they often assume that the quadrotor system is always away from the singularity points and that the quadrotor does not rotate too much. From a practical viewpoint, the trajectory generated by the leader vehicle might be aggressive sometimes. Consequently, the follower quadrotor is most likely exposed to a fall in singularity points. This, in turn, may result in poor tracking performance of the proposed controller. An immediate solution to alleviate this problem is to constrain the variation of the quadrotor states within a specified range, then design a controller such that the quadrotor’s state never transgresses these constraints.
1.2. Contributions and Novelties
Motivated by the earlier works, the main purpose of this work is to tackle the distance-based formation control problem for a group of homogeneous quadrotor vehicles modeled by an undirected graph. Contrary to the current state of the art, limited explicit communication among neighboring quadrotors is mainly motivated to avoid flooding the communication bandwidth in real-time applications, particularly when the number of cooperating quadrotors increases. To keep communication to a minimum, we propose a new distance-based formation control protocol for a group of multiple quadrotors in a leader-follower architecture, where sensor constraints are explicitly addressed in the design. In this architecture, we assume that the target’s relative position with a leader in the group can be shared among other follower quadrotors, while its velocity is made unavailable. To learn this reference velocity, an estimation mechanism is implemented on each quadrotor so that a predefined formation is recovered so the quadrotors maintain a desired distance among their neighboring followers, while avoiding collision and ensuring relative localization. To deal with underactuation of the quadrotor vehicle in the formation, a decentralized intermediary translational control input is designed for relative position tracking, by which a bounded control thrust and bounded desired orientation are extracted. Then, a control torque for each individual quadrotor is designed to track the desired orientation while avoiding overturn of the quadrotors.
In comparison with existing works, the main contributions of this paper are listed as follows: (1)Compared to [6, 26, 27], our controller is completely designed independently from a global reference coordinate and does not require the local frame of all vehicles to be aligned. Only the distance requirements are used with local sensors to ensure formation maintenance(2)Our controller is decentralized and ensures simultaneously collision avoidance and self-localization. As opposed to [6, 28], the control input for the translational dynamics is not designed based on a negative gradient of specific local/global potential functions; it is designed using a barrier Lyapunov function ensuring convergence to the appropriate distances with predefined performances(3)The output constraints on the orientation angles are handled using the integral barrier Lyapunov (iBL) function  in the control design, thus relaxing the assumption on small orientation angles in [30–32] and therefore ensuring that the quadrotor will never overturn(4)Our design is modular in the sense that the stability of the cooperative system is easily shown by the fact that the closed-loop system is regarded as cascaded systems in which the position dynamic subsystem is the driven subsystem while the orientation subsystem is the driving subsystem
The paper is organized as follows. The upcoming section introduces basic concepts of rigid graph theory and results on the stability of cascade systems. Section 3 models the dynamic equations of motion for an individual quadrotor and presents the decentralized distance-based formation control objectives. The controllers for the translational and rotational dynamics of the quadrotors in the group formation are introduced in Section 4. The main result and the convergence analysis for the proposed controller are provided in Section 5. Numerical simulations are presented in Section 6. Finally, the conclusion is presented in Section 7.
In this section, we present concepts of rigid graph theory for the control formulation taken from  and a lemma from  for the stability analysis of nonautonomous nonlinear systems, in cascade, which will be useful to establish the stability of the whole closed-loop system.
An undirect graph with pair is defined as a set of vertices connected with a set of undirected edges such that if the edge in ,then . The number of edges that amounts for the connection between the vertices is . The set of neighbors of vertex is defined as such that . A framework of is the pair , where is the stack vector containing the position of all quadrotors. The rigidity function associated with is denoted by and is given by , where denotes the Euclidian norm. The rigidity matrix is defined as . It can be shown similarly to  that . It follows that is infinitesimally rigid  in if the corresponding graph has at least edges. An isometry of is a bijective map satisfying . Two frameworks are said to be isomorphic in if they are related by isometry. The set of all frameworks that is isomorphic to is denoted by . Frameworks and are equivalent if and are congruent if . If the infinitesimally rigid frameworks and are equivalent but not congruent, they are called ambiguous. The collection of frameworks which are ambiguous to the infinitesimally rigid frameworks is denoted by .
Lemma 1 (see ). Given a vector and let be the vector of ones, then .
Lemma 2. (see ). If the framework is minimally and infinitesimally rigid, the the matrix is positive definite.
Definition 3. A continuous function is said to belong to class if it is strictly increasing and . It is said to belong to class if and . A continuous function is said to belong to class if, for each fixed , the mapping belongs to class with respect to and, for each fixed , the mapping is decreasing with respect to s and as .
Lemma 4. (see ). Consider the following cascade system: If the equilibrium is GAS/LES and the equilibrium is LES and assume also that there exists a constant and a function , differentiable at where If in addition there exists a positive semidefinite radially unlimited function and some positive constants and such that then the equilibrium point is LES.
3. Problem Formulation
The proposed formation scheme is to coordinate the motion of quadrotors under the leader-follower architecture. The leader quadrotor in the group is indexed by , while the rest of the follower vehicles update their state with locally available information. The formation control scheme to be addressed in this paper will be formulated as formation maneuvering and target interception problems as in . Hereafter, the interaction among the vehicles is modeled by undirected graphs where is the set of vertices (quadrotors) and is the set of edges representing the connectivity among the vehicles. Define the set of neighbors of the -th quadrotor as .
3.1. Quadrotor Model Dynamics
To develop the quadrotor’s equations of motion, we first defined two reference frames. The first one is the earth-fixed inertial frame defined as , and the second is a body-fixed frame denoted by whose origin is located at the quadrotor’s center of gravity (CG) as shown in Figure 1.
Let be the three-dimensional position of the i-th quadrotor and the rotation matrix parameterized with respect to the three Euler angles with , and being the roll, pitch, and yaw angles, respectively, and where the special Euclidean group is the group of orthogonal matrices with a determinant of one, i.e., . Following the standard results, the associated orientation dynamics are governed by where
The kinematic and dynamic equations of motion of the i-th quadrotor are given by where is the linear velocity vector in inertial coordinates, , denotes the total mass of the quadrotor , is the inertia matrix with respect to the frame , is the unit vector, and is the gravitational acceleration. The thrust magnitude is denoted as , and is the moment vector; both are considered as the control inputs of the quadrotor. The matrix is a skew-symmetric matrix containing the elements of the body-fixed angular velocity . Denote by and the unstructured uncertainties in the translational dynamics and the rotational dynamics of a quadrotor, respectively. We further assume that the uncertainties and are small time varying and bounded, that is, where and are known constants. From the dynamic equations (7)–(9), it is worth noting that the quadrotor is an underactuated system with only one degree of freedom for the thrust actuation in the body frame, while the attitude dynamics are fully actuated and can be used to drive the thrust force to any desired orientation.
3.2. Control Objective
The leader-follower approach considered in this paper assumes a group of quadrotor vehicles with the -th vehicle set to be the leader which can merely measure the target’s relative position . Additionally, it is also assumed that the only information that the quadrotors can sense is exclusively acquired from their local sensors. The control objective is to design controllers for the group of vehicles to maintain interdistance specification among the vehicles to attain a rigid desired formation, while guaranteeing formation maintenance and intervehicle collision, that is, if the desired formation is modeled by the framework , which is assumed to be infinitesimally and minimally rigid, where and are the collection of the targets’ positions. Also, considering that the actual formation of quadrotors is modeled by , then the formation control problem boils down to ensure the following:
Mathematically speaking, the control objective reduces to design a desired thrust magnitude for the system dynamics of each quadrotor vehicle (7) such that where is a bounded, continuous function designating the translational velocity of the formation and is a positive limit upper-bound. An important issue in the formation of multiple quadrotor vehicles is to ensure safe navigation by avoiding collision among interacting vehicles. To this end, the distance that separates the quadrotors should be kept greater than a safety zone. Similarly, the quadrotors should also be retained within the connectivity network to achieve the group tasks. The decentralized controller should also satisfy the following constraint: where and are prescribed functions.In addition to the translational motion of the quadrotor, an essential part of the control is the attitude innermost layer of the controller. While the position controller calculates the desired direction of the thrust vector, the attitude controller will ensure the alignment of the real thrust vector with the desired one. A second control objective is to ensure a fast control loop for the attitude innermost layer. Thus, our goal is to satisfy the following limit: where represents the desired attitude that is calculated based on the desired trust vector which will be derived later in the next section.
4. Control Design
4.1. Decentralized Control Design for the Position Outer-Loop
In this section, we will use the backstepping technique to derive a decentralized cooperative controller for the position control loop in the sense that each quadrotor requires the relative position of its neighbors and its own velocity, while ensuring collision avoidance among the vehicles. For that purpose, the vector term will be considered as the control input signal for the cooperative position-tracking task. It is common to assign to that input signal a desired value that we wish to reach in the final stabilization process. denotes the desired orientation of the follower quadrotor while represents the direction of the force amplitude
Assumption 5. The actuator dynamics of the follower quadrotor are negligible with respect to the rigid body dynamics of the quadrotor.
Due to the dynamics consideration, Assumption 5 reveals that the desired thrust magnitude can be instantaneously reached by the force control input ; therefore, it is straightforward to conclude that . Hence, the first two equations in (7) can be rewritten in terms of and as follows: where the coupling term which can be written as , where is the rotation matrix error. Thus, system (15) can be viewed as a nominal system, perturbed by a bounded term of the orientation error, which is assumed to converge to zero faster than the closed-loop translational dynamics by ensuring that or, in other words, by designing an attitude control law that ensures that converges to as time goes to infinity. For this reason and to design the position control law, we will make the following assumption.
Assumption 6. Since the closed-loop attitude dynamics is faster than the translational dynamics, then it is logical to consider that .
Remark 7. It is important to mention that Assumption 6 is only useful to help for the design of the control law position without explicitly accounting for the orientation tracking error in the translational dynamics. The stability analysis of the resulting translational error dynamics with the coupling term will be investigated in detail later on in Section 5.
To help for the development of the decentralized cooperative controller, we first define the following state variable: where from the definition of the relative distance, it can be easily seen that which can be shown to be constrained as where is a performance function chosen similar to , such that it enforces transient and steady state performance specifications on the state variable . As such, , where are performance specifications and . Owing to the decreasing property of the performance function, one can see that and serve as the maximum allowed overshoot and the minimum allowed undershot of , respectively. The control design proceeds as follows:
Step 1. It is important to transform the control problem with the constraint (18) into an unconstrained one. To this end, the following barrier function is defined; it is a subtle treatment for ensuring collision avoidance and connectivity maintenance :
Clearly, from definition (19), if the variable is bounded, then remains within the interval for all ; therefore, the constraints in (18) are satisfied. Furthermore, it is straightforward to show that , then when , we have .
For the dynamic system (20), design the following velocity profile: where is a design parameter, denotes the minimum eigenvalue, and is the estimate of the target velocity in which position is only available to the leader vehicle. The estimate of the target velocity is the output of the following filter designed in a similar way as in : where and , and are design parameters, , and is the interception error between the leader quadrotor and the target, with .
Step 2. In this step, define the velocity tracking error as
Note that the term is temporarily ignored at this stage of the design. It will, however, be investigated in a later stage of the stability analysis. It becomes easy from (25) to select the thrust force that stabilizes to the origin. It is also important to point out that the total thrust force to be designed must be limited such that actuators’ physical limitations are not violated. By defining the saturated input and the nominal input as , then this unsaturated input is designed as where k1i and k2i are design parameters and is an auxiliary state system generated as where , is a small number; , are design constants; , and . More discussion about the auxiliary system to overcome the saturation effect of the control input can be found in . is the estimate of the disturbance , generated using the following disturbance observer : where is s a symmetric and positive definite matrix.
The following proposition summarizes the distributed cooperative control for the translational dynamics of the set of quadrotors in the formation.
Proposition 8. Consider the formation of a group of quadrotors, given the translational dynamics for each quadrotor in group (15), under Assumptions 5 and 6, if the nominal thrust magnitude and its direction are selected according to (30) and (31) where is the estimate of the target velocity using filter (23), then the equilibrium of the closed-loop system is globally asymptotically stable (GAS). Furthermore, the velocity vector of the leader quadrotor uniformly semiglobally practical asymptotically converges to .
Proof. The proof proceeds in two folds: In a first stage, we show that the desired formation is attained with prescribed transient performance, while avoiding interagent collision and connectivity intermittent by ensuring that the leader vehicle tracks an estimate of the target’s state quite accurately. In a second stage, we show that the estimate of the target’s velocity converges to a vicinity of its true value. To proceed, we first define the following candidate Lyapunov function:
where . Differentiating along the solutions of (20) results in
where , , and . Let , then adding and subtracting the term and substituting the expression , from the velocity protocol (22), using Lemmas 1 and 2, we obtain
where and is a parameter chosen such that . Note that the second term on the right-hand side of inequality (34) will be dealt with in a further step of the backstepping procedure. In the sequel, let us consider the following positive definite Lyapunov function:
where represents the estimation error of the disturbances affecting the transitional dynamics of the follower quadrotor. Take the time derivative of (35) along the solutions of (25), (27), (28), and (34) and substituting the control (30) and (31) we obtain
where . From (36), it is clear that ; therefore, is decreasing for all from which it can be concluded that , and are all bounded. Thus, there exists a positive constant such that , which from (19), by noting that , we have . That is, remains within the compact set of , from which it is straightforward to conclude that relative distance performance among the quadrotors in group (18) is guaranteed. Additionally, from (36), the equilibrium point is exponentially stable for all .
Thus, it can be concluded that . Also, from (22), it can be deduced that .
The second stage of the proof is to show that the estimate of the leader’s velocity converges to a neighborhood of the target’s true velocity. In this direction, following , a Lyapunov function candidate is proposed as , where . The time derivative of along the solutions of filter (23) yields Since we showed that , is bounded and so is , also by construction, is bounded. It follows that there exists a constant such that; therefore, where . Letting be a given positive constant, we obtain for the following bound for since ; therefore, ; this shows that the solution of filter (23) is uniform semipractically asymptotically stable . This completes the proof.
4.2. Control Design for the Attitude Inner-Loop
In this stage, the last two equations of (9) are considered. The moment vector will be designed to globally asymptotically and locally exponentially stabilize the tracking error and the errors between the virtual controls of the roll and pitch angles and their actual values at the origin.
Step 1. Controlling the yaw angle.
The desired orientation can be deduced from the expression of the rotation matrix (2) using its Euler angle parameterization subject to the constraint given by the specification of the desired yaw angle extracted from the translational system as shown in Figure 2. Indeed, in Figure 2, the angle represents the desired angle associated with the vector projected on the - plan. Clearly, when matches with , the yaw rate angle converges to zero. Following , the desired yaw angle is calculated by extracting yaw rate information from the cross- and inner-product of the desired velocity and the actual velocity as follows: which implies that . Therefore, the desired yaw angle can be obtained by the following expression:
Note also that the time derivative of can be calculated as indicated in (43). Therefore, we can write where is the angular velocity error, with yet to be determined later. and is the stabilizing function, which will be chosen such that it cancels out unneeded terms in (42). As such, the stabilizing function can be chosen as follows: where is a positive gain. Substituting (44) into (42) results in the following closed-loop dynamics:
Clearly, if the last term of (45) is exponentially vanishing, then the tracking error is also exponentially stable; this can be seen by associating the quadratic Lyapunov function , whose time derivative along the solutions of (45) gives where . Hence, the virtual control law (44) renders the closed-loop (45) system input-to-state stability (ISS) from to . Therefore, if is globally exponentially stable GES, then so is.
Remark 9. It has been shown in  that overturn of the quadrotor can only be avoided if the last component of is ensured to be strictly positive. This constraint further requires that (with is a vector that contains the first two component of ) exponentially decays to zero in order to guarantee that and exponentially. Since , it is sufficient to show through an adequate control law that
Remark 10. The virtual control (44) is well defined as long as ; this singularity is avoided as long as the term is strictly positive or alternatively
Step 2. Controlling the pitch and roll angles.
The pitch and roll angles are contained in the column vector and are denoted by the vector . The time derivative of along the solutions of the first equation of (9) and (5) gives the following dynamics: where and the matrix can be obtained by a long but simple calculation as
The matrix is invertible, due to the fact that , which implies that as long as the singularity (i.e., ) is avoided, the matrix is nonsingular. The control objective is therefore to ensure on top of forcing the pitch and roll angles to track their immediate desired values and to also prevent them from violating the constraint that leads to singularity. In other words, the state variable is required to remain in the set for all To address this problem, the integral barrier Lyapunov function iBLF  will be employed to guarantee nonviolation of the aforementioned constraint.
Motivated by the work in , in order to design the virtual control , the following integral barrier Lyapunov candidate functional is proposed: where . It can be seen that is positive definite and continuously differentiable and satisfies the decrescent condition in the set such that for , we have
Based on the definition in (49), the time derivative of the iBLF along the solutions of (47) yields where and with the vector 1 being all components 1. We can obtain the stabilizing function as where is a positive gain to be designed. Under (52), the time derivative of is given by at this step according to (53); the time derivative of becomes negative definite only when exponentially converges to zero. The design of the force moment is not finished yet. One more step is required to ensure that exponentially converges to zero.
Step 3. Force moment control design.
The angular velocity error dynamics based on the second equation of (9) has the following form:
At this stage, the control input for the force moment can be readily designed to compensate for the coupling terms created in the previous steps and the external perturbation to which the quadrotor is subject. To this end, let the control input be designed as follows: