Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006
ThA02.6
A Cooperative Control Structure for UAV’s Executing a Cooperative Ground Moving Target Engagement (CGMTE) Scenario Eric J. Barth, Member, IEEE
Abstract—This paper presents a cooperative control structure appropriate for cooperative tasks requiring a high degree of tightly coupled coordinated action by multiple dynamic vehicles. The method is based on the formation of a cooperative control Lyapunov function (CCLF). The CCLF is a positive definite function involving the states of the cooperating vehicles and provides a metric of the team’s performance useful for higher level assignment algorithms. Low level control laws are obtained by separating the CCLF into multiple agent control functions (ACFs). Forcing the time derivative of the ACFs to be negative definite gives rise to individual control laws for each agent and assures decrement of the CCLF and therefore convergence of the team. The cooperative ground moving target engagement (CGMTE) scenario is utilized as a case study. Simulation results are shown.
cooperative control, significant issues with regard to both theory and implementation still exist. One such issue is the apparent disconnect between assignment and low-level control. Assignment is with regard to the selection of available vehicles and the selection of a sequence of available tasks. Ideally a solution technique capable of finding the optimal assignment is employed. Assignment is typically approached as a static optimization problem by considering Euclidean distances between vehicles and targets (tasks). Low-level control is with regard to achieving a desired path or trajectory of each individual vehicle, toward way-points dictated by the assignment algorithm, by using feedback control on-board the vehicle.
I. INTRODUCTION
Assignment and low-level control can be decoupled for non-cooperative tasks by considering only “flyable” trajectories as dictated by the kinematics that result from the system dynamics of a vehicle such as a UAV. However, for cooperative or joint tasks that involve a high degree of precision and coordinated action in time, the dynamics cannot be decoupled from the planning and assignment. Given that the planned trajectories must depend very closely on the states of those vehicles being cooperated with, in addition to their kinematic “flyable” trajectories, the path planning becomes increasingly dependent on the accuracy of the dynamic models involved. This added degree of precision associated with joint tasks essentially converts the problem from one requiring decisions to be made at discrete points in time, to one that requires a continuous decision process. Such a continuous decision process is best addressed within the context of feedback control. The approach presented in this paper addresses the combined path planning and feedback control requirements associated with one particular example joint task. As will be seen, the method also generates a team metric that will be useful for higher level assignment algorithms.
T
HERE are two classes of robotic cooperative control problems: those concerning manipulators and those concerning mobile robots. The most straightforward examples of cooperative control of mobile robots involve groups or teams of autonomous vehicles cooperating to achieve a common goal. The “cooperative” aspect of the problem implies that no benefit is derived by the team if only a single vehicle performs a task, and that a high degree of tightly coupled, coordinated action is required. Such tasks are often termed “joint tasks” in military circles. Interest in cooperative control of mobile robots has been heightened in recent years mainly due to future planned scenarios involving multiple Unmanned Aerial Vehicles (UAVs). Typical military scenarios include the Cooperative Ground Moving Target Engagement (CGMTE) scenario, and Suppression of Enemy Air Defenses (SEAD) missions, among others. Typical non-military scenarios include coordinated search and rescue either at sea or on/over land, and cooperative forest and brush fire suppression, among others. Despite the heightened research activity in the area of Manuscript received September 23, 2005. This work was supported in part by an ASEE Faculty Fellowship at AFRL/VACA, Wright Patterson Air Force Base. E. J. Barth is with the Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235-1592 USA, (phone: 615-3221893; fax: 615-343-6925; e-mail:
[email protected]).
1-4244-0210-7/06/$20.00 ©2006 IEEE
II. THE COOPERATIVE CONTROL LYAPUNOV FUNCTION (CCLF) APPROACH A significant challenge that faces path planning for nonholonomic vehicles is the turning rate constraint. This constraint is obvious given that non-holonomic vehicles such as aircraft, passenger cars, and wheeled mobile robots have limited turning radii. The early work of Dubins [11]
2183
established interest in the static optimization problem of trajectory or path planning for a single non-holonomic vehicle with minimum turning radius constraints. Such turning rate input constraints result in non-trivial optimal path lengths given an initial and final position and orientation. Murray [23] and Tilbury [38] (sinusoids), Leonard [18], Bloch [6, 7, 8], and Nair [25] (controlled Lagrangians), Nair [24] and Åström [1] (energy approaches), and others [17, 40] have also tackled various aspects of the constrained input problem. These approaches are motivated to find the optimal path in closed-form. The approach taken in the work presented here will be to treat such turning rate constraints as input constraints within a feedback structure. That is, a closed-form optimal path will not be sought but rather a control law will continuously seek to minimize a tracking metric. In instances where the input constraint prohibs further reduction of this metric, the feedback system will in effect wait until such an opportunity arises. The assignment of multiple available vehicles to various tasks has been approached by many including Passino [30], Beard [3, 4], Liu [20], Chandler [9], Schumacher [36], Pachter [28], and others [22, 31, 21, 26], but typically the problem is approached from a non-dynamical systems point of view. Cooperative path planning via mixed integer linear programming is a popular non-dynamical systems approach [34, 37]. Various other approaches cast the problem in terms of a team metric. Such approaches for assignment type problems include: satisficing approaches [32, 29, 12, 10, 16], Lyapunov functionals [15], Lyapunov certificates [39], probability of loss [5], behavior-based approaches [35, 2], consensus approaches [27, 33], biologically inspired strategies [13, 19], and auction methods [14]. Despite all these approaches, a significant challenge remains in relating any such cooperative team metric useful to an assignment algorithm to the dynamics of the vehicles. The unification of vehicle dynamics and real-time cooperative path planning and assignment remains a difficult challenge. The approach taken here casts the problem in terms of a team metric that is not only useful in terms of assignment, but can be used directly to obtain the control laws for each vehicle.
CCLF (the generalized velocity toward the goal) is designed to be a particular analytical function, then the generalized time to reach the goal can be solved in closed form. In this manner the CCLF not only specifies the required control inputs for the vehicles, it serves as an optimal return function (ORF) that indicates the current cost of assigning the vehicles involved to a particular task. In terms of assignment (single tour), various team tasks and their associated ORF’s can be evaluated in order to select the optimal task for the team from among a discrete set of possible team tasks (optimal in the sense of the ORF plus the cost to switch tasks). Various combinations of different vehicles available to be assigned to a task (the assignment problem) can then be selected by continuously evaluating ORF’s for each combination. To illustrate the approach, a Cooperative Ground Moving Target Engagement (CGMTE) scenario will be considered. Two cases will be considered – the first case will involve cooperatively acquiring the location of a stationary target, and the second will involve a moving target. The ability of the method to provide real-time robust path planning and control will be illustrated by both cases. The ability of the method to continuously re-plan will be seen in the second case. Both cases will also illustrate the utility of the CCLF in defining the role of each member or agent of the cooperative team. III. CGMTE – STATIONARY TARGET In order to illustrate how the CCLF can be used to separate the cooperative control task into “agents” representing the individual vehicles, and how each agent can be symmetric (i.e. look identical to other agents) and exercise a local control law (from the ACF) that contributes constructively along with other ACF’s to minimizing the global cost (from the CCLF), consider the following example. Consider the simplified problem of two cooperating non-holonomic vehicles that must establish a
The approach taken here is to formulate a cooperative control Lyapunov function (CCLF) that assigns a positive definite metric to the current performance of a team of vehicles in achieving a given joint task. The CCLF will be a function of the states of the vehicles in the team such that the control inputs of each vehicle appear in the time derivative of the CCLF. In a manner akin to sliding mode control, where the time derivative of a positive definite control Lyapunov function is designed to be negative definite and of a particular form, the time derivative of the CCLF will be prescribed. If the CCLF is viewed as a generalized distance to a goal, and the time derivative of the 2184
UAV2 Sensor Footprint for UAV1 Sensor Footprint for UAV2 Velocity vector of ground target UAV1 Sensor detection error ellipses Doppler Visibility Cone
Fig. 1. Cooperative Ground Moving Target Engagement (CGMTE) problem showing the cooperative requirements of two UAV’s. The UAV’s must be within the Doppler visibility cone, with their sensor footprints over the target, and with the error detection ellipses of each sensor contributing to a more precise and accurate localization of the target than achievable by either UAV individually.
particular orbit around a target with a particular angular separation between the two vehicles. This problem is considered as a simplified first step toward solving the Cooperative Ground Moving Target Engagement (CGMTE) problem. The scenario considered is shown in Figure 1. In such a CGMTE problem, it is necessary for more than one UAV to cooperatively track the location of a ground target. The UAVs must cooperatively overlay their sensor footprints so that their respective sensor detection error ellipses overlap in such a way as to locate the target with sufficient accuracy. The problem also requires that each UAV stay within a range such that their Doppler radar systems obtain an adequately detectable relative velocity between themselves and the target. This requirement is shown by the “Doppler visibility cone” in Figure 1. To simplify the problem for consideration as a cooperative control problem, the sensor overlay requirement (maintaining a 90 degree sensor error ellipse overlay) will be considered, but the Doppler visibility requirement will be dropped. Consider a two dimensional representation of the problem where the planar dynamics of each vehicle traveling at constant speed are given by the so-called Dubin’s car model [11]:
x v cos(T) y v sin(T) T u
(1c)
To establish the required positions of the two vehicles cooperatively, three error metrics are required. The first two metrics pertain to each vehicle being at a prescribed radius rd from the ground target. Additionally, in order to place the vehicles at 90 degrees of each other for the proper sensor fusion, the distances between the vehicles must be maintained as rd 2 . To satisfy the first two requirements, we wish to drive the following error to zero for each vehicle ( e1 and e2 ):
where r
(r12 rd2 ) and e2
(r22 rd2 )
(2)
1
2rr 2r 2 ( xx yy ) 2 2( x 2 y 2 ) 2( xy xy )u
er
( x2 x1 ) 2 ( y2 y1 ) 2 2rd2
(6)
Note the following relative separation error dynamics:
2( x1 x2 )( x1 x 2 ) 2( y1 y2 )( y1 y 2 )
er er
2( x1 x 2 ) 2 2( x1 x2 )( y1u1 y 2 u 2 ) 2( y1 y 2 ) 2 2( y1 y 2 )( x1u1 x 2 u 2 )
(7) (8)
Consider the following cooperative control Lyapunov function (CCLF) candidate:
V
1 2 2 1
s 12 s22
kr 2
sr2
(9)
where driving this value to zero will establish the following three invariant manifolds which in turn specify desired first order stable error dynamics regarding e1 , e2 , and er with a relative weighting of k r : s1 e1 O1e1 , s2 e2 O 2 e2 ,
sr er O r er . In order to drive the CCLF to zero, take the derivative with respect to time and then force it to be negative definite. V
s1s1 s2 s2 k r sr sr
(10)
A substitution of appropriate relations given by the equations above results in:
V
f1 b1u1 f 2 b2 u 2 2 f r br1u1 br 2 u 2
(11)
where f1 and b1 are functions solely of states regarding vehicle 1, and f 2 and b2 are functions solely of states regarding vehicle 2: f1
For each individual vehicle, note the following error dynamics in terms of the system dynamics:
2rr 2rd rd
(5)
To satisfy the coupling requirement for the vehicles being a certain distance apart, we wish to drive the relative separation distance error to zero:
x2 y2 .
e
(4)
where r ( x 2 y 2 ) 2 ( xx yy ) , and the input u appears in the 2rr term of Equation (4):,
(1a) (1b)
where x and y are the Cartesian coordinates of the vehicle, T is the heading angle, v is the velocity of the vehicle (assumed constant), and u is the control variable representing the heading angle rate of change.
e1
e 2r 2 2rr 2rd2 2rd rd
(3)
s1[2r12 2O1r1r1 2r12 ( x1 x1 y1 y1 ) 2
2( x12 y12 )] b1 2s1 ( x1 y1 x1 y1 ) f2
s 2 [2r22 2O 2 r2 r2 2r22 ( x2 x 2 y 2 y 2 ) 2 2( x 22 y 22 )]
2185
(12) (13) (14)
b2
2s2 ( x 2 y2 x2 y 2 )
(15)
Functions f r , br1 and br 2 contain the states of both vehicles and represent the coupling introduced between the vehicles by virtue of the desired error dynamics prescribed by invariant manifolds s1 , s2 and sr . These coupling functions are given by the following: (16)
br1
k r sr [2( x1 x 2 ) 2 2( y1 y 2 ) 2 O r er ] k r sr [2( x1 x2 )( y1 ) 2( y1 y2 )( x1 )]
br 2
k r sr [2( x1 x2 )( y 2 ) 2( y1 y2 )( x 2 )]
(18)
2 fr
(17)
The f and b notation has been adopted due to the fact that these functions express the dynamics of the overall cooperative system in a state space representation of the desired cooperative task. We now wish to ensure that V as given by Equation (11) is negative definite by dividing this task up in some kind of symmetric way between the two vehicles and in a way so that a vehicle can utilize information about the other vehicle (in the form of states, a combination of states, or a subset of states) in the same manner that the other vehicle (identical and doing the identical task) uses information about it. Likewise we could consider the asymmetric case where one vehicle has more responsibility for the task than the other, but this is not currently under consideration (although such considerations would be appropriate for heterogeneous teams of vehicles). We will assume that each vehicle has full state feedback of its own states. One sufficient condition for accomplishing this is to separate V into two symmetric contributions and then make each contribution negative definite. The sum of two negative definite functions is in turn a negative definite function. Consider the following separation of V into two symmetric contributions:
V
the following: f1 f r (b1 br1 )u1
K1 b1 br1
1 2 2 1
s
kr 4
sr2 (20a)
Choose u 2 to enforce the following: f 2 f r (b2 br 2 )u 2
K 2 b2 br 2
1 2
s 22
kr 4
sr2 (20b)
The right hand sides of Equations (20) are negative definite by choice of positive coefficients K1 and K 2 , selected to account for modeling error bounds in the typical sliding mode control sense. This gives rise to the following control laws in the typical sliding mode control law form:
u1
u2
( f1 f r ) K1 sgn(b1 br1 ) (b1 br1 )
1 2 2 1
( f2 fr ) K 2 sgn(b2 br 2 ) (b2 br 2 )
1 2
s
s22
kr 4
sr2
kr 4
(21a)
sr2 (21b)
f1 f r (b1 br1 )u1
(19b)
f 2 f r (b2 br 2 )u 2
(19c)
K1
f1 f r (b1 br1 )u1 f 2 f r (b2 br 2 )u 2 Vehicle 2 responsible for enforcing this portion negative definite
(19a)
V1 V2
2
T 2 , x2 and y 2 by utilizing vehicle 2’s known dynamic equations, Equations (1a). Therefore vehicle 1 can exploit the fact that it knows the dynamic behavior of vehicle 2 in order to construct an observer (in general this may also require knowledge of vehicle 2’s control u2 ). Once the required information is obtained, each vehicle can enforce the following relationships in order to make their respective contributions to V negative definite. Choose u1 to enforce
Also note (from the V term of Equation (19)) that each vehicle only acts based on its contribution to the positive definite scalar metric of the overall global error given by V (this must be the case given that u1 cannot influence 12 s22 and vice-versa). A typical run of this control law formulation is shown in Figures 2 and 3 (values used were: O1 O 2 2S / 200 , O r 2S / 50 , kr 3 , v 1,
Vehicle 1 responsible for enforcing this portion negative definite
V1 V
y 2 . An important point general to cooperative control among vehicles with knowledge about the other vehicles is the following: vehicle 1 has the ability to use its knowledge of vehicle 2’s dynamics in order to obtain x 2 and y 2 from
The division of V into these contributions constitutes the formation of the time derivative of the Agent Control Functions (ACF’s). In order for each vehicle to make its contributed ACF rate negative definite, each needs information regarding the other vehicle as dictated by functions f r , br1 (for vehicle 1). Discussing this information sharing from the point of view of vehicle 1’s responsibilities toward Equation (19a), it will need the following information about vehicle 2: T 2 , x2 , x 2 , y 2 and
K2
0.3 , rd
5 , with u1 , u 2 [1, 1] ).
Figure 2 illustrates the two vehicles cooperatively establishing the required orbit around a stationary ground target, located at the origin, from random initial positions and orientations. Figure 3 shows a snapshot of the positions of the two vehicles after they have converged on the cooperative orbit.
2186
e1 e1
20
2( x1 xT )( x1 x T ) 2( y1 yT )( y1 y T )
2( x1 x T ) 2( y1 y T ) 2 2[( y1 yT ) x1 ( x1 xT ) y1 ] u1 a1
15
(23)
2
b1
(24)
a1 b1 u1
10
where terms a1 and b1 have been defined for notational compactness. Note that the input term appears in equation (24). Likewise, vehicle 2’s error, and derivative thereof, in maintaining a prescribed radius of rd can be stated as:
y
5 0 -5 -10
-20 -20
( x2 xT ) 2 ( y2 yT ) 2 rd2 2( x2 xT )( x2 x T ) 2( y2 yT )( y 2 y T )
e2
-15
e2 -15
-10
-5
0 x
5
10
15
e2
20
(26)
2( x 2 x T ) 2 2( y 2 y T ) 2 2[( y 2 yT ) x 2 ( x2 xT ) y 2 ] u 2 (27) a b 2
Fig. 2: Control performance of the CCLF/ACF approach in cooperatively establishing an orbit of two vehicles around a target (0,0) with a specified radial distance and a specified angular separation (relative distance). The plot shows the trajectory of the two vehicles in establishing the cooperative orbit.
(25)
2
a2 b2 u 2
The relative separation error dynamics are given by the following:
( x1 x2 ) 2 ( y1 y2 ) 2 2rd2 2( x1 x2 )( x1 x 2 ) 2( y1 y2 )( y1 y 2 )
er
10
er
8
er
6 4
2
(29)
2
2( x1 x 2 ) 2( y1 y 2 ) 2[( y1 y 2 ) x1 ( x1 x2 ) y1 ] u1 ar
b r1
2[( x1 x 2 ) y 2 ( y1 y 2 ) x 2 ] u 2
2
(28)
(30)
b r2
0
a r b1r u1 b2 r u 2
-2 -4
The CCLF that combines all three objectives is given as before as:
-6 -8 -10 -10
-8
-6
-4
-2
0
2
4
6
8
V
10
Fig. 3: Steady-state control performance of the CCLF/ACF approach in cooperatively establishing an orbit of two vehicles around a target (0,0) with a specified radial distance and a specified angular separation (relative distance). The plot shows a snapshot of the desired relative angular separation of 90 degrees after the vehicles have converged to the desired cooperative orbit.
s1
To consider cooperative tracking of a moving target within the context of the CCLF method, only the objectives need to be changed. Denoting the target position (or estimate thereof) by xT and yT , vehicle 1’s error in maintaining a
Derivatives thereof are the following:
kr 2
sr2
(31)
s1 s2 sr
(22)
(32)
s2
e1 O1e1 e2 O 2 e2
sr
er O r er
(34)
(33)
Derivatives of these relationships are given by the following:
prescribed radius of rd can be stated as:
( x1 xT ) 2 ( y1 yT ) 2 rd2
s12 12 s22
with the following sliding surfaces defining stable error dynamics specified by the constants O1 , O 2 , and O r :
IV. CGMTE – MOVING TARGET
e1
1 2
e1 O1e1 e2 O 2 e2
a1 b1u1 O1e1 a2 b2u 2 O 2 e2
ar br1u1 br 2u2 O r er
(35) (36) (37)
The CCLF can be divided into two symmetric ACF’s accordingly:
2187
kr 2 1 2 1 2 12 s12 12 s22 k4r sr2 1 2 s 2 4 sr 2 s
V
V1
(38)
25
V2
20 15
The derivative of the positive definite CCLF is given by:
10
V
V1 V2
5
(39)
0
where,
-5 -10
V1
s1 ( a1 O1e1 ) k2 sr ( ar O r er ) ( s1b1 k r sr br1 ) u1 r
-15
(40)
g1
f1
-20
f1 g1u1 V2
-25 -25
s2 ( a2 O 2 e2 ) k2 sr (a r O r er ) ( s2b2 k r sr br 2 ) u 2 r
-15
-10
-5
0
5
10
15
20
25
(41)
g2
f2
-20
Fig. 5: Cooperative tracking of a moving target executing sudden turns.
f 2 g 2u 2
Forcing these to be negative definite according to the following prescriptions,
25 20
V1
k1 g1 V1
(42)
V2
k 2 g 2 V2
(43)
15 10 5
results in the following control laws for each agent: u1 u2
0 -5
f 1 k1 sgn( g1 ) V1 g1
(44)
f2 k 2 sgn( g 2 ) V2 g2
(45)
-10 -15 -20 -25 -25
A simulation of this moving target scenario is shown in Figures 4, 5 and 6.
25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 -30
-25
-20
-15
-10
-5
0
5
10
15
20
25
Fig. 4: Cooperative tracking of a target moving at constant speed and constant turning rate.
-20
-15
-10
-5
0
5
10
15
20
25
Fig. 6: Cooperative tracking of a moving target executing sudden turns.
Figure 4 shows two vehicles cooperatively tracking a target moving at a constant speed and with a constant turning rate. It is interesting to note the loiter maneuver executed by one of the vehicles while waiting for the second vehicle, which started farther away, to arrive on the scene. This loiter behavior is a consequence of the methodology but was not explicitly specified. This and other complex but logical behaviors are an advantage of the method. Figures 5 and 6 show scenarios where the target turns suddenly, as might happen if a truck or tank turns at an intersection of two roads. Due to the difference in the starting positions and orientations of the two vehicles, Figure 5 shows the two vehicles orbiting the target always in a counter-clockwise sense, whereas Figure 6 shows the vehicles changing from clockwise to counter-clockwise near the third turn executed by the target vehicle. This illustrates the continuous replanning quality of the method. This is logical given that the vehicles do not know future turns of the target vehicle, but 2188
rather are continually reassessing the best course of action based on current target states. In all three figures, the final positions of the two cooperating vehicles can be observed as having achieved the required relative separation orientation of 90 degrees with respect to the target.
[4]
[5]
[6]
V. CONCLUSION A cooperative control method based on a cooperative control Lyapunov function (CCLF) was presented. As a case study for the performance of the control method, the Cooperative Ground Moving Target Engagement scenario was used to formulate the controllers for the two cooperating non-holonomic vehicles. Cases involving both stationary and moving targets were considered.
[7]
[8]
[9]
It was observed that the method provides a metric of the team’s current performance that results in an agile continuous re-planning quality. Given that circumstances in the battlesphere are apt to change suddenly, this seems an appropriate feature. By providing a team metric, the method should also lend itself well to high-level assignment algorithms. Such algorithms often use simple Euclidean distances to evaluate costs. The CCLF value associated with forming different teams of vehicles pursuing different targets would be an appropriate replacement for such cost estimates. Furthermore, the CCLF is based on the dynamics not only of the individual vehicles, but also on the dynamics associated with the whole team relative to the goal. These and other features provide promise for relating low and mid level planning together for cooperative control scenarios. Various observations regarding possible extensions to the cooperative control structure presented here can also be made. By splitting the CCLF into two symmetric ACF’s, the role of each agent was explicitly defined. Although this was performed in a symmetric sense here, heterogeneous cooperating vehicles may require an asymmetric separation of the CCLF. The method would allow for this extension. The method would also allow for ACF’s which do not decrease monotonically in time – as long as the CCLF still does. This would suggest that individual sacrifice (temporary increase of an agent’s cost metric given by its ACF) for the benefit of the team can be included within the proposed cooperative control structure.
[10] [11]
[12]
[13] [14] [15]
[16] [17] [18] [19] [20]
[21]
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