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Formation Control of Holonomic Air-Bearing Robots Using Potential Field Interaction Laws

John Camp*, Edward LeMaster†, Scott Nortman‡, and Sherry Cordova§ Lockheed Martin Advanced Technology Center, Palo Alto CA 94304

In many distributed sensing applications such as search and rescue, environmental monitoring, planetary exploration and space-based radio interferometry, sensor coverage of an area of interest is paramount, while the precise positioning of each sensor element is unnecessary. This paper presents the results from a series of emergent behavior experiments in which autonomous robotic agents follow simple potential field interaction laws to produce organized swarms useful for such distributed sensing applications. Emergent behavior refers to the quality possessed by multi-agent systems in which complex, and often organized, behavior emerges at one scale as a result of simple behavior exhibited at smaller scales. Such systems can be robust to changing environmental conditions and the failure of individual agents, even though they lack any form of centralized control. The experimental system consists of a number of small holonomic robots floating on an air-bearing surface. In addition to experimental results, simulation results for large-scale swarms, collectives, and target environments are also reported.

I. Introduction This paper presents results from a series of formation-flying experiments in which groups of simulated

spacecraft coordinate their motions using simple potential field interaction laws. Simulation results for both 2D and 3D experiments are presented along with experimental validation of some 2D simulation results. Formation flying experiments were conducted on a family of air-bearing robots in the Distributed Space Systems Laboratory (DSSL) at Lockheed Martin’s Advanced Technology Center located in Palo Alto, CA. These robots, shown in Fig. 1, are functional analogues for typical spacecraft systems. Each robot is equipped with on-board compressed CO2 for flotation and thrusting, a reaction wheel for heading control, a camera system for heading and position determination, and an onboard computer for navigation and control calculations and wireless inter-robot communication.

In this work, the method by which the robots determine their desired positions and motions is through the use of artificially-calculated and imposed potential fields, the resulting ‘force’ from which is physically realized using the robot actuators. Potential fields are a useful method for computing the desired interaction forces between multiple robots for a number of different reasons. Physical relevance is one, since potentials describe the nature of electrical and gravitational fields. Additionally, conservation of energy arguments can provide a number of useful stability proofs. Of greater importance, potential fields add linearly, making it easy to superimpose the effects of many objects. These properties make potentials valuable for distributed systems, since the interactions can be computed locally by each agent using only knowledge of the relative locations of nearby or otherwise significant neighbors. Potential field interactions are therefore easily scalable to a large number of agents, enabling completely non-centralized formation control.

The use of potential fields for robot control is by no means new. Early work such as that by Khatib [Khatib] and Volpe and Khosla [Volpe] focused on the motion of manipulators or solitary robots through an obstacle field defined by relative potentials, and laid some of the groundwork on the relevant mathematics. More recent work has expanded the field of study to include multiple robot systems for both manipulation [Song] and general flocking or swarming behaviors [Leonard][Olfati 1][Olfati 2].

* Cooperative Control Lead, Precision Pointing and Controls, Bldg. 201, 3251 Hanover St. † DSSL Lead, Precision Pointing and Controls, Bldg. 201, 3251 Hanover St. ‡ Electronics Engineer, Precision Pointing and Controls, Bldg. 201, 3251 Hanover St. § Senior Controls Engineer, Precision Pointing and Controls, Bldg. 201, 3251 Hanover St.

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Copyright © 2004 by Lockheed Martin Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Likewise, laboratory experiments involving cooperative mobile robots are becoming more commonplace [Spletzer][Clark]. Many of these testbeds involve non-holonomic wheeled vehicles because of the simplicity of construction and the relative ease of control: motion is relatively well predicted by odometry and steering angles, and once stopped the vehicles tend to stay put. In contrast, many of the behaviors necessary for the command and control of groups of cooperating spacecraft in orbit are best demonstrated using holonomic, and preferably free-floating, robots. The DSSL testbed uses several such free-flying robots to experimentally validate a number of different control strategies, including the potential-field based controllers described in this paper.

II. Potential fields Figure 2 shows a schematic of the DSSL robots in the lab. All robots within a given swarm have identical

potential fields fixed in their respective body frames. The fact that all robots carry the same field ensures that every pair of robots shares an equal and opposite ’internal‘ force, defined to be proportional to the gradient of the potential field at each robot’s position.

Figure 1. The DSSL robots

Figure 2. Potential field schematic

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Equation (1) describes the radial interaction force experienced by the ith robot due to the potential field u(x) fixed in the jth robot. Here, the ’λhat’ is the unit vector from the jth to the ith robot and xji is the distance between the jth and the ith robot, as shown in Fig. 2.

( )

jiji

ji dxxdu

F λ̂−=v

(1)

The net force on each robot due to potential field interactions is simply the sum of each pair’s internal interaction force. Equation (2) shows the net force on the i’th robot, where n is the number of robots in the formation. This provides a simple and scalable method for calculating the gradient of the net field experienced by each robot.

( )

−Σ=

−= jiji

nj

neti dx

xduF λ̂

1:1

v (2)

In each experiment, the potential field governing radial inter-agent forcing is defined as a polynomial of order N scaled by a decaying exponential resulting in small interaction forces at large relative displacements. Equation (3) shows the general form of this axisymmetric potential field,

( ) ( ) xCN

NN NeCxCxCxu 1...121

+−− +++= (3)

where the positive x axis lies in the direction from the agent “carrying” the potential field to the agent “experiencing” the field, and N is dictated by the number of constraints on the desired field. The polynomial terms allow one to create localized ‘forces’ of varying complexity, while the exponential mimics the effect of many real potentials and insures that agents only exert an appreciable influence on each other if they are in the general vicinity.

The results shown in this paper utilize a potential fields of the form (3) with N=1 or 2. While using a higher-order polynomial gives the designer greater freedom in designing agent interactions, care must be taken that each additional degree of freedom does not introduce an undesirable behavior. This is most easily done by keeping the order of the polynomial at the minimum that is necessary for the desired interaction, in this case one. The specific form of the potential field for N=1 is given by Eq. (4).

( ) ( ) xCeCxCxu 321

−+= (4)

The process of calculating the desired potential field has two-steps, first to identify a generalized field with the desired overall properties and then to scale this generalized potential to the hardware or the physical situation at hand.

A. Generalized Potential Fields The 3 free parameters in Eq. (4) allow for 3 constraints to be imposed on the desired potential field. The three

constraints chosen at this stage are given below, and yield the field illustrated in Fig. 3:

1 @ ,0)( minmin == xxdxdu

(5)

( ) 1minmin −== uxu (6)

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( ) 00 uu = (7)

In this general potential field, a local minimum of unit depth is located at a unit distance. This is the preferred distance of each agent with respect to every other agent. The use of a first order polynomial guarantees that this minimum is also a global minimum. Additionally, the height of the potential field at x=0 is set to u0. Constraining u0 to be greater than zero allows for some additional level of collision avoidance control for lightly damped cases.

One can see from Eqs. (4) and (7) that the constraint on uo gives

02 uC = (8)

Looking to the derivative of u(x) at x=xmin=1 one can write C1 in terms of C2 and C3.

( ) ( )( ) 011

33231 =−−= −CeCCCC

dxdu

(9)

therefore,

( )3

321 1 C

CCC−

= (10)

Substituting Eqs. (8) and (10) into Eq. (4) we have,

( ) ( )xCeux

CCuxu 3

03

30

1−

+

−

= (11)

Equation (11) and constraint Eq. (6) yield an equation strictly in terms of C3,

( ) 11 min0

3

30 3 −==

+

−

− ueuCCu C (12)

or,

( ) 011

30

3

30 =+

+

−

−CeuCCu

(13)

The nonlinearity of Eq. (13) requires that numerical methods be used to obtain C3. In the example shown in Fig.

3, where u0=0.5, a Newton-Raphson search was used to calculate C3. Figure 4 shows the full axisymmetric generalized field produced utilizing a potential field of the type shown in

Fig. 3. Note that this field constrains radial motion between two agents, but leaves them free to rotate azimuthally about each other. If this behavior is undesirable, additional nonaxisymmetric potentials or damping terms may be added to constrain this degree of freedom.

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B. Scaling and Physical Realization The final step in calculating an experimentally useful potential field involves modifying the generalized potential

field calculated above in order to achieve the desired scale for realization on the physical system. Firstly the distance from each robot to its potential well must be scaled from a unit value to that which is desired

in the experimental system. To achieve the proper distance to the potential well, a change of variables is made such that,

( ) 0min =

′xdxdu lab

(14)

The scaling between x and ‘xprime’ is given by the ratio,

lablab xxx

xx

minmin

min 1==′

(15)

so,

labxxxmin

′= (16)

Equation (2) then becomes,

( )

′−

+

′=′

labxxC

lab eCxxCxu min

3

2min

1 (17)

Finally, the curvature of the potential field at the potential well can be thought of as a proportional control gain (kp) for small displacements from the well’s minimum. Given that kp dictates the dynamics of the potential field about the local minimum, it is often desirable to assign a particular value to kp. To achieve a desired kp an additional gain is introduced to scale the entire field after the final well location has been determined. The final potential field to be used in the lab can then be expressed as,

Figure 4. Potential field surface plot Figure 3. Inter-Agent potential field interactionlaw

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( )

′−

+

′=′′

labxxC

lab eCxxCKxu min

3

2min

1 (18)

where K is selected such that,

( )

p

lab

kxdxud =′

′2min

2

(19)

C. Target Tracking The basic potential fields outlined above govern the relative spacing between numerous agents. These agents can

be induced to ‘swarm’ around one or more target objects by assigning an additional potential field to each target, which attracts the mobile tracking agents.

In multi-target situations, a high-level collective assigns each tracker to a particular target swarm. In the scenarios studied at the DSSL, this collective assigns trackers in equal numbers to each target swarm utilizing a sorting algorithm that minimizes the system’s maximum tracker-to-target distance. This collective continually reassess the appropriateness of the target swarms, enabling it to reassign trackers in real-time as dictated by the movement of both targets and trackers. An example grouping is shown in Fig. 5.

In normal operation, a tracker only feels the

influence of other trackers and the target within its assigned swarm. To avoid inter-swarm collisions, each tracker is assigned a keep-out radius, shown in Fig. 6, within which it experiences the repulsive effects of trackers/targets from other swarms.

A potential field of the form given in Eq. (3) is assigned to each target. Figure 7 shows the relative magnitudes of example tracker and target potential fields.

Figure 5. Example collective target grouping

Figure 7. Target and tracker potential fields

Figure 6. Collision avoidance keep-out regions

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D. Damping Potential fields are conservative by nature, which can cause undesirable

oscillations in the system. To ensure formation settling, damping was added to the system through internal radial and tangential virtual dashpots. These inter-agent dashpots are shown schematically in Fig. 8. To implement these dashpots, a virtual ‘internal’ force is applied between each agent pair that is proportional to, and opposite in direction to, their relative radial and tangential velocities.

III. Simulation

A. 2D Potential Field Swarming In order to evaluate the effectiveness of these potential functions for developing emergent behaviors, a number

of simulations were conducted using the DSSL robots as a guide. These robots were modeled as point masses free to move holonomically and frictionlessly in a plane.

Figure 9 shows the most basic simulated swarming behavior using a potential field of the form given in Eq. (3). The robots start in random positions, and then travel down the gradients of their virtual potential fields to arrive at their final positions in the formation. In the case of five robots, a pentagon is the most stable configuration of the formation. Another stable formation possibility is a square with a single robot in the center.

B. 2D Target Selection/Tracking Additional simulations were conducted of the target-tracking behavior described in Section 2.3. In the first

example below (Fig.s 10, 11 and 12), two stationary targets, designated by boxes, are surrounded by a cloud of randomly-spaced tracking agents. A high-level collective assigns each agent to a particular target, based upon their relative proximity. At steady-state, each target is surrounded by a uniform distribution of tracking agents.

Figure 8. Dashpot schematic

Figure 9. Trajectories from 2D swarming simulation

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The second example shows how the real-time target selection algorithm and inter-group collision avoidance scheme interact to dynamically reassign groups in the face of unanticipated target maneuvering. In this example, shown in Fig. 13, the targets’ movement forces certain trackers to initiate inter-group collision avoidance. This, in turn, skews each target grouping enough to force a reassignment of certain trackers. The shading of the original target groups is retained to reveal the reassignment.

Figure 13. Collision avoidance and target reassignment

Figure 12. Full trajectories from 2D target-tracking simulation

Figure 10. Initial conditions of 2D target-tracking simulation

Figure 11. Final conditions of 2D target-tracking simulation

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Figure 13a shows the two target groups at the onset of inter-group collision avoidance action. The approximate target velocities are expressed by the associated vectors.

Figure 13b shows the two groups during the interaction period. At this point the individual swarms have deformed and rotated. It is at this instant that the collective target assignment algorithm decides to reassign one tracker from each group.

Figure 13c shows the reassigned target groups after they have separated and settled. The shading of the original target groups has been retained to reveal the adjustment made during the groups’ interaction.

C. 3D Simulation Three-dimensional formations were also explored using potential fields of the form given in Eq. (3). In these

cases, an example of which is shown in Fig. 14, hollow spherical distributions were desired in order to reflect a geometry useful for certain space-based radio interferometry missions.

To achieve a hollow spherical distribution, a virtual potential field g(x) of the form given in Eq. (3) was fixed to

a point in space that would become the center of the distribution. Each mobile agent experienced radial forcing due to this potential field and tangential forcing due to a separate inter-agent potential field of the form shown in Eq. (20).

( ) ( ) xeCxg −= 1 (20)

Members of the formation experience only the tangential component of these repulsive forces. This additional constraint allows for the uniform distribution of agents on the sphere, while decoupling the tangential and radial behavior.

IV. Experimental System Overview Figure 15 shows three of the robots used in the DSSL to explore and validate technologies and concepts such as

swarming, coordinated motion, and formation flying and control. Each robot is equipped with on-board CO2 for floatation and thrusting, a thin film air-bearing ‘foot’ of porous material, a reaction wheel for heading control, a vertically-oriented color camera for position and heading determination, and four solenoid air-jet thrusters. A custom-made electronics board is equipped with an Intel 586 processor, 802.11 wireless card and a high current driver for each thruster solenoid valve. Each robot is about 33 cm tall, and has a mass of approximately 3.5 kg. With these sizes and capabilities, the robots are useful testbeds for developing technology for nanosatellite clusters and other small, distributed robot systems.

Figure 14a. 3D hollow spherical swarmingdistribution

Figure 14b. 3D hollow spherical swarmingdistribution, with surface showing potentialwell location

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V. Experimental Results During a series of experiments, stable formations of three robots were achieved using potential fields of the form

shown in Eq. (18). For three robots, stable formations are triangular in shape. For the experiments presented here, robot position and heading were determined onboard each robot using its

color camera and a pair of red and green pseudo-stars created by directing lasers toward the laboratory ceiling. A Kalman filter was employed to calculate both robot position and velocity, and after each position/velocity calculation, the robot communicated this updated information directly to all other robots via wireless Ethernet.

Formation control was completely non-centralized, with each robot calculating its own virtual net force using its prescribed virtual potential field/dashpot, and the knowledge of relative robot positions and velocities gained from the direct inter-robot communication described above. Each robot also seeks to maintain a fixed inertial orientation based on feedback from the onboard camera and a simple PID control loop.

Figure 16 shows an example of such a formation. In this example all robots were at initially at rest in random positions, move into position based upon the virtual force exerted by the neighboring potential fields, and finish in a stable triangular formation. The initial conditions are indicated by the open circles, while the final positions are given by the closed dots. The dotted paths show instantaneous position measurements.

Figure 15. DSSL robots in the lab

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In a second example, seen in Fig. 17, one robot was given a nonzero initial velocity. After a short period of time

to allow the formation to settle, the expected triangular formation arises.

VI. Summary The usefulness of potential fields for coordinating the behavior of multi-agent systems has been known for some

time. In this paper we have shown a systematic method for calculating smooth potential fields with an arbitrary number of constraints on field shape. Utilizing potential fields of this form we have simulated the performance of swarms of free-flying self-coordinating spacecraft, and experimentally validated some of these simulation results on the DSSL’s free-flying air-bearing robots.

Additionally we have applied such potential field interaction laws, along with a high level collective, in a novel approach to distributed target tracking. This approach uses real-time target selection algorithms to optimally distribute an arbitrary number of tracking agents around an arbitrary number of dynamic targets.

Figure 16. Experimental results, full trajectory Figure 17. Experimental results, full trajectory

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11) References [Khatib] Khatib, Oussama, “Real-Time Obstacle Avoidance for Manipulators and Mobile Robots”, International Journal of Robotics Research, Vol. 5, No. 1, Spring 1986, pp. 90-98. [Volpe] Volpe, Richard, and Khosla, Pradeep, “Manipulator Control with Superquadric Artificial Potential Functions: Theory and Experiment”, IEEE Transactions on Systems, Man, and Cybernetics, 20(6), Nov/Dec 1990. [Song] Song, Peng, and Kumar, Vijay, “A Potential Field Based Approach to Multi-Robot Manipulation”, ICRA 2002. [Leonard] Leonard, Naomi, and Fiorelli, Edward, “Virtual Leaders, Artificial Potentials and Coordinated Control of Groups”, Proc. 40th IEEE Conf. On Decision and Control, 2001, pp. 2968-2973. [Olfati 1] Olfati-Saber, Reza, and Murray, Richard, “Distributed Cooperative Control of Multiple Vehicle Formations Using Structural Potential Functions”, 15th IFAC World Congress, Jul 2002. [Olfati 2] Olfati-Saber, Reza, and Murray, Richard, “Flocking with Obstacle Avoidance: Cooperation with Limited Communication in Mobile Networks”, 42nd IEEE Conference on Decision and Control, 2003. [Spletzer] Spletzer, J., et.al., “Cooperative Localization and Control for Mulit-Robot Manipulation”, Source. [Clark] Clark, Christopher, et.al., “Motion Planning for Multiple Mobile Robots using Dynamic Networks”, Proceedings of the 2003 International Conference on Robotics and Automation, May 2003.