AbstractâThis work investigates a multithreat containment problem where a team of autonomous robots cooperate to contain occurring threats. The robots aim ...
Potential Field Based Approach for Multi-threat Containment With Cooperative Robots Bhushan Mehendale
Shanchieh Jay Yang
Department of Computer Engineering Rochester Institute of Technology Rochester, NY 14623 Email: mehendale(at)bhushan(dot)in
Department of Computer Engineering Rochester Institute of Technology Rochester, NY 14623 Email: jay(dot)yang(at)rit(dot)edu
Abstract— This work investigates a multithreat containment problem where a team of autonomous robots cooperate to contain occurring threats. The robots aim to form circles around multiple threats that appear at random times and expire after a certain time. Containing single static threats has been studied, yet extending it to the dynamic regime with threat lifetime overlapping presents additional complex challenges. An algorithm named MUltiple Threat Containment Algorithm (MUTCA) is proposed and analyzed in this work. MUTCA is designed for simplicity and based on Quadratic Artificial Potential Functions (QAPF) and the use of random walks. Multiple QAPFs are carefully designed to allow robots to move toward the threats, to avoid collisions, and to surround the threats evenly. This work investigates the capabilities and limitations of autonomous robots utilizing MUTCA via simulation using MAHESHDAS - a simulator developed to simulate autonomous robots. Application specific success rates are examined for various scenarios and robot sensor capabilities. MUTCA has been found to work well with sufficient robots that have moderate capabilities, but is sensitive to application specific parameters.
I. I NTRODUCTION The miniaturization of modern processors, sensors, and mechanical components has made it possible to utilize a large number of cost-effective and autonomous robots to accomplish tasks not possible by a single powerful robot. For example, a search and destroy mission in a wide spread terrain may be more efficiently accomplished by a group of robots. A key challenge in this paradigm is to have individual robots make decisions of its own actions with only limited or zero interaction with nearby robots. This work considers robots that can only passively sense nearby robots and target threats. The goal of the robots is to move quickly toward the detected threats and encircle them for a sufficient period of time. The threat may appear at random times, at random locations, and have a lifetime beyond which there is no need to contain them. We refer to this problem as the multithreat containment problem. Containing a single static threat has been investigated by Song and Kumar in [1] and Unsal and Bay in [2]. In their work, the threat has a fixed location and an infinite lifetime. The objective of their studies is to determine an efficient and robust algorithm allowing the robots to autonomously move to encircle the threat without colliding with each other. The multithreat containment problem is different in two aspects.
First, a robot may detect multiple threats and need to make its own decision (with passive local sensing) on where it should go, hoping all threats will be contained by the entire robot team. Second, the threats may dynamically appear and expire at random locations. Thus, the robots cannot swarm into one or few areas and abandon the rest. To address these additional challenges, yet with a low complexity algorithm, we adopt the concept of Artificial Potential Function (APF). The APF in robotics is typically used to model artificial attraction and repulsion forces perceived by the robots, and has been applied for various problems. The authors of [1] use this concept to surround and move a target to a pre-defined destination. They have also investigated the stability of different attraction and repulsion functions. A trailer-like system being driven to a goal while avoiding obstacles using APF is discussed in [3]. The authors of [4] discuss an evolutionary APF to avoid local minima in the aggregate potential field. APFs have also been studied as swarm algorithms, such as those presented in [5] and [6] with stability analysis. A discussion of ‘social potential fields’ is presented in [7] where the authors compare several different APFs, and provide guidelines to design application specific APFs. Their work suggests that designing an APF for an nbody system achieving a certain behavior can be ‘polynomial space hard.’ The authors in [8] explain the characteristics of APFs and relate the forces acting on robots due to a simple quadratic potentials following Hooke’s Spring Law. This work adopts a combination of quadratic APFs to model the attraction and repulsion forces from the threats and the repulsion forces between the robots to solve the multithreat containment problem. Our work draws its inspiration from the multiple-threat encirclement approach taken in [2] and the robot team shape formation using APFs presented in [9]. II. P ROBLEM S TATEMENT Consider n robots arbitrarily situated on a finite 2D region of interest. Each robot occupies a physical circular space of radius rr , weighs m, has a maximum speed of v, a maximum angular speed of τ , and equipped with two types of sensors. The two sensor types are for detecting the locations of other robots and the locations of the threats. Let rs be the range, within which the sensors can detect the corresponding objects
III. M ULTITHREAT C ONTAINMENT A LGORITHM (MUTCA) To address the challenges of the multithreat containment problem, this work adopts the concept of Artificial Potential Functions (APFs). APFs are used to represent potential induced by threats and robots. Each robot periodically scan its surrounding and determines the relative locations of threats and robots within a proximity of radius rs . These detected threats and robots form an artificial potential field P~i for the observing robot i, which acts as a Newtonian particle moving according to the aggregate force F~i = −∆P~i . An energy loss ratio (α) will be assumed to ensure that each robot will stabilized at the local or global minima. Note that each robot may have a unique view of its proximity, and this view may change as the robot moves. The changes will be made periodically according to the sufficiently small interrupt period (δ) of the sensing devices used. Each robot will utilize this virtual aggregate and changing force to move and contain dynamically occurring threats. The algorithm is referred to as the Multithreat Containment Algorithm (MUTCA). A. Quadratic Artificial Potential Functions Quadratic Artificial Potential Functions (QAPF) are chosen for this work for their simplicity and effectiveness [7]. The QAPF induced by a threat (or robot) j in its general form, P (xj ) = k(xj − d)2 , k, d > 0, shall have robots at distance xj away from j gradually settle at a distance d away from it. Three QAPFs are developed to attract robots toward the threats (Prt (x)), to prevent inter-robot collision (Prr (x)), and to spread the robots around the threats (Pspread (x)). 1) Threat-Robot QAPF: Let drt be the target distance the robots aim to encircle each threat. When the robots approach a threat (x > drt ), they should gradually stabilized at the distance of drt from the threat. On the other hand, when
robots are getting too close to the threat (x ≤ drt ), they should rapidly go away from the threat. In order to achieve this behavior, a more aggressive repulsion QAPF and a less aggressive attraction QAPF are combined to form the overall Prt (x), which is given below (with krtr > krta > 0) and illustrated in Figure 1. krtr (x − drt )2 , x ≤ drt , Prt (x) = (1) krta (x − drt )2 , x > drt ,
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with close to 100% accuracy. Our approach does not require both sensor types to use the same rs , but assumes so in this work for simplicity. The threats may occur anywhere in the finite 2D region and occupies a physical circular space of radius rt . In order to contain a threat, the robots may temporarily move out of the finite 2D region, but the center of the threat may not go beyond the boundary of the region. The threats may occur following an arbitrary arrival process and have a constant or random lifetime. The multithreat containment problem is to have the robots dynamically move and form a circular shape around each occurring threat. The circle should be centered at the threat and has a radius of drt . The robots need to surround the threat for a sufficient amount of time before the threat lifetime expires. The key challenge of the problem is to have each individual robot (which is only capable of sensing and locomotion) to dynamically choose the threat and move to surround it. In the process of containing a threat, a robot needs to move toward the threat, avoid colliding with other robots and the threats, and be part of an evenly spread circumference of the containment circle.
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rt Distance from threat (x) Fig. 1. The threat-robot QAPF, Prt (x), is designed to attract robots (with sensing range rs ) to be at a target distance of drt to the threat (rt ).
2) Inter-Robot QAPF: The purpose of the inter-robot QAPF Prr (x) is to avoid robot collisions. As a result, only repulsion is needed. The quadratic function is designed to have a smaller effect than the Prt (x) so that a robot will not be pushed away from a threat or pushed in to collide with a threat by other robots around the threat. Meanwhile, when two robots are indeed very close to each other, they need to aggressively move away from each other even in the presence of threat attraction force. Consequently, a robot will be repelled by another robot only when it is within a distance of drep < drr from it, where Prr (drr ) = 0. The function Prr (x) is given below and shown in Figure 2. krr (x − drr )2 , x ≤ drep , Prr (x) = (2) 0, x > drep , 3) Spreading QAPF: When robots get closer to a threat, they are supposedly to repel each other and evenly spread out to encircle the threat. The use of a small drep allows many robots to surround a threat together. However, if there are only few robots in the proximity of a threat, these robots may be clustered on one side of the threat instead of spreading evenly around the threat. In this case, the inter-robot QAPF needs to be modified with a larger drep,s and a larger drr,s , allowing robots to perceive the repulsion forces from more distant robots and be repelled to a larger inter-robot distance. MUTCA has each robot enters a ‘spreading mode’ when it enters the proximity of a threat, i.e., within 80%-120% of drt from the threat and sense less than two robots within drep .
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entire region of interest will be covered. MUTCA adopts the Random Walk (RW) mobility model, which closely resembles Brownian Motion in molecules [10]. In the RW mode, a robot will choose a random direction and move with full speed for t seconds. After t, or when the robot encounters an edge of the field, it changes its direction. In this work we assume t = 20 seconds. The robot will enter normal containment mode whenever it detects force again. A high level flowchart of MUTCA with all the aforementioned components is given in Figure 3. MUTCA is implemented and simulated in a customized simulator MAHESHDAS - a flexible event-driven simulator for collaborative robots [11].
Distance from other robot (x) start
Fig. 2. The inter-robot QAPF, Prr , is designed to repel robots from each other to avoid collision. The robots will perceive the repulsion force when getting closer than drep < drr .
In the spreading mode, a spreading force is perceived by the robot and the corresponding QAPF function is given below with drr,s > drr , drep,s > drep , and krr,s < krr as compared to (2). The reason for a smaller krr,s is for a milder QAPF in this spreading mode. krr,s (x − drr,s )2 , x ≤ drep,s , Pspread (x) = (3) 0, x > drep,s ,
sense & detect threats and neighbors calculate forces due to threats and robots
am I in spreading mode? request environment to go to new position
The three QAPFs described in the previous section make up the primary component of MUTCA. The exact parameters used are determined through experiments and are based on the application specific parameter, drt , and the robot size, mass, speed, sensor range as well as the interrupt interval. Note that the inter-robot QAPF aims to avoid collision. In rare cases, however, many combined forces may push a robot into another. The robots, therefore, need to be capable of detecting imminent collision and override the QAPF functions in such cases. In our simulation study, we implement an imminent collision detection component to model this typical robot capability. Furthermore, in the simulation environment, the robots may be forced to move out of the boundary of a finite 2D region. Our simulated algorithm accounts for such special cases. If a robot moves out of the boundary to contain a threat (which can only occur within the confined 2D region), then no special actions will be taken. For other cases, however, the robot will be stopped at the boundary until other robots are cleared out. A critical part of the multithreat containment problem is that the threats may dynamically appear at arbitrary locations of the 2D region. If the robots do not move when they perceive no force after a threat expires, it is possible that the robots cannot sense a later threat situated at a remote location. To overcome this limitation, MUTCA introduces a component called zero-force movement. The objective of the zero-force movement is to ensure that the robots will randomly move in a predefined region when no forces are detected, so that the
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All robots follow the sequence shown in this flowchart.
IV. S IMULATION R ESULTS AND D ISCUSSION A. Simulation Setup MUTCA is tested with simulated robots in a 10m×10m 2D field. Table I shows the default values used for the physical robot parameters, the MUTCA algorithm parameters, and the simulated environment parameters. The threats occur uniformly at random places in the 2D field and arrive according to a Poisson process with an average rate of 0.01 (s−1 ). Each threat has a constant lifetime of 60 seconds. These default values gives on average 0.6 simultaneous threats in the 2D field. Threats overlapping in their lifetime produce the challenge of containing multiple threats both spatially and across the time domain. Note that we do not claim that uniform locations, Poisson arrivals, or constant lifetime are representative of a specific application. The choices of these models are merely for simulation experiments to exhibit the capability and limitations of a robot team utilizing MUTCA. Different arrival rates, number of robots, robot speeds, sensing ranges, threat lifetime, and target
Default Value 0.05m π/s 0.1m/s 1kg 0.1s 3m Default Value (0.67, 0.052, 0.75m) (0.021, 1.0m, 0.3m) (0.01, 2.5m, 1.6m) 15% Default Value 10m×10m 7200s 0.05m 0.01/s 60s 50
TABLE I D EFAULT S IMULATION PARAMETERS
process, leading to a better success rate. Given the default threat arrival rate and lifetime settings, 45-50 robots seem to be sufficient with only marginal performance improvements beyond which. The reason for the success rate to cap off is due to that robots reaching equilibrium around the threat may be occasional disturbed due to new occurring threats or robots passing by. The overall dynamics present a much more complex environment than the static threat containment problem for which MUTCA exhibits 100% succcess rate. % Threats Successfully Contained
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Physical Robot Parameter Robot radius (rr ) Max angular velocity (τ ) Max velocity (v) Mass of robot (m) Interrupt Interval (δ) Sensing Range (rs ) MUTCA Parameter (krta , krtr , drt ) (krr , drr , drep ) (krr,s , drr,s , drep,s ) Energy loss ratio Environment Parameter 2D field dimensions Simulation time Threat radius (rt ) Average threat arrival rate Constant lifetime of threat Number of robots
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containment radius are experimented to test the robot team’s containment capability. To test the capability of the robot team, one needs to define the criteria for a successful containment. The criteria need to be defined both spatially and in the time domain. Spatially, we consider an imaginary worst case scenario where at least one robot will have shorter or equal travel distance to any point on the circumference of the containment circle as the threat does (which is the radius of the containment circle). This scenario leads to a minimum of three robots forming a equilateral triangle on the perimeter circumference. In other √ words, any two robots can have no more than a spacing of 3· drt . In addition to the maximum spacing, a robot is considered containing a given threat if it is situated within a distance of 1.25 · drt from that threat - it is unreasonable to require the robots stay perfectly ‘on’ the circumference of the containment circle throughout the lifetime of the threat. With respect to the threat lifetime, we define that a threat need to be contained by robots at least 80% of its lifetime. B. Simulation Results Simulation of MUTCA for containing a static threat anywhere in the 2D field has shown 100% success rate as long as sufficient robots can sense the threat. Threat occurrence and expiration in different regions of the field, however, have shown to cause less than perfect success rate. Below we vary one factor at a time and keep other variables at the default values. 1) Success versus Number of Robots: Figure 4 shows how the percentage of success increases as the number of robots increases from 5 to 70. The changes seems to taper off between the 80%−90% region. An 80% or more success is evident if 45 or more robots are used. Few robots clearly are not sufficient to contain the threats with the default setting. As the number of robots increases, more robots participate in the containment
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Fig. 4. The percentage of success as the number of robots increases. Each data point was averaged over 7 runs with an error bar corresponding to ±1 standard deviation.
2) Success versus Threats Arrival Rates: Our default settings of threat arrival rate and lifetime results in an average of 0.6 threats on the field. As we increase the arrival rate and keep the threat lifetime constant at 60 seconds, the average number of threats on the field increases. Figure 5 shows a clear decline in the success rate as the threat arrival rate increases. The trend is the same regardless of the number of robots used. At first the decline in the success rate seems to be largely because of the fact that insufficient number of robots are trying to contain more and more threats that appear ‘simultaneously’ in different regions of the field. It turns out that the key to this declining trend is more due to the ‘rapidness’ of threats appearing in the field, rather than the static number of threats in the field. The next set of results clarifies this point further. 3) Success versus Lifetime of Threat: Increasing the lifetime of the threats also increases the average number of threats in the field. In fact, the threat occurrence follows M/D/∞ model and, thus, the arrival rate and the lifetime both have linear effect on the average number of threats. Figure 6 shows that the success rates, for different number of robots, actually increases for the most part as the threat lifetime increases contrary to the trend exhibited in Figure 5. This suggests that the average number of threats simultaneously situated in the field is not the critical factor for this dynamic and multiple threat containment problem. It is the arrivals (as suggested in the previous subsection) and expirations of the threats that causes the instability of the entire system and makes the
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of very large formation radii, a run with drt = 2.5m was done. An extremely large formation radius does not seem to affect the success given our default setting.
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Fig. 5. The percentage of success as the threat arrival rate increases. Each data point was averaged over 5 runs with an error bar corresponding to ±1 standard deviation.
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problem challenging. Note that as the threat lifetime becomes too short, the robots may not be able to get to a threat, even with its maximum speed, before it expires. The robots’ sensing range and maximum speed must allow sufficient travel time for them to reach the neighborhood of detectable threats. % success (50 Robots) %success (15 Robots)
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Fig. 6. The percentage of success as the threat lifetime increases. Each data point was averaged over 5 runs with an error bar corresponding to ±1 standard deviation.
4) Success versus Containment Radius (drt ): We further examine the effect of the threat containment radius (drt ). Logically, a larger drt allows shorter robot travel time and more robots joining each containment process within the threat lifetime. Moreover, our success definition will give a less stringent criterion as the drt increases. Figure 7 clearly shows the benefit of using an increasing drt . Note that our MUTCA parameters are calibrated for a formation radius of 0.75m. As a result, there is a big improvement in success when the drt changes from 0.45m to 0.75m. The success rate continues to improve and reaches 95% with 50 robots when drt = 1.35m, as compared to 83% when drt = 0.75m. To examine the effect
Fig. 7. The percentage of success as the containment radius (drt ) increases. Each data point was averaged over 5 runs with an error bar corresponding to ±1 standard deviation.
5) Success versus Sensing Range (rs ): A logical choice to improve the containment success rate is to improve the robot sensor capability. We examine the effect of sensing range (rs ), which defines the distance within which a robot can detect other robots and threats. Recall that the QAPFs are defined assuming rs = 3m. Particularly, it is unreasonable to set rs < drep,s . In order to keep other parameters constant, the sensing range was increased from 1.75m to 5.0m. Figure 8 exhibits the changes in success rate as rs increases in this range. Note that the increase in sensing range has a different effect when there are different numbers of robots in the field. When there are few robots (20 in this experiment), increasing rs greatly helps improve the chance the robots detecting the threats and hence the containment success rate. The benefit diminishes as rs reaches 2.5m and beyond. This is because the robots randomly move in the 2D field, and 2.5m to 3m is sufficient for the robots to detect occurring threats, though not necessary contain them successfully. With more robots (e.g., 50), however, excessive sensing range causes too many robots reacting to each single threat. This may lead to congestion as robots move to contain the threat, and consequently, a worse success rate. V. C ONCLUDING R EMARKS An algorithm called MUTCA has been proposed to tackle the dynamic multiple threat containment problem. MUTCA is designed with the target platform being inexpensive robots that have only local sensing and locomotion capabilities. MUTCA utilizes a combination of QAPFs and the zero-force random walk to perform threat containment in an autonomous manner. Despite its simplicity, simulation results suggest that MUTCA may achieve a success rate up to 85-95% in containing dynamically arriving threats. This success rate is achievable
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Fig. 8. The percentage of success as the robot’s sensing range (ds ) increases. Each data point was averaged over 5 runs with an error bar corresponding to ±1 standard deviation.
with reasonable number of robots that have moderate sensing range and locomotion capabilities. Because of its simplicity, however, MUTCA cannot guarantee 100% success rate, except in the case of a single static threat. In fact, our results suggest that it is the dynamics in the time domain (caused by threat arrivals), but not that in the spatial domain, that cause the imperfect success rate. Improvement of MUTCA may require additional software complexity for better robot intelligence, or additional hardware resources that enable interrobot communication. One possible approach for optimization of the QAPFs used in MUTCA may be the use of Lyapunov functions, which were used for a single static threat case as described in [6]. This is an open problem since not only the robots (the Newtonian particles) are moving, but the threats also have a birth-death process affecting the potential field. Another aspect of the multithreat containment problem is the energy needed for the robots to continuously monitor and contain newly arriving threats. As the premise is built on a large number of costeffective robots, energy sources will certainly become an issue. Therefore, in addition to its complexity, the algorithm perhaps should be designed to induce as few movements as possible, as the locomotion component will likely consume energy the fastest. R EFERENCES [1] P. Song and V. Kumar, “A potential field based approach to multi-robot manipulation,” in Proceedings of the IEEE International Conference on Robotics and Automation, vol. 2, 2002, pp. 1217–1222. [2] C. Unsal and J. S. Bay, “Spatial self-organisation in large populations of mobile robots.” in Proceedings of the IEEE International Symposium on Intelligent Control, 1994, pp. 249–254. [3] T. A. Vidal-Calleja, M. Velasco-Villa, and E. Aranda-Bricaire, “Artificial potential fields for trailer-like systems,” in Proceedings of the 10th Latin American Congress on Automatic Control, Guadalajara, 2002. [4] P. Vadakkepat, C. Kay, and M.-L. Wang, “Evolutionary artificial potential fields and their application in real time robot path planning,” in Proceedings of the 2000 Congress on Evolutionary Computation, vol. 1, 2000, pp. 256–263.
[5] Y. Liu and K. M. Passino, “Swarm intelligence: Literature overview,” Ohio State University, Department of Electrical Engineering, Columbus, OH, Tech. Rep., March 2000. [6] V. Gazi, “Swarm aggregations using artificial potentials and slidingmode control,” IEEE Transactions on Robotics and Automation, vol. 21, no. 6, pp. 1208–1214, 2005. [7] J. H. Reif and H. Wang, “Social potential fields: A distributed behavioral control for autonomous robots,” Robotics and Autonomous Systems, no. 27, pp. 171–194, 1999. [8] R. Volpe and P. Khosla, “Manipulator control with superquadric artificial potential functions: Theory and experiments,” IEEE Transactions on Systems, Man and Cybernetics, vol. 20, no. 6, pp. 1423–1436, 1990. [9] W. M. Spears, R. Heil, and D. Zarzhitsky, “Artificial physics for mobile robot formations,” in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, vol. 3, 2005, pp. 2287–2292. [10] F. Bai and A. Helmy, A Survey of Mobility Modeling and Analysis in Wireless Adhoc Networks. Kluwer Academic Publishers, 2004. [11] B. U. Mehendale, “Potential field based approach for multi-threat containment with cooperative robots,” Master’s thesis, Rochester Institute of Technology, Department of Computer Engineering, Rochester, NY, September 2006, http://hdl.handle.net/1850/2626.
