Multi-Robot Rendezvous in Unknown Environments or What to do When You’re Lost at the Zoo

Gregory Dudek and Nicholas Roy Centre for Intelligent Machines and School of Computer Science McGill University 3480 University St., Montreal, Qu´ebec, Canada H3A 2A7

Abstract geometry, wireless transmission technology, power con- siderations and atmospheric conditions (or water con- We consider the problem of rendezvous between ditions for underwater agents) all contribute to fairly two robots exploring an unknown environment. short communication limits. In the absence of sophis- That is, how can two autonomous exploring agents that cannot communicate with one another over ticated satellite receivers or high power devices, a com- long distances meet if they start exploring at dif- mon constraint for successful communication is main- ferent locations in an unknown environment. The taining “line-of-sight” between agents, a constraint that intended application is collaborative map explo- is rarely satisfied in the real world. However, multi- ration. agent robot systems for the majority of real-life appli- Ours is the first work to formalize the characteris- tics of the rendezvous problem, and we approach cations enjoy substantial speed gains only with some it by proposing several alternative algorithms that level of communication, when compared with single- the robots could use in attempting to rendezvous agent systems or multi-agent systems that do not com- quickly while continuing to explore. The algo- municate. (Balch & Arkin 1994). Many distributed- rithms are based on the assumption that poten- agent algorithms, for instance dynamic path-planning, tial rendezvous locations, called landmarks, can be selected by the robots as they explore; these assume and rely upon instantaneous, infinite bandwidth locations are based on a distinctiveness measure communication between agents at all times in order to computed with an arbitrary sensor. achieve promised performance levels (Brumitt & Stentz We consider the performance of our proposed algo- 1996). rithms analytically with respect to both expected- Looking to biology, some simple algorithms are easily and worst-case behaviour. We then examine their behaviour under a wider set of conditions in sim- observed. Most animals rely upon well-known common ulation. meeting points, such as a beehive, or a watering hole. In an unknown environment, however, such an absolute reference point is almost impossible. A common strat- Introduction egy has one agent (eg. a child lost at the zoo) wait to A rendezvous is a meeting between two or more agents be found by other agents (eg. desperate parents). As at an appointed place and time, for example, when two we shall see, such simple strategies can perform poorly people meet at a familiar restaurant. The problem of under many conditions. rendezvous is ubiquitous in nature. Migratory animals The problem that is discussed in this paper is how to must be able to meet to share information about food. determine the best strategy for a successful rendezvous Non-social animals must be able to find each other dur- between two agents in optimal time, and ours is the ing mating season. Humans are equally familiar with first work to formalize the characteristics of this prob- the problem of rendezvous, as any family whose mem- lem. We will consider how this rendezvous task can bers become separated at a large zoo or mall well knows. be efficiently accomplished under various assumptions Multi-agent robot systems also have an inherent need about the environment and the perceptual abilities of for the ability of inter-agent rendezvous. the agents involved. In particular we are interested in The ability to meet facilitates localisation, allows col- the problem of rendezvous in the context of multi-robot laborative map exploration and has a plethora of other exploration using video or sonar sensing. In practice, advantages, but most importantly allows communica- the particular sensing modality has numerous pragmatic tion. Most existing hardware agents are only capable implications, a major factor being the range at which of communication over short distances. Environmental the agents can either recognize one another or land- marks in the environment. In the context of a general of rendezvous, but focussed only on how to merge maps rendezvous strategy we will, however, consider a generic once the rendezvous has occurred. It is worth noting “abstract” sensor that allows the agents to recognize that map fusion is also closely related to the generic one another when they are sufficiently close together image-registration problem. and which allows them to evaluate any point in space as to its suitability as a rendezvous point. The Rendezvous Problem We have separated the rendezvous problem into two sep- Background arate sub-problems. The first is how to select points in Several authors have considered interaction and coor- the environment for potential rendezvous. These points dination between multiple mobile robots. Arkin and will be referred to as landmarks. The second sub- his colleagues, in particular, have considered issues of problem that is the major focus of this work, is how to co-ordinated motion as well as collaborative behaviour select which point, out of the set of potential rendezvous with minimal pre-planning (Arkin & Hobbs 1992; Balch points, to visit at the assigned time. The context of the & Arkin 1994). His work, however, has not focussed on rendezvous is an unknown environment, with no shared exploration or the use of deliberative strategies to allow spatial information between agents, and no communica- robots to meet. tion until rendezvous. We are examining the rendezvous There has been considerable work in addressing the problem in terms of minimizing time to rendezvous un- problems of communication between agents in a mul- der various conditions such as sensor noise, environment tiple agent system, however the majority of work has size, etc.. Determining when a rendezvous is necessary been to maximise efficiency and minimise complex- in the framework of another task is a task-dependent ity (Mataric 1992) (et al. 1992). Mataric has looked problem, and is outside the scope of this paper. at models of collaborative behaviour between mobile We start by briefly considering some properties of robots (Mataric 1992), and observed that the form of good distinctiveness measures. We will then present communication plays an important role in how collabo- several rendezvous strategies, consider their statistical rative actions proceed. In this work, we deal with how properties analytically and simulate their behaviour nu- to facilitate that communication by allowing the robots merically. We conclude with a discussion of the results. to meet. Yoshida et al. addressed the problem of how to reduce Landmark Selection a global communication network to local communica- As an agent travels throughout the environment, every tion, in order to minimizing information complexity (et visited location is evaluated by the agent in terms of al. 1995). However, there has not been much research in its uniqueness. The assumption is that distinctive loca- overcoming communication limitations, except by lim- tions (with respect to some sensor-based computation) iting the scope of the system to some area (such as a serve as locations that both robots can independently factory or a port) where communication between agents select as good landmarks. This notion of a landmark can be guaranteed by some global co-ordinator. also serves as the basis of the topological mapping strat- The selection of distinctive locations in a simple 2- egy proposed by Kuipers (Kuipers & Byun 1991). We D environment has been considered previously in the refer to the scalar measure of suitability as a rendezvous context of map-making (Kuipers & Byun 1991), in point as distinctiveness: D(x, y), or more generally, which distinctive locations were determined by hill- for a pose vector q we can define D(q). This is implic- climbing, that is, by local gradient ascent over some itly a function of sensor data f(q), so we have D(f(q)). function of the sensor output. More generally, lo- Although the agent’s sensor may not return scalar val- cal maxima in some continuous property of the en- ues, some scalar suitability measure can be usually be vironment would seem to present an opportunity for computed from the sensor. Some intuitive examples of converting a metric environment representation into a environmental attributes that might serve as distinc- graph-like or topological one (Chatila & Laumond 1985; tiveness measures are: symmetry, distance to the near- Dudek et al. 1991). est obstacle, or altitude (for 3D surfaces – for example The problem of map generation from co-operative humans might select hill tops). multi-agent exploration was discussed and implemented In order for two robots to agree on a good landmark, by Ishioka et al. (Ishioka, Hiraki, & Anzai 1993). Their they must have similar perceptions of the environment work is a canonical example of the potential applications or be able to convert their percepts into a common in- of the technique presented in this paper, in which co- termediate form. In the extreme case of two agents operative heterogeneous robots generated maps of un- with dramatically different sensing modalities, there is known environments. They did not discuss the problem essentially no way for them to rendezvous based on the recognition of environmental characteristics. Sensor a point robot capable of arbitrary motion within free noise can play a similar problematic role. We model this space. aspect of the problem by parameterizing the extent to The distinctiveness measure typically used for re- which the two agents can reliably obtain the same mea- search in this area, in the context of mobile robotics surement of distinctiveness at the same location. With and sonar-based perception, is the mean distance re- full generality, we can consider one of the two agents as turned by the sonar ring, which essentially uses the en- the reference perceiver (the arbiter of good taste) with closed space as the value of the landmark - the bigger a percept D1(x, y) = D(x, y) while the second robot the room, the better a landmark. Note that errors due obtains a sensor measurement which can be viewed as to specularity with respect to sonar make the physical noisy with respect to that of the first robot: interpretation of the measurement ambiguous.

D (x, y) = (1 δ)D(x, y) + δη(x, y), (1) Rendezvous Strategies 2 − In order to estimate the effectiveness of alternative where η(x, y) is a noise process and δ specifies the extent strategies for rendezvous, we have identified key at- to which both robots sense (or perceive) the same thing. tributes that must be formalised. Important attributes If both robots have exactly the same perceptions of the of the rendezvous problem are: environment we have δ = 0. In the context of this formalism, η(x, y) combines both intrinsic sensor noise Similarities — the reproducibility of the perceptions and any differences in the type of sensor used. • between agents (do they sense the same attributes, The possible distinctiveness measures are heavily de- and do they even use the same sensors), pendent on the types of sensor the robots have at their Landmark Commonality — the extent of overlap be- disposal. Because the robot assigns a value to every • point, a good modality is one that allows the distinctive- tween the spatial ranges of the agents (this may ness to be defined at any location in the environment, change with time), and for which there exists some metric that can order Synchronisation — the level of synchronisation be- the resultant landmarks in the environment in terms • tween the agents (for example, can they agree to meet of distinctiveness. This ordering allows the landmarks at high noon). to be ranked in terms of their likelihood to lead to a successful rendezvous. Landmark Cardinality — the number n of landmarks Certain generic properties apply to suitable land- • selected by each agent. marks and the distinctiveness function D(x, y) indepen- dent of the sensing modality. If the distinctiveness func- Implicit in the description of these attributes are cer- tion is smooth and has few local extrema or inflection tain assumptions. It is assumed that all agents share points, then it may be possible to define highly stable some degree of synchronization - that is, all agents can and mutually agreed-upon landmarks with great ease agree on when rendezvous attempts should be made. using gradient ascent. However, although this strategy However, this synchronisation may contain noise, which is attractive in principle, we believe that in many real will be dealt with shortly. The second assumption is environments, sensor noise, occlusion and other factors that all agents have the same landmark set cardinality may make the “distinctiveness surfaces” highly non- - they all attempt rendezvous over the same number of convex and thus complicate the process. landmarks (even if they are not using identically the The distinctiveness function and the associated land- same landmarks in their sets). Finally, it is assumed marks should be stable over time and should not depend that all agents are performing the same task, and using on the trajectory or history of the robot. For example, the same rendezvous strategies. the “Northern-most” point in the already-explored en- We have developed several fundamental strategies for vironment is a poor choice since if, for example, the assuring a rendezvous. As we will see, the best strategy explored area of each robot is circular, then two robots depends on several properties of the robots and of the will only have the same “northern-most” point if the environment. In the simplest, idealized, noise-free case, environment is highly constrained or if the explored re- each robot should select the location in the environment gions are very similar. that is the most distinctive. Given 100% landmark com- In this paper, we will neglect issues of navigation and monality and the absence of noise, they will select the assume an agent can always accurately reach a desired same location. Each robot should navigate to this loca- goal in the environment. While our framework can ac- tion and wait for the other robot(s) to arrive. At such a comodate navigational error, it is outside the scope of time, they could fuse their maps and suitably partition this paper. For concreteness, the reader can imagine any remaining exploration to be done. The problem with this idealized scenario in practice is that due to case the best strategy is simply to have one robot visit sensor variations, or disjoint landmark sets, they may every landmark, and have the other robot sit and wait not agree on where the ideal landmark is situated. for it. However, this is also an unrealistically pessimistic Our formalization of the rendezvous problem takes scenario. If the robots have been constructed to facil- the key attributes mentioned above into account as fol- itate rendezvous, they are likely to have a somewhat lows: common perception of the environment and to have some commonality in their explored areas. In reality, 1. Sensor noise — the distinctiveness measure observed the robots will probably experience some limited sen- by the two robots is unlikely to match perfectly. This sor noise, minimal dissimilarities, some asynchrony, and is expressed by the constant δ and leads to strategies partial but not complete landmark commonality. So, that must effectively consider a larger number of can- the best strategy takes these factors into account, and didate rendezvous landmarks since a single guaran- chooses a series of landmarks to visit in some intelligent teed candidate may not be determined reliably. The way. issue of non-repeatability of sensor readings due to We are interested in strategies that would permit noise is not relevant in the context of this abstract a robot to interleave exploration and attempted ren- model and is not considered here. dezvous so that if the rendezvous fails, the robots can continue their work and the strategies remain robust 2. Sensor dissimilarities — the two robots may not mea- even in the face of a complete inability to find their as- sure the landmarks the same as each other. As illus- sociates. Below, we describe four alternative rendezvous trated in Eq. 1, so long as we do not consider issues strategies: these form exemplars of what we believe are of repeatability, sensor differences can be modelled as two key representative algorithm classes. a form of noise. 1. Deterministic Algorithms – Given the same set of 3. Asynchrony — when two robots are attempting to landmarks, these algorithms will always create the meet at the same landmark, the rendezvous may fail same ordering of landmarks. because one could not reach the landmark in a dy- namic environment, or even more likely, one robot Sequential – One robot picks a landmark and could not reach the landmark in time, and the other • waits there for the other robot, which visits ev- moved on. This asynchrony is referred to in this pa- ery landmark in turn. If the second robot has vis- per as the probability that a given meet at a common ited every landmark without encountering the first landmark will fail. This effect leads to a need for robot, the first robot moves to another landmark it strategies that may re-visit the same landmarks re- has not yet visited. peatedly to compensate for missed meetings. Smart-sequential – Each pairwise combination of • landmarks known to a robot is assigned a “good- 4. Non-identical Landmark sets — the robots may have explored different areas, and will have selected dif- ness” value. This value is the product of the dis- tinctiveness of the pair. The list of landmark pairs ferent landmarks that are not in the common region is sorted by this product, and one side of each pair (assuming such a common region exists at all). This 2 is modelled formally as the number d of landmarks is discarded, leaving an ordered list of n landmarks from a set of n. The robot then visits the land- out of a total set of n that are not common to the robots. The effect of the non-commonality is that marks in this order. both robots must consider a larger number of candi- 2. Probabilistic – The landmarks are sorted with re- date landmarks, since any given subset of landmarks spect to their distinctiveness and then assigning a selected by one robot may not be known to the other likelihood of visitation pi for landmark i as a func- robot. tion of its rank in the sorted list i.e pi = f(i). The algorithm probabilistically selects a landmark to visit, We will show that the choice of an appropriate ren- using p for each landmark. dezvous strategy depends on the extent to which the i robots have found the same set of landmarks, the Exponential – The likelihood of visiting the i th amount of sensor noise (or, equivalently, the similarity • best landmark is αei. − of the sensors) and the reliability of the robots being at Random – On each attempted visit, each robot a mutually selected rendezvous point and detecting one • selects a landmark at random and goes there. another. In the presence of large amounts of sensor noise, the Each of these methods has particular advantages and landmark selection will be essentially random, in which disadvantages. The sequential method is simple, but makes no effort to account for relative likelihoods, or mon ordering between agents of the same landmarks asynchrony. In view of the potential shortcomings of can be reliably determined. The first assessment is the sequential method, we have proposed an alternative the algorithmic time complexity, i.e., the expected time method, the probabilistic method, that has an increased to rendezvous, for the three algorithms in the limit chance of compensating for a missed rendezvous and of δ = 1. The expected time to rendezvous is the also attempts to compensate for small variations in the maximum number of unsuccessful rendezvous attempts, respective rankings of the landmarks selected by the two where the probability of no success on the next attempt robots. For instance, the distinctiveness of each land- is greater than or equal to 50%. For a landmark set of mark could be the same, which would lead to a uni- size n, the probability of any single, random rendezvous form random visitation strategy. The probability dis- attempt being unsuccessful is: tribution f() could be a linear function of value, or, if n 1 we assume that the amount of sensor noise is low, an P = − (2) unsuccessful n exponential strategy. However, if sensor noise is high and the two agents do not share the same ordering of If the asynchrony rate is accounted for, then the prob- landmarks, then the agents may be forced into revisit- ability of an attempt being unsuccessful rises to ing the incorrect landmarks much too often. A good n 1 + j compromise between these two methods is the smart- Punsuccessful = − (3) n sequential method. The advantage of this method is that, if δ is low, landmark combinations with high val- These equations give rise to table 1. The first column ues are explored before landmark combinations where refers to both robots having the same set of landmarks. one landmark has a very high value, and the other has The second column considers the scenario where the a relatively low value. This leads to an increased prob- robots may fail to get to the appointed landmark at the ability of meeting even with substantial asynchrony. same time (or fail to notice one another). This proba- The smart-sequential method is tantamount to guessing bility is the asynchrony, j. The third column deals with where the other robot might be, given relatively similar the case where d of each robot’s n landmarks are not in but not identical landmark rankings. the other robot’s landmark set. In the deterministic sequential algorithm, the ex- Behaviour - Analytical results pected time of the simplest case (identical landmark sets, no asynchrony), is very straightforward. One agent sits at a landmark, and the other agent visits every land- Algorithm Simple Async. < 100% Comm. mark in turn until they meet - on average n/2 land- 1 1 1 Random log ( n ) log ( n ) n2 marks. However, in the presence of asynchrony, ad- 2 n 1 2 n 1+j log2( ) − − n2 n+d − 1 ditional sweeps of all n landmarks will have to be per- 1 − − log d n log j n d n Sequential n/2 2 + j 2 + n formed. To find the expected number, k such additional 1 1 − − d log d sweeps, we use Smart-seq. n n + j log j n + n ≈ n 0.5 = jk (4) Figure 1: Expected case behaviour. The columns de- noting that each extra sweep i of k will reduce the note the ideal case, the case where the asynchrony j = 0 probability of failure, and k such sweeps must reduce 6 and the case where the landmark sets are not identical, the probability of failure to 50%. Thus, on average 1 but each agent has d non-common landmarks. j log− j sweeps during the rendezvous will fail due to asyn- chrony. Similarly, for non-identical landmark sets, addi- We can make an analytical assessment of the perfor- tional sweeps of n landmarks will have to be performed mance of the deterministic rendezvous algorithms, com- 1 − log d d n pared to the random algorithm baseline. If there is no on average n times. noise, no asynchrony, and 100% landmark commonality, In the worst case, the performance time complexity is then all of the algorithms which use the distinctiveness much more straightforward. For the probabilistic algo- measure to sort landmarks will lead to a rendezvous af- rithms, such as the random strategy, or whenever asyn- ter only one attempt (i.e., both robots will go straight to chrony is an issue, the worst-case is always ( ), be- the mutually agreed upon best landmark.). The random cause a meet can never be guaranteed. SimilarlyO ∞, a ren- algorithm can never assure a rendezvous but will have dezvous can never be guaranteed if any asynchrony is a small, equal probability of leading to a rendezvous on present, and so for j = 0, the worst case for all algo- every attempt. rithms is ( ). 6 More interesting is the performance of the algorithms However,O the∞ deterministic algorithms are guaranteed in the limit of high noise, δ = 1, such that no com- to terminate when j = 0. In the worst case, the two Noise vs. Time 2 120 algorithms terminate in n iterations when they share Sequential no common landmarks. At this point, both agents can Smart-Sequential 100 Exponential determine that they cannot meet, and continue explo- Random ration. If, however, the agents share identical landmark sets, the sequential algorithm has a much lower worst- 80 case complexity than the smart-sequential strategy, be- cause one agent is guaranteed to visit every landmark 60 in the other agent’s set in n iterations. Time 40 Algorithm Simple Async. < 100% Comm. Random 20 Sequential ∞n ∞ nd∞ ∞ 0 Smart Seq. n2 n n2 (n d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 − ∞ − − Noise Figure 2: Worst case behaviour. Columns as in Fig. 1. Figure 3: Baseline Performance - Time to Rendezvous as a function of Noise-level δ Numerical Simulation In order to determine the behaviour of the various algo- the face of high noise (i.e., δ > 0.2) , which concurs with rithms with increasing noise, the algorithms were tested the analytical result. Clearly, exponential is a very frag- in numerical simulation. Rather than simulating an ac- ile function, failing catastrophically with noise, δ > 0.2. 1 tual exploration , two agents were modelled as having Noise vs. Time, 100 Landmarks 200 already explored an unknown area, and collected a set "Random" of landmarks. The distinctiveness values of the ordered 180 "Exponential" "Sequential" landmarks were generated with a linear function, and 160 "Smart-Sequential" then the random noise δ as developed in equation 1 was 140 applied to the two sets. 120 The visitation strategy was then executed on the two 100 ordered sets of landmarks, creating a (potentially in- Time finite) sequence of landmarks for each agent to visit. 80

The sequence was terminated at the first instance where 60 both agents had a landmark at the same position in the 40 sequence, corresponding to a successful rendezvous. At δ = 0, the two ordered sets were identical, and the deter- 20 0 ministic algorithms (sequential and smart-sequential) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 were guaranteed to generate sequences of length 1. The Noise length of the sub-sequences until rendezvous was used Figure 4: Performance with 100 Landmarks, ideal case as a measure of time until successful rendezvous. The cardinality of the landmark set was 50 landmarks, un- Figure 4 shows the performance of the algorithms less otherwise specified. with a larger landmark set cardinality. Unsurprisingly, The baseline simulation shows the performance of the performance of the algorithms scales appropriately four algorithms in the face of increasing noise. The four with landmark set size. algorithms are the deterministic sequential and smart- In the face of asynchrony, however, the algorithms sequential algorithms, and weighted probabilistic dis- exhibit less intuitive behaviour. Asynchrony, again, is tributions with exponential and linear probability func- the probability that a particular rendezvous succeeded. tions. Recall that the exponential probabilistic func- The simulation (which creates landmark sequences) im- tion, for example, would have an exponentially higher plemented asynchrony as the probability that a partic- probability of visiting the best landmark over any other. ular sequence element could be used. Even if the pair There is no asynchrony present, and the landmark sets of landmark sequences contained the same landmark at were completely common. Figure 3 shows that the se- identical positions, the sequence may not have termi- quential algorithm is the best performer, especially in nated there, because the asynchrony probability pre- 1This was carried out as well, but is not reported here vented the first pair of matching landmarks in sequence due to lack of space. from being compared, as if the robots had failed to ren- dezvous successfully despite attempting to do so at the rapidly in the case of high noise (δ > 0.25). The expo- same location at the same time. nential algorithm essentially forces the robot to return

Noise vs. Time, 50% Asynchrony Rate to the same landmark over and over again, which is 200 "Random" the correct strategy when asynchrony is high. However, 180 "Exponential" when noise is high, the odds that the recurrent land- "Sequential" 160 "Smart-Sequential" mark is the wrong one increase, and the deterministic

140 algorithms, which do not return to the same landmark as often, perform better. 120

100 Noise vs. Time, Non-identical Landmark Sets Time 120 80 Sequential Smart Sequential 60 100 Exponential Random 40 80 20

0 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Noise

40 Figure 5: Performance with 50% Asynchrony rate

20 Figure shows the performance of the algorithms given a 50% asynchrony rate, or a 50% probability of suc- 0 0 0.2 0.4 0.6 0.8 1 cessfully making a rendezvous. In this case, the smart- Noise sequential and exponential algorithms out-perform the sequential strategy, because the sequential form suffers Figure 7: Performance with 25% non-commonality in from having to visit every other landmark before being landmark sets able to return to the landmark that failed on a par- ticular iteration, whereas the other two algorithms can Noise vs. Time, 50% Asynchrony Rate, Non-identical Landmark Sets 200 return to landmarks relatively quickly. However, once "Random" 180 "Exponential" noise dominates the values, (δ > 0.5) the sequential al- "Sequential" gorithm becomes faster because it does not rely heavily 160 "Smart-Sequential" on particular landmark values - it is not returning to 140 the same landmark over and over again. 120

Noise vs. Time, 80% Asynchrony Rate 100 300 Time 80 Sequential Smart Sequential 250 60 Exponential Random 40 200 20

150 0

Time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Noise

100 Figure 8: Performance with non-identical landmark

50 sets, and 50% Asynchrony rate Finally, Figures 7 and 8 show performance for maps 0 with only 75% of the landmarks in common. The per- 0 0.2 0.4 0.6 0.8 1 Noise formance with non-identical landmark sets (akin to non- isomorphic maps) is very similar to performance under Figure 6: Performance with 80% Jitter rate low- to medium-asynchrony. The smart-sequential al- gorithm performs better with low noise because it can Even more interesting in the case of very high asyn- return to landmarks faster than sequential, but in the chrony, Figure 6 shows that the exponential probabilis- case of high noise (δ > 0.35), returning to landmarks too tic function outperforms the deterministic algorithms in frequently can be costly, and the sequential algorithm the face of low noise (0.5 < δ < 0.25), but again fails again dominates. Conclusion noise. The pure sequential strategy is preferable when asynchrony is low, since without asynchrony a meet In this paper we have described the new problem of per- is assured after n visits by avoiding visiting combina- forming rendezvous between two exploring robots in an tions that might otherwise compensate for asynchrony. unknown environment. We are specifically interested in With substantial levels of both asynchrony and moder- multi-robot environment exploration, where communi- ate noise the stochastic search strategy is preferable to cation is limited to short range. This is the first paper deterministic ones: a phenomenon we are investigating to describe a formalism for this problem, and we have further. presented an analytical and numerical analysis of some Issues for future consideration are the role of land- solutions to the rendezvous problem. Although we have mark positions and the robot in selecting where to visit. primarily dealt with two agents, the algorithms we have In practice, it may be preferable to visit a nearby land- presented could readily be adapted to larger collections mark for rendezvous, although the consequences of such of agents, or swarms. heuristics are not obvious in all cases. A further issue is We have shown that certain algorithms for performing how to select appropriate distinctiveness measures with a rendezvous in an unknown environment are especially different sensor systems. good or bad under certain system and environment con- References ditions. These factors include sensor noise, lack of com- Arkin, R. C., and Hobbs, J. D. 1992. Dimensions of com- monality in the regions explored by the robots, and the munication and social organization in multi-agent robotic possibility of the robots missing a scheduled rendezvous. systems. In Proc. SJB 92. 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