Optimal Backbone Generation for Robotic Relay Networks

Optimal Backbone Generation for Robotic Relay Networks Ying Zhang Palo Alto Research Center Inc. 3333 Coyote Hill Rd Palo Alto, CA 9434, USA Emails: yzhang@parc.com Mark Quilling Lockheed Martin Space Systems Center 1111 Lockheed Martin Way Sunnyvale, CA, 9489, USA Email: mark.l.quilling@lmco.com Abstract There has been a growing interest in robotic relay networks for applications in dangerous and hazardous environments. In particular, a team of robots with onboard radios deployed in an unknown environment would coordinate to form an ad-hoc relay network that maximizes the connectivity to transmit data from a set of unknown data s to a gateway. In this paper, we present a problem and solutions to minimize the number of stationary relay nodes given a set of unknown but stationary s. The goal is to maximize the number of free robots to search for more unknown s to extend the connectivity of the network, while keeping the communications to the existing connected s. We present two types of algorithms: OLQ (Optimal Link Quality) and AST (Approximate Steiner Tree), and analyze their performance in various routing metrics. The algorithms have been developed in a real robotic platform and demonstrated in multi-floor indoor environments. keywords: robotic relay networks, spanning trees, approximate Steiner trees, optimal backbones I. INTRODUCTION There has been a growing interest in robotic relay networks for applications in dangerous and hazardous environments. One of the challenging research issues is to develop strategies for robots to form a reliable communication relay network to support human operations in an indoor Non Line Of Sight (NLOS) environment such as buildings, rubbles, caves and other urban terrains. In particular, a team of small and lowcost robots (Fig. 1) with onboard radios deployed in an unknown environment would coordinate to form an ad-hoc relay network that maintains the connectivity to transmit data from a set of unknown data s to a gateway. In this paper, we focus on a problem of establishing an optimal backbone for a relay network for the connected data s in a sense that the number of robots used in the backbone is minimized. The goal is for more robots to search for unknown s and connect as many s as possible so that the overall data throughput in a given period of time is maximized. The motion strategy of searching for data s is out of the scope of this paper. The robots in this study are icreates and LANdroids (Fig. 2) from irobot Corporation (www.irobot.com), with SR71 USB radios. A version of OLSR (Optimal Link State Routing: www.olsr.org/) is used as the protocol for data transmissions from the data s to the gateway. The system uses Fig. 1. A robotic team, consisting of LANdroids, each of which is equipped with an onboard radio. neighbor discovery in OLSR and propagates the neighboring connectivity to build a global radio matrix, i.e., a pairwise radio connectivity matrix of the connected radios. We have developed a protocol to disseminate the radio matrix to each node that guarantees that such a radio matrix is consistent for all the nodes in the connected network. This consistency is important for global solution consistency with local computation. The protocol consists of two phases. In the first phase, all nodes broadcast hello packets a couple of times while averaging received radio signal strength for each neighbor. Phase one ends when no more packets received after certain time, by which time, each node has a neighbor table with signal strengths from all its neighbors. The second phase disseminates the neighbor table to form a radio matrix. Each node assembles all the pairwise connections it has received and broadcasts the list. The second phase ends when no new connections are received. Our application does not involve hundred thousands of nodes as many other sensor network applications, therefore this protocol works well for our purpose. Using such a radio matrix, each node will compute the optimal backbone. Since all nodes use the same data, the optimal solution is the same. The nodes that do not belong to the backbone are free to move for searching other radios in the environment. 978-1-4577-638-7 /11/$26. 211 IEEE

II. PROBLEM DEFINITIONS Fig. 2. icreate (left) and LANdroid (right) from irobot Corporation Building a backbone for a subset of nodes in a network using minimum number of relay nodes is a Steiner Tree problem [4]. We propose two algorithms for computing such backbones: OLQ (Optimal Link Quality) uses minimum spanning tree heuristics that produces backbones with the best link quality defined by signal strength, AST (Approximate Steiner Tree) uses shortest path heuristics that produces backbones with minimum number of hops from s to the gateway. Variations of the algorithms are also proposed to further improve the performance metrics. Various research has been done for creating optimal spanning trees in wireless ad hoc networks for transmitting data to a gateway, e.g., work in [7] focused on routing with minimum power given data s and transmission frequency, using approximate algorithms of Steiner trees. The requirement for this work is different since we are more interested using minimum number of relay nodes, while keeping the number of hops small and signal strength high. The AST algorithm, however, is similar to ISTH in [7]. The OLQ algorithm uses the minimum spanning tree algorithm, with weights as a function of signal strength. The contributions of this paper is on the application of these algorithms, not the algorithms of themselves. There also has been work on optimization for mobile backbone networks [1][6], however, the focus of their research is on how to place the mobile nodes among the static nodes to achieve connectivity and maximum throughput. Similar work has been done in [2] to control the mobility of robots for minimum power consumption in terms of both communication and motion power usage. In our case, we try to find out which nodes should stay in place and which can move, not how to control the mobility of the mobile nodes, and we do not assume robots have the knowledge of their geographical locations. There has been work in building spanning tree for maximum throughput in P2P live media streaming [5], which maximize minimum node throughput. Here we assume the data rate is much smaller than the maximum throughput a node can support, therefore there is no restriction on the number of children as in [5]. The rest of the paper is organized as follows. Section II describes the problem with assumptions and performance metrics. Section III presents the Optimal Link Quality algorithm and Section IV develops the Approximate Steiner Tree algorithm. Section V analyzes performance of both algorithms via simulations and Section VI illustrates a real test scenario in a multi-floor office environment. Section VII concludes the paper. We consider a set of unknown but static data s S in an environment, each of which is transmitting data to a gateway g at a constant rate. Given a set of robotic relay nodes R, our ultimate goal is to control the mobility of relay nodes in a way so that all data s can reach the gateway. We assume, however, the total data rate from all the data s is less than the maximum throughput that a node can support, i.e., there is no limitation on the number of s a node can relay from. This may not seem scaleable, but it is the case in our real scenarios. We have about 1-15 relay nodes, 3-1 data s, and one gateway node. Each data sends 1.5 Mega bits data per second to the gateway as soon as they are deployed. We do not assume robots or data s location-awareness. Given an initial deployment scenario of the robots, only a subset of data s are connected to the gateway through the robotic relay nodes. In order to search for more data s, robots have to explore the environment to locate the additional radio s. We decompose this problem into two subproblems: (1) which robots can move and which should stay in place in order to relay the connected data s, and (2) for those can move, how to explore the environment in an efficient way. This paper focuses on the first problem. We formulate the first problem as an optimal backbone generation problem. Given a graph G = V,E, where V = R S {g} and E is the radio matrix, such that E(u, v) is the signal strength from node u to v. If v cannot receive radio signal from u, E(u, v) =. In our real platform experiments, the non-zero signal strength ranges from 16 to 7, with larger values indicate higher signal strengths. The connections with small signal strengths are not very reliable and may cause large data loss. We say a link exists from u to v, if and only if E(u, v) L where L is a threshold of the signal strength for reliable data transmission. A data s S is connected to the gateway g, if and only if there is a path from s to g in G. Most routing protocols require bidirectional links. In practice, links are asymmetric, i.e., E(u, v) E(v, u). To get symmetric links from asymmetric links, instead of averaging the two signal strength from both directions, we set E(u, v) and E(v, u) to be the minimum signal strength of the two directions, so that one directional links are removed and the routing protocol is more robust. The problem of backbone generation is to find a tree T G, such that T = V,E, with V V and E E, and V includes g and all data s in S that are connected to g and a subset of relay nodes in R (Fig. 3), which we call backbone nodes. In addition, we prefer links in the backbone with high signal strengths and the number of hops from data s to the gateway small. In summary, we consider three objectives: (1) minimum number of backbone nodes, (2) maximum signal strength for backbone links, and (3) minimum hops from s to the gateway, with the order that (1) has higher priority than (2) and (3). This three objectives will be used as performance metrics for comparing different algorithms.

Fig. 3. Backbone example: thick lines indicate backbone links The primary goal, i.e., finding a spanning tree covering a subset of nodes with minimum number of relay nodes (or minimum weight assuming each link has the same weight) is a Steiner Tree problem, which is NP-hard [3]. We developed two types of approximate algorithms: Optimal Link Quality (OLQ) uses the minimum spanning tree heuristics and Approximate Steiner Tree (AST) uses the shortest path heuristics. III. OPTIMAL LINK QUALITY ALGORITHM We associate link weight w(u, v) to each link E(u, v), the lower the weight, the higher the quality. Let L be the threshold of signal strengths, below which, the data loss is high, and O be the value of signal strength, above which, the data loss is low, and M is the maximum signal strength, L O M. The link weight is defined as follows: { α(e(u, v) O) if E(u, v) O w(u, v) = (1) β(o E(u, v)) if E(u, v) <O where α> and β>, and α(m O) β(o L). (2) A minimum spanning tree rooted at g is then generated given the weights on the links. The algorithm first orders all the weights from minimum to maximum. Starting from the gateway g T, each time a node v is selected to add to T if u T and v T and w(u, v) is the smallest among all the possible choices of u and v, link (u, v) is then added to T. This weight function prefers links close to the optimal value O, because with this value, two nodes are not too close to each other, while link quality is good enough. If signal strength is below the critical threshold L, its weight would be larger than the weights of links whose signal strength are above L given relation (2). Links with signal strength below threshold are not chosen unless nothing else can be used for building the tree. After building the minimum spanning tree, we repeatedly remove relay nodes that are leaves, until nothing can be removed. What left is the backbone which connects existing s to the gateway. What removed are free nodes that can be used for exploring more data s. Theorem 1: Let L m be the minimum signal strength in the backbone tree generated from the weight function defined by Eq. (1). If L m <L, the network is not connected. Proof: Let L m <Lbe the signal strength of a link between u and v, i.e., (u, v) in the spanning tree, there is no path from u to v with signal strength at or above L. This can be proved by contradiction. If there exists a path from u to v with the minimal signal strength of the links in the path L, weights of these links are smaller than the weight of (u, v), therefore, those links would be used instead of link (u, v) for the backbone. Note that if O = M, the algorithm would build a spanning tree with strongest minimum signal strength, which may use more relay nodes. On the other hand if O is close to L, the algorithm would build a tree that prefers low signal strength, resulting possibly less relay nodes. From our real platform experiments, we found that 23 <O<3 seems to work the best, such that the links chosen are reliable while keeping the number of backbone nodes small. Theorem 2: If O = M, the minimum signal strength of the backbone tree is maximized among all possible backbone trees. Proof: Links with the highest signal strength will be picked to build the tree. Note that when O < M, this algorithm may generate path with larger number of hops using weaker links closer to optimal weights. A variation of the spanning tree algorithm for this problem is as follows: Starting from the gateway g T, each time a node v is selected to add to T if u T and v T and w(u, v) is the smallest among all the possible choices of u and v. Instead of adding link (u, v) to T, link (u, t) is added to T given that E(u, t) is the highest among all nodes of t T. We call this variation OLQm. IV. APPROXIMATE STEINER TREE ALGORITHM Similar to OLQ, we associate a link weight w(u, v) to each link E(u, v), 1 if E(u, v) L E(u,v) w(u, v) = n (3) if E(u, v) <L E(u,v) where n = V is the total number of nodes. The algorithm first computes the shortest path from every node v to the gateway g using the weight functions given by Eq. (3). Let d(v, g) be the distance of the shortest path from v to g. For each node v, let p(v) be a penalty to select v, which is a small constant c, c< 1 L,ifv is not a parent of a node yet. Initially, p(v) is c for all nodes. Order all nodes in increasing order with respect to their shortest distance to the gateway. For every node v in that order, select a neighbor u to be its parent, if and only if (1) d(u, g) <d(v, g) and (2) d(v, u) +d(u, g) +p(u) is minimum. Then set p(u) =since u is a parent (of v). The algorithm finishes when every node except g has a parent.

After repeatedly removing relay nodes that are leaves, what left is a backbone tree. This algorithm prefers to select a parent that is already a parent given the same shortest path distance, resulting possibly less relay nodes in the backbone. This weight function, Eq. (3), has a good property: Theorem 3: For any node, the path with minimum distance to the gateway has the number of hops to the gateway bounded by a constant factor of the minimum number of hops. In M particular, the constant is k L, where L and M are minimum and maximum signal strength, respectively. Proof: Let h be the minimum number of hops with distance D and H be the number of hops with minimum distance d, from a node to the gateway, i.e., h H and d D. Also, assume m is the maximum and l is the minimum signal strength in H this network, m d and D h H l. Therefore, m h l, i.e., H m l h.letk = m l, since m M and l L, M k L. Since the penalty for non-parent nodes p(v) is a small constant < 1 L, it prefers parent-sharing without losing too much on link quality, or adding one extra hop. We can also incorporate the information of the existing routing structure to reduce the time of re-routing. To achieve this, we let p(v) be the penalty function of node v: p(v) is c if it is not a relay node in the current routing and is otherwise. The link threshold L is sometimes hard to set. If L is set too low, it may create a tree with weak links. On the other hand, if it is too high, one may use many relay nodes and the network may be fragmented. We can alleviate this problem by given a range [L min,l max ] and set L min L L max as follows: First run OLQ with O = M and find out the weakest link in the tree with signal strength L m. Then set L by Eq. (4). L m if L min L m L max L = L min if L m <L min (4) L max if L m >L max L min indicates the lower bound, below which there is a large data loss, and L max is set to trade-off the quality of links and the number of relay nodes. In our real platform experiments, L [18, 22] works good. Theorem 4: The backbone will connect all nodes if and only if L m L min. Proof: Let L m be the signal strength between u and v. If L m <L min, there is no connection between u and v with signal strength at or above L min, therefore it is not connected. Otherwise, there will be a connection above or at L L m between any two nodes. V. PERFORMANCE ANALYSIS We have analyzed the performance of these two algorithms with their variations, using simulations of radio networks: OLQ: Optimal Link Quality; OLQm: Optimal Link Quality with maximum strength link extension; AST: Approximate Steiner Tree without penalty to new patents; 25 2 15 1 5 5 1 gateway 15 2 3 2 1 1 2 3 Fig. 4. Example of a network and backbone using AST. Red lines indicate walls that divide the area AST: Approximate Steiner Tree with penalty to new parents. For each algorithm, we compute three performance metrics: BNs: the number of backbone nodes from data s to the gateway; mss: the minimum signal strength of all the links used in the backbone; MHs: the maximum number of hops from any data s to the gateway. The simulation creates an indoor environment 4m by 4m in size, with walls dividing the areas. A gateway is placed at an entrance, and four data s are located at four corners. About 4+ robotic nodes are deployed at the leftlower corner of the area. A simple radio model is used to create the radio matrices, i.e., pair-wise signal strengths. Signals across each wall are reduced by a constant factor and signals less than a threshold are cut off and set as. Signal strengths range from 16 to 7 to match our real experiment data. Fig. 4 shows a scenario, where lines indicate communication pairs, thick lines are branches of the backbone tree. By running the algorithms, the system is able to figure out which robots to act as backbone nodes and which can go to explore the environment to find the fourth data. One hundred random test cases are generated to run all four algorithms. For OLQs, we have set O =25, α =1and β =2.ForASTs,wehavesetL =2and c =.25. The performance metrics for the four algorithms are shown in Fig. 5, Fig. 6, and Fig. 7, respectively. These figures are histograms, so that X-axes are the performance measurements of that performance metric and Y-axes are number of occurrences for such a measurement. The plots indicate rough probability distributions for the given metrics. They are more informed than simply average and standard deviation numbers. Here is the summary of the performance of the four algorithms.

3 OLQ 3 OLQm 4 OLQ 4 OLQm 2 2 3 3 2 2 1 1 1 1 1 15 2 1 15 2 5 1 15 2 5 1 15 2 5 AST 5 AST 6 AST 6 AST 4 3 4 3 4 4 2 1 2 1 2 2 6 8 1 12 14 6 8 1 12 14 5 5.5 6 6.5 7 5 5.5 6 6.5 7 Fig. 5. Histograms of the number of backbone nodes (BNs) Fig. 7. Histograms of the maximum number of hops (MHs) 4 3 2 1 OLQ 15 2 25 3 6 4 2 AST 2 25 3 Fig. 6. 4 3 2 1 OLQm 15 2 25 3 6 4 2 AST 2 25 3 Histograms of the minimum signal strength (mss) For the number of backbone nodes, our main objective, AST is the best, which is slightly better than AST. OLQm is a lot better than OLQ, however, both perform a lot worse than ASTs. For the minimum signal strength, OLQm has the best profile, which is better than OLQ, both are better than ASTs. For the maximum number of hops from s, similar to the number of backbone nodes, AST is slightly better than AST. OLQm is a lot better than OLQ, however, both perform a lot worse than ASTs. From those performance data, we see that AST is a better choice for this application. AST and OLQ can be combined in the sense that OLQ can be used to set L forast,aswe described in Section IV. VI. REAL EXPERIMENTS We have implemented these algorithms in C++/Linux on microprocessors in icreates and LANdroids, with SR71 USB radios (Fig. 2), as part of the software suite for the LANdroids platform. We have tested the system in multiple multi-floor buildings. Fig. 8 is a scenario where total of 18 nodes are deployed in two floors of an office building, 1 gateway, 4 data s, and 13 robotic relay nodes. This figure shows the result of backbone using AST. In this case, there are 6 backbone nodes, and 7 free nodes. The maximum number of hops from a to the gateway is 4. And the minimum signal strength in the backbone is 31, which is way above the minimum threshold. From those experiments, we learn that the signal strengths have little to do with distance between two nodes; they are related to the locations of nodes in a very complex way (signal amplifications and reflections in indoor environments) and transmitter/receiver battery levels. The only reliable information is the radio matrix itself, which our algorithms are based upon. VII. CONCLUSIONS We have presented in this paper some algorithms for creating a backbone using minimum number of robotic relay nodes, which is important so that more robots can be used for exploring the environment to search for unknown s. Analysis in simulations has shown that AST is the best algorithm in this aspect, in terms of the performance metrics we care about. We have implemented these algorithms in real robotic platforms and tested in various indoor multifloor environments. Although there has been related work on spanning trees for networks, our problem has some special features of its own. Since the robotic relay networks are relatively new, we have not seen existing algorithms for our application to compare our algorithms with. Motion planning and control have also been developed to use with the backbone generation, however, it is out of the scope of this paper.

Fig. 8. Scenario: the backbone is shown in dark thick lines and connectivity is shown in thin lines. Nodes with dark borders are backbone nodes and others are free nodes. Numbers by the data s show the number of hops to the gateway through the backbone. More variations of these algorithms can be developed. For example, for AST, adding a general penalty function giving the number of existing children, in which case, one encourages sharing parents up to a maximum number and discourages further children if throughput is limited. Another approach is to simultaneously optimize all three performance metrics, for example, discovering the value of L in AST that is best overall for all three metrics. We can also use as many existing relay nodes as possible to avoid re-routing, which will reduce data loss significantly. These algorithms are independent to any type of routing protocols one may use for data communications. ACKNOWLEDGEMENT This work was sponsored by the Defense Advanced Project Research Agency (DARPA) contract #FA865-8-C-7814, LANdroids (http://www.darpa.mil/i2o/programs/ld/ld.asp). REFERENCES [1] E. M. Craparo, J. P. How, and E. Modiano, Optimization of mobile backbone networks: Improved algorithms and approximation, in Proceedings of American Control Conference, 28. [2] F. El-Moukaddem, E. Torng, and G. Xing, Mobile relay configuration in data-intensive wireless sensor networks, in IEEE International Conference on Mobile Ad-hoc and Sensor Systems, 29. [3] M. R. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., 199. [4] E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM Applied Mathematics, vol. 16, pp. 1 29, 1968. [5] X. Lu, Q. Wu, R. Li, and Y. Lin, On tree construction of super peers for hybrid P2P live media streaming, in Proceedings of International Conference on Computer Communication Networks, 21. [6] A. Srinivas, G. Zussman, and E. Modiano, Construction and maintenance of wireless mobile backbone networks, IEEE/ACM Transactions on Networking, 29. [7] G. Xing, C. Lu, Y. Zhang, Q. Huang, and R. Pless, Minimum power configuration for wireless communication in sensor networks, ACM Transactions in Sensor Networks, vol. 3, 27.