A Guidance Law for a Mobile Robot for Coverage Applications: A Limited Information Approach

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1 Proceedings of the Third International Conference on Adances in Control and Optimization of Dynamical Systems, Indian Institute of Technology anpur, India, March 1-15, 014 Frcsh1.6 A Guidance Law for a Mobile Robot for Coerage Applications: A Limited Information Approach Twinle Tripathy Arpita Sinha Interdisciplinary Programme in Systems and Control Engineering, IIT Bombay, Mumbai , India twinle.tripathy@sc.iitb.ac.in, asinha@sc.iitb.ac.in Abstract: In this paper a guidance law has been proposed to sole applications related to coerage of a pre-defined area using a mobile robot. In many situations, it may not be possible to measure all the information due to practical limitations or from computational point of iew. So, we hae used only range information to sole the aboe specified problem. The target point is considered to be stationary in this particular case. Now, for any initial condition for which the elocity ector of the robot is perpendicular to the line of sight with respect to the stationary target, the autonomous agent displays interesting behaiours which hae been analysed theoretically. All the analytical results obtained here hae been alidated through MATLAB simulations also. eywords: Coerage problems, minimal information, autonomous mobile robots, unicycle model, root locus. 1. INTRODUCTION Oer the past few decades the problem of coerage of an area by a mobile robot has been a ery challenging problem. It finds seeral applications these days. Most common applications would be acuum cleaning, painting, humanitarian demining, foraging, farm irrigation and so on. Some of the more crucial applications would be monitor the area around a stationary point lie a building. The coerage problem refers to the problem of guiding an autonomous agent to coer an unnown or predefined area. Seeral techniues hae been proposed to achiee coerage, which ary depending on the application. Sureillance based systems, discussed in Dhillon et al. 00), Lai et al. 007) and Wang et al. 008), use wireless sensor networs to sole the coerage problem. The sensor nodes are capable of behaing autonomously. In Dhillon et al. 00) and Wang et al. 008), the coerage achieed is aerage, but more emphasis is laid on the placement of sensors for effectie coerage; Dhillon et al. 00) hae based their wor on fixed sensor nodes. Lai et al. 007) discuss sureillance systems based on wireless sensor networs. In Lai et al. 007), the deployed sensors are diided into disjoint subsets of sensors, or sensor coers, such that each sensor coer can coer all targets and wor in turns. A sureillance system based on wireless sensor networs has been discussed by Yan et al. 00). The system proides a differentiated sureillance system which deploys sensors nodes in appropriate areas depending on the degree of sureillance reuired. But, the wireless sensor networs suffer from the problem of energy reuirements which increase with the number of sensor nodes used in the networ. There are seeral algorithms which are used to coer unnown areas. For example, a lot of wor has been done on map based coerage techniues see Wong et al. 00), Zelinsy et al. 199), Stachniss et al. 00) and Rutishauser et al. 009) and references therein) which wor by diiding the area into cells and then assigning alues to the cells based on the presence of obstacles or free space. Wong et al. 00) used topological maps to sole the coerage problem and their results are erified by simulation tests which show that oer 99% of the surface is coered. The topological maps, represent the enironment as graphs where landmars are nodes, and edges represent the connectiity between the landmars. The algorithm achiees coerage with single mobile agent. Zelinsy et al. 199) also use maps for coerage but with a different approach. The solution ensures complete coerage by the use of a distance transform path planning methodology. The solution was simulated and implemented on an autonomous robot called Yamabico to gie satisfactory performance. Stachniss et al. 00) introduced the concept of coerage maps where each cell of a gien grid corresponds to the amount of a cell which is coered by an obstacle; the coerage maps are improement of occupancy grids which are based on the assumption that each cell is either occupied or free. The model presented in the paper allows updation of the coerage maps upon input obtained from sensors. Rutishauser et al. Rutishauser et al. 009) sole the problem of collaboratie coerage using a swarm of networed miniature robots again by the use of grid based methods. For a multi-agent system, the problem of coerage, addressed by Bataia et al. 009), has been soled with high accuracy for a semi- structured enironment. The approach has the adantage of an operator driing the outline of a desired coerage area as input to a coerage generation algorithm. Acar et al. 001) hae 47

2 Tx T, y T ) R max ρr T R min 1 ρ)r VTx V T, y V T ) α θ Fig. 1. Region to be coered Fig.. Coerage obtained by an Autonomous Agent soled the coerage problem by the decomposition of the enironment, by the use of oronoi diagrams. In this paper we hae proposed a guidance strategy that uses only range information. The transmission of minimal information reduces the bandwidth reuirement. This leads to energy conseration. Priliminary results on the guidance strategy hae been done by Tripathy et al. 01). In the wor by Tripathy et al. 01), we hae shown that the coerage strategy is stable for eery initial condition. The problem discussed in this paper is formulated as follows. The autonomous agent is modeled as an unicycle. We assume, the region to be coered, is circular or annular around the target as shown in Fig. 1. The agent uses only distance information to coer the annular region. The annular region is specified by R min and R max. It is shown that R min and R max depend on the initial conditions. Using this, we hae deried: The trajectory of the agent under the initial condition for which the elocity ector is perpendicular to the line of sight with respect to the target. The initial conditions necessary to achiee coerage of an entire circular dis of any radius. This paper is organized as follows. In Section and, we hae analysed the problem for a particular situation. Simulations results are presented in Section 4 and Section 5 concludes the paper and discusses the areas on which further wor can be done in future.. COVERAGE STRATEGY The paper addresses the coerage problem as shown in Fig. 1. Let the target be at x T, y T ) and is stationary at any instant t > 0, position of agent is gien by xt), yt)), Ax, y) Fig.. Engagement geometry Reference as shown in Fig.. The inematics of agent is as shown below: ẋt) = cos αt) 1) ẏt) = sin αt) ) αt) = ut) ) where is the linear elocity, which is a constant for our case, αt) is the heading angle and ut) is the control input. In Fig., rt) is the line-of-sightlos) distance and θt) is the LOS angle. Since only range measurement is feasible, agent nows only instantaneous rt) and not θt). Let V T denote a irtual point, called the irtual target which is at a distance ρrt) from the target along the LOS, where ρ 0, 1). In the control law, the lateral acceleration of the autonomous agent is made proportional to the distance between the irtual target and the agent, gien by x V T xt)) + y V T yt)) ) 1/ = 1 ρ)rt). So, the algorithm is defined in terms of a M t) as: a M t) = h1 ρ)rt) 4) αt) = a M t) = rt) 5) where h is the constant of proportionality between a M and the distance, 1 ρ)r, and = h1 ρ). The parameter h is a system gain. The concept of the irtual target increases the flexibility of the algorithm as we get another parameter ρ to be aried as needed. The algorithm generates arious coerage patterns surrounding the stationary target. One of the patterns generated is shown in Fig.. In the next section, the control algorithm is analysed for area coerage applications. Consider the Fig., the LOS between the agent and the target can be charaterised as follows: r = ṙ = cosα θ) 6) θ = r θ = sinα θ) 7) Let us define φ = α θ. By combining 5)-7) and soling the integration, we establish a relationship between r and φ as, r sin φ + r = r 0 sin φ 0 + r 0 8) In the aboe euation, r 0 and φ 0 are the initial conditions. We define a ariable which contains all the terms corresponding to the initial conditions, such that, 48

3 r Bφ = π) point A and thereafter it decreases as r < 0 in the path A D C. Hence, R max occurs at Aφ = π ) and R min occurs at Cφ = π ). Cφ = π ) Fig. 4. θ, r ) plot C B r D A Fig. 5. Engagement trajectory 1 in θ, r ) space θ Dφ = 0) r0 sinφ 0 ) = θ Aφ = π ) r B C A Fig. 6. Engagement trajectory in θ, r ) space ) + r0 9) r r sin φ + = 0 10) By setting 6) to zero, we obsere that r can tae its maximum and minimum alues that is R min and R max when φ euals either π or π. Next we loo into how the inematics aries in the θ, r ) plane. Upon suaring and adding 6) and 7), we find that the instantaneous θ and r lie on a circle in the θ, r ) plane as shown in Fig. 4. By the use of 6) and 7), we can calculate the alue of φ corresponding to each set of instantaneous r and θ. Point A corresponds to θ, r ) =, 0), thus, φ = π. Similarly, we get that at B, φ = π, at C, φ = π and at D, φ = 0. Hence, the clocwise moement about the circle shown in Fig. 4 gies the direction of increasing alues φ. Combining 5) - 7), we get, dφ dt = + r ) 11) r The rate of change of φ with respect to time aries depending on the sign of. Considering the two cases separately, we hae: Case 1: > 0 The aboe euation indicates that for > 0 we get φ > 0. This corresponds to clocwise moement in the θ, r ) plane as shown in Fig. 5. Now, r > 0 in the path C B A which means instantaneous r increases in this path upto θ Case : 0 r and θ are continuous functions of φ. By re-arranging 10), we get sin φ = r ) r. Hence, from 7) for all alues of 0, θ 0. This indicates that the engagement geometry can exist only in the first and fourth uadrants in the θ, r ) plane. So, the only feasible inflexion point is φ = π. Hence, considering the continuity of the θ and r functions, we conclude that the θ, r ) plot oscillates about φ = π as shown in Fig. 6. Now, r > 0 in the path A B A which means instantaneous r increases in this path till the second time it reaches point A. Thereafter, it decreases as r < 0 in the path A C A till it comes to the point A for the second time. Hence, in this case, both R max and R min occur at Aφ = π ). Thus, by using the coerage control algorithm gien in 5), we get two types of engagement trajectories in r, θ ) plane, depending on or the initial conditions r 0, φ 0 ). By differentiating 6) and 7), we get: r = θ φ 1) θ = r φ 1) At the points B and C shown in Fig. 6, r and θ become zero simultaneously, which implies φ = 0. Using this condition in 11), we get the following result for the instantaneous alue of r: r = ) 1 14) where = ). Substituting 14) in 10) and further soling, we get an expression of instantaneous φ as, sin φ = 4 ) 1 15) En. 15) gies a bound on = ) as sine function can tae alues between -1 to 1. Hence, the minimum alue of can be obtained by setting sin φ = 1, which gies = =. Let us denote the lower alue of by: br =. Thus, coerage becomes feasible for negatie alues of only if br 0. Next we address two problems: If we are gien an initial condition, what will be the coerage area and if we are gien an area to be coered, what should be the initial conditions to achiee the coerage? Gien an initial condition r 0, φ 0 ), we hae the following result: Theorem 1. Tripathy et al. 01)) Gien a set of initial conditions r 0, φ 0 ) for an agent with inematics 1)-) and a control law gien by 5), coerage will be achieed for a region gien by the maximum radius R max and minimum radius R min which are gien as: R max = az 1 + bz 16) R min = az + bz 4 17) 49

4 Theorem. Tripathy et al. 01)) For r 0 = and φ 0 = π, the trajectory of an agent with inematics 1)- ) and a control law gien by 5), is a circle of radius. Ax, y) α φ = π θ Tx T, y T ) Reference The detailed proof of this theorem has been coered in the paper by Tripathy et al. 01). Proposition. If φ 0 = π and r 0 [0, ], then R min = r 0. Fig. 7. Agent and target initial engagement for φ 0 = π where 1 + i a =, if br 0 1, if 0 < < 1 i b =, if br 0 1, if 0 < < z 1 = ) z = ) 1 4 z if br 0 z = ) if 0 < < z 1 if br 0 z 4 = ) if 0 < <. The detailed proof of the theorem is gien in the wor by Tripathy et al. 01).. AREA OF COVERAGE In this section we hae considered the special case when the initial conditions are selected such that φ 0 = π. In physical sense, this is the configuration when that the initial heading angle is perpendicular to the LOS between the agent and the stationary target as shown in fig. 7). Critical Initial Position: Critical initial position refers to that alue of r 0, φ 0 ) for which R max and R min are eual. This maes the agent moe on a circular path around the stationary target for all time. Also, it is interesting to note that for these initial coordinates, we get R max = R min = r 0. Proof. Substituting φ 0 = π in 9), we find out that for r 0 [0, ] 0 and otherwise > 0. Let us consider the case r 0 [0, ]. From Section, we now that for 0, R max and R min occur at φ = π. Now, we substitute φ 0 = π in 8) and sole it to get the following roots: r 1 = r 0 18) r = r 0 + r0 + 19) 4 One of the roots is R max and the other one is R min. The negatie roots hae been ignored. r 1 r = r 0 1 r ) From 0) it can be shown that r 1 r 0 for r 0. Hence, R min = r 0 for r 0 [0, ]. Proposition 4. If φ 0 = π and r 0 [, ], then R max = r 0. Proof. In the proof gien for Proposition, we hae seen that for r 0 [0, ] 0. Hence, in this region both R max and R min occur at φ 0 = π. So, substituting the alue of φ 0 in 8), we get the roots 18) and 19). Again, we hae ignored the negatie root. Using 0), it can be shown that r 1 r 0 for r 0 [, ]. Hence, R max = r 0 for r 0 [, ]. Theorem 5. The agent coers a maximum radius of r 0 if it starts at r 0 with heading φ 0 = π. Proof. For r 0 [, ], it follows directly from Proposition 4 that R max = r 0. From Section, we now that R max always occurs at φ 0 = π irrespectie of the sign of. Substituting the alue of φ 0 in 8), we get the positie roots as, r 1 = r 0 1) ) 1 r = ) ) Using simple calculations, it can be shown that r 1 > r for eery r 0 >. Hence, we conclude that for r 0, R max = r

5 π and r0 [0, ], the trajectory of the agent is bound by a circle of radius. Corollary 6. Gien φ0 = Proof. In Proposition 4, we found out that for r0 p [, ] Rmax = r0. Hence, Rmax becomes maximum when r0 is maximum. Hence,in the gien range the maximum alue of Rmax is. For the case when p r0 [0, ], the expression for Rmax is gien by 19). The first deriatie of the expression is gien by 1 1 r0 r0 + 1 ). The first deriatie is always negatie, so Rmax decreases with respect to r0. Hence, Rmax is maximum for r0 = 0. Using 19), the maximum alue of Fig. 8. Trajectory plot: r0 = 45, φ0 = Fig. 9. r ersus t : r0 = 45, φ0 = 110π 180. Since, for the gien range of r0 the maximum alue of Rmax has an upper bound of, the trajectory is also bounded by a circle of radius. Theorem 7. To coer an entire region of radius, the Rmax is eual to agent must hae an initial condition gien by d, π ) or π 0, ) and = d. Proof. In Corollary 6, we deried that the maximum alue of Rmax is eual to which occurs at r0 = 0 and. Using 18) and 19), we calculated that for r0 = 0 and, the alue of Rmin is eual to zero. This proes that the agent coers the entire circular dis of radius π π for the initial configurations, ) and 0, ). 110π 180 Let us consider a circle of radius d which is desired to be coered. The alue of the controller gain should be chosen such that d =. Hence, as explained in the aboe paragraph, for = d the agent should start at π π d, ) and 0, ) in order to coer a circular area of radius d. 4. SIMULATION RESULTS In this section, we alidate the results obtained in Section through MATLAB simulations. The simulations are performed with the parameters h = 0.01, ρ = 0. and = 5. The control algorithm deeloped in 5) leads to formation of patterns around the target ensuring coerage of the area surrounding the target. 4.1 Critical Initial Position p For initial conditions, π ), the trajectory of the agent is circular with the target as the centre as gien by Theorem. This result has been erified by simulation as shown in fig. 10) 4. Range of Rmax and Rmin In Proposition, we deried p that Rmin aries linearly with respect to r0 for r0 [0, ]. This can be erified by the fig. 11 which has been obtained using MATLAB Fig. 10. Agent trajectory for critical initial position simulation. Similarly in Proposition 4, we found outthat p Rmax aries linearly with respect to r0 for r0 [, ]. This result has been shown in fig CONCLUSION In the paper, we hae analysed the control law 5) under initial conditions such that the elocity ector is perpendicular to the line of sight with respect to the stationary target. For this initial heading, the agent exhibits seeral interesting behaiours. For the critical initial initial condition gien in Theorem, the agent circles around the target. Hence, the agent can continuously monitor the target using only range information. The radius of the circular trajectory depends on the controller gain and speed and, hence, can be specified by the user. In Theorem 7, it has been shown that by adjusting the controller gain, 441

6 Fig. 11. Variation of Rmin with respect to r0 Fig. 1. Variation of Rmax with respect to r0 Acar, E. U., Choset, H., and Atar, P. N., Complete Sensor-based Coerage with Extended-range Detectors: A Hierarchical Decomposition in Terms of Critical Points and Voronoi Diagrams, Proceedings of the IEEW/RSJ International Conference on Intelligent Robots and Systems, , 001 Dhillon, S. S. and Charabarty,. Sensor Placement for Effectie Coerage and Sureillance in Distributed Sensor Networs Proceedings of IEEE Wireless Communications and Networing Conference, , 00. Lai, C. C., Ting, C. C., and o, R. S. An Effectie Genetic Algorithm to Improe Wireless Sensor Networ Lifetime for Large-Scale Sureillance Applications IEEE Congress on Eolutionary Computation, 5158, 007. Wang, W., Sriniasan, V., Wang, B., and Chua,. C. Coerage for Target Localization in Wireless Sensor Networs IEEE Transactions on Wireless Communications, Vol. 7, No., , 008. Yan, T., He, T., and Stanoic, J. A. Differentiated Sureillance for Sensor Networs First ACM Conference on Embedded Networed Sensor Systems, 51-6, 00. Tripathy, T. and Sinha, A. Guidance of an autonomous agent for coerage applications using range only measurement Proceedings of AIAA, Guidance, Naigation & Control Conference, 01, the agent can be made to coer an entire dis. The wor can be extended to find out the effect of the controller gain and speed. Also, more wor can be done to mae the controller robust. Since, the agent uses only onerange) parameter s data for the control law, the noise associated with the system reduces. The guidance law also uses minimal information for coerage. This leads to cost saing. It also minimises the energy consumption which is one of the major issues of today! REFERENCES Wong, S. C. and MacDonald, B. A. A topological coerage algorithm for mobile robots Proceedings of the IEEW/RSJ International Conference on Intelligent Robots and Systems,, , 00. Zelinsy, A., Jaris, R. A., Byrne, J. C., and Yuta, S. Planning paths of complete coerage of an unstructured enironment by a mobile robots Proceedings of International Conference on Adanced Robotics, 5-58, 199. Stachniss, C. and Burgard, W. Mapping and Exploration with Mobile Robots using Coerage Maps Proceedings of the IEEW/RSJ International Conference on Intelligent Robots and Systems, Vol. 1, , 00. Rutishauser, S., Correll, N., and Martinoli, A. Collaboratie coerage using a swarm of networed miniature robots Robotics and Autonomous Systems, Vol. 57, 51755, 009. Bataia, P. H., Roth, S. A., and Singh, S. Autonomous Coerage Operations In Semi-Structured Outdoor Enironments Proceedings of the IEEW/RSJ International Conference on Intelligent Robots and Systems, ,