PURDUE UNIVERSITY GRADUATE SCHOOL Thesis Acceptance

Transcription

1 Graduate School ETD Form 9 (01/07) PURDUE UNIVERSITY GRADUATE SCHOOL Thesis Acceptance This is to certify that the thesis prepared By Yongguo Mei Entitled Energy-Efficient Mobile Robots Complies with University regulations and meets the standards of the Graduate School for originality and quality For the degree of Doctor of Philosophy Final examining committee members Y. H. Lu, Chair C. S. G. Lee Y. C. Hu G. T. Chiu Approved by Major Professor(s): Y. H. Lu Approved by Head of Graduate Program: V. Balakrishnan Date of Graduate Program Head's Approval: 04/20/2007

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3 ENERGY-EFFICIENT MOBILE ROBOTS A Dissertation Submitted to the Faculty of Purdue University by Yongguo Mei In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2007 Purdue University West Lafayette, Indiana

4 UMI Number: UMI Microform Copyright 2008 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI

5 To my wife Xiaoying and my son Isaac. ii

6 iii ACKNOWLEDGMENTS First of all, I would like to thank my adviser, Professor Yung-Hsiang Lu, for his help during my PhD study. Without his encouragement and generous financial support, I could not be able to reach this point to finish my degree. My thanks also go to my other committee members, Professor C. S. George Lee, Professor Y. Charlie Hu and Professor George Chiu. They have given me many insightful guidances for my research. Other members in our HELPS (High Efficiency, Low Power System) group deserve my many thanks. Le Cai, ChangJiu Xian, Eddie Pettis, Doug Herbert and Jeff Brateman have listened to many of my presentations over and over again and helped me sharpen my ideas a lot. Changjiu cooperated with me in the research of multirobot coordination for sensor network maintenance and we had a great success. I also thank Her-Jay Chang, Saumitra Das, Nirut Naksuk in our robot group and many other friends for many good discussions and cooperations. For my family members, I thank my wife for her persistent support, my son for bringing my life a lot of joy, and my parents for their selfless help. Finally, I would like to thank National Science Foundation (NSF IIS ) and Purdue Graduate School (Bilsland Dissertation Fellowship) for their financial support for my PhD research.

7 iv TABLE OF CONTENTS Page LIST OF TABLES vii LIST OF FIGURES viii ABSTRACT xii 1 INTRODUCTION Background Single-Robot Systems Power Consumption of Mobile Robots Motion Planning in Known Environment Robot Exploration Multi-Robot Systems Fleet Size Robot Deployment Multi-Robot Coordination for Sensor Network Maintenance Thesis Contribution RELATED WORK Background Energy-Efficient Motion Planning Fleet Size and Robot Deployment Multi-Robot Coordination for Sensor Network Maintenance Our Contributions ENERGY-EFFICIENT MOTION PLANNING Introduction Power Modeling Energy Sources and Consumers

8 v Page Motion Power Sensing Power Microcontroller and Embedded Computer s Power Robots Power Models Motion Planning in Known Environment Coverage Problem in An Open Area Energy Efficiency of Different Routes Simulations Robot Exploration Motivating Examples Energy-Efficient Exploration Simulations and Results FLEET SIZE AND ROBOT DEPLOYMENT Introduction Fleet Size Problem Statement Determining Fleet Size Case Studies Robot Deployment Robot Deployment Problem Deployment Strategy Environments with Obstacles Case Studies MULTI-ROBOT COORDINATION FOR SENSOR NETWORK MAINTE- NANCE Introduction Problem Statement Coordination Algorithms

9 vi Page Centralized Manager Algorithm Fixed Distributed Manager Algorithm Dynamic Distributed Manager Algorithm Performance Analysis Random Variables Fixed Algorithm Centralized Manager Algorithm Dynamic Algorithm Overhead Comparison Simulations Simulation Setup Geographic Routing and Location Service Simulation Results SUMMARY Energy-Efficient Motion Planning Fleet size and Deployment Multi-Robot Coordination for Sensor Network Maintenance Future Work LIST OF REFERENCES VITA

10 vii Table LIST OF TABLES Page 1.1 Overview of my thesis Deployment algorithm Deployment for covering m 2 within three hours with PPRK Deployment for covering m 2 within three hours with Pioneer Overhead comparison of the three coordination algorithms with 9 maintenance robots. Motion overhead is represented by the average traveling distance per failure. Message overhead is represented by both the message passing distance and the location update area. The value of l is the square root of the size of average area per robot

11 viii Figure LIST OF FIGURES Page 1.1 PPRK Pioneer 3DX Power measurementing for PPRK Power measurementing for Pioneer 3DX A common component architecture of a robot Power consumption of a DC motor at different angular velocities Energy efficiencies at different angular velocities A robot s velocity can be represented by (V x, V y, Ω). The value of Ω is the angular velocity when the robot is turning The robot of PPRK The robot s velocity (V x, V y ) is controlled by the three wheels velocities Motion power at different speeds of PPRK The robot of Pioneer 3DX Motion power at different speeds of Pioneer 3DX Sonar sensors power consumption (a) scan lines, (b) square spiral, and (c) spiral Energy efficiencies of three paths with different covering area Energy efficiencies of the scan lines with different heights (h) Energy efficiencies of three paths with different velocities Time to cover 100m 2 with different velocities (a) Utility-based target selection. (b) Orientation-based target selection Two routes R1 and R2 connecting location A with location B. R1 is shorter than R2, but consumes more energy because R1 has more stops and turns (a) Free and obstacle cells. (b) R: robot s current location; area enclosed by dash line show the robot s sensing range

12 ix Figure Page 3.19 Frontier cells and target selection. Our algorithm lists the 8 frontier cells in the number order: cell 1, cell 2,..., cell 8, starting from robot s left direction and following the clockwise order Closely move along a wall Transform a grid cell map into a graph for energy-efficient motion planning. The circles are vertices, and the solid lines with arrows represent directed edges. The vertices represent the robot s states. The weight of one edge is the energy needed for the robot traveling from one state to another state. For example, in the figure w1 represent the energy needed for moving from state < i, j, 45 > (NE) to state < i+1, j +1, 135 > (NW) Shortest and energy-efficient paths. The black squares are obstacle cells (a) Our target selection method. (b) Choose the widest frontier (a) Our target selection method. (b) Choose the widest frontier (a) Our target selection method. (b) Choose the widest frontier Coverage ratio at different steps for (a) Figure 3.23, (b) Figure 3.24, and (c) Figure Mobile Robots for Pickup Requests and Time Intervals, e N+1 is the Last Request Satisfying Probability with Unlimited Energy Satisfying Probability with Limited Energy Satisfying Interval Satisfying Probability Satisfying Interval Satisfying Probability (a) A scanline-covering route. (b) Three robots are unloaded at A. The starting locations are A, B, and C. The segments AB and AC represent the dispersing overhead. The second robot runs out of energy and stops at E The area is covered by a group of 12 robots. The areas with subscripts from 1 to 12 are the areas covered by these robots. The areas are symmetric to the unloading location A A deployment with three groups

13 x Figure Page 4.12 Grid lines, cells and neighbor cells Four examples of environments filled with probabilistic obstacles with obstacle density 8%: (a) generated by random model; (b), (c), and (d) generated by clustering model with different (k 1, k 2 ) Scanline covering an environment with obstacles Obstacles and traveling steps Distance ratio verse obstacle density Traveling distance with different energy, PPRK Fleet size and speed management, PPRK Fleet size verses speed for Pioneer 3DX to cover different areas, time =3 hours, energy = J Area covered by different number of robots with different ratios of height and width (6 hours 25000J PPRK) Fleet size verses area for PPRK, time =3 hours, energy = 25000J Fleet size verses area for PPRK, time =6 hours, energy = 25000J Fleet size verses area for Pioneer 3DX, time =3 hours, energy = J Fleet size verses area for Pioneer 3DX, time =6 hours, energy = J Fleet size verses area for PPRK, time =3 hours, energy = 25000J, area m Fleet size verses area for Pioneer 3DX, time =3 hours, energy = J area: m An overview of sensor replacement using robots. The circles represent sensor nodes and squares represent robots. A guardian node detects a guardee s failure (shown by a cross over the circle) and then reports the failure to a robot. A robot moves to the failure location and replaces the failed node An scenario in the centralized manager algorithm A sensor node s behavior in the centralized algorithm The manager robot s behavior in the centralized algorithm A maintainer robot s behavior in the centralized algorithm Partition an area into (a) squares (b) hexagons. The small squares represent the robots in each subarea, and the small circles indicate failures... 99

14 xi Figure Page 5.7 A sensor node s behavior in the fixed algorithm A robot s behavior in the fixed algorithm Voronoi graphs. (a) original Voronoi graph; a failure happens at S inside R1 s subarea. (b) After R1 moves to S, the Voronoi graph changes; the original graph is shown by dashed lines. The shading area shows the area that the robot needs to update its location A sensor node s behavior in the dynamic algorithm A robot behavior in the dynamic algorithm Average robot traveling distance at different number of robots n. The value of l is the the square root of the size of one subarea The average robot traveling distance as a function of the number of robots The average message passing hops per failure The average number of transmissions for location update per failure The average replacement time per failure The average traveling distances with different number of robots. The failures are 9 times more likely happened in one half of the area than in the other half

15 xii ABSTRACT Mei, Yongguo Ph.D., Purdue University, May, Energy-Efficient Mobile Robots. Major Professor: Yung-Hsiang Lu. Mobile robots can be used in many applications, such as carpet cleaning, search and rescue, exploration, and entertainment. Robots usually carry limited energy and thus energy conservation is important. To our best knowledge, this is the first research project to study energy-efficient mobile robots in a systematic way. In this thesis, I first develop power models for two different types of robots, and then focus on five related problems: motion planning, exploration, fleet size, deployment, and sensor network maintenance. The first two problems consider single-robot systems, and the latter three focus on multi-robot systems. Corresponding to the five individual problems, my contributions are the following. (1) My research compares energy consumptions of three different routes and presents my method of energy-efficient motion planning. (2) This thesis investigates an orientation-based exploration algorithm that can reduce repeated coverage and save traveling distance and energy consumption. (3) I study the fleet size problem to determine the number of robots for pickup and delivery tasks under energy and time constraints. (4) This study develops three robot deployment strategies to minimize the deployment overhead and use fewer robots to cover an area. (5) This thesis designs three different algorithms for coordinating robots with sensor networks to automate sensor network maintenance and consider motion and communication overhead.

16 1 1. INTRODUCTION 1.1 Background Mobile robots can be used in many different applications. Robots can automatically clean carpet or mow lawn without human intervention. Some robots can entertain people by dancing, skiing and talking [1]. Robots can also be used in pickup and delivery, search and rescue, surveillance, coverage, and exploration. The United Nations Economic Commission predicts that home robots will surge sevenfold by 2007 [2]. An article from BusinessWeek July 19, 2004 foresees a $60 billion annual market before 2010 [3]. Mobile robots are usually powered by batteries. Therefore, energy constraints are critical to mobile robots. Honda humanoid robots can keep walking for only 30 minutes with a fully charged battery pack [4]; energy constraints are the most important challenge for mobile robots. Makimoto et al. in a plenary speech of ISLPED 2003 [5] predicted that Robots will be a next driver of our industry and The robot will provide the biggest challenges for the low power electronics in the future. Meanwhile, time constraints are usually considered in many applications of mobile robots. For example, when using robots to search and rescue survivors after a disaster, the survivors have to be reached within 24 hours; otherwise, the chance of survival diminishes quickly. In a dynamic environment, the robot needs to detect obstacles and react in time. Therefore, many real-time issues need to be considered. Energy and time can be conflicting constraints. For example, a robot can travel at a high speed and reach the destination earlier (meeting the time constraint). However, fuel efficiency (meters per Joule) can drop dramatically at a high speed and the vehicle may run out of fuel (failing the energy constraint). Our study considers both constraints together.

17 2 This thesis focuses on energy-efficient mobile robots and can be divided into two parts: single-robot systems and multi-robot systems. For single robot systems, we have investigated three topics: power modeling, motion planning and exploration. Robot exploration can be regarded as motion planning in an unknown environment. We present all the three topics in chapter 3. For multi-robot systems, we have studied three topics: fleet size, deployment, and coordination. We present the three topics in two chapters, chapter 4 for fleet size and deployment and chapter 5 for multi-robot coordination. Table 1.1 shows the overview of the whole thesis. In the rest of this chapter, we first summarize the three chapters and then present the contribution of the whole thesis. Table 1.1 Overview of my thesis. Single-robot systems Multi-robot systems Power modeling [6] Fleet size [7] Motion planning [8] Deployment [9 11] Exploration [12] Coordination [13, 14] 1.2 Single-Robot Systems Power Consumption of Mobile Robots A robot usually consists of several components, including motors, sensors, microcontrollers, and embedded computers. Accordingly, the power consumption of a robot can be divided into four major parts: motion, sensing, communication, and computation. To determine the power of a robot, we model each component s power individually. Many factors affect the power consumption. For example, traveling speed affects the motion power. Motors consume more power at a higher rotational speed. As another example, sensing power is affected by how frequently the robots sense the environment. We experimentally measure two different robots power consumptions.

18 3 The first robot is called PPRK developed by Carnegie-Mellon University [15]. The second robot is Pioneer 3DX by Activmedia and it is widely used in research community [16] [17] [18]. The power models are used for later studies. Figures 1.1 and 1.2 show the pictures of the two robots controlled separately by a PDA and a laptop. Figures 1.3 and 1.4 show that we measure the two robots power consumption through a data acquisition (DAQ) card connected with a laptop. There is a box functioning as a weight on the top of the Pioneer 3DX to control the robot s load and a cart to hold the two laptops, one for control and the other for measurement. Fig PPRK. Fig Pioneer 3DX. Fig Power measurementing for PPRK. Fig Power measurementing for Pioneer 3DX.

19 Motion Planning in Known Environment We present a new approach to find energy-efficient motion plans for mobile robots. Motion planning has two goals: finding routes and determining velocities [19]. We model the relationship between motors speed and their power consumption with polynomials. The velocity of the robot is related to its wheels velocities by performing a linear transformation. We use simulations to compare the energy efficiency of different routes at different velocities. The results show that the energy efficiency depends on the covered areas, the peak speeds, and the paths. For a small area, maximum energy efficiency is achieved when the robot moves along straight lines. When the areas become larger, spirals become the most energy-efficient because the robot can move continuously without stopping and turning Robot Exploration Exploration in an unknown area is a basic application of mobile robots. It is also the foundation for many other applications. Current utility-based exploration techniques try to maximize the area covered and minimize the cost such as traveling distances. This exploration strategy is efficient at the beginning when unknown areas are large and close to each other, but inefficient later when the remaining unknown areas are small and distant from each other. Therefore, utility-based strategies are inefficient in terms of energy consumption. Our solution considers orientation information and explores the area in a consistent way according to the relative orientation of the unknown area to the robot. This strategy avoids the situation of small distant unknown areas and reduces the total energy consumption. Simulation results show that our algorithm can save up to 42% energy consumption.

20 5 1.3 Multi-Robot Systems Fleet Size A fundamental question for multi-robot applications is to decide the number of robots needed (i.e., the fleet-size problem ) to accomplish tasks. We provide a probabilistic method to decide the fleet size necessary to serve requests with random arrival times and locations. We consider five factors on which the fleet size depends: available energy, power consumption, service field, request rate, and time constraints. Two metrics are used to measure the system performance: satisfying probability (probability that a new request can be served within the time constraint) and satisfying interval (the average time interval between two consecutive unsatisfied requests). A simplification method is provided to accelerate the computation Robot Deployment Deployment of robots is to transport robots from the place where the robots are stored to the working field. A common scenario is using a carrier to transport the robots into the field and unload them at several locations. The deployment problem is to determine the number of groups unloaded by a carrier, the number of robots in each group and the unloading locations of those robots. We investigate robot deployment for coverage tasks. Both time and energy constraints are considered; the robots carry limited energy and need to finish their tasks before deadlines. This study provides a speed management method to maximize the traveling distance satisfying both energy and time constraints. We use an analytic method to solve the deployment problem and use fewer robots. Our algorithm is called the space partition area coverage algorithm (SPACA). We also provide an approach to consider environments with random obstacles. Environments with obstacles are modeled and an empirical rule over the extra detouring distance is derived from simulations. With this rule, SPACA can be applied into areas with obstacles. Compared with two other heuristics (one

21 6 unloads all the robots in one place, the other unloads an equal number of robots each time), our solution uses 31% fewer robots for open areas and 21% fewer for areas with obstacles Multi-Robot Coordination for Sensor Network Maintenance Sensor networks have been widely studied due to their broad applications. Sensors often are deployed into fields far away from people. When sensors fail, coverage holes are formed. It is desired to automate sensor network maintenance. Existing solutions assume sensors can move and can dynamically adjust their locations so that the coverage holes can be recovered. However, a large number of mobile sensors are expensive. We propose using mobile robots to maintain sensor networks. Only a few robots are needed for a large number of sensors. When sensors fail, robots can move in and replace the sensors with functional ones. We present three different algorithms for coordinating mobile robots and static sensors, and analyze the algorithms properties in terms of motion overhead, communication overhead, and reaction time. 1.4 Thesis Contribution To my best knowledge, this thesis is the first study focusing on energy-efficient mobile robots in a systematic way. I study energy conservation techniques for both single-robot systems and multi-robot systems. For single robot systems, we focus on motion planning that is a fundamental problem for mobile robots. We study energyefficient motion planning from three different aspects: power modeling [6], motion planning in known environments [8], and motion planning in unknown environments or exploration [12]. We build power models based upon the real measurements and consider motion power, sensing power, control power and computation power separately. Based upon our motion power models and assuming the environment is known, three different routes to cover an area are compared in terms of their energy consumption. Simulation results show that the robots speed plays an important role

22 7 in energy-efficient motion plans, and sharp turns is another important factor affecting the motion power. We also study robot exploration. To explore an unknown area, how to select a target to visit next is an important issue. The state-of-the-art of target selection is utility-based, to cover more area with less cost, such as distance. However, this is not optimal in terms of exploration distance and energy consumption. We propose an orientation-based exploration strategy that can greatly avoid repeated coverage and shorten traveling distance and reduce energy consumption. The other part of my thesis focuses on multi-robot coordination in three aspects: fleet size [7], deployment [9 11] and coordination for sensor network maintenance [13, 14]. We first investigate the fleet size problem: to determine the number of robots needed under energy and time constraints. We are the first to consider both energy and time constraints in the fleet size problem. Random variables are used to model the system and a simplification method is provided for efficient computation. Then, we concentrate on robot deployment: the strategy of delivering and unloading a large number of robots into the working field considering the energy and time constraints. We reduce the number of groups and the number of robots in each group so that the robots together can cover more area before the deadline. Both open areas and areas with obstacles are considered. Finally, we study a problem of sensor network maintenance. Coverage holes are left in sensor networks when sensors fail. We are the first to propose using a few robots for maintaining a large sensor network. Robots can move and replace failed sensor nodes with functional ones. Three different coordination algorithms are presented and their motion and communication overhead is analyzed and compared. The rest of this thesis is organized as following. Next chapter focuses on related work. Three chapters on motion planning, fleet size and deployment, and multi-robot coordination are followed. Finally, chapter 6 concludes the thesis.

23 8 2. RELATED WORK 2.1 Background Mobile robots can be used in many different applications. Carpet cleaning [20] and entertainment [1] are two kinds of applications that we may encounter at our homes. Mobile robots can also be used for pickup and delivery tasks [21]. Multiple robots can work together to accomplish a task and communication among them is important for cooperation purposes. Das et al. [22] evaluate three communication protocols in supporting many-to-one communication for mobile robots. Rybski et al. use small robots for reconnaissance and surveillance [23]. Exploration using mobile robots has been studied by many researchers. Zelinsky [24] uses a quad-tree data structure to model the environment and presents an adaptive path planning algorithm for a robot exploring an unknown environment. Batalin et al. [25] present an algorithm to cover an area using markers. Gonzalez et al. [26] design a coverage algorithm combining spiral paths and backtracking to completely cover an area. Taylor et al. [27] model the environment by boundary graphs and present their vision-based path planning algorithms. Mobile robots can also be used in extraterrestrial explorations. Matthies et al. [28] build a small rover as a testbed for Mars exploration. They discuss localization, obstacle detection, and path planning; they also evaluate the performance of the robot executing these operations. In recent years, many researchers investigate simultaneous localization and mapping (SLAM) using mobile robots. SLAM is a problem closely related to exploration. Dellaert et al. [29] present a linear algorithm to detect 2-dimensional structures and motions; this algorithm provides initial estimates for multi-robot SLAM. Chang et al. [30] use a logarithmic map partition method to reduce the computational complexity for SLAM. Mobile robots are also used for search and rescue applications. Davids et al. [31] introduce the application of mobile

24 9 robots in search and rescue in urban areas. Schreiner et al. [32] discuss the research issues in landmine detection using robots. Zhang et al. [33] use a probabilistic method in searching landmines. Their algorithm computes probabilistic distributions of the landmine locations, and uses the distributions to direct the robot s search. Tadokoro et al. [34] investigate the requirements for rescue robots based upon an analysis of an earthquake happened in Kobe, Japan. Baltes et al. [35] demonstrate a binary space partition method for robot rescue; this method is useful for path planning in a dynamic environment. This thesis focuses on energy-efficient mobile robots. There are many studies and publications related to our research. In the rest of this chapter, we separate them into three sections, corresponding to the topics in the following three chapters. 2.2 Energy-Efficient Motion Planning A mobile robot has several major components: sensors, motors, microcontrollers and computers. Some studies analyze energy breakdowns for mobile robots [36] [37]; they investigate the power of sensors, controllers and communication. However, their power models do not consider speed variation. One of them is the paper by Liu et al. [36] on power-aware scheduling algorithms for a Mars rover. Their method considers free solar energy and schedules the rover s operations to use the solar energy and battery efficiently. Some studies concentrate on motion planning to reduce the motors power consumption. For example, improving motors energy efficiency [38] [39] can reduce motion power. After the motors are chosen, motion energy can be saved by choosing routes and setting the velocities. If a robot has to visit several places, the route can be arranged to reduce the total traveled distance (similar to the traveling salesperson problem, TSP) and to avoid sharp turns that require decelerations and accelerations. Sun et al. [40] find energy-efficient paths using topography information of the ground. Barili et al. [41] describe the concept of controlling the velocities to save energy for

25 10 a mobile robot. Their work does not discuss the relationship between path planning and velocity control. Hwang et al. [42] survey the methods for motion planning but do not discuss energy conservation. Katoh et al. [43] present an approach for energy conservation by creating elliptic paths but they focus on space manipulators for flying robots. Some energy-conserving techniques are designed for specific robots. For example, Silva et al. [44] analyze the energy consumption of walking robots by controlling the locomotion variables. Yamasaki et al. [45] develop control algorithms to reduce the energy consumption of humanoid robots. Both work focus on energy consumption of walking robots. Redi et al. [46] reduce the communication energy among a group of robots; the study does not consider the energy consumed by motors. Kim et al. [47] present a control method for industrial manipulators to minimize a weighted time-fuel cost function. They assume that motors power is proportional to the drive torque and do not consider the loss due to armature resistance or mechanical friction. Li et al. [48] show that parallel manipulators are more energy-efficient than serial manipulators in similar configurations. They consider the armature resistance loss and friction loss when they compute the energy consumption. Their calculation does not include the energy consumed by the control circuits. Duleba et al. [49] discuss nonholonomic energy-efficient motion planning based on the Newton algorithm. They take the square of control signal as the objective function; their work is theoretical without experimental validation. Exploration is motion planning in unknown environments. There are many studies on mobile robot exploration. Engelson [50] focuses on passive exploration where the robot motion is controlled by an operator. Chang et al. [51] compare energy and time efficiency of different dispatching algorithms for ant-like robot systems to explore an unknown area. For autonomous robot exploration, the key step is in selecting the next exploration target. Yamauchi [52] proposes frontier-based exploration. The candidates of next targets are along the frontiers between the known area and the unknown area. Frontier-based methods have been used in some later studies [53] [54] [55]. Simmons et al. [54] propose an utility-based approach to coordinate multi-robot

26 11 exploration and mapping. Robots submit their own estimations of the utilities and costs of different frontier cells, and a central control assigns targets to different robots to maximize the total utility. Zlot et al. [55] present a market control architecture for multi-robot exploration to minimize the cost and maximize the utility. Jia et al. [56] propose a cost-efficient motion planning algorithm. The cost can be distance or time. However, their algorithm does not consider the robot s direction. Burgard et al. [53] present a method for coordinating multiple robots in exploration. Their method estimates the utilities of frontier cells in a coordinated way. The more robots move toward the same location, the lower the location s utility is, thus preventing multiple robots moving into the same location. This method requires communication among robots. These studies all adopt a utility-based strategy, selecting the next target that can immediately maximize the utility and minimize the cost. Batalin et al. [25] present a method of using markers for exploration and avoiding the localization problem. The robot drops markers to identify those places that have been explored. However, this method is limited by the number of available markers. Trevai at al. [57] distribute observation points or markers into the environment using reaction-diffusion equations. The exploration task is reduced to traveling through all the observation points. The disadvantage of this method is the requirement of distributing observation points before exploration. 2.3 Fleet Size and Robot Deployment Kirby [58] presents a mathematical model to estimate the fleet sizes and determine the number of vehicles to purchase. The method simply assumes the probabilities of needing N robots are known. Mole [59] considers the periodic variation of the requests and uses dynamic programming to minimize a cost function in determining the fleet size. Huang et al. [60] show several heuristics to estimate the fleet size of automated guided vehicle systems. Their methods consider only one constant travel speed and assume all the vehicles are always available. Some previous studies decide the fleet

27 12 size based on simulation. For example, Lesyna [61] describes a procedure to decide the fleet size by simulations; however, no analytic model is provided. Evans [21] describes a system of multiple mobile robots used in a hospital. Rossetti et al. [62] simulate a mobile robot delivery system used in a clinical laboratory. Some researchers investigate how to coordinate multiple mobile robots. Marapane et al. [63] present a method for motion coordination using computer vision without direct communication among robots. Parker [64] designs an architecture to coordinate multiple robots by adaptively assigning tasks to the robots. The deployment problem is to transport and unload multiple robots into the working field. Deployment is important to multi-robot applications. However, only a few existing studies discuss the deployment problem. Simmons et al. [65] study coordination techniques among a group of robots. They demonstrate the effectiveness of their method by deploying robots from the same location to their individual destinations. Rybski et al. [23] use large ranger robots to transport and deploy small scout robots. The rangers can travel up to 20 kilometers, greatly extending the search range of scouts. Chang et al. [51] study the energy and time properties of different dispatching algorithms for ant-like robot systems. Yamaguchi [66] discusses adaptive formation control for mobile robots to keep their relative positions. Deployment is related to the fleet size problem to determine the number of robots needed for a task. An efficient deployment strategy can reduce the overhead and use fewer robots (i.e. a smaller fleet) to accomplish the same task. 2.4 Multi-Robot Coordination for Sensor Network Maintenance Meguerdichian et al. [67] first address the coverage problem and use Voronoi diagram to establish an optimal polynomial-time algorithm for coverage calculation. For static sensor network, Wang et al. [68] present the Coverage Configuration Protocol (CCP) that determines which sensors should be turned on to achieve the desired coverage and the other sensors can be turned off to save energy. Zhang et al. [69]

28 13 propose the Optimal Geographical Density Control (OGDC) protocol to minimize the overlap of sensing areas of all sensors. Tian et al. [70] propose turning off a node if its sensing area can be completely covered by its neighbors. To avoid the possibility of multiple neighbors turning off and creating a coverage hole, the nodes use a random back-off algorithm before going to sleep. Several studies have considered using robots for initial sensor deployment, for example, [71, 72]. Howard et al. [73] present a deployment algorithm that incrementally deploys nodes based upon information gathered by previously deployed nodes. Zou et al. [74] use virtual forces to determine the positions to redeploy or relocate sensor nodes for better coverage. Cloqueur et al. [75] present a strategy that determines the number of sensors deployed in each step until the desired coverage is achieved. The above techniques are insufficient to handle the case where a coverage hole exists even all sensors around the hole use their maximum sensing ranges. This problem can be solved by relocating some sensors from the densely deployed subarea to the holes to amend the coverage. Wang et al. [76] propose relocating redundant mobile sensors to fill the coverage holes and using a cascading movement method to balance the energy cost and the replacement time of sensor relocation. Ganeriwal et al. [77] propose to make low energy nodes, on predicting a failure, broadcast a Panic-Request message. Nodes with high energy levels respond with the Panic-Reply message if they can move without losing existing coverage. Howard et al. [78] propose a potential-field based approach for self-deployment of mobile sensor networks. In such potential fields, nodes are treated as virtual particles and the virtual force between two nodes is larger if the two nodes are closer. As the virtual forces repel the nodes from each other, the sensors tend to form a uniform distribution in the sensor area without coverage holes. These methods require nodes equipped with motors, steering devices, and GPS. These components are very expensive while a large-scale sensor network usually assumes that the sensors are small and cheap. Meanwhile, higher-capacity batteries may be required for mobile sensors because mobility is also an energy-consuming feature. Recently, some studies have been focusing on hybrid sensor networks with both static

29 14 and mobile sensors [79,80]. Hybrid sensor network can compensate the disadvantages of pure static sensor networks without requiring all sensors to have mobility. Ingelrest et al. [81] describe several broadcasting strategies for hybrid sensor networks in a more general context. However, their strategies primarily optimize broadcasting for fixed infrastructure points. multi-hop 2.5 Our Contributions Our study focuses on energy-efficient mobile robots and is different from previous studies in the following ways. (1) Power Modeling [6]: We build power models for each components of a mobile robot based upon real measurement data. We measure the power consumption when robots move at different speeds. Our study differentiates the robot s speed from the motors speeds. (2) Motion Planning [8]: We compares the power consumption of three different routes when the robot travels at different speeds based upon the motion power model we build. Our study shows the existence of the most-energy-efficient speed at which the robot consumes the minimum energy for traveling across the same distance. This study shows a significant amount of energy consumption of turns and stops. (3) Exploration [12]: This thesis is the first to present orientation-based target selection for robot exploration. Orientation-based target selection determines the next target according to the relative orientations of frontier cells to the robot. Our method can greatly reduce repeated coverage and save significant portions of energy. The efficiency of our method has been validated by simulations in both office-like environments and environments with random obstacles. (4) Fleet size [7]: Our study is the first to consider five factors: customer distribution, request rate, time constraint, traveling speed, and energy capacity,

30 15 together to determine the fleet size. We calculate the probability that a new request can be served within the timing constraint at different fleet sizes. We provide a simplified computation method to reduce computation time and the results are validated by event-driven simulations within 4% errors. (2) Deployment [9 11]: Our study is the first to consider both energy and timing constraints in robot deployment. We classify deployment overhead into three types: unloading overhead, scattering overhead, and overlapping overhead. We design an algorithm to decide the number of groups and the number of robots in each group so that the three types of overhead are reduced. Our study models environments with obstacles and estimate the additional distances the robots need to travel for covering an area. (3) Coordination [13,14]: We are the first to propose using robots for sensor network maintenance. Our study considers both motion overhead and communication overhead. We propose three different coordination algorithms including one centralized and two distributed algorithms for multiple robots to maintain sensor networks. We analyze the properties of the algorithms in terms of motion overhead, communication overhead, and response time.

31 16 3. ENERGY-EFFICIENT MOTION PLANNING 3.1 Introduction This chapter focuses on energy-efficient motion planning. We divide this into three sub-problems: modeling power consumption of mobile robots, motion planning in known environment, and motion planning in unknown environment (also called robot exploration). We first analyze the power consumption of mobile robots, including motion power, sensing power, and computation power. Based upon the motion power model we build, we then study motion planning in known environments. Finally, we propose an orientation-based target selection strategy for energy-efficient robot exploration. 3.2 Power Modeling A robot usually comprises of several components, including motors, sensors, controllers, and embedded computers. The power of a robot can be divided into motion power, sensing power, control power and computation power accordingly. Liu et al. [36] present an energy breakdown table of a Mars rover. Michaud et al. [37] estimate the energy consumption of a rover including the communication power. They assume that the energy consumption of each operation is constant. For instance, moving one meter takes a fixed amount of energy. We use a more accurate approach by measuring the power consumption of two robots, PPRK and Pioneer 3DX, when they move at different speeds. This section first analyzes the power consumption of individual components, and then combines them together.

32 Energy Sources and Consumers Figure 3.1 shows a common architecture for a mobile robot. This robot has five major components: batteries, motors, sensors, microcontrollers, and embedded computers. In this section, we first discuss energy sources and consumers for mobile robots and then analyze their power properties. Embedded computer Batteries Microcontroller Sensors Motors Fig A common component architecture of a robot. The most commonly used energy sources are rechargeable batteries. In some cases, such as the Mars Rovers, solar panels are also used. However, if the solar power is not enough, the batteries may still run out of energy. Lead-acid, Nickel Cadmium/Nickel Metal Hydride (Nicd/NiMH) and Lithium Ion (Li-ion) are three commonly used types of rechargeable batteries. Lead-acid batteries are cheap, have large capacities and can provide high currents. However, they are bulky and need long charging time. Nicd/NiMH batteries have higher energy density than leadacid batteries, and Nicd/NiMH batteries small sizes make them suitable for potable applications, such as cell phones and cameras. Li-ion batteries are only available after 1990 s, and they have even higher energy density. Although the battery technology improves, the energy limitation is still a great challenge. To prolong the operation time, energy-efficient designs are important.

33 18 Motors, sensors, microcontrollers and embedded computers are energy consumers. DC motors transform direct current into mechanical energy, and have been widely used in robots as the actuators. As the robots become more sophisticated, control, sensing, communication and computation consume higher portions of energy. Robots use many kinds of sensors, such as encoders, vision, sonar, laser and infrared rangers. Encoders read the speeds of motors, and provide information for feedback control. Encoders are passive sensors and consume negligible power. A vision sensor can be a video camera, usually consuming less than 1 Watt. A vision system includes a vision sensor and an image processing unit. The image processing unit is the embedded computer. Sonar, infrared and laser rangers are active sensors to position obstacles by the echo waves. Sonar and infrared usually consume a small amount of power, and they can detect only short range obstacles. The laser ranger can detect further than one hundred meters but it consumes much more power, possibly more than 10 Watts. The microcontrollers work with motors and sensors, and provide an API (application programming interface) for the embedded computers. The embedded computers have better computation ability to handle high level tasks, such as motion planning, data processing and communication. The embedded computers receive sensing data from the microcontrollers, determine the next motion and sensing activities, and send commands to the microcontrollers. The microcontrollers execute the commands and control the motors and sensors Motion Power DC Motors Power Model DC motors are frequently adopted in robots so we focus on DC motors in this chapter. Let P m (ω, α) be the power consumption of a motor when it revolves at angular velocity ω with angular acceleration α (α = dω dt ). The value of P m depends on many factors, including the back electric and magnetic fields (EMF), armature

34 19 inductance, and armature resistance. From the theory of electromagnetics, P m (ω, α) of a basic DC motor can be modeled as a second-degree polynomial of ω and α. For many DC motors, the effect of accelerations is negligible [82] [43]. Our experimental data show that a second-degree polynomial is insufficient to closely model a DC motor s power. This is because a typical DC motor also contains an internal control circuit. Figure 3.2 is an example of a DC motor (MS492MH by Mr Robot Inc.) used in the PPRK robot (Figure 3.5). From Figure 3.2, we can see that a sixth-degree polynomial (equation (3.1)) is a better model than a second-degree polynomial. The average relative error is decreased from 3.63% to 1.59%. We chose a sixth-degree polynomial model because it can closely approximate the experimental data without significantly more computation. Figure 3.3 shows the energy consumption per radian of the motor. From this figure, we can see that at the beginning the energy efficiency increases but it decreases at the end as velocity increases. Power (w) Measurement data Second degree polynomial fitting Sixth degree polynomial fitting Angular velocity (rad/s) Energy consumption per radian (J/rad) Measurement data Second degree polynomial fitting Sixth degree polynomial fitting Angular velocity (rad/s) Fig Power consumption of a DC motor at different angular velocities. Fig Energy efficiencies at different angular velocities. P m (ω) = ω ω ω ω ω ω (3.1)

35 20 Robot s Velocity and Power Consumption A robot usually has multiple motors and each motor drives a wheel. A wheel s velocity is the product of the wheel s radius r and the controlling motor s angular velocity ω. The velocities of a robot s wheels are not equivalent to the velocity of the robot itself. Consider a robot on a two-dimensional surface. We can represent its velocity by three variables: V x, V y, and Ω, as illustrated in Figure 3.4. In this figure, the robot is represented by a triangle. The three-element vector < V x, V y, Ω > represents the robot s linear and angular velocities (when the robot is turning). We use upper-case letters (V x, V y, and Ω) to represent the velocity of the robot and lowercase letters for the motors. If the robot is in three-dimensional space, we need to add v z and two more angular velocities. (Vx,Vy) Ω Fig A robot s velocity can be represented by (V x, V y, Ω). The value of Ω is the angular velocity when the robot is turning. Omnidirectional robots are one particular type of robots that can change directions at their current locations [83] [84]. Figure 3.5 shows an example of a three-wheel omnidirectional robot called PPRK [15], and Figure 3.6 show the robot s speeds and the wheels speeds. The three wheels are mounted at distance b from the center. By adjusting the wheels velocities, the robot can move in any direction. If a robot is omnidirectional, it can turn (Ω 0) without moving (V x = V y = 0). Let v 1, v 2, and v 3 be the velocities of the three wheels. The relationship between the robot s velocity and the wheels velocities can be expressed by equation (3.2).

36 21 V1 (Vx,Vy) Ω V3 b V2 Fig The robot of PPRK. Fig The robot s velocity (V x, V y ) is controlled by the three wheels velocities. v 1 v 2 v b V = 1 x 3 b V 2 2 y Ω b (3.2) This type of relationship is not restricted to the robot shown in Figure 3.6. Linear transformations are also used in other types of robots, such as a differential robot [85] or a four-wheel omnidirectional robot [83]. Suppose a robot has k motors and the velocity of the i th wheel is v i. We can find a transformation to describe the relationship between v i and < V x, V y, Ω >. This relationship is called the robot s manipulator Jacobian [86]. We control the robot s velocity < V x V y Ω > by adjusting the motors velocity < v 1 v 2... v k >.

37 22 v 1 v 2 V x v 3 = J V y... Ω v k (3.3) Let v(t) =< V x (t), V y (t), Ω(t) > be the robot s velocity at time t. The i th wheel s velocity is v i (t) = J i,1 V x (t) + J i,2 V y (t) + J i,3 Ω(t). Suppose all wheels have the same radius. The motors angular velocities is the wheels velocities divided by the radius, r, of the wheels. The robot s power consumption is the sum of all k motors power: k i=1 P m ( v i(t) r, 1 r dv i (t) ) (3.4) dt Simplified Motion Power Model In many scenarios, robots move along straight lines. Along the straight lines, wheels usually move at the same direction as the robots, and we can treat robot s speed the same as motor s speed. In these cases, we can build simplified motion power model p m (v, a), where v and a are linear speed and acceleration. The power consumption of the robot is the sum of the mechanical output power and the transforming loss from the electrical power to the mechanical power. Let m be the robot s mass, and the ground friction constant be µ. When the robot travels with a linear speed of v and an acceleration of a, it needs traction force of m(a + gµ). Therefore, the output mechanical power is m(a + gµ)v, where g is the gravity constant. The motion power can be modeled as a function of the speed, the acceleration and the mass: p m (v, a) = p l + m(a + gµ)v, (3.5) where p m is the motion power, and p l is the transforming loss.

38 23 For DC motors, the power loss consists of armature loss, internal mechanical loss, and eddy-current loss. The power loss increases as the speed increases, but this relationship is non-linear. For example, the eddy-current loss increases by the square of the speed [87]. In this chapter, we use polynomial functions to model p l. If we consider only the speed, the motion power efficiency can be defined as the inverse of the energy per unit distance, or v. p m(v) Sensing Power Sensing power varies from different sensors and sensing frequencies. We can denote the sensing frequency by f s. For video cameras, it is the number of frames per second; for laser rangers, it is the firing frequency. We use a linear function to model the power consumption of sensors: p s (f s ) = c s0 + c s1 f s, (3.6) where p s is the sensing power, c s0 and c s1 are two positive constant coefficients. Their values depend on the used sensors Microcontroller and Embedded Computer s Power The microcontroller periodically sends commands to motors and sensors, polls sensors readings, and communicates with the embedded computer. The microcontroller usually runs continuously such that the power consumption of the microcontroller can be modeled by a constant. The embedded computer is more complex than the microcontroller. Many studies have been devoted into simulation-based methods to estimate computers power consumption [88] [89] [90]. The power consumption of the embedded computer may vary significantly for running different programs.

39 Robots Power Models Power Model of PPRK Power (W) Speed (m/s) Fig Motion power at different speeds of PPRK. PPRK has three DC motors, three infrared sensors, and one microcontroller; the robot is powered by four 1.2V and one 9V NiMH batteries. The power consumption of the PPRK robot is divided into three parts: motion, sensing and control. We use a data acquisition (DAQ) card from National Instrument Inc. as shown in Figure 1.3 to measure the current and the voltage. The DAQ is connected to a laptop computer so that the measurement facility can follow the robot while it is moving. In our experiments, we do not change the sensing frequency. The total power of sensing and control is measured constant at 0.998W. We control the robot movement at a constant speed along the y-axis direction (V x = Ω = 0) and measure the motion power (the total power of the three DC motors). The robot carries no additional load and moves on an indoor flat surface. The results are shown in Figure 3.7. The power increases slowly before speed 0.1m/s, but it increases super-linearly when the speed is larger than 0.1m/s. The maximum speed of the PPRK robot v m is 0.117m/s, and the maximum power is 1.025W. The optimal speed is v o = 0.110m/s, at which the power is 0.869W. Including the sensing and control power, the energy efficiency at

40 25 speed v o is = m/J. The energy efficiency at the maximum speed v m is = m/J or 1.9% lower than that at speed v o. We fit the data by a fourth-degree polynomial and the power model (including the sensing and control power) is: p(v) = v v v v (3.7) Notice that Figure 3.7 is the motion power of the PPRK robot while Figure 3.2 is only motion power of one DC motor of the PPRK robot. We use a fourth-degree polynomial instead of a sixth-degree polynomial in Equation (3.1) because fourthdegree polynomial is enough to approximate the power consumption here with less that 2% average relative error. Power Model of Pioneer 3DX Fig The robot of Pioneer 3DX. The Pioneer 3DX weighs about 20 pounds, and can carry at most a 50-pound load. The robot is powered by a 12V Lead-acid rechargeable battery. The robot has two DC motors driving two wheels. The DC motors are assembled with encoders. The robot has two arrays of sonar sensors (Figure 3.8), one in the front and one in the

41 26 rear. Each array has 8 transducers. A Hitachi-8S microcontroller is used to control motors and sensors, and it communicates with an embedded computer through a serial port. The microcontroller is managed by a real-time operating system called AROS. Motion power (W) pound load fitting line no load fitting line Speed (mm/s) Fig Motion power at different speeds of Pioneer 3DX We measure the motion power of the robot when the robot moves straight forward at constant speeds. We also change the load of the robot. Figure 3.9 shows the motion power of the robot runs at different speeds. In this figure, the lower set of data and the fitting line are for the robot without load; the upper set of data is for robot with the 20-pound load. The motion power increases linearly as the speed increases. The two motion power models (without load and with the 20-pound load, respectively) are: p m (v) = v (3.8) p m (v) = v (3.9)

42 B Sensing power (W) A Sensing frequency (Hz) Fig Sonar sensors power consumption. Figure 3.10 shows the total power consumption of the two sonar arrays at different sensing frequencies. The sonar sensing power model is: p s (f s ) = f s. (3.10) The static power is 0.51 W; 76.9% of the total sensing power when the sensing frequency is 40 Hz. The power consumption increases as the sensing frequency increases; the power consumption at 10Hz (point A) is 38.2% lower than that at 100Hz (point B). The power consumption of the microcontroller is very stable at 4.6W from our measurements. The power consumption of the embedded computer is estimated by dividing the battery capacity by the time the computer can run with a fully charged battery when running different programs. The power of embedded computer is in the range between 8 W to 15 W. We assume the robot uses a fixed sensing frequency (25Hz), the power of the microcontroller is constant at 4.6 W, and the power of the embedded computer is constant at 12 W. The robot moves without a load. The power of this robot is

43 v. Therefore, the power model of this robot is: p(v) = v (3.11) Since the power is a linear function of the speed, the maximum energy-efficient speed v o is equal to the maximum speed v m. 3.3 Motion Planning in Known Environment Energy conservation can be achieved in several ways, for example, using energyefficient motors, finding better routes, or avoiding frequent accelerations and turns. A motion plan includes a path plan and a velocity schedule. The path plan specifies the route from a source to one or more destinations. The velocity schedule decides the accelerations, velocities, and decelerations along the route. In this chapter, our goal is determining which motion plans are more energy-efficient. We compare the energy consumption of different paths at different velocities. The energy consumption of different paths is calculated by considering turns, accelerations, and peak velocities. Our study is both theoretical and practical. We formulate a general problem in motion planning and use the PPRK robot to experimentally validate our approach. We use an experiment-validated simulator and show up to 51% energy savings Coverage Problem in An Open Area 2nl h 2l 2l 2nl (ρ, θ) 2l 2nl (a) (b) (c) Fig (a) scan lines, (b) square spiral, and (c) spiral.

44 29 We consider a task to cover an open area, such as a robot for automatic floor cleaning. Suppose a robot that can clean a square tile of area (2l) 2 (l from the robot s center to each side). The robot can clean the area in different ways. In this chapter, we compare three cleaning strategies: scan lines, spirals, and square spirals. Figure 3.11 shows these strategies. Using the scan-line strategy, the robot starts from one corner and moves at a constant velocity until it reaches the boundary. Then, the robot turns 90, moves 2l, turns 90, and moves at a constant velocity again. The distance between two scan lines is 2l so that the robot can cover the whole area because the robot s sensing radius is l. With the spiral strategy, the robot follows a curve whose radius continuously increases. Using the polar coordinate, the radius linearly increases as the angle increases. When the robot moves around an angle of 2π, the radius increases by 2l. as ρ = 2l 2π θ = l π θ. Therefore, the radius ρ can be expressed Square spirals are similar to spirals: the robot moves around the center and the traveled distance increases after the robot moves 2π around the center. The difference is that the robot moves along straight lines, not curves, for square spirals and have 90 turns. Notice that the robot starts from the center for the spirals and the square spirals. In contrast, in the scan lines the robot starts from one corner. Also, the covered shapes are different. Since our focus is calculating the energy per unit area, we do not consider these differences further. We calculate the energy consumed by the robot for covering a planar area. The energy efficiency of a motion plan is defined as the size of covered area per unit energy: energy efficiency = size of an area energy to cover the area (3.12) Even though we compare only three strategies, our approach is general and applicable to other motion plans. We calculate the robot s energy by dividing it into three parts: moving at a constant velocity, accelerating and decelerating, and turning.

45 Energy Efficiency of Different Routes The robot s power consumption model is shown by equation (3.4). With this power model, we can calculate the energy consumption of the three different routes. Scan Lines Suppose the height of the scan lines is h and the width is 2nl, as shown in Figure 3.11 (a). This path can be divided into 2n + 1 line segments: n + 1 of length h and n of length 2l. The robot has to accelerate and decelerate once for each segment so it accelerates 2n + 1 times. The robot also decelerates 2n + 1 times. The robot turns 2n times, 90 each time. The scan lines in Figure 3.11 (a) show the movement of the robot s center. Because the robot covers a distance of l on each side, the covered area is (2nl + 2l) (h + 2l). First, we calculate the energy consumption while the robot moves at a constant velocity along straight lines (Ω = 0). We can find the energy consumed for each line segment. Let < 0, S, 0 > be the velocity when the robot moves upward vertically in Figure 3.11 (a). The value of S is the peak speed of the robot. The motion for the other directions (downward vertically and horizontally) can be calculated in the same way. Suppose the robot uses a constant acceleration, < 0, A, 0 >, to reach the velocity. It takes S A time to accelerate and the robot moves S2 2A during acceleration. For simplicity, we assume the deceleration is the opposite of the acceleration, namely < 0, A, 0 >. The robot also moves S2 2A moves at the constant velocity across distance h S2 A during deceleration. Consequently, the robot for n + 1 vertical segments. For the n horizontal segments, the robot moves across distance 2l S2 A at a constant velocity, < S, 0, 0 >, if the robot starts from the right corner. The total distance is (n + 1) (h S2 S2 ) + n (2l ). The time to travel this distance is 1 [(n + A A S 1)(h S2 A S2 ) + n(2l )]. The power of the motors can be computed by using Formula A (3.4). Let P S be the power when the robot moves at the constant velocity. Let J i be

46 31 < J i,1, J i,2, J i,3 > and 1 r J i < 0, S, 0 > T be the angular velocity of the i th motor. The total energy can be computed by If h < S2 A P S = k P m ( 1J r i < 0, S, 0 > T, 0) i=1 E 1 = P S 1 S (3.13) S2 S2 [(n + 1)(h ) + n(2l )] A A S2 or 2l <, the robot does not travel at a constant velocity; instead, the A robot consumes only acceleration, deceleration, and turning energy. To calculate the energy consumed during acceleration or decelerating, we have to find the instantaneous velocity of the motors. Suppose acceleration starts at time zero, at time t, the robot s velocity is < 0, At, 0 >. The i th motor s angular velocity is 1 r J i < 0, At, 0 > T and the angular acceleration is 1 r J i < 0, A, 0 > T. Its power consumption is P m ( 1 r J i < 0, At, 0 > T, 1 r J i < 0, A, 0 > T ). Let P A (t) be the power of all motors at time t, then P A (t) = k i=1 P m ( 1 r J i < 0, At, 0 > T, 1 r J i < 0, A, 0 > T ). Similarly, we can define the power during deceleration P D (t) = k P m ( 1J r i < 0, S At, 0 > T, 1J r i < 0, A, 0 > T ) i=1 The total energy for acceleration and deceleration is E 2 = (2n + 1) S A 0 P A (t)dt E 3 = (2n + 1) S A 0 P D (t)dt Last, we compute the energy consumed for turning. (3.14) We can apply the same technique by dividing it into three parts: turning at a constant angular velocity, accelerating the angular velocity, and decelerating the angular velocity. Let Ω be the robot s constant angular velocity and Λ be the angular acceleration. The total energy for turning can be calculated using the same approach as shown in Formulas (3.13) and (3.14). We need to make four adjustments: (i) replace linear speed S by angular speed Ω and linear acceleration A by angular acceleration Λ, (ii) change the traveled distances (h and 2l) into the turned angle ( π ), (iii) change 2n + 1 to 2n since there 2 are only 2n turns, and (iv) add them together. Let P Ω, P Λ, P be the robot s power consumption at the constant angular speed, the constant angular acceleration, and

47 32 the constant angular deceleration. Let E 4 be the energy for turning. They can be calculated by the following equations. P Ω = k P m ( 1J r i < 0, 0, Ω > T, 0) i=1 P Λ (t) = k P m ( 1J r i < 0, 0, Λt > T, 1J r i < 0, 0, Λ > T ) i=1 P (t) = k P m ( 1J r i < 0, 0, Ω Λt > T, 1J r i < 0, 0, Λ > T ) i=1 E 4 = P Ω 1 Ω [(2n)( π 2 Ω2 Λ )] + (2n) Ω Λ 0 P Λ (t)dt + (2n) Ω Λ 0 P (t)dt (3.15) The energy efficiency is the covered area divided by the total energy: area energy = (2nl + 2l)(h + 2l) E 1 + E 2 + E 3 + E 4 (3.16) Square Spirals A similar procedure can be used to calculate the energy efficiency of square spirals. In Figure 3.11 (c), the covered area is (2nl + 2l) 2. Notice that the covered area of square spirals are always square. This is different from scan lines because scan lines cover rectangular areas. There are 2n + 1 line segments and 2n turns. The robot accelerates 2n + 1 times and decelerates 2n + 1 times. The only difference between the scan lines and the square spirals is the length of the line segments: for the square spiral, the lengths gradually increase and are 2l, 2l, 4l, 4l, 6l, 6l,..., 2nl, 2nl, and 2nl. Therefore, E 1 for the square spiral is written as follows: E 1 = P S S2 n [(2nl S A ) + 2 (2jl S2 )] (3.17) A j=1

48 33 Spirals A different approach is needed for computing the energy efficiency for spirals. The robot is constantly turning (Ω 0). Let Θ be the angle the robot moves around the center. The length of the path is The covered area is Θ 0 ρdθ = Θ 0 l π θdθ = l 2π Θ2 (3.18) 2l 2 + Θ 1 (ρ + Θ 2π 2 l)2 dθ = 2l 2 + Θ 1 ( l θ + Θ 2π 2 π l)2 dθ = 2l 2 + πl2 + l2 Θ 2 3 π (3.19) The area 2l 2 is added for half of the covered tile when the robot stops (the other half tile is included in the integration). The robot moves at velocity < V x, V y, Ω >. To make a fair comparison, we assume the robot s linear speed, V 2 x + V 2 y, is S. When the robot is at location (ρ, θ) using the polar coordinate, the curvature K is π(π2 ρ 2 +2l 2 ) [91]. The robot s angular velocity Ω is KS. The total energy is the (π 2 ρ 2 +l 2 ) 3 2 integration of all motors power along the spiral Simulations We use simulations to compare the energy efficiency of different scenarios. The simulation use a power model measured from PPRK. Our simulator has been validated by comparing the simulated results with the data from real measurement with 96% accuracy (shown in Figure 3.2). For our robot, the distance from each wheel to the robot s center b is 0.12m. The radius of each wheel r is 0.02m. We use the following parameters for our simulations: the covered length l is 0.3m., the height of scan line h is 8m, the constant speed S is 0.08m/s, the acceleration A is 0.2m/s 2, the constant angular velocity Ω is 2rad/s, the angular acceleration Λ is rad/s2. The energy for

49 34 moving one meter along a straight line is 9.34J. The energy for turning 90 degree is 2.35J. These numbers are obtained from the measurement. Figure 3.12 shows the energy efficiency of the three routes. When the covered area is small, scan lines have the best efficiency because the robot moves along one straight line without turning. As the area increases, spirals become better. The reason is that scan lines need to decelerate, turn, and accelerate many times when the covered area is large. For a scan line, the robot has to turn 2n times in order to cover area (2nl+2l) (h+2l). For a large covered area, the efficiency of square spirals is between the efficiency of scan lines and spirals. This is because the robot travels increasingly longer straight lines between turns as the covered area of a square spiral grows. Thus, the overhead of turning decreases. This figure demonstrates the importance of an analytic approach for comparing different paths. Figure 3.13 shows the efficiency of scan lines for different h when the covered area is 100m 2. When the area is fixed, increasing h reduces the number of decelerations, turns, and accelerations. As the figure indicates, the efficiency can improves 50% from 0.042m 2 /J to 0.063m 2 /J Scan lines Square spiral Spiral 0.07 Energy efficiency (m 2 /J) 0.06 Energy efficiency (m 2 /J) Area (m 2 ) Scan line height, h (m) Fig Energy efficiencies of three paths with different covering area Fig Energy efficiencies of the scan lines with different heights (h)

50 35 The energy efficiency is also affected by velocities. Figure 3.14 shows the energy efficiency at different velocities S for an area of 100m 2. When S is small, it takes much longer time for the robot to cover this area. Figure 3.15 shows the time to cover this area. As S increases, the efficiency quickly improves. For spirals, when the velocity S increases from 0.05m/s to 0.1m/s, the energy efficiency increases 51.7% from 0.046m 2 /J to 0.07m 2 /J. When S is too large, however, the efficiency decreases again. This can be understood from Figure 3.2. The power increases rapidly as the angular velocity of the motor increases above 5 rad/s. Energy efficiency (m 2 /J) Scan lines Square spiral Spiral Robot s linear velocity, S (m/s) Time to cover 100 m 2 (s) x 104 Scan lines Square spiral Spiral Robot s linear velocity, S (m/s) Fig Energy efficiencies of three paths with different velocities Fig Time to cover 100m 2 with different velocities While these figures compare the energy efficiency of scenarios with specific parameters, our approach is general and can be applied to any robot whose motion can be described through linear transformation of individual motors. 3.4 Robot Exploration Exploration is a basis for many other applications including mapping, search, rescue, reconnaissance, hazard detection, and carpet cleaning [31] [92] [93]. Exploration in an unknown area is to identify the locations of obstacles, objects, and free space

51 36 by sensing the environment. For example, to map an unknown area, robots need to explore the area. For another example, to search and rescue survivors after a disaster, robots have to explore the area to find survivors. Robots usually carry limited energy, such as batteries; thus energy conservation is an important problem for mobile robots. This section focuses on energy-efficient robot exploration. To our knowledge, this is the first study for energy-efficient robot exploration in an environment with random or structured obstacles, such as walls. In exploration, the robot senses the environment while moving. The robot accumulates the information from sensor data and constructs a map of the environment incrementally. At any moment, the robot needs to decide the next target to explore based upon the partial information the robot has already acquired about the environment. This is called target selection and is a fundamental problem in exploration. Target selection determines the exploring sequence of different locations, and directly affects the exploration time and the energy consumption. In general, the next target is selected from frontier cells along the border between the known area and the unknown area [52]. Many existing studies select the next target based on the utilities and costs of the frontier cells [53] [54] [55]. The utility of one frontier cell is estimated based upon the size of the new area that can be potentially covered at the frontier cell. The cost can be the traveling distance or the energy consumption. This strategy usually can cover more new area at the beginning. However, to fully cover an area, the robot has to visit some places with small unknown areas later. This often results in repeated coverage, long exploration time and large energy consumption. Ideally, we want no repeated coverage and no crossover along the exploration path. This requires selecting targets based upon the locations of frontier cells. We presents an approach for energy-efficient robot exploration. Our method is divided into two major steps as the two main contributions of this section. (a) It is an orientation-based method for target selection. Different from the utility-based strategy, our method chooses the next target based on the robot s direction and relative location of frontier cells. Our target selection method can greatly reduce repeated

52 37 (a) (b) Fig (a) Utility-based target selection. (b) Orientation-based target selection. coverage, and thus shorten the exploration distance and save energy. (b) It can find the most energy-efficient path from the current location to the next target. Different from many existing studies that choose the shortest route, our method estimates the energy consumption and chooses the most energy-efficient route. This requires a dedicated motion planning algorithm that is different from the algorithms planning the shortest path. We conduct extensive simulations to compare our method with one existing method that chooses the next target based upon the utility of frontier cells. Simulation results show that our method is effective in reducing repeated coverage and saving energy. Our method can also shorten the traveling distance Motivating Examples Target Selection Figures 3.16(a) and 3.16(b) show two exploration routes of the same robot exploring the same area. The routes are generated by our simulator to be described later. Inside this area, there are three rooms with small openings at the doors. The robot starts from the upper left corner and stops until the whole area is covered by the robot s sensors. In Figure 3.16 (a), the robot selects the next target to maximize the

53 38 A R2 R1 B Fig Two routes R1 and R2 connecting location A with location B. R1 is shorter than R2, but consumes more energy because R1 has more stops and turns. utility, therefore skipping small openings at the doors or corners at the beginning. However, the robot has to visit the rooms or corners later to fully cover this area. There are many crossovers along the path, resulting in repeated coverage. In Figure 3.16(b), the robot selects the next target based on the relative directions of the frontier cells to the robot. When the robot moves, it always chooses the next target from the frontier cells on the left side first. If there is no frontier cell in the left side, it will choose frontier cells in the front. The priorities of the frontier cells depend on their relative directions to the robot, starting from robot s left side and following the clockwise sequence: left, front, right, and back. The essence of this strategy ensures that the robot s left side has been explored when the robot explores unknown areas in the front. Adopting this strategy, the robot in Figure 3.16(b) visits the rooms earlier than the robot in Figure 3.16(a). The strategy of Figure 3.16(b) is better because the path has no crossing point, and the path length is shorter. This example shows the importance of orientation-based target selection. Energy-Efficient Motion Planning Figure 3.17 shows the two routes from location A to location B. The gray area represents obstacles. Route R1 comprises of ten short line segments, while route R2 has three long line segments. R1 has a shorter distance with more stops and

54 39 turns than R2. Stops and turns cause acceleration and deceleration that consume a significant amount of energy. Hence, R1 may be shorter but consume more energy. This shows the difference between the short-distance paths and energy-efficient paths Energy-Efficient Exploration Problem Definition Exploration is to cover a 2-dimensional area by a robot s sensors. We use a grid cell map to represent this area. Each cell is a 1 1 unit of square. Each cell is either free or occupied by an obstacle. Obstacle cells are inaccessible to the robot and impenetrable to the sensors. The robot can move from one cell to one of its eight neighbors, if both cells are free. A cell can be represented by its two coordinates (i, j), where i and j are two nonnegative integers. We count a robot s movement from one cell to one of its neighbors as one step. In Figure 3.18 (a), there are eight free cells and one obstacle cell. From the cell in the center, the robot can travel to seven of the eight neighbors as illustrated by the arrows. At step 0, the robot starts from the initial location and senses the environment. The robot is equipped with sensors and the sensing range is a circle with a radius of d s. This radius is also called the sensing distance. The robot moves and updates the explored map until all the accessible area has been explored. At each step, the robot s state can be represented by its location (i, j) and direction θ, and we denote robot s state at step k as State(k) =< i(k), j(k), θ(k) >. The exploration trajectory is a link of the robot s states at each step State(0),..., State(k),... The energy-efficient exploration problem is to determine a trajectory to explore the whole accessible area with minimum energy. There are two sub-problems involved: target selection and motion planning. Target selection is to select a cell from frontier cells for the robot to explore next. Frontier cells are explored free cells along the border between the explored area and the unexplored area. Motion planning is to plan a viable path from the current cell to the target cell. Since both the current

55 40 Obstacle (2,4) (2,2) R (3,2) (4,4) (1,1) (3,1) (a) (b) Fig (a) Free and obstacle cells. (b) R: robot s current location; area enclosed by dash line show the robot s sensing range. and the target cells are explored free cells, there must exist a viable path within the explored area. The robot continues target selection and motion planning until visiting all accessible areas, and the exploration trajectory is thus generated. Target Selection Target selection determines which frontier cell to explore next. For example, in Figure 3.18 (b), the robot is at location (3, 3), and all cells enclosed by the dash lines have been explored. There is a total of five frontier cells: (2, 2), (3, 1), (3, 2), (2, 4), and (4, 4). Among the five frontier cells, three are connected: (2, 2), (3, 1), and (3, 2). The utility-based method selects a frontier cell that can cover more potentially unexplored area immediately. For example, in Figure 3.18 (b), cell (3, 1) is more distant from obstacles than the other four frontier cells because the closest obstacle cell of cell (3, 1) is of 2 distance away from cell (3, 1) and all other frontier cells has closest obstacle cells of distance 1; thus, the robot can cover more unexplored area after moving to cell (3, 1). A utility-based method selects (3, 1) as the next

56 41 target. However, as we have shown by the motivating example in section 3.4.1, the utility-based method causes more repeated coverage. left clockwise Robot s direction 3 4 Fig Frontier cells and target selection. Our algorithm lists the 8 frontier cells in the number order: cell 1, cell 2,..., cell 8, starting from robot s left direction and following the clockwise order. Our method uses an orientation-based target selection strategy, as described in the following steps: (1) It identifies all the frontier cells that are within the current sensing region. If no such frontier cell exists, go to step (4). (2) The algorithm lists all the frontier cells from step (1) in a clockwise order starting from the robot s left direction. Figure 3.19 illustrates this ordering. The figure shows 25 explored cells, and the area outside this region is unexplored. There are 8 frontier cells as labeled by the numbers. The cell at the robot s left direction is an obstacle cell. The 8 frontier cells are labeled from 1 to 8 in clockwise order; therefore, the list for this figure is cell 1, cell 2,..., cell 8. The head of the list is cell 1. (3) The algorithm picks a frontier cell in the list to satisfy the following two conditions: (a) From the list head to this frontier cell, any one cell and its next cell are neighbors. (b) The distance from the head to this cell is less than 0.7d s. The coefficient 0.7 is chosen according to simulation results. We have tried values of 0.6,

57 and 0.8, and found 0.7 can satisfy the following requirements better than the other two choices. The first condition guarantees that there is no obstacle between the head and the selected target cell along the border. The second condition ensures that when the robot moves to the target, the robot can sense the obstacles in the left side and also some distance (at least 0.3d s ) outside the head of the frontier cell list. This strategy is to make sure that the robot s left side has been explored when the robot proceeds to explore unknown areas in the front, essential to our orientation-based method. If the algorithm does not follow the condition (b), and directly picks the list head as the next target, the robot tends to move too close to obstacles. For example, in Figure 3.20 there are continuous obstacles (a wall). Picking the list head as the next target, the robot moves to a frontier that is a neighbor to the wall and then moves along the wall. Since the sensors can sense a distance of d s > 1, this wastes the sensors capability. (4) If no frontier cell satisfies the two conditions in step (3), no frontier cell is within the current sensing region. Even if there is only one frontier cell in the current sensing range, that cell will be the only cell in the list and will be selected according to the conditions (a) and (b). In this scenario, the algorithm picks the closest frontier cell outside the current sensing range as the next target. (5) If there is no frontier cell at all, then all the accessible area has been explored and the exploration is completed. Motion Planning If the next target is within the robot s current sensing range as in step (1) of our target selection method, the current location and the target are close and a simple motion planning strategy combining turning and straight-line motion is sufficient since the sensor works in a line-of-sight fashion. However, if there is no frontier cell inside the current sensing region, the robot may select a frontier cell far away form

58 43 Fig Closely move along a wall. the current location as the next target, as in step (4) of our target selection method. In this situation, motion planning is important to find an energy-efficient route. Most existing studies plan a shortest path between the current location and the next target using Dijkstra s algorithm. To find shortest paths using Dijkstra s algorithm, the grid cell map can be transformed into a graph in this way: free cells are vertices and an edge exists between two accessible vertices if they are neighbors. The edge has a weight of either 1 or 2 depending on their relative locations, representing the distance between two neighbor cells. The graph is undirected. To generate the energy-efficient paths, we transform the grid map into a graph in a different way. To incorporate the direction information, the vertices in the graph should represent the robot s states that include both locations and directions. Each free cell in the grid map is transformed into 8 vertices, representing the 8 possible robot states at this cell. If the cell is (i, j), the 8 vertices are 8 states: < i, j, 0 >, < i, j, 45 >,..., < i, j, 315 >. We assume the robot uses 45 as the unit for turns, since we only allow the robot to move from one cell to one of its eight neighbors. We can also label these 8 vertices by their directions: N in short of North, NE in short of Northeast. The rest are E, SE, S, SW, W, and NW. Each of these 8 neighbor cells are represented

59 44 by 8 vertices, as 8 possible leaving states at that cell. Figure 3.21 shows two cells (i, j) and its Northeast neighbor (i + 1, j + 1) with 8 vertices for each. NW N NE <i, j, 180> NW <i, j, 135> W E SW <i, j, 225> Current cell(i, j) w1 N NE S SE W SW S E SE Northeast cell (i+1, j+1) Fig Transform a grid cell map into a graph for energy-efficient motion planning. The circles are vertices, and the solid lines with arrows represent directed edges. The vertices represent the robot s states. The weight of one edge is the energy needed for the robot traveling from one state to another state. For example, in the figure w1 represent the energy needed for moving from state < i, j, 45 > (NE) to state < i + 1, j + 1, 135 > (NW). An edge connects two states. The edge is directed, because the robot may travel from one state to another state but not in the reverse direction. From any state, the robot can only reach 8 other states. For example, from state < i, j, 45 >, the robot can reach its Northeast neighbor < i + 1, j + 1 >, and this neighbor cell has 8 possible leaving states: < i + 1, j + 1, 0 >,..., < i + 1, j + 1, 315 >. However, the robot can not change directly from state < i + 1, j + 1, 0 > to state < i, j, 45 >. There are 8 edges that start from < i, j, 45 > and end at < i + 1, j + 1, 0 >,..., < i+1, j +1, 315 >, respectively. In Figure 3.21, the solid lines with arrows show the 8 edges. The weight of one edge between two states is the energy needed for the robot to move from one state to the other state. We consider the energy for stops and turns if the two states have different directions. For example, the weight of the edge from

60 45 NE of (i, j) to NE of (i + 1, j + 1) is only the energy of traveling a distance of 2, because the robot does not stop or turn. However, the weight of the edge from NE of (i, j) to E of (i + 1, j + 1) includes the energy for traveling distance 2 and the energy for a stop and a turn of 45. After the procedure, we have transformed a grid map into a graph. In such a graph, a path represents a trajectory of the robot and the sum of the weights of the edges along a path represents the energy consumption of that path Simulations and Results Simulation Setup We use the following parameters in our simulations. The cell is a square with each side of one unit length. The robot consumes one unit of energy for traveling one unit of distance. One stop takes an extra energy of 0.5 unit. A turn of 45 takes 0.4 unit of energy. Turns of 90, 135, 180 take 0.6, 0.8 and 1 unit of energy, respectively. The turning process includes three steps: angular acceleration, angular constant speed and angular deceleration; therefore, turning 90 takes less than twice the energy needed for turning 45. These numbers are approximately estimated from our energy measurements for a Pioneer 3-DX robot [16]. The robot s sensing range is a circle with a radius d = 10 units of distance. The unknown areas are rectangles of different sizes. Two types of obstacles are used for simulations: random obstacles and structured obstacles. Figures show random and structured maps in our simulations. In these figures, white cells represent free cells and black cells represent obstacle cells. These areas are bounded by obstacles. Energy-Efficient Motion Planning We compare the paths generated by our energy-efficient motion planning method with the shortest paths. We use random maps for this simulation. Figure 3.22 shows

61 46 S D Fig Shortest and energy-efficient paths. The black squares are obstacle cells. two paths from the source cell S to the destination cell D. The lower one is the shortest path of length 33.87, consuming energy of units. The upper one is the energy-efficient path with a length of 36.21, 6.9% longer than the shortest path. However, the energy-efficient path only consumes energy of units, 7.5% lower than the energy consumption of the shortest path. In a random map with 20% obstacle cells, we compare the distances and the energy consumption of the shortest and the corresponding energy-efficient paths between more than 3800 different pairs of source and destination cells. The results show that on average the energy-efficient paths save 8.4% energy while they are 0.7% longer compared with the shortest paths. This is because the energy-efficient paths have fewer stops and turns that may consume significant amounts of energy. Target Selection and Robot Exploration We run simulations and compare our method with a utility-based method [53], called widest frontier. This method first clusters the frontier cells within the current sensing region into groups. The frontier cells inside one group are close to each other,

62 47 and they are connected as neighbors. We call each group as one frontier. This method chooses the widest frontier, the group with the maximum number of frontier cells, and picks one in the middle as the target cell. If no frontier cell is found in the current region, the algorithm picks one closest frontier cell outside as the next target. This method is based on utility of frontier cells because moving to the middle cell of the widest frontier is likely to cover more new area than any other frontier cell. Figures 3.23 (a) and (b) show two different exploration routes of the same map generated by our target selecting method and the widest frontier method. The route from our method has a length of with total energy consumption of The route from the widest frontier has a length of with total energy consumption of The route from our method is 41.8% shorter in distance and consumes 42.8% less energy. (a) (b) Fig (a) Our target selection method. (b) Choose the widest frontier. Figures 3.24 (a) and (b) show the exploration routes in another map with structured obstacles. The route from our method has a length of with total energy consumption of The route from the widest frontier has a length of with total energy consumption of The route from our method is 38.9% shorter in distance and consumes 38.5% less energy. The above two maps are areas with structured obstacles. Figures 3.25 (a) and (b) show the exploration routes in an area with random obstacles. The route from

63 48 (a) (b) Fig (a) Our target selection method. (b) Choose the widest frontier. our method has a length of with total energy consumption of The route from the widest frontier has a length of with total energy consumption of The route from our method is 10.1% shorter in distance and consumes 9.3% less energy. (a) (b) Fig (a) Our target selection method. (b) Choose the widest frontier. From the above simulation results, we can see that our orientation-based target selection method can reduce repeated coverage, have shorter exploration distances, and consume less energy. We further investigate the coverage ratio. The coverage ratio is the number of explored free cells over the total number of accessible free cells. It is 0 when the robot starts and it is 1 at the end. If we study the intermediate coverage