3D Magnetic Field Mapping in Large-Scale Indoor Environment Using Measurement Robot and Gaussian Processes

Transcription

1 3D Magnetic Field Mapping in Large-Scale Indoor Environment Using Measurement Robot and Gaussian Processes Naoki Akai 1 and Koichi Ozaki 2 Abstract Magnetic fields are used for localization and navigation in the field of robotics. In recent years, because of the spread of mobile devices equipped with magnetic sensors (e.g., smart phones), the use of magnetic fields has been extensive, especially for position tracking of mobile devices. One example application of such tracking is in identifying the position of a person with a mobile device. Development of this application requires a three-dimensional (3D) magnetic map that represents the magnetic distribution of a 3D environment since the device moves around in 3D space. It is, however, difficult to construct a 3D magnetic map of a large-scale environment because measuring the magnetic field is time consuming and expensive. In this paper we propose an efficient method for mapping the 3D magnetic field of a large-scale environment. The method uses a mobile manipulator to measure the 3D magnetic field, enabling 3D magnetic data to be automatically collected. Moreover, the method uses Gaussian processes (GPs) for regression of the magnetic field. In this study, we first evaluate the performance of the GPs and then describe the measurement robot. In an experiment, a 3D magnetic field of an indoor environment is visualized by using this method and the performance of the presented method is demonstrated. I. INTRODUCTION Magnetic fields are used for localization and navigation in the field of robotics [1], [2], [3], [4]. In recent years, because of the spread of mobile devices equipped with magnetic sensors (e.g., smart phones), the use of magnetic fields has been extensive, especially for position tracking of mobile devices [5], [6], [7]. One example application of such tracking is in identifying the position of a person with a mobile device. Development of this application requires a three-dimensional (3D) magnetic map that represents the magnetic distribution of a 3D environment since the device moves around in 3D space. It is, however, difficult to construct a 3D magnetic map of a large-scale environment because measuring the magnetic field is time consuming and expensive. Constructing a magnetic map requires measuring the magnetic field by using a magnetic sensor of known position. Position measurement methods using motion capture have been proposed [8], [9]. Mapping methods based on an individual carrying a hand-held sensor such as an inertial measurement sensor (IMU) have also been proposed [5], [6]. We have proposed a mapping method using a mobile robot that can recognize its own position by using geometric landmarks [10]. Magnetic-field-based simultaneous localization and mapping (SLAM) has also been proposed [11], 1 Naoki Akai is with the Institute of Innovation for Future Society (MIRAI), Nagoya University, Furochou, Chikusa, Nagoya, Japan akai@coi.nagoya-u.ac.jp 2 Koichi Ozaki is with Graduate School of Engineering, Utsunomiya University, Yoto, Utsunomiya, Tochi, Japan ozaki@cc.utsunomiya-u.ac.jp [12], [13]. Although these studies enabled large-scale twodimensional (2D) magnetic fields or small 3D magnetic fields to be mapped, mapping a large-scale 3D magnetic field has not yet been achieved because an efficient 3D magnetic field measurement system has not been developed. In this paper we propose an efficient 3D magnetic field mapping system suitable in a large-scale environment. The system uses a mobile robot with a simple manipulator equipped with magnetic sensors and scans the 3D magnetic field, thereby enabling the robot to automatically collect 3D magnetic field data. It is, however, difficult for the robot to exactly measure all of the magnetic field. Constructing the magnetic map that represents all of the magnetic field in the environment requires an estimate of the unmeasured magnetic fields. In the proposed system, Gaussian processes (GPs) are used and regression of the unmeasured magnetic fields is performed. In this study, we first evaluate the performance of the GPs and then describe the measurement robot. In an experiment, a 3D magnetic field of an indoor environment is visualized by using this system and the performance of the presented system is demonstrated. II. RELATED WORK An indoor magnetic field includes many magnetic fluctuations because of the many magnetized materials in buildings. Stability of the fluctuations over long periods of time has been demonstrated [14]. In other words, these magnetic fluctuations can be used as landmarks. Localization and SLAM methods based on these fluctuations have been proposed. As far as we know, Suksakulchai et al. first proposed a localization method based on the magnetic fluctuations for mobile robots [1]. In this method, a linear magnetic map that represents a magnetic field of a predetermined path is used. Haverinen et al. [3] extended this method by applying Monte Carlo Localization (MCL) [15]. We proposed a navigation method based on a linear magnetic map and achieved autonomous navigation in a cluttered environment [4]. However, navigating a mobile robot using a linear magnetic map is not easy because accurate location cannot be estimated. To achieve accurate localization based on a magnetic map, 2D and/or 3D magnetic maps are required. Rahok et al. achieved 2D magnetic-map-based localization in indoor environments [2]. In their method, the 2D magnetic map is built by combining several linear magnetic maps. It is, however, difficult to construct a large-scale magnetic map since measuring both the position of and the readings from a magnetic sensor is not done. Frassl et al. used motion

2 capture to construct a 2D magnetic map. In their method, a robot equipped with a magnetic sensor sweeps a room and the motion capture measures the position of the robot. Although they efficiently constructed a 2D magnetic map, applying this method to a large-scale environment is not easy because of the restricted range of motion capture. We proposed an efficient method to construct a large-scale 2D magnetic map using magnetic data collected by a mobile robot and GPs [10]. In this method, the robot can identify its own position by using geometric landmarks and scans the magnetic field. Since measuring all of the magnetic field by using the robot is difficult, GPs are applied to estimate unmeasured magnetic fields. We achieved accurate magneticmap-based localization in a large-scale environment, but the applicable dimension of the method is restricted to two dimensions since the robot cannot measure a 3D magnetic field. Also, 2D magnetic-field-based SLAM has also been proposed [11], [12], [13]. However, 3D magnetic-field-based SLAM has not yet been achieved. Wahlstrom et al. proposed a mapping method for a 3D magnetic field [9]. They used motion capture to measure the position of a magnetic sensor and collected 3D magnetic data. GPs were also applied to estimate unmeasured magnetic fields in their method and they were able to model the magnetic field around a magnetic material. In recent years, mobile devices equipped with magnetic sensors have become pervasive and localization methods using such devices have been proposed. Gozick et al. and Grand et al. proposed a localization method based on a magnetic map using a smart phone [5], [6]. IndoorAtlas provides a position tracking service using a smart phone; its position is tracked on the basis of a magnetic map. In these methods, a person with a smart phone first walks an environment to collect magnetic data and the position of the smart phone is calculated by using an IMU. Because only an IMU is used to estimate the position, it is difficult to accurately estimate the location of the smart phone. Accurately collecting 3D magnetic data in a large-scale environment is not an easy task with such methods. A. Preliminaries III. GAUSSIAN PROCESSES Let D = {(x 1, t 1 ), (x 2, t 2 ),,(x N, t N )} be a set of training data, where x R d and t R denote an input state and a target value, respectively [16]. A target value t i which corresponds to an input state x i is denoted as follows: t i = f(x i )+ϵ, (1) where ϵ is the zero mean Gaussian noise with variance σ 2 n. For notational convenience, N input vectors x i are aggregated into a d N matrix X, and N target values are also aggregated into a N column vector t. In the GPs, posterior distributions over functions f including unknown input states are determined from training data D. A key idea of the GPs is the requirement that the function values at different states are correlated, where the covariance between two function values, f(x p ) and f(x q ), depends on the input states x p and x q, respectively. This dependency can be specified via an arbitrary covariance function or kernel k(x p, x q ). In this study, we use a Gaussian kernel [9]: k(x p, x q ) = σf 2 exp ( x p x q 2 ) 2l 2, (2) where σf 2 and l determine the strength of correlation between input states. As can be seen from equation (2), the correlation decreases when the distance between two input states is large. From equations (1) and (2), the covariance of target values t p and t q is given as: cov(t p,t q ) = k(x p, x q )+σ 2 nδ pq, (3) where δ pq denotes the Kronecker delta which has a value of 1 if p = q; otherwise, its value is 0. From the equation, a covariance matrix of the target vector t is denoted as follows: cov(t) = K +σ 2 ni, (4) where I is an identity matrix and K is given as: k(x 1, x 1 ) k(x 1, x 2 )... k(x 1, x N ) k(x 2, x 1 ) k(x 2, x 2 )... k(x 2, x N ) K = (5) k(x N, x 1 ) k(x N, x 2 )... k(x N, x N ) The target vector t is defined as an N-dimensional Gaussian distribution: t N(0,K +σ 2 ni) (6) Our purpose is to determine the posterior distributions over function f at an arbitrary input state x from the training data D. In the GPs, the posterior distribution is given as a Gaussian distribution and its mean µ x and variance σ 2 x are calculated as follows [16]: p(f(x ) x,x, t) = N(f(x );µ x,σ 2 x ) (7) µ x = k T (K +σ 2 ni) 1 t (8) σ 2 x = k(x, x ) k T (K +σ 2 ni) 1 k, (9) where k denotes a covariance vector of the focused input state x and training data X: k = (k(x, x 1 ),k(x, x 2 ),,k(x, x N )) T. (10) From these equations, we can determine the posterior distributions from the training data. B. Hyperparameter estimation An important process for the GPs is to estimate optimized values for the hyperparameters θ = σ 2 f,l,σ2 n. These values significantly affect the estimation results of the posterior distributions. We determine the optimized values by maximizing the log likelihood of p(t X,θ) given as follows [9]: logp(t X,θ) = 1 2 t(k +σ2 ni) 1 t 1 2 log K +σ2 ni n 2 log2π. (11)

3 (a) Manually measured 3D magnetic field Fig. 1. Environment for 3D magnetic field measurement The partial differentiation by a hyperparameterθ j of equation (11) is denoted as follows: logp(t X,θ) = 1 ( θ j 2 tr (K 1 t)(k 1 t) T K ). (12) θ j From equation (2), the partial differentiations by each hyperparameter are given as: ( K pq = 2σ f exp 1 ( ) ) 2 xp x q (13) σ f 2 l ( K pq = σf 2 exp 1 ( ) ) 2 xp x q x p x q 2 l 2 l l 3 (14) (b) 0.4 m x 0.4 m x 0.4 m (c) Estimated result from (b) K pq σ n = 2σ n δ pq. (15) C. Modelling a 3D magnetic map for the GPs In this study, a 3D magnetic map is represented by a voxel map. We first collect the magnetic data by using a measurement robot and record magnetic intensity to a corresponding grid. Three axis magnetic sensors (3DM- DH) are used in this study and the magnetic intensity b is computed by using the following: b = (b x b ref x ) 2 +(b y b ref y ) 2 +(b z b ref z ) 2 (r base r ref ), (16) where b x, b y, and b z denote sensor readings of the magnetic sensors, b ref x, b ref y, b ref z, and r ref denote calibration values of each sensor, and r base denotes a base value for this magnetic intensity. These calibration values are determined in each sensor and the calibration method and results are described in Section VI. The recorded magnetic intensities in the voxels, b i (i = 1, 2,, N), and positions of the voxels, x i = [x i, y i, z i ] T (i = 1, 2,, N), are used as the training data D. The posterior distributions shown in equation (7) are determined in each voxel. IV. VERIFICATION OF ACCURACY OF MAGNETIC FIELD MAPPING BY THE GPS For applying the estimated magnetic map from the GPs (e.g., localization), its estimation accuracy has to be evaluated. We manually measured all of the 3D magnetic field of (d) 0.6 m x 0.6 m x 0.6 m (f) 0.8 m x 0.8 m 0.8 m (h) 1.0 m x 1.0 m 1.0 m Fig. 2. (e) Estimated result from (d) (g) Estimated result from (f) (i) Estimated result from (i) Manually measured and estimated magnetic fields the environment shown in Fig. 1 for the evaluation. Ranges of x, y, and z axes are 3.0, 3.0, and 2.2 m, respectively, and the magnetic field was measured every 0.2 m along each axis; namely, the magnetic fields of 2816 points were measured. Figure 2(a) shows a visualization result of the

4 TABLE I HYPERPARAMETERS USED IN THIS STUDY σ f l σ n Fig. 4. System architecture Fig. 3. Measurement robot 3D magnetic field. Some magnetic fluctuations can be seen from the result. We selected some measured data and verified the estimation accuracy. Figures 2(b), (d), (f), and (h) show the selected magnetic data, where the captions indicated the range of the selected interval. Figures 2(c), (e), (g), and (i) show the estimation result in each case. Table I lists values of the hyperparameters used in this study. Magnetic fluctuations included in the manually measured magnetic field can be seen in all cases. Table II lists comparison results of the estimated magnetic fields with the manually measured result. Used data number means the number of selected magnetic data in each case. The standard deviation of the magnetic sensors we used is 0.01 G. These results indicate that, if distances of measurement points of a magnetic field are kept to within 0.6 m, the GPs can estimate with enough accuracy since the GPs determine variances of estimated values. A. Hardware V. MEASUREMENT ROBOT Figure 3 shows a platform for measuring a 3D magnetic field. The robot is constructed from two modules: a mobile module and a manipulator module. The mobile module is equipped with two driving wheels and a LIDAR (UTM- 30LX) on a 3D swinging mechanism (3D LIDAR). Rotations of the driving wheels are measured by the encoders. The 3D LIDAR detects 3D geometric objects and enables the robot to recognize its own position in an environment. The manipulator module is equipped with a wooden stick whose length is 1.2 m. Two magnetic sensors are attached to the stick. The manipulator has two degrees of freedom, and its yaw and pitch can be controlled. The manipulator cyclically scans a 3D magnetic field, but it may collide with geometric objects in a narrow space. To prevent such collisions, the robot recognizes the 3D surroundings by using the 3D LIDAR and controls the motion of the stick. The distance between the magnetic sensors is 0.6 m. This means that magnetic fields between the sensors can be accurately estimated by the GPs (as was ascertained from the above-mentioned experiment). Moreover, since the stick continuously moves around, magnetic fields surrounding the sensors can also be accurately estimated. In other words, an estimated 3D magnetic map from 3D magnetic data collected by the robot has enough accuracy. Also, the robot can measure magnetic fields from 0.2 m to 2.1 m above the ground surface. This means that the 3D magnetic map can be applied to track a person. B. System architecture Figure 4 shows the system architecture of the robot. The main object of the robot is to collect 3D magnetic data. To collect the magnetic data, it is necessary for the robot to know its own position in the environment. A 3D geometric map is built beforehand and localization is done by comparing the map and 3D geometric objects detected by the 3D LIDAR. The localization result is used in magnetic data collection and path planning processes. The 3D LIDAR readings are also used for planning the motion of the manipulator to avoid collision with geometric objects. C. 3D geometric-map-based localization A particle filtering algorithm is used for the 3D geometric map-based localization [15]. The state of the robot x = [x, y, θ] T is represented by the following conditional distribution. p(x t z 1:t, u 1:t ) = ηp(z t x t ) p(x t x t 1, u t )p(x t 1 z t 1, u t 1 )dx t 1, (17) where z 1:t and u 1:t denote, respectively, an observation of the 3D LIDAR and the encoder readings from time 1 to t. Also, η is a normalizer coefficient and p(z t x t ) and p(x t x t 1, u t ) denote observation and motion models, respectively. Although we use 3D geometric map, state of the robot is represented in a 2D space. This is because that we assumed that the ground surface is flat.

5 TABLE II VERIFICATION RESULTS OF THE ESTIMATION ACCURACY Selected interval [m] Used data number Average of the errors [G] Standard deviation of the errors [G] 0.4 m x 0.4 m x 0.4 m m x 0.6 m x 0.6 m m x 0.8 m x 0.8 m m x 1.0 m x 1.0 m The motion model of the robot is given by the equation: x t x t 1 d t cosθ t 1 y t = y t 1 + d t sinθ t 1, (18) θ t θ t 1 θ t where u t = [ d t, θ t ] T denotes the amount of translational and rotational movement measured by the encoders from t 1 to t. The 3D LIDAR detects 3D points p i = [x i, y i, z i ] T (z t = [p t,1, p t,2,, p t,k ] T ) and the observation model of the sensor readings is given by the following: p(z t x t ) = exp{ κ where d map i K i=1 (d map i (x t ) d obs i (p t,i ))}, (19) (x t ) and d map i (p t,i ) denote, respectively, the expected range data of the i-th degree from state x t on the 3D geometric map and the range data of the point p t,i. Also, κ denotes a variance variable, which is predetermined. D. Navigation in the environment The robot navigates the environment to collect 3D magnetic data. By using an autonomous data collection strategy, the robot can automatically collect the data. However, developing such a strategy is not easy. In this study, we manually control the robot by using a controller and delete the pathplanning process from the system architecture. E. Collecting 3D magnetic data Figure 5 shows an image of the 3D magnetic field measurement. The colored dots and the white stick represent geometric objects detected by the 3D LIDAR and the wooden stick. If the stick will be collide with an object, the direction of motion of the manipulator is reversed. Positions of the magnetic sensors are computed on the basis of a localization result that is executed by the mobile module. VI. EXPERIMENT A. Calibration of magnetic sensors For each sensor to measure the same value of the magnetic intensity, sensor calibration is necessary. We manually rotate each magnetic sensor around all three axes randomly with as little translation as possible [17]. These magnetic data form a sphere, and the center of a fitted sphere can be used to estimate the three biases. The center b ref x, b ref y, and b ref z and radius r ref are determined by the least-squares method. Table III lists the calibration results for the magnetic sensors. Sensors 1 and 2 represent the sensors attached to the top and middle of the wooden stick, respectively (see Fig. 3). The r ref of the Sensor 1 is used as r base shown in equation (16). Fig. 5. Measurement of a 3D magnetic field. The colored dots and white stick shown in the bottom figure represent geometric objects detected by the 3D LIDAR and the wooden stick. If the stick will be collide with an object, the direction of movement of the stick is reversed. TABLE III CALIBRATION RESULTS FOR THE MAGNETIC SENSORS r ref [G] b ref x [G] b ref y [G] b ref z [G] Sensor Sensor B. Magnetic data collection and mapping Figure 6 shows a 3D geometric map of an experimental environment; the color dots represent geometric landmarks. We collected 3D magnetic data by using the robot, and the collected magnetic data are shown in Fig. 7. Colors of the voxels indicate the level of magnetic intensity. The size of the voxels was set to 0.2 m and magnetic data were recorded for 6556 voxels. Collecting the 3D magnetic data required 1 hour, including both the geometric map building and magnetic data collection. In contrast, it took 8 hours to measure all of the 3D magnetic fields of the environment shown in Fig. 1. The

6 robot to efficiently collect 3D magnetic data. It was, however, difficult to measure all of the magnetic field using the robot. Therefore, we used Gaussian processes to estimate unmeasured magnetic fields. We also verified the performance of the GPs and demonstrated that the GPs can be applied to the magnetic field estimation with sufficient accuracy. In the experiment, we built a 3D magnetic map of an actual indoor environment whose volume was approximately 1000 m 3. Finally, we concluded that the presented system is useful for constructing a large-scale 3D magnetic map. ACKNOWLEDGMENT This study was supported by KAKENHI Fig. 6. 3D geometric map of an experimental environment volume of the environment is approximately 20 m 3. The volume of the experimental environment shown in Fig. 6 is approximately 1000 m 3 ; thus 400 hours are required if we manually measure all of the magnetic fields. This result highlights the effectiveness of the robot in collecting 3D magnetic data. Figure 8 shows the estimated 3D magnetic map obtained by the GPs. Figures 8(a) shows the magnetic intensity scale and other figures show hierarchical magnetic maps. The heights listed in each caption represent the height above the ground surface. Differences of the magnetic distribution at different heights can be seen. These results demonstrate the importance of a 3D magnetic map for tracking a mobile device equipped with a magnetic sensor and that the estimated 3D magnetic map provides a solution for the tracking system. Our future work will entail developing a tracking system capable of tracking a person with a mobile device. The presented system has one serious problem: The GPs require long computation times. In this experiment, it took approximately 30 hours for Core i5 CPU GPs to estimate the unmeasured magnetic fields. If we apply GPUs to the calculation, the GPs may perform much faster, so this is also a subject of our future work. Although the GPs required a long computation time, it still is significantly shorter than what would be required to manually measure the field. Therefore, we believe that the presented system will be useful for constructing a large-scale 3D magnetic map 1. VII. CONCLUSION Although magnetic-field-based localization methods have been developed, a 3D magnetic-field-based localization method was not yet been proposed because an efficient system for measuring a 3D magnetic field has not yet been developed. In this study we successfully constructed a 3D magnetic map of a large-scale indoor environment and visualized the 3D magnetic field. To construct the 3D magnetic map, we used a mobile robot with a simple manipulator equipped with magnetic sensors to enable the 1 The related video can be found at watch?v=fwl46wce-sm REFERENCES [1] S. Suksakulchai, S. Thongchai, D.M. Wilkes, and K. Kawamura. Mobile robot localization using an electronic compass for corridor environment, IEEE Int. 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7 Fig. 7. Collected 3D magnetic data. The bird s-eye view shows a overview of the data and environment; other figures show views from viewpoints 1, 2, and 3 depicted in the bird s-eye view. (a) Magnetic intensity scale Fig. 8. (b) 0.2 m (c) 0.4 m (d) 0.6 m (e) 0.8 m (f) 1.0 m (g) 1.2 m (h) 1.4 m (i) 1.6 m (j) 1.8 m (k) 2.0 m (l) 2.2 m (m) 2.4 m Estimated 3D magnetic map. These figures hierarchically represent the 3D magnetic map; captions indicate the height from the ground surface.