Separation of Position and Direction Information of Robots by a Product Model of Self-Organizing Map and Neural Gas

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1 Systems and Computers in Japan, Vol. 36, No. 11, 2005 Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J87-D-II, No. 7, July 2004, pp Separation of Position and Direction Information of Robots by a Product Model of Self-Organizing Map and Neural Gas Akira Date 1 and Koji Kurata 2 1 National Institute of Information and Communications Technology, Kyoto, Japan 2 Faculty of Engineering, University of the Ryukyus, Okinawa, Japan SUMMARY This paper proposes a neural network model which receives visual inputs from a robot moving freely in a room, and extracts its position and direction information separately. The model has three-dimensional structure in which two-dimensional neural fields (unit planes) are arranged in a toroidal form. The learning algorithm takes a form of neural gas in each unit plane and self-organizing map (SOM) in the toroidal direction. It is shown by computer simulation that position and view direction of the robot can be extracted from visual inputs, and the effectiveness of the learning algorithm is verified Wiley Periodicals, Inc. Syst Comp Jpn, 36(11): 1 11, 2005; Published online in Wiley InterScience ( DOI /scj Key words: self-organization; SOM; neural gas; information separation; neurocomputing. 1. Introduction The mechanism which acquires the structure of the external world by self-organization is interesting in connection with the study of biological systems and the study of Contract grant sponsor: Supported in part by a Grant-in-Aid ( , ) from the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT). robots. Usually, a human in a familiar scene is aware of his or her own position and view direction. This is the result of learning in which sequences of visual signals are given repeatedly. So far there have been many studies about self-organization mechanism by which from visual inputs position and view direction are identified. It is known, in particular, that neurons called place cells exist in the hippocampus of the rat, and respond selectively when the rat comes to a particular position [1]. Furthermore, the environment map is considered useful for guidance and action generation of mobile robots [2 4], and various methods have been proposed to extract the structure of the external world from the sensor information. Consider the situation in which a robot is moving around in a room. The input from a vision sensor, such as an image, is determined depending on the position and direction of the robot. In order to acquire the position and direction of the robot from only the vision information, it is necessary to separate two kinds of information the position in the room (2D information) and the direction (1D information) from the higher-dimensional information, that is, the image. The self-organizing map (SOM) of Kohonen can be used to summarize such higher-dimensional information and to extract important variables [5 7]. We have shown recently that by carefully utilizing an SOM, the position and direction information of the robot can be separated [8]. The method differs from self-organization using multiple maps, which we have been investigating [9, 10], but is a method Wiley Periodicals, Inc.

2 close to ICA (independent component analysis) using SOM, which was proposed by Pajunen (1996) [11]. It should be noted that SOM is not suited for learning a signal space with a topology which is different from the topology of the SOM, or a signal space with a large shape difference, even though the dimension and topology are the same. Thus, it is difficult for the model in Ref. 8 to acquire by self-organization the structure of a room with obstacles, or a room which is not square. If the shape of the room is known beforehand, it suffices to match the SOM structure beforehand, but then it does not make sense to use self-organization. Neural gas [12] is a self-organizing model which can deal with input signals with various topological structures. Considering the position and direction information of a robot located in a room, however, although the shape of the room (the movable range of the robot) is unknown, it is clear that the direction information has toroidal topology. Consequently, this paper proposes a model in which the learning algorithms of two self-organizing models are combined. That is, a SOM of toroidal structure is used in self-organization of the robot direction, and a neural gas is used in self-organization of the positional information in order to deal with the unknown room shape. The learning of the position and direction of the robot is a problem which is interesting in the sense that the structure of the signal space is neither completely unknown nor completely known. It is the direct product of the known signal space (direction) and the unknown signal space (position). In such a problem, SOM should be applied to the known part and the neural gas should be applied to the unknown part. In other words, a model combining the two as the direct product, as is proposed in this paper, is expected to be useful. Below, the reference vector of the i-th unit is written as m i. The dimension of m i is the same as that of the input signal x given to the SOM. The learning algorithm is as follows. (SOM1) Initial values of the reference vectors m i, i = 1,..., n are set, in which n is the number of units in the SOM. (SOM2) An input vector x is generated. (SOM3) The unit which has the reference vector closest to the input (called the winner) is determined: (SOM4) Learning of the reference vectors m i, i = 1,..., n is performed by the following formulas: Here, r i is the position of the i-th unit in the array (neural field), and α is a positive constant representing the rate of learning. (SOM5) The procedure goes back to SOM2 and is repeated. Learning occurs near the winner. It is called neighborhood learning, and h ci is called the neighborhood function. The neighborhood function in the above formula is a Gaussian function, and the positive constant σ determines the spread of the neighborhood. The norm in Eq. (1) is the Euclidean distance in the input signal space, and the norm in Eq. (3) is the Euclidean distance in the unit array. (1) (2) (3) 2.2. Learning algorithm of neural gas 2. Self-Organizing Algorithms The learning algorithms of SOM and neural gas are briefly summarized as follows Learning algorithm of Kohonen s SOM SOM is a self-organizing model proposed by Kohonen [5 7]. SOM consisting of a two-dimensional (2D) array of units is often used; it can be a mathematical model that describes the formation of the functional map in the sensory field of the brain cortex in terms of a certain learning algorithm [13, 14]. The point is to summarize the higher-dimensional data and to extract only the two most important variables. A 2D array is not essential: a 1D or 3D array can be used whenever necessary. Each unit retains an n-dimensional vector called the reference vector. The SOM performs neighborhood learning. In contrast, the neural gas performs ranking learning. The neural gas has the feature that it can handle signal spaces of various shapes, since the neighborhood relation among the units is defined after learning. The learning algorithm is as follows. (GAS1) Initial values for reference vectors m i, i = 1,..., n are set. (GAS2) An input signal x is generated. (GAS3) Euclidean distance between the reference vector m i of each unit to the input signal x is determined. The units are ordered in increasing order of distance. Below it is assumed that the i-th unit is ranked as the s i -th. (GAS4) Learning of the reference vectors m i, i = 1,..., n is performed by the following formula: (4) 2

3 (GAS5) The procedure goes back to GAS2 and is repeated. The amount of learning is larger for units with reference vectors closer to the input signal x. g(s) is a function that determines the amount of learning. It may be set as follows, for example: (5) model, the dimension is too high, and it takes a long time for learning. We divide the inputs from 60 sensors into 15 4 blocks, and the signal is compressed to a 16-dimensional vector x = (x 1,..., x 16 ) to be used as input signal to the model: where R is a real number such that 0 R < External Environment and Input Signal to Robot Consider the situation in which a robot is placed in a square room (Fig. 1). The position of the robot is described as (u, v), 0 u, v 1. Let the surrounding walls be painted with different colors, such as blue, black, white, and red. It is assumed that the robot has 60 vision sensors ξ i, i = 1,..., 60, equally dividing the whole environment around the robot into 60 sectors. Each sensor can ascertain the color of the wall in front of it. The information from each sensor is represented as follows, depending on the input signal (blue, black, white and red): Each input is represented as a 240-dimensional vector of binary components. If it is used directly as an input to the (6) x 1, for example, represents the number of sensors among the first 15 that are facing the blue wall. 4. Separation of Direction and Position Information By receiving the 16-dimensional input signal x, this study is intended to use a self-organizing model to separate the position (2D) and direction θ (1D) information simultaneously. Since the input signal has essentially a 3D structure, modeling by the 3D SOM and also modeling by the SOM and neural gas are performed. In either case, the neural field has the 3D structure shown in Fig. 2. In this study, 49 units are placed on the unit plane formed by the r 1 and r 2 axes, and 20 units are placed in the unit column formed by the r 3 axis, making 980 units in all. In the case of the 3D SOM, 7 7 units are placed to form a grid on each unit plane. In the case of the product model composed of a neural gas and SOM, 49 units exist Fig. 1. A robot with sensors in a room (top view). Fig. 2. Three-dimensional structure of the model with periodic boundary condition. 3

4 independently in each unit plane. In order to embed the direction information of the robot, a toroidal configuration is used in the r 3 direction. There is a greater change in the input signal x when the robot changes its direction than when it changes its position. For this reason, the length in the r 3 direction is made longer than in the r 1 and r 2 directions. It is assumed that the robot moves at random from corner to corner in the room during learning, but the position and direction of the robot never change at the same time. In other words, there is a constraint that the position or the direction, but not both, changes from the present state to the next state. The learning algorithm for each model is summarized below Self-organization by 3D-SOM Let the position of the i-th unit in the neural field be (r 1i, r 2i, r 3i ). The learning algorithm in this case differs from the general learning algorithm described in Section 2 in providing the time-series input signal. (RN1) The reference vectors m i, i = 1,..., n are initialized. (RN2) The position and direction of the robot are set at random, and a 16-dimensional input signal x is generated. (RN3) The winner c which has the reference vector closest to x is determined. Then each reference vector m i, i = 1,..., n performs neighborhood learning as follows: (7) where r i = (r 1i, r 2i, r 3i ) is the position of the i-th unit in the 3D array (neural field). Let the position of the winner be r c = (r 1c, r 2c, r 3c ). (RN4) The position or the direction of the robot is changed randomly. The procedure goes to RN5 in the former case and to RN6 in the latter case. The input signal x is updated. (RN5) The new winner is determined from the unit plane r 3 = r 3c to which the winner c belongs, and neighborhood learning is performed. The procedure goes to RN7. (RN6) The new winner is determined from the unit column (r 1, r 2 ) = (r c1, r c2 ) to which the winner c belongs, and neighborhood learning is performed. (RN7) The procedure goes back to RN4. Learning is repeated until a stable state is obtained Self-organization by neural gas and SOM (SG1) The reference vectors m i, i = 1,..., n are initialized. (8) (SG2) The input signal x (the position and direction of the robot) is set at random. (SG3) The winner c with the reference vector closest to the input x is determined, and the following learning is performed, in which ranking learning and neighborhood learning are combined: Here r 3i is the position of the i-th unit on the r 3 axis. R is a real number such that 0 R < 1. s is the ranking in the unit plane to which the i-th unit belongs. (SG4) The position or the direction of the robot is changed randomly. The procedure goes to SG5 in the former case and to SG6 in the latter case. The input signal x is updated. (SG5) The new winner is chosen from the unit plane r 3 = r 3c to which the winner c belongs, and learning by Eqs. (9) and (10) is performed. The procedure goes to SG7. (SG6) The new winner is chosen from the unit column (r 1, r 2 ) = (r c1, r c2 ) to which the winner c belongs, and learning by Eqs. (9) and (10) is performed. (SG7) The procedure goes back to SG4. Learning is repeated until a stable state is obtained Selectivity of reference vector The input signal x has 16 dimensions, but it is a function of the position (u, v) and the direction θ, and thus x has essentially 3D structure. By examining the response of the system to various input signals, it can be ascertained whether the reference vectors of the units are organized so that the positional (2D) and directional θ (1D) information can be separated. The position selectivity of the i-th unit is studied as follows. The inputs are given to the model in accordance with the probability distribution of the input signal x. In the case of the SOM, the result is weighted by h ci obtained at that trial (note that h ci is a function of x). In the case of GAS, g(s i ) is used as the weight. The selectivity of the i-th unit for the position in the room is defined as follows: Here x = x (u, v, θ). In practice, the value is determined with appropriate steps for u, v, and θ. When a large number of input signals x are given, the adjacency between units is defined on the basis of the grid in the SOM. In the GAS, any pair of units which is ranked as the first and the second at least once are defined as adjacent to each other. (9) (10) (11) 4

5 Fig. 3. Self-organizing processes (3D-SOM) of preferred positions of units in the 12th unit layer. 5. Computer Simulation 5.1. Square room without an obstacle Self-organization by 3D SOM The robot is moved at random in a room without any obstacle. Figures 3 to 5 show the progress of self-organization. The learning coefficient α is set as α t = 0.8(1.0 t/t) , so that it decreases with time. In the expression, t is the time during learning, and T is the total learning time. In Fig. 3, for 49 units in the unit plane (r 3 = 12), the selectivity of each reference vector in the (u, v) plane, on which the robot is movable, is shown by a point. The results for units in the adjacency relation are connected by segments. The outer frame indicates the movable area of the robot. In practice, the range of u, v(0 u, v 1) is divided in steps of 0.05, for both u and v, and the range of θ ( π θ < π) is divided in steps of π/18. Then, the input signals are formed with a uniform distribution and the selectivity of the units is determined. The left, center, and right of Fig. 3 show the situations for t = 10, 10000, and , respectively. It is evident that the map is gradually constructed with the progress of learning. For the unit planes other than the 12th, almost the same results as in the right figure of Fig. 3 are obtained. In order to show how similar positional selectivity is formed in different layers, Fig. 4 shows the position selectivity of all 980 units. The position selectivity of each unit is shown by a point, and the positional selectivities of units which are adjacent in each unit column are connected by lines. It can be seen that when learning is completed (right figure of Fig. 4), units in each unit column have position selectivity at almost the same position. Along the r 3 axis, on the other hand, the selectivity regarding the direction in which the robot is facing is self-organized. Figure 5 shows the result, which is a plot of the directional selectivities of all units. At the initial stage of learning, 49 units belonging to each unit plane (r 3 = constant) exhibit diversified selectivity. It can be seen, however, that the units exhibit selectivity to a particular direction with the progress of learning. All 49 units belonging to the unit plane r 3 = 5, for example, have Fig. 4. Self-organizing processes of preferred positions of all units. Preferred positions of nearby units in the same unit column are linked. 5

6 Fig. 5. Self-organizing processes of preferred directions of all units. selectivity to the 180 direction (in Fig. 5, the results for 49 units are plotted with almost complete overlap). Thus, it can be seen that each unit has its own selectivity near the state of the robot in a particular position and facing in a particular direction. It is also seen that directional invariance for the robot is realized in the unit plane, and that positional invariance is acquired in the unit column by self-organization Self-organization by a product model of SOM and neural gas The learning coefficient α is set as α t = 0.3(1.0 t/t) The parameter σ t is set as σ t = 1.414(1.0 t/t) + 0.1, so that it decreases with time so long as σ t > 0.5 is satisfied. Otherwise, we set σ t = 0.5. The parameter R is set as follows, as a function of σ t : (12) As in the case of 3D SOM, each unit exhibits a selectivity for both the position and the direction after learning. Figures 6 to 8 show the situation Room with obstacles We studied whether or not the model presented in the previous section can perform self-organization in a room with obstacles. It is assumed that although the robot cannot enter the area occupied by obstacles, the height of obstacles is lower than the vision sensor, so that the obstacle itself is not detected by the sensor. Figure 9 shows the results for three cases with an obstacle. The robot cannot enter the rectangular area drawn in the room. In 3D SOM, it can be the case that the reference vector is located in the area of the signal space for which no input can be given due to the obstacle (the unit in Fig. 9, for example, with positional selectivity at the center of the room). Even though the system tries to cover the whole room in spite of the obstacle, it can happen that the map is folded in the area in which the shape of the room changes, due to the effect of neighborhood learning (lower center panel of Fig. 9). In the case of the product model of SOM and neural gas, on the other hand, self-organization is performed so that the whole area except the areas of obstacles is completely covered (Fig. 10). Fig. 6. Self-organizing processes of preferred positions of units in the 7th unit layer by a product model of SOM and neural gas. 6

7 Fig. 7. Self-organizing processes of preferred positions of all units. Preferred positions of nearby units in the same unit column are linked. 6. Discussion In all of the results, self-organization of units is realized so that the position information of the robot is represented in the unit plane, and the directional information is represented in the unit column. This is mostly due to the effect of the constraints on the selection of the winner in the next step, as is shown in RN5 and RN6, as well as SG5 and SG6 in the learning algorithms. Self-organization is difficult if this constraint is removed. That is, the robot in self-organization utilizes information as to whether it has rotated or shifted position, in addition to vision information. The following observations are made on the selectivity of the units. For the direction of the robot, the self-organization is such that 360 is uniformly covered in the unit column. It can be seen, on the other hand, that the representation of the position information in the unit plane is inclined by approximately 45 (right panel of Fig. 3, which is the case of 3D SOM in the room without obstacles). Because of this property, the receptive field of the unit for the position is slightly spread near the corners of the room, which degrades the accuracy of position detection. This is a property which does not depend on the initial value of the reference vector. A similar result is obtained when the product model of a neural gas and SOM is used. This property seems to be due to the fact that the input information from the vision sensor changes less near the corners of the room than in other areas. The fifth color was placed at a corner of the room, and an experiment was performed. It was verified that the reference vector moved toward the corner (Fig. 11). In addition, when there is a fine texture on the walls, not just at the corners, the reference vector tends to concentrate in that area. However, detailed analysis of this property is left for the future. In the case of 3D SOM, it can happen that the map is peculiarly deformed and a node is formed in an area in which the robot cannot enter, except for a room with a simple structure. In that case, if the robot is guided on the basis of the constructed map, it may collide with the obstacle. In the product model of SOM and neural gas, self-organization is performed so that only the area without obstacles is completely covered. When the structure of the input signal space is known beforehand, the SOM can be used to form the position selectivity. However, when the input signal space has higher or unknown dimension or can have a complex shape, neural gas has advantages over SOM. In Fig. 8. Self-organizing processes of preferred directions of all units. 7

8 Fig. 9. Self-organizing processes (3D-SOM) in the room with obstacles. this study, the product model of a neural gas and SOM is used effectively and the information is successfully separated. This is based on the idea that the signal is a combination of two kinds of information, one of which is the direction, which is continuous toroidal information, and the other of which is the position, which is 2D but whose shape as a whole is unknown. In Section 5, the success of the computer simulation is the result of 1,000,000 learning trials. This is a large number, but actually it is not necessary for the robot to move around 1,000,000 times to acquire the experience. If the first input 2000 signals are memorized, for example, the robot can continue iterative learning in the inside. A relatively long time is required for the learning itself, mostly due to the ranking computation in the neural gas. For a model with the dimension described in this paper, the map was constructed in 15 minutes by 1,000,000 learning trials (Pentium 4 processor 2.80 GHz). That is, a minimum number of samples is required in constructing the map for the robot to experience, but 1,000,000 learning trials is not unrealistic for the robot. In this paper, we considered a situation in which the robot moves in a room surrounded by walls painted in four colors. When real image data are used, learning can be achieved by this model if a signal with somewhat smaller dimensions is input into the model. In extracting rough 8

9 Fig. 10. Self-organizing processes (a product model of SOM and neural gas) in the room with obstacles. Fig. 11. A self-organized map when an additional color is put on the corner. information such as its own direction, highly precise input is not required, in contrast to the case of object recognition, since relatively low frequency components in the image can play a role (see, for example, Section 3.1 in Ref. 15). Consequently, the low-dimensional input might be sufficient for applications in real environments. The purpose of this study is to explore the possibility of applying the neural network model to information processing. It is not intended to give a faithful representation from the biological viewpoint. Since no cell is provided to output the separated information, the information is not separated as a neural network model. As the information processing algorithm, however, the separated information 9

10 appears as the two indices of the winner cells, which can be utilized. The same idea is used in a study in which SOM is applied to ICA [11]. After learning is completed, position and direction of the robot can be identified by receiving the visual input and calculating the winner in the 3D neural field. The navigation by the robot [16], for example, in the search for the shortest route from one location to another, is reduced to determination of the shortest route in the graph specified by the neural gas [17]. In the model used in this approach, it is expected that the reference vectors retained by units will be distributed uniformly in the room. Therefore, the length of each edge in the graph can be approximately set as 1. When the shortest route is determined, it is possible to determine which direction should be taken in progressing from one node to the next node by comparing the reference vectors of the two nodes. A more detailed discussion of this point is left for the future. 7. Conclusions This paper has proposed a method of separating the position and direction information of a robot by combining a self-organizing map (SOM) and a neural gas. The effectiveness of the method was demonstrated by computer simulation. Acknowledgment. This work was partly supported by a Grant-in-Aid ( , ) from the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT). REFERENCES 1. Knierim JJ, Kudrimoti HS, McNaughton BL. Place cells, head direction cells, and the learning of landmark stability. J Neurosci 1995;15: Asada M. Vision for action acquisition of robot. J IEEJ 1995;115: Shimazu T, Ishikawa M. Navigation of a mobile robot by self-organization. Tech Rep IEICE 2000;NC Asoh H. Learning of machine and learning of human: From the statistical viewpoint. J Soc Artif Intell 2003;18: Kohonen T. Self-organized formation of topology correct feature maps. Biol Cybern 1982;43: Kohonen T. Self-organizing maps, 3rd ed. Springer- Verlag; Kurata K. Basis of SOM. In Hodaka H, Fujimura K, Yamakawa K, editors. Examples of self-organizing map applications. Kaibundo Co.; p Kurata K, Oshiro N. Information separation of position and direction of a robot by self-organizing map. Proc 8th International Symposium on Artificial Life and Robotics, p , Shirakura J, Kurata K. Locking of self-organizing multiple maps by weak similarity of input information. Trans IEICE 1998;J81-D-II: Mitsutake M, Norita H, Wada K, Kurata K. Separate extraction of two kinds of information by multiple map. Brain and Neural Networks 1999;6: Pajunen P. Nonlinear independent component analysis by self-organizing maps. Artificial Neural Networks (ICANN 96), p Martinetz T, Berkovich S, Schulten K. Neural-gas network for vector quantization and its application to time-series prediction. IEEE Trans Neural Networks 1993;4: Kohonen T. Physiological interpretation of the selforganizing map algorithm. Neural Networks 1993;6: Kurata K. Neural fields and topographic mapping. Imago 1994;5: Madokoro H, Sato K, Ishii M. Acquisition of world image and self-localization using sequential view images. Trans IEICE 2000;J83-D-II: Tanaka T, Nishida K, Kurita T. Navigation of mobile robot using neural-gas and reinforcement learning. Tech Rep IEICE 2002;NC Dijkstra EW. A note on two problems in connexion with graphs. Numerische Mathematik 1959;1:

11 AUTHORS (from left to right) Akira Date (member) received his B.E. and M.E. degrees in communication and systems engineering from the University of Electro-Communications in 1991 and 1993 and Ph.D. degree in computer science from Tokyo University of Agriculture and Technology in He spent a postdoctoral period at the Division of Applied Mathematics, Brown University, Providence, RI ( ), and served as a staff scientist at the RIKEN Brain Science Institute ( ) and a researcher at the Communications Research Laboratory ( ). Since 2004 he has been a member of the Department of Computer Science and Systems Engineering at the University of Miyazaki, where he is currently an associate professor. His research interests include representation and computation in biological and mechanical systems. He is a member of the Japan Neuroscience Society, the Japanese Neural Network Society, and the Society for Neuroscience. Koji Kurata (member) received his B.S. degree from the Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo, in 1981 and completed the M.E. program in In 1984, before completion of the doctoral program, he was appointed a research associate in the Department of Mathematical Engineering and Instrumentation Physics. He became a lecturer in the Department of Biological Engineering, Osaka University, in He was an exchange researcher at Ruhr University, Germany, for a half year in He served as a vice-president of the Japanese Neural Network Society in He has been an associate professor in the Department of Mechanical Systems, University of the Ryukyus, since He is engaged in research on neural network models, pattern formation, and mathematical morphology. He received a Research Award from the Japanese Neural Network Society in