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A New Learning Algorithm to Control an Autonomous Biped Robot

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Abstract:-In this paper an adaptive neural-fuzzy walking control of an autonomous biped robot is proposed. This control system uses a feed forward neural network based on nonlinear regression. The general regression neural network is used to
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    A New Learning Algorithm to Control an Autonomous Biped Robot JOÃO PAULO FERREIRA Institute Superior of Engineering Polytechnic Institute of Coimbra Institute of Systems and Robotics - Coimbra Quinta da Nora, 3040-228 Coimbra PORTUGAL TITO G. AMARAL V. FERNÃO PIRES Superior School of Technology Polytechnic Institute of Setúbal Institute of Systems and Robotics - Coimbra R. Vale Chaves, Estefanilha, 2914-508 Setúbal PORTUGAL MANUEL CRISÓSTOMO A. PAULO COIMBRA Institute of Systems and Robotics University of Coimbra Polo II, 3030-290 Coimbra PORTUGAL  Abstract:  - In this paper an adaptive neural-fuzzy walking control of an autonomous biped robot is proposed. This control system uses a feed forward neural network based on nonlinear regression. The general regression neural network is used to construct the base of an adaptive neuro-fuzzy system. The membership functions used in the antecedent part of the fuzzy system are asymmetric and with varying shapes which is less common in the fuzzy modelling literature. The neural network uses a new iterative grid partition method for the initial structure identification of the controller parameters.   The proposed new approach is tested with a biped robot developed in the Institute of Systems and Robotics of the University of Coimbra, Portugal. Comparison results are done between the proposed method and the ANFIS tool provided in the fuzzy MATLAB toolbox. This robot uses an inverted pendulum for his balance. Therefore, the control system uses the balance of the gaits for the correction of the lateral and longitudinal angle of the pendulum. This robot can walk in horizontal, sloping planes, and to go up and down stairs. Mathematical models of the static and dynamic walking of the biped robot are also presented. With these models it is possible to simulate the movement of the robot and test the control algorithms. The effectiveness of the proposed control system is demonstrated by simulation and experimental tests. Key-Words:  - Neuro-fuzzy Systems, Biped robot, Control, Inference, Learning, Zero Moment Point. 1 Introduction In the last years many biped walks have been presented and studied. These robots with legs are a very attractive locomotion style since they can circulate in hostile lands, characterized by obstacles, stairs and inclined  planes. To control the biped robot several mathematical models have been proposed [1]. Another approach to control the biped robot was the use of the gait running over the data of the human kinematics [2]. A locomotion pattern generated by a passive interaction of the gravity and inertia in a descending inclined ramp was also proposed [3]. With the objective of extending the application of the method “Energy-Optimal Walking” to the ascending locomotion in ramps, Rostomi [4], and Roussel [5], have proposed generation methods of gaits  by minimizing the energy consumption. However, the investigation developed until this moment does not    consider the robot stability. Since biped robot is easily knocked down, it is necessary to take the stability into account for the determination of the gait. Zheng [6] has  proposed a method of synthesis of the gait using the static stability. Chevallerreau [7] lifted the discussion concerning the dynamic stability through the analysis of the reaction force between the base of the foot and the soil; unhappily the defined reference trajectory doesn't assure that the restriction of stability is satisfied. To assure the dynamic stability in a biped robot, standard methods to gait synthesis based on zero moment point (ZMP) have been proposed [8] [9] [10]. To improve the walking robustness we propose an adaptive neural-fuzzy walking control for the autonomous biped robot. There is an evident complementary between neural and fuzzy theory. The fuzzy knowledge-based models, proposed by Zadeh [11], consist on the idea of acquiring and customizing a set of rules from an expert and designing an inference mechanism, which operates on such rules without employing precise quantitative analyses. However, these models have been shown to fail in real-world  problems, since it is very difficult for an expert to  provide a sufficient set of rules that control the whole  process. Neural networks can be an adaptive approach that considers consistent parameterised models whose  parameters are adapted by minimizing an objective function over a sample of training data. In this  perspective, the proposed control system combines the fuzzy expert knowledge and neural network into an adaptive neuro-fuzzy inference system. The initial parameter values of the network are obtained using a new iterative grid partition method. Then, the parameters of the network are adapted using a supervised learning algorithm derived from the neural network theory. The learning process is not knowledge  based, but data driven defining the membership functions (MF) and the fuzzy rules of the fuzzy inference system. Finally, the fuzzy system works without the neural network. 2 Robot System Description The mechanical structure of the biped robot is formed  by joints of the hip, knee, and ankle, for each leg. Another joint controlling an inverted pendulum is used in order to balance the structure. Seven servomotors are used in the robot. The structure is made of acrylic and aluminum, with 1.7 Kg weight, and 50 cm height. In figure 1 it is presented the biped robot that was developed. The robot was designed to walk in horizontal and inclined planes, and to go up and down stairways, having a maximum step speed of approximately 1 cm/s. Fig. 1 - Implemented robot. The project possesses innovative characteristics, namely at the level of autonomy of the robot, which has a radio link to the controller PC. 3 System Modelling 3.1 Kinematics The Denavit-Hartenberg method was used to describe the kinematics characteristics of the autonomous biped robot. The method consists in assign a coordinate frame to each link of the robotic chain. 3.2 Static System The static and dynamic models of the biped robot are  based on the calculation of ZMP [12]. The static system has in consideration the weight of the several bodies that constitute the robot, despising completely the translation and rotation movements. In order to develop the static and dynamic models of the biped robot, its links where approximated to  parallelepipeds as shown in figure 2. The system is composed of 8 rigid bodies. Fig. 2 - Biped robot model. The ZMP of the system is obtained by equations (1).    ∑∑ == = 7070 ... iiiii zmp gm xgm x  and ∑∑ == = 7070 ... iiiii zmp gm ygm y  (1) 3.3 Dynamics System Dynamic modelling of the biped robot is a very complex task. Therefore, some simplification concepts are made in order to simplify the mathematical point of view of the robot behaviour in its locomotion. Three concepts that allow to simplify and to synthesize the system modeling will be presented. 3.3.1 Mass Center The movement of the biped robot is very complex,  because it is composed by several links with different geometry and specific mass. Therefore, the mass center is generally one of the used concepts. This concept considers that several parts of the body are joined in  just one. Thus, the mass center (C.M.) is defined by: ∑∑ ⋅= iimiir imCM  R !!  (2) where r  i  is the position of the body i in relation to a referential and m i  is the mass of the body i. 3.3.2 Stability Margin Zero Moment Point is defined as the point in the soil where the sum of every moment of force is null. In figure 3, the minimum distance between ZMP and the  border of the stable area is denominated by margin of stability and it can be considered as an indicator of the quality of the stability. Fig. 3 - Definition of stability margin. In case ZMP is inside the frame of the contact  polygon (stable area) between the foot and the soil, it can be affirmed that the biped robot is stable. If the distance between ZMP and the border of the stable area of the polygon is maximum, that is to say when the coordinates of ZMP are closed to the ones of the center of the stable area, it can be affirmed that the  biped robot presents a high stability. 3.3.3 Inertia In the dynamics of rigid bodies, case rotation movement exists, the moment of inertia should be taken into account. It is necessary to characterize the distribution of mass of a rigid  body in a complete way, if the rotation about an arbitrary axis is considered, being that characterization given by the inertia matrix, which is referred to a coordinate system {A} −−−−−−=  zz zy zx yz yy yx xz xy xx A  I  I  I  I  I  I  I  I  I  I   (3) where I xx , I yy , I zz  are the main moments of inertia and I xy , I xz , I yx , I yz , I zx , I zy  are the mass of the inertia  products. Since the inertia products are null and the rotations of the robot's joints around the x and y axes, do not exist, there’s no need to calculate I z . The obtained moments of inertia for the developed biped robot are shown in the table 1. For a robot with four or more legs it is possible to consider the static stability that uses the center of gravity, but for a biped robot it is necessary to have the dynamic stability into account; the calculation of ZMP allows counting that aspect. Table 1 - Physical Characteristics of the Biped Robot. Biped I x ( ×××× 10 -4  kgm 2 ) I y ( ×××× 10 -4  kgm 2 ) Foot 0,0813 0,1413 Shank 0,8188 0,7721 Thigh 0,8188 0,7721 Body 1,4167 0,6467 Pendulum 5,0725 5,1050 In opposite to the static system, the dynamic system has into account the weight, and the translation and rotation movements of the links. In this case, the ZMP of the system, is obtained by equations (4) and (5). ( )( ) ∑=−∑=−∑=−∑=−= 70707070 igi zimiiyiy I ii zi ximii xgi zim zmp x """"""  α   (4) ( )( ) ∑=−∑=+∑=−∑=−= 70707070 igi zimiixix I ii zi yimii ygi zim zmp y """"""  α   (5) 4 Gaits  In this work 4 different gaits were conceived with the goal to approach the gaits that are considered to be the most frequent in the human being locomotion. These gaits are moving forwards and backwards in horizontal  planes and in inclined planes. To project the gaits used in this work, a trajectory for the foot in movement was defined, considering the following conditions:    • The robots walking must be like a human; • At the beginning and at the end of the trajectory, the height of the foot is null, relative to the ground; • At 50% of the trajectory the height of the foot is maximum. With these presuppositions the following normalized equation for the trajectory of the foot was considered: ) 1,0 4 2 ≤≤−⋅=  x y x x y  (6) To obtain the final trajectory it is needed to know the height (A) and the length (C) of the trajectory, resulting 0 0 24 C  X  AY C  X C  X  AY  ≤≤≤≤−⋅⋅=              (7) where  y AY   ⋅=  and  xC  X   ⋅= . To walk in horizontal planes the trajectory given by the equation (7) is applied. To walk in inclined planes the trajectory is determined through the rotation of the  positions given by the equation (7), in agreement with the inclination angle. 5 Adaptive Neuro-Fuzzy Controller To control the biped motion, an adaptive neuro-fuzzy inference controller is proposed. In the proposed method, the neuro-fuzzy system learns with the training data set derived from the expert knowledge of the  biped motion control. After the adaptive neuro-fuzzy controller is trained, the controller can be used to control the biped motion as shown in figure 4 a). In the training phase, the structure of the vector of each data element is {fix ankle, fix knee, fix hip, ankle, knee, hip, pendulum, δ  a  , δ  b  , δ  c  , δ  d   } , where d cba and   δ δ δ δ  ,,  represent the motions reference step length, the floor type, the step rate, and inclination degree, respectively. To control the biped robot in its different motions, one adaptive neuro-fuzzy controller is used with four inputs and seven outputs. The output values are the angles of the joints of the walking robot. 5.1 Neuro-Fuzzy Architecture Figure 4 b) shows the four-layered architecture for the adaptive neuro-fuzzy controller. There are m  nodes in layer 1 representing the inputs in the fuzzy system and n  nodes in layer 2 corresponding to each rule. There are two nodes in layer 3 and only one node in layer 4. As it happens in the generalised regression neural network configuration, the number of nodes in the summation layer is equal to the nodes of the output layer plus one node [13].  Layer 1:  This layer, called the input layer  , consists in the input data derived from the m  input variables ( ) m j x  j ...,,1 = .  Layer 2:  In this layer, called the  pattern layer  , the number of nodes it is the same as the number of rules in the fuzzy inference system. Each node is a special type of radial basis function processor and represents a rule where the output of each basis function unit  ji  xT   is given by the following equation: ( ) ( )  ji jm j ji  x LX  xT  )(1 = ∏= . (8) The firing strength of th i  rule is obtained by taking the multiplicative T-norm in the antecedent part of the rule. ( )  ji j  x LX  )(  is the membership function of the linguistic label of the fuzzy set )( i j  A . For the multiplicative T-norm in the antecedent part of the rule, a generalized Gaussian membership function was chosen according to equation (9) where balc ijij ,,,  are the antecedent parameters. For each membership function the central position ( ij c ), the shape of the function ( ij l ), and bandwidth of the function ( ba , ) are tuned. a)  b)   Fig. 4 - Proposed adaptive neuro-fuzzy controller. a)  block diagram of the controller; b) neuro-fuzzy structure.   Control InputsOutputsRefª. input data fuzzy sets, fuzzy rules output data Y(X) X 1 X m f (x) 1 T (x) 1   DST (x)nf (x)n InputLayerPatternLayerSummationLayerOutputLayer    Since the values of these parameters change, the  bell-shaped functions also change, thus exhibiting various forms of membership functions. The antecedent fuzzy membership functions can be both symmetric and asymmetric shapes. This type of membership functions is less common in the literature of fuzzy modelling where the symmetry is usually assumed. In the consequent part the first order Sugeno rule was used, described by the following equation mimii ji  xb xbb x f   +++= ... 110  (10) where )(  ji  x f   defines a locally valid model on the support of the Cartesian product of fuzzy sets constituting the antecedent part [14].  Layer 3:  This layer, called the summation layer  , has two different types of processing units: the summation units and the division unit. The function of the summation units is essentially the same as the function of the output units in the radial basis function network (RBFN). The division unit only sums the weighted activations of the pattern units without using any activation function. ( ) ≥〈〈       +       −−       +≤〈       −−−≤〈       +       −−−       +≤〈       −−≤= b xb xbcbc xblbc xcbcc xlc xcaac xclca xaaca xla x x LX   j jijlij jijij jijlijij jijij jijlij jijijij jlij jij j ji j ijijijij ,02,*2,*12,*12,*,0 )(  (9)  Layer 4:  This layer, called the output layer  , consists in a simple division of the signal coming from the summation unit by the signal coming from the division unit. For a neuro-fuzzy controller with multiple outputs, each output value is obtained by equation (11). ( ) ( ) ∑∑ == = ni jini jk i jik   xT  x f  xT   xu 11 )()( . (11) where k   ( ) Pk  ,...,1 =  represents the current output and P  is the number of controller outputs. Thus an adaptive network was constructed that is functionally equivalent to a type-3 fuzzy inference system (Takagi and Sugeno fuzzy if-then rules). 5.2 Initial Structure Identification In a fuzzy inference system, the consequent part describes the behaviour within a given region via various consequents. The consequent part can be an output M.F. (Mamdani and Tsukamoto fuzzy models), a constant or a linear equation (zero-order or first-order Sugeno models, respectively). The antecedents of fuzzy rules separate the input-space into a number of local fuzzy regions. In a conventional fuzzy inference system, an expert who is familiar with the system to be controlled decides the number of rules in the antecedent part. In the proposed method, no expert is required. The number of membership functions assigned to each input variable is chosen using a new iterative grid partition method to partitioning the universe of discourses of the antecedents and form the antecedent part of fuzzy rules. The iterative method used for partitioning the universe of discourse is based on a changing grid  partition using only one iteration for each grid partition shape. For each iteration the objective function that we want to minimize is computed. Then, changing the number of membership functions assigned to the input variables, the same process is repeated using different grid partitions of the input space (see figure 5 a)). For each partition, the membership functions will take different shapes. An example is showed in figure 5 a), where there are six membership functions assigned to an input variable. In figures (i) to (iii) and in figures (iv) to (vi), the partition of the input space is the same  but the membership functions change theirs shapes. The grid partition of the input space changes from figures (i) to (iii), to figures (iv) to (vi). The minimum value of the objective function defines the number of membership functions used for each input and also the correct values of the antecedent parameters. After the computation of the antecedent part is completed, a good approximation of the desired performance index without the training stage can be obtained. The non-iterative grid partition method has problems when a moderately large number of inputs are presented. Using this iterative method, for a given number of inputs, the number of the corresponding membership functions and the parameter values are obtained. This iterative grid partition method can permit to substantially reduce the training stage, depending on the desired precision for the system. The main difference between this method and the classical fuzzy grid partition technique is that the last method divides the input space in equal parts and in the  proposed method the input space is divided in different  parts. The shape of the membership functions can also  be different from the triangular shape mostly used in the standard fuzzy partition technique. In the proposed method, the correct partition is obtained minimizing the objective function. The operating range for each input variable is obtained applying the maximum and
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