Humanoid Robotics Modeling by Dynamic Fuzzy Neural Network

Transcription

1 Proceedings of International Joint Conference on Neural Networks, Orlando, Florida, USA, August 1-17, 7 umanoid Robotics Modeling by Dynamic Fuzzy Neural Network Zhe Tang, Meng Joo Er, and Geok See Ng Abstract Motion planning is an essential task for humanoid robots. owever, it is still very challenging to obtain good motion performance in humanoid motion planning, because of its high DOFs (degree of freedoms, variable mechanical structure and nonlinearity. In humanoid motion planning, the motion performance can be given only after one whole cycle motion is completed. This is a demanding condition for motion planning on either real robots or simulation platform. In this paper, a DFNN (dynamic neural fuzzy network is adopted to model humanoid robots for motion planning. The inputs of DFNN are parameters which determine motion of humanoid robots. The output is evaluation of humanoid motion performance. The DFNN after training can give evaluation of motion performance immediately once the parameters are determined. The DFNN models not only the dynamics of robots, also the motion planning method. Therefore, the DFNN stores two kind of knowledge: the mapping be-tween parameters and humanoid motion, the mapping between humanoid motion and motion performance. I. INTRODUCTION Motion planning is a basic task in humanoid research. For any kinds of motions planning (e.g. walking, running, turning, a dynamic model is necessary to give the performance of humanoid motion. There are many approaches to humanoid system modeling. These approaches can be roughly classified into two categories, mathematical model and nonmathematical model. The classical modeling methods (e.g. link-based model, inverted pendulum model are categorized into mathematical modeling approach. owever, the mathematical model could not give exact dynamics of humanoid due to its complexity. The intelligent control techniques (e.g. Neural Network, Fuzzy Logic, Genetic Algorithm have been used for humanoid model as non-mathematical approach. Zhou [1] proposed a reinforcement learning-based neuro-fuzzy scheme to synthesize trunk trajectories of the humanoid robot, while all other joint trajectories are prescribed. The proposed gait synthesis can combine the linguistic knowledge of fuzzy logic with the learning strength of NN. Juang [] presented a fuzzy neural network method to synthesize the humanoid walking gait. The method uses a fuzzy modeling neural network controller with the BTT (back-propagation through time algorithm. The fuzzy modeling neural network overcomes the uncertainty of network size in the conventional neural network learning scheme. Fukuda, et.al [3] used ZMP (zero moment point as stabilization index, and select the best configuration with recurrent neural network. Recurrent networks are trained Zhe Tang and Meng Joo Er are with Intelligent Systems Centre 5 Nanyang Drive, 7th Storey, Research Techno Plaza, BoderX Block, Nanyang Technological University Singapore TangZhe, Emjer@ntu.edu.sg; Prof Geok See Ng is with School of Computer Engineering, Nanyang Technological University Singapore AS- GSNG@ntu.edu.sg by GAs which have the self-adaptive mutation. Nagasaka et.al [] gave a GA based approach for the motion acquisition problem which treats a real robot with a complicated body and a noisy environment by taking the case of visually guided swing motion by vision-equipped two-armed bipedal robot. A new approach to humanoid motion planning is proposed in this paper, which combines both mathematical and nonmathematical model of humanoid. Firstly, an IPM (inverted pendulum model based motion planning is given. In this planning, robot stability is guaranteed given constraints of IPM. Secondly, a DFNN (dynamic neural fuzzy network is used to give humanoid motion performance evaluation. The input of DFNN is key parameters which represent a motion planning. The output is evaluation of motion performance. Because the performance of humanoid motion can be given only after one cycle motion is completed, the available training samples for DFNN is limited. The objective of DFNN is to learn the relationship between the parameters of motion planning and its performance. Only walking motion is addressed in this paper, other motions can be applied in a similar way. This paper is organized as follows: Section gives a walking planning based on IPM. By the mathematical model of IPM, the stability this planning method is achieved. But the performance of this walking planning is not given explicitly. Therefore, a DFNN is used to approximate the relationship between the walking planning and motion performance, which is described in section 3. Simulation results are presented in section. Conclusions and references are given finally. II. UMANOID WALKING PLANNING BASED ON IPM IPM has been used as dynamic model of humanoid robots by many researchers [5-7]. The main advantage of the IPM is its simplicity and closed analytic solution for the trajectories. To use IPM as a model of humanoid robot, the following ideal assumptions are made: 1 a concentrated mass of robot is resided at hip robot motions in the sagittal and frontal planes are independent 3 the COG (center of gravity position in the z-axis is constant The IPM adopted for humanoid motion is an enhanced IPM, which takes the angular momentum into account. Therefore, the ground force vector does not have to pass through the COG. Its advantage is that the IPM could generate various patterns of humanoid motion. Goswami[8] had investigated different stability criteria for humanoid robots. It is shown that ZMP and COP (center of pressure are equivalent while keeping motion stability. ZMP can be X/7/$5. 7 IEEE

2 Fig. 1. z (F x,f z ZMP=aX+b cx+d+zmp The relationship between COG, ZMP and reaction force considered as the center of ground reaction force. Under assumption 3, the level reaction force is linear to the position of COG. Therefore, according to the definition of ZMP, the ZMP is linear to COG. The relationship between ZMP, COG and reaction force is depicted in Fig.1. (F x, F z is reaction force. The position of COG is(x,, the position of ZMP is (ax + b,, and the normal vector of ground force is parallel to vector (cx + d,. The dynamics of the IPM can be given by F x : F z = ẍ : ( z + g = (cx + d : (1 Because z =, we can get X ẍ = g(cx + d/ ( A. IPM for umanoid Motion in the Sagittal Plane One step of humanoid walking consists of two phases, single support and double support phase. The dynamics equation of the IPM in the sagittal plane is given by x gs : ( z gs + g = x gs (c xs a 1 xa : x gd : ( z gd + g = x gd (c a 1 : where x gs and x gd are the trajectories of the IPM s COG during the single support phase and double support phase respectively. c xs, a xs,c and a are the parameters determining linear relationship between the ZMP and COG of IPM. Solving above equation, we get the COG motion as follows x (3 x gs (t = C xs1 coshs t xs + C xs sinhs t xs x gd (t = C 1 coss t + C sins t ( With boundary conditions x gd ( = L xs, x gs (T s = L xs, x gd ( = L, x gd (T d = L. Coefficient values are obtained, L xs + L xs + cosh( T s S xs C xs1 = L xs, C xs = sinh( T s S xs, L + L + cos( T d C 1 = L, C = S sin( T d S Where S xs = ( 1, S = c xs g (c 1, T s g a xs a is the single support time, T d is the double support time, L xs and L are parameters defining the range of IPM motion. The motion in the frontal plane can be derived similarly. B. Stability Constraints To achieve dynamic walking, ZMP is usually used as a stability criterion for humanoid robots.the control objective of the gait synthesis for biped dynamic walking can be described as P zmp = (z zmp, y zmp, S (5 where(z zmp, y zmp, is the coordinate of ZMP with respect to O-XYZ. S is the domain of the supporting area. During the single support time, the support area is the support foot area; during the double support phase, the support area is the convex area defined by the two feet. By combining the ZMP constraints (Equ.5 and the motion of IPM (Equ., the following parameter constraints are acquired L xs a xs L fs / (6 L a L fs + L s Where L fs is the length of foot print, L s is step length. The above constraints guarantee the motion stability during the single-support phase. Because the boundary conditions restrict the motion transition between different phases, motion stability during the double-support phase is achieved as long as the motion stability during the single-support phase is guaranteed. The IPM-based walking planning is summarized as follows: Firstly, robot walking parameters are identified, the foot, hip parameters and step length determine the robot support area. Secondly, the motion of IPM, which approximates the COG of a humanoid robot, is generated. To ensure stable walking, the ZMP of the humanoid must be kept in the robot support area. The third step of walking planning is to give two legs motion. Finally, the angular trajectories of the humanoid s 1 DOFs can be calculated through robot inverse kinematics constraints. These angular trajectories are feasible walking gait to control a real robot. This walking planning based on IPM can guarantee stability of walking, but the performance of walking is unclear. III. DFNN FOR UMANOID WALKING PLANNING DFNN with a hierarchical on-line self-organizing learning algorithm was developed in [9]. Many successful applications of DFNN have been published [1], [11].The architecture of the DFNN is depicted in. It consists of 5 layers. Layer 1: Input layer. Each node represents an input linguistic variable.

3 Fig.. Architecture of DFNN Layer : Each node represents a membership function [ ] µ ij = exp (x i c ij i = 1,, r j = 1,, u (7 where c ij, σ j are the center and width of the Gaussian function respectively. Layer 3: each node represents a possible IF-part for fuzzy rules. The jth rule R j is [ ] [ ] r i=1 R j = exp (x i c ij = exp (X C j j = 1,, u (8 Layer : This layer is to normalize the previous layer outputs. N j = R j u j=1 R j Layer 5: This layer is output layer. y(x = (9 u w j N j (1 j=1 For the TSK model, w j can be expressed as follows: w j = k j + k j1 x k jr x r (11 then, eq.(1 has its more compact form where Y = W Ψ (1 W = [k 1,, k u, k 11,, k u1,, k 1r,, k ur ] (13 Ψ = a 11 a 1n a u1 a un a 11 x 11 a 1n x 1n a u1 x 11 a un x 1n a 11 x r1 a 1n x rn a u1 x r1 a un x rn (1 where a jk is the normalized output from layer, j = 1,, u, k = 1,, n, n is the length of the processing data. The optimal coefficient vector W can be easily solved by the well-known linear least square (LLS method W = Y (Ψ T Ψ 1 Ψ T (15 The main advantage of DFNN lies in its capabilities of reconstruction and selecting fuzzy rules automatically. It can start learning from zero RBF units or zero neurons. DFNN is normally used for a function or system approximation. Once a sample pattern input is fed to DFNN, it may reconstruct network to store system information given from the sample data. The above walking pattern planning is determined by many parameters. The basic requirement of the planning is to ensure stability or to keep the ZMP in the support area. Because the ZMP of a humanoid is linear to the COG of IPM, the movement of IPM is limited to a certain range defined by the support area of humanoid robot. There are many possible walking patterns under this stability condition. igh level control of a humanoid may need to select a walking pattern with optimal performance. This evaluation can be given by DFNN, which is trained by sample data. Two key parameters (L xs and D s are selected to describe the walking pattern, where L xs defines the movement range of the COG of IPM, step length (D s determines the support area of a humanoid robot. The performance of humanoid motion is evaluated by two standards, namely torque consumption and ZMP error, which is described by the following equation (u Tc τ T τdt + (1 u Tc e dt min (16 where T c is walking cycle, τ is the torque vector of the robot. e is the error of the desired robot ZMP and calculated ZMP, small e value means wide stability region, u is a coefficient, which balances the weight of the two objectives. The torque in the IPM can be measured by the reaction force. In this paper, DFNN is used to approximate the relationship between the humanoid walking planning and the walking performance. It is very difficult to give an accurate mathematical model to describe the dynamics of a humanoid. ence, the performance of humanoid motion is usually

4 evaluated by observation. But the performance of humanoid walking can be given only after one walking cycle is completed. This requirement leads to limited sample patterns. By learning these limited sample patterns, the DFNN could give an approximate performance evaluation value of any feasible humanoid walking planning. IV. SIMULATION RESULTS For the sample data, 6 walking gaits are selected to train the DFNN. These 6 walking data are at different IPM movement range (L xs and different step length (D s. Under the stability and mechanical constraints, the L xs and D s should be in.(cm < L xs < 3.5(cm and 5(cm < D s < 7.5(cm. Fig.3 shows that four rules were generated in the DFNN after training 6 samples. Fig. gives the root mean square error of the training process. After the training process, the DFNN generates a smooth plane (Fig.5 to give the performance evaluation for all feasible walking planning. To show the evaluation of walking performance, Fig.6 and Fig.7 give the ZMP and reaction force trajectories of the walking pattern at L xs =3(cm and D s =5(cm. Fig.8 is the 3D simulation of walking planning. Root Mean Square Error Sample Patterns Fig.. Performance of DFNN Rule Number Sample Patterns Fig. 5. Evaluation of walking pattern given by GFNN Fig. 3. Rule generation V. CONCLUSIONS It is still an open issue for optimal humanoid walking planning, because it is almost impossible to give a mathematical model to describe the dynamics of a humanoid robot exactly. ence, intelligent techniques have found wide applications in humanoid modeling. But intelligent techniques are normally time-consuming for learning the knowledge of a system. The walking planning presented in this paper combines both mathematical modeling and non-mathematical modeling for humanoid walking planning, which is able to take advantages of the two approaches. REFERENCES Y (cm Walking Direction CoG ZMP X (cm [1] C. Zhou, Nero-fuzzy gait synthesis with reinforcement learning for a biped walking robot, Soft Computing, Springer-Verlag, Berlin- eidelberg,, pp Fig. 6. ZMP of walking planning

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