% as the dataset to train the first ANFIS network. %% Similarly, the second ANFIS network will be trained with X and Y% coordinates as input and corresponding |theta2| values as output. The% matrix |data2| contains the |x-y-theta2| dataset required to train the% second ANFIS network. Therefore |data1| will be used as the dataset to% train the second ANFIS network. %% |anfis| is the function that is used to train an ANFIS network. There are% several syntaxes to the function. If called with the following syntax,% |anfis| automatically creates a Sugeno-type FIS and trains it using the% training data passed to the function.anfis1 = anfis(data1, 7, 150, [0,0,0,0]); % train first ANFIS networkanfis2 = anfis(data2, 6, 150, [0,0,0,0]); % train second ANFIS network%%% The first parameter to |anfis| is the training data, the second parameter% is the number of membership functions used to characterize each input and% output, the third parameter is the number of training <#23 epochs> and% the last parameter is the options to display progress during training.% The values for number of epochs and the number of membership functions% have been arrived at after a fair amount of experimentation with% different values.%% The toolbox comes with GUI's that helps build and experiment with ANFIS% networks.%% |anfis1| and |anfis2| represent the two trained ANFIS networks that% will be deployed in the larger control system.%% Once the training is complete, the two ANFIS networks would have learned% to approximate the angles (|theta1, theta2|) as a function of the% coordinates (|x, y|). One advantage of using the fuzzy approach is that% the ANFIS network would now approximate the angles for coordinates that% are similar but not exactly the same as it was trained with. For example,% the trained ANFIS networks are now capable of approximating the angles% for coordinates that lie between two points that were included in the% training dataset. This will allow the final controller to move the arm% smoothly in the input space. % % We now have two trained ANFIS networks which are ready to be deployed% into the larger system that will utilize these networks to control the% robotic arms.% %% Validating the ANFIS networks% % Having trained the networks, an important follow up step is to validate the% networks to determine how well the ANFIS networks would perform inside% the larger control system.%% Since this demo problem deals with a two-joint robotic arm whose inverse% kinematics formulae can be derived, it is possible to test the answers% that the ANFIS networks produce with the answers from the derived% formulae.%% Let's assume that it is important for the ANFIS networks to have low% errors within the operating range |0<x<2| and |8<y<10|.%x = 0:0.1:2; % x coordinates for validationy = 8:0.1:10; % y coordinates for validation%%% The |theta1| and |theta2| values are deduced mathematically from the x% and y coordinates using inverse kinematics formulae.[X, Y] = meshgrid(x,y);c2 = (X.^2 + Y.^2 - l1^2 - l2^2)/(2*l1*l2);s2 = sqrt(1 - c2.^2);THETA2D = atan2(s2, c2); % theta2 is deducedk1 = l1 + l2.*c2;k2 = l2*s2;THETA1D = atan2(Y, X) - atan2(k2, k1); % theta1 is deduced%%% <matlab:edit('traininv') Click here for unvectorized code>%%% |THETA1D| and |THETA2D| are the variables that hold the values of% |theta1| and |theta2| deduced using the inverse kinematics formulae.%% |theta1| and |theta2| values predicted by the trained anfis networks% are obtained by using the command |evalfis| which evaluates a FIS for the% given inputs.%% Here, |evalfis| is used to find out the FIS outputs for the same x-y% values used earlier in the inverse kinematics formulae.XY = [X(:) Y(:)];THETA1P = evalfis(XY, anfis1); % theta1 predicted by anfis1THETA2P = evalfis(XY, anfis2); % theta2 predicted by anfis2%%% Now, we can see how close the FIS outputs are with respect to the% deduced values.theta1diff = THETA1D(:) - THETA1P;theta2diff = THETA2D(:) - THETA2P;subplot(2,1,1);plot(theta1diff);ylabel('THETA1D - THETA1P')title('Deduced theta1 - Predicted theta1')subplot(2,1,2);plot(theta2diff);ylabel('THETA2D - THETA2P')title('Deduced theta2 - Predicted theta2')%%% The errors are in the |1e-3| range which is a fairly good number for the% application it is being used in. However this may not be acceptable for% another application, in which case the parameters to the |anfis| function% may be tweaked until an acceptable solution is arrived at. Also, other% techniques like input selection and alternate ways to model the problem% may be explored.%% Building a solution around the trained ANFIS networks%% Now given a specific task, such as robots picking up an object in an% assembly line, the larger control system will use the trained ANFIS% networks as a reference, much like a lookup table, to determine what the% angles of the arms must be, given a desired location for the tip of the% arm. Knowing the desired angles and the current angles of the joints, % the system will apply force appropriately on the joints of the arms to% move them towards the desired location.%% The GUI demo |invkine| demonstrates how the two trained ANFIS networks% have been used to trace an ellipse in the input space. invkine%%% The two ANFIS networks used in the demo have been pre-trained and are% deployed into a larger system that controls the tip of the two-joint% robot arm to trace an ellipse in the input space.%% The ellipse to be traced can be moved around. Move the ellipse to a% slightly different location and observe how the system responds by moving% the tip of the robotic arm from its current location to the closest% point on the new location of the ellipse. Also observe that the system% responds smoothly as long as the ellipse to be traced lies within the 'x'% marked spots which represent the data grid that was used to train the% networks. Once the ellipse is moved outside the range of data it was% trained with, the ANFIS networks respond unpredictably. This emphasizes% the importance of having relevant and representative data for training.% Data must be generated based on the expected range of operation to avoid% such unpredictability and instability issues.%%%% Conclusion% This demo illustrated using ANFIS to solve an inverse kinematics problem.% Fuzzy logic has also found numerous other applications in other areas of% technology like non-linear control, automatic control, signal processing,% system identification, pattern recognition, time series prediction, data% mining, financial applications etc.,%% Explore other demos and the documentation for more insight into fuzzy% logic and its applications.%%% Glossary% *ANFIS* - Adaptive Neuro-Fuzzy Inference System. a technique for% automatically tuning Sugeno-type inference systems based on training data.%% *membership functions* - a function that specifies the degree to which a% given input belongs to a set or is related to a concept.%% *input space* - it is a term used to define the range of all possible% values%% *FIS* - Fuzzy Inference System. The overall name for a system that uses% fuzzy reasoning to map an input space to an output space.%% *epochs* - 1 epoch of training represents one complete presentation of% all the samples/datapoints/rows of the training dataset to the FIS. The% inputs of each sample are presented and the FIS outputs are computed% which are compared with the desired outputs to compute the error between% the two. The parameters of the membership functions are then tuned to% reduce the error between the desired output and the actual FIS output.%displayEndOfDemoMessage(mfilename)##### SOURCE END #####-->   </body></html>