[optimal] = ccl_learna_alpha (Un, X, options)
Learning state dependent projection matrix N(q) for problem with the form
Un = N(q) * F(q) where N is a state dependent projection matrix
F is some policy
Input:
X State of the system
Un Control of the system generated with the form Un(q) = N(q) * F(q)
where N(q)=I-pinv(A(q))'A(q) is the projection matrix that projects
F(q) unto the nullspace of A(q). N(q) can be state dependent, but
it should be generated in a consistent way.
Output:
optimal A model for the projection matrix
optimal.f_proj(q) A function that predicts N(q) given q0001 function [optimal] = ccl_learna_alpha (Un, X, options) 0002 % [optimal] = ccl_learna_alpha (Un, X, options) 0003 % 0004 % Learning state dependent projection matrix N(q) for problem with the form 0005 % Un = N(q) * F(q) where N is a state dependent projection matrix 0006 % F is some policy 0007 % 0008 % Input: 0009 % 0010 % X State of the system 0011 % Un Control of the system generated with the form Un(q) = N(q) * F(q) 0012 % where N(q)=I-pinv(A(q))'A(q) is the projection matrix that projects 0013 % F(q) unto the nullspace of A(q). N(q) can be state dependent, but 0014 % it should be generated in a consistent way. 0015 % 0016 % Output: 0017 % optimal A model for the projection matrix 0018 % optimal.f_proj(q) A function that predicts N(q) given q 0019 0020 0021 0022 0023 % CCL: A MATLAB library for Constraint Consistent Learning 0024 % Copyright (C) 2007 Matthew Howard 0025 % Contact: matthew.j.howard@kcl.ac.uk 0026 % 0027 % This library is free software; you can redistribute it and/or 0028 % modify it under the terms of the GNU Lesser General Public 0029 % License as published by the Free Software Foundation; either 0030 % version 2.1 of the License, or (at your option) any later version. 0031 % 0032 % This library is distributed in the hope that it will be useful, 0033 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0034 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 0035 % Lesser General Public License for more details. 0036 % 0037 % You should have received a copy of the GNU Library General Public 0038 % License along with this library; if not, write to the Free 0039 % Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 0040 0041 % essential parameters 0042 model.dim_b = options.dim_b ; % dimensionality of the gaussian kernel basis 0043 model.dim_r = options.dim_r ; % dimensionality of the end effector 0044 model.dim_x = size(X, 1) ; % dimensionality of input 0045 model.dim_u = size(Un,1) ; % dimensionality of output Un = N(X) * F(X) where X 0046 model.dim_t = model.dim_u - 1 ; % dimensionality of each constraint parameters 0047 dim_n = size(X,2) ; % number of training points 0048 0049 % choose a method for generating the centres for gaussian kernel. A 0050 % grid centre is usually adequate for a 2D problem. For higher 0051 % dimensionality, kmeans centre normally performs better 0052 if model.dim_x < 3 0053 model.dim_b = floor(sqrt(model.dim_b))^2 ; 0054 centres = ccl_math_gc (X, model.dim_b) ; % generate centres based on grid 0055 else 0056 centres = ccl_math_kc (X, model.dim_b) ; % generate centres based on K-means 0057 end 0058 variance = mean(mean(sqrt(ccl_math_distances(centres, centres))))^2 ; % set the variance as the mean distance between centres 0059 model.phi = @(x) ccl_basis_rbf ( x, centres, variance ); % gaussian kernel basis function 0060 BX = model.phi(X) ; % K(X) 0061 0062 optimal.nmse = 10000000 ; % initialise the first model 0063 model.var = sum(var(Un,0,2)) ;% variance of Un 0064 0065 % The constraint matrix consists of K mutually orthogonal constraint vectors. 0066 % At the k^{th} each iteration, candidate constraint vectors are 0067 % rotated to the space orthogonal to the all ( i < k ) constraint 0068 % vectors. At the first iteration, Rn = identity matrix 0069 Rn = cell(1,dim_n) ; 0070 for n = 1 : dim_n 0071 Rn{n} = eye(model.dim_u) ; 0072 end 0073 0074 % The objective functions is E(Xn) = A(Xn) * Rn(Xn) * Un. For faster 0075 % computation, RnUn(Xn) = Rn(Xn)*Un(Xn) is pre-caldulated to avoid 0076 % repeated calculation during non-linear optimisation. At the first iteration, the rotation matrix is the identity matrix, so RnUn = Un 0077 RnUn = Un ; 0078 0079 Iu = eye(model.dim_u) ; 0080 for alpha_id = 1:model.dim_r 0081 model.dim_k = alpha_id ; 0082 model = ccl_learna_sa (BX, RnUn, model ) ; % search the optimal k^(th) constraint vector 0083 theta = [pi/2*ones(dim_n, (alpha_id-1)), (model.w{alpha_id}* BX)' ] ; % predict constraint parameters 0084 for n = 1: dim_n 0085 Rn{n} = ccl_math_rotmat (theta(n,:), Rn{n}, model, alpha_id) ; % update rotation matrix for the next iteration 0086 RnUn(:,n) = Rn{n} * Un(:,n) ; % rotate Un ahead for faster computation 0087 end 0088 % if the k^(th) constraint vector reduce the fitness, then the 0089 % previously found vectors are enough to describe the constraint 0090 if (model.nmse > optimal.nmse) && (model.nmse > 1e-3) 0091 break ; 0092 else 0093 optimal = model ; 0094 end 0095 end 0096 optimal.f_proj = @(q) ccl_learna_pred_proj_alpha (optimal, q, Iu) ; % a function to predict the projection matrix 0097 fprintf('\t Found %d constraint vectors with residual error = %4.2e\n', optimal.dim_k, optimal.nmse) ; 0098 end