function [new_config, new_ctrm, new_norms]  = achilles151_km(sp_coords,Theta)

% This code represents the kinematic model created by 'Build_data_frame' as a function.

%%% INPUTS %%%
% sp_coords: a 4 x 1 vector for global position, R3, and rotation, [-pi,pi], in the spatial frame.
% Theta: a vector of N x 1 angles, where N is the number of rotation axes.
%%% Outputs %%%
% new_config: a point cloud from the surface of the kinematic model in the new configuration.
% new_ctrm: a new center of mass estimate for the kinematic model.

% new_norms: a new norm associated with the model point is calculated to reflect the new pose.

%%% Sampled meshes for each segment
I1  = [  0.002116  0.123018  0.142047 -0.034704  0.054550 -0.052080  0.088305  0.055177 -0.031362  0.092904  0.036652 -0.000831 -0.018535 -0.012983  0.116474  0.007827  0.039835  0.139757  0.131890 -0.004120 -0.031499 -0.039567  0.011204  0.022794  0.067180 -0.024628  0.039602 -0.044219  0.023092  0.084819  0.010857  0.090479  0.035322 -0.006991  0.077146  0.081567  0.023092  0.099137  0.087928  0.042413  0.029033  0.010155  0.023261  0.018381  0.039490  0.085426 -0.046766 -0.058751  0.114021 -0.062647 -0.062878 -0.002438  0.046526 -0.037529  0.112431  0.119624 -0.038684  0.077689  0.117590  0.067562  0.006754 -0.018226  0.021061  0.085561  0.023350  0.088175  0.121584  0.116656  0.104537  0.127567 -0.008064  0.051859 -0.028386 -0.023155  0.033481  0.013829  0.099915 -0.040911  0.127772 -0.028913  0.066453  0.070181  0.102732  0.033724  0.127081  0.077527  0.013516 -0.036427  0.058647  0.099855 -0.024559  0.137443 -0.012994  0.049492 -0.023741  0.062802  0.061318  0.114284  0.065687  0.075986  0.047043  0.108945  0.113129  0.140084  0.103456  0.111317 -0.063161  0.029556 -0.042877  0.129395  0.093263  0.074012  0.047951  0.063873  0.011520  0.119009  0.049571 -0.048834  0.095815  0.121642  0.072413 -0.027205  0.119534  0.018024  0.088393  0.129539 -0.032961 -0.027925  0.117100  0.021895  0.119362 -0.036400  0.047160  0.104923  0.009355  0.006922  0.069836  0.049026  0.025429  0.001183  0.053467  0.048552  0.134832 -0.037312  0.130439  0.036180 -0.026527  0.094216  0.038349  0.061293  0.085310  0.119738  0.052864  0.133579 -0.005945 -0.035331  0.015970 -0.001080  0.004814 -0.034217  0.056184  0.036252 -0.031476  0.088902  0.051092 -0.008171  0.040914 -0.020051  0.066136  0.009368  0.095592  0.058913 -0.013254 -0.047091  0.038093  0.063854 -0.006119  0.112257  0.011686  0.127218  0.064935  0.028828  0.061283  0.140353  0.042750  0.067122  0.067087  0.112375  0.139305  0.041052 -0.058185  0.033989  0.048262  0.093888  0.100591 -0.013320  0.138809 -0.020088 -0.000566  0.043875 -0.033884  0.121959 -0.001491 -0.052414  0.009566  0.061368 -0.011480  0.081106  0.093530  0.004257  0.087367 -0.059410  0.093669  0.031749  0.107651  0.093780  0.038979  0.138409 -0.039484 -0.058770  0.081720  0.004712  0.008292  0.030619  0.055928  0.002548  0.129593 -0.006882  0.040888  0.014047  0.079737  0.117770 -0.029817  0.066723  0.038446  0.038210 -0.016858  0.102790  0.140595  0.131488 -0.008701  0.012198 -0.034626  0.140437  0.065526 -0.035762 -0.040602  0.140176  0.044253 -0.038946  0.067657  0.029714  0.121304  0.049432  0.109680 -0.016915
         0.067511  0.026890 -0.042517 -0.111135 -0.126537  0.118385 -0.109135 -0.039025  0.118303 -0.114196 -0.136745  0.103735 -0.121043 -0.126732 -0.030055  0.061746 -0.129117  0.012255  0.053339 -0.083782  0.031114 -0.118269 -0.087448 -0.089885  0.089097  0.117659  0.010274  0.126094 -0.009533 -0.107676 -0.127485 -0.102966  0.062644 -0.144801 -0.004947 -0.007613 -0.108637 -0.113076 -0.105694  0.103238 -0.112250 -0.124572 -0.123679 -0.136785 -0.074145 -0.134017 -0.122084  0.112254 -0.080342  0.109493  0.100423  0.038186 -0.141343 -0.122569 -0.073053 -0.038552 -0.131025 -0.073784 -0.078135  0.051331  0.054725  0.068681  0.066041 -0.134509 -0.138440 -0.052065 -0.035867 -0.096469 -0.037203 -0.030718 -0.096824  0.079565 -0.121195 -0.099535 -0.108165 -0.111618 -0.010616 -0.124178 -0.072651 -0.131848 -0.066222 -0.038104 -0.117590  0.012589 -0.045946 -0.133562 -0.138331  0.089881  0.010969 -0.120144  0.099217  0.043387 -0.126051 -0.141107 -0.112443  0.075931 -0.131317 -0.022255 -0.026432  0.058747 -0.138760 -0.006968 -0.089446  0.004743 -0.071459 -0.077748  0.118169 -0.120388  0.079969  0.045888 -0.048160 -0.054489  0.086424 -0.080364 -0.016795 -0.025303 -0.038956  0.100960 -0.022216 -0.087889  0.078207 -0.123380 -0.077796 -0.130315 -0.063681  0.039786 -0.116077  0.086371 -0.053042  0.084658  0.007046  0.079382 -0.125446  0.030433 -0.068935  0.020653 -0.027347  0.093602  0.011185 -0.143093 -0.026934  0.100660 -0.066300 -0.116650 -0.047086 -0.131109  0.041976 -0.106455 -0.132057  0.064509 -0.134677 -0.092072 -0.126291  0.049873 -0.122904 -0.138649  0.082869 -0.129219  0.048611 -0.124330  0.081376 -0.058592 -0.121375 -0.065545 -0.086089 -0.082370 -0.053510  0.102249 -0.005513  0.077557  0.085474  0.024192 -0.131846  0.093600  0.103601 -0.056221 -0.126045 -0.103665 -0.019536 -0.071436 -0.084733 -0.068578  0.098590 -0.006388 -0.142636 -0.045848 -0.006078  0.073861 -0.020397 -0.121430  0.096747 -0.008609 -0.001759 -0.105613  0.080975  0.084066 -0.003901 -0.114024 -0.013136  0.069429 -0.143940 -0.025321  0.115530  0.120434  0.116006 -0.085161 -0.123264  0.092353 -0.128210  0.051686  0.072187  0.117635 -0.128062 -0.131506 -0.100340 -0.040558 -0.088157 -0.027816  0.116759  0.107608  0.054248  0.028545  0.073559  0.105181 -0.093810 -0.067994  0.054603 -0.131209 -0.004576 -0.120545 -0.119127 -0.031036 -0.121849 -0.038041 -0.120186 -0.059114  0.057985 -0.116144 -0.049451 -0.020520  0.033779 -0.001248 -0.137735 -0.017349 -0.122946 -0.121362  0.044031 -0.029232 -0.131805  0.116392  0.013914 -0.132945 -0.091629 -0.114952  0.077907 -0.143987
         0.360103  0.262227  0.208380  0.261640  0.289600  0.251901  0.271344  0.385235  0.229847  0.255713  0.247568  0.271586  0.349180  0.300072  0.305164  0.363023  0.287248  0.212407  0.227415  0.398436  0.408740  0.300799  0.372949  0.360665  0.251811  0.228943  0.357793  0.217906  0.388212  0.279578  0.297154  0.277573  0.334822  0.201960  0.325145  0.325257  0.343268  0.246218  0.277564  0.233527  0.331599  0.312215  0.312370  0.244142  0.365568  0.220457  0.398123  0.293873  0.272578  0.305588  0.325705  0.394892  0.219648  0.384721  0.285509  0.297283  0.216018  0.329471  0.269026  0.311383  0.371595  0.370581  0.344513  0.199615  0.236880  0.341475  0.293816  0.224138  0.329292  0.280042  0.384660  0.288199  0.368703  0.402204  0.332733  0.345361  0.312025  0.385918  0.249505  0.219760  0.350722  0.378523  0.214292  0.364310  0.276894  0.240902  0.233270  0.339150  0.336450  0.225598  0.307439  0.216730  0.275860  0.220415  0.382995  0.286444  0.264287  0.306031  0.372711  0.297161  0.236670  0.296985  0.261020  0.232663  0.302555  0.280659  0.262149  0.317449  0.366145  0.243277  0.338256  0.359098  0.275715  0.335617  0.407795  0.300128  0.388554  0.315629  0.338980  0.214929  0.274075  0.355464  0.265791  0.281663  0.329554  0.247834  0.224216  0.344550  0.295249  0.302964  0.273063  0.366604  0.297002  0.283384  0.401933  0.393553  0.371244  0.253212  0.374313  0.209267  0.379862  0.234016  0.225508  0.395776  0.266789  0.277557  0.399418  0.266228  0.272432  0.305452  0.210471  0.224431  0.291620  0.228512  0.324896  0.220058  0.315261  0.286423  0.378699  0.326266  0.280905  0.384419  0.375441  0.327095  0.342184  0.406738  0.385118  0.293040  0.334546  0.334123  0.222022  0.333827  0.228259  0.336680  0.236565  0.367310  0.303906  0.215641  0.410695  0.255005  0.328699  0.378926  0.226484  0.225135  0.209196  0.373422  0.334172  0.227597  0.239338  0.307007  0.334731  0.374285  0.350456  0.268172  0.230755  0.342468  0.246393  0.374642  0.415413  0.315555  0.223149  0.293505  0.234151  0.243560  0.224571  0.331848  0.325091  0.221133  0.205454  0.376399  0.270285  0.261349  0.205669  0.275658  0.254009  0.344637  0.351516  0.243558  0.253058  0.306435  0.295909  0.393819  0.343276  0.238267  0.326261  0.413501  0.231895  0.238794  0.360206  0.327214  0.268234  0.302288  0.369104  0.380724  0.310958  0.382298  0.382523  0.202737  0.214512  0.269367  0.401611  0.394531  0.219633  0.217193  0.283734  0.385030  0.398332  0.232128  0.272920  0.257200  0.326198  0.267686  0.240611  0.308842  0.217730  0.206881
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N1  = [  0.437713  0.891797  1.130441 -0.358363  0.258879  0.076382  0.733334  0.501157  0.170203  0.735552  0.028135  0.236848  0.309848 -0.063563  0.993228  0.511081  0.140764  1.139025  0.972136  0.611105 -0.008537 -0.670222  0.559188  0.456055  0.345812  0.056039  0.789309  0.210885  0.644943  0.726807  0.010856  0.750398  0.584725 -0.003910  0.538629  0.543050  0.464960  0.779202  0.747847  0.348313  0.517501  0.037108  0.194930 -0.094480  0.468084  0.612534  0.217546  0.120792  0.828655  0.223723  0.112495  0.443631  0.069129  0.502330  0.873858  1.021144 -0.999963  0.630948  0.886711  0.714389  0.524713  0.277220  0.486131  0.646651 -0.009132  0.789372  1.023104  0.968717  0.996793  1.056760  0.631198  0.386904  0.396671  0.614816  0.570158  0.380541  0.728967  0.319326  1.015784 -0.373886  0.631658  0.763358  0.897566  0.806927  1.046625  0.492585 -0.163682  0.165810  0.795937  0.848825  0.265578  1.119961 -0.366537  0.072094  0.578385  0.405454  0.304168  0.925552  0.413418  0.627220  0.046052  0.785891  0.855174  1.133097  0.822856  0.864569  0.082801  0.351937 -0.062941  0.946114  0.827592  0.647072  0.317495  0.618606  0.611358  0.995762  0.564986  0.057744  0.605838  0.965087  0.447279  0.367060  0.888655  0.007875  0.672438  0.936619 -0.710956  0.152098  0.990098  0.362258  0.883739  0.050336  0.230033  0.859187  0.625825  0.698186  0.533899  0.312393  0.782574  0.025845  0.448306  0.363013  1.058553  0.509557  1.065341  0.043095  0.038818  0.732061  0.045264  0.554778  0.597988  0.965117  0.392888  0.993114  0.056184 -0.829583  0.362870 -0.008896  0.602999  0.036989  0.369664  0.576416  0.433015  0.672947  0.602876  0.611053  0.521167  0.196347  0.616182  0.413246  0.535122  0.794332 -0.442986  0.177609  0.343992  0.510236 -0.031601  0.935316  0.539014  0.996836  0.585539  0.501718  0.399437  1.140212  0.094815  0.630734  0.617133  0.707727  1.131649  0.406112  0.077930  0.620301  0.772456  0.738916  0.562903  0.290485  1.067518  0.515835  0.552669  0.456082  0.171803  1.054062  0.116610  0.155463  0.262788  0.581972  0.050649  0.430674  0.827246  0.522216  0.545632  0.086552  0.827386  0.038664  0.790033  0.891936  0.529865  1.113900  0.027321  0.282303  0.669814  0.695977  0.417453  0.285215  0.598185  0.629669  0.911753 -0.479603  0.615288  0.259684  0.707669  0.994523  0.395240  0.525556  0.479365  0.503932  0.166874  0.896026  1.120365  1.056299  0.319161  0.671431 -0.074550  1.139767  0.517888  0.504097 -0.049449  1.136878  0.102033  0.027859  0.784112  0.008611  0.869291  0.557835  0.674483 -0.107675
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%%% Ordered transformation matrices
%Root Node
% Global rotation operator
w1a = [0.999790 -0.020328 -0.002402]';
q1a = [0.0, 0.0, 0.0]';
t1b = twist_matrix_fast(w1a,q1a,sp_coords(4));

% Global translation operator
t1a = twist_matrix_fast(sp_coords(1:3),0,0);

w1c = [-0.920157 0.011244 -0.391388]';
q1c = [-0.007332 -0.027377 0.355014]';
t1c = twist_matrix_fast(w1c,q1c,Theta(1));

w1d = [0.391128 0.072769 -0.917455]';
q1d = [-0.007332 -0.027377 0.355014]';
t1d = twist_matrix_fast(w1d,q1d,Theta(2));

w1e = [0.018165 -0.997285 -0.071356]';
q1e = [-0.007332 -0.027377 0.355014]';
t1e = twist_matrix_fast(w1e,q1e,Theta(3));

w4a = [0.060252 -0.996755 -0.053372]';
q4a = [0.080728 -0.010994 0.148456]';
t4a = twist_matrix_fast(w4a,q4a,Theta(4));

w4b = [-0.261804 0.035817 -0.964456]';
q4b = [0.080728 -0.010994 0.148456]';
t4b = twist_matrix_fast(w4b,q4b,Theta(5));

w4c = [-0.963239 -0.072083 0.258796]';
q4c = [0.080728 -0.010994 0.148456]';
t4c = twist_matrix_fast(w4c,q4c,Theta(6));

w5a = [-0.007657 -0.999357 -0.035035]';
q5a = [0.013719 -0.001827 -0.098396]';
t5a = twist_matrix_fast(w5a,q5a,Theta(7));

w5b = [-0.007188 0.035090 -0.999358]';
q5b = [0.013719 -0.001827 -0.098396]';
t5b = twist_matrix_fast(w5b,q5b,Theta(8));

w5c = [-0.999945 0.007400 0.007452]';
q5c = [0.013719 -0.001827 -0.098396]';
t5c = twist_matrix_fast(w5c,q5c,Theta(9));

w6a = [0.207089 -0.977666 -0.035818]';
q6a = [0.012823 0.002548 -0.222986]';
t6a = twist_matrix_fast(w6a,q6a,Theta(10));

w6b = [0.375414 0.113222 -0.919916]';
q6b = [0.012823 0.002548 -0.222986]';
t6b = twist_matrix_fast(w6b,q6b,Theta(11));

w6c = [-0.903426 -0.177058 -0.390477]';
q6c = [0.012823 0.002548 -0.222986]';
t6c = twist_matrix_fast(w6c,q6c,Theta(12));

w7a = [0.101648 -0.991553 -0.080557]';
q7a = [0.041231 0.011116 -0.292597]';
t7a = twist_matrix_fast(w7a,q7a,Theta(13));

w7b = [-0.907033 -0.059114 -0.416888]';
q7b = [0.041231 0.011116 -0.292597]';
t7b = twist_matrix_fast(w7b,q7b,Theta(14));

w7c = [-0.408605 -0.115444 0.905381]';
q7c = [0.041231 0.011116 -0.292597]';
t7c = twist_matrix_fast(w7c,q7c,Theta(15));

%%% Construct kinematic chains of transformations
% Generate new_config and new_norms
U1 = t1a*t1b*t1c*t1d*t1e*I1;
V1 = t1a*t1b*t1c*t1d*t1e*N1;
V1 = V1-U1;

U4 = t1a*t1b*t1c*t1d*t1e*t4a*t4b*t4c*I4;
V4 = t1a*t1b*t1c*t1d*t1e*t4a*t4b*t4c*N4;
V4 = V4-U4;

U5 = t1a*t1b*t1c*t1d*t1e*t4a*t4b*t4c*t5a*t5b*t5c*I5;
V5 = t1a*t1b*t1c*t1d*t1e*t4a*t4b*t4c*t5a*t5b*t5c*N5;
V5 = V5-U5;

U7 = t1a*t1b*t1c*t1d*t1e*t4a*t4b*t4c*t5a*t5b*t5c*t6a*t6b*t6c*t7a*t7b*t7c*I7;
V7 = t1a*t1b*t1c*t1d*t1e*t4a*t4b*t4c*t5a*t5b*t5c*t6a*t6b*t6c*t7a*t7b*t7c*N7;
V7 = V7-U7;

%%% Construct complete update
new_config = [U1(1:3,:) U4(1:3,:) U5(1:3,:) U7(1:3,:)];
new_norms  = [V1(1:3,:) V4(1:3,:) V5(1:3,:) V7(1:3,:)];

% Update center of mass estimate
new_ctrm = mean(new_config,2);

%%% Twist matrix function
function [twist] = twist_matrix_fast(w,q,theta)
% Overview:
% This code generates an exponential map matrix for rotation and translation
% using the Rodrigues' formula for the matrix exponential, from "A Mathematical 
% Introduction to Robotic Manipulation" by Murray, Li, and Sastry (1994) p. 28. 
% The equation for the exponential map for twists is from Ibid. 42.
%
% This code has very little error checking to increase speed.
%
% [twist] = twist_matrix_fast(w,q,theta)
% INPUTS %
% w: a 3d vector pointing: 1) in the direction of translation or 2) 
%      along the axis of rotation relative to the origin.
% q: a 3d point along the axis of rotation, providing the offset
% theta: the amount of rotation, in radians
%
% OUTPUTS %
% twist: the resulting matrix for left multiplication
%
% I treat 2 cases: 1) pure translation and 2) pure rotation
if numel(q) < 3 && (q == 0 || theta == 0)
    
    % we want column vectors
    if size(w,1)<size(w,2)
        w = w';
    end
    
    % 1) pure translation
	% pure translation in absolute frame
	twist = [eye(3) w;zeros(1,3) 1];
else
    % 2) pure rotation twist
    % correct for case where w is the origin
    if sum(w) == 0
        w = q;
        W = 0;
    else
        W = 1;
    end
    
    % Generate cross product operator so(3)
    w_hat = [0   -w(3) w(2);
             w(3) 0   -w(1);
            -w(2) w(1) 0];
    
    v = -w_hat*q;
    % Generate rotation matrix using the Rodrigues' equation pg. 28
    Id3x3  = eye(3);
    w_hat2 = w*w'-W*Id3x3;
    R      = Id3x3+w_hat*sin(theta)+w_hat2*(1-cos(theta));
    
    % Generate relative frame twist, pg. 42
    twist  = [R (Id3x3-R)*w_hat*v+w*w'*v*theta;
              zeros(1,3) 1];
end
