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  = [ -0.051271 -0.052816 -0.036424 -0.054299 -0.047031 -0.052571 -0.054048 -0.034020 -0.031514 -0.031854 -0.043003 -0.042422 -0.043472 -0.035456 -0.038316 -0.020580 -0.018291 -0.019685 -0.032041 -0.018969 -0.018170 -0.018170 -0.026458 -0.030519 -0.018608 -0.032150 -0.014767 -0.029807 -0.019357 -0.013237 -0.031083 -0.026854 -0.002882 -0.018023 -0.010334 -0.002731 -0.007361 -0.000384 -0.013129 -0.008629  0.004784 -0.009670 -0.008294 -0.019027 -0.019819  0.003872  0.016028  0.016118  0.009348  0.009977  0.019903 -0.000783  0.011509  0.007247  0.002241  0.002107  0.017294  0.021515  0.007602  0.010805  0.008432 -0.000134  0.003560  0.009543  0.033867  0.015546  0.037847  0.037656  0.021977  0.015545  0.012786  0.025544  0.032102  0.016670  0.026782  0.036778  0.028284  0.027830  0.017184  0.013705  0.033607  0.037498  0.051211  0.051113  0.044130  0.044126  0.052580  0.046488  0.032418  0.034696  0.050954  0.036826  0.035963  0.032846  0.037029  0.050999  0.032630  0.049810  0.032339  0.067899  0.067711  0.071316  0.085362  0.080085  0.068131  0.081410  0.067147  0.077930  0.085404  0.078826  0.079925  0.070139  0.080418  0.068603  0.086872  0.078726  0.086295  0.076794  0.089463  0.086636  0.064205  0.072027  0.088680  0.078218  0.088228  0.104685  0.105465  0.094062  0.113689  0.117412  0.110108  0.106327  0.132212  0.138684  0.132004  0.129330  0.127535  0.135395  0.139828  0.140273  0.134979  0.136572  0.133639  0.123242  0.119445  0.128718  0.121793  0.110048  0.102418  0.101841  0.109236  0.108655  0.117050  0.103434  0.104572  0.107982  0.119364  0.112835  0.110353  0.114785  0.117668  0.103360  0.102572  0.125060  0.125188  0.103310  0.096817  0.097004  0.083151  0.070678  0.062112  0.071728  0.085345  0.063460  0.054090  0.054592  0.060589  0.046454  0.051247  0.091196  0.120285  0.136127  0.139524  0.140390  0.141786  0.141887  0.140148  0.140071  0.137205  0.139983  0.124399  0.126915  0.129455  0.138491  0.099924  0.101026  0.109641  0.059232  0.054822  0.058759  0.090167  0.093123  0.097070  0.094736  0.070282  0.094340  0.062915  0.056103  0.055162  0.093404  0.060453  0.071946  0.026620  0.020996  0.025859  0.027315  0.059936  0.061711  0.002638  0.044590  0.012456  0.039228  0.048970  0.042316 -0.013532 -0.017402 -0.005815  0.024893  0.027241  0.026501  0.026695  0.026806  0.009110  0.010272  0.009718  0.006520 -0.043031 -0.038798 -0.008082 -0.008098 -0.041133 -0.036344 -0.032069 -0.044682 -0.009753 -0.036923 -0.037855 -0.025237 -0.026258 -0.059513 -0.024229 -0.026369 -0.027659 -0.039078 -0.060718 -0.060646
         0.021240  0.040402  0.049024  0.080929  0.072373  0.088297  0.098611  0.089170  0.097315  0.104254  0.114784  0.116742  0.119561  0.120252 -0.139536 -0.143548 -0.139566 -0.129135 -0.114079 -0.120464 -0.119971 -0.125211 -0.128284 -0.125215 -0.124590 -0.124733 -0.122079 -0.124024 -0.117334 -0.109777 -0.110515 -0.097995  0.007006  0.023729  0.048064  0.053664  0.068566  0.070802  0.083938  0.088216  0.094602  0.103489  0.110147  0.117005  0.117866  0.115634 -0.142842 -0.139837 -0.136550 -0.133658 -0.133022 -0.128786 -0.126453 -0.125211 -0.122991 -0.119135 -0.111799 -0.105017 -0.099476 -0.090513 -0.083409 -0.080627 -0.069717 -0.020486 -0.036085 -0.014109 -0.010414  0.019462  0.020077  0.029520  0.006440  0.044415  0.052832  0.071371  0.076705  0.082830  0.091972  0.099916  0.107526  0.102109  0.105228 -0.141875 -0.140227 -0.137579 -0.134958 -0.131734 -0.129188 -0.123623 -0.120623 -0.115938 -0.105539 -0.101966 -0.093146 -0.080495 -0.076051 -0.066173 -0.058879 -0.048063 -0.041947 -0.038000 -0.047413 -0.038510 -0.049787 -0.074082 -0.066548 -0.091046 -0.082162 -0.107227 -0.097778 -0.110077 -0.125557 -0.121795 -0.128052 -0.139350 -0.131247 -0.138077  0.089220  0.089076  0.081455  0.074074  0.068167  0.055478  0.032336  0.019914 -0.007540 -0.008448  0.033479  0.062767  0.057365  0.058863  0.077105  0.073382  0.055590  0.032825  0.033444  0.016054 -0.002479  0.017043  0.016132 -0.005226 -0.066861 -0.058733 -0.057146 -0.067729 -0.061165 -0.030828 -0.024835 -0.014224 -0.020762 -0.034716 -0.053891 -0.034992 -0.041127 -0.062950 -0.087031 -0.091944 -0.077726 -0.100403 -0.096628 -0.099559 -0.094023 -0.116245 -0.116656 -0.083358 -0.086132 -0.096987 -0.085200 -0.051133 -0.022428 -0.023611 -0.024655 -0.006649 -0.018118  0.002301  0.021477  0.047179  0.060434  0.027333 -0.003095 -0.021137 -0.006313 -0.030263 -0.022829 -0.026153 -0.040624 -0.047537 -0.003676 -0.051022 -0.047094 -0.034952 -0.075778 -0.077898 -0.012221  0.022977  0.083633  0.072445  0.041292  0.073140  0.081298  0.085085  0.085441  0.087740 -0.120594 -0.115234 -0.118535 -0.100100 -0.108948 -0.092331 -0.085551 -0.061423 -0.063120 -0.055567 -0.068060 -0.073961 -0.084097 -0.106571 -0.129587 -0.138692  0.117372  0.089874  0.086295  0.066889 -0.007536 -0.027332 -0.086008 -0.105311 -0.110139 -0.116775 -0.129145 -0.132833 -0.135810 -0.138839  0.110279  0.091271  0.076948 -0.011632 -0.122835 -0.128462 -0.122359 -0.123082 -0.116971 -0.110285 -0.110292 -0.102635 -0.130250 -0.142318 -0.137914  0.118072  0.116661  0.117406  0.094952  0.085193  0.049462 -0.135036  0.096892  0.125296
         0.416005  0.401193  0.393604  0.365098  0.375083  0.350893  0.326697  0.339404  0.318031  0.295278  0.274009  0.254895  0.238091  0.223947  0.204591  0.213180  0.217179  0.225972  0.228550  0.244182  0.260600  0.277866  0.290969  0.299764  0.307353  0.329083  0.339158  0.357714  0.365252  0.372276  0.397745  0.409404  0.406655  0.411502  0.390252  0.380289  0.365682  0.356467  0.342777  0.322598  0.292540  0.280014  0.258686  0.238143  0.221093  0.230322  0.211577  0.227137  0.240202  0.255900  0.265652  0.290275  0.302645  0.309206  0.321777  0.337713  0.342567  0.348338  0.363520  0.369149  0.381463  0.397796  0.409808  0.413914  0.402637  0.400647  0.371271  0.359100  0.376751  0.382082  0.390182  0.362115  0.348871  0.339089  0.317952  0.292859  0.277293  0.257369  0.242885  0.264484  0.235246  0.217144  0.227406  0.243539  0.257404  0.273318  0.281312  0.300424  0.314872  0.320389  0.317519  0.337536  0.348836  0.363712  0.365330  0.365363  0.387094  0.383458  0.404326  0.380341  0.371626  0.377629  0.348707  0.326866  0.348504  0.307049  0.330059  0.290777  0.292865  0.285205  0.255708  0.278949  0.249966  0.226935  0.231425  0.217975  0.226342  0.243585  0.249325  0.268080  0.298298  0.303930  0.300748  0.313994  0.319177  0.303514  0.282182  0.275770  0.257142  0.249825  0.221051  0.242634  0.222430  0.228507  0.247548  0.256607  0.267459  0.246370  0.227244  0.220028  0.222967  0.230286  0.248611  0.269166  0.284812  0.277410  0.295251  0.303562  0.329087  0.335574  0.310572  0.320553  0.301765  0.311219  0.279190  0.266131  0.266318  0.240030  0.256522  0.232640  0.205415  0.207593  0.224968  0.220716  0.234042  0.266295  0.294071  0.330015  0.352532  0.365850  0.372872  0.330472  0.341174  0.332767  0.340086  0.329277  0.310895  0.347660  0.347801  0.343476  0.280800  0.254148  0.236824  0.226508  0.211156  0.207935  0.237202  0.216874  0.242363  0.239052  0.258162  0.206598  0.270032  0.236095  0.220352  0.254043  0.274564  0.295077  0.282179  0.270726  0.237687  0.218632  0.235570  0.250773  0.284610  0.275624  0.304568  0.327457  0.338483  0.326361  0.361560  0.360618  0.381273  0.379038  0.364503  0.341709  0.272770  0.235801  0.225835  0.269268  0.308713  0.325173  0.353747  0.388547  0.409248  0.385805  0.363164  0.328394  0.288488  0.268151  0.251925  0.235830  0.241136  0.295260  0.335034  0.407327  0.393139  0.344500  0.329704  0.324182  0.300782  0.261475  0.257288  0.241977  0.237415  0.219288  0.219182  0.220402  0.246924  0.262357  0.318142  0.349456  0.392787  0.214469  0.335798  0.233847
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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
