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isam.m
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function [x, xlin, Rp,bp,qp] = isam(x,xlin, xprior,Pprior, Rp,bp,qp, curr_step, dt, nR, v_m, omega_m, sigma_v, sigma_w, nT, dim_target,PHIT, QTd, zr, Rr, zt, Rt, zl, Rl, nL)
% % incremental smoothing and mapping (iSAM) for CLATT (cooperative localization and target tracking)
%% parameters/constants
global gRB
J = [0 -1; 1 0];
epsilon = 1e-4;
itermax = 10; %max number of iterations
nrelin = 10; %frequency of relinearization
nddt = 3;
ddt = dt/nddt;
Q = [ sigma_v^2 0; 0 sigma_w^2 ]; %odometry noise
nsteps = curr_step; %number of total time steps
dofR = 3*nR;
dofT = dim_target*nT;
dofk = dofR+dofT;
% % first robot state is known, so excluded from the state, while first target state has prior!
% state = [robots-targets; robots-targets;...]
x0 = x(1:dofR,1);
xlin0 = xlin(1:dofR,1);
x = x(dofR+1:end,1);
xlin = xlin(dofR+1:end,1);
%% Gauss-Newton
if mod(curr_step,nrelin) && curr_step>2 %incremental QR update
% NOTE: run batch update for the first step in order to easily get Rp, bp, qp by QR (though not necessary)
ndim = dofT + dofk*(nsteps-1); %first robot pose is known, hence not included in the state
if gRB==3
nmeas = dofT + 3*nR*(nsteps-1)+dim_target*nT*(nsteps-1) + 2*nR*nR* (nsteps-1) + 2*nR*nT*(nsteps-1); %may be converative!!!
else
nmeas = dofT + 3*nR*(nsteps-1)+dim_target*nT*(nsteps-1) + nR*nR* (nsteps-1) + nR*nT*(nsteps-1); %may be converative!!!
end
A = sparse(nmeas,ndim); %jacobian
b = zeros(nmeas,1); %residual
k = curr_step; %only compute jacobians for current time step
% % proprioception measurements % %
% robot propagation
for ell = 1:nR
if k==2 %this actuall is the first robot in the state
x1k = xlin0(3*ell-2:3*ell,1); %known xlin0 or x0
P1k = zeros(3);
for i=1:nddt
xk = [ x1k(1) + v_m(ell,k-1)*ddt*cos(x1k(3));
x1k(2) + v_m(ell,k-1)*ddt*sin(x1k(3));
pi_to_pi(x1k(3) + omega_m(ell,k-1)*ddt) ];
F1k = [ 1 0 -v_m(ell,k-1)*ddt*sin(x1k(3));
0 1 v_m(ell,k-1)*ddt*cos(x1k(3));
0 0 1 ];
Gk = [ ddt*cos(x1k(3)) 0;
ddt*sin(x1k(3)) 0;
0 ddt ];
P1k = F1k*P1k*F1k'+Gk*Q*Gk';
x1k = xk;
end
Qk = P1k;
L = chol(Qk,'lower');
invQk = (L'\eye(size(L)))*(L\eye(size(L)));
sqrtQk = sqrtm(invQk);
indr2 = dofT + [3*ell-2:3*ell];
indxm =dofT + [3*ell-2:3*ell ];
Fk = - eye(3);
ak = xlin(indr2,1) - xk;
ak(3) = pi_to_pi(ak(3));
b(indxm,1) = sqrtQk'*ak ;
A(indxm,indr2) = sqrtQk'*Fk ;
elseif k>2
indr1 = dofT + dofk*(k-3)+ [3*(ell)-2 : 3*(ell)] ; %note the first pose (k=1) is excluded in x, i.e., x_k actually corresponds to xinit_k+1
indr2 = dofT + dofk*(k-2)+[3*(ell)-2 : 3*(ell)] ;
indxm = dofT + dofk*(k-2)+[ 3*ell-2:3*ell ];
x1k = xlin(indr1,1);
P1k = zeros(3);
for i=1:nddt
xk = [ x1k(1) + v_m(ell,k-1)*ddt*cos(x1k(3));
x1k(2) + v_m(ell,k-1)*ddt*sin(x1k(3));
pi_to_pi(x1k(3) + omega_m(ell,k-1)*ddt) ]; %note the odometry actually is from time k
F1k = [1 0 -v_m(ell,k-1)*ddt*sin(x1k(3));
0 1 v_m(ell,k-1)*ddt*cos(x1k(3));
0 0 1];
Gk = [ddt*cos(x1k(3)) 0;
ddt*sin(x1k(3)) 0;
0 ddt];
P1k = F1k*P1k*F1k'+Gk*Q*Gk';
x1k = xk;
end
Qk = P1k;
L = chol(Qk,'lower');
invQk = (L'\eye(size(L)))*(L\eye(size(L)));
sqrtQk = sqrtm(invQk);
F1k = [ eye(2) J*(xlin(indr2(1:2),1)-xlin(indr1(1:2),1)); zeros(1,2) 1 ]; %beware of sign
Fk = - eye(3);
ak = xlin(indr2,1) - xk;
ak(3) = pi_to_pi(ak(3));
b(indxm,1) = sqrtQk'*ak ;
A(indxm,indr1) = sqrtQk'*F1k ;
A(indxm,indr2) = sqrtQk'*Fk ;
end
end
% target propagation
for ell = 1:nT
L = chol(QTd(:,:,ell,k-1),'lower');
invQk = (L'\eye(size(L)))*(L\eye(size(L)));
sqrtQk = sqrtm(invQk);
indt1 = dofT + dofk*(k-3)+ 3*nR +[dim_target*(ell-1)+1:dim_target*ell];
indt2 = dofT + dofk*(k-2)+ 3*nR +[dim_target*(ell-1)+1:dim_target*ell];
indxm = dofT + dofk*(k-2)+ 3*nR +[dim_target*(ell-1)+1:dim_target*ell];
F1k = PHIT(:,:,ell,k-1);
Gk = - eye(dim_target);
ak = xlin(indt2,1) - F1k*x(indt1,1);
b(indxm,1) = sqrtQk'*ak ;
A(indxm, indt1) = sqrtQk'*F1k ;
A(indxm, indt2) = sqrtQk'*Gk ;
end
% % exterocpetive measurements % %
% robot-to-robot
imeas = 0;
for ell = 1:nR
for i = 1:size(zr(ell,:,:,k),3)
if zr(ell,3,i,k)<=0, continue, end
Ri = Rr{ell,k}(2*i-1:2*i, 2*i-1:2*i);
sqrtRk = sqrtm(inv(Ri));
imeas = imeas+1;
indr1 = dofT + dofk*(k-2)+[3*ell-2:3*ell];
indr2 = dofT + dofk*(k-2)+[3*zr(ell,3,i,k)-2 : 3*zr(ell,3,i,k)];
C = [cos(xlin(indr1(3))) -sin(xlin(indr1(3))); sin(xlin(indr1(3))) cos(xlin(indr1(3))) ];
i_p_r = C'*(xlin(indr2(1:2))-xlin(indr1(1:2)));
rho = norm(i_p_r);
th = atan2(i_p_r(2),i_p_r(1));
zhat = [rho; th];
if gRB==3
indxm = dofT + dofk*(nsteps-1) + 2*nR*nR*(k-2) + [ 2*imeas-1:2*imeas ] ;
r = [ zr(ell,1,i,k)-zhat(1,1); pi_to_pi(zr(ell,2,i,k)-zhat(2,1)) ] ; %beware of sign
Htem = [ 1/norm(i_p_r)*i_p_r'; 1/norm(i_p_r)^2*i_p_r'*J' ];
H1 = - Htem* C'*[ eye(2) J*(xlin(indr2(1:2)) - xlin(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,1)];
elseif gRB==2
indxm = dofT + dofk*(nsteps-1) + nR*nR*(k-2) + imeas ;
r = pi_to_pi(zr(ell,2,i,k)-zhat(2,1)) ; %beware of sign
Htem = [ 1/norm(i_p_r)*i_p_r'; 1/norm(i_p_r)^2*i_p_r'*J' ];
H1 = - Htem* C'*[ eye(2) J*(xlin(indr2(1:2)) - xlin(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,1)];
H1 = H1(2,:);
H2 = H2(2,:);
sqrtRk = sqrtRk(2,2);
elseif gRB==1
indxm = dofT + dofk*(nsteps-1) + nR*nR*(k-2) + imeas ;
r = zr(ell,1,i,k)-zhat(1,1); %beware of sign
Htem = [ 1/norm(i_p_r)*i_p_r'; 1/norm(i_p_r)^2*i_p_r'*J' ];
H1 = - Htem* C'*[ eye(2) J*(xlin(indr2(1:2)) - xlin(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,1)];
H1 = H1(1,:);
H2 = H2(1,:);
sqrtRk = sqrtRk(1,1);
end
A(indxm, [indr1, indr2]) = [sqrtRk'*H1 , sqrtRk'*H2] ;
b(indxm,1) = sqrtRk'*r;
end
end
% robot-to-target
imeas = 0;
for ell = 1:nR
for i = 1:size(zt(ell,:,:,k),3)
if zt(ell,3,i,k)<=0, continue, end
Ri = Rt{ell,k}(2*i-1:2*i, 2*i-1:2*i);
sqrtRk = sqrtm(inv(Ri));
imeas = imeas+1;
indr1 = dofT + dofk*(k-2)+ [3*ell-2:3*ell];
indr2 = dofT + dofk*(k-2)+ 3*nR+ [dim_target*(zt(ell,3,i,k)-1)+1 : dim_target*zt(ell,3,i,k)];
C = [cos(xlin(indr1(3))) -sin(xlin(indr1(3))); sin(xlin(indr1(3))) cos(xlin(indr1(3))) ];
i_p_t = C'*(xlin(indr2(1:2))-xlin(indr1(1:2)));
rho = norm(i_p_t);
th = atan2(i_p_t(2),i_p_t(1));
zhat = [rho; th];
if gRB==3
indxm = dofT + dofk*(nsteps-1) + 2*nR*nR*(nsteps-1) + 2*nR*nT*(k-2) + [2*imeas-1:2*imeas] ;
r = [ zt(ell,1,i,k)-zhat(1,1); pi_to_pi(zt(ell,2,i,k)-zhat(2,1)) ] ; %beware of sign
Htem = [ 1/norm(i_p_t)*i_p_t'; 1/norm(i_p_t)^2*i_p_t'*J' ];
H1 = - Htem* C'*[ eye(2) J*(xlin(indr2(1:2)) - xlin(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,dim_target-2)];
elseif gRB==2
indxm = dofT + dofk*(nsteps-1) + nR*nR*(k-2) + imeas ;
r = pi_to_pi(zt(ell,2,i,k)-zhat(2,1)) ; %beware of sign
Htem = [ 1/norm(i_p_t)*i_p_t'; 1/norm(i_p_t)^2*i_p_t'*J' ];
H1 = - Htem* C'*[ eye(2) J*(xlin(indr2(1:2)) - xlin(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,dim_target-2)];
H1 = H1(2,:);
H2 = H2(2,:);
sqrtRk = sqrtRk(2,2);
elseif gRB==1
indxm = dofT + dofk*(nsteps-1) + nR*nR*(k-2) + imeas ;
r = zt(ell,1,i,k)-zhat(1,1); %beware of sign
Htem = [ 1/norm(i_p_t)*i_p_t'; 1/norm(i_p_t)^2*i_p_t'*J' ];
H1 = - Htem* C'*[ eye(2) J*(xlin(indr2(1:2)) - xlin(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,dim_target-2)];
H1 = H1(1,:);
H2 = H2(1,:);
sqrtRk = sqrtRk(1,1);
end
A(indxm, [indr1, indr2]) = [sqrtRk'*H1 , sqrtRk'*H2] ;
b(indxm,1) = sqrtRk'*r;
end
end
% robot-to-landmark
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% reuse previously-computed jacobian and residual, and augment by the new ones
qp = [qp, length(qp)+1:length(qp)+dofk];
Ak = A(:,qp); % reordering, qp, which was obtained from batch step
if 1 %ISNEWNODE
A = sparse([ Rp, zeros(length(Rp),dofk) ; Ak ]); %match the dimension, as new robot and target states added
else
A = sparse([Rp; Ak]);
end
b = [bp; b];
% solve linear system using QR
if 0% standard QR
[Qp,Rp] = qr(A,0);
bp = Qp'*b ;
else % sparse QR with reordering
[m, n] = size(A);
[V,beta,p,R] = cs_qr(A) ;
d = cs_qleft(V, beta, p, b) ;
Rp = R(1:n,1:n);
bp = d(1:n,1);
end
delta_x = Rp \ bp;
delta_x(qp) = delta_x; %back to original ordering
ndx1 = norm(delta_x);
% % *linearization points, xlin* updated by deltax to be the new estimate!!!
x = xlin + delta_x;
else %%%batch every *nrelin* steps (including first step)%%%
for iter = 1:itermax
ndim = dofT + dofk*(nsteps-1); %first robot pose is known, hence not included in the state
if gRB==3
nmeas = dofT + 3*nR*(nsteps-1)+dim_target*nT*(nsteps-1) + 2*nR*nR* (nsteps-1) + 2*nR*nT*(nsteps-1); %may be converative!!!
else
nmeas = dofT + 3*nR*(nsteps-1)+dim_target*nT*(nsteps-1) + nR*nR* (nsteps-1) + nR*nT*(nsteps-1); %may be converative!!!
end
A = sparse(nmeas, ndim); %jacobian
b = zeros(nmeas,1); %residual
% % prior for the first target's state
xTprior = xprior(3*nR+[1:dofT],1);
PTprior = Pprior(3*nR+[1:dofT] , 3*nR+[1:dofT]);
sqrtQk = real(sqrtm(inv(PTprior)));
ak = xTprior - x(1:dofT,1);
Gk = eye(dofT);
indt2 = 1:dofT;
indxm = 1:dofT;
b(indxm,1) = sqrtQk'*ak ;
A(indxm, indt2) = sqrtQk'*Gk ;
% % all measurements
for k = 2:nsteps %because first robot pose is known
% % proprioception measurements % %
% robot propagation
for ell = 1:nR
if k==2 %this actuall is the first robot in the state
x1k = x0(3*ell-2:3*ell,1); %xprior(3*ell-2:3*ell,1); %x0 should be same as xprior
P1k = zeros(3);
for i=1:nddt
xk = [ x1k(1) + v_m(ell,k-1)*ddt*cos(x1k(3));
x1k(2) + v_m(ell,k-1)*ddt*sin(x1k(3));
pi_to_pi(x1k(3) + omega_m(ell,k-1)*ddt) ];
F1k = [ 1 0 -v_m(ell,k-1)*ddt*sin(x1k(3));
0 1 v_m(ell,k-1)*ddt*cos(x1k(3));
0 0 1 ];
Gk = [ ddt*cos(x1k(3)) 0;
ddt*sin(x1k(3)) 0;
0 ddt ];
P1k = F1k*P1k*F1k'+Gk*Q*Gk';
x1k = xk;
end
Qk = P1k;
%sqrtQk = real(sqrtm(inv(Qk)));
L = chol(Qk,'lower');
invQk = (L'\eye(size(L)))*(L\eye(size(L)));
sqrtQk = sqrtm(invQk);
indr2 = dofT + [3*ell-2:3*ell];
indxm =dofT + [3*ell-2:3*ell ];
Fk = - eye(3);
ak = x(indr2,1) - xk;
ak(3) = pi_to_pi(ak(3));
b(indxm,1) = sqrtQk'*ak ;
A(indxm,indr2) = sqrtQk'*Fk ;
elseif k>2
indr1 = dofT + dofk*(k-3)+ [3*(ell)-2 : 3*(ell)] ; %note the first pose (k=1) is excluded in x, i.e., x_k actually corresponds to xinit_k+1
indr2 = dofT + dofk*(k-2)+[3*(ell)-2 : 3*(ell)] ;
indxm = dofT + dofk*(k-2)+[ 3*ell-2:3*ell ];
x1k = x(indr1,1);
P1k = zeros(3);
for i=1:nddt
xk = [ x1k(1) + v_m(ell,k-1)*ddt*cos(x1k(3));
x1k(2) + v_m(ell,k-1)*ddt*sin(x1k(3));
pi_to_pi(x1k(3) + omega_m(ell,k-1)*ddt) ]; %note the odometry actually is from time k
F1k = [1 0 -v_m(ell,k-1)*ddt*sin(x1k(3));
0 1 v_m(ell,k-1)*ddt*cos(x1k(3));
0 0 1];
Gk = [ddt*cos(x1k(3)) 0;
ddt*sin(x1k(3)) 0;
0 ddt];
P1k = F1k*P1k*F1k'+Gk*Q*Gk';
x1k = xk;
end
Qk = P1k;
L = chol(Qk,'lower');
invQk = (L'\eye(size(L)))*(L\eye(size(L)));
sqrtQk = sqrtm(invQk);
F1k = [ eye(2) J*(x(indr2(1:2),1)-x(indr1(1:2),1)); zeros(1,2) 1 ];%beware of sign
Fk = - eye(3);
ak = x(indr2,1) - xk;
ak(3) = pi_to_pi(ak(3));
b(indxm,1) = sqrtQk'*ak ;
A(indxm,indr1) = sqrtQk'*F1k ;
A(indxm,indr2) = sqrtQk'*Fk ;
end
end
% target propagation
for ell = 1:nT
indt1 = dofT + dofk*(k-3)+ 3*nR +[dim_target*(ell-1)+1:dim_target*ell];
indt2 = dofT + dofk*(k-2)+ 3*nR +[dim_target*(ell-1)+1:dim_target*ell];
indxm = dofT + dofk*(k-2)+ 3*nR +[dim_target*(ell-1)+1:dim_target*ell];
L = chol(QTd(:,:,ell,k-1),'lower');
invQk = (L'\eye(size(L)))*(L\eye(size(L)));
sqrtQk = sqrtm(invQk);
F1k = PHIT(:,:,ell,k-1);
Gk = - eye(dim_target);
ak = x(indt2,1) - F1k*x(indt1,1);
b(indxm,1) = sqrtQk'*ak ;
A(indxm, indt1) = sqrtQk'*F1k ;
A(indxm, indt2) = sqrtQk'*Gk ;
end
% % exterocpetive measurements % %
% robot-to-robot
imeas = 0;
for ell = 1:nR
for i = 1:size(zr(ell,:,:,k),3)
if zr(ell,3,i,k)<=0, continue, end
Ri = Rr{ell,k}(2*i-1:2*i, 2*i-1:2*i);
sqrtRk = sqrtm(inv(Ri));
imeas = imeas+1;
indr1 = dofT + dofk*(k-2)+[3*ell-2:3*ell];
indr2 = dofT + dofk*(k-2)+[3*zr(ell,3,i,k)-2 : 3*zr(ell,3,i,k)];
C = [cos(x(indr1(3))) -sin(x(indr1(3))); sin(x(indr1(3))) cos(x(indr1(3))) ];
i_p_r = C'*(x(indr2(1:2))-x(indr1(1:2)));
rho = norm(i_p_r);
th = atan2(i_p_r(2),i_p_r(1));
zhat = [rho; th];
if gRB==3
indxm = dofT + dofk*(nsteps-1) + 2*nR*nR*(k-2) + [ 2*imeas-1:2*imeas ] ;
r = [ zr(ell,1,i,k)-zhat(1,1); pi_to_pi(zr(ell,2,i,k)-zhat(2,1)) ] ; %beware of sign
Htem = [ 1/norm(i_p_r)*i_p_r'; 1/norm(i_p_r)^2*i_p_r'*J' ];
H1 = - Htem* C'*[ eye(2) J*(x(indr2(1:2)) - x(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,1)];
elseif gRB==2
indxm = dofT + dofk*(nsteps-1) + nR*nR*(k-2) + imeas ;
r = pi_to_pi(zr(ell,2,i,k)-zhat(2,1)); %beware of sign
Htem = [ 1/norm(i_p_r)*i_p_r'; 1/norm(i_p_r)^2*i_p_r'*J' ];
H1 = - Htem* C'*[ eye(2) J*(x(indr2(1:2)) - x(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,1)];
H1 = H1(2,:);
H2 = H2(2,:);
sqrtRk = sqrtRk(2,2);
elseif gRB==1
indxm = dofT + dofk*(nsteps-1) + nR*nR*(k-2) + imeas ;
r = zr(ell,1,i,k)-zhat(1,1); %beware of sign
Htem = [ 1/norm(i_p_r)*i_p_r'; 1/norm(i_p_r)^2*i_p_r'*J' ];
H1 = - Htem* C'*[ eye(2) J*(x(indr2(1:2)) - x(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,1)];
H1 = H1(1,:);
H2 = H2(1,:);
sqrtRk = sqrtRk(1,1);
end
A(indxm, [indr1, indr2]) = [sqrtRk'*H1 , sqrtRk'*H2] ;
b(indxm,1) = sqrtRk'*r;
end
end
% robot-to-target
imeas = 0;
for ell = 1:nR
for i = 1:size(zt(ell,:,:,k),3)
if zt(ell,3,i,k)<=0, continue, end
Ri = Rt{ell,k}(2*i-1:2*i, 2*i-1:2*i);
sqrtRk = sqrtm(inv(Ri));
imeas = imeas+1;
indr1 = dofT + dofk*(k-2)+ [3*ell-2:3*ell];
indr2 = dofT + dofk*(k-2)+ 3*nR+ [dim_target*(zt(ell,3,i,k)-1)+1: dim_target*zt(ell,3,i,k)];
C = [cos(x(indr1(3))) -sin(x(indr1(3))); sin(x(indr1(3))) cos(x(indr1(3))) ];
i_p_t = C'*(x(indr2(1:2))-x(indr1(1:2)));
rho = norm(i_p_t);
th = atan2(i_p_t(2),i_p_t(1));
zhat = [rho; th];
if gRB==3
indxm = dofT + dofk*(nsteps-1) + 2*nR*nR*(nsteps-1) + 2*nR*nT*(k-2) + [2*imeas-1:2*imeas] ;
r = [ zt(ell,1,i,k)-zhat(1,1); pi_to_pi(zt(ell,2,i,k)-zhat(2,1)) ] ; %beware of sign
Htem = [ 1/norm(i_p_t)*i_p_t'; 1/norm(i_p_t)^2*i_p_t'*J' ];
H1 = - Htem* C'*[ eye(2) J*(x(indr2(1:2)) - x(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,dim_target-2)];
elseif gRB==2
indxm = dofT + dofk*(nsteps-1) + nR*nR*(nsteps-1) + nR*nT*(k-2) + imeas ;
r = pi_to_pi(zt(ell,2,i,k)-zhat(2,1)) ; %beware of sign
Htem = [ 1/norm(i_p_t)*i_p_t'; 1/norm(i_p_t)^2*i_p_t'*J' ];
H1 = - Htem* C'*[ eye(2) J*(x(indr2(1:2)) - x(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,dim_target-2)];
H1 = H1(2,:);
H2 = H2(2,:);
sqrtRk = sqrtRk(2,2);
elseif gRB==1
indxm = dofT + dofk*(nsteps-1) + nR*nR*(nsteps-1) + nR*nT*(k-2) +imeas ;
r = zt(ell,1,i,k)-zhat(1,1); %beware of sign
Htem = [ 1/norm(i_p_t)*i_p_t'; 1/norm(i_p_t)^2*i_p_t'*J' ];
H1 = - Htem* C'*[ eye(2) J*(x(indr2(1:2)) - x(indr1(1:2))) ];
H2 = Htem* C'*[eye(2) zeros(2,dim_target-2)];
H1 = H1(1,:);
H2 = H2(1,:);
sqrtRk = sqrtRk(1,1);
end
A(indxm, [indr1, indr2]) = [sqrtRk'*H1 , sqrtRk'*H2] ;
b(indxm,1) = sqrtRk'*r;
end
end
% robot-to-landmark
end%k
%%%%%%%%%%%%%%%%%%%%%
% % solve linear system : Ax = b
if 0 % standard QR
[Qp,Rp] = qr(A,0);
bp = Qp'*b ;
delta_x = Rp \ bp;
qp = 1:length(Rp); %no reordering
else % sparse QR with reordering
[~,n] = size(A);
qp = 1:n;
[V,beta,p,R,qp] = cs_qr(A) ; % fill-in reduced reordering qp
d = cs_qleft(V, beta, p, b) ;
Rp = R(1:n,1:n);
bp = d(1:n,1);
delta_x = Rp \ bp;
delta_x(qp) = delta_x ;
end
%%
x = x + delta_x;
xlin = x;
ndx1 = norm(delta_x);
if ndx1<epsilon, break, end
end%end iteration
end
% % augment the first robot/target state
x = [x0; x];
xlin = [xlin0; xlin];