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estCq.m
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estCq.m
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function [Cq,Cqd,Qt]=estCq(z,const,body)
%% ESTCQ: Establish the Jacobian of the constraints
%Inputs: z - The extended state vector
% const - The structured array const
% body - The structured array body
%Output: Cq - The partial derivative of C wrt q
% Cqd - The partial derivative of C wrt qdot (equals the partial
% derivative of the "hidden" constraint wrt q)
% Qt - The partial derivative of C wrt time (with negative sign)
%Call: [Cq,Cqd,Qt]=estCq(z,const,body)
%Copyright: Thomas Abrahamsson, Chalmers, Sweden
%Written: 2009-03-27
%Modified: 2009-05-19 (Corrected Cq for revolute joint)
%Modified: 2009-07-01 (Corrected Cq and Cqd for translational, sliding and
% universal joints)
%% Initiate
nb=body.nb;
nceq=const.nceq;
nj=const.nj;
nq=7*nb;
Cq=zeros(nceq,nq);Cqd=Cq;Qt=sparse(zeros(2*nq+nceq,1));
%% Establish the matrices for joint constraints
% ncdone=nb;
ncdone=0;
for J=1:nj
ctype=const.type{J};
body1=const.body1{J};
body2=const.body2{J};
dofs1=(body1-1)*7+[1:7];
dofs2=(body2-1)*7+[1:7];
if strcmpi(ctype(1:3),'FIX')
ei =z((body1-1)*7+[4:7]);Ei=este(ei);
eid=z((nb+body1-1)*7+[4:7]);Eid=este(eid);
rib=const.r1{J};
if body2>0
ej =z((body2-1)*7+[4:7]);ejd=z((nb+body2-1)*7+[4:7]);
else
ej=[1 0 0 0]';ejd=[0 0 0 0 ]';
end
Ej=este(ej);Ejd=este(ejd);
Cq(ncdone+[1:6],dofs1)= ...
[zeros(3,3) -2*Eid*tilde4(rib);
zeros(3,3) -2*Eid ];
Cqd(ncdone+[1:6],dofs1)= ...
[eye(3,3) -2*Ei*tilde4(rib) ;
zeros(3,3) 2*Ei ];
if body2>0
rjb=const.r2{J};
Cq(ncdone+[1:6],dofs2)= ...
[zeros(3,3) 2*Ejd*tilde4(rjb) ;
zeros(3,3) 2*Ejd ];
Cqd(ncdone+[1:6],dofs2)= ...
[-eye(3,3) 2*Ej*tilde4(rjb) ;
zeros(3,3) -2*Ej ];
end
ncdone=ncdone+6;
elseif strcmpi(ctype(1:3),'REV')
ei =z((body1-1)*7+[4:7]);Ei=este(ei);Eib=este(ei,1);
eid=z((nb+body1-1)*7+[4:7]);Eid=este(eid);Eibd=este(eid,1);
eic=const.e1{J};Aic=esta('eulerp',eic);Aic=Aic(:,1:2);
rib=const.r1{J};
if body2>0
ej =z((body2-1)*7+[4:7]);
ejd=z((nb+body2-1)*7+[4:7]);
else
ej=[1 0 0 0]';
ejd=[0 0 0 0 ]';
end
Ej=este(ej);Ejd=este(ejd);
Cq(ncdone+[1:5],dofs1)= ...
[zeros(3,3) -2*Eid*tilde4(rib) ;
zeros(2,3) -2*Aic'*(Eibd+2*Eib*tilde4(Ej*ejd,1))];% Corrected
Cqd(ncdone+[1:5],dofs1)= ...
[eye(3,3) -2*Ei*tilde4(rib) ;
zeros(2,3) 2*Aic'*Eib];
if body2>0
rjb=const.r2{J};
Cq(ncdone+[1:5],dofs2)= ...
[zeros(3,3) 2*Ejd*tilde4(rjb) ;
zeros(2,3) 2*Aic'*Ei*Eib'*Ejd];
Cqd(ncdone+[1:5],dofs2)= ...
[-eye(3,3) 2*Ej*tilde4(rjb) ;
zeros(2,3) -2*Aic'*Ei*Eib'*Ej];
end
ncdone=ncdone+5;
elseif strcmpi(ctype(1:3),'TRA')
Rid=z((nb+body1-1)*7+[1:3]);
ei =z((body1-1)*7+[4:7]);Ai=esta('eulerp',ei);Ei=este(ei);Eib=este(ei,1);
eid=z((nb+body1-1)*7+[4:7]);Eid=este(eid);
eic=const.e1{J};Aic=esta('eulerp',eic);Aic=Aic(:,1:2);
ej =z((body2-1)*7+[4:7]);Aj=esta('eulerp',ej);Ejb=este(ej,1);
ejc=const.e2{J};Ajc=esta('eulerp',ejc);Ajc=Ajc(:,1:2);
rib=const.r1{J};rjb=const.r2{J};
if body2>0
Rjd=z((nb+body2-1)*7+[1:3]);
ej =z((body2-1)*7+[4:7]);
ejd=z((nb+body2-1)*7+[4:7]);
else
Rjd=zeros(3,1);
ej=[1 0 0 0]';
ejd=zeros(4,1);
end
Ej=este(ej);Ejd=este(ejd);
Cq(ncdone+[1:5],dofs1)= ...
[zeros(2,3) Aic'*(Eib*tilde4(Rid,1)+Omega34(eid,rib)) ;
zeros(3,3) -2*Eid ];
Cqd(ncdone+[1:5],dofs1)= ...
[Aic'*[Eib*Ei' -2*Ai'*Ei*tilde4(rib)] ;
zeros(3,3) 2*Ei ];
if body2>0
rjb=const.r2{J};
Cq(ncdone+[1:5],dofs2)= ...
[zeros(2,3) Ajc'*(Ejb*tilde4(Rjd,1)+Omega34(ejd,rjb)) ;
zeros(3,3) 2*Ejd ];
Cqd(ncdone+[1:5],dofs2)= ...
[-Ajc'*[Ejb*Ej' -2*Aj'*Ej*tilde4(rjb)];
zeros(3,3) -2*Ej ];
end
ncdone=ncdone+5;
elseif strcmpi(ctype(1:3),'SPH')
ei =z((body1-1)*7+[4:7]);Ei=este(ei);
eid=z((nb+body1-1)*7+[4:7]);Eid=este(eid);
rib=const.r1{J};
if body2>0
ej =z((body2-1)*7+[4:7]);ejd=z((nb+body2-1)*7+[4:7]);
else
ej=[1 0 0 0]';ejd=[0 0 0 0 ]';
end
Ej=este(ej);Ejd=este(ejd);
Cq(ncdone+[1:3],dofs1)= ...
[zeros(3,3) -2*Eid*tilde4(rib)];
Cqd(ncdone+[1:3],dofs1)= ...
[eye(3,3) -2*Ei*tilde4(rib)];
if body2>0
rjb=const.r2{J};
Cq(ncdone+[1:3],dofs2)= ...
[zeros(3,3) 2*Ejd*tilde4(rjb)];
Cqd(ncdone+[1:3],dofs2)= ...
[-eye(3,3) 2*Ej*tilde4(rjb)];
end
ncdone=ncdone+3;
elseif strcmpi(ctype(1:3),'SLI')
Rid=z((nb+body1-1)*7+[1:3]);
ei =z((body1-1)*7+[4:7]);Ei=este(ei);Eib=este(ei,1);
eid=z((nb+body1-1)*7+[4:7]);Eid=este(eid);Eibd=este(eid,1);
eic=const.e1{J};Aic=esta('eulerp',eic);Aic12=Aic(:,1:2);Aic3=Aic(:,3);
ejc=const.e2{J};Ajc=esta('eulerp',ejc);Ajc3=Ajc(:,3);
rib=const.r1{J};rjb=const.r2{J};
if body2>0
Rjd=z((nb+body2-1)*7+[1:3]);
ej =z((body2-1)*7+[4:7]);
ejd=z((nb+body2-1)*7+[4:7]);
else
Rjd=zeros(3,1);
ej=[1 0 0 0]';
ejd=zeros(4,1);
end
Ej=este(ej);Ejd=este(ejd);
Cq(ncdone+[1:3],dofs1)= ...
[zeros(1,3) Aic3'*(Eib*tilde4(Rid,1)+Omega34(eid,rib)) ;
zeros(2,3) -2*Aic12'*(Eibd+Eib*tilde4(Ej*ejd,1))];
Cqd(ncdone+[1:3],dofs1)= ...
[Aic3'*[Ai' -2*Ai'*Ei*tilde4(rib)] ;
zeros(2,3) 2*Aic12'*Eib];
if body2>0
rjb=const.r2{J};
Cq(ncdone+[1:3],dofs2)= ...
[zeros(1,3) Ajc3'*(Ejb*tilde4(Rjd,1)+Omega34(ejd,rjb)) ;
zeros(2,3) 2*Aic12'*Ai'*Ejd];
Cqd(ncdone+[1:3],dofs2)= ...
[-Ajc3'*[Aj' -2*Aj'*Ej*tilde4(rjb)];
zeros(2,3) -2*Aic12'*Ai'*Ej];
end
ncdone=ncdone+3;
elseif strcmpi(ctype(1:3),'UNI')
ei =z((body1-1)*7+[4:7]);Ei=este(ei);Eib=este(ei,1);
eid=z((nb+body1-1)*7+[4:7]);Eid=este(eid);Eibd=este(eid,1);
eic=const.e1{J};Aic=esta('eulerp',eic);Aic=Aic(:,3);
ejc=const.e2{J};Ajc=esta('eulerp',ejc);Ajc=Ajc(:,3);
rib=const.r1{J};
if body2>0
ej =z((body2-1)*7+[4:7]);ejd=z((nb+body2-1)*7+[4:7]);
else
ej=[1 0 0 0]';ejd=[0 0 0 0 ]';
end
Ej=este(ej);Ejb=este(ej,1);Ejd=este(ejd);Ejbd=este(ejd,1);
Cq(ncdone+[1:4],dofs1)= ...
[zeros(3,3) -2*Eid*tilde4(rib);
zeros(1,3) -2*Aic'*Eibd ];
Cqd(ncdone+[1:4],dofs1)= ...
[eye(3,3) -2*Ei*tilde4(rib) ;
zeros(1,3) 2*Aic'*Eib ];
if body2>0
rjb=const.r2{J};
Cq(ncdone+[1:4],dofs2)= ...
[zeros(3,3) 2*Ejd*tilde4(rjb) ;
zeros(1,3) 2*Ajc'*Ejbd ];
Cqd(ncdone+[1:4],dofs2)= ...
[-eye(3,3) 2*Ej*tilde4(rjb) ;
zeros(1,3) -2*Ajc'*Ejb ];
end
ncdone=ncdone+4;
end
end
%% Euler parameter constraints
for J=1:nb
dofs=(J-1)*7+[4:7];
Cqd(J+ncdone,dofs)=2*z((J-1)*7+[4:7])';
Cq(J+ncdone,dofs)=2*z((nb+J-1)*7+[4:7])';
end
%%
Cq=sparse(Cq);
Cqd=sparse(Cqd);
function O34=Omega34(e,r)
%% Omega34: The O34 operator for 3D and 4D vectors to 3x4 matrix
%Inputs: e - A 4-element vector
% r - A 3-element vector
%Output: O34 - O34 3x4 matrix
%Call: Omega34=O34(e,r)
%Copyright: Thomas Abrahamsson, Chalmers, Sweden
%Written: 2009-04-06
e0=e(1);e1=e(2);e2=e(3);e3=e(4);
r1=r(1);r2=r(2);r3=r(3);
O34=-2*[ (+e3*r2-e2*r3) (-e2*r2-e3*r3) (+e1*r2+e0*r3) (-e0*r2+e1*r3);
(-e3*r1+e1*r3) (+e2*r1-e0*r3) (-e1*r1-e3*r3) (+e0*r1+e2*r3);
(+e2*r1-e1*r2) (+e3*r1+e0*r2) (-e0*r1+e3*r2) (-e1*r1-e2*r2)];