function [qdd_out,tau_out] = HD( model, fd, q, qd, qdd, tau, f_ext, grav_accn) % HD Articulated-Body Hybrid Dynamics Algorithm % [qdd_out,tau_out]=HD(model,fd,q,qd,qdd,tau,f_ext,grav_accn) calculates % the hybrid dynamics of a kinematic tree using the articulated-body % algorithm. fd is an array of boolean values such that fd(i)==1 if joint % i is a forward-dynamics joint, and fd(i)==0 otherwise. If fd(i)==1 then % tau(i) contains the given force at joint i, and the value of qdd(i) is % ignored; and if fd(i)==0 then qdd(i) contains the given acceleration at % joint i, and the value of tau(i) is ignored. Likewise, if fd(i)==1 then % qdd_out(i) contains the calculated acceleration at joint i, and % tau_out(i) contains the given force copied from tau(i); and if fd(i)==0 % then tau_out(i) contains the calculated force and qdd_out(i) the given % acceleration copied from qdd(i). Thus, the two output vectors are always % fully instantiated. f_ext is a cell array specifying external forces % acting on the bodies. If f_ext == {} then there are no external forces; % otherwise, f_ext{i} is a spatial force vector giving the force acting on % body i, expressed in body i coordinates. Empty cells in f_ext are % interpreted as zero forces. grav_accn is a 3D vector expressing the % linear acceleration due to gravity. The arguments f_ext and grav_accn % are optional, and default to the values {} and [0,0,-9.81], respectively, % if omitted. if nargin < 8 a_grav = [0;0;0;0;0;-9.81]; else a_grav = [0;0;0;grav_accn(1);grav_accn(2);grav_accn(3)]; end external_force = ( nargin > 6 && length(f_ext) > 0 ); for i = 1:model.NB [ XJ, S{i} ] = jcalc( model.pitch(i), q(i) ); vJ = S{i}*qd(i); Xup{i} = XJ * model.Xtree{i}; if model.parent(i) == 0 v{i} = vJ; c{i} = zeros(6,1); else v{i} = Xup{i}*v{model.parent(i)} + vJ; c{i} = crm(v{i}) * vJ; end if fd(i) == 0 c{i} = c{i} + S{i} * qdd(i); end IA{i} = model.I{i}; pA{i} = crf(v{i}) * model.I{i} * v{i}; if external_force && length(f_ext{i}) > 0 pA{i} = pA{i} - f_ext{i}; end end for i = model.NB:-1:1 if fd(i) == 0 if model.parent(i) ~= 0 Ia = IA{i}; pa = pA{i} + IA{i}*c{i}; IA{model.parent(i)} = IA{model.parent(i)} + Xup{i}' * Ia * Xup{i}; pA{model.parent(i)} = pA{model.parent(i)} + Xup{i}' * pa; end else U{i} = IA{i} * S{i}; d{i} = S{i}' * U{i}; u{i} = tau(i) - S{i}'*pA{i}; if model.parent(i) ~= 0 Ia = IA{i} - U{i}/d{i}*U{i}'; pa = pA{i} + Ia*c{i} + U{i} * u{i}/d{i}; IA{model.parent(i)} = IA{model.parent(i)} + Xup{i}' * Ia * Xup{i}; pA{model.parent(i)} = pA{model.parent(i)} + Xup{i}' * pa; end end end for i = 1:model.NB if model.parent(i) == 0 a{i} = Xup{i} * -a_grav + c{i}; else a{i} = Xup{i} * a{model.parent(i)} + c{i}; end if fd(i) == 0 qdd_out(i,1) = qdd(i); tau_out(i,1) = S{i}'*(IA{i}*a{i} + pA{i}); else qdd_out(i,1) = (u{i} - U{i}'*a{i})/d{i}; tau_out(i,1) = tau(i); a{i} = a{i} + S{i}*qdd_out(i); end end