function tau = ID( model, q, qd, qdd, f_ext, grav_accn ) % ID Inverse Dynamics via Recursive Newton-Euler Algorithm % ID(model,q,qd,qdd,f_ext,grav_accn) calculates the inverse dynamics of a % kinematic tree via the recursive Newton-Euler algorithm. q, qd and qdd % are vectors of joint position, velocity and acceleration variables; and % the return value is a vector of joint force variables. 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 < 6 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 > 4 && 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; a{i} = Xup{i} * -a_grav + S{i}*qdd(i); else v{i} = Xup{i}*v{model.parent(i)} + vJ; a{i} = Xup{i}*a{model.parent(i)} + S{i}*qdd(i) + crm(v{i})*vJ; end f{i} = model.I{i}*a{i} + crf(v{i})*model.I{i}*v{i}; if external_force && length(f_ext{i}) > 0 f{i} = f{i} - f_ext{i}; end end for i = model.NB:-1:1 tau(i,1) = S{i}' * f{i}; if model.parent(i) ~= 0 f{model.parent(i)} = f{model.parent(i)} + Xup{i}'*f{i}; end end