function qdd = FDabp( model, q, qd, tau, f_ext, grav_accn ) % FDabp Forward Dynamics (planar) via Articulated-Body Algorithm. % FDabp(model,q,qd,tau,f_ext,grav_accn) calculates the forward dynamics of % a kinematic tree via the articulated-body algorithm, evaluated using % planar vectors. q, qd and tau are vectors of joint position, velocity % and force variables; and the return value is a vector of joint % acceleration 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 planar 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 2D vector expressing the % linear acceleration due to gravity in the x-y plane. The arguments f_ext % and grav_accn are optional, and default to zero (i.e., {} and [0 0], % respectively) if omitted. if nargin < 6 a_grav = [0;0;0]; else a_grav = [0;grav_accn(1);grav_accn(2)]; end external_force = ( nargin > 4 && length(f_ext) > 0 ); for i = 1:model.NB [ XJ, S{i} ] = jcalcp( model.jcode(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(3,1); else v{i} = Xup{i}*v{model.parent(i)} + vJ; c{i} = crmp(v{i}) * vJ; end IA{i} = model.I{i}; pA{i} = crfp(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 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 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 qdd(i,1) = (u{i} - U{i}'*a{i})/d{i}; a{i} = a{i} + S{i}*qdd(i); end