%JTRAJ Compute a joint space trajectory
%
% [Q,QD,QDD] = JTRAJ(Q0, QF, M) is a joint space trajectory Q (MxN) where the joint
% coordinates vary from Q0 (1xN) to QF (1xN).  A quintic (5th order) polynomial is used 
% with default zero boundary conditions for velocity and acceleration.  
% Time is assumed to vary from 0 to 1 in M steps.  Joint velocity and 
% acceleration can be optionally returned as QD (MxN) and QDD (MxN) respectively.
% The trajectory Q, QD and QDD are MxN matrices, with one row per time step,
% and one column per joint.
%
% [Q,QD,QDD] = JTRAJ(Q0, QF, M, QD0, QDF) as above but also specifies
% initial QD0 (1xN) and final QDF (1xN) joint velocity for the trajectory.
%
% [Q,QD,QDD] = JTRAJ(Q0, QF, T) as above but the number of steps in the
% trajectory is defined by the length of the time vector T (Mx1).
%
% [Q,QD,QDD] = JTRAJ(Q0, QF, T, QD0, QDF) as above but specifies initial and 
% final joint velocity for the trajectory and a time vector.
%
% Notes::
% - When a time vector is provided the velocity and acceleration outputs
%   are scaled assumign that the time vector starts at zero and increases
%   linearly.
%
% See also QPLOT, CTRAJ, SerialLink.jtraj.




% Copyright (C) 1993-2017, by Peter I. Corke
%
% This file is part of The Robotics Toolbox for MATLAB (RTB).
% 
% RTB is free software: you can redistribute it and/or modify
% it under the terms of the GNU Lesser General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
% 
% RTB is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU Lesser General Public License for more details.
% 
% You should have received a copy of the GNU Leser General Public License
% along with RTB.  If not, see <http://www.gnu.org/licenses/>.
%
% http://www.petercorke.com

function [qt,qdt,qddt] = jtraj(q0, q1, tv, qd0, qd1)
    if length(tv) > 1
        tscal = max(tv);
        t = tv(:)/tscal;
    else
        tscal = 1;
        t = (0:(tv-1))'/(tv-1); % normalized time from 0 -> 1
    end

    q0 = q0(:);
    q1 = q1(:);

    if nargin == 3
        qd0 = zeros(size(q0));
        qd1 = qd0;
    elseif nargin == 5
        qd0 = qd0(:);
        qd1 = qd1(:);
    else
        error('incorrect number of arguments')
    end

    % compute the polynomial coefficients
    A = 6*(q1 - q0) - 3*(qd1+qd0)*tscal;
    B = -15*(q1 - q0) + (8*qd0 + 7*qd1)*tscal;
    C = 10*(q1 - q0) - (6*qd0 + 4*qd1)*tscal;
    E = qd0*tscal; % as the t vector has been normalized
    F = q0;

    tt = [t.^5 t.^4 t.^3 t.^2 t ones(size(t))];
    c = [A B C zeros(size(A)) E F]';
    
    qt = tt*c;

    % compute optional velocity
    if nargout >= 2
        c = [ zeros(size(A)) 5*A 4*B 3*C  zeros(size(A)) E ]';
        qdt = tt*c/tscal;
    end

    % compute optional acceleration
    if nargout == 3
        c = [ zeros(size(A))  zeros(size(A)) 20*A 12*B 6*C  zeros(size(A))]';
        qddt = tt*c/tscal^2;
    end
