control_act.m

function [u, ut, uc, U_corr_history, q_pred] = control_act(t, q, sim_data)   
    dc = decouple_matrix(q, sim_data);
    ut = utrack(t, q, sim_data);
    [uc, U_corr_history, q_pred] = ucorr(t, q, sim_data);

    ut = dc*ut;
    %uc = dc*uc;
    %uc = zeros(2,1);

    u = ut+uc;
    % saturation
    u = min(sim_data.SATURATION, max(-sim_data.SATURATION, u));
end

function [u_corr, U_corr_history, q_pred] = ucorr(t, q, sim_data)
    pred_hor = sim_data.PREDICTION_HORIZON;
    SATURATION = sim_data.SATURATION;
    PREDICTION_SATURATION_TOLERANCE = sim_data.PREDICTION_SATURATION_TOLERANCE;
    tc = sim_data.tc;

    u_corr = zeros(2,1);
    U_corr_history = zeros(2,1,sim_data.PREDICTION_HORIZON);
    q_act = q;
    u_track_pred=zeros(2,1, pred_hor);
    T_inv_pred=zeros(2,2, pred_hor);

    q_pred = [];

    s_ = SATURATION - ones(2,1)*PREDICTION_SATURATION_TOLERANCE;
    if eq(pred_hor, 0)
        return
    elseif eq(pred_hor, 1)
        H = eye(2);
        f = zeros(2,1);
        T_inv = decouple_matrix(q_act, sim_data);
        ut = utrack(t, q_act, sim_data);
        %A = [T_inv; -T_inv];        
        A = [eye(2); -eye(2)];

        d = T_inv*ut;
        b = [s_-d;s_+d];
        
        % solve qp problem
        options = optimoptions('quadprog', 'Display', 'off');
        u_corr = quadprog(H, f, A, b, [],[],[],[],[],options);
        
        q_pred = q_act;
        U_corr_history(:,:,1) = u_corr;
        return
    else
        %if pred_hor > 1
            % move the horizon over 1 step and add trailing zeroes
            U_corr_history = cat(3, sim_data.U_corr_history(:,:, 2:end), zeros(2,1));
        %end
    
        %disp('start of simulation')
        % for each step in the prediction horizon, integrate the system to
        % predict its future state
    
        for k = 1:pred_hor
            % start from the old (known) state
    
            % compute the inputs, based on the old state
    
            % u_corr is the prediction done at some time in the past, as found in U_corr_history
            u_corr_ = U_corr_history(:, :, k);
            % u_track can be computed from q
            t_ = t + tc * (k-1);
            u_track_ = utrack(t_, q_act, sim_data);
            
            T_inv = decouple_matrix(q_act, sim_data);
            % compute inputs (v, w)/(wr, wl)
            u_ = T_inv * u_track_ + u_corr_;
            

            % if needed, map (wr, wl) to (v, w) for unicicle
            if eq(sim_data.robot, 1)
                u_ = diffdrive_to_uni(u_, sim_data);
            end
    
            % integrate unicycle
            theta_new = q_act(3) + tc*u_(2);
            % compute the state integrating with euler
            %x_new = q_act(1) + tc*u_(1) * cos(q_act(3));
            %y_new = q_act(2) + tc*u_(1) * sin(q_act(3));
            % compute the state integrating via runge-kutta
            x_new = q_act(1) + tc*u_(1) * cos(q_act(3) + 0.5*tc*u_(2));
            y_new = q_act(2) + tc*u_(1) * sin(q_act(3) + 0.5*tc*u_(2));
    
            q_new = [x_new; y_new; theta_new];
    
            % save history
            q_pred = [q_pred; q_new'];
            u_track_pred(:,:,k) = u_track_;
            T_inv_pred(:,:,k) = T_inv;
    
            % Prepare old state for next iteration
            q_act = q_new;
        end
    
        %{
            Now setup the qp problem
            It needs:
            - Unknowns, u_corr at each timestep. Will be encoded as a vector of
            vectors, in which every two elements is a u_j
            i.e. (u_1; u_2; u_3; ...; u_C) = (v_1; w_1; v_2, w_2; v_3, w_3; ...
            ; v_C, w_C)
            It is essential that the vector stays a column, so that u'u is the
            sum of the squared norms of each u_j
    
            - Box constraints: a constraint for each timestep in the horizon.
            Calculated using the predicted state and inputs. They need to be
            put in matrix (Ax <= b) form
        %}
    
        % box constrains
        % A becomes sort of block-diagonal
        % A will be at most PREDICTION_HORIZON * 2 * 2 (2: size of T_inv; 2:
        % accounting for T_inv and -T_inv) by PREDICTION_HORIZON (number of
        % vectors in u_corr times the number of elements [2] in each vector)
        b_deq = [];
        for k=1:pred_hor
            T_inv = T_inv_pred(:,:,k);
            u_track = u_track_pred(:,:,k);
            d = T_inv*u_track;
            b_deq = [b_deq; s_ - d; s_ + d];
        end
        
        A_deq = kron(eye(pred_hor), [eye(2); -eye(2)]);
        %A_deq
        %b_deq
        % unknowns
       
        % squared norm of u_corr. H must be identity,
        % PREDICTION_HORIZON*size(u_corr)
        H = eye(pred_hor*2)*2;
        % no linear terms
        f = zeros(pred_hor*2, 1);
    
        % solve qp problem
        options = optimoptions('quadprog', 'Display', 'off');
        U_corr = quadprog(H, f, A_deq, b_deq, [],[],[],[],[],options);
        %U_corr = lsqnonlin(@(pred_hor) ones(pred_hor, 1), U_corr_history(:,:,1), [], [], A_deq, b_deq, [], []);
    
        % reshape the vector of vectors to be an array, each element being
        % u_corr_j as a 2x1 vector
        % and add the prediction at t_k+C
        U_corr_history = reshape(U_corr, [2,1,pred_hor]);
        % first result is what to do now
        u_corr=U_corr_history(:,:, 1);
    end
end

function u_track = utrack(t, q, sim_data)    
    ref_s = double(subs(sim_data.ref, t));
    dref_s = double(subs(sim_data.dref, t));
    
    f = feedback(q, sim_data.b);
    err = ref_s - f;
    u_track = dref_s + sim_data.K*err;
end

function q_track = feedback(q, b)
    q_track = [q(1) + b*cos(q(3)); q(2) + b*sin(q(3))  ];
end

function T_inv = decouple_matrix(q, sim_data)
    theta = q(3);

    st = sin(theta);
    ct = cos(theta);
    
    b = sim_data.b;

    if eq(sim_data.robot, 0)
        T_inv = [ct, st; -st/b, ct/b];
    elseif eq(sim_data.robot, 1)
        r = sim_data.r;
        d = sim_data.d;
        
        T_inv = [2*b*ct - d*st, d*ct + 2*b*st ; 2*b*ct + d*st, -d*ct+2*b*st] / (2*b*r);
    end
end