Identification of the Stewart Platform using Simscape

1. Identification
2. Cartesian Plot
3. From a force to force sensor
4. From a force applied in the leg to the displacement of the leg
5. Transmissibility
6. Compliance
7. Inertial
8. identifyPlant

1 Identification

The hexapod structure and Sample structure are initialized.

initializeHexapod();
initializeSample();

G = identifyPlant();

2 Cartesian Plot

From a force applied in the Cartesian frame to a displacement in the Cartesian frame.

figure;
hold on;
bode(G.G_cart(1, 1));
bode(G.G_cart(3, 3));
hold off;

3 From a force to force sensor

figure;
hold on;
bode(G.G_forc(1, 1));
bode(G.G_forc(2, 2));
bode(G.G_forc(3, 3));
bode(G.G_forc(4, 4));
bode(G.G_forc(5, 5));
bode(G.G_forc(6, 6));
hold off;

figure;
hold on;
bode(G.G_forc(1, 1));
bode(G.G_forc(1, 2));
bode(G.G_forc(1, 3));
bode(G.G_forc(1, 4));
bode(G.G_forc(1, 5));
bode(G.G_forc(1, 6));
hold off;

4 From a force applied in the leg to the displacement of the leg

figure;
hold on;
bode(G.G_legs(1, 1));
bode(G.G_legs(2, 2));
bode(G.G_legs(3, 3));
bode(G.G_legs(4, 4));
bode(G.G_legs(5, 5));
bode(G.G_legs(6, 6));
hold off;

figure;
hold on;
bode(G.G_legs(1, 1));
bode(G.G_legs(1, 2));
bode(G.G_legs(1, 3));
bode(G.G_legs(1, 4));
bode(G.G_legs(1, 5));
bode(G.G_legs(1, 6));
hold off;

5 Transmissibility

figure;
hold on;
bode(G.G_tran(1, 1));
bode(G.G_tran(2, 2));
bode(G.G_tran(3, 3));
hold off;

figure;
hold on;
bode(G.G_tran(4, 4));
bode(G.G_tran(5, 5));
bode(G.G_tran(6, 6));
hold off;

figure;
hold on;
bode(G.G_tran(1, 1));
bode(G.G_tran(2, 1));
bode(G.G_tran(3, 1));
hold off;

6 Compliance

From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.

figure;
hold on;
bode(G.G_comp(1, 1));
bode(G.G_comp(2, 2));
bode(G.G_comp(3, 3));
hold off;

7 Inertial

From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.

figure;
hold on;
bode(G.G_iner(1, 1));
bode(G.G_iner(2, 2));
bode(G.G_iner(3, 3));
hold off;

8 identifyPlant

function [sys] = identifyPlant(opts_param)

We use this code block to pass optional parameters.

%% Default values for opts
opts = struct();

%% Populate opts with input parameters
if exist('opts_param','var')
    for opt = fieldnames(opts_param)'
        opts.(opt{1}) = opts_param.(opt{1});
    end
end

We defined the options for the linearize command. Here, we just identify the system at time \(t = 0\).

options = linearizeOptions;
options.SampleTime = 0;

We define the name of the Simulink File used to identification.

mdl = 'stewart';

Then we defined the input/output of the transfer function we want to identify.

%% Inputs
io(1) = linio([mdl, '/F'],  1, 'input'); % Cartesian forces
io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion

%% Outputs
io(5) = linio([mdl, '/Dm'],  1, 'output'); % Relative Motion
io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
io(8) = linio([mdl, '/Xm'],  1, 'output'); % Absolute motion of platform

The linearization is run.

G = linearize(mdl, io, 0);

We defined all the Input/Output names of the identified transfer function.

G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
                'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
                'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
                'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
                'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
                'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
                'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};

We split the transfer function into sub transfer functions and we compute their minimum realization.

sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
sys.G_forc = minreal(G({'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_legs = minreal(G({'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_tran = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
sys.G_comp = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
% sys.G_all  = minreal(G);

end