Identification of the Stewart Platform using Simscape
+
+
+
+Table of Contents
+ +
+
+
+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
+
+