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fb81352caf
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merged
| Author | SHA1 | Date | |
|---|---|---|---|
| 43eafb2c8f | |||
| e753cb6e75 | |||
| 6876adf4f7 | |||
| 28c4fbe083 | |||
| a48225c18b | |||
| 42de69005d | |||
| 647b360342 |
@@ -6,7 +6,7 @@
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@@ -1645,5 +1645,16 @@
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@@ -1,45 +1,48 @@
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function [granite] = initializeGranite(args)
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function [granite] = initializeGranite(args)
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arguments
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args.type char {mustBeMember(args.type,{'rigid', 'flexible', 'none'})} = 'flexible'
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args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3]
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args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = [5e9; 5e9; 5e9; 2.5e7; 2.5e7; 1e7] % [N/m]
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args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = [4.0e5; 1.1e5; 9.0e5; 2e4; 2e4; 1e4] % [N/(m/s)]
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args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m]
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args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m]
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args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m]
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args.sample_pos (1,1) double {mustBeNumeric} = 0.8 % Height of the measurment point [m]
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end
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granite = struct();
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switch args.type
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case 'none'
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granite.type = 0;
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case 'rigid'
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granite.type = 1;
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case 'flexible'
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granite.type = 2;
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end
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granite.density = args.density; % [kg/m3]
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granite.STEP = 'granite.STEP';
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granite.sample_pos = args.sample_pos; % [m]
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granite.K = args.K; % [N/m]
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granite.C = args.C; % [N/(m/s)]
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if exist('./mat', 'dir')
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if exist('./mat/nass_model_stages.mat', 'file')
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save('mat/nass_model_stages.mat', 'granite', '-append');
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else
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save('mat/nass_model_stages.mat', 'granite');
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arguments
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args.type char {mustBeMember(args.type,{'rigid', 'flexible', 'none'})} = 'flexible'
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args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3]
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args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = [5e9; 5e9; 5e9; 2.5e7; 2.5e7; 1e7] % [N/m]
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args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = [4.0e5; 1.1e5; 9.0e5; 2e4; 2e4; 1e4] % [N/(m/s)]
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args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m]
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args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m]
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args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m]
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args.sample_pos (1,1) double {mustBeNumeric} = 0.775 % Height of the measurment point [m]
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end
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elseif exist('./matlab', 'dir')
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if exist('./matlab/mat/nass_model_stages.mat', 'file')
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save('matlab/mat/nass_model_stages.mat', 'granite', '-append');
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else
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save('matlab/mat/nass_model_stages.mat', 'granite');
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granite = struct();
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switch args.type
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case 'none'
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granite.type = 0;
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case 'rigid'
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granite.type = 1;
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case 'flexible'
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granite.type = 2;
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end
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granite.density = args.density; % [kg/m3]
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granite.STEP = 'granite.STEP';
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% Z-offset for the initial position of the sample with respect to the granite top surface.
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granite.sample_pos = args.sample_pos; % [m]
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granite.K = args.K; % [N/m]
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granite.C = args.C; % [N/(m/s)]
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if exist('./mat', 'dir')
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if exist('./mat/nass_model_stages.mat', 'file')
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save('mat/nass_model_stages.mat', 'granite', '-append');
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else
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save('mat/nass_model_stages.mat', 'granite');
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end
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elseif exist('./matlab', 'dir')
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if exist('./matlab/mat/nass_model_stages.mat', 'file')
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save('matlab/mat/nass_model_stages.mat', 'granite', '-append');
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else
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save('matlab/mat/nass_model_stages.mat', 'granite');
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end
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end
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end
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@@ -1,34 +1,35 @@
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function [ground] = initializeGround(args)
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function [ground] = initializeGround(args)
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arguments
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args.type char {mustBeMember(args.type,{'none', 'rigid'})} = 'rigid'
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args.rot_point (3,1) double {mustBeNumeric} = zeros(3,1) % Rotation point for the ground motion [m]
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end
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ground = struct();
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switch args.type
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case 'none'
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ground.type = 0;
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case 'rigid'
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ground.type = 1;
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end
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ground.shape = [2, 2, 0.5]; % [m]
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ground.density = 2800; % [kg/m3]
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ground.rot_point = args.rot_point;
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if exist('./mat', 'dir')
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if exist('./mat/nass_model_stages.mat', 'file')
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save('mat/nass_model_stages.mat', 'ground', '-append');
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else
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save('mat/nass_model_stages.mat', 'ground');
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arguments
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args.type char {mustBeMember(args.type,{'none', 'rigid'})} = 'rigid'
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args.rot_point (3,1) double {mustBeNumeric} = zeros(3,1) % Rotation point for the ground motion [m]
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end
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elseif exist('./matlab', 'dir')
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if exist('./matlab/mat/nass_model_stages.mat', 'file')
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save('matlab/mat/nass_model_stages.mat', 'ground', '-append');
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else
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save('matlab/mat/nass_model_stages.mat', 'ground');
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ground = struct();
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switch args.type
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case 'none'
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ground.type = 0;
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case 'rigid'
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ground.type = 1;
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end
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ground.shape = [2, 2, 0.5]; % [m]
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ground.density = 2800; % [kg/m3]
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ground.rot_point = args.rot_point;
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if exist('./mat', 'dir')
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if exist('./mat/nass_model_stages.mat', 'file')
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save('mat/nass_model_stages.mat', 'ground', '-append');
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else
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save('mat/nass_model_stages.mat', 'ground');
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end
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elseif exist('./matlab', 'dir')
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if exist('./matlab/mat/nass_model_stages.mat', 'file')
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save('matlab/mat/nass_model_stages.mat', 'ground', '-append');
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else
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save('matlab/mat/nass_model_stages.mat', 'ground');
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end
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end
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end
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@@ -1,107 +1,108 @@
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function [micro_hexapod] = initializeMicroHexapod(args)
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function [micro_hexapod] = initializeMicroHexapod(args)
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arguments
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args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
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% initializeFramesPositions
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args.H (1,1) double {mustBeNumeric, mustBePositive} = 350e-3
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||||
args.MO_B (1,1) double {mustBeNumeric} = 270e-3
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||||
% generateGeneralConfiguration
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||||
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
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args.FR (1,1) double {mustBeNumeric, mustBePositive} = 175.5e-3
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args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180)
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args.MH (1,1) double {mustBeNumeric, mustBePositive} = 45e-3
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args.MR (1,1) double {mustBeNumeric, mustBePositive} = 118e-3
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args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180)
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% initializeStrutDynamics
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args.Ki (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e7*ones(6,1)
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args.Ci (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.4e3*ones(6,1)
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% initializeCylindricalPlatforms
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args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 10
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args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
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args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 207.5e-3
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args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 10
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args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
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args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3
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% initializeCylindricalStruts
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args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1
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args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
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args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
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args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1
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args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
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args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
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% inverseKinematics
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args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
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args.ARB (3,3) double {mustBeNumeric} = eye(3)
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end
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, ...
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'H', args.H, ...
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'MO_B', args.MO_B);
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stewart = generateGeneralConfiguration(stewart, ...
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'FH', args.FH, ...
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'FR', args.FR, ...
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'FTh', args.FTh, ...
|
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'MH', args.MH, ...
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'MR', args.MR, ...
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'MTh', args.MTh);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, ...
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'K', args.Ki, ...
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'C', args.Ci);
|
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|
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stewart = initializeJointDynamics(stewart, ...
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'type_F', 'universal_p', ...
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'type_M', 'spherical_p');
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|
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stewart = initializeCylindricalPlatforms(stewart, ...
|
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'Fpm', args.Fpm, ...
|
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'Fph', args.Fph, ...
|
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'Fpr', args.Fpr, ...
|
||||
'Mpm', args.Mpm, ...
|
||||
'Mph', args.Mph, ...
|
||||
'Mpr', args.Mpr);
|
||||
|
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stewart = initializeCylindricalStruts(stewart, ...
|
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'Fsm', args.Fsm, ...
|
||||
'Fsh', args.Fsh, ...
|
||||
'Fsr', args.Fsr, ...
|
||||
'Msm', args.Msm, ...
|
||||
'Msh', args.Msh, ...
|
||||
'Msr', args.Msr);
|
||||
|
||||
stewart = computeJacobian(stewart);
|
||||
|
||||
stewart = initializeStewartPose(stewart, ...
|
||||
'AP', args.AP, ...
|
||||
'ARB', args.ARB);
|
||||
|
||||
stewart = initializeInertialSensor(stewart, 'type', 'none');
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
stewart.type = 0;
|
||||
case 'rigid'
|
||||
stewart.type = 1;
|
||||
case 'flexible'
|
||||
stewart.type = 2;
|
||||
end
|
||||
|
||||
micro_hexapod = stewart;
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'micro_hexapod', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'micro_hexapod');
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
% initializeFramesPositions
|
||||
args.H (1,1) double {mustBeNumeric, mustBePositive} = 350e-3
|
||||
args.MO_B (1,1) double {mustBeNumeric} = 270e-3
|
||||
% generateGeneralConfiguration
|
||||
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
|
||||
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 175.5e-3
|
||||
args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180)
|
||||
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 45e-3
|
||||
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 118e-3
|
||||
args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180)
|
||||
% initializeStrutDynamics
|
||||
args.Ki (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e7*ones(6,1)
|
||||
args.Ci (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.4e3*ones(6,1)
|
||||
% initializeCylindricalPlatforms
|
||||
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 10
|
||||
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
|
||||
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 207.5e-3
|
||||
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 10
|
||||
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
|
||||
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3
|
||||
% initializeCylindricalStruts
|
||||
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
|
||||
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
|
||||
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
|
||||
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
|
||||
% inverseKinematics
|
||||
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
||||
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'micro_hexapod', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'micro_hexapod');
|
||||
|
||||
stewart = initializeStewartPlatform();
|
||||
|
||||
stewart = initializeFramesPositions(stewart, ...
|
||||
'H', args.H, ...
|
||||
'MO_B', args.MO_B);
|
||||
|
||||
stewart = generateGeneralConfiguration(stewart, ...
|
||||
'FH', args.FH, ...
|
||||
'FR', args.FR, ...
|
||||
'FTh', args.FTh, ...
|
||||
'MH', args.MH, ...
|
||||
'MR', args.MR, ...
|
||||
'MTh', args.MTh);
|
||||
|
||||
stewart = computeJointsPose(stewart);
|
||||
|
||||
stewart = initializeStrutDynamics(stewart, ...
|
||||
'K', args.Ki, ...
|
||||
'C', args.Ci);
|
||||
|
||||
stewart = initializeJointDynamics(stewart, ...
|
||||
'type_F', 'universal_p', ...
|
||||
'type_M', 'spherical_p');
|
||||
|
||||
stewart = initializeCylindricalPlatforms(stewart, ...
|
||||
'Fpm', args.Fpm, ...
|
||||
'Fph', args.Fph, ...
|
||||
'Fpr', args.Fpr, ...
|
||||
'Mpm', args.Mpm, ...
|
||||
'Mph', args.Mph, ...
|
||||
'Mpr', args.Mpr);
|
||||
|
||||
stewart = initializeCylindricalStruts(stewart, ...
|
||||
'Fsm', args.Fsm, ...
|
||||
'Fsh', args.Fsh, ...
|
||||
'Fsr', args.Fsr, ...
|
||||
'Msm', args.Msm, ...
|
||||
'Msh', args.Msh, ...
|
||||
'Msr', args.Msr);
|
||||
|
||||
stewart = computeJacobian(stewart);
|
||||
|
||||
stewart = initializeStewartPose(stewart, ...
|
||||
'AP', args.AP, ...
|
||||
'ARB', args.ARB);
|
||||
|
||||
stewart = initializeInertialSensor(stewart, 'type', 'none');
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
stewart.type = 0;
|
||||
case 'rigid'
|
||||
stewart.type = 1;
|
||||
case 'flexible'
|
||||
stewart.type = 2;
|
||||
end
|
||||
|
||||
micro_hexapod = stewart;
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'micro_hexapod', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'micro_hexapod');
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'micro_hexapod', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'micro_hexapod');
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@@ -1,54 +1,57 @@
|
||||
function [ry] = initializeRy(args)
|
||||
function [ry] = initializeRy(args)
|
||||
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
args.Ry_init (1,1) double {mustBeNumeric} = 0
|
||||
end
|
||||
|
||||
ry = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
ry.type = 0;
|
||||
case 'rigid'
|
||||
ry.type = 1;
|
||||
case 'flexible'
|
||||
ry.type = 2;
|
||||
end
|
||||
|
||||
% Ry - Guide for the tilt stage
|
||||
ry.guide.density = 7800; % [kg/m3]
|
||||
ry.guide.STEP = 'Tilt_Guide.STEP';
|
||||
|
||||
% Ry - Rotor of the motor
|
||||
ry.rotor.density = 2400; % [kg/m3]
|
||||
ry.rotor.STEP = 'Tilt_Motor_Axis.STEP';
|
||||
|
||||
% Ry - Motor
|
||||
ry.motor.density = 3200; % [kg/m3]
|
||||
ry.motor.STEP = 'Tilt_Motor.STEP';
|
||||
|
||||
% Ry - Plateau Tilt
|
||||
ry.stage.density = 7800; % [kg/m3]
|
||||
ry.stage.STEP = 'Tilt_Stage.STEP';
|
||||
|
||||
ry.z_offset = 0.58178; % [m]
|
||||
|
||||
ry.Ry_init = args.Ry_init; % [rad]
|
||||
|
||||
ry.K = [3.8e8; 4e8; 3.8e8; 1.2e8; 6e4; 1.2e8];
|
||||
ry.C = [1e5; 1e5; 1e5; 3e4; 1e3; 3e4];
|
||||
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'ry', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'ry');
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
args.Ry_init (1,1) double {mustBeNumeric} = 0
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'ry', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'ry');
|
||||
|
||||
ry = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
ry.type = 0;
|
||||
case 'rigid'
|
||||
ry.type = 1;
|
||||
case 'flexible'
|
||||
ry.type = 2;
|
||||
end
|
||||
|
||||
% Ry - Guide for the tilt stage
|
||||
ry.guide.density = 7800; % [kg/m3]
|
||||
ry.guide.STEP = 'Tilt_Guide.STEP';
|
||||
|
||||
% Ry - Rotor of the motor
|
||||
ry.rotor.density = 2400; % [kg/m3]
|
||||
ry.rotor.STEP = 'Tilt_Motor_Axis.STEP';
|
||||
|
||||
% Ry - Motor
|
||||
ry.motor.density = 3200; % [kg/m3]
|
||||
ry.motor.STEP = 'Tilt_Motor.STEP';
|
||||
|
||||
% Ry - Plateau Tilt
|
||||
ry.stage.density = 7800; % [kg/m3]
|
||||
ry.stage.STEP = 'Tilt_Stage.STEP';
|
||||
|
||||
% Z-Offset so that the center of rotation matches the sample center;
|
||||
ry.z_offset = 0.58178; % [m]
|
||||
|
||||
ry.Ry_init = args.Ry_init; % [rad]
|
||||
|
||||
ry.K = [3.8e8; 4e8; 3.8e8; 1.2e8; 6e4; 1.2e8];
|
||||
ry.C = [1e5; 1e5; 1e5; 3e4; 1e3; 3e4];
|
||||
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'ry', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'ry');
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'ry', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'ry');
|
||||
end
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
@@ -1,45 +1,47 @@
|
||||
function [rz] = initializeRz(args)
|
||||
function [rz] = initializeRz(args)
|
||||
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
end
|
||||
|
||||
rz = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
rz.type = 0;
|
||||
case 'rigid'
|
||||
rz.type = 1;
|
||||
case 'flexible'
|
||||
rz.type = 2;
|
||||
end
|
||||
|
||||
% Spindle - Slip Ring
|
||||
rz.slipring.density = 7800; % [kg/m3]
|
||||
rz.slipring.STEP = 'Spindle_Slip_Ring.STEP';
|
||||
|
||||
% Spindle - Rotor
|
||||
rz.rotor.density = 7800; % [kg/m3]
|
||||
rz.rotor.STEP = 'Spindle_Rotor.STEP';
|
||||
|
||||
% Spindle - Stator
|
||||
rz.stator.density = 7800; % [kg/m3]
|
||||
rz.stator.STEP = 'Spindle_Stator.STEP';
|
||||
|
||||
rz.K = [7e8; 7e8; 2e9; 1e7; 1e7; 1e7];
|
||||
rz.C = [4e4; 4e4; 7e4; 1e4; 1e4; 1e4];
|
||||
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'rz', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'rz');
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'rz', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'rz');
|
||||
|
||||
rz = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
rz.type = 0;
|
||||
case 'rigid'
|
||||
rz.type = 1;
|
||||
case 'flexible'
|
||||
rz.type = 2;
|
||||
end
|
||||
|
||||
% Spindle - Slip Ring
|
||||
rz.slipring.density = 7800; % [kg/m3]
|
||||
rz.slipring.STEP = 'Spindle_Slip_Ring.STEP';
|
||||
|
||||
% Spindle - Rotor
|
||||
rz.rotor.density = 7800; % [kg/m3]
|
||||
rz.rotor.STEP = 'Spindle_Rotor.STEP';
|
||||
|
||||
% Spindle - Stator
|
||||
rz.stator.density = 7800; % [kg/m3]
|
||||
rz.stator.STEP = 'Spindle_Stator.STEP';
|
||||
|
||||
rz.K = [7e8; 7e8; 2e9; 1e7; 1e7; 1e7];
|
||||
rz.C = [4e4; 4e4; 7e4; 1e4; 1e4; 1e4];
|
||||
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'rz', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'rz');
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'rz', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'rz');
|
||||
end
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
@@ -1,69 +1,71 @@
|
||||
function [ty] = initializeTy(args)
|
||||
function [ty] = initializeTy(args)
|
||||
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
end
|
||||
|
||||
ty = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
ty.type = 0;
|
||||
case 'rigid'
|
||||
ty.type = 1;
|
||||
case 'flexible'
|
||||
ty.type = 2;
|
||||
end
|
||||
|
||||
% Ty Granite frame
|
||||
ty.granite_frame.density = 7800; % [kg/m3] => 43kg
|
||||
ty.granite_frame.STEP = 'Ty_Granite_Frame.STEP';
|
||||
|
||||
% Guide Translation Ty
|
||||
ty.guide.density = 7800; % [kg/m3] => 76kg
|
||||
ty.guide.STEP = 'Ty_Guide.STEP';
|
||||
|
||||
% Ty - Guide_Translation12
|
||||
ty.guide12.density = 7800; % [kg/m3]
|
||||
ty.guide12.STEP = 'Ty_Guide_12.STEP';
|
||||
|
||||
% Ty - Guide_Translation11
|
||||
ty.guide11.density = 7800; % [kg/m3]
|
||||
ty.guide11.STEP = 'Ty_Guide_11.STEP';
|
||||
|
||||
% Ty - Guide_Translation22
|
||||
ty.guide22.density = 7800; % [kg/m3]
|
||||
ty.guide22.STEP = 'Ty_Guide_22.STEP';
|
||||
|
||||
% Ty - Guide_Translation21
|
||||
ty.guide21.density = 7800; % [kg/m3]
|
||||
ty.guide21.STEP = 'Ty_Guide_21.STEP';
|
||||
|
||||
% Ty - Plateau translation
|
||||
ty.frame.density = 7800; % [kg/m3]
|
||||
ty.frame.STEP = 'Ty_Stage.STEP';
|
||||
|
||||
% Ty Stator Part
|
||||
ty.stator.density = 5400; % [kg/m3]
|
||||
ty.stator.STEP = 'Ty_Motor_Stator.STEP';
|
||||
|
||||
% Ty Rotor Part
|
||||
ty.rotor.density = 5400; % [kg/m3]
|
||||
ty.rotor.STEP = 'Ty_Motor_Rotor.STEP';
|
||||
|
||||
ty.K = [2e8; 1e8; 2e8; 6e7; 9e7; 6e7]; % [N/m, N*m/rad]
|
||||
ty.C = [8e4; 5e4; 8e4; 2e4; 3e4; 1e4]; % [N/(m/s), N*m/(rad/s)]
|
||||
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'ty', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'ty');
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'ty', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'ty');
|
||||
|
||||
ty = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
ty.type = 0;
|
||||
case 'rigid'
|
||||
ty.type = 1;
|
||||
case 'flexible'
|
||||
ty.type = 2;
|
||||
end
|
||||
|
||||
% Ty Granite frame
|
||||
ty.granite_frame.density = 7800; % [kg/m3] => 43kg
|
||||
ty.granite_frame.STEP = 'Ty_Granite_Frame.STEP';
|
||||
|
||||
% Guide Translation Ty
|
||||
ty.guide.density = 7800; % [kg/m3] => 76kg
|
||||
ty.guide.STEP = 'Ty_Guide.STEP';
|
||||
|
||||
% Ty - Guide_Translation12
|
||||
ty.guide12.density = 7800; % [kg/m3]
|
||||
ty.guide12.STEP = 'Ty_Guide_12.STEP';
|
||||
|
||||
% Ty - Guide_Translation11
|
||||
ty.guide11.density = 7800; % [kg/m3]
|
||||
ty.guide11.STEP = 'Ty_Guide_11.STEP';
|
||||
|
||||
% Ty - Guide_Translation22
|
||||
ty.guide22.density = 7800; % [kg/m3]
|
||||
ty.guide22.STEP = 'Ty_Guide_22.STEP';
|
||||
|
||||
% Ty - Guide_Translation21
|
||||
ty.guide21.density = 7800; % [kg/m3]
|
||||
ty.guide21.STEP = 'Ty_Guide_21.STEP';
|
||||
|
||||
% Ty - Plateau translation
|
||||
ty.frame.density = 7800; % [kg/m3]
|
||||
ty.frame.STEP = 'Ty_Stage.STEP';
|
||||
|
||||
% Ty Stator Part
|
||||
ty.stator.density = 5400; % [kg/m3]
|
||||
ty.stator.STEP = 'Ty_Motor_Stator.STEP';
|
||||
|
||||
% Ty Rotor Part
|
||||
ty.rotor.density = 5400; % [kg/m3]
|
||||
ty.rotor.STEP = 'Ty_Motor_Rotor.STEP';
|
||||
|
||||
ty.K = [2e8; 1e8; 2e8; 6e7; 9e7; 6e7]; % [N/m, N*m/rad]
|
||||
ty.C = [8e4; 5e4; 8e4; 2e4; 3e4; 1e4]; % [N/(m/s), N*m/(rad/s)]
|
||||
|
||||
if exist('./mat', 'dir')
|
||||
if exist('./mat/nass_model_stages.mat', 'file')
|
||||
save('mat/nass_model_stages.mat', 'ty', '-append');
|
||||
else
|
||||
save('mat/nass_model_stages.mat', 'ty');
|
||||
end
|
||||
elseif exist('./matlab', 'dir')
|
||||
if exist('./matlab/mat/nass_model_stages.mat', 'file')
|
||||
save('matlab/mat/nass_model_stages.mat', 'ty', '-append');
|
||||
else
|
||||
save('matlab/mat/nass_model_stages.mat', 'ty');
|
||||
end
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
@@ -1,5 +1,3 @@
|
||||
% Matlab Init :noexport:ignore:
|
||||
|
||||
%% ustation_1_kinematics.m
|
||||
|
||||
%% Clear Workspace and Close figures
|
||||
@@ -25,76 +23,6 @@ colors = colororder;
|
||||
%% Frequency Vector
|
||||
freqs = logspace(log10(10), log10(2e3), 1000);
|
||||
|
||||
% Micro-Station Kinematics
|
||||
% <<ssec:ustation_kinematics>>
|
||||
|
||||
% Each stage is described by two frames; one is attached to the fixed platform $\{A\}$ while the other is fixed to the mobile platform $\{B\}$.
|
||||
% At "rest" position, the two have the same pose and coincide with the point of interest ($O_A = O_B$).
|
||||
% An example of the tilt stage is shown in Figure ref:fig:ustation_stage_motion.
|
||||
% The mobile frame of the translation stage is equal to the fixed frame of the tilt stage: $\{B_{D_y}\} = \{A_{R_y}\}$.
|
||||
% Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the spindle: $\{B_{R_y}\} = \{A_{R_z}\}$.
|
||||
|
||||
% #+name: fig:ustation_stage_motion
|
||||
% #+caption: Example of the motion induced by the tilt-stage $R_y$. "Rest" position in shown in blue while a arbitrary position in shown in red. Parasitic motions are here magnified for clarity.
|
||||
% [[file:figs/ustation_stage_motion.png]]
|
||||
|
||||
% The motion induced by a positioning stage can be described by a homogeneous transformation matrix from frame $\{A\}$ to frame $\{B\}$ as explain in Section ref:ssec:ustation_kinematics.
|
||||
% As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in eqref:eq:ustation_translation_stage_errors.
|
||||
|
||||
% \begin{equation}\label{eq:ustation_translation_stage_errors}
|
||||
% {}^A\mathbf{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) =
|
||||
% \left[ \begin{array}{ccc|c}
|
||||
% & & & D_x \\
|
||||
% & \mathbf{R}_x(\theta_x) \mathbf{R}_y(\theta_y) \mathbf{R}_z(\theta_z) & & D_y \\
|
||||
% & & & D_z \cr
|
||||
% \hline
|
||||
% 0 & 0 & 0 & 1
|
||||
% \end{array} \right]
|
||||
% \end{equation}
|
||||
|
||||
% The homogeneous transformation matrix corresponding to the micro-station $\mathbf{T}_{\mu\text{-station}}$ is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation eqref:eq:ustation_transformation_station.
|
||||
|
||||
% \begin{equation}\label{eq:ustation_transformation_station}
|
||||
% \mathbf{T}_{\mu\text{-station}} = \mathbf{T}_{D_y} \cdot \mathbf{T}_{R_y} \cdot \mathbf{T}_{R_z} \cdot \mathbf{T}_{\mu\text{-hexapod}}
|
||||
% \end{equation}
|
||||
|
||||
% $\mathbf{T}_{\mu\text{-station}}$ represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite.
|
||||
|
||||
% If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), $\mathbf{T}_{\mu\text{-station}}$ then represent the pose setpoint of the sample with respect to the granite.
|
||||
% The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation eqref:eq:ustation_transformation_matrices_stages.
|
||||
|
||||
% \begin{equation}\label{eq:ustation_transformation_matrices_stages}
|
||||
% \begin{align}
|
||||
% \mathbf{T}_{D_y} &= \begin{bmatrix}
|
||||
% 1 & 0 & 0 & 0 \\
|
||||
% 0 & 1 & 0 & D_y \\
|
||||
% 0 & 0 & 1 & 0 \\
|
||||
% 0 & 0 & 0 & 1
|
||||
% \end{bmatrix} \quad
|
||||
% \mathbf{T}_{\mu\text{-hexapod}} =
|
||||
% \left[ \begin{array}{ccc|c}
|
||||
% & & & D_{\mu x} \\
|
||||
% & \mathbf{R}_x(\theta_{\mu x}) \mathbf{R}_y(\theta_{\mu y}) \mathbf{R}_{z}(\theta_{\mu z}) & & D_{\mu y} \\
|
||||
% & & & D_{\mu z} \cr
|
||||
% \hline
|
||||
% 0 & 0 & 0 & 1
|
||||
% \end{array} \right] \\
|
||||
% \mathbf{T}_{R_z} &= \begin{bmatrix}
|
||||
% \cos(\theta_z) & -\sin(\theta_z) & 0 & 0 \\
|
||||
% \sin(\theta_z) & \cos(\theta_z) & 0 & 0 \\
|
||||
% 0 & 0 & 1 & 0 \\
|
||||
% 0 & 0 & 0 & 1
|
||||
% \end{bmatrix} \quad
|
||||
% \mathbf{T}_{R_y} = \begin{bmatrix}
|
||||
% \cos(\theta_y) & 0 & \sin(\theta_y) & 0 \\
|
||||
% 0 & 1 & 0 & 0 \\
|
||||
% -\sin(\theta_y) & 0 & \cos(\theta_y) & 0 \\
|
||||
% 0 & 0 & 0 & 1
|
||||
% \end{bmatrix}
|
||||
% \end{align}
|
||||
% \end{equation}
|
||||
|
||||
|
||||
%% Stage setpoints
|
||||
Dy = 1e-3; % Translation Stage [m]
|
||||
Ry = 3*pi/180; % Tilt Stage [rad]
|
||||
|
||||
@@ -1,5 +1,3 @@
|
||||
% Matlab Init :noexport:ignore:
|
||||
|
||||
%% ustation_2_modeling.m
|
||||
|
||||
%% Clear Workspace and Close figures
|
||||
@@ -25,18 +23,6 @@ colors = colororder;
|
||||
%% Frequency Vector
|
||||
freqs = logspace(log10(10), log10(2e3), 1000);
|
||||
|
||||
% Comparison with the measured dynamics
|
||||
% <<ssec:ustation_model_comp_dynamics>>
|
||||
|
||||
% The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions.
|
||||
% The obtained FRFs were then projected at the CoM of each stage.
|
||||
|
||||
% To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the Simscape model and compared with the measurements in Figure ref:fig:ustation_comp_com_response.
|
||||
|
||||
% Even though there is some similarity between the model and the measurements (similar overall shapes and amplitudes), it is clear that the Simscape model does not accurately represent the complex micro-station dynamics.
|
||||
% Tuning the numerous model parameters to better match the measurements is a highly non-linear optimization problem that is difficult to solve in practice.
|
||||
|
||||
|
||||
%% Indentify the model dynamics from the 3 hammer imapcts
|
||||
% To the motion of each solid body at their CoM
|
||||
|
||||
@@ -134,75 +120,6 @@ leg.ItemTokenSize(1) = 15;
|
||||
xlim([10, 200]);
|
||||
ylim([1e-6, 1e-1])
|
||||
|
||||
% Micro-station compliance
|
||||
% <<ssec:ustation_model_compliance>>
|
||||
|
||||
% As discussed in the previous section, the dynamics of the micro-station is complex, and tuning the multi-body model parameters to obtain a perfect match is difficult.
|
||||
|
||||
% When considering the NASS, the most important dynamical characteristics of the micro-station is its compliance, as it can affect the plant dynamics.
|
||||
% Therefore, the adopted strategy is to accurately model the micro-station compliance.
|
||||
|
||||
% The micro-station compliance was experimentally measured using the setup illustrated in Figure ref:fig:ustation_compliance_meas.
|
||||
% Four 3-axis accelerometers were fixed to the micro-hexapod top platform.
|
||||
% The micro-hexapod top platform was impacted at 10 different points.
|
||||
% For each impact position, 10 impacts were performed to average and improve the data quality.
|
||||
|
||||
% #+name: fig:ustation_compliance_meas
|
||||
% #+caption: Schematic of the measurement setup used to estimate the compliance of the micro-station. The top platform of the positioning hexapod is shown with four 3-axis accelerometers (shown in red) are on top. 10 hammer impacts are performed at different locations (shown in blue).
|
||||
% [[file:figs/ustation_compliance_meas.png]]
|
||||
|
||||
% To convert the 12 acceleration signals $a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]$ to the acceleration expressed in the frame $\{\mathcal{X}\}$ $a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix $\mathbf{J}_a$ is written based on the positions and orientations of the accelerometers eqref:eq:ustation_compliance_acc_jacobian.
|
||||
|
||||
% \begin{equation}\label{eq:ustation_compliance_acc_jacobian}
|
||||
% \mathbf{J}_a = \begin{bmatrix}
|
||||
% 1 & 0 & 0 & 0 & 0 &-d \\
|
||||
% 0 & 1 & 0 & 0 & 0 & 0 \\
|
||||
% 0 & 0 & 1 & d & 0 & 0 \\
|
||||
% 1 & 0 & 0 & 0 & 0 & 0 \\
|
||||
% 0 & 1 & 0 & 0 & 0 &-d \\
|
||||
% 0 & 0 & 1 & 0 & d & 0 \\
|
||||
% 1 & 0 & 0 & 0 & 0 & d \\
|
||||
% 0 & 1 & 0 & 0 & 0 & 0 \\
|
||||
% 0 & 0 & 1 &-d & 0 & 0 \\
|
||||
% 1 & 0 & 0 & 0 & 0 & 0 \\
|
||||
% 0 & 1 & 0 & 0 & 0 & d \\
|
||||
% 0 & 0 & 1 & 0 &-d & 0
|
||||
% \end{bmatrix}
|
||||
% \end{equation}
|
||||
|
||||
% Then, the acceleration in the cartesian frame can be computed using eqref:eq:ustation_compute_cart_acc.
|
||||
|
||||
% \begin{equation}\label{eq:ustation_compute_cart_acc}
|
||||
% a_{\mathcal{X}} = \mathbf{J}_a^\dagger \cdot a_{\mathcal{L}}
|
||||
% \end{equation}
|
||||
|
||||
% Similar to what is done for the accelerometers, a Jacobian matrix $\mathbf{J}_F$ is computed eqref:eq:ustation_compliance_force_jacobian and used to convert the individual hammer forces $F_{\mathcal{L}}$ to force and torques $F_{\mathcal{X}}$ applied at the center of the micro-hexapod top plate (defined by frame $\{\mathcal{X}\}$ in Figure ref:fig:ustation_compliance_meas).
|
||||
|
||||
% \begin{equation}\label{eq:ustation_compliance_force_jacobian}
|
||||
% \mathbf{J}_F = \begin{bmatrix}
|
||||
% 0 & -1 & 0 & 0 & 0 & 0\\
|
||||
% 0 & 0 & -1 & -d & 0 & 0\\
|
||||
% 1 & 0 & 0 & 0 & 0 & 0\\
|
||||
% 0 & 0 & -1 & 0 & -d & 0\\
|
||||
% 0 & 1 & 0 & 0 & 0 & 0\\
|
||||
% 0 & 0 & -1 & d & 0 & 0\\
|
||||
% -1 & 0 & 0 & 0 & 0 & 0\\
|
||||
% 0 & 0 & -1 & 0 & d & 0\\
|
||||
% -1 & 0 & 0 & 0 & 0 & -d\\
|
||||
% -1 & 0 & 0 & 0 & 0 & d
|
||||
% \end{bmatrix}
|
||||
% \end{equation}
|
||||
|
||||
% The equivalent forces and torques applied at center of $\{\mathcal{X}\}$ are then computed using eqref:eq:ustation_compute_cart_force.
|
||||
|
||||
% \begin{equation}\label{eq:ustation_compute_cart_force}
|
||||
% F_{\mathcal{X}} = \mathbf{J}_F^t \cdot F_{\mathcal{L}}
|
||||
% \end{equation}
|
||||
|
||||
% Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame $\{\mathcal{X}\}$ to the 6 linear/angular accelerations of the top platform expressed with respect to $\{\mathcal{X}\}$.
|
||||
% These FRFs were then used for comparison with the Simscape model.
|
||||
|
||||
|
||||
% Positions and orientation of accelerometers
|
||||
% | *Num* | *Position* | *Orientation* | *Sensibility* | *Channels* |
|
||||
% |-------+------------+---------------+---------------+------------|
|
||||
@@ -345,13 +262,6 @@ end
|
||||
% FRF_cartesian = inv(Ja) * FRF * inv(Jf)
|
||||
FRF_cartesian = pagemtimes(Ja_inv, pagemtimes(G_raw, Jf_inv));
|
||||
|
||||
|
||||
|
||||
% The compliance of the micro-station multi-body model was extracted by computing the transfer function from forces/torques applied on the hexapod's top platform to the "absolute" motion of the top platform.
|
||||
% These results are compared with the measurements in Figure ref:fig:ustation_frf_compliance_model.
|
||||
% Considering the complexity of the micro-station compliance dynamics, the model compliance matches sufficiently well for the current application.
|
||||
|
||||
|
||||
%% Identification of the compliance of the micro-station
|
||||
|
||||
% Initialize simulation with default parameters (flexible elements)
|
||||
@@ -379,12 +289,12 @@ Gm.OutputName = {'Dx', 'Dy', 'Dz', 'Drx', 'Dry', 'Drz'};
|
||||
%% Extracted FRF of the compliance of the micro-station in the Cartesian frame from the Simscape model
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, abs(squeeze(FRF_cartesian(1,1,:))), '-', 'color', [colors(1,:), 0.5], 'DisplayName', '$D_x/F_x$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(2,2,:))), '-', 'color', [colors(2,:), 0.5], 'DisplayName', '$D_y/F_y$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(3,3,:))), '-', 'color', [colors(3,:), 0.5], 'DisplayName', '$D_z/F_z$ - Measured')
|
||||
plot(f, abs(squeeze(freqresp(Gm(1,1), f, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(2,2), f, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(3,3), f, 'Hz'))), '--', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$ - Model')
|
||||
plot(f, abs(squeeze(FRF_cartesian(1,1,:))), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5, 'DisplayName', '$D_x/F_x$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(2,2,:))), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5, 'DisplayName', '$D_y/F_y$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(3,3,:))), '-', 'color', [colors(3,:), 0.5], 'linewidth', 2.5, 'DisplayName', '$D_z/F_z$ - Measured')
|
||||
plot(f, abs(squeeze(freqresp(Gm(1,1), f, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(2,2), f, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(3,3), f, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$ - Model')
|
||||
hold off;
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
@@ -396,12 +306,12 @@ xticks([20, 50, 100, 200, 500])
|
||||
%% Extracted FRF of the compliance of the micro-station in the Cartesian frame from the Simscape model
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, abs(squeeze(FRF_cartesian(4,4,:))), '-', 'color', [colors(1,:), 0.5], 'DisplayName', '$R_x/M_x$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(5,5,:))), '-', 'color', [colors(2,:), 0.5], 'DisplayName', '$R_y/M_y$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(6,6,:))), '-', 'color', [colors(3,:), 0.5], 'DisplayName', '$R_z/M_z$ - Measured')
|
||||
plot(f, abs(squeeze(freqresp(Gm(4,4), f, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$R_x/M_x$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(5,5), f, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '$R_y/M_y$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(6,6), f, 'Hz'))), '--', 'color', colors(3,:), 'DisplayName', '$R_z/M_z$ - Model')
|
||||
plot(f, abs(squeeze(FRF_cartesian(4,4,:))), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5, 'DisplayName', '$R_x/M_x$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(5,5,:))), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5, 'DisplayName', '$R_y/M_y$ - Measured')
|
||||
plot(f, abs(squeeze(FRF_cartesian(6,6,:))), '-', 'color', [colors(3,:), 0.5], 'linewidth', 2.5, 'DisplayName', '$R_z/M_z$ - Measured')
|
||||
plot(f, abs(squeeze(freqresp(Gm(4,4), f, 'Hz'))), '-', 'color', colors(1,:), 'DisplayName', '$R_x/M_x$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(5,5), f, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$R_y/M_y$ - Model')
|
||||
plot(f, abs(squeeze(freqresp(Gm(6,6), f, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$R_z/M_z$ - Model')
|
||||
hold off;
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
|
||||
@@ -1,5 +1,3 @@
|
||||
% Matlab Init :noexport:ignore:
|
||||
|
||||
%% ustation_3_disturbances.m
|
||||
|
||||
%% Clear Workspace and Close figures
|
||||
@@ -25,13 +23,6 @@ colors = colororder;
|
||||
%% Frequency Vector
|
||||
freqs = logspace(log10(10), log10(2e3), 1000);
|
||||
|
||||
% Ground Motion
|
||||
|
||||
% The ground motion was measured by using a sensitive 3-axis geophone[fn:11] placed on the ground.
|
||||
% The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function.
|
||||
% The obtained ground motion displacement is shown in Figure ref:fig:ustation_ground_disturbance.
|
||||
|
||||
|
||||
%% Compute Floor Motion Spectral Density
|
||||
% Load floor motion data
|
||||
% velocity in [m/s] is measured in X, Y and Z directions
|
||||
@@ -69,24 +60,6 @@ xlim([0, 5]); ylim([-2, 2])
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
|
||||
% Ty Stage
|
||||
|
||||
% To measure the positioning errors of the translation stage, the setup shown in Figure ref:fig:ustation_errors_ty_setup is used.
|
||||
% A special optical element (called a "straightness interferometer"[fn:9]) is fixed on top of the micro-station, while a laser source[fn:10] and a straightness reflector are fixed on the ground.
|
||||
% A similar setup was used to measure the horizontal deviation (i.e. in the $x$ direction), as well as the pitch and yaw errors of the translation stage.
|
||||
|
||||
% #+name: fig:ustation_errors_ty_setup
|
||||
% #+caption: Experimental setup to measure the flatness (vertical deviation) of the translation stage
|
||||
% [[file:figs/ustation_errors_ty_setup.png]]
|
||||
|
||||
% Six scans were performed between $-4.5\,mm$ and $4.5\,mm$.
|
||||
% The results for each individual scan are shown in Figure ref:fig:ustation_errors_dy_vertical.
|
||||
% The measurement axis may not be perfectly aligned with the translation stage axis; this, a linear fit is removed from the measurement.
|
||||
% The remaining vertical displacement is shown in Figure ref:fig:ustation_errors_dy_vertical_remove_mean.
|
||||
% A vertical error of $\pm300\,nm$ induced by the translation stage is expected.
|
||||
% Similar result is obtained for the $x$ lateral direction.
|
||||
|
||||
|
||||
%% Ty errors
|
||||
% Setpoint is in [mm]
|
||||
% Vertical error is in [um]
|
||||
@@ -110,30 +83,9 @@ legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
%% Measurement of the linear (vertical) deviation of the Translation stage - Remove best linear fit
|
||||
figure;
|
||||
plot(ty_errors.setpoint, ty_errors.ty_z - straight_line, 'k-')
|
||||
xlabel('$D_y$ position [mm]'); ylabel('Vertical error [$\mu$m]');
|
||||
xlabel('$D_y$ position [mm]'); ylabel('Residual vertical error [$\mu$m]');
|
||||
xlim([-5, 5]); ylim([-0.4, 0.4]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:ustation_errors_dy
|
||||
% #+caption: Measurement of the linear (vertical) deviation of the Translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).
|
||||
% #+attr_latex: :options [htbp]
|
||||
% #+begin_figure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical}Measured vertical error}
|
||||
% #+attr_latex: :options {0.49\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.95\linewidth
|
||||
% [[file:figs/ustation_errors_dy_vertical.png]]
|
||||
% #+end_subfigure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing linear fit}
|
||||
% #+attr_latex: :options {0.49\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.95\linewidth
|
||||
% [[file:figs/ustation_errors_dy_vertical_remove_mean.png]]
|
||||
% #+end_subfigure
|
||||
% #+end_figure
|
||||
|
||||
|
||||
%% Convert the data to frequency domain
|
||||
% Suppose a certain constant velocity scan
|
||||
delta_ty = (ty_errors.setpoint(end) - ty_errors.setpoint(1))/(length(ty_errors.setpoint) - 1); % [mm]
|
||||
@@ -154,40 +106,6 @@ pxx_dy_dz = mean(pxx_dy_dz')';
|
||||
% is a reasonable assumption (and verified in practice)
|
||||
pxx_dy_dx = pxx_dy_dz;
|
||||
|
||||
% Spindle
|
||||
|
||||
% To measure the positioning errors induced by the Spindle, a "Spindle error analyzer"[fn:7] is used as shown in Figure ref:fig:ustation_rz_meas_lion_setup.
|
||||
% A specific target is fixed on top of the micro-station, which consists of two sphere with 1 inch diameter precisely aligned with the spindle rotation axis.
|
||||
% Five capacitive sensors[fn:8] are pointing at the two spheres, as shown in Figure ref:fig:ustation_rz_meas_lion_zoom.
|
||||
% From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translations and rotations $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ of the target can be estimated.
|
||||
|
||||
% #+name: fig:ustation_rz_meas_lion_setup
|
||||
% #+caption: Experimental setup used to estimate the errors induced by the Spindle rotation (\subref{fig:ustation_rz_meas_lion}). The motion of the two reference spheres is measured using 5 capacitive sensors (\subref{fig:ustation_rz_meas_lion_zoom})
|
||||
% #+attr_latex: :options [htbp]
|
||||
% #+begin_figure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoF metrology}
|
||||
% #+attr_latex: :options {0.49\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_rz_meas_lion.jpg]]
|
||||
% #+end_subfigure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_rz_meas_lion_zoom}Zoom on the metrology system}
|
||||
% #+attr_latex: :options {0.49\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_rz_meas_lion_zoom.jpg]]
|
||||
% #+end_subfigure
|
||||
% #+end_figure
|
||||
|
||||
% A measurement was performed during a constant rotational velocity of the spindle of 60rpm and during 10 turns.
|
||||
% The obtained results are shown in Figure ref:fig:ustation_errors_spindle.
|
||||
% A large fraction of the radial (Figure ref:fig:ustation_errors_spindle_radial) and tilt (Figure ref:fig:ustation_errors_spindle_tilt) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle.
|
||||
% This is displayed by the dashed circle.
|
||||
% After removing the best circular fit from the data, the vibrations induced by the Spindle may be viewed as stochastic disturbances.
|
||||
% However, some misalignment between the "point-of-interest" of the sample and the rotation axis will be considered because the alignment is not perfect in practice.
|
||||
% The vertical motion induced by scanning the spindle is in the order of $\pm 30\,nm$ (Figure ref:fig:ustation_errors_spindle_axial).
|
||||
|
||||
|
||||
%% Spindle Errors
|
||||
% Errors are already converted to x,y,z,Rx,Ry
|
||||
% Units are in [m] and [rad]
|
||||
@@ -226,33 +144,6 @@ xlim([-35, 35]); ylim([-35, 35]);
|
||||
xticks([-30, -15, 0, 15, 30]);
|
||||
yticks([-30, -15, 0, 15, 30]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:ustation_errors_spindle
|
||||
% #+caption: Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis.
|
||||
% #+attr_latex: :options [htbp]
|
||||
% #+begin_figure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_radial}Radial errors}
|
||||
% #+attr_latex: :options {0.33\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_errors_spindle_radial.png]]
|
||||
% #+end_subfigure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_axial}Axial error}
|
||||
% #+attr_latex: :options {0.33\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_errors_spindle_axial.png]]
|
||||
% #+end_subfigure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_tilt}Tilt errors}
|
||||
% #+attr_latex: :options {0.33\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_errors_spindle_tilt.png]]
|
||||
% #+end_subfigure
|
||||
% #+end_figure
|
||||
|
||||
|
||||
%% Remove the circular fit from the data
|
||||
[x0, y0, R] = circlefit(spindle_errors.Dx, spindle_errors.Dy);
|
||||
|
||||
@@ -279,14 +170,6 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
|
||||
[pxx_rz_dx, ~ ] = pwelch(spindle_errors.Dx_err, win, Noverlap, Nfft, Fs);
|
||||
[pxx_rz_dy, ~ ] = pwelch(spindle_errors.Dy_err, win, Noverlap, Nfft, Fs);
|
||||
|
||||
% Sensitivity to disturbances
|
||||
% <<ssec:ustation_disturbances_sensitivity>>
|
||||
|
||||
% To compute the disturbance source (i.e. forces) that induced the measured vibrations in Section ref:ssec:ustation_disturbances_meas, the transfer function from the disturbance sources to the stage vibration (i.e. the "sensitivity to disturbances") needs to be estimated.
|
||||
% This is achieved using the multi-body model presented in Section ref:sec:ustation_modeling.
|
||||
% The obtained transfer functions are shown in Figure ref:fig:ustation_model_sensitivity.
|
||||
|
||||
|
||||
%% Extract sensitivity to disturbances from the Simscape model
|
||||
% Initialize stages
|
||||
initializeGround();
|
||||
@@ -361,13 +244,6 @@ ylim([1e-10, 1e-7]);
|
||||
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
|
||||
% Obtained disturbance sources
|
||||
% <<ssec:ustation_disturbances_results>>
|
||||
|
||||
% From the measured effect of disturbances in Section ref:ssec:ustation_disturbances_meas and the sensitivity to disturbances extracted from the Simscape model in Section ref:ssec:ustation_disturbances_sensitivity, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated.
|
||||
% The obtained power spectral density of the disturbances are shown in Figure ref:fig:ustation_dist_sources.
|
||||
|
||||
|
||||
%% Compute the PSD of the equivalent disturbance sources
|
||||
pxx_rz_fx = pxx_rz_dx./abs(squeeze(freqresp(Gd('Dx', 'Frz_x'), f_rz, 'Hz'))).^2;
|
||||
pxx_rz_fy = pxx_rz_dy./abs(squeeze(freqresp(Gd('Dy', 'Frz_y'), f_rz, 'Hz'))).^2;
|
||||
@@ -435,38 +311,6 @@ xlim([1, 200]); ylim([1e-3, 1e3]);
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
|
||||
|
||||
|
||||
% #+name: fig:ustation_dist_sources
|
||||
% #+caption: Measured spectral density of the micro-station disturbance sources. Ground motion (\subref{fig:ustation_dist_source_ground_motion}), translation stage (\subref{fig:ustation_dist_source_translation_stage}) and spindle (\subref{fig:ustation_dist_source_spindle}).
|
||||
% #+attr_latex: :options [htbp]
|
||||
% #+begin_figure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_ground_motion}Ground Motion}
|
||||
% #+attr_latex: :options {0.33\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_dist_source_ground_motion.png]]
|
||||
% #+end_subfigure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_translation_stage}Translation Stage}
|
||||
% #+attr_latex: :options {0.33\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_dist_source_translation_stage.png]]
|
||||
% #+end_subfigure
|
||||
% #+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_spindle}Spindle}
|
||||
% #+attr_latex: :options {0.33\textwidth}
|
||||
% #+begin_subfigure
|
||||
% #+attr_latex: :width 0.9\linewidth
|
||||
% [[file:figs/ustation_dist_source_spindle.png]]
|
||||
% #+end_subfigure
|
||||
% #+end_figure
|
||||
|
||||
% The disturbances are characterized by their power spectral densities, as shown in Figure ref:fig:ustation_dist_sources.
|
||||
% However, to perform time domain simulations, disturbances must be represented by a time domain signal.
|
||||
% To generate stochastic time-domain signals with a specific power spectral densities, the discrete inverse Fourier transform is used, as explained in [[cite:&preumont94_random_vibrat_spect_analy chap. 12.11]].
|
||||
% Examples of the obtained time-domain disturbance signals are shown in Figure ref:fig:ustation_dist_sources_time.
|
||||
|
||||
|
||||
%% Compute time domain disturbance signals
|
||||
initializeDisturbances();
|
||||
load('nass_model_disturbances.mat');
|
||||
@@ -497,5 +341,5 @@ plot(Dw.t, 1e6*Dw.y, 'DisplayName', '$D_{yf}$');
|
||||
plot(Dw.t, 1e6*Dw.z, 'DisplayName', '$D_{zf}$');
|
||||
xlabel('Time [s]'); ylabel('Amplitude [$\mu$m]')
|
||||
xlim([0, 1]); ylim([-0.6, 0.6])
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
|
||||
@@ -1,5 +1,3 @@
|
||||
% Matlab Init :noexport:ignore:
|
||||
|
||||
%% ustation_4_experiments.m
|
||||
|
||||
%% Clear Workspace and Close figures
|
||||
@@ -25,17 +23,6 @@ colors = colororder;
|
||||
%% Frequency Vector
|
||||
freqs = logspace(log10(10), log10(2e3), 1000);
|
||||
|
||||
% Tomography Experiment
|
||||
% <<sec:ustation_experiments_tomography>>
|
||||
|
||||
% To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm.
|
||||
% Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section ref:sec:ustation_disturbances.
|
||||
% A radial offset of $\approx 1\,\mu m$ between the "point-of-interest" and the spindle's rotation axis is introduced to represent what is experimentally observed.
|
||||
% During the 10 second simulation (i.e. 10 spindle turns), the position of the "point-of-interest" with respect to the granite was recorded.
|
||||
% Results are shown in Figure ref:fig:ustation_errors_model_spindle.
|
||||
% A good correlation with the measurements is observed both for radial errors (Figure ref:fig:ustation_errors_model_spindle_radial) and axial errors (Figure ref:fig:ustation_errors_model_spindle_axial).
|
||||
|
||||
|
||||
%% Tomography experiment
|
||||
% Sample is not centered with the rotation axis
|
||||
% This is done by offsetfing the micro-hexapod by 0.9um
|
||||
@@ -100,16 +87,6 @@ xlim([0,2]); ylim([-40, 40]);
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
|
||||
% Raster Scans with the translation stage
|
||||
% <<sec:ustation_experiments_ty_scans>>
|
||||
|
||||
% A second experiment was performed in which the translation stage was scanned at constant velocity.
|
||||
% The translation stage setpoint is configured to have a "triangular" shape with stroke of $\pm 4.5\, mm$.
|
||||
% Both ground motion and translation stage vibrations were included in the simulation.
|
||||
% Similar to what was performed for the tomography simulation, the PoI position with respect to the granite was recorded and compared with the experimental measurements in Figure ref:fig:ustation_errors_model_dy_vertical.
|
||||
% A similar error amplitude was observed, thus indicating that the multi-body model with the included disturbances accurately represented the micro-station behavior in typical scientific experiments.
|
||||
|
||||
|
||||
%% Translation stage latteral scans
|
||||
set_param(conf_simulink, 'StopTime', '2');
|
||||
|
||||
|
||||
Binary file not shown.
@@ -12,5 +12,12 @@
|
||||
\setabbreviationstyle[acronym]{long-short}
|
||||
\setglossarystyle{long-name-desc}
|
||||
|
||||
\usepackage{amssymb}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{cases}
|
||||
\usepackage{empheq}
|
||||
|
||||
\makeindex
|
||||
\makeglossaries
|
||||
|
||||
\usepackage{bm}
|
||||
|
||||
@@ -2,7 +2,6 @@
|
||||
\usepackage{enumitem}
|
||||
|
||||
\usepackage{caption,tabularx,booktabs}
|
||||
\usepackage{bm}
|
||||
|
||||
\usepackage{xpatch} % Recommanded for biblatex
|
||||
\usepackage[ % use biblatex for bibliography
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
Binary file not shown.
@@ -1,4 +1,4 @@
|
||||
% Created 2024-11-06 Wed 18:35
|
||||
% Created 2025-04-07 Mon 16:40
|
||||
% Intended LaTeX compiler: pdflatex
|
||||
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
|
||||
|
||||
@@ -7,13 +7,13 @@
|
||||
\bibliography{simscape-micro-station.bib}
|
||||
\author{Dehaeze Thomas}
|
||||
\date{\today}
|
||||
\title{Simscape Model - Micro Station}
|
||||
\title{Multi-Body Model - Micro Station}
|
||||
\hypersetup{
|
||||
pdfauthor={Dehaeze Thomas},
|
||||
pdftitle={Simscape Model - Micro Station},
|
||||
pdftitle={Multi-Body Model - Micro Station},
|
||||
pdfkeywords={},
|
||||
pdfsubject={},
|
||||
pdfcreator={Emacs 29.4 (Org mode 9.6)},
|
||||
pdfcreator={Emacs 30.1 (Org mode 9.7.26)},
|
||||
pdflang={English}}
|
||||
\usepackage{biblatex}
|
||||
|
||||
@@ -23,7 +23,6 @@
|
||||
\tableofcontents
|
||||
|
||||
\clearpage
|
||||
|
||||
From the start of this work, it became increasingly clear that an accurate micro-station model was necessary.
|
||||
|
||||
First, during the uniaxial study, it became clear that the micro-station dynamics affects the nano-hexapod dynamics.
|
||||
@@ -43,18 +42,16 @@ Disturbances affecting the positioning accuracy also need to be modeled properly
|
||||
To do so, the effects of these disturbances were first measured experimental and then injected into the multi-body model (Section \ref{sec:ustation_disturbances}).
|
||||
|
||||
To validate the accuracy of the micro-station model, ``real world'' experiments are simulated and compared with measurements in Section \ref{sec:ustation_experiments}.
|
||||
|
||||
|
||||
\chapter{Micro-Station Kinematics}
|
||||
\label{sec:ustation_kinematics}
|
||||
The micro-station consists of 4 stacked positioning stages (Figure \ref{fig:ustation_cad_view}).
|
||||
From bottom to top, the stacked stages are the translation stage \(D_y\), the tilt stage \(R_y\), the rotation stage (Spindle) \(R_z\) and the positioning hexapod.
|
||||
Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex. complex dynamics.
|
||||
Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,width=\linewidth]{figs/ustation_cad_view.png}
|
||||
\caption{\label{fig:ustation_cad_view}CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple). On top of these four stages, a solid part (shown in green) will be replaced by the stabilization stage.}
|
||||
\caption{\label{fig:ustation_cad_view}CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple).}
|
||||
\end{figure}
|
||||
|
||||
There are different ways of modeling the stage dynamics in a multi-body model.
|
||||
@@ -65,7 +62,6 @@ s can be tuned separately for each DoF.
|
||||
The ``controlled'' DoF of each stage (for instance the \(D_y\) direction for the translation stage) is modeled as infinitely rigid (i.e. its motion is imposed by a ``setpoint'') while the other DoFs have limited stiffness to model the different micro-station modes.
|
||||
\section{Motion Stages}
|
||||
\label{ssec:ustation_stages}
|
||||
|
||||
\paragraph{Translation Stage}
|
||||
|
||||
The translation stage is used to position and scan the sample laterally with respect to the X-ray beam.
|
||||
@@ -75,7 +71,6 @@ It was later replaced with a stepper motor and lead-screw, as the feedback contr
|
||||
An optical linear encoder is used to measure the stage motion and for controlling the position.
|
||||
|
||||
Four cylindrical bearings\footnote{Ball cage (N501) and guide bush (N550) from Mahr are used.} are used to guide the motion (i.e. minimize the parasitic motions) and have high stiffness.
|
||||
|
||||
\paragraph{Tilt Stage}
|
||||
|
||||
The tilt stage is guided by four linear motion guides\footnote{HCR 35 A C1, from THK.} which are placed such that the center of rotation coincide with the X-ray beam.
|
||||
@@ -99,14 +94,12 @@ To precisely control the \(R_y\) angle, a stepper motor and two optical encoders
|
||||
\captionof{figure}{\label{fig:ustation_ry_stage}Tilt Stage}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\paragraph{Spindle}
|
||||
|
||||
Then, a rotation stage is used for tomography experiments.
|
||||
It is composed of an air bearing spindle\footnote{Made by LAB Motion Systems.}, whose angular position is controlled with a 3 phase synchronous motor based on the reading of 4 optical encoders.
|
||||
|
||||
Additional rotary unions and slip-rings are used to be able to pass electrical signals, fluids and gazes through the rotation stage.
|
||||
|
||||
\paragraph{Micro-Hexapod}
|
||||
|
||||
Finally, a Stewart platform\footnote{Modified Zonda Hexapod by Symetrie.} is used to position the sample.
|
||||
@@ -128,7 +121,6 @@ It can also be used to precisely position the PoI vertically with respect to the
|
||||
\captionof{figure}{\label{fig:ustation_hexapod_stage}Micro Hexapod}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\section{Mathematical description of a rigid body motion}
|
||||
\label{ssec:ustation_motion_description}
|
||||
In this section, mathematical tools\footnote{The tools presented here are largely taken from \cite{taghirad13_paral}.} that are used to describe the motion of positioning stages are introduced.
|
||||
@@ -176,21 +168,17 @@ The \emph{orientation} of a rigid body is the same at all its points (by definit
|
||||
Hence, the orientation of a rigid body can be viewed as that of a moving frame attached to the rigid body.
|
||||
It can be represented in several different ways: the rotation matrix, the screw axis representation, and the Euler angles are common descriptions.
|
||||
|
||||
The rotation matrix \({}^A\mathbf{R}_B\) is a \(3 \times 3\) matrix containing the Cartesian unit vectors of frame \(\{\mathbf{B}\}\) represented in frame \(\{\mathbf{A}\}\) \eqref{eq:ustation_rotation_matrix}.
|
||||
The rotation matrix \({}^A\bm{R}_B\) is a \(3 \times 3\) matrix containing the Cartesian unit vectors \([{}^A\hat{\bm{x}}_B,\ {}^A\hat{\bm{y}}_B,\ {}^A\hat{\bm{z}}_B]\) of frame \(\{\bm{B}\}\) represented in frame \(\{\bm{A}\}\) \eqref{eq:ustation_rotation_matrix}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_rotation_matrix}
|
||||
{}^A\mathbf{R}_B = \left[ {}^A\hat{\mathbf{x}}_B | {}^A\hat{\mathbf{y}}_B | {}^A\hat{\mathbf{z}}_B \right] = \begin{bmatrix}
|
||||
u_{x} & v_{x} & z_{x} \\
|
||||
u_{y} & v_{y} & z_{y} \\
|
||||
u_{z} & v_{z} & z_{z}
|
||||
\end{bmatrix}
|
||||
{}^A\bm{R}_B = \left[ {}^A\hat{\bm{x}}_B | {}^A\hat{\bm{y}}_B | {}^A\hat{\bm{z}}_B \right]
|
||||
\end{equation}
|
||||
|
||||
Consider a pure rotation of a rigid body (\(\{\bm{A}\}\) and \(\{\bm{B}\}\) are coincident at their origins, as shown in Figure \ref{fig:ustation_rotation}).
|
||||
The rotation matrix can be used to express the coordinates of a point \(P\) in a fixed frame \(\{A\}\) (i.e. \({}^AP\)) from its coordinate in the moving frame \(\{B\}\) using Equation \eqref{eq:ustation_rotation}.
|
||||
|
||||
\begin{equation} \label{eq:ustation_rotation}
|
||||
{}^AP = {}^A\mathbf{R}_B {}^BP
|
||||
{}^AP = {}^A\bm{R}_B {}^BP
|
||||
\end{equation}
|
||||
|
||||
|
||||
@@ -198,17 +186,17 @@ For rotations along \(x\), \(y\) or \(z\) axis, the formulas of the correspondin
|
||||
|
||||
\begin{subequations}\label{eq:ustation_rotation_matrices_xyz}
|
||||
\begin{align}
|
||||
\mathbf{R}_x(\theta_x) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta_x) & -\sin(\theta_x) \\ 0 & \sin(\theta_x) & \cos(\theta_x) \end{bmatrix} \\
|
||||
\mathbf{R}_y(\theta_y) &= \begin{bmatrix} \cos(\theta_y) & 0 & \sin(\theta_y) \\ 0 & 1 & 0 \\ -\sin(\theta_y) & 0 & \cos(\theta_y) \end{bmatrix} \\
|
||||
\mathbf{R}_z(\theta_z) &= \begin{bmatrix} \cos(\theta_z) & -\sin(\theta_z) & 0 \\ \sin(\theta_z) & \cos(\theta_x) & 0 \\ 0 & 0 & 1 \end{bmatrix}
|
||||
\bm{R}_x(\theta_x) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta_x) & -\sin(\theta_x) \\ 0 & \sin(\theta_x) & \cos(\theta_x) \end{bmatrix} \\
|
||||
\bm{R}_y(\theta_y) &= \begin{bmatrix} \cos(\theta_y) & 0 & \sin(\theta_y) \\ 0 & 1 & 0 \\ -\sin(\theta_y) & 0 & \cos(\theta_y) \end{bmatrix} \\
|
||||
\bm{R}_z(\theta_z) &= \begin{bmatrix} \cos(\theta_z) & -\sin(\theta_z) & 0 \\ \sin(\theta_z) & \cos(\theta_z) & 0 \\ 0 & 0 & 1 \end{bmatrix}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Sometimes, it is useful to express a rotation as a combination of three rotations described by \(\mathbf{R}_x\), \(\mathbf{R}_y\) and \(\mathbf{R}_z\).
|
||||
Sometimes, it is useful to express a rotation as a combination of three rotations described by \(\bm{R}_x\), \(\bm{R}_y\) and \(\bm{R}_z\).
|
||||
The order of rotation is very important\footnote{Rotations are non commutative in 3D.}, therefore, in this study, rotations are expressed as three successive rotations about the coordinate axes of the moving frame \eqref{eq:ustation_rotation_combination}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_rotation_combination}
|
||||
{}^A\mathbf{R}_B(\alpha, \beta, \gamma) = \mathbf{R}_u(\alpha) \mathbf{R}_v(\beta) \mathbf{R}_c(\gamma)
|
||||
{}^A\bm{R}_B(\alpha, \beta, \gamma) = \bm{R}_u(\alpha) \bm{R}_v(\beta) \bm{R}_c(\gamma)
|
||||
\end{equation}
|
||||
|
||||
Such rotation can be parameterized by three Euler angles \((\alpha,\ \beta,\ \gamma)\), which can be computed from a given rotation matrix using equations \eqref{eq:ustation_euler_angles}.
|
||||
@@ -220,7 +208,6 @@ Such rotation can be parameterized by three Euler angles \((\alpha,\ \beta,\ \ga
|
||||
\gamma &= \text{atan2}(-R_{12}/\cos(\beta),\ R_{11}/\cos(\beta))
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
\paragraph{Motion of a Rigid Body}
|
||||
|
||||
Since the relative positions of a rigid body with respect to a moving frame \(\{B\}\) attached to it are fixed for all time, it is sufficient to know the position of the origin of the frame \(O_B\) and the orientation of the frame \(\{B\}\) with respect to the fixed frame \(\{A\}\), to represent the position of any point \(P\) in the space.
|
||||
@@ -228,24 +215,24 @@ Since the relative positions of a rigid body with respect to a moving frame \(\{
|
||||
Therefore, the pose of a rigid body can be fully determined by:
|
||||
\begin{enumerate}
|
||||
\item The position vector of point \(O_B\) with respect to frame \(\{A\}\) which is denoted \({}^AP_{O_B}\)
|
||||
\item The orientation of the rigid body, or the moving frame \(\{B\}\) attached to it with respect to the fixed frame \(\{A\}\), that is represented by \({}^A\mathbf{R}_B\).
|
||||
\item The orientation of the rigid body, or the moving frame \(\{B\}\) attached to it with respect to the fixed frame \(\{A\}\), that is represented by \({}^A\bm{R}_B\).
|
||||
\end{enumerate}
|
||||
|
||||
The position of any point \(P\) of the rigid body with respect to the fixed frame \(\{\mathbf{A}\}\), which is denoted \({}^A\mathbf{P}\) may be determined thanks to the \emph{Chasles' theorem}, which states that if the pose of a rigid body \(\{{}^A\mathbf{R}_B, {}^AP_{O_B}\}\) is given, then the position of any point \(P\) of this rigid body with respect to \(\{\mathbf{A}\}\) is given by Equation \eqref{eq:ustation_chasles_therorem}.
|
||||
The position of any point \(P\) of the rigid body with respect to the fixed frame \(\{\bm{A}\}\), which is denoted \({}^A\bm{P}\) may be determined thanks to the \emph{Chasles' theorem}, which states that if the pose of a rigid body \(\{{}^A\bm{R}_B, {}^AP_{O_B}\}\) is given, then the position of any point \(P\) of this rigid body with respect to \(\{\bm{A}\}\) is given by Equation \eqref{eq:ustation_chasles_therorem}.
|
||||
|
||||
\begin{equation} \label{eq:ustation_chasles_therorem}
|
||||
{}^AP = {}^A\mathbf{R}_B {}^BP + {}^AP_{O_B}
|
||||
{}^AP = {}^A\bm{R}_B {}^BP + {}^AP_{O_B}
|
||||
\end{equation}
|
||||
|
||||
While equation \eqref{eq:ustation_chasles_therorem} can describe the motion of a rigid body, it can be written in a more convenient way using \(4 \times 4\) homogeneous transformation matrices and \(4 \times 1\) homogeneous coordinates.
|
||||
The homogeneous transformation matrix is composed of the rotation matrix \({}^A\mathbf{R}_B\) representing the orientation and the position vector \({}^AP_{O_B}\) representing the translation.
|
||||
The homogeneous transformation matrix is composed of the rotation matrix \({}^A\bm{R}_B\) representing the orientation and the position vector \({}^AP_{O_B}\) representing the translation.
|
||||
It is partitioned as shown in Equation \eqref{eq:ustation_homogeneous_transformation_parts}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_homogeneous_transformation_parts}
|
||||
{}^A\mathbf{T}_B =
|
||||
{}^A\bm{T}_B =
|
||||
\left[ \begin{array}{ccc|c}
|
||||
& & & \\
|
||||
& {}^A\mathbf{R}_B & & {}^AP_{O_B} \\
|
||||
& {}^A\bm{R}_B & & {}^AP_{O_B} \\
|
||||
& & & \cr
|
||||
\hline
|
||||
0 & 0 & 0 & 1
|
||||
@@ -259,12 +246,12 @@ Then, \({}^AP\) can be computed from \({}^BP\) and the homogeneous transformatio
|
||||
=
|
||||
\left[ \begin{array}{ccc|c}
|
||||
& & & \\
|
||||
& {}^A\mathbf{R}_B & & {}^AP_{O_B} \\
|
||||
& {}^A\bm{R}_B & & {}^AP_{O_B} \\
|
||||
& & & \cr
|
||||
\hline
|
||||
0 & 0 & 0 & 1
|
||||
\end{array} \right]
|
||||
\left[ \begin{array}{c} \\ {}^BP \\ \cr \hline 1 \end{array} \right] \quad \Rightarrow \quad {}^AP = {}^A\mathbf{R}_B {}^BP + {}^AP_{O_B}
|
||||
\left[ \begin{array}{c} \\ {}^BP \\ \cr \hline 1 \end{array} \right] \quad \Rightarrow \quad {}^AP = {}^A\bm{R}_B {}^BP + {}^AP_{O_B}
|
||||
\end{equation}
|
||||
|
||||
One key advantage of homogeneous transformation is that it can easily be generalized for consecutive transformations.
|
||||
@@ -278,29 +265,28 @@ Frame \(\{A\}\) represents the initial location, frame \(\{B\}\) is an intermedi
|
||||
\end{figure}
|
||||
|
||||
Furthermore, suppose the position vector of a point \(P\) of the rigid body is given in the final location, that is \({}^CP\) is given, and the position of this point is to be found in the fixed frame \(\{A\}\), that is \({}^AP\).
|
||||
Since the locations of the rigid body are known relative to each other, \({}^CP\) can be transformed to \({}^BP\) using \({}^B\mathbf{T}_C\) using \({}^BP = {}^B\mathbf{T}_C {}^CP\).
|
||||
Similarly, \({}^BP\) can be transformed into \({}^AP\) using \({}^AP = {}^A\mathbf{T}_B {}^BP\).
|
||||
Since the locations of the rigid body are known relative to each other, \({}^CP\) can be transformed to \({}^BP\) using \({}^B\bm{T}_C\) using \({}^BP = {}^B\bm{T}_C {}^CP\).
|
||||
Similarly, \({}^BP\) can be transformed into \({}^AP\) using \({}^AP = {}^A\bm{T}_B {}^BP\).
|
||||
|
||||
Combining the two relations, Equation \eqref{eq:ustation_consecutive_transformations} is obtained.
|
||||
This shows that combining multiple transformations is equivalent as to compute \(4 \times 4\) matrix multiplications.
|
||||
|
||||
\begin{equation}\label{eq:ustation_consecutive_transformations}
|
||||
{}^AP = \underbrace{{}^A\mathbf{T}_B {}^B\mathbf{T}_C}_{{}^A\mathbf{T}_C} {}^CP
|
||||
{}^AP = \underbrace{{}^A\bm{T}_B {}^B\bm{T}_C}_{{}^A\bm{T}_C} {}^CP
|
||||
\end{equation}
|
||||
|
||||
Another key advantage of homogeneous transformation is the easy inverse transformation, which can be computed using Equation \eqref{eq:ustation_inverse_homogeneous_transformation}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_inverse_homogeneous_transformation}
|
||||
{}^B\mathbf{T}_A = {}^A\mathbf{T}_B^{-1} =
|
||||
{}^B\bm{T}_A = {}^A\bm{T}_B^{-1} =
|
||||
\left[ \begin{array}{ccc|c}
|
||||
& & & \\
|
||||
& {}^A\mathbf{R}_B^T & & -{}^A \mathbf{R}_B^T {}^AP_{O_B} \\
|
||||
& {}^A\bm{R}_B^{\intercal} & & -{}^A \bm{R}_B^{\intercal} {}^AP_{O_B} \\
|
||||
& & & \cr
|
||||
\hline
|
||||
0 & 0 & 0 & 1 \\
|
||||
\end{array} \right]
|
||||
\end{equation}
|
||||
|
||||
\section{Micro-Station Kinematics}
|
||||
\label{ssec:ustation_kinematics}
|
||||
|
||||
@@ -320,50 +306,51 @@ The motion induced by a positioning stage can be described by a homogeneous tran
|
||||
As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in \eqref{eq:ustation_translation_stage_errors}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_translation_stage_errors}
|
||||
{}^A\mathbf{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) =
|
||||
{}^A\bm{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) =
|
||||
\left[ \begin{array}{ccc|c}
|
||||
& & & D_x \\
|
||||
& \mathbf{R}_x(\theta_x) \mathbf{R}_y(\theta_y) \mathbf{R}_z(\theta_z) & & D_y \\
|
||||
& \bm{R}_x(\theta_x) \bm{R}_y(\theta_y) \bm{R}_z(\theta_z) & & D_y \\
|
||||
& & & D_z \cr
|
||||
\hline
|
||||
0 & 0 & 0 & 1
|
||||
\end{array} \right]
|
||||
\end{equation}
|
||||
|
||||
The homogeneous transformation matrix corresponding to the micro-station \(\mathbf{T}_{\mu\text{-station}}\) is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation \eqref{eq:ustation_transformation_station}.
|
||||
The homogeneous transformation matrix corresponding to the micro-station \(\bm{T}_{\mu\text{-station}}\) is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation \eqref{eq:ustation_transformation_station}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_transformation_station}
|
||||
\mathbf{T}_{\mu\text{-station}} = \mathbf{T}_{D_y} \cdot \mathbf{T}_{R_y} \cdot \mathbf{T}_{R_z} \cdot \mathbf{T}_{\mu\text{-hexapod}}
|
||||
\bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}}
|
||||
\end{equation}
|
||||
|
||||
\(\mathbf{T}_{\mu\text{-station}}\) represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite.
|
||||
\(\bm{T}_{\mu\text{-station}}\) represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite.
|
||||
|
||||
If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), \(\mathbf{T}_{\mu\text{-station}}\) then represent the pose setpoint of the sample with respect to the granite.
|
||||
If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), \(\bm{T}_{\mu\text{-station}}\) then represents the pose setpoint of the sample with respect to the granite.
|
||||
The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation \eqref{eq:ustation_transformation_matrices_stages}.
|
||||
The setpoints are \(D_y\) for the translation stage, \(\theta_y\) for the tilt-stage, \(\theta_z\) for the spindle, \([D_{\mu x},\ D_{\mu y}, D_{\mu z}]\) for the micro-hexapod translations and \([\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]\) for the micro-hexapod rotations.
|
||||
|
||||
\begin{equation}\label{eq:ustation_transformation_matrices_stages}
|
||||
\begin{align}
|
||||
\mathbf{T}_{D_y} &= \begin{bmatrix}
|
||||
\bm{T}_{D_y} &= \begin{bmatrix}
|
||||
1 & 0 & 0 & 0 \\
|
||||
0 & 1 & 0 & D_y \\
|
||||
0 & 0 & 1 & 0 \\
|
||||
0 & 0 & 0 & 1
|
||||
\end{bmatrix} \quad
|
||||
\mathbf{T}_{\mu\text{-hexapod}} =
|
||||
\bm{T}_{\mu\text{-hexapod}} =
|
||||
\left[ \begin{array}{ccc|c}
|
||||
& & & D_{\mu x} \\
|
||||
& \mathbf{R}_x(\theta_{\mu x}) \mathbf{R}_y(\theta_{\mu y}) \mathbf{R}_{z}(\theta_{\mu z}) & & D_{\mu y} \\
|
||||
& \bm{R}_x(\theta_{\mu x}) \bm{R}_y(\theta_{\mu y}) \bm{R}_{z}(\theta_{\mu z}) & & D_{\mu y} \\
|
||||
& & & D_{\mu z} \cr
|
||||
\hline
|
||||
0 & 0 & 0 & 1
|
||||
\end{array} \right] \\
|
||||
\mathbf{T}_{R_z} &= \begin{bmatrix}
|
||||
\bm{T}_{R_z} &= \begin{bmatrix}
|
||||
\cos(\theta_z) & -\sin(\theta_z) & 0 & 0 \\
|
||||
\sin(\theta_z) & \cos(\theta_z) & 0 & 0 \\
|
||||
0 & 0 & 1 & 0 \\
|
||||
0 & 0 & 0 & 1
|
||||
\end{bmatrix} \quad
|
||||
\mathbf{T}_{R_y} = \begin{bmatrix}
|
||||
\bm{T}_{R_y} = \begin{bmatrix}
|
||||
\cos(\theta_y) & 0 & \sin(\theta_y) & 0 \\
|
||||
0 & 1 & 0 & 0 \\
|
||||
-\sin(\theta_y) & 0 & \cos(\theta_y) & 0 \\
|
||||
@@ -371,7 +358,6 @@ The transformation matrices for the translation stage, tilt stage, spindle, and
|
||||
\end{bmatrix}
|
||||
\end{align}
|
||||
\end{equation}
|
||||
|
||||
\chapter{Micro-Station Dynamics}
|
||||
\label{sec:ustation_modeling}
|
||||
In this section, the multi-body model of the micro-station is presented.
|
||||
@@ -381,7 +367,7 @@ The inertia of the solid bodies and the stiffness properties of the guiding mech
|
||||
The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section \ref{ssec:ustation_model_comp_dynamics}).
|
||||
|
||||
As the dynamics of the nano-hexapod is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station.
|
||||
To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the Simscape model (Section \ref{ssec:ustation_model_compliance}).
|
||||
To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section \ref{ssec:ustation_model_compliance}).
|
||||
\section{Multi-Body Model}
|
||||
\label{ssec:ustation_model_simscape}
|
||||
|
||||
@@ -396,7 +382,7 @@ External forces can be used to model disturbances, and ``sensors'' can be used t
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/ustation_simscape_stage_example.png}
|
||||
\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbances forces for all DoF can be included}
|
||||
\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included}
|
||||
\end{figure}
|
||||
|
||||
Therefore, the micro-station is modeled by several solid bodies connected by joints.
|
||||
@@ -408,7 +394,7 @@ The obtained 3D representation of the multi-body model is shown in Figure \ref{f
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,width=0.8\linewidth]{figs/ustation_simscape_model.jpg}
|
||||
\caption{\label{fig:ustation_simscape_model}3D view of the micro-station Simscape model}
|
||||
\caption{\label{fig:ustation_simscape_model}3D view of the micro-station multi-body model}
|
||||
\end{figure}
|
||||
|
||||
The ground is modeled by a solid body connected to the ``world frame'' through a joint only allowing 3 translations.
|
||||
@@ -422,6 +408,7 @@ The springs and dampers values were first estimated from the joint/stage specifi
|
||||
The spring values are summarized in Table \ref{tab:ustation_6dof_stiffness_values}.
|
||||
|
||||
\begin{table}[htbp]
|
||||
\caption{\label{tab:ustation_6dof_stiffness_values}Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by ``-''. The frames in which the 6-DoF joints are defined are indicated in figures found in Section \ref{ssec:ustation_stages}}
|
||||
\centering
|
||||
\begin{tabularx}{\linewidth}{Xcccccc}
|
||||
\toprule
|
||||
@@ -434,19 +421,16 @@ Spindle & \(700\,N/\mu m\) & \(700\,N/\mu m\) & \(2\,kN/\mu m\) & \(10\,Nm/\mu\t
|
||||
Hexapod & \(10\,N/\mu m\) & \(10\,N/\mu m\) & \(100\,N/\mu m\) & \(1.5\,Nm/rad\) & \(1.5\,Nm/rad\) & \(0.27\,Nm/rad\)\\
|
||||
\bottomrule
|
||||
\end{tabularx}
|
||||
\caption{\label{tab:ustation_6dof_stiffness_values}Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by ``-''. The frames in which the 6-DoF joints are defined are indicated in figures found in Section \ref{ssec:ustation_stages}}
|
||||
|
||||
\end{table}
|
||||
|
||||
\section{Comparison with the measured dynamics}
|
||||
\label{ssec:ustation_model_comp_dynamics}
|
||||
|
||||
The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions.
|
||||
The obtained FRFs were then projected at the CoM of each stage.
|
||||
|
||||
To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the Simscape model and compared with the measurements in Figure \ref{fig:ustation_comp_com_response}.
|
||||
To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure \ref{fig:ustation_comp_com_response}.
|
||||
|
||||
Even though there is some similarity between the model and the measurements (similar overall shapes and amplitudes), it is clear that the Simscape model does not accurately represent the complex micro-station dynamics.
|
||||
Even though there is some similarity between the model and the measurements (similar overall shapes and amplitudes), it is clear that the multi-body model does not accurately represent the complex micro-station dynamics.
|
||||
Tuning the numerous model parameters to better match the measurements is a highly non-linear optimization problem that is difficult to solve in practice.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@@ -468,9 +452,8 @@ Tuning the numerous model parameters to better match the measurements is a highl
|
||||
\end{center}
|
||||
\subcaption{\label{fig:ustation_comp_com_response_ry_z}Tilt, $z$ response}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_comp_com_response}FRFs between the hammer impacts on the translation stage and the measured stage acceleration expressed at its CoM. Comparison of the measured and extracted FRFs from the Simscape model. Different directions are computed for different stages.}
|
||||
\caption{\label{fig:ustation_comp_com_response}FRFs between the hammer impacts on the translation stage and the measured stage acceleration expressed at its CoM. Comparison of the measured and extracted FRFs from the multi-body model. Different directions are computed for different stages.}
|
||||
\end{figure}
|
||||
|
||||
\section{Micro-station compliance}
|
||||
\label{ssec:ustation_model_compliance}
|
||||
|
||||
@@ -490,10 +473,10 @@ For each impact position, 10 impacts were performed to average and improve the d
|
||||
\caption{\label{fig:ustation_compliance_meas}Schematic of the measurement setup used to estimate the compliance of the micro-station. The top platform of the positioning hexapod is shown with four 3-axis accelerometers (shown in red) are on top. 10 hammer impacts are performed at different locations (shown in blue).}
|
||||
\end{figure}
|
||||
|
||||
To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]\) to the acceleration expressed in the frame \(\{\mathcal{X}\}\) \(a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]\), a Jacobian matrix \(\mathbf{J}_a\) is written based on the positions and orientations of the accelerometers \eqref{eq:ustation_compliance_acc_jacobian}.
|
||||
To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]\) to the acceleration expressed in the frame \(\{\mathcal{X}\}\) \(a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]\), a Jacobian matrix \(\bm{J}_a\) is written based on the positions and orientations of the accelerometers \eqref{eq:ustation_compliance_acc_jacobian}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_compliance_acc_jacobian}
|
||||
\mathbf{J}_a = \begin{bmatrix}
|
||||
\bm{J}_a = \begin{bmatrix}
|
||||
1 & 0 & 0 & 0 & 0 &-d \\
|
||||
0 & 1 & 0 & 0 & 0 & 0 \\
|
||||
0 & 0 & 1 & d & 0 & 0 \\
|
||||
@@ -512,13 +495,13 @@ To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1
|
||||
Then, the acceleration in the cartesian frame can be computed using \eqref{eq:ustation_compute_cart_acc}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_compute_cart_acc}
|
||||
a_{\mathcal{X}} = \mathbf{J}_a^\dagger \cdot a_{\mathcal{L}}
|
||||
a_{\mathcal{X}} = \bm{J}_a^\dagger \cdot a_{\mathcal{L}}
|
||||
\end{equation}
|
||||
|
||||
Similar to what is done for the accelerometers, a Jacobian matrix \(\mathbf{J}_F\) is computed \eqref{eq:ustation_compliance_force_jacobian} and used to convert the individual hammer forces \(F_{\mathcal{L}}\) to force and torques \(F_{\mathcal{X}}\) applied at the center of the micro-hexapod top plate (defined by frame \(\{\mathcal{X}\}\) in Figure \ref{fig:ustation_compliance_meas}).
|
||||
Similar to what is done for the accelerometers, a Jacobian matrix \(\bm{J}_F\) is computed \eqref{eq:ustation_compliance_force_jacobian} and used to convert the individual hammer forces \(F_{\mathcal{L}}\) to force and torques \(F_{\mathcal{X}}\) applied at the center of the micro-hexapod top plate (defined by frame \(\{\mathcal{X}\}\) in Figure \ref{fig:ustation_compliance_meas}).
|
||||
|
||||
\begin{equation}\label{eq:ustation_compliance_force_jacobian}
|
||||
\mathbf{J}_F = \begin{bmatrix}
|
||||
\bm{J}_F = \begin{bmatrix}
|
||||
0 & -1 & 0 & 0 & 0 & 0\\
|
||||
0 & 0 & -1 & -d & 0 & 0\\
|
||||
1 & 0 & 0 & 0 & 0 & 0\\
|
||||
@@ -535,11 +518,11 @@ Similar to what is done for the accelerometers, a Jacobian matrix \(\mathbf{J}_F
|
||||
The equivalent forces and torques applied at center of \(\{\mathcal{X}\}\) are then computed using \eqref{eq:ustation_compute_cart_force}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_compute_cart_force}
|
||||
F_{\mathcal{X}} = \mathbf{J}_F^t \cdot F_{\mathcal{L}}
|
||||
F_{\mathcal{X}} = \bm{J}_F^{\intercal} \cdot F_{\mathcal{L}}
|
||||
\end{equation}
|
||||
|
||||
Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame \(\{\mathcal{X}\}\) to the 6 linear/angular accelerations of the top platform expressed with respect to \(\{\mathcal{X}\}\).
|
||||
These FRFs were then used for comparison with the Simscape model.
|
||||
These FRFs were then used for comparison with the multi-body model.
|
||||
|
||||
The compliance of the micro-station multi-body model was extracted by computing the transfer function from forces/torques applied on the hexapod's top platform to the ``absolute'' motion of the top platform.
|
||||
These results are compared with the measurements in Figure \ref{fig:ustation_frf_compliance_model}.
|
||||
@@ -558,21 +541,20 @@ Considering the complexity of the micro-station compliance dynamics, the model c
|
||||
\end{center}
|
||||
\subcaption{\label{fig:ustation_frf_compliance_Rxyz_model}Compliance in rotation}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_frf_compliance_model}Compliance of the micro-station expressed in frame \(\{\mathcal{X}\}\). The measured FRFs are display by solid lines, while the FRFs extracted from the multi-body models are shown by dashed lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed.}
|
||||
\caption{\label{fig:ustation_frf_compliance_model}Compliance of the micro-station expressed in frame \(\{\mathcal{X}\}\). The measured FRFs are display by translucent lines, while the FRFs extracted from the multi-body models are shown by opaque lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed.}
|
||||
\end{figure}
|
||||
|
||||
\chapter{Estimation of Disturbances}
|
||||
\label{sec:ustation_disturbances}
|
||||
The goal of this section is to obtain a realistic representation of disturbances affecting the micro-station.
|
||||
These disturbance sources are then used during time domain simulations to accurately model the micro-station behavior.
|
||||
The focus on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances.
|
||||
Such disturbances include ground motions and vibrations induces by scanning the translation stage and the spindle.
|
||||
The focus is on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances.
|
||||
Such disturbances include ground motions and vibrations induce by scanning the translation stage and the spindle.
|
||||
|
||||
In the multi-body model, stage vibrations are modeled as internal forces applied in the stage joint.
|
||||
In practice, disturbance forces cannot be directly measured.
|
||||
Instead, the vibrations of the micro-station's top platform induced by the disturbances were measured (Section \ref{ssec:ustation_disturbances_meas}).
|
||||
|
||||
To estimate the equivalent disturbance force that induces such vibration, the transfer functions from disturbance sources (i.e. forces applied in the stages' joint) to the displacements of the micro-station's top platform with respect to the granite are extracted from the Simscape model (Section \ref{ssec:ustation_disturbances_sensitivity}).
|
||||
To estimate the equivalent disturbance force that induces such vibration, the transfer functions from disturbance sources (i.e. forces applied in the stages' joint) to the displacements of the micro-station's top platform with respect to the granite are extracted from the multi-body model (Section \ref{ssec:ustation_disturbances_sensitivity}).
|
||||
Finally, the obtained disturbance sources are compared in Section \ref{ssec:ustation_disturbances_results}.
|
||||
\section{Disturbance measurements}
|
||||
\label{ssec:ustation_disturbances_meas}
|
||||
@@ -583,7 +565,7 @@ The tilt stage and the micro-hexapod also have positioning errors; however, they
|
||||
Therefore, from a control perspective, they are not important.
|
||||
\paragraph{Ground Motion}
|
||||
|
||||
The ground motion was measured by using a sensitive 3-axis geophone\footnote{A 3-Axis L4C geophone manufactured Sercel was used.} placed on the ground.
|
||||
The ground motion was measured by using a sensitive 3-axis geophone shown in Figure \ref{fig:ustation_geophone_picture} placed on the ground.
|
||||
The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function.
|
||||
The obtained ground motion displacement is shown in Figure \ref{fig:ustation_ground_disturbance}.
|
||||
|
||||
@@ -600,7 +582,6 @@ The obtained ground motion displacement is shown in Figure \ref{fig:ustation_gro
|
||||
\captionof{figure}{\label{fig:ustation_geophone_picture}(3D) L-4C geophone}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\paragraph{Ty Stage}
|
||||
|
||||
To measure the positioning errors of the translation stage, the setup shown in Figure \ref{fig:ustation_errors_ty_setup} is used.
|
||||
@@ -610,7 +591,7 @@ A similar setup was used to measure the horizontal deviation (i.e. in the \(x\)
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/ustation_errors_ty_setup.png}
|
||||
\caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the flatness (vertical deviation) of the translation stage}
|
||||
\caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the straightness (vertical deviation) of the translation stage}
|
||||
\end{figure}
|
||||
|
||||
Six scans were performed between \(-4.5\,mm\) and \(4.5\,mm\).
|
||||
@@ -635,7 +616,6 @@ Similar result is obtained for the \(x\) lateral direction.
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_errors_dy}Measurement of the linear (vertical) deviation of the Translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).}
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Spindle}
|
||||
|
||||
To measure the positioning errors induced by the Spindle, a ``Spindle error analyzer''\footnote{The Spindle Error Analyzer is made by Lion Precision.} is used as shown in Figure \ref{fig:ustation_rz_meas_lion_setup}.
|
||||
@@ -688,7 +668,6 @@ The vertical motion induced by scanning the spindle is in the order of \(\pm 30\
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_errors_spindle}Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis.}
|
||||
\end{figure}
|
||||
|
||||
\section{Sensitivity to disturbances}
|
||||
\label{ssec:ustation_disturbances_sensitivity}
|
||||
|
||||
@@ -717,11 +696,10 @@ The obtained transfer functions are shown in Figure \ref{fig:ustation_model_sens
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_model_sensitivity}Extracted transfer functions from disturbances to relative motion between the micro-station's top platform and the granite. The considered disturbances are the ground motion (\subref{fig:ustation_model_sensitivity_ground_motion}), the translation stage vibrations (\subref{fig:ustation_model_sensitivity_ty}), and the spindle vibrations (\subref{fig:ustation_model_sensitivity_rz}).}
|
||||
\end{figure}
|
||||
|
||||
\section{Obtained disturbance sources}
|
||||
\label{ssec:ustation_disturbances_results}
|
||||
|
||||
From the measured effect of disturbances in Section \ref{ssec:ustation_disturbances_meas} and the sensitivity to disturbances extracted from the Simscape model in Section \ref{ssec:ustation_disturbances_sensitivity}, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated.
|
||||
From the measured effect of disturbances in Section \ref{ssec:ustation_disturbances_meas} and the sensitivity to disturbances extracted from the multi-body model in Section \ref{ssec:ustation_disturbances_sensitivity}, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated.
|
||||
The obtained power spectral density of the disturbances are shown in Figure \ref{fig:ustation_dist_sources}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@@ -748,7 +726,7 @@ The obtained power spectral density of the disturbances are shown in Figure \ref
|
||||
|
||||
The disturbances are characterized by their power spectral densities, as shown in Figure \ref{fig:ustation_dist_sources}.
|
||||
However, to perform time domain simulations, disturbances must be represented by a time domain signal.
|
||||
To generate stochastic time-domain signals with a specific power spectral densities, the discrete inverse Fourier transform is used, as explained in \cite[chap. 12.11]{preumont94_random_vibrat_spect_analy}.
|
||||
To generate stochastic time-domain signals with a specific power spectral density, the discrete inverse Fourier transform is used, as explained in \cite[chap. 12.11]{preumont94_random_vibrat_spect_analy}.
|
||||
Examples of the obtained time-domain disturbance signals are shown in Figure \ref{fig:ustation_dist_sources_time}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@@ -772,7 +750,6 @@ Examples of the obtained time-domain disturbance signals are shown in Figure \re
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_dist_sources_time}Generated time domain disturbance signals. Ground motion (\subref{fig:ustation_dist_source_ground_motion_time}), translation stage (\subref{fig:ustation_dist_source_translation_stage_time}) and spindle (\subref{fig:ustation_dist_source_spindle_time}).}
|
||||
\end{figure}
|
||||
|
||||
\chapter{Simulation of Scientific Experiments}
|
||||
\label{sec:ustation_experiments}
|
||||
To fully validate the micro-station multi-body model, two time-domain simulations corresponding to typical use cases were performed.
|
||||
@@ -804,8 +781,7 @@ A good correlation with the measurements is observed both for radial errors (Fig
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_errors_model_spindle}Simulation results for a tomography experiment at constant velocity of 60rpm. The comparison is made with measurements for both radial (\subref{fig:ustation_errors_model_spindle_radial}) and axial errors (\subref{fig:ustation_errors_model_spindle_axial}).}
|
||||
\end{figure}
|
||||
|
||||
\section{Raster Scans with the translation stage}
|
||||
\section{Scans with the translation stage}
|
||||
\label{sec:ustation_experiments_ty_scans}
|
||||
|
||||
A second experiment was performed in which the translation stage was scanned at constant velocity.
|
||||
@@ -819,9 +795,8 @@ A similar error amplitude was observed, thus indicating that the multi-body mode
|
||||
\includegraphics[scale=1]{figs/ustation_errors_model_dy_vertical.png}
|
||||
\caption{\label{fig:ustation_errors_model_dy_vertical}Vertical errors during a constant-velocity scan of the translation stage. Comparison of the measurements and simulated errors.}
|
||||
\end{figure}
|
||||
|
||||
\chapter*{Conclusion}
|
||||
\label{sec:uniaxial_conclusion}
|
||||
\label{sec:ustation_conclusion}
|
||||
|
||||
In this study, a multi-body model of the micro-station was developed.
|
||||
It was difficult to match the measured dynamics obtained from the modal analysis of the micro-station.
|
||||
@@ -830,6 +805,5 @@ After tuning the model parameters, a good match with the measured compliance was
|
||||
|
||||
The disturbances affecting the sample position should also be well modeled.
|
||||
After experimentally estimating the disturbances (Section \ref{sec:ustation_disturbances}), the multi-body model was finally validated by performing a tomography simulation (Figure \ref{fig:ustation_errors_model_spindle}) as well as a simulation in which the translation stage was scanned (Figure \ref{fig:ustation_errors_model_dy_vertical}).
|
||||
|
||||
\printbibliography[heading=bibintoc,title={Bibliography}]
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user