+
+
Control of the NASS with Voice coil actuators
+Table of Contents
+-
+
- 1. Initialization +
- 2. Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\) + + +
- 3. High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\) + + +
- 4. On the usefulness of the High Authority Control loop / Linearization loop + + +
- 5. Primary Controller in the task space - \(\bm{K}_\mathcal{X}\) + + +
- 6. Simulation +
- 7. Results + + +
+The goal here is to study the use of a voice coil based nano-hexapod. +That is to say a nano-hexapod with a very small stiffness. +
+ + +
+
Figure 1: Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs
+1 Initialization
++We initialize all the stages with the default parameters. +
+initializeGround(); +initializeGranite(); +initializeTy(); +initializeRy(); +initializeRz(); +initializeMicroHexapod(); +initializeAxisc(); +initializeMirror(); ++
+The nano-hexapod is a voice coil based hexapod and the sample has a mass of 1kg. +
+initializeNanoHexapod('actuator', 'lorentz'); +initializeSample('mass', 1); ++
+We set the references that corresponds to a tomography experiment. +
+initializeReferences('Rz_type', 'rotating', 'Rz_period', 1); ++
initializeDisturbances(); ++
initializeController('type', 'cascade-hac-lac'); ++
initializeSimscapeConfiguration('gravity', true); ++
+We log the signals. +
+initializeLoggingConfiguration('log', 'all'); ++
Kx = tf(zeros(6)); +Kl = tf(zeros(6)); +Kiff = tf(zeros(6)); ++
2 Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)
+ +2.1 Identification
++Let’s first identify the plant for the IFF controller. +
+%% Name of the Simulink File +mdl = 'nass_model'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors + +%% Run the linearization +G_iff = linearize(mdl, io, 0); +G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; +G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}; ++
2.2 Plant
+ +
+2.3 Root Locus
++As seen in the root locus (Figure 3, no damping can be added to modes corresponding to the resonance of the micro-station. +
+ ++However, critical damping can be achieve for the resonances of the nano-hexapod as shown in the zoomed part of the root (Figure 3, left part). +The maximum damping is obtained for a control gain of \(\approx 70\). +
+ + + +
+
+3 High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)
+ +3.1 Identification of the damped plant
++Let’s identify the dynamics from \(\bm{\tau}^\prime\) to \(d\bm{\mathcal{L}}\) as shown in Figure 1. +
+ +%% Name of the Simulink File +mdl = 'nass_model'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Micro-Station'], 3, 'output', [], 'Dnlm'); io_i = io_i + 1; % Leg Displacement + +%% Run the linearization +Gl = linearize(mdl, io, 0); +Gl.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; +Gl.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}; ++
+There are some unstable poles in the Plant with very small imaginary parts. +These unstable poles are probably not physical, and they disappear when taking the minimum realization of the plant. +
+isstable(Gl) +Gl = minreal(Gl); +isstable(Gl) ++
3.2 Obtained Plant
++The obtained dynamics is shown in Figure 5. +
+ ++Few things can be said on the dynamics: +
+-
+
- the dynamics of the diagonal elements are almost all the same +
- the system is well decoupled below the resonances of the nano-hexapod (1Hz) +
- the dynamics of the diagonal elements are almost equivalent to a critically damped mass-spring-system with some spurious resonances above 50Hz corresponding to the resonances of the micro-station +
+3.3 Controller Design and Loop Gain
++As the plant is well decoupled, a diagonal plant is designed. +
+ +wc = 2*pi*5; % Bandwidth Bandwidth [rad/s] + +h = 2; % Lead parameter + +Kl = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead + (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead + (s + 2*pi*10)/s * ... % Weak Integrator + (s + 2*pi*1)/s * ... % Weak Integrator + 1/(1 + s/2/pi/10); % Low pass filter after crossover + +% Normalization of the gain of have a loop gain of 1 at frequency wc +Kl = Kl.*diag(1./diag(abs(freqresp(Gl*Kl, wc)))); ++
4 On the usefulness of the High Authority Control loop / Linearization loop
++Let’s see what happens is we omit the HAC loop and we directly try to control the damped plant using the measurement of the sample with respect to the granite \(\bm{\mathcal{X}}\). +
+ ++We can do that in two different ways: +
+ + + +
+
Figure 6: IFF control + primary controller in the task space
+
+
Figure 7: HAC-LAC control architecture in the frame of the legs
+4.1 Identification
+initializeController('type', 'hac-iff'); ++
%% Name of the Simulink File +mdl = 'nass_model'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Controller/HAC-IFF/Kx'], 1, 'input'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror + +%% Run the linearization +G = linearize(mdl, io, 0); +G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; ++
isstable(G)
+G = -minreal(G);
+isstable(G)
+
+4.2 Plant in the Task space
++The obtained plant is shown in Figure +
+ +Gx = G*inv(nano_hexapod.J'); ++
4.3 Plant in the Leg’s space
+Gl = nano_hexapod.J*G;
+
+5 Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)
+ +5.1 Identification of the linearized plant
++We know identify the dynamics between \(\bm{r}_{\mathcal{X}_n}\) and \(\bm{r}_\mathcal{X}\). +
+%% Name of the Simulink File +mdl = 'nass_model'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Controller/Cascade-HAC-LAC/Kx'], 1, 'input'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror + +%% Run the linearization +Gx = linearize(mdl, io, 0); +Gx.InputName = {'rL1', 'rL2', 'rL3', 'rL4', 'rL5', 'rL6'}; +Gx.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; ++
+As before, we take the minimum realization. +
+isstable(Gx)
+Gx = -minreal(Gx);
+isstable(Gx)
+
+5.2 Obtained Plant
+5.3 Controller Design
+wc = 2*pi*200; % Bandwidth Bandwidth [rad/s] + +h = 2; % Lead parameter + +Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... + (s + 2*pi*10)/s * ... + (s + 2*pi*100)/s * ... + 1/(1+s/2/pi/500); % For Piezo +% Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * (s + 2*pi*10)/s * (s + 2*pi*1)/s ; % For voice coil + +% Normalization of the gain of have a loop gain of 1 at frequency wc +Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc)))); ++
6 Simulation
+load('mat/conf_simulink.mat'); +set_param(conf_simulink, 'StopTime', '2'); ++
+And we simulate the system. +
+sim('nass_model'); ++
cascade_hac_lac_lorentz = simout; +save('./mat/cascade_hac_lac.mat', 'cascade_hac_lac_lorentz', '-append'); ++
7 Results
+7.1 Load the simulation results
+load('./mat/experiment_tomography.mat', 'tomo_align_dist'); +load('./mat/cascade_hac_lac.mat', 'cascade_hac_lac', 'cascade_hac_lac_lorentz'); ++
7.2 Control effort
+7.3 Load the simulation results
+n_av = 4; +han_win = hanning(ceil(length(cascade_hac_lac.Em.En.Data(:,1))/n_av)); ++
t = cascade_hac_lac.Em.En.Time; +Ts = t(2)-t(1); + +[pxx_ol, f] = pwelch(tomo_align_dist.Em.En.Data, han_win, [], [], 1/Ts); +[pxx_ca, ~] = pwelch(cascade_hac_lac.Em.En.Data, han_win, [], [], 1/Ts); +[pxx_vc, ~] = pwelch(cascade_hac_lac_lorentz.Em.En.Data, han_win, [], [], 1/Ts); ++
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