test-bench-nano-hexapod/test-bench-nano-hexapod.org

Nano-Hexapod - Test Bench

• Introduction
• Encoders fixed to the Struts
  • Introduction
  • Identification of the dynamics
    • Load Data
    • Spectral Analysis - Setup
    • DVF Plant
    • IFF Plant
  • Comparison with the Simscape Model
    • Introduction
    • Dynamics from Actuator to Force Sensors
    • Dynamics from Actuator to Encoder
  • Integral Force Feedback
    • Plant
    • Root Locus
    • Gains
    • Experimental Results

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This report is also available as a pdf.

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\clearpage

Introduction   ignore

In this document, the dynamics of the nano-hexapod shown in Figure fig:picture_bench_granite_nano_hexapod is identified.

Here are the documentation of the equipment used for this test bench:

• Voltage Amplifier: PiezoDrive PD200
• Amplified Piezoelectric Actuator: Cedrat APA300ML
• DAC/ADC: Speedgoat IO313
• Encoder: Renishaw Vionic and used Ruler
• Interferometers: Attocube

\definecolor{instrumentation}{rgb}{0, 0.447, 0.741}
\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098}

\begin{tikzpicture}
  % Blocs
  \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics};
  \coordinate[] (inputF)  at (nano_hexapod.west);
  \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$);
  \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$);

  \node[block, left= 0.8 of inputF,  fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks};
  \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200)   {PD200};
  \node[DAC,   left= 0.8 of PD200,   fill=instrumentation!20!white] (F_DAC)   {DAC};
  \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack};
  \node[ADC,   right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC)  {ADC};
  \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder};

  % Connections and labels
  \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west);
  \draw[->] (F_DAC.east)   -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west);
  \draw[->] (PD200.east)   -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west);
  \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$};

  \draw[->] (outputF)       -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$};
  \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west);
  \draw[->] (Fm_ADC.east)   -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$};

  \draw[->] (outputL)      -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$};
  \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$};

  % Nano-Hexapod
  \begin{scope}[on background layer]
    \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {};
    \node[above] at (system.north) {Nano-Hexapod};
  \end{scope}
\end{tikzpicture}

	Unit	Matlab	Vector	Elements
Control Input (wanted DAC voltage)	[V]	u	$\bm{u}$	$u_i$
DAC Output Voltage	[V]	u	$\tilde{\bm{u}}$	$\tilde{u}_i$
PD200 Output Voltage	[V]	ua	$\bm{u}_a$	$u_{a,i}$
Actuator applied force	[N]	tau	$\bm{\tau}$	$\tau_i$
Strut motion	[m]	dL	$d\bm{\mathcal{L}}$	$d\mathcal{L}_i$
Encoder measured displacement	[m]	dLm	$d\bm{\mathcal{L}}_m$	$d\mathcal{L}_{m,i}$
Force Sensor strain	[m]	epsilon	$\bm{\epsilon}$	$\epsilon_i$
Force Sensor Generated Voltage	[V]	taum	$\tilde{\bm{\tau}}_m$	$\tilde{\tau}_{m,i}$
Measured Generated Voltage	[V]	taum	$\bm{\tau}_m$	$\tau_{m,i}$
Motion of the top platform	[m,rad]	dX	$d\bm{\mathcal{X}}$	$d\mathcal{X}_i$
Metrology measured displacement	[m,rad]	dXm	$d\bm{\mathcal{X}}_m$	$d\mathcal{X}_{m,i}$

Encoders fixed to the Struts

Introduction

In this section, the encoders are fixed to the struts.

Identification of the dynamics

Load Data

meas_data_lf = {};

for i = 1:6
    meas_data_lf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_lf.mat', i), 't', 'Va', 'Vs', 'de')};
    meas_data_hf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_hf.mat', i), 't', 'Va', 'Vs', 'de')};
end

Spectral Analysis - Setup

% Sampling Time [s]
Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

And we get the frequency vector.

[~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts);

i_lf = f < 250; % Points for low frequency excitation
i_hf = f > 250; % Points for high frequency excitation

DVF Plant

First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure fig:enc_struts_dvf_coh).

%% Coherence
coh_dvf_lf = zeros(length(f), 6, 6);
coh_dvf_hf = zeros(length(f), 6, 6);

for i = 1:6
    coh_dvf_lf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
    coh_dvf_hf(:, :, i) = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
end

Obtained coherence for the DVF plant

Then the 6x6 transfer function matrix is estimated (Figure fig:enc_struts_dvf_frf).

%% DVF Plant
G_dvf_lf = zeros(length(f), 6, 6);
G_dvf_hf = zeros(length(f), 6, 6);

for i = 1:6
    G_dvf_lf(:, :, i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
    G_dvf_hf(:, :, i) = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
end

Measured FRF for the DVF plant

IFF Plant

First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure fig:enc_struts_iff_coh).

%% Coherence
coh_iff_lf = zeros(length(f), 6, 6);
coh_iff_hf = zeros(length(f), 6, 6);

for i = 1:6
    coh_iff_lf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
    coh_iff_hf(:, :, i) = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts);
end

Obtained coherence for the IFF plant

Then the 6x6 transfer function matrix is estimated (Figure fig:enc_struts_iff_frf).

%% IFF Plant
G_iff_lf = zeros(length(f), 6, 6);
G_iff_hf = zeros(length(f), 6, 6);

for i = 1:6
    G_iff_lf(:, :, i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
    G_iff_hf(:, :, i) = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts);
end

Measured FRF for the IFF plant

Comparison with the Simscape Model

Introduction   ignore

In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.

Dynamics from Actuator to Force Sensors

n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Fm'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

Giff = 20*exp(-s*Ts)*linearize(mdl, io, 0.0, options);

Diagonal elements of the IFF Plant

Off diagonal elements of the IFF Plant

Dynamics from Actuator to Encoder

n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders

Gdvf = 20*exp(-s*Ts)*linearize(mdl, io, 0.0, options);

Diagonal elements of the DVF Plant

Off diagonal elements of the DVF Plant

TODO Integral Force Feedback

Plant

Root Locus

Gains

Experimental Results