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<title>Nanopositioning system with force feedback for high-performance tracking and vibration control - My digital brain</title>
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—
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Thomas Dehaeze
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" /><meta name="description" content="Tags Sensor Fusion, Force Sensors Reference (Fleming, 2010) Author(s) Fleming, A. Year 2010 Summary:
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The noise generated by a piezoelectric force sensor is much less than a capacitive sensor Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor Noise of a piezoelectric force sensor IFF with a piezoelectric stack actuator and piezoelectric force sensor A force sensor is used as a displacement sensor below the frequency of the first zero Sensor fusion architecture with a capacitive sensor and a force sensor and using complementary filters Virtual sensor fusion architecture (called low-frequency bypass) Analog implementation of the control strategies to avoid quantization noise, finite resolution and sampling delay Model of a multi-layer monolithic piezoelectric stack actuator" />
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<h1 class="post-title">Nanopositioning system with force feedback for high-performance tracking and vibration control</h1>
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<h2 class="post-toc-title">Contents</h2>
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<ul>
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<li><a href="#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator">Model of a multi-layer monolithic piezoelectric stack actuator</a></li>
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<li><a href="#dynamics-of-a-piezoelectric-force-sensor">Dynamics of a piezoelectric force sensor</a></li>
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<li><a href="#noise-of-a-piezoelectric-force-sensor">Noise of a piezoelectric force sensor</a></li>
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<dl>
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<dt>Tags</dt>
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<dd><a href="/zettels/sensor_fusion/">Sensor Fusion</a>, <a href="/zettels/force_sensors/">Force Sensors</a></dd>
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<dt>Reference</dt>
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<dd><sup id="c823f68dd2a72b9667a61b3c046b4731"><a href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup></dd>
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<dt>Author(s)</dt>
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<dd>Fleming, A.</dd>
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<dt>Year</dt>
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<dd>2010</dd>
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</dl>
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<p>Summary:</p>
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<ul>
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<li>The noise generated by a piezoelectric force sensor is much less than a capacitive sensor</li>
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<li>Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor</li>
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<li>Noise of a piezoelectric force sensor</li>
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<li>IFF with a piezoelectric stack actuator and piezoelectric force sensor</li>
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<li>A force sensor is used as a displacement sensor below the frequency of the first zero</li>
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<li>Sensor fusion architecture with a capacitive sensor and a force sensor and using complementary filters</li>
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<li>Virtual sensor fusion architecture (called low-frequency bypass)</li>
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<li>Analog implementation of the control strategies to avoid quantization noise, finite resolution and sampling delay</li>
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</ul>
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<h2 id="model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator">Model of a multi-layer monolithic piezoelectric stack actuator</h2>
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<p><a id="orgf7e4ab9"></a></p>
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<figure>
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<img src="/ox-hugo/fleming10_piezo_model.png"
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alt="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model"/> <figcaption>
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<p>Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model</p>
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</figcaption>
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</figure>
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<p>The actuator experiences an internal stress in response to an applied voltage.
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This stress is represented by the voltage dependent force \(F_a\) and is related to free displacement by
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\[ \Delta L = \frac{F_a}{k_a} \]</p>
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<ul>
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<li>\(\Delta L\) is the change in actuator length in [m]</li>
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<li>\(k_a\) is the actuator stiffness in [N/m]</li>
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</ul>
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<p>The developed force \(F_a\) is related to the applied voltage by:
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\[ \Delta L = d_{33} n V_a \]</p>
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<ul>
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<li>\(d_{33}\) is the piezoelectric strain constant in [m/V]</li>
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<li>\(n\) is the number of layers</li>
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<li>\(V_a\) is the applied voltage in [V]</li>
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</ul>
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<p>Combining the two equations, we obtain:
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\[ F_a = d_{33} n k_a V_a \]</p>
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<p>The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V].
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We denote this constant by \(g_a\) and:
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\[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]</p>
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<h2 id="dynamics-of-a-piezoelectric-force-sensor">Dynamics of a piezoelectric force sensor</h2>
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<p>Piezoelectric force sensors provide a high sensitivity and bandwidth with low noise at high frequencies.</p>
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<p>If a <strong>single wafer</strong> of piezoelectric material is sandwiched between the actuator and platform:
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\[ D = d_{33} T \]</p>
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<ul>
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<li>\(D\) is the amount of generated charge per unit area in \([C/m^2]\)</li>
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<li>\(T\) is the stress in \([N/m^2]\)</li>
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<li>\(d_{33}\) is the piezoelectric strain constant in \([m/V] = [C/N]\)</li>
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</ul>
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<p>The generated charge is then
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\[ q = d_{33} F_s \]</p>
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<p>If an <strong>n-layer</strong> piezoelectric transducer is used as a force sensor, the generated charge is then:
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\[ q = n d_{33} F_s \]</p>
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<hr>
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<p>We can use a <strong>charge amplifier</strong> to measure the force \(F_s\).</p>
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<figure>
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<img src="/ox-hugo/fleming10_charge_ampl_piezo.png"
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alt="Figure 2: Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \(q\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \(V_s\) equal to \(-q/C_s\)"/> <figcaption>
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<p>Figure 2: Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \(q\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \(V_s\) equal to \(-q/C_s\)</p>
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</figcaption>
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</figure>
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<p>The output voltage \(V_s\) is equal to
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\[ V_s = -\frac{q}{C_s} = -\frac{n d_{33}F_s}{C_s} \]
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that is, the scaling between the force and voltage is \(-\frac{n d_{33}F_s}{C_s}\ [V/N]\) .</p>
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<hr>
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<p>We can also use a voltage amplifier.
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In that case, the generated charge is deposited on the transducer’s internal capacitance.</p>
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<p>The open-circuit voltage of a piezoelectric force sensor is:
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\[ V_s = \frac{n d_{33} F_s}{C} \]</p>
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<ul>
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<li>\(C\) is the transducer capacitance defined by \(C = n \epsilon_T A / h\) in [F]
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<ul>
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<li>\(A\) is the area in \([m^2]\)</li>
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<li>\(h\) is the layer thickness in [m]</li>
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<li>\(\epsilon_T\) is the dielectric permittivity under a constant stress in \([F/m]\)</li>
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</ul>
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</li>
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</ul>
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<p>We obtain
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\[ V_s = g_s F_s, \quad g_s = \frac{n d_{33}}{C} \]</p>
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<h2 id="noise-of-a-piezoelectric-force-sensor">Noise of a piezoelectric force sensor</h2>
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<p>As piezoelectric sensors have a capacitive source impedance, the sensor noise density \(N_{V_s}(\omega)\) is primarily due to current noise \(i_n\) reacting the capacitive source impedance:
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\[ N_{V_s}(\omega) = i_n \frac{1}{C \omega} \]</p>
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<ul>
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<li>\(N_{V_s}\) is the measured noise in \(V/\sqrt{\text{Hz}}\)</li>
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<li>\(i_n\) is the current noise in \(A/\sqrt{\text{Hz}}\)</li>
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<li>\(C\) is the capacitance of the piezoelectric in \(F\)</li>
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</ul>
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<p>The current noise density of a general purpose LM833 FET-input op-amp is \(0.5\ pA/\sqrt{\text{Hz}}\).
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The capacitance of a piezoelectric stack is typically between \(1 \mu F\) and \(100 \mu F\).</p>
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<h1 id="bibliography">Bibliography</h1>
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<p><a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., <em>Nanopositioning system with force feedback for high-performance tracking and vibration control</em>, IEEE/ASME Transactions on Mechatronics, <em>15(3)</em>, 433–447 (2010). <a href="http://dx.doi.org/10.1109/tmech.2009.2028422">http://dx.doi.org/10.1109/tmech.2009.2028422</a> <a href="#c823f68dd2a72b9667a61b3c046b4731">↩</a></p>
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<li><a href="/zettels/actuators/">Actuators</a></li>
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<li><a href="/zettels/force_sensors/">Force Sensors</a></li>
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