5.7 KiB
+++ title = "Nanopositioning system with force feedback for high-performance tracking and vibration control" author = ["Dehaeze Thomas"] draft = false +++
- Tags
- [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Force Sensors]({{< relref "force_sensors.md" >}})
- Reference
- (Fleming 2010)
- Author(s)
- Fleming, A.
- Year
- 2010
Summary
- 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
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="<span class="figure-number">Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
The actuator experiences an internal stress in response to an applied voltage. This stress is represented by the voltage dependent force \(F_a\) and is related to free displacement by \[ \Delta L = \frac{F_a}{k_a} \]
- \(\Delta L\) is the change in actuator length in [m]
- \(k_a\) is the actuator stiffness in [N/m]
The developed force \(F_a\) is related to the applied voltage by: \[ \Delta L = d_{33} n V_a \]
- \(d_{33}\) is the piezoelectric strain constant in [m/V]
- \(n\) is the number of layers
- \(V_a\) is the applied voltage in [V]
Combining the two equations, we obtain: \[ F_a = d_{33} n k_a V_a \]
The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V]. We denote this constant by \(g_a\) and: \[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]
Dynamics of a piezoelectric force sensor
Piezoelectric force sensors provide a high sensitivity and bandwidth with low noise at high frequencies.
If a single wafer of piezoelectric material is sandwiched between the actuator and platform: \[ D = d_{33} T \]
- \(D\) is the amount of generated charge per unit area in \([C/m^2]\)
- \(T\) is the stress in \([N/m^2]\)
- \(d_{33}\) is the piezoelectric strain constant in \([m/V] = [C/N]\)
The generated charge is then \[ q = d_{33} F_s \]
If an n-layer piezoelectric transducer is used as a force sensor, the generated charge is then: \[ q = n d_{33} F_s \]
We can use a charge amplifier to measure the force \(F_s\).
{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="<span class="figure-number">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 \(-q/C_s\)" >}}
The output voltage \(V_s\) is equal to \[ V_s = -\frac{q}{C_s} = -\frac{n d_{33}F_s}{C_s} \] that is, the scaling between the force and voltage is \(-\frac{n d_{33}F_s}{C_s}\ [V/N]\) .
We can also use a voltage amplifier. In that case, the generated charge is deposited on the transducer's internal capacitance.
The open-circuit voltage of a piezoelectric force sensor is: \[ V_s = \frac{n d_{33} F_s}{C} \]
- \(C\) is the transducer capacitance defined by \(C = n \epsilon_T A / h\) in [F]
- \(A\) is the area in \([m^2]\)
- \(h\) is the layer thickness in [m]
- \(\epsilon_T\) is the dielectric permittivity under a constant stress in \([F/m]\)
We obtain \[ V_s = g_s F_s, \quad g_s = \frac{n d_{33}}{C} \]
Noise of a piezoelectric force sensor
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: \[ N_{V_s}(\omega) = i_n \frac{1}{C \omega} \]
- \(N_{V_s}\) is the measured noise in \(V/\sqrt{\text{Hz}}\)
- \(i_n\) is the current noise in \(A/\sqrt{\text{Hz}}\)
- \(C\) is the capacitance of the piezoelectric in \(F\)
The current noise density of a general purpose LM833 FET-input op-amp is \(0.5\ pA/\sqrt{\text{Hz}}\). The capacitance of a piezoelectric stack is typically between \(1 \mu F\) and \(100 \mu F\).
Tested feedback control strategies
{{< figure src="/ox-hugo/fleming10_fb_control_strats.png" caption="<span class="figure-number">Figure 3: Comparison of: (a) basic integral control. (b) direct tracking control. (c) dual-sensor feedback. (d) low frequency bypass" >}}