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content/zettels/jacobian.md
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## Jacobian Matrices of a Parallel Manipulator {#jacobian-matrices-of-a-parallel-manipulator}
From ([Taghirad 2013](#org1eb570f)):
From ([Taghirad 2013](#orgdcd348b)):
> The Jacobian matrix not only reveals the **relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities**, it also constructs the transformation needed to find the **actuator forces from the forces and moments acting on the moving platform**.
([Merlet 2006](#org77b3718))
([Merlet 2006](#org3c93a40))
## Computing the Jacobian Matrix {#computing-the-jacobian-matrix}
How to derive the Jacobian matrix is well explained in chapter 4 of ([Taghirad 2013](#org1eb570f)) ([notes]({{< relref "taghirad13_paral" >}})).
How to derive the Jacobian matrix is well explained in chapter 4 of ([Taghirad 2013](#orgdcd348b)) ([notes]({{< relref "taghirad13_paral" >}})).
Consider parallel manipulator shown in Figure [1](#orgf2877e3) (it represents a Stewart platform).
Consider parallel manipulator shown in Figure [1](#orgea8dcb1) (it represents a Stewart platform).
Kinematic loop closures are:
@@ -50,7 +50,7 @@ By taking the time derivative, we obtain the following **Velocity Loop Closures*
{}^A\hat{\bm{s}}\_i {}^A\bm{v}\_p + ({}^A\bm{b}\_i \times \hat{\bm{s}}\_i) {}^A\bm{\omega} = \dot{l}\_i \label{eq:velocity\_loop\_closure}
\end{equation}
<a id="orgf2877e3"></a>
<a id="orgea8dcb1"></a>
{{< figure src="/ox-hugo/jacobian_geometry.png" caption="Figure 1: Example of parallel manipulator with defined frames and vectors" >}}
@@ -83,7 +83,7 @@ And therefore \\(\bm{J}\\) then **depends only** on:
- \\({}^A\hat{\bm{s}}\_i\\) the orientation of the limbs
- \\({}^A\bm{b}\_i\\) the position of the joints with respect to \\(O\_B\\) and express in \\(\\{\bm{A}\\}\\).
For the platform in Figure [1](#orgf2877e3), we have:
For the platform in Figure [1](#orgea8dcb1), we have:
\begin{equation}
\begin{bmatrix} \dot{l}\_1 \\ \dot{l}\_2 \\ \dot{l}\_3 \\ \dot{l}\_4 \\ \dot{l}\_5 \\ \dot{l}\_6 \end{bmatrix} =
@@ -112,8 +112,9 @@ in which \\(\bm{\tau} = [f\_1, f\_2, \cdots, f\_6]^T\\) is the vector of actuato
Note that it is here assumed that the forces are static and **along the limb axis** \\(\hat{\bm{s}}\_i\\).
## Bibliography {#bibliography}
<a id="org77b3718"></a>Merlet, Jean-Pierre. 2006. “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots.”
<a id="org3c93a40"></a>Merlet, Jean-Pierre. 2006. “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots.”
<a id="org1eb570f"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.
<a id="orgdcd348b"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.

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content/zettels/piezoelectric_actuators.md
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### Model {#model}
A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#orga50fca3)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#org4089875)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
Basically, it can be represented by a spring \\(k\_a\\) with the force source \\(F\_a\\) in parallel.
@@ -56,14 +56,14 @@ Some manufacturers propose "raw" plate actuators that can be used as actuator /
## Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators}
The Amplified Piezo Actuators principle is presented in ([Claeyssen et al. 2007](#orgc2229f2)):
The Amplified Piezo Actuators principle is presented in ([Claeyssen et al. 2007](#orge4dbf99)):
> The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
> The flatter is the actuator, the higher is the amplification.
A model of an amplified piezoelectric actuator is described in ([Lucinskis and Mangeot 2016](#org661d95e)).
A model of an amplified piezoelectric actuator is described in ([Lucinskis and Mangeot 2016](#orga7e7177)).
<a id="org8c43728"></a>
<a id="org22709f8"></a>
{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms <sup id=\"d9e8b33774f1e65d16bd79114db8ac64\"><a href=\"#ling16_enhan_mathem_model_displ_amplif\" title=\"Mingxiang Ling, Junyi Cao, Minghua Zeng, Jing Lin, \&amp; Daniel J Inman, Enhanced Mathematical Modeling of the Displacement Amplification Ratio for Piezoelectric Compliant Mechanisms, {Smart Materials and Structures}, v(7), 075022 (2016).\">ling16_enhan_mathem_model_displ_amplif</a></sup>" >}}
@@ -155,43 +155,43 @@ For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolut
### Electrical Capacitance {#electrical-capacitance}
The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org538bacc)).
The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org38927da)).
This is due to the fact that voltage amplifier has a limitation on the deliverable current.
[Voltage Amplifier]({{< relref "voltage_amplifier" >}}) with high maximum output current should be used if either high bandwidth is wanted or piezoelectric stacks with high capacitance are to be used.
<a id="org538bacc"></a>
<a id="org38927da"></a>
{{< figure src="/ox-hugo/piezoelectric_capacitance_voltage_max.png" caption="Figure 2: Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance" >}}
## Piezoelectric actuator experiencing a mass load {#piezoelectric-actuator-experiencing-a-mass-load}
When the piezoelectric actuator is supporting a payload, it will experience a static deflection due to its finite stiffness \\(\Delta l\_n = \frac{mg}{k\_p}\\), but its stroke will remain unchanged (Figure [3](#org8d008aa)).
When the piezoelectric actuator is supporting a payload, it will experience a static deflection due to its finite stiffness \\(\Delta l\_n = \frac{mg}{k\_p}\\), but its stroke will remain unchanged (Figure [3](#org35604e1)).
<a id="org8d008aa"></a>
<a id="org35604e1"></a>
{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="Figure 3: Motion of a piezoelectric stack actuator under external constant force" >}}
## Piezoelectric actuator in contact with a spring load {#piezoelectric-actuator-in-contact-with-a-spring-load}
Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its maximum stroke \\(\Delta L\\) is less than its free stroke \\(\Delta L\_f\\) (Figure [4](#orgf006168)):
Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its maximum stroke \\(\Delta L\\) is less than its free stroke \\(\Delta L\_f\\) (Figure [4](#org2f55c26)):
\begin{equation}
\Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e}
\end{equation}
<a id="orgf006168"></a>
<a id="org2f55c26"></a>
{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="Figure 4: Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}
For piezo actuators, force and displacement are inversely related (Figure [5](#org4b9d568)).
For piezo actuators, force and displacement are inversely related (Figure [5](#orgf384614)).
Maximum, or blocked, force (\\(F\_b\\)) occurs when there is no displacement.
Likewise, at maximum displacement, or free stroke, (\\(\Delta L\_f\\)) no force is generated.
When an external load is applied, the stiffness of the load (\\(k\_e\\)) determines the displacement (\\(\Delta L\_A\\)) and force (\\(\Delta F\_A\\)) that can be produced.
<a id="org4b9d568"></a>
<a id="orgf384614"></a>
{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="Figure 5: Relation between the maximum force and displacement" >}}
@@ -199,13 +199,14 @@ When an external load is applied, the stiffness of the load (\\(k\_e\\)) determi
## Driving Electronics {#driving-electronics}
Piezoelectric actuators can be driven either using a voltage to charge converter or a [Voltage Amplifier]({{< relref "voltage_amplifier" >}}).
Limitations of the electronics is discussed in the book [Design, modeling and control of nanopositioning systems]({{< relref "fleming14_desig_model_contr_nanop_system#electrical-considerations" >}}).
## Bibliography {#bibliography}
<a id="orgc2229f2"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” _Ferroelectrics_ 351 (1):3–14. <https://doi.org/10.1080/00150190701351865>.
<a id="orge4dbf99"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” _Ferroelectrics_ 351 (1):3–14. <https://doi.org/10.1080/00150190701351865>.
<a id="orga50fca3"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
<a id="org4089875"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
<a id="org661d95e"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”
<a id="orga7e7177"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”

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content/zettels/reference_books.md
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## Here are my favorite books {#here-are-my-favorite-books}
([Steinbuch and Oomen 2016](#orgf417be1))
([Taghirad 2013](#org5d52649))
([Lurie 2012](#org55fc1e1))
([Skogestad and Postlethwaite 2007](#orgc1de88b))
([Schmidt, Schitter, and Rankers 2014](#orgb0fd6be))
([Preumont 2018](#orgf335f1e))
([Leach 2014](#orgcac846b))
([Ewins 2000](#orgff1b332))
([Leach and Smith 2018](#orga27fe16))
([Horowitz 2015](#orgf44e740))
([Steinbuch and Oomen 2016](#orgb1557ba))
([Taghirad 2013](#org82a60a2))
([Lurie 2012](#org1239999))
([Skogestad and Postlethwaite 2005](#org73832af))
([Schmidt, Schitter, and Rankers 2014](#orgc7f4ff4))
([Preumont 2018](#orgf92c7c5))
([Leach 2014](#org830c619))
([Ewins 2000](#orga0a3ec2))
([Leach and Smith 2018](#orgc115008))
([Horowitz 2015](#org7915565))
## Bibliography {#bibliography}
<a id="orgff1b332"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
<a id="orga0a3ec2"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
<a id="orgf44e740"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
<a id="org7915565"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
<a id="orgcac846b"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
<a id="org830c619"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
<a id="orga27fe16"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
<a id="orgc115008"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
<a id="org55fc1e1"></a>Lurie, B. J. 2012. _Classical Feedback Control : with MATLAB and Simulink_. Boca Raton, FL: CRC Press.
<a id="org1239999"></a>Lurie, B. J. 2012. _Classical Feedback Control : with MATLAB and Simulink_. Boca Raton, FL: CRC Press.
<a id="orgf335f1e"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<a id="orgf92c7c5"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<a id="orgb0fd6be"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
<a id="orgc7f4ff4"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
<a id="orgc1de88b"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="org73832af"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2005. _Multivariable Feedback Control: Analysis and Design - Second Edition_. John Wiley.
<a id="orgf417be1"></a>Steinbuch, Maarten, and Tom Oomen. 2016. “Model-Based Control for High-Tech Mechatronics Systems.” CRC Press/Taylor & Francis.
<a id="orgb1557ba"></a>Steinbuch, Maarten, and Tom Oomen. 2016. “Model-Based Control for High-Tech Mechatronics Systems.” CRC Press/Taylor & Francis.
<a id="org5d52649"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.
<a id="org82a60a2"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.