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content/article/kwakernaak93_robus_contr_h_tutor_paper.md
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@ -8,7 +8,7 @@ Tags
: [H Infinity Control]({{< relref "h_infinity_control" >}}), [Weighting Functions]({{< relref "weighting_functions" >}})
Reference
: ([Kwakernaak 1993](#org7c5d045))
: ([Kwakernaak 1993](#orge60c373))
Author(s)
: Kwakernaak, H.
@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org7c5d045"></a>Kwakernaak, Huibert. 1993. “Robust Control and H$infty$-Optimization - Tutorial Paper.” _Automatica_ 29 (2):255–73. <https://doi.org/10.1016/0005-1098(93)>90122-a.
<a id="orge60c373"></a>Kwakernaak, Huibert. 1993. “Robust Control and H$infty$-Optimization - Tutorial Paper.” _Automatica_ 29 (2):255–73. <https://doi.org/10.1016/0005-1098(93)>90122-a.

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content/book/albertos04_multiv_contr_system.md
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+++
title = "Multivariable control systems: an engineering approach"
author = ["Thomas Dehaeze"]
draft = false
draft = true
+++
Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
: ([Albertos and Antonio 2004](#orgbcb0991))
: ([Albertos and Antonio 2004](#orga1617be))
Author(s)
: Albertos, P., & Antonio, S.
@ -77,10 +77,10 @@ where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#org335191d), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#orgd447864), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth.
<a id="org335191d"></a>
<a id="orgd447864"></a>
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
@ -129,4 +129,4 @@ If some of the vectors in \\(V\\) (input directions) have a significant componen
## Bibliography {#bibliography}
<a id="orgbcb0991"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
<a id="orga1617be"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.

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content/book/du10_model_contr_vibrat_mechan_system.md
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+++
title = "Modeling and control of vibration in mechanical systems"
author = ["Thomas Dehaeze"]
draft = false
draft = true
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Du and Xie 2010](#orge4a6b7f))
: ([Du and Xie 2010](#orga475b60))
Author(s)
: Du, C., & Xie, L.
@ -536,4 +536,4 @@ Year
## Bibliography {#bibliography}
<a id="orge4a6b7f"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.
<a id="orga475b60"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.

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content/book/du19_multi_actuat_system_contr.md
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+++
title = "Multi-stage actuation systems and control"
author = ["Thomas Dehaeze"]
description = "Proposes a way to combine multiple actuators (short stroke and long stroke) for control."
keywords = ["Control", "Mechatronics"]
draft = false
+++
@ -9,7 +11,7 @@ Tags
Reference
: ([Du and Pang 2019](#orge6fd258))
: ([Du and Pang 2019](#org2403f17))
Author(s)
: Du, C., & Pang, C. K.
@ -81,9 +83,9 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
#### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org5b375a0)) and the microactuator.
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org4cc1c22)) and the microactuator.
<a id="org5b375a0"></a>
<a id="org4cc1c22"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
@ -96,18 +98,18 @@ There characteristics are shown on table [1](#table--tab:microactuator).
Performance comparison of microactuators
</div>
| | Elect. | PZT | Thermal |
|-------------|-----------------------------------------------|-----------------------------------------------|----------------------------|
| TF | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
| \\(\tau\\) | \\(<\SI{0.1}{ms}\\) | \\(<\SI{0.05}{ms}\\) | \\(>\SI{0.1}{ms}\\) |
| \\(omega\\) | \\(1-\SI{2}{kHz}\\) | \\(20-\SI{25}{kHz}\\) | \\(>\SI{15}{kHz}\\) |
| | Elect. | PZT | Thermal |
|--------------|-----------------------------------------------|-----------------------------------------------|----------------------------|
| TF | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
| \\(\tau\\) | \\(<\SI{0.1}{ms}\\) | \\(<\SI{0.05}{ms}\\) | \\(>\SI{0.1}{ms}\\) |
| \\(\omega\\) | \\(1-\SI{2}{kHz}\\) | \\(20-\SI{25}{kHz}\\) | \\(>\SI{15}{kHz}\\) |
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#orgeb3a161), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
A typical closed-loop control system is shown on figure [2](#org3b2af5e), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="orgeb3a161"></a>
<a id="org3b2af5e"></a>
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
@ -117,7 +119,7 @@ A typical closed-loop control system is shown on figure [2](#orgeb3a161), where
Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
<a id="org1f9ca75"></a>
<a id="org9af6d44"></a>
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
@ -143,7 +145,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#orgd6f5782).
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org0221f39).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@ -151,7 +153,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
- \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration
<a id="orgd6f5782"></a>
<a id="org0221f39"></a>
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
@ -173,14 +175,14 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs.
Another type of control scheme is the **parallel structure** as shown in figure [5](#org3fd07ea).
Another type of control scheme is the **parallel structure** as shown in figure [5](#org9edcb9b).
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
<a id="org3fd07ea"></a>
<a id="org9edcb9b"></a>
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
@ -190,7 +192,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org7277927) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org24873cb) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@ -200,11 +202,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
<a id="org7277927"></a>
<a id="org24873cb"></a>
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [1](#org2fdb8dc) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
Equation [1](#orga734f85) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as
\begin{equation}
@ -217,16 +219,16 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
As seen in figure [7](#org038c9b4), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
As seen in figure [7](#orgb5c1410), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
<a id="org038c9b4"></a>
<a id="orgb5c1410"></a>
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
The sensitivity functions are shown in figure [8](#orged3fc33), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
The sensitivity functions are shown in figure [8](#orgd91ec4c), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
<a id="orged3fc33"></a>
<a id="orgd91ec4c"></a>
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
@ -259,13 +261,13 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#orgd7a95ee) shows the control structure for the three-stage actuation system.
Figure [9](#org4bda714) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
<a id="orgd7a95ee"></a>
<a id="org4bda714"></a>
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
@ -294,15 +296,15 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
The open-loop frequency responses of the three stages are shown on figure [10](#org50938c4).
The open-loop frequency responses of the three stages are shown on figure [10](#orgded6e76).
<a id="org50938c4"></a>
<a id="orgded6e76"></a>
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
The obtained sensitivity function is shown on figure [11](#org47b7e75).
The obtained sensitivity function is shown on figure [11](#orgde9819c).
<a id="org47b7e75"></a>
<a id="orgde9819c"></a>
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
@ -317,7 +319,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
For the three-stage control architecture as shown on figure [9](#orgd7a95ee), the position error is
For the three-stage control architecture as shown on figure [9](#org4bda714), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by
@ -333,11 +335,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system}
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#orgc297a81)).
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#orga3b472d)).
The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org77f79f5) and
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org442b5f7) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@ -346,7 +348,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
The open-loop transfer function of the overall system is
\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
<a id="orgc297a81"></a>
<a id="orga3b472d"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
@ -358,9 +360,9 @@ The control signals and the positions of the three actuators are
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*}
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org10ab429)).
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org5311716)).
<a id="org10ab429"></a>
<a id="org5311716"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
@ -672,4 +674,4 @@ As a more advanced concept, PZT elements being used as actuator and sensor simul
## Bibliography {#bibliography}
<a id="orge6fd258"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
<a id="org2403f17"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.

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+++
title = "Modal testing: theory, practice and application"
title = "Modal Testing: Theory, Practice and Application"
author = ["Thomas Dehaeze"]
description = "Reference book for Modal Testing"
keywords = ["system identification", "modal testing"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}}), [Modal Analysis]({{< relref "modal_analysis" >}})
Reference
: ([Ewins 2000](#org15876a9))
: ([Ewins 2000](#orgd25baff))
Author(s)
: Ewins, D.
@ -159,9 +161,9 @@ Indeed, we shall see later how these predictions can be quite detailed, to the p
The main measurement technique studied are those which will permit to make **direct measurements of the various FRF** properties of the test structure.
The type of test best suited to FRF measurement is shown in figure [1](#org8f0a8f0).
The type of test best suited to FRF measurement is shown in figure [1](#orge16f202).
<a id="org8f0a8f0"></a>
<a id="orge16f202"></a>
{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@ -231,11 +233,11 @@ Thus there is **no single modal analysis method**, but rater a selection, each b
One of the most widespread and useful approaches is known as the **single-degree-of-freedom curve-fit**, or often as the **circle fit** procedure.
This method uses the fact that **at frequencies close to a natural frequency**, the FRF can often be **approximated to that of a single degree-of-freedom system** plus a constant offset term (which approximately accounts for the existence of other modes).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see figure [2](#org703f940)).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see figure [2](#org5c5d54f)).
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
<a id="org703f940"></a>
<a id="org5c5d54f"></a>
{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@ -270,10 +272,10 @@ Even though the same overall procedure is always followed, there will be a **dif
Theoretical foundations of modal testing are of paramount importance to its successful implementation.
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#orga0bcee3).
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#org2de3899).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
<a id="orga0bcee3"></a>
<a id="org2de3899"></a>
{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@ -295,7 +297,7 @@ Thus our response model will consist of a set of **frequency response functions
<div class="important">
<div></div>
As indicated in figure [3](#orga0bcee3), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
As indicated in figure [3](#org2de3899), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
</div>
@ -315,10 +317,10 @@ Three classes of system model will be described:
</div>
The basic model for the SDOF system is shown in figure [4](#org863f8fd) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The basic model for the SDOF system is shown in figure [4](#org2c2a70c) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
<a id="org863f8fd"></a>
<a id="org2c2a70c"></a>
{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@ -394,9 +396,9 @@ which is a single mode of vibration with a complex natural frequency having two
- **An imaginary or oscillatory part**
- **A real or decay part**
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#org777e04b)
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#orgbae45c5)
<a id="org777e04b"></a>
<a id="orgbae45c5"></a>
{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@ -427,7 +429,7 @@ which is now complex, containing both magnitude and phase information:
All structures exhibit a degree of damping due to the **hysteresis properties** of the material(s) from which they are made.
A typical example of this effect is shown in the force displacement plot in figure [1](#orgf0a4ea9) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
A typical example of this effect is shown in the force displacement plot in figure [1](#orgf870454) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
The maximum energy stored corresponds to the elastic energy of the structure at the point of maximum deflection.
The damping effect of such a component can conveniently be defined by the ratio of these two:
\\[ \tcmbox{\text{damping capacity} = \frac{\text{energy lost per cycle}}{\text{maximum energy stored}}} \\]
@ -440,13 +442,13 @@ The damping effect of such a component can conveniently be defined by the ratio
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------|----------------------------------------|------------------------------------------|
| <a id="orgf0a4ea9"></a> Material hysteresis | <a id="org2134b3d"></a> Dry friction | <a id="org82f9a69"></a> Viscous damper |
| <a id="orgf870454"></a> Material hysteresis | <a id="orged6e3ed"></a> Dry friction | <a id="org368575f"></a> Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
It may be described very roughly by the simple **dry friction model** shown in figure [1](#org2134b3d).
It may be described very roughly by the simple **dry friction model** shown in figure [1](#orged6e3ed).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in figure [1](#org82f9a69).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in figure [1](#org368575f).
Because the relationship is linear between force and velocity, it is necessary to suppose harmonic motion, at frequency \\(\omega\\), in order to construct a force-displacement diagram.
The resulting diagram shows the nature of the approximation provided by the viscous damper model and the concept of the **effective or equivalent viscous damping coefficient** for any of the actual phenomena as being which provides the **same energy loss per cycle** as the real thing.
@ -567,7 +569,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [3](
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
| <a id="orgf6df26a"></a> Receptance FRF | <a id="org58db881"></a> Mobility FRF | <a id="org1c64176"></a> Accelerance FRF |
| <a id="org1eeee38"></a> Receptance FRF | <a id="org2c0cdc1"></a> Mobility FRF | <a id="orgd6f921d"></a> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@ -590,7 +592,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|------------------------------------------------|------------------------------------------------|
| <a id="orgb3efe5a"></a> Real part | <a id="org6c0e23c"></a> Imaginary part |
| <a id="orgeaffcf5"></a> Real part | <a id="org7e4b6c1"></a> Imaginary part |
| width=\linewidth | width=\linewidth |
@ -598,7 +600,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
It can be seen from the expression of the inverse receptance \eqref{eq:dynamic_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
Figure [5](#org0339be1) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
Figure [5](#org8eb6352) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
<a id="table--fig:inverse-frf"></a>
<div class="table-caption">
@ -608,7 +610,7 @@ Figure [5](#org0339be1) shows an example of a plot of a system with a combinatio
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|---------------------------------------------|-----------------------------------------------|
| <a id="org0339be1"></a> Mixed | <a id="org03893ea"></a> Viscous |
| <a id="org8eb6352"></a> Mixed | <a id="org69817b0"></a> Viscous |
| width=\linewidth | width=\linewidth |
@ -625,7 +627,7 @@ The missing information (in this case, the frequency) must be added by identifyi
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|------------------------------------------------------|---------------------------------------------------------|
| <a id="org6baff82"></a> Viscous damping | <a id="orgfadfd34"></a> Structural damping |
| <a id="org8896e35"></a> Viscous damping | <a id="org2b2e557"></a> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@ -1130,9 +1132,9 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
</div>
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [6](#org64f75be)).
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [6](#org081c1b9)).
<a id="org64f75be"></a>
<a id="org081c1b9"></a>
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@ -1147,7 +1149,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
| <a id="orgf1cdf2d"></a> Almost-real mode | <a id="orgf730e4e"></a> Complex Mode | <a id="orgdb80ebd"></a> Measure of complexity |
| <a id="org0bdcc92"></a> Almost-real mode | <a id="orgd1143ca"></a> Complex Mode | <a id="org11d773f"></a> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1157,7 +1159,7 @@ There exist few indicators of the modal complexity.
The first one, a simple and crude one, called **MCF1** consists of summing all the phase differences between every combination of two eigenvector elements:
\\[ \text{MCF1} = \sum\_{j=1}^N \sum\_{k=1 \neq j}^N (\theta\_{rj} - \theta\_{rk}) \\]
The second measure is shown on figure [7](#orgdb80ebd) where a polygon is drawn around the extremities of the individual vectors.
The second measure is shown on figure [7](#org11d773f) where a polygon is drawn around the extremities of the individual vectors.
The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode, and is defined as **MCF2**.
@ -1253,7 +1255,7 @@ We write \\(\alpha\_{11}\\) the point FRF and \\(\alpha\_{21}\\) the transfer FR
It can be seen that the only difference between the point and transfer receptance is in the sign of the modal constant of the second mode.
Consider the first point mobility (figure [9](#orgc4a7fb9)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
Consider the first point mobility (figure [9](#org0ce9b7d)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
On a logarithmic plot, this produces the antiresonance characteristic which reflects that of the resonance.
<a id="table--fig:mobility-frf-mdof"></a>
@ -1264,10 +1266,10 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|---------------------------------------------------|------------------------------------------------------|
| <a id="orgc4a7fb9"></a> Point FRF | <a id="org55cabb4"></a> Transfer FRF |
| <a id="org0ce9b7d"></a> Point FRF | <a id="org3719638"></a> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [9](#org55cabb4), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
For the plot in figure [9](#org3719638), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
##### FRF modulus plots for MDOF systems {#frf-modulus-plots-for-mdof-systems}
@ -1283,13 +1285,13 @@ If they have apposite signs, there will not be an antiresonance.
##### Bode plots {#bode-plots}
The resonances and antiresonances are blunted by the inclusion of damping, and the phase angles are no longer exactly \\(\SI{0}{\degree}\\) or \\(\SI{180}{\degree}\\), but the general appearance of the plot is a natural extension of that for the system without damping.
Figure [7](#org6fd9292) shows a plot for the same mobility as appears in figure [9](#orgc4a7fb9) but here for a system with added damping.
Figure [7](#org0893e81) shows a plot for the same mobility as appears in figure [9](#org0ce9b7d) but here for a system with added damping.
Most mobility plots have this general form as long as the modes are relatively well-separated.
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
<a id="org6fd9292"></a>
<a id="org0893e81"></a>
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@ -1298,9 +1300,9 @@ This condition is satisfied unless the separation between adjacent natural frequ
Each of the frequency response of a MDOF system in the Nyquist plot is composed of a number of SDOF components.
Figure [10](#org4caa691) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
Figure [10](#org03e786a) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [10](#org5060284) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [10](#orgfec7087) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
<a id="table--fig:nyquist-frf-plots"></a>
<div class="table-caption">
@ -1310,10 +1312,10 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|------------------------------------------|---------------------------------------------|
| <a id="org4caa691"></a> Point receptance | <a id="org5060284"></a> Transfer receptance |
| <a id="org03e786a"></a> Point receptance | <a id="orgfec7087"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [11](#org7d25d6c) and [11](#org9e70037), we show corresponding data for **non-proportional** damping.
In the two figures [11](#org8620a5c) and [11](#org75aa0d4), we show corresponding data for **non-proportional** damping.
In this case, a relative phase has been introduced between the first and second elements of the eigenvectors: of \\(\SI{30}{\degree}\\) in mode 1 and of \\(\SI{150}{\degree}\\) in mode 2.
Now we find that the individual modal circles are no longer "upright" but are **rotated by an amount dictated by the complexity of the modal constants**.
@ -1325,7 +1327,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|-----------------------------------------------------|--------------------------------------------------------|
| <a id="org7d25d6c"></a> Point receptance | <a id="org9e70037"></a> Transfer receptance |
| <a id="org8620a5c"></a> Point receptance | <a id="org75aa0d4"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
@ -1481,7 +1483,7 @@ Examples of random signals, autocorrelation function and power spectral density
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
| <a id="org5355634"></a> Time history | <a id="org618edae"></a> Autocorrelation Function | <a id="org363a29a"></a> Power Spectral Density |
| <a id="orgff9751a"></a> Time history | <a id="orga003731"></a> Autocorrelation Function | <a id="org0e8108d"></a> Power Spectral Density |
| width=\linewidth | width=\linewidth | width=\linewidth |
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@ -1566,8 +1568,8 @@ The existence of two equations presents an opportunity to **check the quality**
There are difficulties to implement some of the above formulae in practice because of noise and other limitations concerned with the data acquisition and processing.
One technique involves **three quantities**, rather than two, in the definition of the output/input ratio.
The system considered can best be described with reference to figure [13](#table--fig:frf-determination) which shows first in [13](#org400650f) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#org7285276) is given a more detailed and representative model of the system which is used in a modal test.
The system considered can best be described with reference to figure [13](#table--fig:frf-determination) which shows first in [13](#orgd67883e) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#orgc7a70ce) is given a more detailed and representative model of the system which is used in a modal test.
<a id="table--fig:frf-determination"></a>
<div class="table-caption">
@ -1577,7 +1579,7 @@ Then in [13](#org7285276) is given a more detailed and representative model of t
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|------------------------------------------|--------------------------------------------------|
| <a id="org400650f"></a> Basic SISO model | <a id="org7285276"></a> SISO model with feedback |
| <a id="orgd67883e"></a> Basic SISO model | <a id="orgc7a70ce"></a> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@ -1597,7 +1599,7 @@ where \\(v\\) is a third signal in the system.
##### Derivation of FRF from MIMO data {#derivation-of-frf-from-mimo-data}
A diagram for the general n-input case is shown in figure [8](#org8f4df84).
A diagram for the general n-input case is shown in figure [8](#orga1854d9).
We obtain two alternative formulas:
@ -1608,7 +1610,7 @@ We obtain two alternative formulas:
In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
<a id="org8f4df84"></a>
<a id="orga1854d9"></a>
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@ -1878,9 +1880,9 @@ The experimental setup used for mobility measurement contains three major items:
2. **A transduction system**. For the most part, piezoelectric transducer are used, although lasers and strain gauges are convenient because of their minimal interference with the test object. Conditioning amplifiers are used depending of the transducer used
3. **An analyzer**
A typical layout for the measurement system is shown on figure [9](#org7f3a496).
A typical layout for the measurement system is shown on figure [9](#org96acfaa).
<a id="org7f3a496"></a>
<a id="org96acfaa"></a>
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@ -1934,21 +1936,21 @@ However, we need a direct measurement of the force applied to the structure (we
The shakers are usually stiff in the orthogonal directions to the excitation.
This can modify the response of the system in those directions.
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [10](#orge1056cd).
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [10](#orgf80f52f).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
<a id="orge1056cd"></a>
<a id="orgf80f52f"></a>
{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
The support of shaker is also of primary importance.
The setup shown on figure [14](#org2ce9b2d) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
The setup shown on figure [14](#orgad52c50) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
Figure [14](#orgaf570a9) shows an alternative configuration in which the shaker itself is supported.
Figure [14](#orgfdb8113) shows an alternative configuration in which the shaker itself is supported.
It may be necessary to add an additional inertia mass to the shaker in order to generate sufficient excitation forces at low frequencies.
Figure [14](#orgc943938) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
Figure [14](#org2d3f7b0) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
<a id="table--fig:shaker-mount"></a>
<div class="table-caption">
@ -1958,7 +1960,7 @@ Figure [14](#orgc943938) shows an unsatisfactory setup. Indeed, the response mea
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
| <a id="org2ce9b2d"></a> Ideal Configuration | <a id="orgaf570a9"></a> Suspended Configuration | <a id="orgc943938"></a> Unsatisfactory |
| <a id="orgad52c50"></a> Ideal Configuration | <a id="orgfdb8113"></a> Suspended Configuration | <a id="org2d3f7b0"></a> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1973,10 +1975,10 @@ The magnitude of the impact is determined by the mass of the hammer head and its
The frequency range which is effectively excited is controlled by the stiffness of the contacting surface and the mass of the impactor head: there is a resonance at a frequency given by \\(\sqrt{\frac{\text{contact stiffness}}{\text{impactor mass}}}\\) above which it is difficult to deliver energy into the test structure.
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#orgb47b9bd).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#orgb47b9bd).
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#org9b1d7fe).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#org9b1d7fe).
<a id="orgb47b9bd"></a>
<a id="org9b1d7fe"></a>
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@ -2005,9 +2007,9 @@ By suitable design, such a material may be incorporated into a device which **in
#### Force Transducers {#force-transducers}
The force transducer is the simplest type of piezoelectric transducer.
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#org930ef4e)).
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#org493b0fd)).
<a id="org930ef4e"></a>
<a id="org493b0fd"></a>
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@ -2016,11 +2018,11 @@ There exists an undesirable possibility of a cross sensitivity, i.e. an electric
#### Accelerometers {#accelerometers}
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#orga075bcf)).
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#org17d53ed)).
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
<a id="orga075bcf"></a>
<a id="org17d53ed"></a>
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@ -2056,9 +2058,9 @@ However, they cannot be used at such low frequencies as the charge amplifiers an
The correct installation of transducers, especially accelerometers is important.
There are various means of fixing the transducers to the surface of the test structure, some more convenient than others.
Some of these methods are illustrated in figure [15](#orge053903).
Some of these methods are illustrated in figure [15](#org1468fa8).
Shown on figure [15](#org1b85602) are typical high frequency limits for each type of attachment.
Shown on figure [15](#org7bb2dd9) are typical high frequency limits for each type of attachment.
<a id="table--fig:transducer-mounting"></a>
<div class="table-caption">
@ -2068,7 +2070,7 @@ Shown on figure [15](#org1b85602) are typical high frequency limits for each typ
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|-----------------------------------------------------|------------------------------------------------------------|
| <a id="orge053903"></a> Attachment methods | <a id="org1b85602"></a> Frequency response characteristics |
| <a id="org1468fa8"></a> Attachment methods | <a id="org7bb2dd9"></a> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@ -2153,9 +2155,9 @@ That however requires \\(N\\) to be an integral power of \\(2\\).
Aliasing originates from the discretisation of the originally continuous time history.
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#org91dbe3e).
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#orgeaeb967).
<a id="org91dbe3e"></a>
<a id="orgeaeb967"></a>
{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
@ -2172,7 +2174,7 @@ This is illustrated on figure [16](#table--fig:effect-aliasing).
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|--------------------------------------------------|-----------------------------------------------------|
| <a id="orgb4560b8"></a> True spectrum of signal | <a id="orgd413cee"></a> Indicated spectrum from DFT |
| <a id="org3e3d162"></a> True spectrum of signal | <a id="org65765c2"></a> Indicated spectrum from DFT |
| width=\linewidth | width=\linewidth |
The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@ -2193,12 +2195,12 @@ Leakage is a problem which is a direct **consequence of the need to take only a
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|--------------------------------------|----------------------------------------|
| <a id="orgd54be6b"></a> Ideal signal | <a id="org95b6cdc"></a> Awkward signal |
| <a id="org7c07e86"></a> Ideal signal | <a id="orgdd714e0"></a> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [17](#table--fig:leakage).
In the first case (figure [17](#orgd54be6b)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case (figure [17](#org95b6cdc)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
In the first case (figure [17](#org7c07e86)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case (figure [17](#orgdd714e0)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
As a result, the spectrum produced for this case does not indicate the single frequency which the original time signal possessed.
Energy has "leaked" into a number of the spectral lines close to the true frequency and the spectrum is spread over several lines.
@ -2216,14 +2218,14 @@ Leakage is a serious problem in many applications, **ways of avoiding its effect
Windowing involves the imposition of a prescribed profile on the time signal prior to performing the Fourier transform.
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#org105c7d0).
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#orga7ce3a7).
<a id="org105c7d0"></a>
<a id="orga7ce3a7"></a>
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
The analyzed signal is then \\(x^\prime(t) = x(t) w(t)\\).
The result of using a window is seen in the third column of figure [15](#org105c7d0).
The result of using a window is seen in the third column of figure [15](#orga7ce3a7).
The **Hanning and Cosine Taper windows are typically used for continuous signals**, such as are produced by steady periodic or random vibration, while the **Exponential window is used for transient vibration** applications where much of the important information is concentrated in the initial part of the time record.
@ -2239,7 +2241,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<a id="org4b52cde"></a>
<a id="org009a00a"></a>
##### Increasing transform size {#increasing-transform-size}
@ -2263,9 +2265,9 @@ The common solution to the need for finer frequency resolution is to zoom on the
There are various ways of achieving this result.
The easiest way is to use a frequency shifting process coupled with a controlled aliasing device.
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on figure [18](#orgfeb63a7), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on figure [18](#orgfbb1177), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
If we apply a band-pass filter to the signal, as shown on figure [18](#org94b4dd9), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see figure [16](#org0cfcb53)).
If we apply a band-pass filter to the signal, as shown on figure [18](#orgf8735d6), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see figure [16](#orga5b098f)).
<a id="table--fig:frequency-zoom"></a>
<div class="table-caption">
@ -2275,10 +2277,10 @@ If we apply a band-pass filter to the signal, as shown on figure [18](#org94b4dd
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|------------------------------------------------|------------------------------------------|
| <a id="orgfeb63a7"></a> Spectrum of the signal | <a id="org94b4dd9"></a> Band-pass filter |
| <a id="orgfbb1177"></a> Spectrum of the signal | <a id="orgf8735d6"></a> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="org0cfcb53"></a>
<a id="orga5b098f"></a>
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@ -2348,9 +2350,9 @@ For instance, the typical FRF curve has large region of relatively slow changes
This is the traditional method of FRF measurement and involves the use of a sweep oscillator to provide a sinusoidal command signal with a frequency that varies slowly in the range of interest.
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#orgd1e88bf).
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#org3a0fa7e).
<a id="orgd1e88bf"></a>
<a id="org3a0fa7e"></a>
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@ -2466,9 +2468,9 @@ where \\(v(t)\\) is a third signal in the system, such as the voltage supplied t
It is known that a low coherence can arise in a measurement where the frequency resolution of the analyzer is not fine enough to describe adequately the very rapidly changing functions such as are encountered near resonance and anti-resonance on lightly-damped structures.
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [18](#org2d9ba99).
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [18](#orga2df003).
<a id="org2d9ba99"></a>
<a id="orga2df003"></a>
{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
@ -2509,9 +2511,9 @@ For the chirp and impulse excitations, each individual sample is collected and p
##### Burst excitation signals {#burst-excitation-signals}
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [19](#org63d0501)).
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [19](#orgb1cdd01)).
<a id="org63d0501"></a>
<a id="orgb1cdd01"></a>
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@ -2526,22 +2528,22 @@ In the case of burst random, however, each individual burst will be different to
##### Chirp excitation {#chirp-excitation}
The chirp consist of a short duration signal which has the form shown in figure [20](#org3d7182f).
The chirp consist of a short duration signal which has the form shown in figure [20](#org3e96514).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
<a id="org3d7182f"></a>
<a id="org3e96514"></a>
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
##### Impulsive excitation {#impulsive-excitation}
The hammer blow produces an input and response as shown in the figure [21](#orgee86d4a).
The hammer blow produces an input and response as shown in the figure [21](#org7d16186).
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
<a id="orgee86d4a"></a>
<a id="org7d16186"></a>
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@ -2549,9 +2551,9 @@ The frequency content of the hammer blow is dictated by the **materials** involv
However, it should be recorded that in the region below the first cut-off frequency induced by the elasticity of the hammer tip structure contact, the spectrum of the force signal tends to be **very flat**.
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#org2914aa8)).
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#org465da50)).
<a id="org2914aa8"></a>
<a id="org465da50"></a>
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@ -2624,9 +2626,9 @@ and so **what is required is the ratio of the two sensitivities**:
The overall sensitivity can be more readily obtained by a calibration process because we can easily make an independent measurement of the quantity now being measured: the ratio of response to force.
Suppose the response parameter is acceleration, then the FRF obtained is inertance which has the units of \\(1/\text{mass}\\), a quantity which can readily be independently measured by other means.
Figure [23](#org1f3d9fc) shows a typical calibration setup.
Figure [23](#org3793510) shows a typical calibration setup.
<a id="org1f3d9fc"></a>
<a id="org3793510"></a>
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@ -2639,9 +2641,9 @@ Thus, frequent checks on the overall calibration factors are strongly recommende
It is very important the ensure that the force is measured directly at the point at which it is applied to the structure, rather than deducing its magnitude from the current flowing in the shaker coil or other similar **indirect** processes.
This is because near resonance, the actual applied force becomes very small and is thus very prone to inaccuracy.
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#org5a54bfb).
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#org8b94b73).
<a id="org5a54bfb"></a>
<a id="org8b94b73"></a>
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@ -2696,9 +2698,9 @@ There are two problems to be tackled:
1. measurement of rotational responses
2. generation of measurement of rotation excitation
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [25](#org7c44a7c).
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [25](#org9d4e788).
<a id="org7c44a7c"></a>
<a id="org9d4e788"></a>
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@ -2714,12 +2716,12 @@ The principle of operation is that by measuring both accelerometer signals, the
This approach permits us to measure half of the possible FRFs: all those which are of the \\(X/F\\) and \\(\Theta/F\\) type.
The others can only be measured directly by applying a moment excitation.
Figure [26](#org69e6665) shows a device to simulate a moment excitation.
Figure [26](#org65dc42a) shows a device to simulate a moment excitation.
First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneous force \\(F\_0 = F\_1\\) and a moment \\(M\_0 = -F\_1 l\_1\\).
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
<a id="org69e6665"></a>
<a id="org65dc42a"></a>
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@ -3043,8 +3045,8 @@ Then, each PRF is, simply, a particular combination of the original FRFs, and th
On example of this form of pre-processing is shown on figure [19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in figure [20](#table--fig:PRF-measured) for the case of real measured test data.
The second plot [19](#org1966197) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [19](#orgf7309a1) shows the genuine modes distinct from the computational modes.
The second plot [19](#org21a5511) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [19](#org57a90c0) shows the genuine modes distinct from the computational modes.
<div class="important">
<div></div>
@ -3071,7 +3073,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
| <a id="org85391fe"></a> FRF | <a id="org1966197"></a> Singular Values | <a id="orgf7309a1"></a> PRF |
| <a id="org866353d"></a> FRF | <a id="org21a5511"></a> Singular Values | <a id="org57a90c0"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
@ -3082,7 +3084,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
| <a id="org410e27f"></a> FRF | <a id="org6b7d854"></a> Singular Values | <a id="orgc6a23d0"></a> PRF |
| <a id="org071dcb1"></a> FRF | <a id="org61b6872"></a> Singular Values | <a id="org760fea0"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -3114,7 +3116,7 @@ The **Complex mode indicator function** (CMIF) is defined as
</div>
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in figure [27](#org08ac181):
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in figure [27](#org405ffa2):
- **Natural frequencies are indicated by large values of the first CMIF** (the highest of the singular values)
- **double or multiple modes by simultaneously large values of two or more CMIF**.
@ -3124,7 +3126,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
- the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
<a id="org08ac181"></a>
<a id="org405ffa2"></a>
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@ -3197,7 +3199,7 @@ In this method, it is assumed that close to one local mode, any effects due to t
This is a method which works adequately for structures whose FRF exhibit **well separated modes**.
This method is useful in obtaining initial estimates to the parameters.
The peak-picking method is applied as follows (illustrated on figure [28](#orgd1dacfd)):
The peak-picking method is applied as follows (illustrated on figure [28](#org017fa0f)):
1. First, **individual resonance peaks** are detected on the FRF plot and the maximum responses frequency \\(\omega\_r\\) is taken as the **natural frequency** of that mode
2. Second, the **local maximum value of the FRF** \\(|\hat{H}|\\) is noted and the **frequency bandwidth** of the function for a response level of \\(|\hat{H}|/\sqrt{2}\\) is determined.
@ -3219,7 +3221,7 @@ The peak-picking method is applied as follows (illustrated on figure [28](#orgd1
It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
Only real modal constants and thus real modes can be deduced by this method.
<a id="orgd1dacfd"></a>
<a id="org017fa0f"></a>
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@ -3244,7 +3246,7 @@ In the case of a system assumed to have structural damping, the basic function w
\end{equation}
since the only effect of including the modal constant \\({}\_rA\_{jk}\\) is to scale the size of the circle by \\(|{}\_rA\_{jk}|\\) and to rotate it by \\(\angle {}\_rA\_{jk}\\).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org0c46692).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org6e0c9b9).
<a id="table--fig:modal-circle-figures"></a>
<div class="table-caption">
@ -3254,7 +3256,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org0c46692)
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|----------------------------------------|--------------------------------------------------------------------|
| <a id="org0c46692"></a> Properties | <a id="orga918af0"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <a id="org6e0c9b9"></a> Properties | <a id="orgbc7eafe"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@ -3292,7 +3294,7 @@ It may also be seen that an **estimate of the damping** is provided by the sweep
\end{equation}
Suppose now we have two specific points on the circle, one corresponding to a frequency \\(\omega\_b\\) below the natural frequency and the other one \\(\omega\_a\\) above the natural frequency.
Referring to figure [21](#orga918af0), we can write:
Referring to figure [21](#orgbc7eafe), we can write:
\begin{equation}
\begin{aligned}
@ -3358,7 +3360,7 @@ The sequence is:
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using \eqref{eq:estimate_damping_sweep_rate}.
A typical example is shown on figure [29](#org96a13a2).
A typical example is shown on figure [29](#orgb1f1c40).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate_damping}.
Then, we can choose the damping estimate to be the mean value.
@ -3368,7 +3370,7 @@ The sequence is:
5. **Determine modal constant modulus and argument**.
The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
<a id="org96a13a2"></a>
<a id="orgb1f1c40"></a>
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
@ -3482,8 +3484,8 @@ We need to introduce the concept of **residual terms**, necessary in the modal a
The first occasion on which the residual problem is encountered is generally at the end of the analysis of a single FRF curve, such as by the repeated application of an SDOF curve-fit to each of the resonances in turn until all modes visible on the plot have been identified.
At this point, it is often desired to construct a theoretical curve (called "**regenerated**"), based on the modal parameters extracted from the measured data, and to overlay this on the original measured data to assess the success of the curve-fit process.
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#org398d4d8).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on figure [22](#org8ee9d90).
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#org3cda8ae).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on figure [22](#org92dafed).
<a id="table--fig:residual-modes"></a>
<div class="table-caption">
@ -3493,7 +3495,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|--------------------------------------------|-----------------------------------------|
| <a id="org398d4d8"></a> without residual | <a id="org8ee9d90"></a> with residuals |
| <a id="org3cda8ae"></a> without residual | <a id="org92dafed"></a> with residuals |
| width=\linewidth | width=\linewidth |
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@ -3522,9 +3524,9 @@ The three terms corresponds to:
2. the **high frequency modes** not identified
3. the **modes actually identified**
These three terms are illustrated on figure [30](#org473ef14).
These three terms are illustrated on figure [30](#org8849a18).
<a id="org473ef14"></a>
<a id="org8849a18"></a>
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
@ -3818,7 +3820,7 @@ with
</div>
The composite function \\(HH(\omega)\\) can provide a useful means of determining a single (average) value for the natural frequency and damping factor for each mode where the individual functions would each indicate slightly different values.
As an example, a set of mobilities measured are shown individually in figure [23](#org1ee7063) and their summation shown as a single composite curve in figure [23](#orgfb3f6a3).
As an example, a set of mobilities measured are shown individually in figure [23](#org433946d) and their summation shown as a single composite curve in figure [23](#orgd3aaebe).
<a id="table--fig:composite"></a>
<div class="table-caption">
@ -3828,7 +3830,7 @@ As an example, a set of mobilities measured are shown individually in figure [23
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|-------------------------------------------|-----------------------------------------|
| <a id="org1ee7063"></a> Individual curves | <a id="orgfb3f6a3"></a> Composite curve |
| <a id="org433946d"></a> Individual curves | <a id="orgd3aaebe"></a> Composite curve |
| width=\linewidth | width=\linewidth |
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@ -4382,11 +4384,11 @@ There are basically two choices for the graphical display of a modal model:
##### Deflected shapes {#deflected-shapes}
A static display is often adequate for depicting relatively simple mode shapes.
Measured coordinates of the test structure are first linked as shown on figure [31](#org0dcf72a) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [31](#org0dcf72a) (b).
Measured coordinates of the test structure are first linked as shown on figure [31](#org873fbad) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [31](#org873fbad) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
<a id="org0dcf72a"></a>
<a id="org873fbad"></a>
{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
@ -4395,16 +4397,16 @@ It is customary to select the largest eigenvector element and to scale the whole
##### Multiple frames {#multiple-frames}
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on figure [31](#org0dcf72a) (c).
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on figure [31](#org873fbad) (c).
Some indication of the motion of the structure can be obtained, and the points of zero motion (nodes) can be clearly identified.
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#org0dcf72a) (d).
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#org873fbad) (d).
Complex modes do not, in general, exhibit fixed nodal points.
##### Argand diagram plots {#argand-diagram-plots}
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of figure [31](#org0dcf72a) (e) and (f).
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of figure [31](#org873fbad) (e) and (f).
Although there is no attempt to show the physical deformation of the actual structure in this format, the complexity of the mode shape is graphically displayed.
@ -4427,11 +4429,11 @@ We then tend to interpret this as a motion which is purely in the x-direction wh
The second problem arises when the **grid of measurement points** that is chosen to display the mode shapes is **too coarse in relation to the complexity of the deformation patterns** that are to be displayed.
This can be illustrated using a very simple example: suppose that our test structure is a straight beam, and that we decide to use just three response measurements points.
If we consider the first six modes of the beam, whose mode shapes are sketched in figure [32](#org843940c), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
If we consider the first six modes of the beam, whose mode shapes are sketched in figure [32](#orgd0cec90), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
<a id="org843940c"></a>
<a id="orgd0cec90"></a>
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@ -4478,11 +4480,11 @@ However, it must be noted that there is an important **limitation to this proced
<div></div>
As an example, suppose that FRF data \\(H\_{11}\\) and \\(H\_{21}\\) are measured and analyzed in order to synthesize the FRF \\(H\_{22}\\) initially unmeasured.
The predict curve is compared with the measurements on figure [24](#orga9d477f).
The predict curve is compared with the measurements on figure [24](#org257d04e).
Clearly, the agreement is poor and would tend to indicate that the measurement/analysis process had not been successful.
However, the synthesized curve contained only those terms relating to the modes which had actually been studied from \\(H\_{11}\\) and \\(H\_{21}\\) and this set of modes did not include **all** the modes of the structure.
Thus, \\(H\_{22}\\) **omitted the influence of out-of-range modes**.
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in figure [24](#orgc3d79ab).
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in figure [24](#org82dd447).
</div>
@ -4494,7 +4496,7 @@ The inclusion of these two additional terms (obtained here only after measuring
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|--------------------------------------------------------|-----------------------------------------------------------|
| <a id="orga9d477f"></a> Using measured modal data only | <a id="orgc3d79ab"></a> After inclusion of residual terms |
| <a id="org257d04e"></a> Using measured modal data only | <a id="org82dd447"></a> After inclusion of residual terms |
| width=\linewidth | width=\linewidth |
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@ -4546,10 +4548,10 @@ If the **transmissibility** is measured during a modal test which has a single e
</div>
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [33](#orgf71911f).
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [33](#orgcaddc1e).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="orgf71911f"></a>
<a id="orgcaddc1e"></a>
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@ -4570,7 +4572,7 @@ The fact that the excitation force is not measured is responsible for the lack o
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|---------------------------------------------------------|-------------------------------------------------------|
| <a id="org8c4c8e9"></a> Conventional modal test setup | <a id="orge759ea9"></a> Base excitation setup |
| <a id="orge8ea85c"></a> Conventional modal test setup | <a id="org5a6dcef"></a> Base excitation setup |
| height=4cm | height=4cm |
@ -4614,4 +4616,4 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
## Bibliography {#bibliography}
<a id="org15876a9"></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="orgd25baff"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.

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content/book/fleming14_desig_model_contr_nanop_system.md
View File

@ -1,15 +1,16 @@
+++
title = "Design, modeling and control of nanopositioning systems"
author = ["Thomas Dehaeze"]
draft = true
description = "Talks about various topics related to nano-positioning systems."
keywords = ["Control", "Metrology", "Flexible Joints"]
draft = false
+++
Tags
:
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([Fleming and Leang 2014](#org6bfb955))
: ([Fleming and Leang 2014](#org9f0983e))
Author(s)
: Fleming, A. J., & Leang, K. K.
@ -18,523 +19,484 @@ Year
: 2014
## 1 Introduction {#1-introduction}
## Introduction {#introduction}
### 1.1 Introduction to Nanotechnology {#1-dot-1-introduction-to-nanotechnology}
### Introduction to Nanotechnology {#introduction-to-nanotechnology}
### 1.2 Introduction to Nanopositioning {#1-dot-2-introduction-to-nanopositioning}
### Introduction to Nanopositioning {#introduction-to-nanopositioning}
### 1.3 Scanning Probe Microscopy {#1-dot-3-scanning-probe-microscopy}
### Scanning Probe Microscopy {#scanning-probe-microscopy}
### 1.4 Challenges with Nanopositioning Systems {#1-dot-4-challenges-with-nanopositioning-systems}
### Challenges with Nanopositioning Systems {#challenges-with-nanopositioning-systems}
#### 1.4.1 Hysteresis {#1-dot-4-dot-1-hysteresis}
#### Hysteresis {#hysteresis}
#### 1.4.2 Creep {#1-dot-4-dot-2-creep}
#### Creep {#creep}
#### 1.4.3 Thermal Drift {#1-dot-4-dot-3-thermal-drift}
#### Thermal Drift {#thermal-drift}
#### 1.4.4 Mechanical Resonance {#1-dot-4-dot-4-mechanical-resonance}
#### Mechanical Resonance {#mechanical-resonance}
### 1.5 Control of Nanopositioning Systems {#1-dot-5-control-of-nanopositioning-systems}
### Control of Nanopositioning Systems {#control-of-nanopositioning-systems}
#### 1.5.1 Feedback Control {#1-dot-5-dot-1-feedback-control}
#### Feedback Control {#feedback-control}
#### 1.5.2 Feedforward Control {#1-dot-5-dot-2-feedforward-control}
#### Feedforward Control {#feedforward-control}
### 1.6 Book Summary {#1-dot-6-book-summary}
### Book Summary {#book-summary}
#### 1.6.1 Assumed Knowledge {#1-dot-6-dot-1-assumed-knowledge}
#### Assumed Knowledge {#assumed-knowledge}
#### 1.6.2 Content Summary {#1-dot-6-dot-2-content-summary}
#### Content Summary {#content-summary}
### References {#references}
## 2 Piezoelectric Transducers {#2-piezoelectric-transducers}
## Piezoelectric Transducers {#piezoelectric-transducers}
### 2.1 The Piezoelectric Effect {#2-dot-1-the-piezoelectric-effect}
### The Piezoelectric Effect {#the-piezoelectric-effect}
### 2.2 Piezoelectric Compositions {#2-dot-2-piezoelectric-compositions}
### Piezoelectric Compositions {#piezoelectric-compositions}
### 2.3 Manufacturing Piezoelectric Ceramics {#2-dot-3-manufacturing-piezoelectric-ceramics}
### Manufacturing Piezoelectric Ceramics {#manufacturing-piezoelectric-ceramics}
### 2.4 Piezoelectric Transducers {#2-dot-4-piezoelectric-transducers}
### Piezoelectric Transducers {#piezoelectric-transducers}
### 2.5 Application Considerations {#2-dot-5-application-considerations}
### Application Considerations {#application-considerations}
#### 2.5.1 Mounting {#2-dot-5-dot-1-mounting}
### Response of Piezoelectric Actuators {#response-of-piezoelectric-actuators}
#### 2.5.2 Stroke Versus Force {#2-dot-5-dot-2-stroke-versus-force}
### Modeling Creep and Vibration in Piezoelectric Actuators {#modeling-creep-and-vibration-in-piezoelectric-actuators}
#### 2.5.3 Preload and Flexures {#2-dot-5-dot-3-preload-and-flexures}
#### 2.5.4 Electrical Considerations {#2-dot-5-dot-4-electrical-considerations}
#### 2.5.5 Self-Heating Considerations {#2-dot-5-dot-5-self-heating-considerations}
### 2.6 Response of Piezoelectric Actuators {#2-dot-6-response-of-piezoelectric-actuators}
#### 2.6.1 Hysteresis {#2-dot-6-dot-1-hysteresis}
#### 2.6.2 Creep {#2-dot-6-dot-2-creep}
#### 2.6.3 Temperature Dependence {#2-dot-6-dot-3-temperature-dependence}
#### 2.6.4 Vibrational Dynamics {#2-dot-6-dot-4-vibrational-dynamics}
#### 2.6.5 Electrical Bandwidth {#2-dot-6-dot-5-electrical-bandwidth}
### 2.7 Modeling Creep and Vibration in Piezoelectric Actuators {#2-dot-7-modeling-creep-and-vibration-in-piezoelectric-actuators}
### 2.8 Chapter Summary {#2-dot-8-chapter-summary}
### Chapter Summary {#chapter-summary}
### References {#references}
## 3 Types of Nanopositioners {#3-types-of-nanopositioners}
## Types of Nanopositioners {#types-of-nanopositioners}
### 3.1 Piezoelectric Tube Nanopositioners {#3-dot-1-piezoelectric-tube-nanopositioners}
### Piezoelectric Tube Nanopositioners {#piezoelectric-tube-nanopositioners}
#### 3.1.1 63mm Piezoelectric Tube {#3-dot-1-dot-1-63mm-piezoelectric-tube}
#### 63mm Piezoelectric Tube {#63mm-piezoelectric-tube}
#### 3.1.2 40mm Piezoelectric Tube Nanopositioner {#3-dot-1-dot-2-40mm-piezoelectric-tube-nanopositioner}
#### 40mm Piezoelectric Tube Nanopositioner {#40mm-piezoelectric-tube-nanopositioner}
### 3.2 Piezoelectric Stack Nanopositioners {#3-dot-2-piezoelectric-stack-nanopositioners}
### Piezoelectric Stack Nanopositioners {#piezoelectric-stack-nanopositioners}
#### 3.2.1 Phyisk Instrumente P-734 Nanopositioner {#3-dot-2-dot-1-phyisk-instrumente-p-734-nanopositioner}
#### Phyisk Instrumente P-734 Nanopositioner {#phyisk-instrumente-p-734-nanopositioner}
#### 3.2.2 Phyisk Instrumente P-733.3DD Nanopositioner {#3-dot-2-dot-2-phyisk-instrumente-p-733-dot-3dd-nanopositioner}
#### Phyisk Instrumente P-733.3DD Nanopositioner {#phyisk-instrumente-p-733-dot-3dd-nanopositioner}
#### 3.2.3 Vertical Nanopositioners {#3-dot-2-dot-3-vertical-nanopositioners}
#### Vertical Nanopositioners {#vertical-nanopositioners}
#### 3.2.4 Rotational Nanopositioners {#3-dot-2-dot-4-rotational-nanopositioners}
#### Rotational Nanopositioners {#rotational-nanopositioners}
#### 3.2.5 Low Temperature and UHV Nanopositioners {#3-dot-2-dot-5-low-temperature-and-uhv-nanopositioners}
#### Low Temperature and UHV Nanopositioners {#low-temperature-and-uhv-nanopositioners}
#### 3.2.6 Tilting Nanopositioners {#3-dot-2-dot-6-tilting-nanopositioners}
#### Tilting Nanopositioners {#tilting-nanopositioners}
#### 3.2.7 Optical Objective Nanopositioners {#3-dot-2-dot-7-optical-objective-nanopositioners}
#### Optical Objective Nanopositioners {#optical-objective-nanopositioners}
### References {#references}
## 4 Mechanical Design: Flexure-Based Nanopositioners {#4-mechanical-design-flexure-based-nanopositioners}
## Mechanical Design: Flexure-Based Nanopositioners {#mechanical-design-flexure-based-nanopositioners}
### 4.1 Introduction {#4-dot-1-introduction}
### Introduction {#introduction}
### 4.2 Operating Environment {#4-dot-2-operating-environment}
### Operating Environment {#operating-environment}
### 4.3 Methods for Actuation {#4-dot-3-methods-for-actuation}
### Methods for Actuation {#methods-for-actuation}
### 4.4 Flexure Hinges {#4-dot-4-flexure-hinges}
### Flexure Hinges {#flexure-hinges}
#### 4.4.1 Introduction {#4-dot-4-dot-1-introduction}
#### Introduction {#introduction}
#### 4.4.2 Types of Flexures {#4-dot-4-dot-2-types-of-flexures}
#### Types of Flexures {#types-of-flexures}
#### 4.4.3 Flexure Hinge Compliance Equations {#4-dot-4-dot-3-flexure-hinge-compliance-equations}
#### Flexure Hinge Compliance Equations {#flexure-hinge-compliance-equations}
#### 4.4.4 Stiff Out-of-Plane Flexure Designs {#4-dot-4-dot-4-stiff-out-of-plane-flexure-designs}
#### Stiff Out-of-Plane Flexure Designs {#stiff-out-of-plane-flexure-designs}
#### 4.4.5 Failure Considerations {#4-dot-4-dot-5-failure-considerations}
#### Failure Considerations {#failure-considerations}
#### 4.4.6 Finite Element Approach for Flexure Design {#4-dot-4-dot-6-finite-element-approach-for-flexure-design}
#### Finite Element Approach for Flexure Design {#finite-element-approach-for-flexure-design}
### 4.5 Material Considerations {#4-dot-5-material-considerations}
### Material Considerations {#material-considerations}
#### 4.5.1 Materials for Flexure and Platform Design {#4-dot-5-dot-1-materials-for-flexure-and-platform-design}
#### Materials for Flexure and Platform Design {#materials-for-flexure-and-platform-design}
#### 4.5.2 Thermal Stability of Materials {#4-dot-5-dot-2-thermal-stability-of-materials}
#### Thermal Stability of Materials {#thermal-stability-of-materials}
### 4.6 Manufacturing Techniques {#4-dot-6-manufacturing-techniques}
### Manufacturing Techniques {#manufacturing-techniques}
### 4.7 Design Example: A High-Speed Serial-Kinematic Nanopositioner {#4-dot-7-design-example-a-high-speed-serial-kinematic-nanopositioner}
### Design Example: A High-Speed Serial-Kinematic Nanopositioner {#design-example-a-high-speed-serial-kinematic-nanopositioner}
#### 4.7.1 State-of-the-Art Designs {#4-dot-7-dot-1-state-of-the-art-designs}
#### State-of-the-Art Designs {#state-of-the-art-designs}
#### 4.7.2 Tradeoffs and Limitations in Speed {#4-dot-7-dot-2-tradeoffs-and-limitations-in-speed}
#### Tradeoffs and Limitations in Speed {#tradeoffs-and-limitations-in-speed}
#### 4.7.3 Serial- Versus Parallel-Kinematic Configurations {#4-dot-7-dot-3-serial-versus-parallel-kinematic-configurations}
#### Serial- Versus Parallel-Kinematic Configurations {#serial-versus-parallel-kinematic-configurations}
#### 4.7.4 Piezoactuator Considerations {#4-dot-7-dot-4-piezoactuator-considerations}
#### Piezoactuator Considerations {#piezoactuator-considerations}
#### 4.7.5 Preloading Piezo-Stack Actuators {#4-dot-7-dot-5-preloading-piezo-stack-actuators}
#### Preloading Piezo-Stack Actuators {#preloading-piezo-stack-actuators}
#### 4.7.6 Flexure Design for Lateral Positioning {#4-dot-7-dot-6-flexure-design-for-lateral-positioning}
#### Flexure Design for Lateral Positioning {#flexure-design-for-lateral-positioning}
#### 4.7.7 Design of Vertical Stage {#4-dot-7-dot-7-design-of-vertical-stage}
#### Design of Vertical Stage {#design-of-vertical-stage}
#### 4.7.8 Fabrication and Assembly {#4-dot-7-dot-8-fabrication-and-assembly}
#### Fabrication and Assembly {#fabrication-and-assembly}
#### 4.7.9 Drive Electronics {#4-dot-7-dot-9-drive-electronics}
#### Drive Electronics {#drive-electronics}
\*\*\*\*0 Experimental Results
#### 4.7.10 Experimental Results {#4-dot-7-dot-10-experimental-results}
### 4.8 Chapter Summary {#4-dot-8-chapter-summary}
### Chapter Summary {#chapter-summary}
### References {#references}
## 5 Position Sensors {#5-position-sensors}
## Position Sensors {#position-sensors}
### 5.1 Introduction {#5-dot-1-introduction}
### Introduction {#introduction}
### 5.2 Sensor Characteristics {#5-dot-2-sensor-characteristics}
### Sensor Characteristics {#sensor-characteristics}
#### 5.2.1 Calibration and Nonlinearity {#5-dot-2-dot-1-calibration-and-nonlinearity}
#### Calibration and Nonlinearity {#calibration-and-nonlinearity}
#### 5.2.2 Drift and Stability {#5-dot-2-dot-2-drift-and-stability}
#### Drift and Stability {#drift-and-stability}
#### 5.2.3 Bandwidth {#5-dot-2-dot-3-bandwidth}
#### Bandwidth {#bandwidth}
#### 5.2.4 Noise {#5-dot-2-dot-4-noise}
#### Noise {#noise}
#### 5.2.5 Resolution {#5-dot-2-dot-5-resolution}
#### Resolution {#resolution}
#### 5.2.6 Combining Errors {#5-dot-2-dot-6-combining-errors}
#### Combining Errors {#combining-errors}
#### 5.2.7 Metrological Traceability {#5-dot-2-dot-7-metrological-traceability}
#### Metrological Traceability {#metrological-traceability}
### 5.3 Nanometer Position Sensors {#5-dot-3-nanometer-position-sensors}
### Nanometer Position Sensors {#nanometer-position-sensors}
#### 5.3.1 Resistive Strain Sensors {#5-dot-3-dot-1-resistive-strain-sensors}
#### Resistive Strain Sensors {#resistive-strain-sensors}
#### 5.3.2 Piezoresistive Strain Sensors {#5-dot-3-dot-2-piezoresistive-strain-sensors}
#### Piezoresistive Strain Sensors {#piezoresistive-strain-sensors}
#### 5.3.3 Piezoelectric Strain Sensors {#5-dot-3-dot-3-piezoelectric-strain-sensors}
#### Piezoelectric Strain Sensors {#piezoelectric-strain-sensors}
#### 5.3.4 Capacitive Sensors {#5-dot-3-dot-4-capacitive-sensors}
#### Capacitive Sensors {#capacitive-sensors}
#### 5.3.5 MEMs Capacitive and Thermal Sensors {#5-dot-3-dot-5-mems-capacitive-and-thermal-sensors}
#### MEMs Capacitive and Thermal Sensors {#mems-capacitive-and-thermal-sensors}
#### 5.3.6 Eddy-Current Sensors {#5-dot-3-dot-6-eddy-current-sensors}
#### Eddy-Current Sensors {#eddy-current-sensors}
#### 5.3.7 Linear Variable Displacement Transformers {#5-dot-3-dot-7-linear-variable-displacement-transformers}
#### Linear Variable Displacement Transformers {#linear-variable-displacement-transformers}
#### 5.3.8 Laser Interferometers {#5-dot-3-dot-8-laser-interferometers}
#### Laser Interferometers {#laser-interferometers}
#### 5.3.9 Linear Encoders {#5-dot-3-dot-9-linear-encoders}
#### Linear Encoders {#linear-encoders}
### 5.4 Comparison and Summary {#5-dot-4-comparison-and-summary}
### Comparison and Summary {#comparison-and-summary}
### 5.5 Outlook and Future Requirements {#5-dot-5-outlook-and-future-requirements}
### Outlook and Future Requirements {#outlook-and-future-requirements}
### References {#references}
## 6 Shunt Control {#6-shunt-control}
## Shunt Control {#shunt-control}
### 6.1 Introduction {#6-dot-1-introduction}
### Introduction {#introduction}
### 6.2 Shunt Circuit Modeling {#6-dot-2-shunt-circuit-modeling}
### Shunt Circuit Modeling {#shunt-circuit-modeling}
#### 6.2.1 Open-Loop {#6-dot-2-dot-1-open-loop}
#### Open-Loop {#open-loop}
#### 6.2.2 Shunt Damping {#6-dot-2-dot-2-shunt-damping}
#### Shunt Damping {#shunt-damping}
### 6.3 Implementation {#6-dot-3-implementation}
### Implementation {#implementation}
### 6.4 Experimental Results {#6-dot-4-experimental-results}
### Experimental Results {#experimental-results}
#### 6.4.1 Tube Dynamics {#6-dot-4-dot-1-tube-dynamics}
#### Tube Dynamics {#tube-dynamics}
#### 6.4.2 Amplifier Performance {#6-dot-4-dot-2-amplifier-performance}
#### Amplifier Performance {#amplifier-performance}
#### 6.4.3 Shunt Damping Performance {#6-dot-4-dot-3-shunt-damping-performance}
#### Shunt Damping Performance {#shunt-damping-performance}
### 6.5 Chapter Summary {#6-dot-5-chapter-summary}
### Chapter Summary {#chapter-summary}
### References {#references}
## 7 Feedback Control {#7-feedback-control}
## Feedback Control {#feedback-control}
### 7.1 Introduction {#7-dot-1-introduction}
### Introduction {#introduction}
### 7.2 Experimental Setup {#7-dot-2-experimental-setup}
### Experimental Setup {#experimental-setup}
### 7.3 PI Control {#7-dot-3-pi-control}
### PI Control {#pi-control}
### 7.4 PI Control with Notch Filters {#7-dot-4-pi-control-with-notch-filters}
### PI Control with Notch Filters {#pi-control-with-notch-filters}
### 7.5 PI Control with IRC Damping {#7-dot-5-pi-control-with-irc-damping}
### PI Control with IRC Damping {#pi-control-with-irc-damping}
### 7.6 Performance Comparison {#7-dot-6-performance-comparison}
### Performance Comparison {#performance-comparison}
### 7.7 Noise and Resolution {#7-dot-7-noise-and-resolution}
### Noise and Resolution {#noise-and-resolution}
### 7.8 Analog Implementation {#7-dot-8-analog-implementation}
### Analog Implementation {#analog-implementation}
### 7.9 Application to AFM Imaging {#7-dot-9-application-to-afm-imaging}
### Application to AFM Imaging {#application-to-afm-imaging}
\*\*\*0 Repetitive Control
### 7.10 Repetitive Control {#7-dot-10-repetitive-control}
\*\*\*\*0.1 Introduction
\*\*\*\*0.2 Repetitive Control Concept and Stability Considerations
#### 7.10.1 Introduction {#7-dot-10-dot-1-introduction}
\*\*\*\*0.3 Dual-Stage Repetitive Control
\*\*\*\*0.4 Handling Hysteresis
#### 7.10.2 Repetitive Control Concept and Stability Considerations {#7-dot-10-dot-2-repetitive-control-concept-and-stability-considerations}
\*\*\*\*0.5 Design and Implementation
\*\*\*\*0.6 Experimental Results and Discussion
#### 7.10.3 Dual-Stage Repetitive Control {#7-dot-10-dot-3-dual-stage-repetitive-control}
#### 7.10.4 Handling Hysteresis {#7-dot-10-dot-4-handling-hysteresis}
#### 7.10.5 Design and Implementation {#7-dot-10-dot-5-design-and-implementation}
#### 7.10.6 Experimental Results and Discussion {#7-dot-10-dot-6-experimental-results-and-discussion}
### 7.11 Summary {#7-dot-11-summary}
\*\*\*1 Summary
### References {#references}
## 8 Force Feedback Control {#8-force-feedback-control}
## Force Feedback Control {#force-feedback-control}
### 8.1 Introduction {#8-dot-1-introduction}
### Introduction {#introduction}
### 8.2 Modeling {#8-dot-2-modeling}
### Modeling {#modeling}
#### 8.2.1 Actuator Dynamics {#8-dot-2-dot-1-actuator-dynamics}
#### Actuator Dynamics {#actuator-dynamics}
#### 8.2.2 Sensor Dynamics {#8-dot-2-dot-2-sensor-dynamics}
#### Sensor Dynamics {#sensor-dynamics}
#### 8.2.3 Sensor Noise {#8-dot-2-dot-3-sensor-noise}
#### Sensor Noise {#sensor-noise}
#### 8.2.4 Mechanical Dynamics {#8-dot-2-dot-4-mechanical-dynamics}
#### Mechanical Dynamics {#mechanical-dynamics}
#### 8.2.5 System Properties {#8-dot-2-dot-5-system-properties}
#### System Properties {#system-properties}
#### 8.2.6 Example System {#8-dot-2-dot-6-example-system}
#### Example System {#example-system}
### 8.3 Damping Control {#8-dot-3-damping-control}
### Damping Control {#damping-control}
### 8.4 Tracking Control {#8-dot-4-tracking-control}
### Tracking Control {#tracking-control}
#### 8.4.1 Relationship Between Force and Displacement {#8-dot-4-dot-1-relationship-between-force-and-displacement}
#### Relationship Between Force and Displacement {#relationship-between-force-and-displacement}
#### 8.4.2 Integral Displacement Feedback {#8-dot-4-dot-2-integral-displacement-feedback}
#### Integral Displacement Feedback {#integral-displacement-feedback}
#### 8.4.3 Direct Tracking Control {#8-dot-4-dot-3-direct-tracking-control}
#### Direct Tracking Control {#direct-tracking-control}
#### 8.4.4 Dual Sensor Feedback {#8-dot-4-dot-4-dual-sensor-feedback}
#### Dual Sensor Feedback {#dual-sensor-feedback}
#### 8.4.5 Low Frequency Bypass {#8-dot-4-dot-5-low-frequency-bypass}
#### Low Frequency Bypass {#low-frequency-bypass}
#### 8.4.6 Feedforward Inputs {#8-dot-4-dot-6-feedforward-inputs}
#### Feedforward Inputs {#feedforward-inputs}
#### 8.4.7 Higher-Order Modes {#8-dot-4-dot-7-higher-order-modes}
#### Higher-Order Modes {#higher-order-modes}
### 8.5 Experimental Results {#8-dot-5-experimental-results}
### Experimental Results {#experimental-results}
#### 8.5.1 Experimental Nanopositioner {#8-dot-5-dot-1-experimental-nanopositioner}
#### Experimental Nanopositioner {#experimental-nanopositioner}
#### 8.5.2 Actuators and Force Sensors {#8-dot-5-dot-2-actuators-and-force-sensors}
#### Actuators and Force Sensors {#actuators-and-force-sensors}
#### 8.5.3 Control Design {#8-dot-5-dot-3-control-design}
#### Control Design {#control-design}
#### 8.5.4 Noise Performance {#8-dot-5-dot-4-noise-performance}
#### Noise Performance {#noise-performance}
### 8.6 Chapter Summary {#8-dot-6-chapter-summary}
### Chapter Summary {#chapter-summary}
### References {#references}
## 9 Feedforward Control {#9-feedforward-control}
## Feedforward Control {#feedforward-control}
### 9.1 Why Feedforward? {#9-dot-1-why-feedforward}
### Why Feedforward? {#why-feedforward}
### 9.2 Modeling for Feedforward Control {#9-dot-2-modeling-for-feedforward-control}
### Modeling for Feedforward Control {#modeling-for-feedforward-control}
### 9.3 Feedforward Control of Dynamics and Hysteresis {#9-dot-3-feedforward-control-of-dynamics-and-hysteresis}
### Feedforward Control of Dynamics and Hysteresis {#feedforward-control-of-dynamics-and-hysteresis}
#### 9.3.1 Simple DC-Gain Feedforward Control {#9-dot-3-dot-1-simple-dc-gain-feedforward-control}
#### Simple DC-Gain Feedforward Control {#simple-dc-gain-feedforward-control}
#### 9.3.2 An Inversion-Based Feedforward Approach for Linear Dynamics {#9-dot-3-dot-2-an-inversion-based-feedforward-approach-for-linear-dynamics}
#### An Inversion-Based Feedforward Approach for Linear Dynamics {#an-inversion-based-feedforward-approach-for-linear-dynamics}
#### 9.3.3 Frequency-Weighted Inversion: The Optimal Inverse {#9-dot-3-dot-3-frequency-weighted-inversion-the-optimal-inverse}
#### Frequency-Weighted Inversion: The Optimal Inverse {#frequency-weighted-inversion-the-optimal-inverse}
#### 9.3.4 Application to AFM Imaging {#9-dot-3-dot-4-application-to-afm-imaging}
#### Application to AFM Imaging {#application-to-afm-imaging}
### 9.4 Feedforward and Feedback Control {#9-dot-4-feedforward-and-feedback-control}
### Feedforward and Feedback Control {#feedforward-and-feedback-control}
#### 9.4.1 Application to AFM Imaging {#9-dot-4-dot-1-application-to-afm-imaging}
#### Application to AFM Imaging {#application-to-afm-imaging}
### 9.5 Iterative Feedforward Control {#9-dot-5-iterative-feedforward-control}
### Iterative Feedforward Control {#iterative-feedforward-control}
#### 9.5.1 The ILC Problem {#9-dot-5-dot-1-the-ilc-problem}
#### The ILC Problem {#the-ilc-problem}
#### 9.5.2 Model-Based ILC {#9-dot-5-dot-2-model-based-ilc}
#### Model-Based ILC {#model-based-ilc}
#### 9.5.3 Nonlinear ILC {#9-dot-5-dot-3-nonlinear-ilc}
#### Nonlinear ILC {#nonlinear-ilc}
#### 9.5.4 Conclusions {#9-dot-5-dot-4-conclusions}
#### Conclusions {#conclusions}
### References {#references}
## 10 Command Shaping {#10-command-shaping}
## Command Shaping {#command-shaping}
### 10.1 Introduction {#10-dot-1-introduction}
@ -600,7 +562,7 @@ Year
### References {#references}
## 11 Hysteresis Modeling and Control {#11-hysteresis-modeling-and-control}
## Hysteresis Modeling and Control {#hysteresis-modeling-and-control}
### 11.1 Introduction {#11-dot-1-introduction}
@ -639,7 +601,7 @@ Year
### References {#references}
## 12 Charge Drives {#12-charge-drives}
## Charge Drives {#charge-drives}
### 12.1 Introduction {#12-dot-1-introduction}
@ -681,7 +643,7 @@ Year
### References {#references}
## 13 Noise in Nanopositioning Systems {#13-noise-in-nanopositioning-systems}
## Noise in Nanopositioning Systems {#noise-in-nanopositioning-systems}
### 13.1 Introduction {#13-dot-1-introduction}
@ -821,11 +783,11 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="org1aabb30"></a>
<a id="orgc11b95b"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#org1aabb30) where a voltage source is connected to a piezoelectric actuator.
Consider the electrical circuit shown in Figure [1](#orgc11b95b) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
@ -949,4 +911,4 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
## Bibliography {#bibliography}
<a id="org6bfb955"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
<a id="org9f0983e"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.

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@ -1,6 +1,8 @@
+++
title = "Vibration Simulation using Matlab and ANSYS"
author = ["Thomas Dehaeze"]
description = "Nice techniques to analyze resonant systems with Ansys and Matlab."
keywords = ["Modal Analysis", "FEM"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
Reference
: ([Hatch 2000](#org8ef052d))
: ([Hatch 2000](#org4036e02))
Author(s)
: Hatch, M. R.
@ -21,14 +23,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<a id="orgf6309da"></a>
<a id="org96f8e54"></a>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#orge443794) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in Figure [1](#org97c03ca) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
<div></div>
@ -46,7 +48,7 @@ The steps are:
</div>
<a id="orge443794"></a>
<a id="org97c03ca"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
@ -58,7 +60,7 @@ Because finite element models usually have a very large number of states, an imp
<div class="important">
<div></div>
Figure [2](#org5c37471) shows such process, the steps are:
Figure [2](#orgdbb9ffa) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -71,14 +73,14 @@ Figure [2](#org5c37471) shows such process, the steps are:
</div>
<a id="org5c37471"></a>
<a id="orgdbb9ffa"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations}
Tables [3](#org9e923ac), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption">
@ -127,22 +129,22 @@ Tables [3](#org9e923ac), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="orgcf960ed"></a>
<a id="orgca1a04d"></a>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#org9e923ac) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
Figure [3](#org4819d7f) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="org9e923ac"></a>
<a id="org4819d7f"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
<div class="important">
<div></div>
([Miu 1993](#org6d53cc3)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
([Miu 1993](#orgcda3e53)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
The resonances of the "overhanging appendages" of the constrained system create the zeros.
@ -151,12 +153,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<a id="orgb8f9c89"></a>
<a id="orgc4e6e06"></a>
## Modal Analysis {#modal-analysis}
<a id="orgdd2d2c8"></a>
<a id="orge1af07f"></a>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@ -196,9 +198,9 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#org829b3b4) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
Let's consider the model shown in Figure [4](#orgde2ed42) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="org829b3b4"></a>
<a id="orgde2ed42"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
@ -297,17 +299,17 @@ One then find:
\end{bmatrix}
\end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#orgabe9314), [6](#org4283877) and [7](#orge77cc5c).
Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc0f09b0), [6](#org88e7153) and [7](#org8225e3c).
<a id="orgabe9314"></a>
<a id="orgc0f09b0"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
<a id="org4283877"></a>
<a id="org88e7153"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="orge77cc5c"></a>
<a id="org8225e3c"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
@ -346,9 +348,9 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#org948bae0).
This procedure is schematically shown in Figure [8](#org0f0be39).
<a id="org948bae0"></a>
<a id="org0f0be39"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@ -696,7 +698,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="org4c868eb"></a>
<a id="org027da35"></a>
The procedure to obtain the frequency response from a modal form is as follow:
@ -704,9 +706,9 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#org4e1f260).
This will be applied to the model shown in Figure [9](#orgafc54fa).
<a id="org4e1f260"></a>
<a id="orgafc54fa"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@ -888,9 +890,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
</div>
Figure [10](#org87a6063) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
Figure [10](#orgf64b6e5) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="org87a6063"></a>
<a id="orgf64b6e5"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -899,16 +901,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="org56b254f"></a>
<a id="org39bd7f2"></a>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org684a769).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orgc285575).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="org684a769"></a>
<a id="orgc285575"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
@ -987,7 +989,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="org1a30462"></a>
<a id="org658f39a"></a>
### Model Description {#model-description}
@ -1001,25 +1003,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="org638024a"></a>
<a id="orgcd094f5"></a>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org084a5d0)).
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org97a4ded)).
### Actuator Description {#actuator-description}
<a id="org084a5d0"></a>
<a id="org97a4ded"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#org0cedd1b)).
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orga92b66d)).
<a id="org0cedd1b"></a>
<a id="orga92b66d"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#org0cedd1b) shows the nodes used for the reduced Matlab model.
Figure [13](#orga92b66d) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces.
@ -1087,7 +1089,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
<a id="orgcc9b585"></a>
<a id="org58a3a47"></a>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@ -1202,14 +1204,14 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="org85fa9f4"></a>
<a id="orgf33e1dd"></a>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgc1d7ce0)).
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org59e7525)).
### Actuator Description {#actuator-description}
<a id="orgc1d7ce0"></a>
<a id="org59e7525"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@ -1231,9 +1233,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
In Figure [15](#orgf58efee) are shown the principal nodes used for the model.
In Figure [15](#org5f31090) are shown the principal nodes used for the model.
<a id="orgf58efee"></a>
<a id="org5f31090"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
@ -1352,11 +1354,11 @@ And we note:
G = zn * Gp;
```
<a id="orgeaf5fed"></a>
<a id="orgbe6df95"></a>
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="orgca5d420"></a>
<a id="orgcec939e"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -1454,13 +1456,13 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#org34eb51a)).
Let's plot the frequency of the modes (Figure [18](#org1183b44)).
<a id="org34eb51a"></a>
<a id="org1183b44"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
<a id="orgfccb84f"></a>
<a id="org350c1cb"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@ -1529,7 +1531,7 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
<a id="org4280511"></a>
<a id="orgd64190f"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@ -1873,7 +1875,7 @@ Then, we compute the controllability and observability gramians.
And we plot the diagonal terms
<a id="org4a478d8"></a>
<a id="orgbdc6b3b"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@ -1891,7 +1893,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
<a id="org016532c"></a>
<a id="org2787898"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@ -2137,6 +2139,6 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
## Bibliography {#bibliography}
<a id="org8ef052d"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org4036e02"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org6d53cc3"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
<a id="orgcda3e53"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.

6
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@ -1,5 +1,5 @@
+++
title = "The art of electronics - third edition"
title = "The Art of Electronics - Third Edition"
author = ["Thomas Dehaeze"]
description = "One of the best book in electronics. Cover most topics (both analog and digital)."
keywords = ["electronics"]
@ -10,7 +10,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
Reference
: ([Horowitz 2015](#org7d6347d))
: ([Horowitz 2015](#org8eab88c))
Author(s)
: Horowitz, P.
@ -22,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="org7d6347d"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
<a id="org8eab88c"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.

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@ -1,14 +1,15 @@
+++
title = "Fundamental principles of engineering nanometrology"
author = ["Thomas Dehaeze"]
draft = false
keywords = ["Metrology"]
draft = true
+++
Tags
: [Metrology]({{< relref "metrology" >}})
Reference
: ([Leach 2014](#orgdf2e918))
: ([Leach 2014](#org284df16))
Author(s)
: Leach, R.
@ -90,4 +91,4 @@ This type of angular interferometer is used to measure small angles (less than \
## Bibliography {#bibliography}
<a id="orgdf2e918"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
<a id="org284df16"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.

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@ -1,14 +1,15 @@
+++
title = "Basics of precision engineering - 1st edition"
author = ["Thomas Dehaeze"]
draft = false
keywords = ["Metrology", "Mechatronics"]
draft = true
+++
Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
: ([Leach and Smith 2018](#org4f15d94))
: ([Leach and Smith 2018](#org02e139c))
Author(s)
: Leach, R., & Smith, S. T.
@ -20,4 +21,4 @@ Year
## Bibliography {#bibliography}
<a id="org4f15d94"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
<a id="org02e139c"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.

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@ -1,6 +1,8 @@
+++
title = "Grounding and shielding: circuits and interference"
title = "Grounding and Shielding: Circuits and Interference"
author = ["Thomas Dehaeze"]
description = "Explains in a clear manner what is grounding and shielding and what are the fundamental physics behind these terms."
keywords = ["Electronics"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [Electronics]({{< relref "electronics" >}})
Reference
: ([Morrison 2016](#orgc3a94fb))
: ([Morrison 2016](#org7a49345))
Author(s)
: Morrison, R.
@ -51,7 +53,7 @@ This displacement current flows when charges are added or removed from the plate
### Field representation {#field-representation}
<a id="orgbb971cb"></a>
<a id="orga3615d0"></a>
{{< figure src="/ox-hugo/morrison16_E_field_charge.svg" caption="Figure 1: The force field lines around a positively chaged conducting sphere" >}}
@ -64,18 +66,18 @@ This displacement current flows when charges are added or removed from the plate
### The force field or \\(E\\) field between two conducting plates {#the-force-field-or--e--field-between-two-conducting-plates}
<a id="org0a58e51"></a>
<a id="org82b88ec"></a>
{{< figure src="/ox-hugo/morrison16_force_field_plates.svg" caption="Figure 2: The force field between two conducting plates with equal and opposite charges and spacing distance \\(h\\)" >}}
### Electric field patterns {#electric-field-patterns}
<a id="org2812c15"></a>
<a id="org16f20a9"></a>
{{< figure src="/ox-hugo/morrison16_electric_field_ground_plane.svg" caption="Figure 3: The electric field pattern of one circuit trace and two circuit traces over a ground plane" >}}
<a id="orge3117ef"></a>
<a id="org38210cb"></a>
{{< figure src="/ox-hugo/morrison16_electric_field_shielded_conductor.svg" caption="Figure 4: Field configuration around a shielded conductor" >}}
@ -88,7 +90,7 @@ This displacement current flows when charges are added or removed from the plate
### The \\(D\\) field {#the--d--field}
<a id="orgd76a948"></a>
<a id="org5a4329e"></a>
{{< figure src="/ox-hugo/morrison16_E_D_fields.svg" caption="Figure 5: The electric field pattern in the presence of a dielectric" >}}
@ -148,9 +150,9 @@ In a few elements, the atomic structure is such that atoms align to generate a n
The flow of electrons is another way to generate a magnetic field.
The letter \\(H\\) is reserved for the magnetic field generated by a current.
Figure [6](#org198efb1) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
Figure [6](#org9b0e888) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
<a id="org198efb1"></a>
<a id="org9b0e888"></a>
{{< figure src="/ox-hugo/morrison16_H_field.svg" caption="Figure 6: The \\(H\\) field around a current-carrying conductor" >}}
@ -167,7 +169,7 @@ Ampere's law states that the integral of the \\(H\\) field intensity in a closed
\boxed{\oint H dl = I}
\end{equation}
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#org198efb1), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#org9b0e888), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
Solving for \\(H\\), we obtain
\begin{equation}
@ -179,7 +181,7 @@ And we see that \\(H\\) has units of amperes per meter.
### The solenoid {#the-solenoid}
The magnetic field of a solenoid is shown in Figure [7](#org7535570).
The magnetic field of a solenoid is shown in Figure [7](#orgd3a9cf9).
The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly.
Using Ampere's law \eqref{eq:ampere_law}:
@ -188,7 +190,7 @@ Using Ampere's law \eqref{eq:ampere_law}:
\oint H dl \approx n I l
\end{equation}
<a id="org7535570"></a>
<a id="orgd3a9cf9"></a>
{{< figure src="/ox-hugo/morrison16_solenoid.svg" caption="Figure 7: The \\(H\\) field around a solenoid" >}}
@ -196,10 +198,10 @@ Using Ampere's law \eqref{eq:ampere_law}:
### Faraday's law and the induction field {#faraday-s-law-and-the-induction-field}
When a conducting coil is moved through a magnetic field, a voltage appears at the open ends of the coil.
This is illustrated in Figure [8](#orgd2dee77).
This is illustrated in Figure [8](#org4b2f5c1).
The voltage depends on the number of turns in the coil and the rate at which the flux is changing.
<a id="orgd2dee77"></a>
<a id="org4b2f5c1"></a>
{{< figure src="/ox-hugo/morrison16_voltage_moving_coil.svg" caption="Figure 8: A voltage induced into a moving coil" >}}
@ -237,7 +239,7 @@ The unit of inductance if the henry.
</div>
For the coil in Figure [7](#org7535570):
For the coil in Figure [7](#orgd3a9cf9):
\begin{equation} \label{eq:inductance\_coil}
V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
@ -483,39 +485,39 @@ For example, signals that overload an input stage can produce noise that may loo
### The basic shield enclosure {#the-basic-shield-enclosure}
Consider the simple amplifier circuit shown in Figure [9](#orgd60f7ec) with:
Consider the simple amplifier circuit shown in Figure [9](#org3286d62) with:
- \\(V\_1\\) the input lead
- \\(V\_2\\) the output lead
- \\(V\_3\\) the conducting enclosure which is floating and taken as the reference conductor
- \\(V\_4\\) a signal common or reference conductor
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#orgd60f7ec) (b).
The equivalent circuit is shown in Figure [9](#orgd60f7ec) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#org3286d62) (b).
The equivalent circuit is shown in Figure [9](#org3286d62) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
<a id="orgd60f7ec"></a>
<a id="org3286d62"></a>
{{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="Figure 9: Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#org412bfcb)).
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#org9f3c9db)).
When this connection is made, the feedback is removed and the enclosure no longer couples signals into the feedback structure.
The conductive enclosure is called a **shield**.
Connecting the signal common to the conductive enclosure is called "**grounding the shield**".
This "grounding" usually removed "hum" from the circuit.
<a id="org412bfcb"></a>
<a id="org9f3c9db"></a>
{{< figure src="/ox-hugo/morrison16_grounding_shield_amp.svg" caption="Figure 10: Grounding the shield to limit feedback" >}}
Most practical circuits provide connections to external points.
To see the effect of making a _single_ external connection, open the conductive enclosure and connect the input circuit common to an external ground.
Figure [11](#org5d67d92) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
Figure [11](#orgc4242ae) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
A problem can be caused by an incorrect location of the connection between the cable shield and the enclosure.
In Figure [11](#org5d67d92) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
In Figure [11](#orgc4242ae) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
This conductor being the common ground that might have a resistance \\(R\\) or \\(1\,\Omega\\), this current induced voltage that it added to the transmitted signal.
Our goal in this chapter is to find ways of keeping interference currents from flowing in any input signal conductor.
To remove this coupling, the shield connection to circuit common must be made at the point, where the circuit common connects to the external ground.
This connection is shown in Figure [11](#org5d67d92) (b).
This connection is shown in Figure [11](#orgc4242ae) (b).
This connection keeps the circulation of interference current on the outside of the shield.
There is only one point of zero signal potential external to the enclosure and that is where the signal common connects to an external hardware ground.
@ -540,7 +542,7 @@ It is an issue of using the _right_ ground.
</div>
<a id="org5d67d92"></a>
<a id="orgc4242ae"></a>
{{< figure src="/ox-hugo/morrison16_enclosure_shield_1_2_leads.png" caption="Figure 11: (a) The problem of bringing one lead out of a shielded region. Unwanted current circulates in the signal lead 2. (b) The \\(E\\) field circulate current in the shield, not in the signal conductor." >}}
@ -552,7 +554,7 @@ The power transformer couples fields from the external environment into the encl
The obvious coupling results from capacitance between the primary coil and the secondary coil.
Note that the secondary coil is connected to the circuit common conductor.
<a id="orgb45b4f3"></a>
<a id="org5995e31"></a>
{{< figure src="/ox-hugo/morrison16_power_transformer_enclosure.png" caption="Figure 12: A power transformer added to the circuit enclosure" >}}
@ -564,7 +566,7 @@ Note that the secondary coil is connected to the circuit common conductor.
The basic analog problem is to condition a signal associated with one ground reference potential and transport this signal to a second ground reference potential without adding interference.
<a id="org75ed03f"></a>
<a id="org3228c82"></a>
{{< figure src="/ox-hugo/morrison16_two_ground_problem.svg" caption="Figure 13: The two-circuit enclosures used to transport signals between grounds" >}}
@ -580,7 +582,7 @@ The basic analog problem is to condition a signal associated with one ground ref
### The basic low-gain differential amplifier (forward referencing amplifier) {#the-basic-low-gain-differential-amplifier--forward-referencing-amplifier}
<a id="orge28ae4f"></a>
<a id="org4f33add"></a>
{{< figure src="/ox-hugo/morrison16_low_gain_diff_amp.svg" caption="Figure 14: The low-gain differential amplifier applied to the two-ground problem" >}}
@ -623,11 +625,11 @@ Here are a few rule that will help in analog board layout:
### Feedback theory {#feedback-theory}
<a id="orgbf57c39"></a>
<a id="org4a09d89"></a>
{{< figure src="/ox-hugo/morrison16_basic_feedback_circuit.svg" caption="Figure 15: The basic feedback circuit" >}}
<a id="org795e24d"></a>
<a id="orgf414d06"></a>
{{< figure src="/ox-hugo/morrison16_LR_stabilizing_network.svg" caption="Figure 16: An LR-stabilizing network" >}}
@ -665,7 +667,7 @@ If the resistors are replaced by capacitors, the gain is the ratio of reactances
This feedback circuit is called a **charge converter**.
The charge on the input capacitor is transferred to the feedback capacitor.
If the feedback capacitor is smaller than the transducer capacitance by a factor of 100, then the voltage across the feedback capacitor will be 100 times greater than the open-circuit transducer voltage.
This feedback arrangement is shown in Figure [17](#org964dc8b).
This feedback arrangement is shown in Figure [17](#org74f6090).
The open-circuit input signal voltage is \\(Q/C\_T\\).
The output voltage is \\(Q/C\_{FB}\\).
The voltage gain is therefore \\(C\_T/C\_{FB}\\).
@ -679,11 +681,11 @@ It converts a charge signal to a voltage.
</div>
<a id="org964dc8b"></a>
<a id="org74f6090"></a>
{{< figure src="/ox-hugo/morrison16_charge_amplifier.svg" caption="Figure 17: A basic charge amplifier" >}}
<a id="orgdd200ce"></a>
<a id="orgb9f996c"></a>
{{< figure src="/ox-hugo/morrison16_charge_amplifier_feedback_resistor.svg" caption="Figure 18: The resistor feedback arrangement to control the low-frequency response" >}}
@ -1031,6 +1033,7 @@ To transport RF power without reflections, the source impedance and the terminat
### Shielded and screen rooms {#shielded-and-screen-rooms}
## Bibliography {#bibliography}
<a id="orgc3a94fb"></a>Morrison, Ralph. 2016. _Grounding and Shielding: Circuits and Interference_. John Wiley & Sons.
<a id="org7a49345"></a>Morrison, Ralph. 2016. _Grounding and Shielding: Circuits and Interference_. John Wiley & Sons.

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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@ -1,6 +1,8 @@
+++
title = "Vibration Control of Active Structures - Fourth Edition"
author = ["Thomas Dehaeze"]
description = "Gives a broad overview of vibration control."
keywords = ["Control", "Vibration"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
Reference
: ([Preumont 2018](#org6703487))
: ([Preumont 2018](#orgf75c814))
Author(s)
: Preumont, A.
@ -61,11 +63,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
<a id="orge1596ba"></a>
<a id="org30e8b62"></a>
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#orge1596ba). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The principle of feedback is represented on figure [1](#org30e8b62). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@ -87,12 +89,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
<a id="org8128933"></a>
<a id="org0cb2cac"></a>
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org8128933).
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org0cb2cac).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@ -123,11 +125,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
### The Various Steps of the Design {#the-various-steps-of-the-design}
<a id="orgf360ea4"></a>
<a id="org5fed023"></a>
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#orgf360ea4).
The various steps of the design of a controlled structure are shown in figure [3](#org5fed023).
The **starting point** is:
@ -154,14 +156,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#orgdf35e26)):
From the block diagram of the control system (figure [4](#orgc558cd1)):
\begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
\end{align\*}
<a id="orgdf35e26"></a>
<a id="orgc558cd1"></a>
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
@ -186,12 +188,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3807050).
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgd0ed9cf).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
<a id="org3807050"></a>
<a id="orgd0ed9cf"></a>
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
@ -398,11 +400,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
<a id="org8a88959"></a>
<a id="orgeec9f86"></a>
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#org8a88959)).
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orgeec9f86)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@ -441,9 +443,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgbd3bc07)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org2389144)).
<a id="orgbd3bc07"></a>
<a id="org2389144"></a>
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
@ -457,9 +459,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
</div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org6c053d5).
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org9a738f7).
<a id="org6c053d5"></a>
<a id="org9a738f7"></a>
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
@ -474,9 +476,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div>
By looking at figure [7](#orgbd3bc07), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
By looking at figure [7](#org2389144), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="org63fc16c"></a>
<a id="org52c26c5"></a>
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
@ -486,7 +488,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation}
The corresponding Bode plot is represented in figure [9](#org63fc16c). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
The corresponding Bode plot is represented in figure [9](#org52c26c5). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@ -508,12 +510,12 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#org459d27b)):
The system consists of (see figure [10](#orga1a9b67)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
<a id="org459d27b"></a>
<a id="orga1a9b67"></a>
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
@ -551,9 +553,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orgc64500b)).
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orgc439137)).
<a id="orgc64500b"></a>
<a id="orgc439137"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
@ -583,9 +585,9 @@ with:
</div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1f0c996)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org3b93a8e)).
<a id="org1f0c996"></a>
<a id="org3b93a8e"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
@ -610,7 +612,7 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
<a id="org26133de"></a>
<a id="org7ded49f"></a>
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
@ -619,9 +621,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti
### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgdcf2def).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org82c090c).
<a id="orgdcf2def"></a>
<a id="org82c090c"></a>
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
@ -646,7 +648,7 @@ With:
Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#orgd6dcc43) can be used.
To do so, the bridge circuit as shown on figure [15](#org8e1c5fb) can be used.
We can show that
@ -656,7 +658,7 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
<a id="orgd6dcc43"></a>
<a id="org8e1c5fb"></a>
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
@ -664,9 +666,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org9608d58) lists various effects that are observed in materials in response to various inputs.
Figure [16](#org29efe87) lists various effects that are observed in materials in response to various inputs.
<a id="org9608d58"></a>
<a id="org29efe87"></a>
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
@ -761,7 +763,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgf820772)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org226015b)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
\begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
@ -782,7 +784,7 @@ where
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="orgf820772"></a>
<a id="org226015b"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
@ -802,7 +804,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org8b2066c).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org4316115).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@ -810,7 +812,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
\end{equation}
<a id="org8b2066c"></a>
<a id="org4316115"></a>
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
@ -844,10 +846,10 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#orgc7393d7), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of figure [19](#orgcdbb831), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="orgc7393d7"></a>
<a id="orgcdbb831"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
@ -866,9 +868,9 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2
\end{equation}
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#orgff1c070)).
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org15dd7b6)).
<a id="orgff1c070"></a>
<a id="org15dd7b6"></a>
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
@ -1566,7 +1568,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org288cbbb).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org0c9fed0).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:
@ -1574,7 +1576,7 @@ This approach has the following advantages:
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="org288cbbb"></a>
<a id="org0c9fed0"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
@ -1819,4 +1821,4 @@ This approach has the following advantages:
## Bibliography {#bibliography}
<a id="org6703487"></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="orgf75c814"></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>.

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@ -1,6 +1,8 @@
+++
title = "The design of high performance mechatronics - third revised edition"
author = ["Thomas Dehaeze"]
description = "Awesome book that gives great overview of high performance mechatronic systems"
keywords = ["Metrology", "Mechatronics", "Control"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Reference
: ([Schmidt, Schitter, and Rankers 2020](#orge3bc57b))
: ([Schmidt, Schitter, and Rankers 2020](#org4e5c703))
Author(s)
: Schmidt, R. M., Schitter, G., & Rankers, A.
@ -64,7 +66,7 @@ Year
#### Electric Field {#electric-field}
<a id="org7a198b4"></a>
<a id="org16b370d"></a>
{{< figure src="/ox-hugo/schmidt20_electrical_field.svg" caption="Figure 1: Charges have an electric field" >}}
@ -219,9 +221,9 @@ The relation between the above defined terms is:
##### Mechanical Waves {#mechanical-waves}
The propagation speed value of a mechanical wave is mostly determined by the density and elasticity of the medium.
The wave propagation through an elastic material can be qualitatively explained with the help of a simplified lumped element model, consisting of a chain of springs and bodies as shown in Figure [2](#org06ad50b).
The wave propagation through an elastic material can be qualitatively explained with the help of a simplified lumped element model, consisting of a chain of springs and bodies as shown in Figure [2](#orgaa33fb8).
<a id="org06ad50b"></a>
<a id="orgaa33fb8"></a>
{{< figure src="/ox-hugo/schmidt20_mechanical_wave.svg" caption="Figure 2: Lumped element model of one wavelength of a mechanical wave." >}}
@ -610,7 +612,7 @@ Finally it can be concluded, that these insights help in designing actively cont
### A Walk around the Control Loop {#a-walk-around-the-control-loop}
Figure [3](#orgd6fb11a) shows a basic control loop of a positioning system.
Figure [3](#org6432052) shows a basic control loop of a positioning system.
First, the A/D and D/A converters are used to translate analog signals into time-discrete digital signals and vice versa.
Secondly, the impact locations of several disturbances are shown, which play a large role in determining what reqwuirements the controller needs to fulfil.
The core of the control system is the _plant_, which is the physical system that needs to be controlled.
@ -618,7 +620,7 @@ The core of the control system is the _plant_, which is the physical system that
<a id="table--tab:walk-control-loop"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:walk-control-loop">Table 3</a></span>:
Symbols used in Figure <a href="#orgd6fb11a">10</a>
Symbols used in Figure <a href="#org6432052">10</a>
</div>
| Symbol | Meaning | Unit |
@ -632,14 +634,14 @@ The core of the control system is the _plant_, which is the physical system that
| \\(y\\) | Measured output motion | [m] |
| \\(y\_m\\) | Measurement value | [m] |
<a id="orgd6fb11a"></a>
<a id="org6432052"></a>
{{< figure src="/ox-hugo/schmidt20_walk_control_loop.svg" caption="Figure 3: Block diagram of a motion control system, including feedforward and feedback control." >}}
The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [4](#orgb4ccf73)).
The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [4](#org21aa9c3)).
They interact in both directions in such a way that each element not only determines the input of the next element, but also influences the previous element by its dynamic load.
<a id="orgb4ccf73"></a>
<a id="org21aa9c3"></a>
{{< figure src="/ox-hugo/schmidt20_energy_actuator_system.svg" caption="Figure 4: The energy converting part of a mechatronic system consists of a the amplifier, the actuator and the mechanical structure." >}}
@ -670,7 +672,7 @@ Fortunately the effect is mostly so small that it can be neglected.
#### Overview Feedforward Control {#overview-feedforward-control}
Figure [5](#orgd52bda9) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
Figure [5](#org4b3a329) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
The reference signal \\(r\\) [m] is applied to the controller, which as a reference transfer function \\(C\_{ff}(s)\\) in [N/m].
The output \\(u\\) in [N] of the controller is connected to the input of the motion system, which has a transfer function \\(G(s)\\) in [m/N] giving the output \\(x\\) in [m].
@ -682,7 +684,7 @@ If one would like to achieve perfect control, which means that there is no diffe
G\_{t,ff}(s) = \frac{x}{r} = C\_{ff}(s)G(s) = 1 \quad \Longrightarrow \quad C\_{ff}(s) = G^{-1}(s)
\end{equation}
<a id="orgd52bda9"></a>
<a id="org4b3a329"></a>
{{< figure src="/ox-hugo/schmidt20_feedforward_control_diagram.svg" caption="Figure 5: Block diagram of a feedforward controller motion system with one input and output (SISO)." >}}
@ -716,7 +718,7 @@ The drawbacks and limitations of feedforward control are:
In feedback control the actuator status of the motion system is monitored by a sensor and the controller generates a control action based on the difference between the desired motion (reference signal) and the actuator system status (sensor signal).
The block diagram of Figure [6](#org39408cb) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
The block diagram of Figure [6](#org3bccc77) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
The output \\(x\\) in [m] is the total motion of the plant on all its parts and details, while \\(y\\) is the measured motion with a measured value \\(y\_m\\) measured on a selected location in the plant.
This measured is compared with \\(r\_f\\), which is the reference \\(r\\) after filtering.
The result of this comparison is used as input for the feedback controller.
@ -728,7 +730,7 @@ The transfer function of any input to any output in a closed-loop feedback contr
</div>
<a id="org39408cb"></a>
<a id="org3bccc77"></a>
{{< figure src="/ox-hugo/schmidt20_feedback_control_diagram.svg" caption="Figure 6: Block diagram of a SISO feedback controlled motion system." >}}
@ -793,9 +795,9 @@ Also, some pitfalls have to be dealt with:
#### Model-Based Feedforward Control {#model-based-feedforward-control}
In the following an example of a model-based feedforward controller is introduced.
The measured frequency-response of the scanning unit taken as as an example is shown in Figure [7](#org7477ca7).
The measured frequency-response of the scanning unit taken as as an example is shown in Figure [7](#org77061d7).
<a id="org7477ca7"></a>
<a id="org77061d7"></a>
{{< figure src="/ox-hugo/schmidt20_bode_plot_scanning.svg" caption="Figure 7: Bode plot of a piezoelectric-actuator based scanning unit for nanometer resolution positioning. It shows the measured response (solid line) and the second order model, which is fitted for the low-frewquency system behaviour (dashed line)." >}}
@ -838,10 +840,10 @@ G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\\\
&= \frac{C\_f}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}}
\end{align}
The bode plot of the resulting dynamics is shown in Figure [8](#org6e59765).
The bode plot of the resulting dynamics is shown in Figure [8](#org1f513e4).
The controlled system has low-pass characteristics, rolling of at the scanner's natural frequency.
<a id="org6e59765"></a>
<a id="org1f513e4"></a>
{{< figure src="/ox-hugo/schmidt20_bode_plot_feedfoward_example.svg" caption="Figure 8: Bode plot of the feedforward-controlled scanning unit" >}}
@ -859,9 +861,9 @@ The oscillation caused by each individual step are 180 degrees out of phase and
This method is clearly very different form pole-zero cancellation.
In the frequency domain, these sampled adaptations to the input create a frequency spectrum with a multiple of notch filters at the harmonic of the frequency where these adaptations are applied.
Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [9](#org92e74b3).
Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [9](#orgddd25e4).
<a id="org92e74b3"></a>
<a id="orgddd25e4"></a>
{{< figure src="/ox-hugo/schmidt20_input_shaping_example.svg" caption="Figure 9: Input-shaping control of the triangular scanning signal in a scanning probe microscope." >}}
@ -889,11 +891,11 @@ The limitations of the actuators and electronics in a controlled motion system a
Of at least the levels of Jerk and preferable also Snap should be limited.
The standard method to cope with these limitations involves shaping the input of a mechatronic motion system by means of _trajectory profile generation_ or _path-planning_.
Figure [10](#orgcb6134e) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
Figure [10](#orge30b109) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
A third order trajectory would show a square profile for the jerk indicating an infinite Snap and the round of the acceleration would be gone.
A second order trajectory would show a square acceleration profile with infinite Jerk and sharp edges on the velocity.
<a id="orgcb6134e"></a>
<a id="orge30b109"></a>
{{< figure src="/ox-hugo/schmidt20_trajectory_profile.svg" caption="Figure 10: Figure caption" >}}
@ -913,12 +915,12 @@ Feedback control is more complex and critical to design than feedforward control
In general, a feedback controlled motion system is to perform a certain predetermined motion task defined by the reference input \\(r\\), while reducing the effects of other inputs like external vibrations and noise from the electronics.
All these input signals, whether desired of undesired, are treated by the feedback loop as disturbances and it is the sensitivity of the desired output signal to all input signals that determine the performance of the feedback controller.
<a id="org2c5d10f"></a>
<a id="org39f635c"></a>
{{< figure src="/ox-hugo/schmidt20_feedback_full_simplified.svg" caption="Figure 11: Full and simplified representation of a feedback loop in order to determine the influence of the reference signal and most important disturbance sources on real motion output of the plant \\(x\\), the feedback controller output \\(u\\) and the measured motion output \\(y\\). \\(y\_m = y\\) when the measurement system is set at unity gain and the sensor disturbance is included in the output disturbance." >}}
Several standard sensitivity functions have been defined to quantify the performance of feedback controlled dynamic systems.
There are derived from a simplified version of the generic feedback loop as shown in Figure [11](#org2c5d10f).
There are derived from a simplified version of the generic feedback loop as shown in Figure [11](#org39f635c).
The first simplification is made by approximating the measurement system to have a unity-gain transfer function.
For further simplification the sensor disturbance in the measurement system is included in the output disturbance \\(n\\), thereby defining the output of the system \\(y\\) as the measured output.
With this simplified model, the transfer functions of the different inputs of the system to three relevant output variables in the loop are written down in a set of equations.
@ -999,17 +1001,17 @@ To achieve sufficient robustness against instability in closed-loop feedback con
The condition for robustness of closed-loop stability is that the total phase-lag of the **total feedback-loop**, consisting of the feedback controller in series with the mechatronic system, must be less than 180 degrees in the frequency region of the _unity-gain cross-over frequency_.
The Nyquist plot of the feedback loop, like the example shown in Figure [12](#org5b472b1), is most appropriate to analyze the robustness on stability of a feedback system.
The Nyquist plot of the feedback loop, like the example shown in Figure [12](#org6e48553), is most appropriate to analyze the robustness on stability of a feedback system.
It is an analysis tool that shows the frequency response of the **feedback-loop** combining magnitude and phase in one plot.
In this figure, two graphs are shown, designed for a different purpose.
The first graph from the left shows margin circles related to the capability of the closed-loop feedback controlled system to follow a reference according to the complementary sensitivity.
The second graph shows a margin circle related to the capability of the closed-loop feedback controlled system to suppress disturbances according to the sensitivity function.
<a id="org5b472b1"></a>
<a id="org6e48553"></a>
{{< figure src="/ox-hugo/schmidt20_nyquist_plot_stable.svg" caption="Figure 12: Nyquist plot of the feedback-loop response of a stable feedback controlled motion system. Stability is guaranteed as the \\(-1\\) point is kept at the left hand side of the feedback loop repsonse line upon passing with increased frequency, even though the phase-lag is more than 180 degrees at low frequencies." >}}
Three values are shown in Figure [12](#org5b472b1) related to the robustness of the closed-loop feedback system:
Three values are shown in Figure [12](#org6e48553) related to the robustness of the closed-loop feedback system:
- **The gain margin** determines by which factor the feedback loop gain additionally can increase before the closed-loop goes unstable.
- **The phase margin** determines how much additional phase-lab at the unity-gain cross-over frequency is acceptable before the closed-loop system becomes unstable.
@ -1022,12 +1024,12 @@ Higher margins corresponds to a higher level of damping.
The Nyquist plot has one significant disadvantage as it does not show directly the frequency along the plot.
For that reason many designers prefer to use the Bode plot.
Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [13](#orgb35ccc2).
Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [13](#orgc932364).
In many not too complicated cases, these two margins are sufficient to tune a feedback motion controller.
In more complicated control systems, it remains useful to also use the Nyquist plot as it also gives the Modulus margin.
<a id="orgb35ccc2"></a>
<a id="orgc932364"></a>
{{< figure src="/ox-hugo/schmidt20_phase_gain_margin_bode.svg" caption="Figure 13: The gain and phase margin in the Bode plot" >}}
@ -1150,9 +1152,9 @@ However, analogue controllers have three important disadvantages:
The digital implementation of filters overcome these problems as well as allows more complex algorithm such as adaptive control, real-time optimization, nonlinear control and learning control methods.
In Figure [14](#orgadda653) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
In Figure [14](#org4b95e2a) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
<a id="orgadda653"></a>
<a id="org4b95e2a"></a>
{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="Figure 14: Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
@ -1174,7 +1176,7 @@ Fixed point arithmetic has been favored in the past, because of the less complex
A main drawback is, that the developer must pay attention to truncation, overflow, underflow and round-off errors that occur during mathematical operations.
Fixed points numbers are equally spaced over the whole range, separated by the gap which is denoted by the least significant bit.
The two's complement is the most used format for representing positive and negative numbers.
For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [15](#org43c66f4)).
For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [15](#orgf73916a)).
\\(m\\) denotes the number of integer bits and \\(n\\) denotes the number of fractional bits.
\\(m+n+1=N\\) bits are necessary to store a signed \\(Q\_{m,n}\\) number.
If the binary representation is given, the decimal value can be calculated to:
@ -1183,9 +1185,9 @@ If the binary representation is given, the decimal value can be calculated to:
x = \frac{1}{2^n} \left( -2^{N-1 }b\_{N-1} + \sum\_{i=0}^{N-2} 2^i b\_i \right)
\end{equation}
where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [15](#org43c66f4).
where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [15](#orgf73916a).
<a id="org43c66f4"></a>
<a id="orgf73916a"></a>
{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" caption="Figure 15: Example of a \\(Q\_{m.n}\\) fixed point number representation and a single precision floating point number" >}}
@ -1207,39 +1209,39 @@ x = -1^i M 2^{E-127}
The term \\(E\\) in the exponent is stored as a positive number ranging from \\(0 \le E < 256\\) with 8 bits.
An offset of \\(-127\\) is added in order to allow very small to very large numbers.
The decimal value is normalized, meaning that only one nonzero digit is noted at the left of the decimal point.
The storage register is divided into three groups, as shown in Figure [15](#org43c66f4).
The storage register is divided into three groups, as shown in Figure [15](#orgf73916a).
1 bit represents the sign, the exponent term \\(E\\) is represented by 8 bits, and the mantissa is stored in 23 bits.
#### Digital Filter Theory {#digital-filter-theory}
<a id="orga7dafc8"></a>
<a id="org3a2480f"></a>
{{< figure src="/ox-hugo/schmidt20_s_z_planes.svg" caption="Figure 16: Corresponding points and area in s and z planes" >}}
#### Finite Impulse Response (FIR) Filter {#finite-impulse-response--fir--filter}
<a id="orgec5f41f"></a>
<a id="org4983bdd"></a>
{{< figure src="/ox-hugo/schmidt20_transversal_filter_structure.svg" caption="Figure 17: Transversal filter structure of a FIR filter. The term \\(z^{-1}\\) each represent a sampling period which means that \\(b\_0\\) is the gain of the last sample, \\(b\_1\\) is the gain of the precious sample etcetera." >}}
<a id="orgd8b516c"></a>
<a id="org936becf"></a>
{{< figure src="/ox-hugo/schmidt20_optimized_fir_filter_structure.svg" caption="Figure 18: Optimized FIR filter structure with symmetric filter coefficients" >}}
<a id="orgaebdf57"></a>
<a id="org8ea00c7"></a>
{{< figure src="/ox-hugo/schmidt20_dir_filter_cascaded_sos.svg" caption="Figure 19: Higher-order FIR filter realization with cascade SOS filter structures" >}}
#### Infinite Impulse Response (IIR) Filter {#infinite-impulse-response--iir--filter}
<a id="orgb650cf3"></a>
<a id="org696a8aa"></a>
{{< figure src="/ox-hugo/schmidt20_irr_structure.svg" caption="Figure 20: (a:) IIR structure in DF-1 realization and (b:) IIR structure in DF-2 realization" >}}
<a id="orgb8e19b3"></a>
<a id="orge99cdac"></a>
{{< figure src="/ox-hugo/schmidt20_irr_sos_structure.svg" caption="Figure 21: IIR SOS structure in DF-2 realization" >}}
@ -2235,4 +2237,4 @@ Motion control is essential for Precision Mechatronic Systems and consists of tw
## Bibliography {#bibliography}
<a id="orge3bc57b"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. _The Design of High Performance Mechatronics - Third Revised Edition_. Ios Press.
<a id="org4e5c703"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. _The Design of High Performance Mechatronics - Third Revised Edition_. Ios Press.

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@ -1,14 +1,15 @@
+++
title = "The scientist and engineer's guide to digital signal processing - second edition"
author = ["Thomas Dehaeze"]
draft = false
keywords = ["Signal Processing"]
draft = true
+++
Tags
: [Digital Signal Processing]({{< relref "digital_signal_processing" >}})
Reference
: ([Smith 1999](#org18cc45c))
: ([Smith 1999](#org023917a))
Author(s)
: Smith, S. W.
@ -17,6 +18,7 @@ Year
: 1999
## Bibliography {#bibliography}
<a id="org18cc45c"></a>Smith, Steven W. 1999. _The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition_. California Technical Publishing.
<a id="org023917a"></a>Smith, Steven W. 1999. _The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition_. California Technical Publishing.

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+++
title = "Parallel robots : mechanics and control"
title = "Parallel Robots : Mechanics and Control"
author = ["Thomas Dehaeze"]
description = "Explains clearly fundamentals of parallel robotics such as how to represent motion, kinematics, jacobian and dynamics."
keywords = ["Stewart Platforms", "Mechatronics"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Reference Books]({{< relref "reference_books" >}})
Reference
: ([Taghirad 2013](#org5caa795))
: ([Taghirad 2013](#org2f7e22a))
Author(s)
: Taghirad, H.
@ -22,7 +24,7 @@ PDF version
## Introduction {#introduction}
<a id="org47dbe02"></a>
<a id="orgd6a205f"></a>
This book is intended to give some analysis and design tools for the increase number of engineers and researchers who are interested in the design and implementation of parallel robots.
A systematic approach is presented to analyze the kinematics, dynamics and control of parallel robots.
@ -47,14 +49,14 @@ The control of parallel robots is elaborated in the last two chapters, in which
## Motion Representation {#motion-representation}
<a id="org50588c0"></a>
<a id="orgc8c013e"></a>
### Spatial Motion Representation {#spatial-motion-representation}
Six independent parameters are sufficient to fully describe the spatial location of a rigid body.
Consider a rigid body in a spatial motion as represented in Figure [1](#org25b870d).
Consider a rigid body in a spatial motion as represented in Figure [1](#org830e006).
Let us define:
- A **fixed reference coordinate system** \\((x, y, z)\\) denoted by frame \\(\\{\bm{A}\\}\\) whose origin is located at point \\(O\_A\\)
@ -62,7 +64,7 @@ Let us define:
The absolute position of point \\(P\\) of the rigid body can be constructed from the relative position of that point with respect to the moving frame \\(\\{\bm{B}\\}\\), and the **position and orientation** of the moving frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
<a id="org25b870d"></a>
<a id="org830e006"></a>
{{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
@ -87,7 +89,7 @@ It can be **represented in several different ways**: the rotation matrix, the sc
##### Rotation Matrix {#rotation-matrix}
We consider a rigid body that has been exposed to a pure rotation.
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#orgd6c2a37)).
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org1389111)).
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
@ -109,7 +111,7 @@ in which \\({}^A\hat{\bm{x}}\_B, {}^A\hat{\bm{y}}\_B\\) and \\({}^A\hat{\bm{z}}\
The nine elements of the rotation matrix can be simply represented as the projections of the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) on the unit vectors of frame \\(\\{\bm{A}\\}\\).
<a id="orgd6c2a37"></a>
<a id="org1389111"></a>
{{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
@ -135,7 +137,7 @@ The term screw axis for this axis of rotation has the benefit that a general mot
The screw axis representation has the benefit of **using only four parameters** to describe a pure rotation.
These parameters are the angle of rotation \\(\theta\\) and the axis of rotation which is a unit vector \\({}^A\hat{\bm{s}} = [s\_x, s\_y, s\_z]^T\\).
<a id="orge51f311"></a>
<a id="org43d820e"></a>
{{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
@ -161,7 +163,7 @@ Three other types of Euler angles are consider with respect to a moving frame: t
The pitch, roll and yaw angles are defined for a moving object in space as the rotations along the lateral, longitudinal and vertical axes attached to the moving object.
<a id="orgb51325b"></a>
<a id="orgbd137db"></a>
{{< figure src="/ox-hugo/taghirad13_pitch-roll-yaw.png" caption="Figure 4: Definition of pitch, roll and yaw angles on an air plain" >}}
@ -364,10 +366,10 @@ There exist transformations to from screw displacement notation to the transform
##### Consecutive transformations {#consecutive-transformations}
Let us consider the motion of a rigid body described at three locations (Figure [5](#org2fa078f)).
Let us consider the motion of a rigid body described at three locations (Figure [5](#orgecf9a83)).
Frame \\(\\{\bm{A}\\}\\) represents the initial location, frame \\(\\{\bm{B}\\}\\) is an intermediate location, and frame \\(\\{\bm{C}\\}\\) represents the rigid body at its final location.
<a id="org2fa078f"></a>
<a id="orgecf9a83"></a>
{{< figure src="/ox-hugo/taghirad13_consecutive_transformations.png" caption="Figure 5: Motion of a rigid body represented at three locations by frame \\(\\{\bm{A}\\}\\), \\(\\{\bm{B}\\}\\) and \\(\\{\bm{C}\\}\\)" >}}
@ -430,7 +432,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
## Kinematics {#kinematics}
<a id="orgc37d0be"></a>
<a id="org3382c61"></a>
### Introduction {#introduction}
@ -537,11 +539,11 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
\end{bmatrix}
\end{equation}
<a id="org2b43912"></a>
<a id="orgc4e251a"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="Figure 6: Geometry of a Stewart-Gough platform" >}}
The geometry of the manipulator is shown Figure [6](#org2b43912).
The geometry of the manipulator is shown Figure [6](#orgc4e251a).
#### Inverse Kinematics {#inverse-kinematics}
@ -590,7 +592,7 @@ The complexity of the problem depends widely on the manipulator architecture and
## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
<a id="org52d5bda"></a>
<a id="org569f038"></a>
### Introduction {#introduction}
@ -685,9 +687,9 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
\end{bmatrix}}
\end{equation}
Now consider the general motion of a rigid body shown in Figure [7](#orgfd75b99), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
Now consider the general motion of a rigid body shown in Figure [7](#orgeb7ba17), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
<a id="orgfd75b99"></a>
<a id="orgeb7ba17"></a>
{{< figure src="/ox-hugo/taghirad13_general_motion.png" caption="Figure 7: Instantaneous velocity of a point \\(P\\) with respect to a moving frame \\(\\{\bm{B}\\}\\)" >}}
@ -949,9 +951,9 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
#### Static Forces of the Stewart-Gough Platform {#static-forces-of-the-stewart-gough-platform}
As shown in Figure [8](#org5bff8ab), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
As shown in Figure [8](#org8322c5f), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
<a id="org5bff8ab"></a>
<a id="org8322c5f"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_static_forces.png" caption="Figure 8: Free-body diagram of forces and moments action on the moving platform and each limb of the Stewart-Gough platform" >}}
@ -1108,9 +1110,9 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
#### Stiffness Analysis of the Stewart-Gough Platform {#stiffness-analysis-of-the-stewart-gough-platform}
In this section, we restrict our analysis to a 3-6 structure (Figure [9](#orge167af1)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
In this section, we restrict our analysis to a 3-6 structure (Figure [9](#org8ac48ca)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
<a id="orge167af1"></a>
<a id="org8ac48ca"></a>
{{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
@ -1140,7 +1142,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
## Dynamics {#dynamics}
<a id="org3802547"></a>
<a id="org57ec22a"></a>
### Introduction {#introduction}
@ -1239,7 +1241,7 @@ Linear acceleration of a point \\(P\\) can be easily determined by time derivati
Note that this is correct only if the derivative is taken with respect to a **fixed** frame.<br />
Now consider the general motion of a rigid body, in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and the problem is to find the absolute acceleration of point \\(P\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see Figure [7](#orgfd75b99)).
The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see Figure [7](#orgeb7ba17)).
To determine acceleration of point \\(P\\), we start with the relation between absolute and relative velocities of point \\(P\\):
\begin{equation}
@ -1272,7 +1274,7 @@ For the case where \\(P\\) is a point embedded in the rigid body, \\({}^B\bm{v}\
In this section, the properties of mass, namely **center of mass**, **moments of inertia** and its characteristics and the required transformations are described.
<a id="org7d8eb1c"></a>
<a id="orgd866005"></a>
{{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
@ -1364,7 +1366,7 @@ On the other hand, if the reference frame \\(\\{B\\}\\) has **pure rotation** wi
##### Linear Momentum {#linear-momentum}
Linear momentum of a material body, shown in Figure [11](#orgf0e919a), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
Linear momentum of a material body, shown in Figure [11](#orgfe9356c), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
\begin{equation}
{}^A\bm{G} = \int\_V \frac{d\bm{p}}{dt} \rho dV
@ -1386,14 +1388,14 @@ in which \\({}^A\bm{v}\_C\\) denotes the velocity of the center of mass with res
This result implies that the **total linear momentum** of differential masses is equal to the linear momentum of a **point mass** \\(m\\) located at the **center of mass**.
This highlights the important of the center of mass in dynamic formulation of rigid bodies.
<a id="orgf0e919a"></a>
<a id="orgfe9356c"></a>
{{< figure src="/ox-hugo/taghirad13_angular_momentum_rigid_body.png" caption="Figure 11: The components of the angular momentum of a rigid body about \\(A\\)" >}}
##### Angular Momentum {#angular-momentum}
Consider the solid body represented in Figure [11](#orgf0e919a).
Consider the solid body represented in Figure [11](#orgfe9356c).
Angular momentum of the differential masses \\(\rho dV\\) about a reference point \\(A\\), expressed in the reference frame \\(\\{\bm{A}\\}\\) is defined as
\\[ {}^A\bm{H} = \int\_V \left(\bm{p} \times \frac{d\bm{p}}{dt} \right) \rho dV \\]
in which \\(d\bm{p}/dt\\) denotes the velocity of differential mass with respect to the reference frame \\(\\{\bm{A}\\}\\).
@ -1523,7 +1525,7 @@ With \\(\bm{v}\_{b\_{i}}\\) an **intermediate variable** corresponding to the ve
\bm{v}\_{b\_{i}} = \bm{v}\_{p} + \bm{\omega} \times \bm{b}\_{i}
\end{equation}
As illustrated in Figure [12](#org8ad224c), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
As illustrated in Figure [12](#orga5de3d9), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
The position vector of these two center of masses can be determined by the following equations:
\begin{align}
@ -1531,7 +1533,7 @@ The position vector of these two center of masses can be determined by the follo
\bm{p}\_{i\_2} &= \bm{a}\_{i} + ( l\_i - c\_{i\_2}) \hat{\bm{s}}\_{i}
\end{align}
<a id="org8ad224c"></a>
<a id="orga5de3d9"></a>
{{< figure src="/ox-hugo/taghirad13_free_body_diagram_stewart.png" caption="Figure 12: Free-body diagram of the limbs and the moving platform of a general Stewart-Gough manipulator" >}}
@ -1558,7 +1560,7 @@ We assume that each limb consists of two parts, the cylinder and the piston, whe
We also assume that the centers of masses of the cylinder and the piston are located at a distance of \\(c\_{i\_1}\\) and \\(c\_{i\_2}\\) above their foot points, and their masses are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
Moreover, consider that the pistons are symmetric about their axes, and their centers of masses lie at their midlengths.
The free-body diagrams of the limbs and the moving platforms is given in Figure [12](#org8ad224c).
The free-body diagrams of the limbs and the moving platforms is given in Figure [12](#orga5de3d9).
The reaction forces at fixed points \\(A\_i\\) are denoted by \\(\bm{f}\_{a\_i}\\), the internal force at moving points \\(B\_i\\) are dentoed by \\(\bm{f}\_{b\_i}\\), and the internal forces and moments between cylinders and pistons are denoted by \\(\bm{f}\_{c\_i}\\) and \\(\bm{M\_{c\_i}}\\) respectively.
Assume that the only existing external disturbance wrench is applied on the moving platform and is denoted by \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\).
@ -1586,7 +1588,7 @@ in which \\(m\_{c\_e}\\) is defined as
##### Dynamic Formulation of the Moving Platform {#dynamic-formulation-of-the-moving-platform}
Assume that the **moving platform center of mass is located at the center point** \\(P\\) and it has a mass \\(m\\) and moment of inertia \\({}^A\bm{I}\_{P}\\).
Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in Figure [12](#org8ad224c).
Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in Figure [12](#orga5de3d9).
The Newton-Euler formulation of the moving platform is as follows:
@ -1745,9 +1747,9 @@ in which
##### Forward Dynamics Simulations {#forward-dynamics-simulations}
As shown in Figure [13](#org59a1fc3), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
As shown in Figure [13](#orgf4709a6), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
<a id="org59a1fc3"></a>
<a id="orgf4709a6"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="Figure 13: Flowchart of forward dynamics implementation sequence" >}}
@ -1758,7 +1760,7 @@ The closed-form dynamic formulation of the Stewart-Gough platform corresponds to
In inverse dynamics simulations, it is assumed that the **trajectory of the manipulator is given**, and the **actuator forces required to generate such trajectories are to be determined**.
As illustrated in Figure [14](#orgd3aaf90), inverse dynamic formulation is implemented in the following sequence.
As illustrated in Figure [14](#org88707f9), inverse dynamic formulation is implemented in the following sequence.
The first step is trajectory generation for the manipulator moving platform.
Many different algorithms are developed for a smooth trajectory generation.
For such a trajectory, \\(\bm{\mathcal{X}}\_{d}(t)\\) and the time derivatives \\(\dot{\bm{\mathcal{X}}}\_{d}(t)\\), \\(\ddot{\bm{\mathcal{X}}}\_{d}(t)\\) are known.
@ -1780,7 +1782,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
\bm{\tau} = \bm{J}^{-T} \left( \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) - \bm{\mathcal{F}}\_d \right)
\end{equation}
<a id="orgd3aaf90"></a>
<a id="org88707f9"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
@ -1805,7 +1807,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
## Motion Control {#motion-control}
<a id="org65878c4"></a>
<a id="orgbaafc08"></a>
### Introduction {#introduction}
@ -1826,7 +1828,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
### Controller Topology {#controller-topology}
<a id="org694cde5"></a>
<a id="orgbb6f5ce"></a>
<div class="important">
<div></div>
@ -1871,11 +1873,11 @@ In general, the desired motion of the moving platform may be represented by the
To perform such motion in closed loop, it is necessary to **measure the output motion** \\(\bm{\mathcal{X}}\\) of the manipulator by an instrumentation system.
Such instrumentation usually consists of two subsystems: the first subsystem may use accurate accelerometers, or global positioning systems to calculate the position of a point on the moving platform; and a second subsystem may use inertial or laser gyros to determine orientation of the moving platform.<br />
Figure [15](#org6edb728) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
Figure [15](#orga261474) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
In such a structure, the measured position and orientation of the manipulator is compared to its desired value to generate the **motion error vector** \\(\bm{e}\_\mathcal{X}\\).
The controller uses this error information to generate suitable commands for the actuators to minimize the tracking error.<br />
<a id="org6edb728"></a>
<a id="orga261474"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback.png" caption="Figure 15: The general topology of motion feedback control: motion variable \\(\bm{\mathcal{X}}\\) is measured" >}}
@ -1883,9 +1885,9 @@ However, it is usually much **easier to measure the active joint variable** rath
The relation between the **joint variable** \\(\bm{q}\\) and **motion variable** of the moving platform \\(\bm{\mathcal{X}}\\) is dealt with the **forward and inverse kinematics**.
The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) and \\(\dot{\bm{\mathcal{X}}}\\) is studied through the **Jacobian analysis**.<br />
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [16](#orga6b318d) to implement such a controller.
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [16](#org9af376b) to implement such a controller.
<a id="orga6b318d"></a>
<a id="org9af376b"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_bis.png" caption="Figure 16: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured" >}}
@ -1894,26 +1896,26 @@ As described earlier, this is a **complex task** for parallel manipulators.
It is even more complex when a solution has to be found in real time.<br />
However, as shown herein before, the inverse kinematic analysis of parallel manipulators is much easier to carry out.
To overcome the implementation problem of the control topology in Figure [16](#orga6b318d), another control topology is usually implemented for parallel manipulators.
To overcome the implementation problem of the control topology in Figure [16](#org9af376b), another control topology is usually implemented for parallel manipulators.
In this topology, depicted in Figure [17](#orgf913d55), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
In this topology, depicted in Figure [17](#org2f578c6), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
<a id="orgf913d55"></a>
<a id="org2f578c6"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_ter.png" caption="Figure 17: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured, and the inverse kinematic analysis is used" >}}
Therefore, the **structure and characteristics** of the controller in this topology is totally **different** from that given in the first two topologies.
The **input and output** of the controller depicted in Figure [17](#orgf913d55) are **both in the joint space**.
The **input and output** of the controller depicted in Figure [17](#org2f578c6) are **both in the joint space**.
However, this is not the case in the previous topologies where the input to the controller is the motion error in task space, while its output is in the joint space.
For the topology in Figure [17](#orgf913d55), **independent controllers** for each joint may be suitable.<br />
For the topology in Figure [17](#org2f578c6), **independent controllers** for each joint may be suitable.<br />
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#orgac91dcd).
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#orgbc32f09).
A force distribution block is added which maps the generated wrench in the task space \\(\bm{\mathcal{F}}\\), to its corresponding actuator forces/torque \\(\bm{\tau}\\).
<a id="orgac91dcd"></a>
<a id="orgbc32f09"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_quater.png" caption="Figure 18: The general topology of motion feedback control in task space: the motion variable \\(\bm{\mathcal{X}}\\) is measured, and the controller output generates wrench in task space" >}}
@ -1923,16 +1925,16 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
### Motion Control in Task Space {#motion-control-in-task-space}
<a id="orgcf26437"></a>
<a id="org7ff46c1"></a>
#### Decentralized PD Control {#decentralized-pd-control}
In the control structure in Figure [19](#orgdb03b09), a number of linear PD controllers are used in a feedback structure on each error component.
In the control structure in Figure [19](#orgcb1e65e), a number of linear PD controllers are used in a feedback structure on each error component.
The decentralized controller consists of **six disjoint linear controllers** acting on each error component \\(\bm{e}\_x = [e\_x,\ e\_y,\ e\_z,\ e\_{\theta\_x},\ e\_{\theta\_y},\ e\_{\theta\_z}]\\).
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), in which \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(6 \times 6\\) **diagonal matrices** denoting the derivative and proportional controller gains for each error term.
<a id="orgdb03b09"></a>
<a id="orgcb1e65e"></a>
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
@ -1951,11 +1953,11 @@ The controller gains are generally tuned experimentally based on physical realiz
#### Feed Forward Control {#feed-forward-control}
A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [20](#orgf1d1d54).
A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [20](#orgedf4c09).
This term is generated from the dynamic model of the manipulator in the task space, represented in a closed form by the following equation:
\\[ \bm{\mathcal{F}}\_{ff} = \bm{\hat{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}(\bm{\mathcal{X}}\_d) \\]
<a id="orgf1d1d54"></a>
<a id="orgedf4c09"></a>
{{< figure src="/ox-hugo/taghirad13_feedforward_control_task_space.png" caption="Figure 20: Feed forward wrench added to the decentralized PD controller in task space" >}}
@ -2011,14 +2013,14 @@ By this means, **nonlinear and coupling behavior of the robotic manipulator is s
</div>
General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#org32f9766).
General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#orgbcb908e).
A corrective wrench \\(\bm{\mathcal{F}}\_{fl}\\) is added in a **feedback structure** to the closed-loop system, which is calculated from the Coriolis and centrifugal matrix and gravity vector of the manipulator dynamic formulation.
Furthermore, mass matrix is added in the forward path in addition to the desired trajectory acceleration \\(\ddot{\bm{\mathcal{X}}}\_d\\).
As for the feedforward control, the **dynamics and kinematic parameters of the robot are needed**, and in practice estimates of these matrices are used.<br />
<a id="org32f9766"></a>
<a id="orgbcb908e"></a>
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
@ -2138,14 +2140,14 @@ in which
\\[ \bm{\eta} = \bm{M}^{-1} \left( \tilde{\bm{M}} \bm{a}\_r + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} \right) \\]
is a measure of modeling uncertainty.
<a id="org88907b2"></a>
<a id="org318b379"></a>
{{< figure src="/ox-hugo/taghirad13_robust_inverse_dynamics_task_space.png" caption="Figure 22: General configuration of robust inverse dynamics control implemented in the task space" >}}
#### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
<a id="org0c70404"></a>
<a id="org3fa8e62"></a>
{{< figure src="/ox-hugo/taghirad13_adaptative_inverse_control_task_space.png" caption="Figure 23: General configuration of adaptative inverse dynamics control implemented in task space" >}}
@ -2158,7 +2160,7 @@ If this measurement is available without any doubt, such topologies are among th
However, as explained in Section , in many practical situations measurement of the motion variable \\(\bm{\mathcal{X}}\\) is difficult or expensive, and usually just the active joint variables \\(\bm{q}\\) are measured.
In such cases, the controllers developed in the joint space may be recommended for practical implementation.<br />
To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#orga6b318d).
To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#org9af376b).
In this topology, the controller input is the joint variable error vector \\(\bm{e}\_q = \bm{q}\_d - \bm{q}\\), and the controller output is directly the actuator force vector \\(\bm{\tau}\\), and hence there exists a **one-to-one correspondence between the controller input to its output**.<br />
The general form of dynamic formulation of parallel robot is usually given in the task space.
@ -2217,7 +2219,7 @@ Furthermore, the main dynamic matrices are all functions of the motion variable
Hence, in practice, to find the dynamic matrices represented in the joint space, **forward kinematics** should be solved to find the motion variable \\(\bm{\mathcal{X}}\\) for any given joint motion vector \\(\bm{q}\\).<br />
Since in parallel robots the forward kinematic analysis is computationally intensive, there exist inherent difficulties in finding the dynamic matrices in the joint space as an explicit function of \\(\bm{q}\\).
In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [16](#orga6b318d), and implement control law design in the task space.<br />
In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [16](#org9af376b), and implement control law design in the task space.<br />
However, one implementable alternative to calculate the dynamic matrices represented in the joint space is to use the **desired motion trajectory** \\(\bm{\mathcal{X}}\_d\\) instead of the true value of motion vector \\(\bm{\mathcal{X}}\\) in the calculations.
This approximation significantly reduces the computational cost, with the penalty of having mismatch between the estimated values of these matrices to their true values.
@ -2226,11 +2228,11 @@ This approximation significantly reduces the computational cost, with the penalt
#### Decentralized PD Control {#decentralized-pd-control}
The first control strategy introduced in the joint space consists of the simplest form of feedback control in such manipulators.
In this control structure, depicted in Figure [24](#orge46fe49), a number of PD controllers are used in a feedback structure on each error component.
In this control structure, depicted in Figure [24](#orga3264e5), a number of PD controllers are used in a feedback structure on each error component.
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), where \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(n \times n\\) **diagonal** matrices denoting the derivative and proportional controller gains, respectively.<br />
<a id="orge46fe49"></a>
<a id="orga3264e5"></a>
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
@ -2250,9 +2252,9 @@ To remedy these shortcomings, some modifications have been proposed to this stru
#### Feedforward Control {#feedforward-control}
The tracking performance of the simple PD controller implemented in the joint space is usually not sufficient at different configurations.
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [25](#orgc35f9a0).
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [25](#org4295da5).
<a id="orgc35f9a0"></a>
<a id="org4295da5"></a>
{{< figure src="/ox-hugo/taghirad13_feedforward_pd_control_joint_space.png" caption="Figure 25: Feed forward actuator force added to the decentralized PD controller in joint space" >}}
@ -2293,14 +2295,14 @@ By this means, the **nonlinear and coupling characteristics** of robotic manipul
</div>
The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in Figure [26](#org63ab6b7).
The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in Figure [26](#orgffe7bec).
A corrective torque \\(\bm{\tau}\_{fl}\\) is added in a **feedback** structure to the closed-loop system, which is calculated from the Coriolis and Centrifugal matrix, and the gravity vector of the manipulator dynamic formulation in the joint space.
Furthermore, the mass matrix is acting in the **forward path**, in addition to the desired trajectory acceleration \\(\ddot{\bm{q}}\_q\\).
Note that to generate this term, the **dynamic formulation** of the robot, and its **kinematic and dynamic parameters are needed**.
In practice, exact knowledge of dynamic matrices are not available, and there estimates are used.<br />
<a id="org63ab6b7"></a>
<a id="orgffe7bec"></a>
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
@ -2574,7 +2576,7 @@ Hence, it is recommended to design and implement controllers in the task space,
## Force Control {#force-control}
<a id="org2ffeff1"></a>
<a id="org800fbc9"></a>
### Introduction {#introduction}
@ -2625,12 +2627,12 @@ However, note that the motion control of the robot when the robot is in interact
To follow **two objectives** with different properties in one control system, usually a **hierarchy** of two feedback loops is used in practice.
This kind of control topology is called **cascade control**, which is used when there are **several measurements and one prime control variable**.
Cascade control is implemented by **nesting** the control loops, as shown in Figure [27](#org0555e95).
Cascade control is implemented by **nesting** the control loops, as shown in Figure [27](#org9d6dce7).
The output control loop is called the **primary loop**, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system.
</div>
<a id="org0555e95"></a>
<a id="org9d6dce7"></a>
{{< figure src="/ox-hugo/taghirad13_cascade_control.png" caption="Figure 27: Block diagram of a closed-loop system with cascade control" >}}
@ -2658,32 +2660,32 @@ Consider the force control schemes, in which **force tracking is the prime objec
In such a case, it is advised that the outer loop of cascade control structure is constructed by wrench feedback, while the inner loop is based on position feedback.
Since different types of measurement units may be used in parallel robots, different control topologies may be constructed to implement such a cascade structure.<br />
Consider first the cascade control topology shown in Figure [28](#orgf4cb78e) in which the measured variables are both in the **task space**.
Consider first the cascade control topology shown in Figure [28](#orgeff41f5) in which the measured variables are both in the **task space**.
The inner loop is constructed by position feedback while the outer loop is based on force feedback.
As seen in Figure [28](#orgf4cb78e), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
As seen in Figure [28](#orgeff41f5), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.<br />
<a id="orgf4cb78e"></a>
<a id="orgeff41f5"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
Other alternatives for force control topology may be suggested based on the variations of position and force measurements.
If the force is measured in the joint space, the topology suggested in Figure [29](#org04f708f) can be used.
If the force is measured in the joint space, the topology suggested in Figure [29](#orgc86e7ec) can be used.
In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).<br />
<a id="org04f708f"></a>
<a id="orgc86e7ec"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
Consider the case where the force and motion variables are both measured in the **joint space**.
Figure [30](#org4f8b19a) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
Figure [30](#org49dc251) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
In this topology, it is advised that the force controller is designed in the **task** space, and the Jacobian transpose mapping is used to project the measured actuator force vector into its corresponding wrench in the task space.
However, as the inner loop is constructed in the joint space, the desired motion variable \\(\bm{\mathcal{X}}\_d\\) is mapped into joint space using **inverse kinematic** solution.
Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.<br />
<a id="org4f8b19a"></a>
<a id="org49dc251"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@ -2695,30 +2697,30 @@ In such a case, force tracking is not the primary objective, and it is advised t
Since different type of measurement units may be used in parallel robots, different control topologies may be constructed to implement such cascade controllers.<br />
Figure [31](#org31cfb98) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
Figure [31](#orgdaac91b) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
The inner loop is loop is constructed by force feedback while the outer loop is based on position feedback.
By this means, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed through the primary controller.
However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.<br />
<a id="org31cfb98"></a>
<a id="orgdaac91b"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
Other alternatives for control topology may be suggested based on the variations of position and force measurements.
If the force is measured in the joint space, control topology shown in Figure [32](#org09698d7) can be used.
If the force is measured in the joint space, control topology shown in Figure [32](#orgbbeae5d) can be used.
In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.<br />
<a id="org09698d7"></a>
<a id="orgbbeae5d"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
If the force and motion variables are both measured in the **joint** space, the control topology shown in Figure [33](#orga67daf8) is suggested.
If the force and motion variables are both measured in the **joint** space, the control topology shown in Figure [33](#org1cc6aaa) is suggested.
The inner loop is based on the measured actuator force vector in the joint space \\(\bm{\tau}\\), and the outer loop is based on the measured actuated joint position vector \\(\bm{q}\\).
In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
Thus, independent controllers for each joint may be suitable for this topology.
<a id="orga67daf8"></a>
<a id="org1cc6aaa"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@ -2737,7 +2739,7 @@ Thus, independent controllers for each joint may be suitable for this topology.
### Direct Force Control {#direct-force-control}
<a id="org52d002e"></a>
<a id="org6953135"></a>
{{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
@ -2799,7 +2801,7 @@ Nevertheless, note that Laplace transform is only applicable for **linear time i
<div class="exampl">
<div></div>
Consider an RLC circuit depicted in Figure [35](#org772f157).
Consider an RLC circuit depicted in Figure [35](#org6a4f56c).
The differential equation relating voltage \\(v\\) to the current \\(i\\) is given by
\\[ v = L\frac{di}{dt} + Ri + \int\_0^t \frac{1}{C} i(\tau)d\tau \\]
in which \\(L\\) denote the inductance, \\(R\\) the resistance and \\(C\\) the capacitance.
@ -2815,7 +2817,7 @@ The impedance of the system may be found from the Laplace transform of the above
<div class="exampl">
<div></div>
Consider the mass-spring-damper system depicted in Figure [35](#org772f157).
Consider the mass-spring-damper system depicted in Figure [35](#org6a4f56c).
The governing dynamic formulation for this system is given by
\\[ m \ddot{x} + c \dot{x} + k x = f \\]
in which \\(m\\) denote the body mass, \\(c\\) the damper viscous coefficient and \\(k\\) the spring stiffness.
@ -2828,7 +2830,7 @@ The impedance of the system may be found from the Laplace transform of the above
</div>
<a id="org772f157"></a>
<a id="org6a4f56c"></a>
{{< figure src="/ox-hugo/taghirad13_impedance_control_rlc.png" caption="Figure 35: Analogy of electrical impedance in (a) an electrical RLC circuit to (b) a mechanical mass-spring-damper system" >}}
@ -2846,7 +2848,7 @@ An impedance \\(\bm{Z}(s)\\) is called
</div>
Hence, for the mechanical system represented in Figure [35](#org772f157):
Hence, for the mechanical system represented in Figure [35](#org6a4f56c):
- mass represents inductive impedance
- viscous friction represents resistive impedance
@ -2879,24 +2881,24 @@ In the impedance control scheme, **regulation of the motion-force dynamic relati
Therefore, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed by a primary controller.
However, when there is an interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure may be designed to control the force-motion dynamic relation.<br />
As a possible impedance control scheme, consider the closed-loop system depicted in Figure [36](#org6c87ae0), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
As a possible impedance control scheme, consider the closed-loop system depicted in Figure [36](#orgf321a0b), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
This structure is advised when a desired impedance relation between the force and motion variables is required that consists of desired inductive, resistive, and capacitive impedances.
As shown in Figure [36](#org6c87ae0), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
As shown in Figure [36](#orgf321a0b), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
Moreover, direct force-tracking objective is not assigned in this control scheme, and therefore the desired force trajectory \\(\bm{\mathcal{F}}\_d\\) is absent in this scheme.
However, an auxiliary force trajectory \\(\bm{\mathcal{F}}\_a\\) is generated from the motion control law and is used as the reference for the force tracking.
By this means, no prescribed force trajectory is tracked, while the **motion control scheme would advise a force trajectory for the robot to ensure the desired impedance regulation**.<br />
<a id="org6c87ae0"></a>
<a id="orgf321a0b"></a>
{{< figure src="/ox-hugo/taghira13_impedance_control.png" caption="Figure 36: Impedance control scheme; motion feedback in the outer loop and force feedback in the inner loop" >}}
The required wrench \\(\bm{\mathcal{F}}\\) in the impedance control scheme, is based on inverse dynamics control and consists of three main parts.
In the inner loop, the force control scheme is based on a feedback linearization part in addition to a mass matrix adjustment, while in the outer loop usually a linear motion controller is considered based on the desired impedance requirements.
Although many different impedance structures may be considered as the basis of the control law, in Figure [36](#org6c87ae0), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
Although many different impedance structures may be considered as the basis of the control law, in Figure [36](#orgf321a0b), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
According to Figure [36](#org6c87ae0), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
According to Figure [36](#orgf321a0b), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
\\[ \bm{\mathcal{F}} = \hat{\bm{M}} \bm{M}\_d^{-1} \bm{e}\_F + \bm{\mathcal{F}}\_{fl} \\]
with:
@ -2926,4 +2928,4 @@ However, note that for a good performance, an accurate model of the system is re
## Bibliography {#bibliography}
<a id="org5caa795"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.
<a id="org2f7e22a"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.

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@ -10,18 +10,18 @@ Tags
## 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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### 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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## 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>.
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