@@ -1310,10 +1310,10 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
|  |  |
|------------------------------------------|---------------------------------------------|
-|
Point receptance |
Transfer receptance |
+|
Point receptance |
Transfer receptance |
| width=\linewidth | width=\linewidth |
-In the two figures [11](#org6a1d778) and [11](#org58ff327), we show corresponding data for **non-proportional** damping.
+In the two figures [11](#org7d25d6c) and [11](#org9e70037), 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 +1325,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
|  |  |
|-----------------------------------------------------|--------------------------------------------------------|
-|
Point receptance |
Transfer receptance |
+|
Point receptance |
Transfer receptance |
| width=\linewidth | width=\linewidth |
@@ -1481,7 +1481,7 @@ Examples of random signals, autocorrelation function and power spectral density
|  |  |  |
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
-|
Time history |
Autocorrelation Function |
Power Spectral Density |
+|
Time history |
Autocorrelation Function |
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 +1566,8 @@ 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](#orgc413fea) the traditional single-input single-output model upon which the previous formulae are based.
-Then in [13](#org1dd02c9) 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](#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.
@@ -1577,7 +1577,7 @@ Then in [13](#org1dd02c9) is given a more detailed and representative model of t
|  |  |
|------------------------------------------|--------------------------------------------------|
-|
Basic SISO model |
SISO model with feedback |
+|
Basic SISO model |
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 +1597,7 @@ 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](#orgad01713).
+A diagram for the general n-input case is shown in figure [8](#org8f4df84).
We obtain two alternative formulas:
@@ -1608,7 +1608,7 @@ 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.
-
+
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@@ -1878,9 +1878,9 @@ 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](#org5658974).
+A typical layout for the measurement system is shown on figure [9](#org7f3a496).
-
+
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@@ -1934,21 +1934,21 @@ 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](#org227d83a).
+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).
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.
-
+
{{< 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](#org0d724dc) 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](#org2ce9b2d) 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](#orgf8ba17d) shows an alternative configuration in which the shaker itself is supported.
+Figure [14](#orgaf570a9) 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](#orge6e0404) 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](#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\\).
@@ -1958,7 +1958,7 @@ Figure [14](#orge6e0404) shows an unsatisfactory setup. Indeed, the response mea
|  |  |  |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
-|
Ideal Configuration |
Suspended Configuration |
Unsatisfactory |
+|
Ideal Configuration |
Suspended Configuration |
Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1973,10 +1973,10 @@ 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](#orgf285619).
-A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#orgf285619).
+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).
-
+
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@@ -2005,9 +2005,9 @@ 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](#org7500151)).
+The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#org930ef4e)).
-
+
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@@ -2016,11 +2016,11 @@ 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](#org8d7da26)).
+In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#orga075bcf)).
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\\).
-
+
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@@ -2056,9 +2056,9 @@ 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](#orgf956916).
+Some of these methods are illustrated in figure [15](#orge053903).
-Shown on figure [15](#orga10bae6) are typical high frequency limits for each type of attachment.
+Shown on figure [15](#org1b85602) are typical high frequency limits for each type of attachment.
@@ -2068,7 +2068,7 @@ Shown on figure [15](#orga10bae6) are typical high frequency limits for each typ
|  |  |
|-----------------------------------------------------|------------------------------------------------------------|
-|
Attachment methods |
Frequency response characteristics |
+|
Attachment methods |
Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@@ -2153,9 +2153,9 @@ 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](#org88605d5).
+These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#org91dbe3e).
-
+
{{< 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 +2172,7 @@ This is illustrated on figure [16](#table--fig:effect-aliasing).
|  |  |
|--------------------------------------------------|-----------------------------------------------------|
-|
True spectrum of signal |
Indicated spectrum from DFT |
+|
True spectrum of signal |
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 +2193,12 @@ Leakage is a problem which is a direct **consequence of the need to take only a
|  |  |
|--------------------------------------|----------------------------------------|
-|
Ideal signal |
Awkward signal |
+|
Ideal signal |
Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [17](#table--fig:leakage).
-In the first case (figure [17](#org5c4b348)), 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](#org9371337)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
+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.
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 +2216,14 @@ 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](#org8903f56).
+The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#org105c7d0).
-
+
{{< 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](#org8903f56).
+The result of using a window is seen in the third column of figure [15](#org105c7d0).
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 +2239,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
-
+
##### Increasing transform size {#increasing-transform-size}
@@ -2263,9 +2263,9 @@ 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](#org41c500b), 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](#orgfeb63a7), 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](#orgfb95cfe), 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](#org1dc44b8)).
+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)).
@@ -2275,10 +2275,10 @@ If we apply a band-pass filter to the signal, as shown on figure [18](#orgfb95cf
|  |  |
|------------------------------------------------|------------------------------------------|
-|
Spectrum of the signal |
Band-pass filter |
+|
Spectrum of the signal |
Band-pass filter |
| width=\linewidth | width=\linewidth |
-
+
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@@ -2348,9 +2348,9 @@ 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](#orgbb6a3e8).
+If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#orgd1e88bf).
-
+
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@@ -2466,9 +2466,9 @@ 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](#org7b09a80).
+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).
-
+
{{< 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 +2509,9 @@ 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](#org98bdc4b)).
+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)).
-
+
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@@ -2526,22 +2526,22 @@ 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](#orgb584cc5).
+The chirp consist of a short duration signal which has the form shown in figure [20](#org3d7182f).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
-
+
{{< 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](#org4db89b3).
+The hammer blow produces an input and response as shown in the figure [21](#orgee86d4a).
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.
-
+
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@@ -2549,9 +2549,9 @@ 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](#orgb29cf45)).
+In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#org2914aa8)).
-
+
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@@ -2624,9 +2624,9 @@ 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](#orgddf08d1) shows a typical calibration setup.
+Figure [23](#org1f3d9fc) shows a typical calibration setup.
-
+
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@@ -2639,9 +2639,9 @@ 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](#orgc770be3).
+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).
-
+
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@@ -2696,9 +2696,9 @@ 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](#org672cadb).
+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).
-
+
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@@ -2714,12 +2714,12 @@ 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](#org1e07ef2) shows a device to simulate a moment excitation.
+Figure [26](#org69e6665) 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\\).
-
+
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@@ -3043,8 +3043,8 @@ 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](#org354c67b) 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](#org9b16f79) shows the genuine modes distinct from the computational modes.
+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.
@@ -3071,7 +3071,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
|  |  |  |
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
-|
FRF |
Singular Values |
PRF |
+|
FRF |
Singular Values |
PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -3082,7 +3082,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
|  |  |  |
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
-|
FRF |
Singular Values |
PRF |
+|
FRF |
Singular Values |
PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -3114,7 +3114,7 @@ The **Complex mode indicator function** (CMIF) is defined as
-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](#org1684572):
+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):
- **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 +3124,7 @@ 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**
-
+
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@@ -3197,7 +3197,7 @@ 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](#org00f64f0)):
+The peak-picking method is applied as follows (illustrated on figure [28](#orgd1dacfd)):
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 +3219,7 @@ The peak-picking method is applied as follows (illustrated on figure [28](#org00
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.
-
+
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@@ -3244,7 +3244,7 @@ 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](#orga9233a4).
+A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org0c46692).
@@ -3254,7 +3254,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#orga9233a4)
|  |  |
|----------------------------------------|--------------------------------------------------------------------|
-|
Properties |
\\(\omega\_b\\) and \\(\omega\_a\\) points |
+|
Properties |
\\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@@ -3292,7 +3292,7 @@ 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](#org0ad9bc7), we can write:
+Referring to figure [21](#orga918af0), we can write:
\begin{equation}
\begin{aligned}
@@ -3358,7 +3358,7 @@ 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](#org751f6e5).
+ A typical example is shown on figure [29](#org96a13a2).
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 +3368,7 @@ 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.
-
+
{{< 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 +3482,8 @@ 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](#orgf043f5e).
-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](#org1a33e90).
+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).
@@ -3493,7 +3493,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
|  |  |
|--------------------------------------------|-----------------------------------------|
-|
without residual |
with residuals |
+|
without residual |
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 +3522,9 @@ 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](#orgd0e499e).
+These three terms are illustrated on figure [30](#org473ef14).
-
+
{{< 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 +3818,7 @@ with
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](#org2873c75) and their summation shown as a single composite curve in figure [23](#orge18611d).
+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).
@@ -3828,7 +3828,7 @@ As an example, a set of mobilities measured are shown individually in figure [23
|  |  |
|-------------------------------------------|-----------------------------------------|
-|
Individual curves |
Composite curve |
+|
Individual curves |
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 +4382,11 @@ 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](#org6eef557) (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](#org6eef557) (b).
+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).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
-
+
{{< 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 +4395,16 @@ 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](#org6eef557) (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](#org0dcf72a) (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](#org6eef557) (d).
+It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#org0dcf72a) (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](#org6eef557) (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](#org0dcf72a) (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 +4427,11 @@ 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](#org77a201c), 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](#org843940c), 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**.
-
+
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@@ -4478,11 +4478,11 @@ However, it must be noted that there is an important **limitation to this proced
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](#orgf256093).
+The predict curve is compared with the measurements on figure [24](#orga9d477f).
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](#org15312a1).
+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).
@@ -4494,7 +4494,7 @@ The inclusion of these two additional terms (obtained here only after measuring
|  |  |
|--------------------------------------------------------|-----------------------------------------------------------|
-|
Using measured modal data only |
After inclusion of residual terms |
+|
Using measured modal data only |
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 +4546,10 @@ If the **transmissibility** is measured during a modal test which has a single e
-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](#orga7adcfa).
+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).
This may explain why transmissibilities are not widely used in modal analysis.
-
+
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@@ -4570,7 +4570,7 @@ The fact that the excitation force is not measured is responsible for the lack o
|  |  |
|---------------------------------------------------------|-------------------------------------------------------|
-|
Conventional modal test setup |
Base excitation setup |
+|
Conventional modal test setup |
Base excitation setup |
| height=4cm | height=4cm |
@@ -4611,6 +4611,7 @@ This is accomplished using the above equation in the form:
Because the rank of each pseudo matrix is less than its order, it cannot be inverted and so we are unable to construct stiffness or mass matrix from this approach.
+
## Bibliography {#bibliography}
-
Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
+
Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
diff --git a/content/book/fleming14_desig_model_contr_nanop_system.md b/content/book/fleming14_desig_model_contr_nanop_system.md
index b401103..76afd69 100644
--- a/content/book/fleming14_desig_model_contr_nanop_system.md
+++ b/content/book/fleming14_desig_model_contr_nanop_system.md
@@ -9,7 +9,7 @@ Tags
Reference
-: ([Fleming and Leang 2014](#org53722bc))
+: ([Fleming and Leang 2014](#org6bfb955))
Author(s)
: Fleming, A. J., & Leang, K. K.
@@ -821,11 +821,11 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
-
+
{{< 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](#orgaaa53eb) where a voltage source is connected to a piezoelectric actuator.
+Consider the electrical circuit shown in Figure [1](#org1aabb30) 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.
@@ -946,6 +946,7 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
### References {#references}
+
## Bibliography {#bibliography}
-
Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing.
.
+Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. .
diff --git a/content/book/hatch00_vibrat_matlab_ansys.md b/content/book/hatch00_vibrat_matlab_ansys.md
index 574e0cf..e72c91c 100644
--- a/content/book/hatch00_vibrat_matlab_ansys.md
+++ b/content/book/hatch00_vibrat_matlab_ansys.md
@@ -8,7 +8,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
Reference
-: ([Hatch 2000](#orgf661cf4))
+: ([Hatch 2000](#org8ef052d))
Author(s)
: Hatch, M. R.
@@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
-
+
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](#org3a8e4cc) shows the methodology for analyzing a lightly damped structure using normal modes.
+The diagram in Figure [1](#orge443794) shows the methodology for analyzing a lightly damped structure using normal modes.
@@ -46,7 +46,7 @@ The steps are:
-
+
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
@@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
-Figure [2](#org0a7e8db) shows such process, the steps are:
+Figure [2](#org5c37471) 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 +71,14 @@ Figure [2](#org0a7e8db) shows such process, the steps are:
-
+
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations}
-Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
+Tables [3](#org9e923ac), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
@@ -127,22 +127,22 @@ Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
-
+
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](#orgc82b5d8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
+Figure [3](#org9e923ac) 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\\).
-
+
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
-([Miu 1993](#org849cfe4)) 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](#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.
The resonances of the "overhanging appendages" of the constrained system create the zeros.
@@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
-
+
## Modal Analysis {#modal-analysis}
-
+
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@@ -196,9 +196,9 @@ 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](#orgebf4457) 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](#org829b3b4) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
-
+
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
@@ -297,17 +297,17 @@ One then find:
\end{bmatrix}
\end{equation}
-Virtual interpretation of the eigenvectors are shown in Figures [5](#org520a99d), [6](#org722a9ff) and [7](#org9e25b28).
+Virtual interpretation of the eigenvectors are shown in Figures [5](#orgabe9314), [6](#org4283877) and [7](#orge77cc5c).
-
+
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
-
+
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
-
+
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
@@ -346,9 +346,9 @@ 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](#orgfbabf08).
+This procedure is schematically shown in Figure [8](#org948bae0).
-
+
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@@ -696,7 +696,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
-
+
The procedure to obtain the frequency response from a modal form is as follow:
@@ -704,9 +704,9 @@ 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](#orge102983).
+This will be applied to the model shown in Figure [9](#org4e1f260).
-
+
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@@ -888,9 +888,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
-Figure [10](#org3024448) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
+Figure [10](#org87a6063) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
-
+
{{< 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 +899,16 @@ 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}
-
+
### Introduction {#introduction}
-In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2292476).
+In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org684a769).
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.
-
+
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
@@ -987,7 +987,7 @@ 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}
-
+
### Model Description {#model-description}
@@ -1001,25 +1001,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
-
+
-In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org143e4e8)).
+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)).
### Actuator Description {#actuator-description}
-
+
{{< 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](#orgc294fc5)).
+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)).
-
+
{{< 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](#orgc294fc5) shows the nodes used for the reduced Matlab model.
+Figure [13](#org0cedd1b) 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 +1087,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
-
+
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 +1202,14 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
-
+
-In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org7003388)).
+In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgc1d7ce0)).
### Actuator Description {#actuator-description}
-
+
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@@ -1231,9 +1231,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
-In Figure [15](#org472d510) are shown the principal nodes used for the model.
+In Figure [15](#orgf58efee) are shown the principal nodes used for the model.
-
+
{{< 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 +1352,11 @@ And we note:
G = zn * Gp;
```
-
+
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
-
+
{{< 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 +1454,13 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
-Let's plot the frequency of the modes (Figure [18](#org6e52a4a)).
+Let's plot the frequency of the modes (Figure [18](#org34eb51a)).
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@@ -1529,7 +1529,7 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@@ -1873,7 +1873,7 @@ Then, we compute the controllability and observability gramians.
And we plot the diagonal terms
-
+
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@@ -1891,7 +1891,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@@ -2137,6 +2137,6 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
## Bibliography {#bibliography}
-
Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
+
Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
-
Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
+
Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
diff --git a/content/book/horowitz15_art_of_elect_third_edition.md b/content/book/horowitz15_art_of_elect_third_edition.md
index d77789b..18e1ba8 100644
--- a/content/book/horowitz15_art_of_elect_third_edition.md
+++ b/content/book/horowitz15_art_of_elect_third_edition.md
@@ -1,6 +1,8 @@
+++
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"]
draft = false
+++
@@ -8,7 +10,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
Reference
-: ([Horowitz 2015](#orgfc7b505))
+: ([Horowitz 2015](#org7d6347d))
Author(s)
: Horowitz, P.
@@ -17,6 +19,7 @@ Year
: 2015
+
## Bibliography {#bibliography}
-
Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
+
Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
diff --git a/content/book/leach14_fundam_princ_engin_nanom.md b/content/book/leach14_fundam_princ_engin_nanom.md
index 2c9e6d7..73ab656 100644
--- a/content/book/leach14_fundam_princ_engin_nanom.md
+++ b/content/book/leach14_fundam_princ_engin_nanom.md
@@ -8,7 +8,7 @@ Tags
: [Metrology]({{< relref "metrology" >}})
Reference
-: ([Leach 2014](#org023e404))
+: ([Leach 2014](#orgdf2e918))
Author(s)
: Leach, R.
@@ -90,4 +90,4 @@ This type of angular interferometer is used to measure small angles (less than \
## Bibliography {#bibliography}
-
Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier.
.
+Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. .
diff --git a/content/book/leach18_basic_precis_engin_edition.md b/content/book/leach18_basic_precis_engin_edition.md
index 356733f..f02a6be 100644
--- a/content/book/leach18_basic_precis_engin_edition.md
+++ b/content/book/leach18_basic_precis_engin_edition.md
@@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
-: ([Leach and Smith 2018](#orgdc805b5))
+: ([Leach and Smith 2018](#org4f15d94))
Author(s)
: Leach, R., & Smith, S. T.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
-Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
+Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
diff --git a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
index 83b8960..7471cd8 100644
--- a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
+++ b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
Reference
-: ([Preumont 2018](#org29acb4a))
+: ([Preumont 2018](#org6703487))
Author(s)
: Preumont, A.
@@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
-
+
{{< 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](#orga09f785). 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](#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 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 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
-
+
{{< 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](#org57ee378).
+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 filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@@ -123,11 +123,11 @@ 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}
-
+
{{< 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](#org8ea735d).
+The various steps of the design of a controlled structure are shown in figure [3](#orgf360ea4).
The **starting point** is:
@@ -154,14 +154,14 @@ 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](#orga135390)):
+From the block diagram of the control system (figure [4](#orgdf35e26)):
\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\*}
-
+
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
@@ -186,12 +186,12 @@ 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](#orge835b98).
+A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3807050).
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.
-
+
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
@@ -398,11 +398,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
-
+
{{< 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](#orga21e5bb)).
+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)).
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 +441,9 @@ 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](#org4ad84e0)).
+\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgbd3bc07)).
-
+
{{< 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 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
-By looking at figure [7](#org4ad84e0), 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](#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.
-
+
{{< 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 +486,7 @@ 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](#org6f76f34). 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](#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}\\).
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 +508,12 @@ 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](#orgc872907)):
+The system consists of (see figure [10](#org459d27b)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
-
+
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
@@ -551,9 +551,9 @@ 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](#org4783db3)).
+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)).
-
+
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
@@ -583,9 +583,9 @@ with:
-Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1a21332)).
+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)).
-
+
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
@@ -610,7 +610,7 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
-
+
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
@@ -619,9 +619,9 @@ 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](#orgf2af0aa).
+The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgdcf2def).
-
+
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
@@ -646,7 +646,7 @@ 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](#orgeb5cb84) can be used.
+To do so, the bridge circuit as shown on figure [15](#orgd6dcc43) can be used.
We can show that
@@ -656,7 +656,7 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
-
+
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
@@ -664,9 +664,9 @@ 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](#org1e5bcfc) lists various effects that are observed in materials in response to various inputs.
+Figure [16](#org9608d58) lists various effects that are observed in materials in response to various inputs.
-
+
{{< 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 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
-If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgffdc1af)) 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](#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:
\begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
@@ -782,7 +782,7 @@ 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\\))
-
+
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
@@ -802,7 +802,7 @@ 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](#org890e9f3).
+Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org8b2066c).
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 +810,7 @@ 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}
-
+
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
@@ -844,10 +844,10 @@ 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](#org87aa6cd), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
+Consider the system of figure [19](#orgc7393d7), 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}\\).
-
+
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
@@ -866,9 +866,9 @@ 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](#org12f8fb9)).
+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)).
-
+
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
@@ -1566,7 +1566,7 @@ 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](#orgca40454).
+The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org288cbbb).
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 +1574,7 @@ 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)
-
+
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
@@ -1819,4 +1819,4 @@ This approach has the following advantages:
## Bibliography {#bibliography}
-
Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing.
.
+Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. .
diff --git a/content/book/schmidt14_desig_high_perfor_mechat_revis_edition.md b/content/book/schmidt14_desig_high_perfor_mechat_revis_edition.md
index 520256c..19edaa5 100644
--- a/content/book/schmidt14_desig_high_perfor_mechat_revis_edition.md
+++ b/content/book/schmidt14_desig_high_perfor_mechat_revis_edition.md
@@ -8,7 +8,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Reference
-: ([Schmidt, Schitter, and Rankers 2014](#orgaeaec45))
+: ([Schmidt, Schitter, and Rankers 2014](#orgddac163))
Author(s)
: Schmidt, R. M., Schitter, G., & Rankers, A.
@@ -29,7 +29,7 @@ Section 2.2.2 Force and Motion
> One should however be aware that another non-destructive source of non-linearity is found in a tried important field of mechanics, called _kinematics_.
> The relation between angles and positions is often non-linear in such a mechanism, because of the changing angles, and controlling these often requires special precautions to overcome the inherent non-linearities by linearisation around actual position and adapting the optimal settings of the controller to each position.
-
+
{{< figure src="/ox-hugo/schmidt14_high_low_freq_regions.png" caption="Figure 1: Stabiliby condition and robustness of a feedback controlled system. The desired shape of these curves guide the control design by optimising the lvels and sloppes of the amplitude Bode-plot at low and high frequencies for suppression of the disturbances and of the base Bode-plot in the cross-over frequency region. This is called **loop shaping design**" >}}
@@ -42,6 +42,7 @@ Section 9.3: Mass Dilemma
> A reduced mass requires improved system dynamics that enable a higher control bandwidth to compensate for the increase sensitivity for external vibrations.
+
## Bibliography {#bibliography}
-Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
+Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
diff --git a/content/book/skogestad07_multiv_feedb_contr.md b/content/book/skogestad07_multiv_feedb_contr.md
index 5f89b55..84cc990 100644
--- a/content/book/skogestad07_multiv_feedb_contr.md
+++ b/content/book/skogestad07_multiv_feedb_contr.md
@@ -54,7 +54,7 @@ draft = false
## Introduction {#introduction}
-
+
### The Process of Control System Design {#the-process-of-control-system-design}
@@ -231,7 +231,7 @@ Notations used throughout this note are summarized in tables [1](#table--tab:not
## Classical Feedback Control {#classical-feedback-control}
-
+
### Frequency Response {#frequency-response}
@@ -273,14 +273,14 @@ We note \\(N(\w\_0) = \left( \frac{d\ln{|G(j\w)|}}{d\ln{\w}} \right)\_{\w=\w\_0}
#### One Degree-of-Freedom Controller {#one-degree-of-freedom-controller}
-The simple one degree-of-freedom controller negative feedback structure is represented in Fig. [1](#orgf7f1eea).
+The simple one degree-of-freedom controller negative feedback structure is represented in Fig. [1](#orgc3b3d54).
The input to the controller \\(K(s)\\) is \\(r-y\_m\\) where \\(y\_m = y+n\\) is the measured output and \\(n\\) is the measurement noise.
Thus, the input to the plant is \\(u = K(s) (r-y-n)\\).
The objective of control is to manipulate \\(u\\) (design \\(K\\)) such that the control error \\(e\\) remains small in spite of disturbances \\(d\\).
The control error is defined as \\(e = y-r\\).
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_alt.png" caption="Figure 1: Configuration for one degree-of-freedom control" >}}
@@ -599,18 +599,18 @@ For reference tracking, we typically want the controller to look like \\(\frac{1
We cannot achieve both of these simultaneously with a single feedback controller.
-The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller (Fig. [2](#org8614c12)), rather than operating on their difference \\(r - y\_m\\).
+The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller (Fig. [2](#org79dc25d)), rather than operating on their difference \\(r - y\_m\\).
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="Figure 2: 2 degrees-of-freedom control architecture" >}}
-The controller can be slit into two separate blocks (Fig. [3](#org49e93e8)):
+The controller can be slit into two separate blocks (Fig. [3](#orge988411)):
- the **feedback controller** \\(K\_y\\) that is used to **reduce the effect of uncertainty** (disturbances and model errors)
- the **prefilter** \\(K\_r\\) that **shapes the commands** \\(r\\) to improve tracking performance
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_sep.png" caption="Figure 3: 2 degrees-of-freedom control architecture with two separate blocs" >}}
@@ -681,7 +681,7 @@ Which can be expressed as an \\(\mathcal{H}\_\infty\\):
W\_P(s) = \frac{s/M + \w\_B^\*}{s + \w\_B^\* A}
\end{equation\*}
-With (see Fig. [4](#org3e8c784)):
+With (see Fig. [4](#org8c865c1)):
- \\(M\\): maximum magnitude of \\(\abs{S}\\)
- \\(\w\_B\\): crossover frequency
@@ -689,7 +689,7 @@ With (see Fig. [4](#org3e8c784)):
-
+
{{< figure src="/ox-hugo/skogestad07_weight_first_order.png" caption="Figure 4: Inverse of performance weight" >}}
@@ -723,7 +723,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
-
+
### Introduction {#introduction}
@@ -760,13 +760,13 @@ The main rule for evaluating transfer functions is the **MIMO Rule**: Start from
#### Negative Feedback Control Systems {#negative-feedback-control-systems}
-For negative feedback system (Fig. [5](#orgd170e54)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant:
+For negative feedback system (Fig. [5](#org94ab3db)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant:
- \\(L = G K\\)
- \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\)
- \\(T \triangleq L(I + L)^{-1}\\) is the transfer function from \\(r\\) to \\(y\\)
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_bis.png" caption="Figure 5: Conventional negative feedback control system" >}}
@@ -1124,9 +1124,9 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
### General Control Problem Formulation {#general-control-problem-formulation}
-The general control problem formulation is represented in Fig. [6](#orgf26fedd) (introduced in ([Doyle 1983](#org4d36284))).
+The general control problem formulation is represented in Fig. [6](#org59185be) (introduced in ([Doyle 1983](#org76415c5))).
-
+
{{< figure src="/ox-hugo/skogestad07_general_control_names.png" caption="Figure 6: General control configuration" >}}
@@ -1157,13 +1157,13 @@ Then we have to break all the "loops" entering and exiting the controller \\(K\\
#### Controller Design: Including Weights in \\(P\\) {#controller-design-including-weights-in--p}
-In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) (Fig. [7](#orge0e26e7)).
+In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) (Fig. [7](#org451fe53)).
We consider:
- The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system)
- The weighted or normalized controlled outputs \\(z = W\_z \tilde{z}\\) (where \\(\tilde{z}\\) often consists of the control error \\(y-r\\) and the manipulated input \\(u\\))
-
+
{{< figure src="/ox-hugo/skogestad07_general_plant_weights.png" caption="Figure 7: General Weighted Plant" >}}
@@ -1216,9 +1216,9 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
#### A General Control Configuration Including Model Uncertainty {#a-general-control-configuration-including-model-uncertainty}
-The general control configuration may be extended to include model uncertainty as shown in Fig. [8](#orgda7871a).
+The general control configuration may be extended to include model uncertainty as shown in Fig. [8](#orge2629d4).
-
+
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta.png" caption="Figure 8: General control configuration for the case with model uncertainty" >}}
@@ -1246,7 +1246,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
## Elements of Linear System Theory {#elements-of-linear-system-theory}
-
+
### System Descriptions {#system-descriptions}
@@ -1619,18 +1619,18 @@ RHP-zeros therefore imply high gain instability.
### Internal Stability of Feedback Systems {#internal-stability-of-feedback-systems}
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="Figure 9: Block diagram used to check internal stability" >}}
-Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in Fig. [9](#orgc94e8c5) is **internally stable** if and only if all four closed-loop transfer matrices are stable.
+Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in Fig. [9](#org712f7b7) is **internally stable** if and only if all four closed-loop transfer matrices are stable.
\begin{align\*}
&(I+KG)^{-1} & -K&(I+GK)^{-1} \\\\\\
G&(I+KG)^{-1} & &(I+GK)^{-1}
\end{align\*}
-Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in Fig. [9](#orgc94e8c5) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable.
+Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in Fig. [9](#org712f7b7) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable.
### Stabilizing Controllers {#stabilizing-controllers}
@@ -1797,7 +1797,7 @@ It may be shown that the Hankel norm is equal to \\(\left\\|G(s)\right\\|\_H = \
## Limitations on Performance in SISO Systems {#limitations-on-performance-in-siso-systems}
-
+
### Input-Output Controllability {#input-output-controllability}
@@ -2283,11 +2283,11 @@ Uncertainty in the crossover frequency region can result in poor performance and
### Summary: Controllability Analysis with Feedback Control {#summary-controllability-analysis-with-feedback-control}
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="Figure 10: Feedback control system" >}}
-Consider the control system in Fig. [10](#org506ba18).
+Consider the control system in Fig. [10](#org334c341).
Here \\(G\_m(s)\\) denotes the measurement transfer function and we assume \\(G\_m(0) = 1\\) (perfect steady-state measurement).
@@ -2317,7 +2317,7 @@ Sometimes, the disturbances are so large that we hit input saturation or the req
\abs{G\_d(j\w)} < 1 \quad \forall \w \geq \w\_c
\end{equation\*}
-
+
{{< figure src="/ox-hugo/skogestad07_margin_requirements.png" caption="Figure 11: Illustration of controllability requirements" >}}
@@ -2339,7 +2339,7 @@ The rules may be used to **determine whether or not a given plant is controllabl
## Limitations on Performance in MIMO Systems {#limitations-on-performance-in-mimo-systems}
-
+
### Introduction {#introduction}
@@ -2719,13 +2719,13 @@ The issues are the same for SISO and MIMO systems, however, with MIMO systems th
In practice, the difference between the true perturbed plant \\(G^\prime\\) and the plant model \\(G\\) is caused by a number of different sources.
We here focus on input and output uncertainty.
-In multiplicative form, the input and output uncertainties are given by (see Fig. [12](#org2e2a888)):
+In multiplicative form, the input and output uncertainties are given by (see Fig. [12](#org25f8203)):
\begin{equation\*}
G^\prime = (I + E\_O) G (I + E\_I)
\end{equation\*}
-
+
{{< figure src="/ox-hugo/skogestad07_input_output_uncertainty.png" caption="Figure 12: Plant with multiplicative input and output uncertainty" >}}
@@ -2869,7 +2869,7 @@ However, the situation is usually the opposite with model uncertainty because fo
## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
-
+
### Introduction to Robustness {#introduction-to-robustness}
@@ -2943,11 +2943,11 @@ In most cases, we prefer to lump the uncertainty into a **multiplicative uncerta
G\_p(s) = G(s)(1 + w\_I(s)\Delta\_I(s)); \quad \abs{\Delta\_I(j\w)} \le 1 \, \forall\w
\end{equation\*}
-which may be represented by the diagram in Fig. [13](#orgf804642).
+which may be represented by the diagram in Fig. [13](#org4751b9a).
-
+
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set.png" caption="Figure 13: Plant with multiplicative uncertainty" >}}
@@ -3012,7 +3012,7 @@ This is of course conservative as it introduces possible plants that are not pre
#### Uncertain Regions {#uncertain-regions}
-To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in Fig. [14](#org56df3ef) the Nyquist plots generated by the following set of plants
+To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in Fig. [14](#orgd3a5a8c) the Nyquist plots generated by the following set of plants
\begin{equation\*}
G\_p(s) = \frac{k}{\tau s + 1} e^{-\theta s}, \quad 2 \le k, \theta, \tau \le 3
@@ -3022,7 +3022,7 @@ To illustrate how parametric uncertainty translate into frequency domain uncerta
In general, these uncertain regions have complicated shapes and complex mathematical descriptions
- **Step 2**. We therefore approximate such complex regions as discs, resulting in a **complex additive uncertainty description**
-
+
{{< figure src="/ox-hugo/skogestad07_uncertainty_region.png" caption="Figure 14: Uncertainty regions of the Nyquist plot at given frequencies" >}}
@@ -3041,11 +3041,11 @@ The disc-shaped regions may be generated by **additive** complex norm-bounded pe
\end{aligned}
\end{equation}
-At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in Fig. [15](#org53103ab).
+At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in Fig. [15](#orgbca4972).
@@ -6098,7 +6098,7 @@ A cascade control structure results when either of the following two situations
|  |  |
|-------------------------------------------------------|---------------------------------------------------|
-|
Extra measurements \\(y\_2\\) |
Extra inputs \\(u\_2\\) |
+|
Extra measurements \\(y\_2\\) |
Extra inputs \\(u\_2\\) |
#### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
@@ -6122,7 +6122,7 @@ where in most cases \\(r\_2 = 0\\) since we do not have a degree-of-freedom to c
##### Cascade implementation {#cascade-implementation}
-To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in Fig. [7](#org1d28d6d):
+To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in Fig. [7](#orgd4f245b):
\begin{align\*}
r\_2 &= K\_1(s)(r\_1 - y\_1) \\\\\\
@@ -6132,13 +6132,13 @@ To obtain an implementation with two SISO controllers, we may cascade the contro
Note that the output \\(r\_2\\) from the slower primary controller \\(K\_1\\) is not a manipulated plant input, but rather the reference input to the faster secondary controller \\(K\_2\\).
Cascades based on measuring the actual manipulated variable (\\(y\_2 = u\_m\\)) are commonly used to **reduce uncertainty and non-linearity at the plant input**.
-In the general case (Fig. [7](#org1d28d6d)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
-However, it is common to encounter the situation in Fig. [59](#org0106b04) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of Fig. [7](#org1d28d6d).
+In the general case (Fig. [7](#orgd4f245b)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
+However, it is common to encounter the situation in Fig. [59](#orgc265822) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of Fig. [7](#orgd4f245b).
-With reference to the special (but common) case of cascade control shown in Fig. [59](#org0106b04), the use of **extra measurements** is useful under the following circumstances:
+With reference to the special (but common) case of cascade control shown in Fig. [59](#orgc265822), the use of **extra measurements** is useful under the following circumstances:
- The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase.
If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\)
@@ -6147,7 +6147,7 @@ With reference to the special (but common) case of cascade control shown in Fig.
-
+
{{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="Figure 59: Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
@@ -6175,7 +6175,7 @@ Then \\(u\_2(t)\\) will only be used for **transient control** and will return t
##### Cascade implementation {#cascade-implementation}
-To obtain an implementation with two SISO controllers we may cascade the controllers as shown in Fig. [7](#org6c3853f).
+To obtain an implementation with two SISO controllers we may cascade the controllers as shown in Fig. [7](#org1299e37).
We again let input \\(u\_2\\) take care of the **fast control** and \\(u\_1\\) of the **long-term control**.
The fast control loop is then
@@ -6197,7 +6197,7 @@ It also shows more clearly that \\(r\_{u\_2}\\), the reference for \\(u\_2\\), m
-Consider the system in Fig. [60](#org9e7696a) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
+Consider the system in Fig. [60](#org96ff8c5) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)).
Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\).
The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\).
@@ -6209,7 +6209,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
-
+
{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}}
@@ -6313,7 +6313,7 @@ By partitioning the inputs and outputs, the overall model \\(y = G u\\) can be w
\end{aligned}
\end{equation}
-Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) (Fig. [61](#org06b827b)).
+Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) (Fig. [61](#org5a1c6ed)).
We get:
\begin{equation} \label{eq:partial\_control}
@@ -6324,7 +6324,7 @@ We get:
\end{aligned}
\end{equation}
-
+
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
@@ -6383,7 +6383,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
##### Sequential design of cascade control systems {#sequential-design-of-cascade-control-systems}
-Consider the conventional cascade control system in Fig. [7](#org1d28d6d) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
+Consider the conventional cascade control system in Fig. [7](#orgd4f245b) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
From \eqref{eq:partial_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
@@ -6453,9 +6453,9 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
### Decentralized Feedback Control {#decentralized-feedback-control}
-In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [62](#org4a88902)).
+In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [62](#org534179d)).
-
+
{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
@@ -6855,7 +6855,7 @@ The conditions are also useful in an **input-output controllability analysis** f
## Model Reduction {#model-reduction}
-
+
### Introduction {#introduction}
@@ -7285,4 +7285,4 @@ In such a case, using truncation or optimal Hankel norm approximation with appro
## Bibliography {#bibliography}
-
Doyle, John C. 1983. “Synthesis of Robust Controllers and Filters.” In _The 22nd IEEE Conference on Decision and Control_, 109–14. IEEE.
+
Doyle, John C. 1983. “Synthesis of Robust Controllers and Filters.” In _The 22nd IEEE Conference on Decision and Control_, 109–14. IEEE.
diff --git a/content/book/taghirad13_paral.md b/content/book/taghirad13_paral.md
index 481bdab..1102659 100644
--- a/content/book/taghirad13_paral.md
+++ b/content/book/taghirad13_paral.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Reference Books]({{< relref "reference_books" >}})
Reference
-: ([Taghirad 2013](#org128f66e))
+: ([Taghirad 2013](#org5caa795))
Author(s)
: Taghirad, H.
@@ -22,7 +22,7 @@ PDF version
## Introduction {#introduction}
-
+
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 +47,14 @@ The control of parallel robots is elaborated in the last two chapters, in which
## Motion Representation {#motion-representation}
-
+
### 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](#org5836a85).
+Consider a rigid body in a spatial motion as represented in Figure [1](#org25b870d).
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 +62,7 @@ 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}\\}\\).
-
+
{{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
@@ -87,7 +87,7 @@ 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](#org808f5cb)).
+Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#orgd6c2a37)).
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
@@ -109,7 +109,7 @@ 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}\\}\\).
-
+
{{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
@@ -135,7 +135,7 @@ 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\\).
-
+
{{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
@@ -161,7 +161,7 @@ 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.
-
+
{{< 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 +364,10 @@ 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](#orgda91190)).
+Let us consider the motion of a rigid body described at three locations (Figure [5](#org2fa078f)).
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.
-
+
{{< 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 +430,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
## Kinematics {#kinematics}
-
+
### Introduction {#introduction}
@@ -537,11 +537,11 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
\end{bmatrix}
\end{equation}
-
+
{{< 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](#org3a7000d).
+The geometry of the manipulator is shown Figure [6](#org2b43912).
#### Inverse Kinematics {#inverse-kinematics}
@@ -590,7 +590,7 @@ The complexity of the problem depends widely on the manipulator architecture and
## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
-
+
### Introduction {#introduction}
@@ -685,9 +685,9 @@ 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](#org48af385), 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](#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}\\}\\).
-
+
{{< 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 +949,9 @@ 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](#orga0c6cd5), 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.
+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.
-
+
{{< 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 +1108,9 @@ 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](#org1aa505c)) 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](#orge167af1)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
-
+
{{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
@@ -1140,7 +1140,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
## Dynamics {#dynamics}
-
+
### Introduction {#introduction}
@@ -1239,7 +1239,7 @@ 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.
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](#org48af385)).
+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)).
To determine acceleration of point \\(P\\), we start with the relation between absolute and relative velocities of point \\(P\\):
\begin{equation}
@@ -1272,7 +1272,7 @@ 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.
-
+
{{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
@@ -1364,7 +1364,7 @@ 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](#orgd93c822), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
+Linear momentum of a material body, shown in Figure [11](#orgf0e919a), 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 +1386,14 @@ 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.
-
+
{{< 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](#orgd93c822).
+Consider the solid body represented in Figure [11](#orgf0e919a).
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 +1523,7 @@ 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](#orgd839e6c), 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](#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}\\).
The position vector of these two center of masses can be determined by the following equations:
\begin{align}
@@ -1531,7 +1531,7 @@ 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}
-
+
{{< 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 +1558,7 @@ 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](#orgd839e6c).
+The free-body diagrams of the limbs and the moving platforms is given in Figure [12](#org8ad224c).
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 +1586,7 @@ 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](#orgd839e6c).
+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).
The Newton-Euler formulation of the moving platform is as follows:
@@ -1745,9 +1745,9 @@ in which
##### Forward Dynamics Simulations {#forward-dynamics-simulations}
-As shown in Figure [13](#org7b2216f), 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](#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**.
-
+
{{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="Figure 13: Flowchart of forward dynamics implementation sequence" >}}
@@ -1758,7 +1758,7 @@ 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](#org3acba0f), inverse dynamic formulation is implemented in the following sequence.
+As illustrated in Figure [14](#orgd3aaf90), 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 +1780,7 @@ 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}
-
+
{{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
@@ -1805,7 +1805,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
## Motion Control {#motion-control}
-
+
### Introduction {#introduction}
@@ -1826,7 +1826,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
### Controller Topology {#controller-topology}
-
+
@@ -1871,11 +1871,11 @@ 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.
-Figure [15](#org868a832) 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](#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.
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.
-
+
{{< 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 +1883,9 @@ 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**.
-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](#org373a3c7) 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](#orga6b318d) to implement such a controller.
-
+
{{< 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 +1894,26 @@ 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.
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](#org373a3c7), another control topology is usually implemented for parallel manipulators.
+To overcome the implementation problem of the control topology in Figure [16](#orga6b318d), another control topology is usually implemented for parallel manipulators.
-In this topology, depicted in Figure [17](#org1ae20e4), 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](#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\\).
Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
-
+
{{< 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](#org1ae20e4) are **both in the joint space**.
+The **input and output** of the controller depicted in Figure [17](#orgf913d55) 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](#org1ae20e4), **independent controllers** for each joint may be suitable.
+For the topology in Figure [17](#orgf913d55), **independent controllers** for each joint may be suitable.
-To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#org08e1864).
+To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#orgac91dcd).
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}\\).
-
+
{{< 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 +1923,16 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
### Motion Control in Task Space {#motion-control-in-task-space}
-
+
#### Decentralized PD Control {#decentralized-pd-control}
-In the control structure in Figure [19](#org59c122f), a number of linear PD controllers are used in a feedback structure on each error component.
+In the control structure in Figure [19](#orgdb03b09), 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.
-
+
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
@@ -1951,11 +1951,11 @@ 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](#org48b2525).
+A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [20](#orgf1d1d54).
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) \\]
-
+
{{< 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 +2011,14 @@ By this means, **nonlinear and coupling behavior of the robotic manipulator is s
-General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#org8872c95).
+General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#org32f9766).
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.
-
+
{{< 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 +2138,14 @@ 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.
-
+
{{< 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}
-
+
{{< 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 +2158,7 @@ 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.
-To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#org373a3c7).
+To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#orga6b318d).
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**.
The general form of dynamic formulation of parallel robot is usually given in the task space.
@@ -2217,7 +2217,7 @@ 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}\\).
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](#org373a3c7), and implement control law design in the task space.
+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.
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 +2226,11 @@ 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](#orgecf2422), a number of PD controllers are used in a feedback structure on each error component.
+In this control structure, depicted in Figure [24](#orge46fe49), 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.
-
+
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
@@ -2250,9 +2250,9 @@ 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](#orgdbbacb2).
+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).
-
+
{{< 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 +2293,14 @@ By this means, the **nonlinear and coupling characteristics** of robotic manipul