+
In order to get dominance at low frequencies, the hexapod must be designed so that:
@@ -206,7 +206,7 @@ By satisfying \eqref{eq:cond_stiff}, \\(f\_p\\) can be aligned with the strut fo
At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component:
\\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\]
-
+
To ensure that the control system acts only in the band of frequencies where dominance is retained, the control bandwidth can be selected so that:
@@ -225,7 +225,7 @@ In this case, it is reasonable to use:
\text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
\end{equation}
-
+
By designing the flexure jointed hexapod and its controller so that both \eqref{eq:cond_stiff} and \eqref{eq:cond_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
@@ -271,6 +271,6 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
## Bibliography {#bibliography}
-
McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil.
.
+McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. .
-———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. .
+———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. .
diff --git a/content/article/mcinroy99_dynam.md b/content/article/mcinroy99_dynam.md
index 26bb54e..f06f9eb 100644
--- a/content/article/mcinroy99_dynam.md
+++ b/content/article/mcinroy99_dynam.md
@@ -4,7 +4,7 @@ author = ["Thomas Dehaeze"]
draft = false
+++
-### Backlinks {#backlinks}
+Backlinks:
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
@@ -12,7 +12,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
-: ([McInroy 1999](#org5efb28a))
+: ([McInroy 1999](#orgfc7fa52))
Author(s)
: McInroy, J.
@@ -20,7 +20,7 @@ Author(s)
Year
: 1999
-This conference paper has been further published in a journal as a short note ([McInroy 2002](#org4990a96)).
+This conference paper has been further published in a journal as a short note ([McInroy 2002](#org7752c60)).
## Abstract {#abstract}
@@ -42,22 +42,22 @@ The actuators for FJHs can be divided into two categories:
1. soft (voice coil), which employs a spring flexure mount
2. hard (piezoceramic or magnetostrictive), which employs a compressive load spring.
-
+
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: A general Stewart Platform" >}}
-Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org6b356c7).
+Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org26f1840).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
-
+
{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="Figure 2: The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
Table 1:
- Definition of quantities on Figure
2
+ Definition of quantities on Figure
2
| **Symbol** | **Meaning** |
@@ -74,11 +74,11 @@ In order to provide low frequency passive vibration isolation, the hard actuator
| \\(v\_i = p\_i - q\_i\\) | vector pointing from the bottom to the top |
| \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut |
-It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org4990a96))).
+It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org7752c60))).
This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used.
This allows to reduce considerably the complexity of the model.
-From Figure [2](#org6b356c7) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
+From Figure [2](#org26f1840) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
\begin{equation}
m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i
@@ -168,6 +168,6 @@ In the next section, a connection between the two will be found to complete the
## Bibliography {#bibliography}
-McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. .
+McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. .
-———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. .
+———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. .
diff --git a/content/article/oomen18_advan_motion_contr_precis_mechat.md b/content/article/oomen18_advan_motion_contr_precis_mechat.md
index 1a09bbe..507063b 100644
--- a/content/article/oomen18_advan_motion_contr_precis_mechat.md
+++ b/content/article/oomen18_advan_motion_contr_precis_mechat.md
@@ -8,7 +8,7 @@ Tags
: [Motion Control]({{< relref "motion_control" >}})
Reference
-: ([Oomen 2018](#orga862567))
+: ([Oomen 2018](#org18923fa))
Author(s)
: Oomen, T.
@@ -16,11 +16,11 @@ Author(s)
Year
: 2018
-
+
{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}}
## Bibliography {#bibliography}
-Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. .
+Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. .
diff --git a/content/article/poel10_explor_activ_hard_mount_vibrat.md b/content/article/poel10_explor_activ_hard_mount_vibrat.md
index 21d896e..65571ba 100644
--- a/content/article/poel10_explor_activ_hard_mount_vibrat.md
+++ b/content/article/poel10_explor_activ_hard_mount_vibrat.md
@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Poel 2010](#org0c02ed0))
+: ([Poel 2010](#org913a900))
Author(s)
: van der Poel, G. W.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
-Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. .
+Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. .
diff --git a/content/article/preumont02_force_feedb_versus_accel_feedb.md b/content/article/preumont02_force_feedb_versus_accel_feedb.md
index eb164af..5add28b 100644
--- a/content/article/preumont02_force_feedb_versus_accel_feedb.md
+++ b/content/article/preumont02_force_feedb_versus_accel_feedb.md
@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Preumont et al. 2002](#org1f14c5d))
+: ([Preumont et al. 2002](#org35c3663))
Author(s)
: Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
@@ -26,14 +26,14 @@ The force applied to a **rigid body** is proportional to its acceleration, thus
Thus force feedback and acceleration feedback are equivalent for solid bodies.
When there is a flexible payload, the two sensing options are not longer equivalent.
-- For light payload (Figure [1](#orgbbc4740)), the acceleration feedback gives larger damping on the higher mode.
-- For heavy payload (Figure [2](#orgdf8d8cc)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
+- For light payload (Figure [1](#org93ee8cd)), the acceleration feedback gives larger damping on the higher mode.
+- For heavy payload (Figure [2](#orgec90b36)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
-
+
{{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="Figure 1: Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
-
+
{{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="Figure 2: Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
@@ -48,4 +48,4 @@ The same is true for the transfer function from the force actuator to the relati
## Bibliography {#bibliography}
-Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback Versus Acceleration Feedback in Active Vibration Isolation.” _Journal of Sound and Vibration_ 257 (4):605–13. .
+Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback Versus Acceleration Feedback in Active Vibration Isolation.” _Journal of Sound and Vibration_ 257 (4):605–13. .
diff --git a/content/article/preumont07_six_axis_singl_stage_activ.md b/content/article/preumont07_six_axis_singl_stage_activ.md
index b2e321c..56b1af3 100644
--- a/content/article/preumont07_six_axis_singl_stage_activ.md
+++ b/content/article/preumont07_six_axis_singl_stage_activ.md
@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
-: ([Preumont et al. 2007](#orgce12c3f))
+: ([Preumont et al. 2007](#org3b370bf))
Author(s)
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
@@ -18,34 +18,34 @@ Year
Summary:
-- **Cubic** Stewart platform (Figure [3](#orgb093d6d))
+- **Cubic** Stewart platform (Figure [3](#org0274a43))
- Provides uniform control capability
- Uniform stiffness in all directions
- minimizes the cross-coupling among actuators and sensors of different legs
-- Flexible joints (Figure [2](#org2531c05))
+- Flexible joints (Figure [2](#orge9ceb91))
- Piezoelectric force sensors
- Voice coil actuators
- Decentralized feedback control approach for vibration isolation
-- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgffaf2cb))
+- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgb3c0578))
- The Stewart platform has 6 suspension modes at different frequencies.
Thus the gain of the IFF controller cannot be optimal for all the modes.
It is better if all the modes of the platform are near to each other.
- Discusses the design of the legs in order to maximize the natural frequency of the local modes.
- To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix
-
+
{{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="Figure 1: Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
-
+
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
-
+
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
## Bibliography {#bibliography}
-Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. .
+Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. .
diff --git a/content/article/saxena12_advan_inter_model_contr_techn.md b/content/article/saxena12_advan_inter_model_contr_techn.md
index c1ae1ae..c8dd637 100644
--- a/content/article/saxena12_advan_inter_model_contr_techn.md
+++ b/content/article/saxena12_advan_inter_model_contr_techn.md
@@ -8,7 +8,7 @@ Tags
: [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}})
Reference
-: ([Saxena and Hote 2012](#org7fbcfc6))
+: ([Saxena and Hote 2012](#org0284b71))
Author(s)
: Saxena, S., & Hote, Y.
@@ -87,4 +87,4 @@ The interesting feature regarding IMC is that the design scheme is identical to
## Bibliography {#bibliography}
-Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. .
+Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. .
diff --git a/content/article/sayed01_survey_spect_factor_method.md b/content/article/sayed01_survey_spect_factor_method.md
index e967a9e..086432f 100644
--- a/content/article/sayed01_survey_spect_factor_method.md
+++ b/content/article/sayed01_survey_spect_factor_method.md
@@ -9,7 +9,7 @@ Tags
Reference
-: ([Sayed and Kailath 2001](#org48c8405))
+: ([Sayed and Kailath 2001](#orgaf03ac4))
Author(s)
: Sayed, A. H., & Kailath, T.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
-Sayed, A. H., and T. Kailath. 2001. “A Survey of Spectral Factorization Methods.” _Numerical Linear Algebra with Applications_ 8 (6-7):467–96. .
+Sayed, A. H., and T. Kailath. 2001. “A Survey of Spectral Factorization Methods.” _Numerical Linear Algebra with Applications_ 8 (6-7):467–96. .
diff --git a/content/article/schellekens98_desig_precis.md b/content/article/schellekens98_desig_precis.md
index 50dcf17..34e75a6 100644
--- a/content/article/schellekens98_desig_precis.md
+++ b/content/article/schellekens98_desig_precis.md
@@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
-: ([Schellekens et al. 1998](#org5a7db02))
+: ([Schellekens et al. 1998](#orge27f1a2))
Author(s)
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
-Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” _Cirp Annals_, no. 2:557–86. 63243-0.
+Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” _Cirp Annals_, no. 2:557–86. 63243-0.
diff --git a/content/article/schroeck01_compen_desig_linear_time_invar.md b/content/article/schroeck01_compen_desig_linear_time_invar.md
index ced8f4e..b6a4f4c 100644
--- a/content/article/schroeck01_compen_desig_linear_time_invar.md
+++ b/content/article/schroeck01_compen_desig_linear_time_invar.md
@@ -9,7 +9,7 @@ Tags
Reference
-: ([Schroeck, Messner, and McNab 2001](#orgd39dd43))
+: ([Schroeck, Messner, and McNab 2001](#orgf8182bc))
Author(s)
: Schroeck, S., Messner, W., & McNab, R.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
-Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. .
+Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. .
diff --git a/content/article/sebastian12_nanop_with_multip_sensor.md b/content/article/sebastian12_nanop_with_multip_sensor.md
index 0469753..f4ae452 100644
--- a/content/article/sebastian12_nanop_with_multip_sensor.md
+++ b/content/article/sebastian12_nanop_with_multip_sensor.md
@@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference
-: ([Sebastian and Pantazi 2012](#org3eeb20a))
+: ([Sebastian and Pantazi 2012](#org03bb39f))
Author(s)
: Sebastian, A., & Pantazi, A.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
-Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. .
+Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. .
diff --git a/content/article/souleille18_concep_activ_mount_space_applic.md b/content/article/souleille18_concep_activ_mount_space_applic.md
index 763e291..b77c40a 100644
--- a/content/article/souleille18_concep_activ_mount_space_applic.md
+++ b/content/article/souleille18_concep_activ_mount_space_applic.md
@@ -8,7 +8,7 @@ Tags
: [Active Damping]({{< relref "active_damping" >}})
Reference
-: ([Souleille et al. 2018](#orgaa465de))
+: ([Souleille et al. 2018](#org5546d0c))
Author(s)
: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C.
@@ -23,10 +23,10 @@ This article discusses the use of Integral Force Feedback with amplified piezoel
## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
-Figure [1](#org024c118) shows a picture of the amplified piezoelectric stack.
+Figure [1](#org8634178) shows a picture of the amplified piezoelectric stack.
The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor.
-
+
{{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
@@ -61,38 +61,38 @@ and the control force is given by:
f = F\_s G(s) = F\_s \frac{g}{s}
\end{equation}
-The effect of the controller are shown in Figure [2](#orge334eeb):
+The effect of the controller are shown in Figure [2](#orgcb733df):
- the resonance peak is almost critically damped
- the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
- the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\)
- the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies
-
+
{{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="Figure 2: Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}}
-
+
{{< figure src="/ox-hugo/souleille18_root_locus.png" caption="Figure 3: Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}}
## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators}
-A heavy payload is mounted on a set of three isolators (Figure [4](#orga53fff7)).
+A heavy payload is mounted on a set of three isolators (Figure [4](#org09ac00a)).
The payload consists of two masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz.
-
+
{{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="Figure 4: Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}}
-As shown in Figure [5](#orge070a1d), both the suspension modes and the flexible modes of the payload can be critically damped.
+As shown in Figure [5](#org2dcbc51), both the suspension modes and the flexible modes of the payload can be critically damped.
-
+
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
## Bibliography {#bibliography}
-Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65.
+Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65.
diff --git a/content/article/spanos95_soft_activ_vibrat_isolat.md b/content/article/spanos95_soft_activ_vibrat_isolat.md
index 4a62935..b1bb3c1 100644
--- a/content/article/spanos95_soft_activ_vibrat_isolat.md
+++ b/content/article/spanos95_soft_activ_vibrat_isolat.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Spanos, Rahman, and Blackwood 1995](#orgad76892))
+: ([Spanos, Rahman, and Blackwood 1995](#orgfa35a11))
Author(s)
: Spanos, J., Rahman, Z., & Blackwood, G.
@@ -16,14 +16,14 @@ Author(s)
Year
: 1995
-**Stewart Platform** (Figure [1](#org6d10ec2)):
+**Stewart Platform** (Figure [1](#org04a652d)):
- Voice Coil
- Flexible joints (cross-blades)
- Force Sensors
- Cubic Configuration
-
+
{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
@@ -41,7 +41,7 @@ After redesign of the struts:
- low frequency zero at 2.6Hz but non-minimum phase (not explained).
Small viscous damping material in the cross blade flexures made the zero minimum phase again.
-
+
{{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="Figure 2: Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
@@ -52,13 +52,13 @@ The controller used consisted of:
- first order lag filter to provide adequate phase margin at the low frequency crossover
- a first order high pass filter to attenuate the excess gain resulting from the low frequency zero
-The results in terms of transmissibility are shown in Figure [3](#orgec62915).
+The results in terms of transmissibility are shown in Figure [3](#org3083c42).
-
+
{{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
## Bibliography {#bibliography}
-Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In _Proceedings of 1995 American Control Conference - ACC’95_, nil. .
+Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In _Proceedings of 1995 American Control Conference - ACC’95_, nil. .
diff --git a/content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md b/content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
index f6a4b27..396031c 100644
--- a/content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
+++ b/content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
@@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference
-: ([Stankevic et al. 2017](#org7b0b97a))
+: ([Stankevic et al. 2017](#org5a36a6f))
Author(s)
: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
@@ -19,7 +19,7 @@ Year
- Similar Station than the NASS
- Similar Metrology with fiber based interferometers and cylindrical reference mirror
-
+
{{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
@@ -32,4 +32,4 @@ Year
## Bibliography {#bibliography}
-Stankevic, Tomas, Christer Engblom, Florent Langlois, Filipe Alves, Alain Lestrade, Nicolas Jobert, Gilles Cauchon, Ulrich Vogt, and Stefan Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” _Review of Scientific Instruments_ 88 (5):053703. .
+Stankevic, Tomas, Christer Engblom, Florent Langlois, Filipe Alves, Alain Lestrade, Nicolas Jobert, Gilles Cauchon, Ulrich Vogt, and Stefan Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” _Review of Scientific Instruments_ 88 (5):053703. .
diff --git a/content/article/tang18_decen_vibrat_contr_voice_coil.md b/content/article/tang18_decen_vibrat_contr_voice_coil.md
index 1b72ea2..57e1a89 100644
--- a/content/article/tang18_decen_vibrat_contr_voice_coil.md
+++ b/content/article/tang18_decen_vibrat_contr_voice_coil.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
-: ([Tang, Cao, and Yu 2018](#orge54a32c))
+: ([Tang, Cao, and Yu 2018](#org45ebb6f))
Author(s)
: Tang, J., Cao, D., & Yu, T.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
-Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ 233 (1):132–45. .
+Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ 233 (1):132–45. .
diff --git a/content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md b/content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
index 8843722..e9774d9 100644
--- a/content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
+++ b/content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
@@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Tjepkema, Dijk, and Soemers 2012](#org04bf993))
+: ([Tjepkema, Dijk, and Soemers 2012](#org349e155))
Author(s)
: Tjepkema, D., Dijk, J. v., & Soemers, H.
@@ -49,4 +49,4 @@ There is a compromise between sensor noise and the influence of the sensor size
## Bibliography {#bibliography}
-Tjepkema, D., J. van Dijk, and H.M.J.R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” _Journal of Sound and Vibration_ 331 (4):735–49. .
+Tjepkema, D., J. van Dijk, and H.M.J.R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” _Journal of Sound and Vibration_ 331 (4):735–49. .
diff --git a/content/article/wang12_autom_marker_full_field_hard.md b/content/article/wang12_autom_marker_full_field_hard.md
index 216fcff..e822ae8 100644
--- a/content/article/wang12_autom_marker_full_field_hard.md
+++ b/content/article/wang12_autom_marker_full_field_hard.md
@@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
Reference
-: ([Wang et al. 2012](#orgc06402b))
+: ([Wang et al. 2012](#org187cf70))
Author(s)
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
@@ -28,4 +28,4 @@ It uses calibrated metrology disc and capacitive sensors
## Bibliography {#bibliography}
-Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. .
+Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. .
diff --git a/content/article/wang16_inves_activ_vibrat_isolat_stewar.md b/content/article/wang16_inves_activ_vibrat_isolat_stewar.md
index ad77a69..946944f 100644
--- a/content/article/wang16_inves_activ_vibrat_isolat_stewar.md
+++ b/content/article/wang16_inves_activ_vibrat_isolat_stewar.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
-: ([Wang et al. 2016](#orgf512901))
+: ([Wang et al. 2016](#orgda82aa7))
Author(s)
: Wang, C., Xie, X., Chen, Y., & Zhang, Z.
@@ -25,7 +25,7 @@ Year
The model is compared with a Finite Element model and is shown to give the same results.
The proposed model is thus effective.
-
+
{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
@@ -35,11 +35,11 @@ Combines:
- the FxLMS-based adaptive inverse control => suppress transmission of periodic vibrations
- direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations
-Force Feedback (Figure [2](#org6f9719a)).
+Force Feedback (Figure [2](#orgaf56f94)).
- the force sensor is mounted **between the base and the strut**
-
+
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
@@ -56,4 +56,4 @@ Sorts of HAC-LAC control:
## Bibliography {#bibliography}
-Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. .
+Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. .
diff --git a/content/article/yang19_dynam_model_decoup_contr_flexib.md b/content/article/yang19_dynam_model_decoup_contr_flexib.md
index 4cf9f4f..79506cb 100644
--- a/content/article/yang19_dynam_model_decoup_contr_flexib.md
+++ b/content/article/yang19_dynam_model_decoup_contr_flexib.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
Reference
-: ([Yang et al. 2019](#org1678fd1))
+: ([Yang et al. 2019](#org8fbcee2))
Author(s)
: Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
@@ -25,23 +25,23 @@ Year
The joint stiffness impose a limitation on the control performance using force sensors as it adds a zero at low frequency in the dynamics.
Thus, this stiffness is taken into account in the dynamics and compensated for.
-**Stewart platform** (Figure [1](#org082a4f7)):
+**Stewart platform** (Figure [1](#org006c2df)):
- piezoelectric actuators
-- flexible joints (Figure [2](#org66efbec))
+- flexible joints (Figure [2](#org8725dbf))
- force sensors (used for vibration isolation)
- displacement sensors (used to decouple the dynamics)
- cubic (even though not said explicitly)
-
+
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
-
+
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: Flexible Joints" >}}
-The stiffness of the flexible joints (Figure [2](#org66efbec)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
+The stiffness of the flexible joints (Figure [2](#org8725dbf)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
@@ -105,9 +105,9 @@ In order to apply this control strategy:
- The jacobian has to be computed
- No information about modal matrix is needed
-The block diagram of the control strategy is represented in Figure [3](#orgc6324f9).
+The block diagram of the control strategy is represented in Figure [3](#org820f661).
-
+
{{< figure src="/ox-hugo/yang19_control_arch.png" caption="Figure 3: Control Architecture used" >}}
@@ -121,10 +121,10 @@ Substituting \\(H(s)\\) in the equation of motion gives that:
**Experimental Validation**:
An external Shaker is used to excite the base and accelerometers are located on the base and mobile platforms to measure their motion.
-The results are shown in Figure [4](#org45c63bf).
+The results are shown in Figure [4](#org990744b).
In theory, the vibration performance can be improved, however in practice, increasing the gain causes saturation of the piezoelectric actuators and then the instability occurs.
-
+
{{< figure src="/ox-hugo/yang19_results.png" caption="Figure 4: Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
@@ -136,4 +136,4 @@ In theory, the vibration performance can be improved, however in practice, incre
## Bibliography {#bibliography}
-
Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” _Journal of Sound and Vibration_ 439 (January). Elsevier BV:398–412.
.
+Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” _Journal of Sound and Vibration_ 439 (January). Elsevier BV:398–412. .
diff --git a/content/article/yun20_inves_two_stage_vibrat_suppr.md b/content/article/yun20_inves_two_stage_vibrat_suppr.md
index eb2efb3..bf909e6 100644
--- a/content/article/yun20_inves_two_stage_vibrat_suppr.md
+++ b/content/article/yun20_inves_two_stage_vibrat_suppr.md
@@ -9,7 +9,7 @@ Tags
Reference
-: ([Yun et al. 2020](#orga992678))
+: ([Yun et al. 2020](#orgd3a0930))
Author(s)
: Yun, H., Liu, L., Li, Q., & Yang, H.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
-Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. .
+Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. .
diff --git a/content/article/zhang11_six_dof.md b/content/article/zhang11_six_dof.md
index 0f6a539..ed1a4aa 100644
--- a/content/article/zhang11_six_dof.md
+++ b/content/article/zhang11_six_dof.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Zhang et al. 2011](#org9aff0f7))
+: ([Zhang et al. 2011](#orgcea0d62))
Author(s)
: Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y.
@@ -25,11 +25,11 @@ Year
- **Accelerometers** for active isolation
- Adaptive FIR filters for active isolation control
-
+
{{< figure src="/ox-hugo/zhang11_platform.png" caption="Figure 1: Prototype of the non-cubic stewart platform" >}}
## Bibliography {#bibliography}
-Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In _2011 6th IEEE Conference on Industrial Electronics and Applications_, nil. .
+Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In _2011 6th IEEE Conference on Industrial Electronics and Applications_, nil. .
diff --git a/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md b/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md
index 7284b19..1ad0943 100644
--- a/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md
+++ b/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md
@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Zuo 2004](#org09e672c))
+: ([Zuo 2004](#orgdb2a627))
Author(s)
: Zuo, L.
@@ -26,23 +26,23 @@ Year
> They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
> Note that reaction force actuators can also work with soft mounts or hard mounts.
-
+
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
-
+
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
-
+
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
-
+
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
## Bibliography {#bibliography}
-Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
+Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
diff --git a/content/book/albertos04_multiv_contr_system.md b/content/book/albertos04_multiv_contr_system.md
index 49de51b..3a230ea 100644
--- a/content/book/albertos04_multiv_contr_system.md
+++ b/content/book/albertos04_multiv_contr_system.md
@@ -8,7 +8,7 @@ Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
-: ([Albertos and Antonio 2004](#org10efb7f))
+: ([Albertos and Antonio 2004](#orge811176))
Author(s)
: Albertos, P., & Antonio, S.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
-Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. .
+Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. .
diff --git a/content/book/du10_model_contr_vibrat_mechan_system.md b/content/book/du10_model_contr_vibrat_mechan_system.md
index 8d59a6e..1515be2 100644
--- a/content/book/du10_model_contr_vibrat_mechan_system.md
+++ b/content/book/du10_model_contr_vibrat_mechan_system.md
@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
-: ([Du and Xie 2010](#org500d07c))
+: ([Du and Xie 2010](#orgad87753))
Author(s)
: Du, C., & Xie, L.
@@ -21,4 +21,4 @@ Read Chapter 1 and 3.
## Bibliography {#bibliography}
-Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. .
+Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. .
diff --git a/content/book/du19_multi_actuat_system_contr.md b/content/book/du19_multi_actuat_system_contr.md
index 55d8c68..b4d1bc3 100644
--- a/content/book/du19_multi_actuat_system_contr.md
+++ b/content/book/du19_multi_actuat_system_contr.md
@@ -9,7 +9,7 @@ Tags
Reference
-: ([Du and Pang 2019](#org0a01a88))
+: ([Du and Pang 2019](#org86410b0))
Author(s)
: Du, C., & Pang, C. K.
@@ -68,9 +68,9 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
#### Secondary Actuators {#secondary-actuators}
-We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#orgc878233)) and the microactuator.
+We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#orgf5ea358)) and the microactuator.
-
+
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
@@ -92,9 +92,9 @@ There characteristics are shown on table [1](#table--tab:microactuator).
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
-A typical closed-loop control system is shown on figure [2](#orgb09803d), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
+A typical closed-loop control system is shown on figure [2](#orga949a40), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
-
+
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
@@ -104,7 +104,7 @@ A typical closed-loop control system is shown on figure [2](#orgb09803d), where
Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
-
+
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
@@ -130,7 +130,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes}
-A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#orge247255).
+A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#orgb533d71).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@@ -138,7 +138,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
- \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration
-
+
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
@@ -160,14 +160,14 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs.
-Another type of control scheme is the **parallel structure** as shown in figure [5](#org89db254).
+Another type of control scheme is the **parallel structure** as shown in figure [5](#org3c23d1d).
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
-
+
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
@@ -177,7 +177,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
-The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#orgabba47e) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
+The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org2b9887b) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@@ -187,11 +187,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
-
+
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
-Equation [1](#org22d8276) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
+Equation [1](#orgcd45840) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as
\begin{equation}
@@ -204,16 +204,16 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
-As seen in figure [7](#orgcd97793), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
+As seen in figure [7](#orgec2571e), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
-
+
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
-The sensitivity functions are shown in figure [8](#org36aa853), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
+The sensitivity functions are shown in figure [8](#orgc3be866), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
-
+
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
@@ -246,13 +246,13 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
-Figure [9](#org546c0d3) shows the control structure for the three-stage actuation system.
+Figure [9](#org0a6e6a1) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
-
+
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
@@ -281,15 +281,15 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
-The open-loop frequency responses of the three stages are shown on figure [10](#orgdd821f3).
+The open-loop frequency responses of the three stages are shown on figure [10](#org596d540).
-
+
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
-The obtained sensitivity function is shown on figure [11](#org9e62de2).
+The obtained sensitivity function is shown on figure [11](#orgb011ee0).
-
+
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
@@ -304,7 +304,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
-For the three-stage control architecture as shown on figure [9](#org546c0d3), the position error is
+For the three-stage control architecture as shown on figure [9](#org0a6e6a1), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by
@@ -320,11 +320,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system}
-A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org46f9bdc)).
+A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org13d72d1)).
The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
-with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org8f76227) and
+with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org8cccb61) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@@ -333,7 +333,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
The open-loop transfer function of the overall system is
\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
-
+
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
@@ -345,9 +345,9 @@ The control signals and the positions of the three actuators are
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*}
-The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org8e2cf1b)).
+The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org96a1b82)).
-
+
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
@@ -658,4 +658,4 @@ As a more advanced concept, PZT elements being used as actuator and sensor simul
## Bibliography {#bibliography}
-Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
+Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
diff --git a/content/book/ewins00_modal.md b/content/book/ewins00_modal.md
index 9cb082c..2f5cd6d 100644
--- a/content/book/ewins00_modal.md
+++ b/content/book/ewins00_modal.md
@@ -8,7 +8,7 @@ Tags
: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}}), [Modal Analysis]({{< relref "modal_analysis" >}})
Reference
-: ([Ewins 2000](#org59430c3))
+: ([Ewins 2000](#org8088c4f))
Author(s)
: Ewins, D.
@@ -141,7 +141,7 @@ The main measurement technique studied are those which will permit to make **dir
The type of test best suited to FRF measurement is shown in figure [fig:modal_analysis_schematic](#fig:modal_analysis_schematic).
-
+
{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@@ -215,7 +215,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
-
+
{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@@ -253,7 +253,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#fig:vibration_analysis_procedure).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
-
+
{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@@ -298,7 +298,7 @@ Three classes of system model will be described:
The basic model for the SDOF system is shown in figure [fig:sdof_model](#fig:sdof_model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
-
+
{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@@ -374,7 +374,7 @@ which is a single mode of vibration with a complex natural frequency having two
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [fig:sdof_response](#fig:sdof_response)
-
+
{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@@ -418,7 +418,7 @@ The damping effect of such a component can conveniently be defined by the ratio
|  |  |  |
|-----------------------------------------------|----------------------------------------|------------------------------------------|
-| Material hysteresis | Dry friction | Viscous damper |
+| Material hysteresis | Dry friction | Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
@@ -537,7 +537,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
|  |  |  |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
-| Receptance FRF | Mobility FRF | Accelerance FRF |
+| Receptance FRF | Mobility FRF | Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@@ -560,7 +560,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
|  |  |
|------------------------------------------------|------------------------------------------------|
-| Real part | Imaginary part |
+| Real part | Imaginary part |
| width=\linewidth | width=\linewidth |
@@ -578,7 +578,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
|  |  |
|---------------------------------------------|-----------------------------------------------|
-| Mixed | Viscous |
+| Mixed | Viscous |
| width=\linewidth | width=\linewidth |
@@ -595,7 +595,7 @@ The missing information (in this case, the frequency) must be added by identifyi
|  |  |
|------------------------------------------------------|---------------------------------------------------------|
-| Viscous damping | Structural damping |
+| Viscous damping | Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@@ -1103,7 +1103,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [fig:real_complex_modes](#fig:real_complex_modes)).
-
+
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@@ -1118,7 +1118,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
|  |  |  |
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
-| Almost-real mode | Complex Mode | Measure of complexity |
+| Almost-real mode | Complex Mode | Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1235,7 +1235,7 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
|  |  |
|---------------------------------------------------|------------------------------------------------------|
-| Point FRF | Transfer FRF |
+| Point FRF | Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [fig:mobility_frf_mdof_transfer](#fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
@@ -1260,7 +1260,7 @@ Most mobility plots have this general form as long as the modes are relatively w
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
-
+
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@@ -1281,7 +1281,7 @@ 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 [fig:nyquist_nonpropdamp_point](#fig:nyquist_nonpropdamp_point) and [fig:nyquist_nonpropdamp_transfer](#fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
@@ -1296,7 +1296,7 @@ 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 |
@@ -1450,7 +1450,7 @@ 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.
@@ -1547,7 +1547,7 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
|  |  |
|------------------------------------------|--------------------------------------------------|
-| 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.
@@ -1580,7 +1580,7 @@ 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" >}}
@@ -1852,7 +1852,7 @@ The experimental setup used for mobility measurement contains three major items:
A typical layout for the measurement system is shown on figure [fig:general_frf_measurement_setup](#fig:general_frf_measurement_setup).
-
+
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@@ -1909,7 +1909,7 @@ 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 [fig:shaker_rod](#fig:shaker_rod).
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" >}}
@@ -1930,7 +1930,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
|  |  |  |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
-| Ideal Configuration | Suspended Configuration | Unsatisfactory |
+| Ideal Configuration | Suspended Configuration | Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1948,7 +1948,7 @@ The frequency range which is effectively excited is controlled by the stiffness
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [fig:hammer_impulse](#fig:hammer_impulse).
-
+
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@@ -1979,7 +1979,7 @@ By suitable design, such a material may be incorporated into a device which **in
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 [fig:piezo_force_transducer](#fig:piezo_force_transducer)).
-
+
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@@ -1992,7 +1992,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
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" >}}
@@ -2040,7 +2040,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
|  |  |
|-----------------------------------------------------|------------------------------------------------------------|
-| Attachment methods | Frequency response characteristics |
+| Attachment methods | Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@@ -2127,7 +2127,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
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 [fig:aliasing](#fig:aliasing).
-
+
{{< 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" >}}
@@ -2144,7 +2144,7 @@ This is illustrated on figure [fig:effect_aliasing](#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.
@@ -2165,7 +2165,7 @@ 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 [fig:leakage](#fig:leakage).
@@ -2190,7 +2190,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [fig:windowing_examples](#fig:windowing_examples).
-
+
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
@@ -2211,7 +2211,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
-
+
##### Increasing transform size {#increasing-transform-size}
@@ -2247,10 +2247,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
|  |  |
|------------------------------------------------|------------------------------------------|
-| 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" >}}
@@ -2322,7 +2322,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
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 [fig:sweep_distortions](#fig:sweep_distortions).
-
+
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@@ -2440,7 +2440,7 @@ It is known that a low coherence can arise in a measurement where the frequency
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 [fig:coherence_resonance](#fig:coherence_resonance).
-
+
{{< 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" >}}
@@ -2483,7 +2483,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
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 [fig:burst_excitation](#fig:burst_excitation)).
-
+
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@@ -2502,7 +2502,7 @@ The chirp consist of a short duration signal which has the form shown in figure
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" >}}
@@ -2513,7 +2513,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
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" >}}
@@ -2523,7 +2523,7 @@ However, it should be recorded that in the region below the first cut-off freque
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 [fig:double_hits](#fig:double_hits)).
-
+
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@@ -2598,7 +2598,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
-
+
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@@ -2613,7 +2613,7 @@ This is because near resonance, the actual applied force becomes very small and
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [fig:mass_cancellation](#fig:mass_cancellation).
-
+
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@@ -2670,7 +2670,7 @@ There are two problems to be tackled:
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 [fig:rotational_measurement](#fig:rotational_measurement).
-
+
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@@ -2691,7 +2691,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
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" >}}
@@ -3031,7 +3031,7 @@ 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 |
@@ -3042,7 +3042,7 @@ 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 |
@@ -3084,7 +3084,7 @@ 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)" >}}
@@ -3179,7 +3179,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
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" >}}
@@ -3214,7 +3214,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
|  |  |
|----------------------------------------|--------------------------------------------------------------------|
-| 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:
@@ -3328,7 +3328,7 @@ 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" >}}
@@ -3453,7 +3453,7 @@ 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
@@ -3484,7 +3484,7 @@ The three terms corresponds to:
These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
-
+
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
@@ -3785,7 +3785,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
|  |  |
|-------------------------------------------|-----------------------------------------|
-| 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.
@@ -4332,7 +4332,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
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 [fig:static_display](#fig:static_display) (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" >}}
@@ -4377,7 +4377,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
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" >}}
@@ -4440,7 +4440,7 @@ 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
@@ -4495,7 +4495,7 @@ If the transmissibility is measured during a modal test which has a single excit
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 [fig:transmissibility_plots](#fig:transmissibility_plots).
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" >}}
@@ -4516,7 +4516,7 @@ 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 |
@@ -4559,4 +4559,4 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
## 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 63dbe7c..1df0717 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](#orgb239e00))
+: ([Fleming and Leang 2014](#orga9e1886))
Author(s)
: Fleming, A. J., & Leang, K. K.
@@ -821,15 +821,15 @@ 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](#orgea18894) where a voltage source is connected to a piezoelectric actuator.
+Consider the electrical circuit shown in Figure [1](#orgddc1a2b) 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.
-
+
Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
@@ -902,7 +902,7 @@ For sinusoidal signals, the amplifiers slew rate must exceed:
\\[ SR\_{\text{sin}} > V\_{p-p} \pi f \\]
where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.
-
+
If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).
@@ -948,4 +948,4 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
## 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 ee1a96c..6a59a7a 100644
--- a/content/book/hatch00_vibrat_matlab_ansys.md
+++ b/content/book/hatch00_vibrat_matlab_ansys.md
@@ -12,7 +12,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
Reference
-: ([Hatch 2000](#org6b18ef5))
+: ([Hatch 2000](#org5c6cd54))
Author(s)
: Hatch, M. R.
@@ -25,14 +25,14 @@ 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](#orge43b275) shows the methodology for analyzing a lightly damped structure using normal modes.
+The diagram in Figure [1](#org03ed3a8) shows the methodology for analyzing a lightly damped structure using normal modes.
@@ -50,7 +50,7 @@ The steps are:
-
+
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
@@ -62,7 +62,7 @@ Because finite element models usually have a very large number of states, an imp
-Figure [2](#orgbba8c29) shows such process, the steps are:
+Figure [2](#orgefbc9c9) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@@ -75,14 +75,14 @@ Figure [2](#orgbba8c29) 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](#org8396d39), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
+Tables [3](#org806d457), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
@@ -131,22 +131,22 @@ Tables [3](#org8396d39), [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](#org8396d39) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
+Figure [3](#org806d457) 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](#org34f9302)) 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](#orgb6198fb)) 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.
@@ -155,12 +155,12 @@ 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.
@@ -200,9 +200,9 @@ 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](#orgc6e42d4) 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](#org87075a2) 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" >}}
@@ -301,17 +301,17 @@ One then find:
\end{bmatrix}
\end{equation}
-Virtual interpretation of the eigenvectors are shown in Figures [5](#org92342b4), [6](#org7e90ea4) and [7](#org7b08139).
+Virtual interpretation of the eigenvectors are shown in Figures [5](#org7cb88db), [6](#org3b438be) and [7](#org1b7d8f7).
-
+
{{< 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" >}}
@@ -350,9 +350,9 @@ 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](#org0f8be1f).
+This procedure is schematically shown in Figure [8](#orgb66daef).
-
+
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@@ -700,7 +700,7 @@ 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:
@@ -708,9 +708,9 @@ 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](#org72472d7).
+This will be applied to the model shown in Figure [9](#org18f2291).
-
+
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@@ -892,9 +892,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
-Figure [10](#org4abb32c) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
+Figure [10](#org235691b) 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\\)" >}}
@@ -903,16 +903,16 @@ 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](#org69792ce).
+In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org670dac0).
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" >}}
@@ -991,7 +991,7 @@ 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}
@@ -1005,25 +1005,25 @@ 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](#org84af594)).
+In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#orge365d89)).
### 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](#orge431564)).
+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](#org117f7e6)).
-
+
{{< 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](#orge431564) shows the nodes used for the reduced Matlab model.
+Figure [13](#org117f7e6) 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.
@@ -1086,7 +1086,7 @@ 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.
@@ -1201,14 +1201,14 @@ 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](#org3022bd2)).
+In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org265cca4)).
### Actuator Description {#actuator-description}
-
+
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@@ -1230,9 +1230,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
-In Figure [15](#orga0e8a08) are shown the principal nodes used for the model.
+In Figure [15](#orge18f970) 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" >}}
@@ -1351,11 +1351,11 @@ 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\\)" >}}
@@ -1453,13 +1453,13 @@ G_f = ss(A, B, C, D);
### Simple mode truncation {#simple-mode-truncation}
-Let's plot the frequency of the modes (Figure [18](#org5f90de8)).
+Let's plot the frequency of the modes (Figure [18](#orge6429ee)).
-
+
{{< 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" >}}
@@ -1528,7 +1528,7 @@ dc_gain = abs(xn(i_input, :).*xn(i_output, :))./(2*pi*f0).^2;
[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" >}}
@@ -1872,7 +1872,7 @@ wo = gram(G_m, 'o');
And we plot the diagonal terms
-
+
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@@ -1890,7 +1890,7 @@ 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" >}}
@@ -2135,6 +2135,6 @@ pos_frames = pos([1, i_input, i_output], :);
## 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 b05b3f9..d77789b 100644
--- a/content/book/horowitz15_art_of_elect_third_edition.md
+++ b/content/book/horowitz15_art_of_elect_third_edition.md
@@ -8,7 +8,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
Reference
-: ([Horowitz 2015](#org53bd07f))
+: ([Horowitz 2015](#orgfc7b505))
Author(s)
: Horowitz, P.
@@ -19,4 +19,4 @@ Year
## 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 bf619f1..2cb4021 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](#orgf8626f0))
+: ([Leach 2014](#orgc3e03e3))
Author(s)
: Leach, R.
@@ -89,4 +89,4 @@ 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 d301228..1ba0854 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](#org8b24674))
+: ([Leach and Smith 2018](#org545df46))
Author(s)
: Leach, R., & Smith, S. T.
@@ -19,4 +19,4 @@ 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 fbebb72..a12cd26 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](#org2443fdb))
+: ([Preumont 2018](#orgaa0487d))
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](#org17539d6). 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](#org5636ea9). 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](#orgb6b4033).
+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](#org88ce537).
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](#org0ed85b6).
+The various steps of the design of a controlled structure are shown in figure [3](#org0685157).
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](#org90c3880)):
+From the block diagram of the control system (figure [4](#org23c9634)):
\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](#org3209437).
+A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgc0a0d3d).
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](#org4377aea)).
+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](#org960ee21)).
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](#orgd6e521d)).
+\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgcc3baba)).
-
+
{{< 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](#orgd6e521d), 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](#orgcc3baba), 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](#org145a286). 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](#orga2c2292). 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](#org589d929)):
+The system consists of (see figure [10](#orgd4ab71d)):
- 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](#org29d3a41)).
+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](#org89e3371)).
-
+
{{< 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](#orgfadd8ec)).
+Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1b03971)).
-
+
{{< 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](#org0f65711).
+The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org8d49672).
-
+
{{< 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](#orgf6f982f) can be used.
+To do so, the bridge circuit as shown on figure [15](#org9077cf9) 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](#org4c1156c) lists various effects that are observed in materials in response to various inputs.
+Figure [16](#orga08bcd9) 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](#org9ace96d)) 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](#orge7aeb11)) 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](#orgb03700a).
+Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org62300cf).
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](#orga70a814), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
+Consider the system of figure [19](#orga98ecb7), 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](#orgc632b09)).
+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](#orge87e33b)).
-
+
{{< 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](#orgfaf8470).
+The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgeb43e36).
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" >}}
@@ -1818,4 +1818,4 @@ 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 ae78d48..520256c 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](#org0e91961))
+: ([Schmidt, Schitter, and Rankers 2014](#orgaeaec45))
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**" >}}
@@ -44,4 +44,4 @@ Section 9.3: Mass Dilemma
## 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 388b67b..fbfb2e5 100644
--- a/content/book/skogestad07_multiv_feedb_contr.md
+++ b/content/book/skogestad07_multiv_feedb_contr.md
@@ -8,7 +8,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
-: ([Skogestad and Postlethwaite 2007](#org00db4bb))
+: ([Skogestad and Postlethwaite 2007](#org81e2975))
Author(s)
: Skogestad, S., & Postlethwaite, I.
@@ -19,7 +19,7 @@ Year
## Introduction {#introduction}
-
+
### The Process of Control System Design {#the-process-of-control-system-design}
@@ -190,7 +190,7 @@ Notations used throughout this note are summarized in tables [table:notatio
## Classical Feedback Control {#classical-feedback-control}
-
+
### Frequency Response {#frequency-response}
@@ -239,7 +239,7 @@ 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" >}}
@@ -551,7 +551,7 @@ 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. [fig:classical_feedback_2dof_alt](#fig:classical_feedback_2dof_alt)), 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" >}}
@@ -560,7 +560,7 @@ The controller can be slit into two separate blocks (Fig. [fig:classical_fe
- 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" >}}
@@ -629,7 +629,7 @@ With (see Fig. [fig:performance_weigth](#fig:performance_weigth)):
-
+
{{< figure src="/ox-hugo/skogestad07_uncertainty_disc_generated.png" caption="Figure 15: Disc-shaped uncertainty regions generated by complex additive uncertainty" >}}
@@ -2643,7 +2643,7 @@ To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_
-
+
{{< figure src="/ox-hugo/skogestad07_uncertainty_weight.png" caption="Figure 16: Relative error for 27 combinations of \\(k,\ \tau\\) and \\(\theta\\). Solid and dashed lines: two weights \\(\abs{w\_I}\\)" >}}
@@ -2682,7 +2682,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency (Fig. [fig:neglected_time_delay](#fig:neglected_time_delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
\\[ l\_I(\w) = \begin{cases} \abs{1 - e^{-j\w\theta\_{\text{max}}}} & \w < \pi/\theta\_{\text{max}} \\ 2 & \w \ge \pi/\theta\_{\text{max}} \end{cases} \\]
-
+
{{< figure src="/ox-hugo/skogestad07_neglected_time_delay.png" caption="Figure 17: Neglected time delay" >}}
@@ -2692,7 +2692,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) (Fig. [fig:neglected_first_order_lag](#fig:neglected_first_order_lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
\\[ w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1} \\]
-
+
{{< figure src="/ox-hugo/skogestad07_neglected_first_order_lag.png" caption="Figure 18: Neglected first-order lag uncertainty" >}}
@@ -2709,7 +2709,7 @@ However, as shown in Fig. [fig:lag_delay_uncertainty](#fig:lag_delay_uncert
It is suggested to start with the simple weight and then if needed, to try the higher order weight.
-
+
{{< figure src="/ox-hugo/skogestad07_lag_delay_uncertainty.png" caption="Figure 19: Multiplicative weight for gain and delay uncertainty" >}}
@@ -2749,7 +2749,7 @@ We use the Nyquist stability condition to test for robust stability of the close
&\Longleftrightarrow \quad L\_p \ \text{should not encircle -1}, \ \forall L\_p
\end{align\*}
-
+
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback.png" caption="Figure 20: Feedback system with multiplicative uncertainty" >}}
@@ -2765,7 +2765,7 @@ Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
&\Leftrightarrow \quad \abs{w\_I T} < 1, \ \forall\w \\\\\\
\end{align\*}
-
+
{{< figure src="/ox-hugo/skogestad07_nyquist_uncertainty.png" caption="Figure 21: Nyquist plot of \\(L\_p\\) for robust stability" >}}
@@ -2803,7 +2803,7 @@ And we obtain the same condition as before.
We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig. [fig:inverse_uncertainty_set](#fig:inverse_uncertainty_set)) in which
\\[ G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1} \\]
-
+
{{< figure src="/ox-hugo/skogestad07_inverse_uncertainty_set.png" caption="Figure 22: Feedback system with inverse multiplicative uncertainty" >}}
@@ -2855,7 +2855,7 @@ The condition for nominal performance when considering performance in terms of t
Now \\(\abs{1 + L}\\) represents at each frequency the distance of \\(L(j\omega)\\) from the point \\(-1\\) in the Nyquist plot, so \\(L(j\omega)\\) must be at least a distance of \\(\abs{w\_P(j\omega)}\\) from \\(-1\\).
This is illustrated graphically in Fig. [fig:nyquist_performance_condition](#fig:nyquist_performance_condition).
-
+
{{< figure src="/ox-hugo/skogestad07_nyquist_performance_condition.png" caption="Figure 23: Nyquist plot illustration of the nominal performance condition \\(\abs{w\_P} < \abs{1 + L}\\)" >}}
@@ -2880,7 +2880,7 @@ Let's consider the case of multiplicative uncertainty as shown on Fig. [fig
The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
\\[ L\_p = G\_p K = L (1 + w\_I \Delta\_I) = L + w\_I L \Delta\_I \\]
-
+
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight_bis.png" caption="Figure 24: Diagram for robust performance with multiplicative uncertainty" >}}
@@ -3046,7 +3046,7 @@ with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
This is illustrated in the block diagram of Fig. [fig:uncertainty_state_a_matrix](#fig:uncertainty_state_a_matrix), which is in the form of an inverse additive perturbation.
-
+
{{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}}
@@ -3064,7 +3064,7 @@ We also derived a condition for robust performance with multiplicative uncertain
## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
-
+
### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
@@ -3075,13 +3075,13 @@ where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g.
If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig. [fig:general_control_delta](#fig:general_control_delta). This form is useful for controller synthesis.
-
+
{{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="Figure 26: General control configuration used for controller synthesis" >}}
If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in Fig. [fig:general_control_Ndelta](#fig:general_control_Ndelta).
-
+
{{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}}
@@ -3101,7 +3101,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig. [fig:general_control_Mdelta_bis](#fig:general_control_Mdelta_bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
-
+
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}}
@@ -3153,7 +3153,7 @@ Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nb
|  |  |  |
|----------------------------------------------------|----------------------------------------------------------|-----------------------------------------------------------|
-|
Additive uncertainty |
Multiplicative input uncertainty |
Multiplicative output uncertainty |
+|
Additive uncertainty |
Multiplicative input uncertainty |
Multiplicative output uncertainty |
In Fig. [fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
@@ -3176,7 +3176,7 @@ In Fig. [fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feed
|  |  |  |
|--------------------------------------------------------|------------------------------------------------------------------|-------------------------------------------------------------------|
-|
Inverse additive uncertainty |
Inverse multiplicative input uncertainty |
Inverse multiplicative output uncertainty |
+|
Inverse additive uncertainty |
Inverse multiplicative input uncertainty |
Inverse multiplicative output uncertainty |
##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
@@ -3246,7 +3246,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig. [fig:input_uncertainty_set_feedback_weight](#fig:input_uncertainty_set_feedback_weight).
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
-
+
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 29: System with multiplicative input uncertainty and performance measured at the output" >}}
@@ -3406,7 +3406,7 @@ Where \\(G = M\_l^{-1} N\_l\\) is a left coprime factorization of the nominal pl
This uncertainty description is surprisingly **general**, it allows both zeros and poles to cross into the right-half plane, and has proven to be very useful in applications.
-
+
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}}
@@ -3438,7 +3438,7 @@ where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same di
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in Fig. [fig:block_diagonal_scalings](#fig:block_diagonal_scalings).
This clearly has no effect on stability.
-
+
{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
@@ -3754,7 +3754,7 @@ with the decoupling controller we have:
\\[ \bar{\sigma}(N\_{22}) = \bar{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right| \\]
and we see from Fig. [fig:mu_plots_distillation](#fig:mu_plots_distillation) that the NP-condition is satisfied.
-
+
{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
@@ -3877,7 +3877,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in Fig. [fig:weights_distillation](#fig:weights_distillation).
-
+
{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
@@ -3900,11 +3900,11 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
- Iteration No. 3.
Step 1: The \\(\mathcal{H}\_\infty\\) norm is only slightly reduced. We thus decide the stop the iterations.
-
+
{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
-
+
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
@@ -3912,13 +3912,13 @@ The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\
The objectives of RS and NP are easily satisfied.
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\bar{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
-
+
{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(mu\text{-plots}\\) with \\(\mu\\) \"optimal\" controller \\(K\_3\\)" >}}
To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in Fig. [fig:perturb_s_k3](#fig:perturb_s_k3).
-
+
{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\bar{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
@@ -3973,7 +3973,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
## Controller Design {#controller-design}
-
+
### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
@@ -3993,7 +3993,7 @@ We have the following important relationships:
\end{align}
\end{subequations}
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
@@ -4035,7 +4035,7 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in Fig. [fig:design_trade_off_mimo_gk](#fig:design_trade_off_mimo_gk).
-
+
{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
@@ -4092,7 +4092,7 @@ The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig. [fig:lqg_separation](#fig:lqg_separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
-
+
{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
@@ -4142,7 +4142,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
-
+
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}}
@@ -4163,7 +4163,7 @@ It has the same degree (number of poles) as the plant.
For the LQG-controller, as shown on Fig. [fig:lqg_kalman_filter](#fig:lqg_kalman_filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in Fig. [fig:lqg_integral](#fig:lqg_integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
-
+
{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
@@ -4176,18 +4176,18 @@ Their main limitation is that they can only be applied to minimum phase plants.
### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control}
-
+
#### General Control Problem Formulation {#general-control-problem-formulation}
-
+
There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
Such a general formulation is afforded by the general configuration shown in Fig. [fig:general_control](#fig:general_control).
-
+
{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
@@ -4438,7 +4438,7 @@ The elements of the generalized plant are
\end{array}
\end{equation\*}
-
+
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
@@ -4447,7 +4447,7 @@ Here we consider a tracking problem.
The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
As the regulation problem of Fig. [fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
-
+
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
@@ -4471,7 +4471,7 @@ The elements of the generalized plant are
\end{array}
\end{equation\*}
-
+
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
@@ -4499,7 +4499,7 @@ The focus of attention has moved to the size of signals and away from the size a
Weights are used to describe the expected or known frequency content of exogenous signals and the desired frequency content of error signals.
Weights are also used if a perturbation is used to model uncertainty, as in Fig. [fig:input_uncertainty_hinf](#fig:input_uncertainty_hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
-
+
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}}
@@ -4521,7 +4521,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
w = \begin{bmatrix}d\\r\\n\end{bmatrix},\ z = \begin{bmatrix}z\_1\\z\_2\end{bmatrix}, \ v = \begin{bmatrix}r\_s\\y\_m\end{bmatrix},\ u = u
\end{equation\*}
-
+
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
@@ -4532,7 +4532,7 @@ This problem is a non-standard \\(\hinf\\) optimization.
It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) procedure can be applied where we require the structured singular value:
\\[ \mu(M(j\omega)) < 1, \quad \forall\omega \\]
-
+
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
@@ -4590,7 +4590,7 @@ For the perturbed feedback system of Fig. [fig:coprime_uncertainty_bis](#fi
Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u\\y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement.
-
+
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
@@ -4637,7 +4637,7 @@ It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\)
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic--hinf--loop-shaping-design-procedure}
-
+
Robust stabilization alone is not much used in practice because the designer is not able to specify any performance requirements.
To do so, **pre and post compensators** are used to **shape the open-loop singular values** prior to robust stabilization of the "shaped" plant.
@@ -4650,7 +4650,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
as shown in Fig. [fig:shaped_plant](#fig:shaped_plant).
-
+
{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
@@ -4687,7 +4687,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
-
+
{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
@@ -4713,7 +4713,7 @@ But in cases where stringent time-domain specifications are set on the output re
A general two degrees-of-freedom feedback control scheme is depicted in Fig. [fig:classical_feedback_2dof_simple](#fig:classical_feedback_2dof_simple).
The commands and feedbacks enter the controller separately and are independently processed.
-
+
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
@@ -4724,7 +4724,7 @@ The design problem is illustrated in Fig. [fig:coprime_uncertainty_hinf](#f
The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
-
+
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
@@ -4749,7 +4749,7 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig. [fig:hinf_synthesis_2dof](#fig:hinf_synthesis_2dof).
-
+
{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
@@ -4821,7 +4821,7 @@ where \\(u\_a\\) is the **actual plant input**, that is the measurement at the *
The situation is illustrated in Fig. [fig:weight_anti_windup](#fig:weight_anti_windup), where the actuators are each modeled by a unit gain and a saturation.
-
+
{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
@@ -4869,14 +4869,14 @@ Moreover, one should be careful about combining controller synthesis and analysi
## Controller Structure Design {#controller-structure-design}
-
+
### Introduction {#introduction}
In previous sections, we considered the general problem formulation in Fig. [fig:general_control_names_bis](#fig:general_control_names_bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
-
+
{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
@@ -4911,7 +4911,7 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
Additional layers are possible, as is illustrated in Fig. [fig:control_system_hierarchy](#fig:control_system_hierarchy) which shows a typical control hierarchy for a chemical plant.
-
+
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
@@ -4933,7 +4933,7 @@ However, this solution is normally not used for a number a reasons, included the
|  |  |  |
|--------------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------------------------|
-|
Open loop optimization |
Closed-loop implementation with separate control layer |
Integrated optimization and control |
+|
Open loop optimization |
Closed-loop implementation with separate control layer |
Integrated optimization and control |
### Selection of Controlled Outputs {#selection-of-controlled-outputs}
@@ -5140,7 +5140,7 @@ 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}
@@ -5189,7 +5189,7 @@ 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\\)" >}}
@@ -5239,7 +5239,7 @@ 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" >}}
@@ -5354,7 +5354,7 @@ We get:
\end{aligned}
\end{equation}
-
+
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
@@ -5474,7 +5474,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [fig:decentralized_diagonal_control](#fig:decentralized_diagonal_control)).
-
+
{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
@@ -5861,7 +5861,7 @@ The conditions are also useful in an **input-output controllability analysis** f
## Model Reduction {#model-reduction}
-
+
### Introduction {#introduction}
@@ -6268,4 +6268,4 @@ In such a case, using truncation or optimal Hankel norm approximation with appro
## Bibliography {#bibliography}
-
Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
+
Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
diff --git a/content/book/taghirad13_paral.md b/content/book/taghirad13_paral.md
index d259378..604f86c 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](#orgae7076f))
+: ([Taghirad 2013](#org83723e5))
Author(s)
: Taghirad, H.
@@ -19,7 +19,7 @@ Year
## 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.
@@ -44,7 +44,7 @@ The control of parallel robots is elaborated in the last two chapters, in which
## Motion Representation {#motion-representation}
-
+
### Spatial Motion Representation {#spatial-motion-representation}
@@ -59,7 +59,7 @@ 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" >}}
@@ -84,7 +84,7 @@ 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](#orgbe0ced8)).
+Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org5798f66)).
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
@@ -106,7 +106,7 @@ 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" >}}
@@ -132,7 +132,7 @@ 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" >}}
@@ -158,7 +158,7 @@ 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" >}}
@@ -363,7 +363,7 @@ There exist transformations to from screw displacement notation to the transform
Let us consider the motion of a rigid body described at three locations (Figure [fig:consecutive_transformations](#fig:consecutive_transformations)).
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}\\}\\)" >}}
@@ -426,7 +426,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
## Kinematics {#kinematics}
-
+
### Introduction {#introduction}
@@ -533,7 +533,7 @@ 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" >}}
@@ -586,7 +586,7 @@ The complexity of the problem depends widely on the manipulator architecture and
## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
-
+
### Introduction {#introduction}
@@ -683,7 +683,7 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
Now consider the general motion of a rigid body shown in Figure [fig:general_motion](#fig:general_motion), 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}\\}\\)" >}}
@@ -942,7 +942,7 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
As shown in Figure [fig:stewart_static_forces](#fig:stewart_static_forces), 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" >}}
@@ -1099,7 +1099,7 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
In this section, we restrict our analysis to a 3-6 structure (Figure [fig:stewart36](#fig:stewart36)) 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" >}}
@@ -1129,7 +1129,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
## Dynamics {#dynamics}
-
+
### Introduction {#introduction}
@@ -1260,7 +1260,7 @@ 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" >}}
@@ -1374,7 +1374,7 @@ 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\\)" >}}
@@ -1519,7 +1519,7 @@ 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" >}}
@@ -1733,7 +1733,7 @@ in which
As shown in Figure [fig:stewart_forward_dynamics](#fig:stewart_forward_dynamics), 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" >}}
@@ -1766,7 +1766,7 @@ 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" >}}
@@ -1791,7 +1791,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
## Motion Control {#motion-control}
-
+
### Introduction {#introduction}
@@ -1812,7 +1812,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
### Controller Topology {#controller-topology}
-
+
@@ -1861,7 +1861,7 @@ Figure [fig:general_topology_motion_feedback](#fig:general_topology_motion_feedb
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" >}}
@@ -1871,7 +1871,7 @@ The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) an
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 [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis) to implement such a controller.
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{{< 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" >}}
@@ -1885,7 +1885,7 @@ To overcome the implementation problem of the control topology in Figure [fig:ge
In this topology, depicted in Figure [fig:general_topology_motion_feedback_ter](#fig:general_topology_motion_feedback_ter), 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\\).
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{{< 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" >}}
@@ -1899,7 +1899,7 @@ For the topology in Figure [fig:general_topology_motion_feedback_ter](#fig:gener
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [fig:general_topology_motion_feedback_quater](#fig:general_topology_motion_feedback_quater).
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}\\).
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{{< 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" >}}
@@ -1909,7 +1909,7 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
### Motion Control in Task Space {#motion-control-in-task-space}
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#### Decentralized PD Control {#decentralized-pd-control}
@@ -1918,7 +1918,7 @@ In the control structure in Figure [fig:decentralized_pd_control_task_space](#fi
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.
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{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
@@ -1941,7 +1941,7 @@ A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the
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) \\]
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{{< 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" >}}
@@ -2004,7 +2004,7 @@ Furthermore, mass matrix is added in the forward path in addition to the desired
As for the feedforward control, the **dynamics and kinematic parameters of the robot are needed**, and in practice estimates of these matrices are used.
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{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
@@ -2126,14 +2126,14 @@ 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.
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{{< 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}
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{{< 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" >}}
@@ -2218,7 +2218,7 @@ In this control structure, depicted in Figure [fig:decentralized_pd_control_join
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.
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{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
@@ -2240,7 +2240,7 @@ To remedy these shortcomings, some modifications have been proposed to this stru
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 [fig:feedforward_pd_control_joint_space](#fig:feedforward_pd_control_joint_space).
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{{< 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" >}}
@@ -2288,7 +2288,7 @@ Furthermore, the mass matrix is acting in the **forward path**, in addition to t
Note that to generate this term, the **dynamic formulation** of the robot, and its **kinematic and dynamic parameters are needed**.
In practice, exact knowledge of dynamic matrices are not available, and there estimates are used.
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{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
@@ -2564,7 +2564,7 @@ Hence, it is recommended to design and implement controllers in the task space,
## Force Control {#force-control}
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### Introduction {#introduction}
@@ -2620,7 +2620,7 @@ The output control loop is called the **primary loop**, while the inner loop is
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{{< figure src="/ox-hugo/taghirad13_cascade_control.png" caption="Figure 27: Block diagram of a closed-loop system with cascade control" >}}
@@ -2654,7 +2654,7 @@ As seen in Figure [fig:taghira13_cascade_force_outer_loop](#fig:taghira13_cascad
The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.
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{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
@@ -2662,7 +2662,7 @@ Other alternatives for force control topology may be suggested based on the vari
If the force is measured in the joint space, the topology suggested in Figure [fig:taghira13_cascade_force_outer_loop_tau](#fig:taghira13_cascade_force_outer_loop_tau) can be used.
In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).
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{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
@@ -2673,7 +2673,7 @@ However, as the inner loop is constructed in the joint space, the desired motion
Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.
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{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@@ -2691,7 +2691,7 @@ By this means, when the manipulator is not in contact with a stiff environment,
However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.
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{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
@@ -2699,7 +2699,7 @@ Other alternatives for control topology may be suggested based on the variations
If the force is measured in the joint space, control topology shown in Figure [fig:taghira13_cascade_force_inner_loop_tau](#fig:taghira13_cascade_force_inner_loop_tau) can be used.
In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.
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{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
@@ -2708,7 +2708,7 @@ The inner loop is based on the measured actuator force vector in the joint space
In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
Thus, independent controllers for each joint may be suitable for this topology.
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{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@@ -2727,7 +2727,7 @@ Thus, independent controllers for each joint may be suitable for this topology.
### Direct Force Control {#direct-force-control}
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{{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
@@ -2818,7 +2818,7 @@ The impedance of the system may be found from the Laplace transform of the above