Update Content - 2020-09-21
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@ -8,7 +8,7 @@ Tags
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: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
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Reference
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: ([Preumont 2018](#org09bc150))
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: ([Preumont 2018](#org2443fdb))
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Author(s)
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: Preumont, A.
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@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
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#### Feedback {#feedback}
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<a id="org3a6239d"></a>
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<a id="org17539d6"></a>
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{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
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The principle of feedback is represented on figure [1](#org3a6239d). 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.
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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.
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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.
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In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
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@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
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#### Feedforward {#feedforward}
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<a id="org0c27bf0"></a>
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<a id="orgb6b4033"></a>
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{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
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The method relies on the availability of a **reference signal correlated to the primary disturbance**.
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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](#org0c27bf0).
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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).
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The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
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@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
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### The Various Steps of the Design {#the-various-steps-of-the-design}
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<a id="org4371fad"></a>
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<a id="org0ed85b6"></a>
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{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
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The various steps of the design of a controlled structure are shown in figure [3](#org4371fad).
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The various steps of the design of a controlled structure are shown in figure [3](#org0ed85b6).
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The **starting point** is:
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@ -154,14 +154,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
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### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
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From the block diagram of the control system (figure [4](#orgffc7bae)):
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From the block diagram of the control system (figure [4](#org90c3880)):
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\begin{align\*}
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y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
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z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
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\end{align\*}
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<a id="orgffc7bae"></a>
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<a id="org90c3880"></a>
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{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
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@ -186,12 +186,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response
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It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
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\\(\sigma\_z(0)\\) is then the global RMS response.
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A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3ce8916).
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A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3209437).
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It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
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The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
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<a id="org3ce8916"></a>
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<a id="org3209437"></a>
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{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
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@ -398,11 +398,11 @@ With:
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D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
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\end{equation}
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<a id="org4219cc6"></a>
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<a id="org4377aea"></a>
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{{< 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\\)" >}}
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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](#org4219cc6)).
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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)).
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And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
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\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
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@ -441,9 +441,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
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If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
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\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgd4f4723)).
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgd6e521d)).
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<a id="orgd4f4723"></a>
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<a id="orgd6e521d"></a>
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{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
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@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
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</div>
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If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org5748d4e).
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If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org3cc8875).
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<a id="org5748d4e"></a>
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<a id="org3cc8875"></a>
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{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
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@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
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</div>
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By looking at figure [7](#orgd4f4723), 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.
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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.
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<a id="org0dd5dad"></a>
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{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
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@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
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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)}
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\end{equation}
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The corresponding Bode plot is represented in figure [9](#org0dd5dad). 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}\\).
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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}\\).
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In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
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@ -508,12 +508,12 @@ Two broad categories of actuators can be distinguish:
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A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
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The system consists of (see figure [10](#org576a7f9)):
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The system consists of (see figure [10](#org589d929)):
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- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
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- A coil which is free to move axially
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<a id="org576a7f9"></a>
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<a id="org589d929"></a>
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{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
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@ -551,9 +551,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
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#### Proof-Mass Actuator {#proof-mass-actuator}
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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](#orge7047bc)).
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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)).
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<a id="orge7047bc"></a>
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<a id="org29d3a41"></a>
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{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
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@ -583,9 +583,9 @@ with:
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</div>
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Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#orgf19efd8)).
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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)).
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<a id="orgf19efd8"></a>
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{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
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@ -610,7 +610,7 @@ By using the two equations, we obtain:
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Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
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<a id="org3d1f388"></a>
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{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
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@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti
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### General Electromechanical Transducer {#general-electromechanical-transducer}
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The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org6266d1c).
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The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org0f65711).
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<a id="org6266d1c"></a>
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<a id="org0f65711"></a>
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{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
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@ -646,7 +646,7 @@ With:
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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.
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Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
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To do so, the bridge circuit as shown on figure [15](#org37d6035) can be used.
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To do so, the bridge circuit as shown on figure [15](#orgf6f982f) can be used.
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We can show that
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@ -656,7 +656,7 @@ We can show that
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which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
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<a id="org37d6035"></a>
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<a id="orgf6f982f"></a>
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{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
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@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
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### Smart Materials {#smart-materials}
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Smart materials have the ability to respond significantly to stimuli of different physical nature.
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Figure [16](#org4f66ebd) lists various effects that are observed in materials in response to various inputs.
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Figure [16](#org4c1156c) lists various effects that are observed in materials in response to various inputs.
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{{< 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" >}}
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@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
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</div>
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If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org9f89282)) 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:
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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:
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\begin{equation}
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\begin{bmatrix}Q\\\Delta\end{bmatrix}
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@ -782,7 +782,7 @@ where
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- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
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- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
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<a id="org9f89282"></a>
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{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
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@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
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#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
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Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org1d36e1c).
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Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#orgb03700a).
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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
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@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
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dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
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\end{equation}
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{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
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@ -844,10 +844,10 @@ The ratio between the remaining stored energy and the initial stored energy is
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#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
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Consider the system of figure [19](#org645378f), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
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Consider the system of figure [19](#orga70a814), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
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The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
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<a id="org645378f"></a>
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<a id="orga70a814"></a>
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{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
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@ -866,9 +866,9 @@ And one can see that
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\frac{z^2 - p^2}{z^2} = k^2
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\end{equation}
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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](#org68045c5)).
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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)).
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<a id="org68045c5"></a>
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<a id="orgc632b09"></a>
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{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
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@ -1566,7 +1566,7 @@ Their design requires a model of the structure, and there is usually a trade-off
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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.
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The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgb182ef8).
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The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgfaf8470).
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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.
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This approach has the following advantages:
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@ -1574,7 +1574,7 @@ This approach has the following advantages:
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- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
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- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
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<a id="orgb182ef8"></a>
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{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
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@ -1818,4 +1818,4 @@ This approach has the following advantages:
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## Bibliography {#bibliography}
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<a id="org09bc150"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
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<a id="org2443fdb"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
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content/zettels/singular_value_decomposition.md
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content/zettels/singular_value_decomposition.md
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+++
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title = "Singular Value Decomposition"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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|
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Tags
|
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:
|
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|
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## SVD of a MIMO system {#svd-of-a-mimo-system}
|
||||
|
||||
We are interested by the physical interpretation of the SVD when applied to the frequency response of a MIMO system \\(G(s)\\) with \\(m\\) inputs and \\(l\\) outputs.
|
||||
|
||||
\begin{equation}
|
||||
G = U \Sigma V^H
|
||||
\end{equation}
|
||||
|
||||
\\(\Sigma\\)
|
||||
: is an \\(l \times m\\) matrix with \\(k = \min\\{l, m\\}\\) non-negative **singular values** \\(\sigma\_i\\), arranged in descending order along its main diagonal, the other entries are zero.
|
||||
|
||||
\\(U\\)
|
||||
: is an \\(l \times l\\) unitary matrix. The columns of \\(U\\), denoted \\(u\_i\\), represent the **output directions** of the plant. They are orthonormal.
|
||||
|
||||
\\(V\\)
|
||||
: is an \\(m \times m\\) unitary matrix. The columns of \\(V\\), denoted \\(v\_i\\), represent the **input directions** of the plant. They are orthonormal.
|
||||
|
||||
The input and output directions are related through the singular values:
|
||||
|
||||
\begin{equation}
|
||||
G v\_i = \sigma\_i u\_i
|
||||
\end{equation}
|
||||
|
||||
So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\).
|
||||
Furthermore, since \\(\normtwo{v\_i}=1\\) and \\(\normtwo{u\_i}=1\\), we see that **the singular value \\(\sigma\_i\\) directly gives the gain of the matrix \\(G\\) in this direction**.
|
||||
|
||||
The **largest gain** for any input is equal to the **maximum singular value**:
|
||||
\\[\maxsv(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_1}}{\normtwo{v\_1}} \\]
|
||||
The **smallest gain** for any input direction is equal to the **minimum singular value**:
|
||||
\\[\minsv(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_k}}{\normtwo{v\_k}} \\]
|
||||
|
||||
We define \\(u\_1 = \bar{u}\\), \\(v\_1 = \bar{v}\\), \\(u\_k=\ubar{u}\\) and \\(v\_k = \ubar{v}\\). Then is follows that:
|
||||
\\[ G\bar{v} = \maxsv \bar{u} ; \quad G\ubar{v} = \minsv \ubar{u} \\]
|
||||
|
||||
|
||||
## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
|
||||
|
||||
This is taken from [Preumont's book](preumont18_vibrat_contr_activ_struc_fourt_edition.md).
|
||||
|
||||
The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix:
|
||||
\\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\]
|
||||
With:
|
||||
|
||||
- \\(U\\) and \\(V\\) orthogonal matrices. The columns \\(u\_i\\) and \\(v\_i\\) of \\(U\\) and \\(V\\) are the eigenvectors of the square matrices \\(JJ^T\\) and \\(J^TJ\\) respectively
|
||||
- \\(\Sigma\\) a rectangular diagonal matrix of dimension \\(m \times n\\) containing the square root of the common non-zero eigenvalues of \\(JJ^T\\) and \\(J^TJ\\)
|
||||
- \\(r\\) is the number of non-zero singular values of \\(J\\)
|
||||
|
||||
The pseudo-inverse of \\(J\\) is:
|
||||
\\[ J^+ = V\Sigma^+U^T = \sum\_{i=1}^r \frac{1}{\sigma\_i} v\_i u\_i^T \\]
|
||||
|
||||
The conditioning of the Jacobian is measured by the **condition number**:
|
||||
\\[ c(J) = \frac{\sigma\_{max}}{\sigma\_{min}} \\]
|
||||
|
||||
When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum.
|
||||
This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\).
|
||||
|
||||
<./biblio/references.bib>
|
||||
@ -40,6 +40,9 @@ Tags
|
||||
| Newport | [link](https://www.newport.com/search/?q1=hexapod%3Arelevance%3Acompatibility%3AMETRIC%3AisObsolete%3Afalse%3A-excludeCountries%3AFR%3AnpCategory%3Ahexapods&ajax&text=hexapod) | USA |
|
||||
| Symetrie | [link](https://symetrie.fr/en/hexapods-en/positioning-hexapods/) | France |
|
||||
| CSA Engineering | [link](https://www.csaengineering.com/products-services/hexapod-positioning-systems/hexapod-models.html) | USA |
|
||||
| Aerotech | [link](https://www.aerotech.com/product-catalog/hexapods.aspx) | USA |
|
||||
| SmarAct | [link](https://www.smaract.com/smarpod) | Germany |
|
||||
| Gridbots | [link](https://www.gridbots.com/hexamove.html) | India |
|
||||
|
||||
|
||||
## Stewart Platforms at ESRF {#stewart-platforms-at-esrf}
|
||||
@ -56,36 +59,36 @@ Tags
|
||||
|
||||
Papers by J.E. McInroy:
|
||||
|
||||
- ([O’Brien et al. 1998](#org6990e82))
|
||||
- ([McInroy, O’Brien, and Neat 1999](#org2eaf165))
|
||||
- ([McInroy 1999](#org19d227f))
|
||||
- ([McInroy and Hamann 2000](#org0de51ce))
|
||||
- ([Chen and McInroy 2000](#orgfa121ee))
|
||||
- ([McInroy 2002](#org1ad3b68))
|
||||
- ([Li, Hamann, and McInroy 2001](#orga0bd81f))
|
||||
- ([Lin and McInroy 2003](#org01eae42))
|
||||
- ([Jafari and McInroy 2003](#org9490dc9))
|
||||
- ([Chen and McInroy 2004](#org3bd3a36))
|
||||
- ([O’Brien et al. 1998](#orgb1b8013))
|
||||
- ([McInroy, O’Brien, and Neat 1999](#org06dafcc))
|
||||
- ([McInroy 1999](#orga04f28d))
|
||||
- ([McInroy and Hamann 2000](#org61fde10))
|
||||
- ([Chen and McInroy 2000](#org4b76086))
|
||||
- ([McInroy 2002](#orgc3f19a5))
|
||||
- ([Li, Hamann, and McInroy 2001](#org469fdd4))
|
||||
- ([Lin and McInroy 2003](#org52d73bd))
|
||||
- ([Jafari and McInroy 2003](#orga8e7146))
|
||||
- ([Chen and McInroy 2004](#org97f0e30))
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org3bd3a36"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
|
||||
<a id="org97f0e30"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
|
||||
|
||||
<a id="orgfa121ee"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
|
||||
<a id="org4b76086"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
|
||||
|
||||
<a id="org9490dc9"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
|
||||
<a id="orga8e7146"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
|
||||
|
||||
<a id="org01eae42"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
|
||||
<a id="org52d73bd"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
|
||||
|
||||
<a id="orga0bd81f"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
|
||||
<a id="org469fdd4"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
|
||||
|
||||
<a id="org19d227f"></a>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. <https://doi.org/10.1109/cca.1999.806694>.
|
||||
<a id="orga04f28d"></a>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. <https://doi.org/10.1109/cca.1999.806694>.
|
||||
|
||||
<a id="org1ad3b68"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
|
||||
<a id="orgc3f19a5"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
|
||||
|
||||
<a id="org0de51ce"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
|
||||
<a id="org61fde10"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
|
||||
|
||||
<a id="org2eaf165"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
|
||||
<a id="org06dafcc"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
|
||||
|
||||
<a id="org6990e82"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.
|
||||
<a id="orgb1b8013"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.
|
||||
|
||||
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