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title = "Papers"
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author = ["Thomas Dehaeze"]
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type = "paper"
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draft = false
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Here is the list of papers I took note about.
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title = "Optimized estimator for real-time dynamic displacement measurement using accelerometers"
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author = ["Dehaeze Thomas"]
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draft = true
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Tags
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:
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Reference
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: (<a href="#citeproc_bib_item_1">Abir et al. 2016</a>)
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Author(s)
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: Abir, J., Longo, S., Morantz, P., & Shore, P.
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Year
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: 2016
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Abir, Jonathan, Stefano Longo, Paul Morantz, and Paul Shore. 2016. “Optimized Estimator for Real-Time Dynamic Displacement Measurement Using Accelerometers.” <i>Mechatronics</i> 39: 1–11. doi:<a href="https://doi.org/10.1016/j.mechatronics.2016.07.003">10.1016/j.mechatronics.2016.07.003</a>.</div>
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</div>
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title = "Active structural vibration control: a review"
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author = ["Dehaeze Thomas"]
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draft = false
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+++
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Tags
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:
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Reference
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: (<a href="#citeproc_bib_item_1">Alkhatib and Golnaraghi 2003</a>)
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Author(s)
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: Alkhatib, R., & Golnaraghi, M. F.
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Year
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: 2003
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## Process of designing an active vibration control system {#process-of-designing-an-active-vibration-control-system}
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1. Analyze the structure to be controled
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2. Obtain an idealized mathematical model with FEM or experimental modal analysis
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3. Reduce the model order is necessary
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4. Analyze the resulting model: dynamics properties, types of disturbances, ...
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5. Quantify sensors and actuators requirements. Decide on their types and location
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6. Analyze the impact of the sensors and actuators on the overall dynamic characteristics
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7. Specify performance criteria and stability tradeoffs
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8. Device of the type of control algorythm to be employed and design a controller to meet the specifications
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9. Simulate the resulting controlled system on a computer
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10. If the controller does not meet the requirements, adjust the specifications or modify the type of controller
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11. Choose hardware and software and integrate the components on a pilot plant
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12. Formulate experiments and perform system identification and model updating
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13. Implement controller and carry out system test to evaluate the performance
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## Feedback control {#feedback-control}
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### Active damping {#active-damping}
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The objective is to reduce the resonance peaks of the closed loop transfer function.
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\\[T(s) = \frac{G(s)H(s)}{1+G(s)H(s)}\\]
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Then \\(T(s) \approx G(s)\\) except near the resonance peaks where the amplitude is reduced.
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This method can be realized without a model of the structure with **guaranteed stability**, granted that the actuators and sensors are **collocated**.
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### Model based feedback {#model-based-feedback}
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Objective: keep a control variable (position, velocity, ...) to a desired value in spite of external disturbances \\(d(s)\\).
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We have \\[\frac{y(s)}{d(s)} = \frac{1}{1+G(s)H(s)}\\] so we need large values of \\(G(s)H(s)\\) in the frequency range where the disturbance has considerable effect.
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To do so, we need a mathematical model of the system, then the control bandwidth and effectiveness are restricted by the accuracy of the model.
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Unmodeled structural dynamics may destabilize the system.
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## Feedforward Control {#feedforward-control}
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We need a signal that is correlated to the disturbance. Then feedforward can improve performance over simple feedback control.
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An adaptive filter manipulates the signal correlated to the disturbance and the output is applied to the system by the actuator.
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The filter coefficients are adapted in such a way that an error signal is minimized.
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The idea is to generate a secondary disturbance, which destructively interferes with the effect of the primary distance at the location of the error sensor.
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However, there is no guarantee that the global response is also reduced at other locations.
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The method is considered to be a **local technique**, in contrast to feedback which is global.
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Contrary to active damping which can only reduce the vibration near the resonance, **feedforward control can be effective for any frequency**.
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The major restriction to the application of feedforward adaptive filtering is the accessibility of a reference signal correlated to the disturbance.
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<a id="table--table:comparison-constrol-strat"></a>
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<div class="table-caption">
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<span class="table-number"><a href="#table--table:comparison-constrol-strat">Table 1</a>:</span>
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Comparison of control strategies
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</div>
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| Type of control | Advantages | Disadvantages |
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|--------------------------------|---------------------------------------------|-----------------------------------------------|
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| Active Damping | Simple to implement | Effective only near resonance |
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| | Does not required accurate model | |
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| | Guaranteed stability (collocated) | |
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| Model Based | Global method | Requires accurate model |
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| | Attenuate all disturbance within bandwidth | Required low delay |
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| | | Limited bandwidth |
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| | | Spillover |
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| Feedforward Adaptive filtering | No model is necessary | Error signal required |
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| | Robust to change in plant transfer function | Local method: may amplify vibration elsewhere |
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| | More effective for narrowband disturbance | Large amount of real-time computation |
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## Controllability and Observability {#controllability-and-observability}
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Controllability and Observability are two fundamental qualitave properties of dynamic systems.
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A system is said to be **controllable** if every state vector can be transform to a desirate state in finite time by the application of unconstrained control inputs.
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A system is said to be **observable** at time \\(t\_0\\) if for a state \\(z(t\_0)\\), there is a finite time \\(t\_1>t\_0\\) such that the knowledge of the input \\(u(t)\\) and output \\(y(t)\\) from \\(t\_0\\) to \\(t\_1\\) are sufficient to determine the state \\(z(t\_0)\\).
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## Coordinate Coupling Control {#coordinate-coupling-control}
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Coordinate coupling control (CCC) is an **energy-basded method**.
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The idea is to **transfer the vibrations** from a low or undamped oscilatory system (the plant) to a damped system (the controller).
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This can be implemented passively using tuned mass damper. But the key advantage of this technique is that one can replace the physical absorber with a computer model. The coupling terms can then be selected to maximise the energy transfer.
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## Robust control {#robust-control}
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Robust control concentrates on the **tradeoffs between performance and stability** in the presence of uncertainty in the system model as well as the exogenous inputs to which it is subjected.
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Uncertainty can be divided into four types:
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- parameter errors
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- error in model order
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- neglected disturbances
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- neglected nonlinearities
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The \\(\mathcal{H}\_\infty\\) controller is developed to address uncertainty by systematic means.
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A general block diagram of the control system is shown [Figure 1](#figure--fig:alkhatib03-hinf-control).
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A **frequency shaped filter** \\(W(s)\\) coupled to selected inputs and outputs of the plant is included.
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The outputs of this frequency shaped filter define the error ouputs used to evaluate the system performance and generate the **cost** that will be used in the design process.
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<a id="figure--fig:alkhatib03-hinf-control"></a>
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{{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="<span class='figure-number'>Figure 1: </span>Block diagram for robust control" >}}
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The generalized plan \\(G\\) can be partitionned according to the input-output variables. And we have that the transfer function matrix from \\(d\\) to \\(z\\) is:
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\\[ H\_{z/d} = G\_{z/d} + G\_{z/u} K (I - G\_{y/u} K)^{-1} G\_{y/d} \\]
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This transfer function matrix contains measures of performance and stability robustness.
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The objective of \\(\mathcal{H}\_\infty\\) control is to design an admissible control \\(u(s)=K(s)y(s)\\) such that \\(\\| H\_{z/d} \\|\_\infty\\) is minimum.
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## Optimal Control {#optimal-control}
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The control \\(u(t)\\) is designed to minimize a cost function \\(J\\), given the initial conditions \\(z(t\_0)\\) and \\(\dot{z}(t\_0)\\) subject to the constraint that:
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\begin{align\*}
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\dot{z} &= Az + Bu\\\\
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y &= Cz
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\end{align\*}
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One such cost function appropriate to a vibration control is
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\\[J = 1/2 \int\_{t\_0}^{t\_f} ( z^T A z + u^T R u ) dt\\]
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Where \\(Q\\) and \\(R\\) and positive definite symmetric weighting matrices.
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## State Observers (Estimators) {#state-observers--estimators}
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It is not always possible to determine the entire state variables. There are usualy too many degrees of freedom and only limited measurements.
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The state vector \\(z(t)\\) can be estimated independently of the control problem, and the resulting estimate \\(\hat{z}(t)\\) can be used.
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## Intelligent Structure and Controller {#intelligent-structure-and-controller}
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Intelligent structure would have the capability to:
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- recognize the present dynamic state of its own structure and evaluate the functional performance of the structure
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- identify functional descriptions of external and internal disturbances
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- detect changes in structural properties and changes in external and internal disturbances
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- predict possible future changes in structural properties
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- make intelligent decisions regarding compensations for disturbances and adequately generale actuation forces
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- learn from past performance to improve future actions
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Two main methodologies:
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- artificial neural networks
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- fuzzy logic
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## Adaptive Control {#adaptive-control}
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Adaptive control is frequently used to control systems whose parameters are unknown, uncertain, or slowly varying.
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The design of an adaptive controller involves several steps:
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- selection of a controller structure with adjustable parameters
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- selection of an adaptation law for adjusting those parameters
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- selection of a performance index
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- real-time evaluation of the performance with respect to some desired behavior
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- real-time plant identification and model updating
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- real-time adjustment of the controller parameters to bring the performance closer to the desired behavior
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It essentially consists of a real-time system identification technique integrated with a control algorithm.
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Two different methods
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- **Direct method**: the controller parameters are adjusted directly based on the error between the measured and desired outputs.
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- **Indirect method**: the computations are divided into two consecutive phases. First, the plant model is first estimated in real time. Second, the controller parameters are modified based on the most recent updated plant parameters.
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## Active Control Effects on the System {#active-control-effects-on-the-system}
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<a id="figure--fig:alkhatib03-1dof-control"></a>
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{{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="<span class='figure-number'>Figure 2: </span>1 DoF control of a spring-mass-damping system" >}}
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Consider the control system [Figure 2](#figure--fig:alkhatib03-1dof-control), the equation of motion of the system is:
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\\[ m\ddot{x} + c\dot{x} + kx = f\_a + f \\]
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The controller force can be expressed as: \\(f\_a = -g\_a \ddot{x} + g\_v \dot{x} + g\_d x\\). The equation of motion becomes:
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\\[ (m+g\_a)\ddot{x} + (c+g\_v)\dot{x} + (k+g\_d)x = f \\]
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Depending of the type of signal used, the active control adds/substracts mass, damping and stiffness.
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## Time Delays {#time-delays}
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One of the limits to the performance of active control is the time delay in controllers and actuators. Time delay introduces phase shift, which deteriorates the controller performance or even causes instability in the system.
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## Optimal Placement of Actuators {#optimal-placement-of-actuators}
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The problem of optimizing the locations of the actuators can be more significant than the control law itself.
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If the actuator is placed at the wrong location, the system will require a greater force control. In that case, the system is said to have a **low degree of controllability**.
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Alkhatib, Rabih, and M. F. Golnaraghi. 2003. “Active Structural Vibration Control: A Review.” <i>The Shock and Vibration Digest</i> 35 (5): 367–83. doi:<a href="https://doi.org/10.1177/05831024030355002">10.1177/05831024030355002</a>.</div>
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</div>
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@@ -0,0 +1,183 @@
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+++
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title = "Guidelines for the selection of weighting functions for h-infinity control"
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author = ["Dehaeze Thomas"]
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draft = false
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+++
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Tags
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: [H Infinity Control]({{< relref "h_infinity_control.md" >}})
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Reference
|
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: (<a href="#citeproc_bib_item_1">Bibel and Malyevac 1992</a>)
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Author(s)
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: Bibel, J. E., & Malyevac, D. S.
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Year
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: 1992
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## Properties of feedback control {#properties-of-feedback-control}
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<a id="figure--fig:bibel92-control-diag"></a>
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{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="<span class='figure-number'>Figure 1: </span>Control System Diagram" >}}
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From the [Figure 1](#figure--fig:bibel92-control-diag), we have:
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\begin{align\*}
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y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\
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e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\\
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u(s) &= S(s)K(s) r(s) - S(s)K(s) d(s) - S(s)K(s) n(s)
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\end{align\*}
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With the following definitions
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- \\(L(s) = G(s)K(s)\\) is the **loop transfer matrix**
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- \\(S(s) = [I+G(s)K(s)]^{-1}\\) is the **Sensitivity** function matrix
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- \\(T(s) = [I+G(s)K(s)]^{-1}G(s)K(s)\\) is the **Transmissibility** function matrix
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<div class="cbox">
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\\[ S(s) + T(s) = 1 \\]
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</div>
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<div class="cbox">
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- **Command following**: \\(S=0\\) and \\(T=1\\) => large gains
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- **Disturbance rejection**: \\(S=0\\) => large gains
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- **Sensor noise attenuation**: \\(T\\) small where the noise is concentrated
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- **Control Sensitivity minimization**: \\(K S\\) small
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- **Robustness to modeling errors**: \\(T\\) small in the frequency range of the expected model undertainties
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</div>
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## SISO tradeoff {#siso-tradeoff}
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We want \\(S\\) small for command following and disturbance rejection.
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We want \\(T\\) small to remain insensitive to sensor noise and modeling errors and to reduce control sensitivity.
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However we cannot keep both \\(S\\) and \\(T\\) small as \\(S(s)+T(s)=1\\).
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We must determine some **tradeoff** between the sensitivity and the complementary sensitivity functions.
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Usually, reference signals and disturbances occur at low frequencies, while noise and modeling errors are concentrated at high frequencies. The tradeoff, in a SISO sense, is to make \\(|S(j\omega)|\\) small as low frequencies and \\(|T(j\omega)|\\) small at high frequencies.
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## \\(H\_\infty\\) and weighting functions {#h-infty-and-weighting-functions}
|
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<div class="cbox">
|
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|
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\\(\mathcal{H}\_\infty\\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller's performance and robustness characteristics.
|
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</div>
|
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|
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<a id="figure--fig:bibel92-general-plant"></a>
|
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|
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{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="<span class='figure-number'>Figure 2: </span>\\(\mathcal{H}\_\infty\\) control framework" >}}
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New design framework ([Figure 2](#figure--fig:bibel92-general-plant)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
|
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|
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- \\(w\\): exogenous inputs
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- \\(z\\): regulated performance output
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- \\(u\\): control inputs
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- \\(y\\): measured output variables
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The plant \\(P\\) has two inputs and two outputs, it can be decomposed into four sub-transfer function matrices:
|
||||
\\[P = \begin{bmatrix}P\_{11} & P\_{12} \\\ P\_{21} & P\_{22} \end{bmatrix}\\]
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## Lower Linear Fractional Transformation {#lower-linear-fractional-transformation}
|
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|
||||
The transformation from the input \\(w\\) to the output \\(z\\), \\(T\_{zw}\\) is called the **Lower Linear Fractional Transformation** \\(F\_l (P, K)\\).
|
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|
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<div class="cbox">
|
||||
|
||||
\\[T\_{zw} = F\_l (P, K) = P\_{11} + P\_{12}K (I-P\_{22})^{-1} P\_{21}\\]
|
||||
|
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</div>
|
||||
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The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\| T\_{zw} \\|\_\infty\\) over the space of all realizable controllers \\(K(s)\\) that stabilize the closed-loop system.
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## Weights for inputs/outputs signals {#weights-for-inputs-outputs-signals}
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Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs ([Figure 3](#figure--fig:bibel92-hinf-weights)).
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<a id="figure--fig:bibel92-hinf-weights"></a>
|
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|
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{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="<span class='figure-number'>Figure 3: </span>Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
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|
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The weights on the input and output variables are selected to reflect the spatial and **frequency dependence** of the respective signals and performance specifications.
|
||||
|
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These inputs and output weighting functions are defined as rational, stable and **minimum-phase transfer function** (no poles or zero in the right half plane).
|
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|
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## General Guidelines for Weight Selection: \\(W\_S\\) {#general-guidelines-for-weight-selection-w-s}
|
||||
|
||||
\\(W\_S\\) is selected to reflect the desired **performance characteristics**.
|
||||
The sensitivity function \\(S\\) should have low gain at low frequency for good tracking performance and high gain at high frequencies to limit overshoot.
|
||||
We have to select \\(W\_S\\) such that \\({W\_S}^-1\\) reflects the desired shape of \\(S\\).
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||||
|
||||
<div class="cbox">
|
||||
|
||||
- **Low frequency gain**: set to the inverse of the desired steady state tracking error
|
||||
- **High frequency gain**: set to limit overshoot (\\(0.1\\) to \\(0.5\\) is a good compromise between overshoot and response speed)
|
||||
- **Crossover frequency**: chosen to limit the maximum closed-loop time constant (\\(\omega\_c \approx 1/\tau\\))
|
||||
|
||||
</div>
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||||
|
||||
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||||
## General Guidelines for Weight Selection: \\(W\_T\\) {#general-guidelines-for-weight-selection-w-t}
|
||||
|
||||
We want \\(T\\) near unity for good tracking of reference and near zero for noise suppresion.
|
||||
|
||||
<div class="cbox">
|
||||
|
||||
A high pass weight is usualy used on \\(T\\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:
|
||||
|
||||
- The **crossover frequency** is chosen to **limit the closed-loop bandwidth**
|
||||
- The **high frequency gain** is set high to proide **sensor noise rejection** and high frequency gain attenuation
|
||||
|
||||
</div>
|
||||
|
||||
When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the magnitude of theise weights at the crossover frequency is less that one to not violate \\(S+T=1\\).
|
||||
|
||||
|
||||
## Unmodeled dynamics weighting function {#unmodeled-dynamics-weighting-function}
|
||||
|
||||
Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output ([Figure 4](#figure--fig:bibel92-unmodeled-dynamics)).
|
||||
|
||||
<a id="figure--fig:bibel92-unmodeled-dynamics"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="<span class='figure-number'>Figure 4: </span>Unmodeled dynamics model" >}}
|
||||
|
||||
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown [Figure 5](#figure--fig:bibel92-weight-dynamics).
|
||||
|
||||
<a id="figure--fig:bibel92-weight-dynamics"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="<span class='figure-number'>Figure 5: </span>Example of unmodeled dynamics weight" >}}
|
||||
|
||||
|
||||
## Inputs and Output weighting function {#inputs-and-output-weighting-function}
|
||||
|
||||
It is possible to **weight the control input and actuator rate**.
|
||||
This is used to **prevent actuator saturation** and **limit amplification of sensor noise signals** on the control input signal.
|
||||
|
||||
Typically actuator input weights are constant over frequency and set at the inverse of the saturation limit.
|
||||
|
||||
|
||||
## Order of the weighting functions {#order-of-the-weighting-functions}
|
||||
|
||||
**The order of the optimal controller is equal to the order of the nominal plant model plus the order of the weights**. The complexity of the controller is increase as the order of the weights increases.
|
||||
|
||||
**The order of the weights should be kept reasonably low** to reduce the order of th resulting optimal compensator and avoid potential convergence problems in the DK interactions.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bibel, J. E., and D. S. Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” Naval Surface Warfare Center Dahlgren div va.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,58 @@
|
||||
+++
|
||||
title = "Control of spacecraft and aircraft"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Bryson 1993</a>)
|
||||
|
||||
Author(s)
|
||||
: Bryson, A. E.
|
||||
|
||||
Year
|
||||
: 1993
|
||||
|
||||
|
||||
## 9.2.3 Roll-Off Filters {#9-dot-2-dot-3-roll-off-filters}
|
||||
|
||||
[Spillover Effect]({{< relref "spillover_effect.md" >}})
|
||||
|
||||
> Synthesizing control logic using only one vibration mode means we are consciously **neglecting the higher-order vibration modes**.
|
||||
> When doing this, it is a good idea to insert "roll-off" into the control logic, so that the loop-transfer gain decreases rapidly with frequency beyond the control bandwidth.
|
||||
> This reduces the possibility of destabilizing the unmodelled higher frequency dynamics ("**spillover**").
|
||||
|
||||
|
||||
## 9.5 Robust Compensator Synthesis {#9-dot-5-robust-compensator-synthesis}
|
||||
|
||||
> LQG synthesis using feedback of estimated states will produce almost the same good response as LQR [...] for systems with control system bandwidths that are well below the frequency of the first vibration mode.
|
||||
> However, it may not be true for systems with higher control system bandwidths, even when one or more vibration modes are included in the control design model.
|
||||
|
||||
<!--quoteend-->
|
||||
|
||||
> If a rate sensor is co-located with an actuator on a flexible body, and its signal is fed back to the actuator, all vibration modes are stabilized.
|
||||
> If a rate sensor is not co-located with an actuator on a flexible body, ans its signal is fed back to the actuator, some vibration modes are stabilized and others are destabilized, depending on the location of the sensor relative to the actuator.
|
||||
|
||||
|
||||
## 9.5.2 Low-Authority Control/High-Authority Control [HAC-HAC]({{< relref "hac_hac.md" >}}) {#9-dot-5-dot-2-low-authority-control-high-authority-control-hac-hac--hac-hac-dot-md}
|
||||
|
||||
> [Figure 1](#figure--fig:bryson93-hac-lac) shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
|
||||
> LAC uses a co-located rate sensor to add damping to all the vibratory modes (but not the rigid-body mode).
|
||||
> HAC uses a separated displacement sensor to stabilize the rigid body mode, which slightly decreases the damping of the vibratory modes but not enough to produce instability (called "spillover")
|
||||
|
||||
<a id="figure--fig:bryson93-hac-lac"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="<span class='figure-number'>Figure 1: </span>HAC-LAC control concept" >}}
|
||||
|
||||
> LAC/HAC is usually insensitive to small deviation of the plant dynamics away from the design values, that is, it is **robust** to plant parameter changes.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bryson, Arthur Earl. 1993. <i>Control of Spacecraft and Aircraft</i>. Princeton university press Princeton, New Jersey.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Position control in lithographic equipment"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Multivariable Control]({{< relref "multivariable_control.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Butler 2011</a>)
|
||||
|
||||
Author(s)
|
||||
: Butler, H.
|
||||
|
||||
Year
|
||||
: 2011
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Butler, H. 2011. “Position Control in Lithographic Equipment.” <i>IEEE Control Systems</i> 31 (5): 28–47. doi:<a href="https://doi.org/10.1109/mcs.2011.941882">10.1109/mcs.2011.941882</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,108 @@
|
||||
+++
|
||||
title = "Identification and decoupling control of flexure jointed hexapods"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>)
|
||||
|
||||
Author(s)
|
||||
: Chen, Y., & McInroy, J.
|
||||
|
||||
Year
|
||||
: 2000
|
||||
|
||||
|
||||
## Abstract {#abstract}
|
||||
|
||||
> By exploiting properties of the joint space mass-inertia matrix of flexure jointed hexapods, a new **decoupling method** is proposed.
|
||||
> The new decoupling method, through a **static** input-output mapping, transforms the highly coupled 6 inputs 6 outputs dynamics into 6 independent single-input single-output channels.
|
||||
> Prior decoupling control algorithms imposed severe constraints on the allowable geometry, workspace and payload.
|
||||
> This paper derives a new algorithm which removes these constraints, thus greatly expanding the applications.
|
||||
> Based on the new decoupling algorithm, an **identification algorithm** is introduced to identify the **joint space mass-inertia matrix** using payload acceleration and base forces.
|
||||
> This algorithm can be used for precision payload calibration, thus improving performance and removing the labor required to design the control for different payloads.
|
||||
> The new decoupling algorithm is experimentally compared to earlier techniques.
|
||||
> These experimental results indicate that the new approach is practical, and improves performance.
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control.md" >}})) impose two constraints:
|
||||
|
||||
- the payload mass/inertia matrix is diagonal
|
||||
- the geometry of the platform and attachment of the payload must be carefully chosen
|
||||
|
||||
This limits the applications significantly.
|
||||
|
||||
The algorithm derived herein removes these constraints, thus greatly expanding the potential applications.
|
||||
|
||||
|
||||
## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
|
||||
|
||||
The derivation of the dynamic model is done in (<a href="#citeproc_bib_item_2">McInroy 1999</a>) ([Notes]({{< relref "mcinroy99_dynam.md" >}})).
|
||||
|
||||
<a id="figure--fig:chen00-flexure-hexapod"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="<span class='figure-number'>Figure 1: </span>A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}}
|
||||
|
||||
In the joint space, the dynamics of a flexure jointed hexapod are written as:
|
||||
|
||||
\begin{equation}
|
||||
\vec{f}\_b = \vec{f}\_m - \bm{K}(\vec{l} - \vec{l}\_r) - \bm{B} \dot{\vec{l}}
|
||||
\end{equation}
|
||||
|
||||
\begin{aligned}
|
||||
& \left( {}^U\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} \right) \ddot{\vec{l}} + \\\\
|
||||
& {}^U\_B\bm{R} \bm{J}^T \bm{B} \dot{\vec{l}} + {}^U\_B\bm{R}\bm{J}^T \bm{K}(\vec{l} - \vec{l}\_r) = \\\\
|
||||
& {}^U\_B\bm{R} \bm{J}^T \vec{f}\_m + \vec{\mathcal{F}}\_e + \vec{\mathcal{F}} + \vec{\mathcal{C}} - \\\\
|
||||
& \left( {}^U\_B\bm{R} \bm{J}^T \bm{M}\_s + {}^U\_P\bm{R} {}^P\bm{M}\_x {}^U\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1} \right) \ddot{\vec{q}}\_s
|
||||
\end{aligned}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\bm{J}\\) is the \\(6 \times 6\\) hexapod Jacobian relating payload Cartesian movements, expressed in {P}, to strut length changes in the joint space
|
||||
- \\({}^B\_U\bm{R}\\) is the \\(6 \times 6\\) rotation matrix from the base frame {B} to the universal inertial frame of reference {U} (it consists of two identical \\(3 \times 3\\) rotation matrices forming a block diagonal \\(6 \times 6\\) matrix)
|
||||
- \\(\bm{J}\_c\\) and \\(\bm{J}\_B\\) are \\(6 \times 6\\) Jacobian matrices capturing base motion
|
||||
- \\({}^P\bm{M}\_x\\) is the \\(6 \times 6\\) mass-inertia matrix of the payload found with respect to the payload frame {P}
|
||||
- \\(\bm{M}\_s\\) is a diagonal \\(6 \times 6\\) matrix containing the moving mass of each strut
|
||||
- \\(\bm{B}\\) and \\(\bm{K}\\) are \\(6 \times 6\\) diagonal matrices containing the damping of stiffness, respectively, of each strut
|
||||
- \\(\vec{l}\\) is the \\(6 \times 1\\) vector of strut lengths, and \\(\vec{l}\_r\\) is the constant vector of relaxed strut length
|
||||
- \\(\vec{f}\_b\\) is the vector of forces exerted at the bottom of the strut
|
||||
- \\(\vec{f}\_m\\) is the vector of strut motor forces
|
||||
- \\(\ddot{\vec{q}}\_s\\) is a \\(6 \times 1\\) vector of base accelerations along each strut plus some Coriolis terms
|
||||
- \\(\vec{\mathcal{F}}\_e\\) is a vector of payload exogenous generalized forces
|
||||
- \\(\vec{\mathcal{C}}\\) is a vector containing all the Coriolis and centripetal terms except the Coriolis terms in \\(\ddot{\vec{q}}\_s\\)
|
||||
- \\(\vec{\mathcal{G}}\\) is a vector containing all gravity terms
|
||||
|
||||
\begin{aligned}
|
||||
\bm{M}\_p & \ddot{\vec{p}}\_s + \bm{B} \dot{\vec{p}}\_s + \bm{K} \vec{p}\_s = \vec{f}\_m + \\\\
|
||||
& \bm{M}\_q \ddot{\vec{q}}\_s + \bm{B} \dot{\vec{q}}\_s + \bm{J}^{-T} {}^U\_B\bm{R}^T \vec{\mathcal{F}}\_e
|
||||
\end{aligned}
|
||||
|
||||
where
|
||||
|
||||
- \\(\bm{M}\_p = \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} + \bm{M}\_s\\)
|
||||
- \\(\bm{M}\_q = \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} - \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1}\\)
|
||||
|
||||
\\(\bm{M}\_p\\) and \\(\bm{M}\_q\\) are joint space mass-inertia matrices.
|
||||
|
||||
|
||||
## Decoupling the Dynamics of Flexure Jointed Hexapods {#decoupling-the-dynamics-of-flexure-jointed-hexapods}
|
||||
|
||||
|
||||
## Identification of Joint Space Mass-Inertia Matrix {#identification-of-joint-space-mass-inertia-matrix}
|
||||
|
||||
|
||||
## Experimental Results {#experimental-results}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)</i>. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>McInroy, J. E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Decoupled control of flexure-jointed hexapods using estimated joint-space mass-inertia matrix"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Decoupled Control]({{< relref "decoupled_control.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Chen and McInroy 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Chen, Y., & McInroy, J.
|
||||
|
||||
Year
|
||||
: 2004
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Y., and J. E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” <i>IEEE Transactions on Control Systems Technology</i> 12 (3): 413–21. doi:<a href="https://doi.org/10.1109/tcst.2004.824339">10.1109/tcst.2004.824339</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,92 @@
|
||||
+++
|
||||
title = "Enhanced damping of flexible structures using force feedback"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Active Damping]({{< relref "active_damping.md" >}}), [Integral Force Feedback]({{< relref "integral_force_feedback.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Chesné, Milhomem, and Collette 2016</a>)
|
||||
|
||||
Author(s)
|
||||
: Simon Chesné, Milhomem, A., & Collette, C.
|
||||
|
||||
Year
|
||||
: 2016
|
||||
|
||||
One problem of Integral Force Feedback (IFF) is that the achievable damping decreases at high frequency.
|
||||
A modification of the IFF is proposed in order to significantly increase the damping of **a** selected mode.
|
||||
|
||||
The test system is shown in [Figure 1](#figure--fig:chesne16-2dof-system).
|
||||
|
||||
Classical IFF corresponds to:
|
||||
|
||||
\begin{equation}
|
||||
H(s) = \frac{g}{s}
|
||||
\end{equation}
|
||||
|
||||
<a id="figure--fig:chesne16-2dof-system"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/chesne16_2dof_system.png" caption="<span class='figure-number'>Figure 1: </span>Two DoF system representing a flexible structuer controlled by an active mount" >}}
|
||||
|
||||
The proposed controller, called **alpha controller** is:
|
||||
|
||||
\begin{equation}
|
||||
H(s) = g \frac{s + \alpha}{s^2}
|
||||
\end{equation}
|
||||
|
||||
where \\(\alpha\\) is a parameter.
|
||||
|
||||
A new pair of pole/zero has been introduced.
|
||||
The new pole is located at \\(s = 0\\) and the zeros at \\(s = -\alpha\\).
|
||||
|
||||
For \\(\omega > \alpha\\) the controller is essentially an integrator.
|
||||
For \\(\omega < \alpha\\) the controller is a double integrator.
|
||||
|
||||
Depending on the chosen \\(\alpha\\) we obtain different root locus as shown in [Figure 2](#figure--fig:chesne16-root-locus-alpha).
|
||||
There is an optimal gain \\(\alpha^\star\\) at which the attainable damping of the flexible mode is maximized.
|
||||
|
||||
<a id="figure--fig:chesne16-root-locus-alpha"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/chesne16_root_locus_alpha.png" caption="<span class='figure-number'>Figure 2: </span>Root locus with the alpha controller for different values of \\(\alpha\\)" >}}
|
||||
|
||||
The obtained transmissibility is shown without controller, for classical IFF and for \\(\alpha\\) controller in [Figure 3](#figure--fig:chesne16-transmissibility).
|
||||
|
||||
Using the \\(\alpha\\) controller, the compliance is however degraded a lot.
|
||||
|
||||
<a id="figure--fig:chesne16-transmissibility"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/chesne16_transmissibility.png" caption="<span class='figure-number'>Figure 3: </span>Transmissibility \\(x\_1/x\_0\\)" >}}
|
||||
|
||||
In order to recover the compliance at low frequency, high pass filters can be added to the controller.
|
||||
|
||||
\begin{equation}
|
||||
H(s) = g \frac{s + \alpha}{(s + \beta)^2}
|
||||
\end{equation}
|
||||
|
||||
The condition for stability found here is:
|
||||
|
||||
\begin{equation}
|
||||
\alpha \ge \beta/2
|
||||
\end{equation}
|
||||
|
||||
<div class="sum">
|
||||
|
||||
The active damping of flexible structures with collocated force sensor/actuator pairs have been reviewed in this Note.
|
||||
In the first part of the Note, two limitations of the integral force feedback (IFF) have been discussed, which are the limited damping of flexible modes and the loss of compliance.
|
||||
By slightly modifying the controller, it has been shown that the active damping of a target mode can be significantly increased.
|
||||
Analytical formulas of the optimal parameters have been derived.
|
||||
In the second part, the loss of compliance inherent to IFF has been addressed.
|
||||
It has been shown that, when a high-pass filter is inserted into the IFF controller, the compliance at low frequency can be recovered but the unconditional stability is lost.
|
||||
On the other side, with the new proposed control law, the stability is always guaranteed even when using a high-pass filter.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chesné, S., A. Milhomem, and C. Collette. 2016. “Enhanced Damping of Flexible Structures Using Force Feedback.” <i>Journal of Guidance, Control, and Dynamics</i> 39 (7): 1654–58. doi:<a href="https://doi.org/10.2514/1.g001620">10.2514/1.g001620</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,41 @@
|
||||
+++
|
||||
title = "Amplified piezoelectric actuators: static & dynamic applications"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Claeyssen et al. 2007</a>)
|
||||
|
||||
Author(s)
|
||||
: Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O.
|
||||
|
||||
Year
|
||||
: 2007
|
||||
|
||||
The amplified piezo actuator APA is an external leveraged actuator based on a shell used both for the ceramic **pre stress** and for the ceramic **motion magnification**.
|
||||
|
||||
It is based on low voltage multilayer piezoelectric ceramics (PZT type).
|
||||
In static conditions, their free strain \\(S\_p\\) is typically 0.1% when driven at 150 V.
|
||||
|
||||
The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
|
||||
The flatter is the actuator, the higher is the amplification.
|
||||
|
||||
Piezoceramics can bear large compressive stress but they can not bear tensile forces with a good reliability.
|
||||
The usual way to solve this limitation consists in prestressing the ceramics by maintaining a compressive stress.
|
||||
This introduces another force limit: if the internal dynamic forces are above the prestress, the actuator is endangered because of the ceramic goes in tensile stress and also the ceramic stack looses contact with the shell interface.
|
||||
|
||||
For many APA actuators, the amplitude of maximal applicable external force is close to half the actuator blocked force.
|
||||
|
||||
The maximum dynamic force achievable by the actuator is determined by the prestress.
|
||||
The prestress design allows a peak force equal to half the blocked force.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Claeyssen, F., R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<a href="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,82 @@
|
||||
+++
|
||||
title = "Review of active vibration isolation strategies"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Collette, Janssens, and Artoos 2011</a>)
|
||||
|
||||
Author(s)
|
||||
: Collette, C., Janssens, S., & Artoos, K.
|
||||
|
||||
Year
|
||||
: 2011
|
||||
|
||||
|
||||
## Background and Motivations {#background-and-motivations}
|
||||
|
||||
|
||||
### Passive Isolation Tradeoffs {#passive-isolation-tradeoffs}
|
||||
|
||||
1DoF Equations:
|
||||
|
||||
\begin{equation}
|
||||
\boxed{X(s) = \underbrace{\frac{cs + k}{ms^2 + cs + k}}\_{T\_{wx}(s)} W(s) + \underbrace{\frac{1}{ms^2 + cs + k}}\_{T\_{Fx}(s)} F(s)}
|
||||
\end{equation}
|
||||
|
||||
- \\(T\_{wx}(s)\\) is called the **transmissibility** of the isolator. It characterize the way seismic vibrations \\(w\\) are transmitted to the equipment.
|
||||
- \\(T\_{Fx}(s)\\) is called the **compliance**. It characterize the capacity of disturbing forces \\(F\\) to create motion \\(x\\) of the equipment.
|
||||
|
||||
In order to minimize the vibrations of a sensitive equipment, a general objective to design a good isolator is to minimize both \\(\abs{T\_{wx}}\\) and \\(\abs{T\_{Fx}}\\) in the frequency range of interest.
|
||||
|
||||
To decrease the amplitude of the overshoot at the resonance frequency, **damping** can be increased.
|
||||
The price to pay is degradation of the isolation at high frequency (the roll off becomes \\(-1\\) instead of \\(-2\\)).
|
||||
|
||||
**First Trade-off**: Trade-off between damping and isolation.
|
||||
|
||||
To improve the transmissibility, the resonance frequency can be decreased.
|
||||
However, the systems becomes more sensitive to external force \\(F\\) applied on the equipment.
|
||||
|
||||
**Second trade-off**: Trade-off between isolation and robustness to external force
|
||||
|
||||
|
||||
### Active Isolation {#active-isolation}
|
||||
|
||||
We apply a feedback control.
|
||||
The general expression of the force delivered by the actuator is \\(f = g\_a \ddot{x} + g\_v \dot{x} + g\_p x\\). \\(g\_a\\), \\(g\_v\\) and \\(g\_p\\) are constant gains.
|
||||
|
||||
<a id="table--table:active-isolation"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--table:active-isolation">Table 1</a>:</span>
|
||||
Active isolation techniques
|
||||
</div>
|
||||
|
||||
| **Feedback Signal** | **Effect** | **Applications** |
|
||||
|---------------------|------------------------------------------|------------------|
|
||||
| Acceleration | Add virtual mass | Few |
|
||||
| Velocity | Add virtual dashpot connected to the sky | Sky-Hook Damping |
|
||||
| Position | Add virtual spring connected to the sky | Sky-Hook Spring |
|
||||
|
||||
|
||||
## Practical Realizations {#practical-realizations}
|
||||
|
||||
|
||||
## Sensor Limitations {#sensor-limitations}
|
||||
|
||||
|
||||
## Conclusions {#conclusions}
|
||||
|
||||
<a id="figure--fig:collette11-comp-isolation-strategies"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="<span class='figure-number'>Figure 1: </span>Comparison of Active Vibration Isolation Strategies" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., S. Janssens, and K. Artoos. 2011. “Review of Active Vibration Isolation Strategies.” <i>Recent Patents on Mechanical Engineeringe</i> 4 (3): 212–19. doi:<a href="https://doi.org/10.2174/2212797611104030212">10.2174/2212797611104030212</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,107 @@
|
||||
+++
|
||||
title = "Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Sensor Fusion]({{< relref "sensor_fusion.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Collette and Matichard 2014</a>)
|
||||
|
||||
Author(s)
|
||||
: Collette, C., & Matichard, F.
|
||||
|
||||
Year
|
||||
: 2014
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
[Sensor Fusion]({{< relref "sensor_fusion.md" >}}) is used to combine the benefits of different types of sensors:
|
||||
|
||||
- Relative sensor for DC positioning capability at low frequency
|
||||
- Inertial sensors for isolation at high frequency
|
||||
- Force sensor / collocated sensor to improve the robustness
|
||||
|
||||
|
||||
## Different types of sensors {#different-types-of-sensors}
|
||||
|
||||
In this paper, three types of sensors are used. Their advantages and disadvantages are summarized [Table 1](#table--tab:sensors).
|
||||
|
||||
> Several types of sensors can be used for the feedback control of vibration isolation systems:
|
||||
>
|
||||
> - Feedback control based on **relative motion sensors** (inductive, capactive, ferromagnetic sensors...) typically permits to servo-position a system or platform relative to a reference (e.g. floor or support base), but does not provide isolation from the ground motion.
|
||||
> - Feedback control based on **force sensors** typically lowers the effective natural frequency, and therefore increases the isolation, but sacrifices the systems compliance in doing so.
|
||||
> - Feedback control based on **inertial sensors** (geophones, seismometers, accelerometers...) improves not only the vibration isolation but also the compliance. Inertial sensors are, however, AC coupled and noisy at low frequencies.
|
||||
|
||||
<a id="table--tab:sensors"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:sensors">Table 1</a>:</span>
|
||||
Types of sensors
|
||||
</div>
|
||||
|
||||
| Sensors | Advantages | Disadvantages |
|
||||
|------------------|----------------------------------|---------------------------------------|
|
||||
| Relative motion | Servo-position | No isolation from ground motion |
|
||||
| Force sensors | Improve isolation | Increase compliance |
|
||||
| Inertial sensors | Improve isolation and compliance | AC couple and noisy at high frequency |
|
||||
|
||||
|
||||
## Inertial Control and sensor fusion configurations {#inertial-control-and-sensor-fusion-configurations}
|
||||
|
||||
For a simple 1DoF model, two fusion-sensor configuration are studied. The results are summarized [Table 2](#table--tab:fusion-trade-off).
|
||||
|
||||
<a id="table--tab:fusion-trade-off"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:fusion-trade-off">Table 2</a>:</span>
|
||||
Sensor fusion configurations
|
||||
</div>
|
||||
|
||||
| Low freq. sensor | High freq. sensor | Transmissibility | Compliance | Trade-off |
|
||||
|------------------|-------------------|------------------|------------|----------------------------------------------------|
|
||||
| Inertial | Force sensor | Unchanged | Degraded | Sensor noise filtering / compliance degradation |
|
||||
| Inertial | Relative sensor | Degraded | Unchanged | Isolation in the bandwidth / amplification outside |
|
||||
|
||||
|
||||
## Flexible structure {#flexible-structure}
|
||||
|
||||
Flexibility is added between the inertial sensor and the actuator.
|
||||
Now the sensor and actuator are not collocated anymore and the system is unstable because there is no zero between the two poles.
|
||||
We use sensor fusion to obtain stability at high frequency.
|
||||
|
||||
|
||||
### Inertial and small accelerometer {#inertial-and-small-accelerometer}
|
||||
|
||||
The idea is to use a small accelerometer which is easier to locate near the actuator at high frequency.
|
||||
However, it is important to verify that the noise introduced by the accelerometer does not degrades too much the isolation performance.
|
||||
|
||||
|
||||
### Inertial and force sensor {#inertial-and-force-sensor}
|
||||
|
||||
Here the advantage is that the deformation mode is almost not present in the open-loop transfer function.
|
||||
This simplifies the loop shaping of the controller.
|
||||
|
||||
|
||||
### Inertial and relative sensor {#inertial-and-relative-sensor}
|
||||
|
||||
The relative sensor introduces coupling between both side of the actuator which induces degradation of the isolation at high frequency. However, the compliance remains unchanged at high frequency.
|
||||
|
||||
|
||||
## Conclusion {#conclusion}
|
||||
|
||||
Fusion of inertial instruments with sensors collocated with the actuator permits to increase the feedback control bandwidth of active isolation systems.
|
||||
|
||||
Three types of sensors have been considered for the high frequency part of the fusion:
|
||||
|
||||
- The fusion with a **relative sensor** improves the stability but compromises the transmissibility. It can be of interested for stiff suspension where high frequency isolation can be sacrified to improve stability.
|
||||
- The fusion with an **accelerometre** is used to increase the loop gain. However, as the accelerometer is not dual with the actuator, there is no guaranty stability when the isolation stage is mounted on a flexible support.
|
||||
- The fusion with a **force sensor** can be used to increase the loop gain with little effect on the compliance and passive isolation, provided that the blend is possible and that no active damping of flexible modes is required.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In <i>International Conference on Noise and Vibration Engineering (ISMA2014)</i>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,32 @@
|
||||
+++
|
||||
title = "Sensor fusion methods for high performance active vibration isolation systems"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Collette and Matichard 2015</a>)
|
||||
|
||||
Author(s)
|
||||
: Collette, C., & Matichard, F.
|
||||
|
||||
Year
|
||||
: 2015
|
||||
|
||||
In order to have good stability margins, it is common practice to collocate sensors and actuators. This ensures alternating poles and zeros along the imaginary axis. Then, each phase lag introduced by the poles is compensated by phase lead introduced by the zeroes. This guarantees stability and such system is referred to as **hyperstable**.
|
||||
|
||||
In this paper, we study and compare different sensor fusion methods combining inertial sensors at low frequency with sensors adding stability at high frequency.
|
||||
The stability margins of the controller can be significantly increased with no or little effect on the low-frequency active isolation, provided that the two following conditions are fulfilled:
|
||||
|
||||
- the high frequency sensor and the actuator are dual
|
||||
- there exists a bandwidth where we can superimpose the open loop transfer functions obtained with the two sensors.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” <i>Journal of Sound and Vibration</i> 342: 1–21. doi:<a href="https://doi.org/10.1016/j.jsv.2015.01.006">10.1016/j.jsv.2015.01.006</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,124 @@
|
||||
+++
|
||||
title = "Exploring the pareto fronts of actuation technologies for high performance mechatronic systems"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Csencsics and Schitter 2020</a>)
|
||||
|
||||
Author(s)
|
||||
: Csencsics, E., & Schitter, G.
|
||||
|
||||
Year
|
||||
: 2020
|
||||
|
||||
|
||||
## Abstract {#abstract}
|
||||
|
||||
> This paper proposes a novel method for estimating the limitations of individual actuation technologies for a desired system class based on analytically obtained relations, which can be used to systematically trade off desired range and speed specifications in the design phase.
|
||||
> The method is presented along the example of **fast steering mirrors** with the tradeoff limit curves estimated for the established **piezoelectric**, **lorentz force** and **hybrid reluctance** actuation technologies.
|
||||
|
||||
<a id="figure--fig:csencsics20-fsm-schematic"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/csencsics20_fsm_schematic.png" caption="<span class='figure-number'>Figure 1: </span>Fast Steering Mirror system. The main components are: mirror, actuators, position sensors and suspension system." >}}
|
||||
|
||||
|
||||
## Fast Steering Mirrors {#fast-steering-mirrors}
|
||||
|
||||
|
||||
### Application area and performance specification {#application-area-and-performance-specification}
|
||||
|
||||
<a id="table--tab:fsm-requirements"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:fsm-requirements">Table 1</a>:</span>
|
||||
FSM performance requirements for two application
|
||||
</div>
|
||||
|
||||
| Application | Pointing | Scanning |
|
||||
|-------------------|-----------------|----------|
|
||||
| System Range | large | large |
|
||||
| System Dimensions | arbitrary | compact |
|
||||
| Main objective | dist. rejection | tracking |
|
||||
| Bandwidth | high | high |
|
||||
| Motion amplitude | small | large |
|
||||
| Mover inertia | arbitrary | small |
|
||||
| Precision | high | high |
|
||||
|
||||
|
||||
### Safe operating area {#safe-operating-area}
|
||||
|
||||
The concept of the Safe Operating Area (SOA) relates the frequency of a sinusoidal reference to the maximum admissible scan amplitude that still stays within the limits of the system.
|
||||
|
||||
From [Figure 2](#figure--fig:csencsics20-soa) we can already see that piezo are typically used for system with high bandwidth and small range.
|
||||
|
||||
<a id="figure--fig:csencsics20-soa"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/csencsics20_soa.png" caption="<span class='figure-number'>Figure 2: </span>Measured safe operating area of closed-loop FSM systems with sinusoidal reference signals. Piezo actuated in blue, lorentz force actuated in red and hybrid reluctance actuated in green." >}}
|
||||
|
||||
|
||||
## Limitations of actuator technology {#limitations-of-actuator-technology}
|
||||
|
||||
|
||||
### Piezo actuation {#piezo-actuation}
|
||||
|
||||
Piezo actuated FMS are in general **high stiffness** system, for which the **bandwidth limitation** for feedback control is typically given by the **first mechanical resonance**.
|
||||
|
||||
<a id="figure--fig:csencsics20-typical-piezo-fsm"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/csencsics20_typical_piezo_fsm.png" caption="<span class='figure-number'>Figure 3: </span>Piezo actuated FSM cross section" >}}
|
||||
|
||||
The angular range of the FSM is:
|
||||
|
||||
\begin{equation}
|
||||
\phi = \frac{L/1000}{2 d}
|
||||
\end{equation}
|
||||
|
||||
with \\(L\\) the length of the stack, and d the distance between the stacks and the center of rotation (the factor 1000 is linked to the fact that typical piezo stack have a store equal to 0.1% of their length).
|
||||
|
||||
The first resonance frequency is:
|
||||
|
||||
\begin{equation}
|
||||
f\_{PZA} = \frac{1}{2\pi L}\sqrt{\frac{3E}{\rho\_\text{piezo}}}
|
||||
\end{equation}
|
||||
|
||||
with \\(E\\) the elastic modulus and \\(\rho\_\text{piezo}\\) the density of the piezo material.
|
||||
|
||||
As the resonance limits the achievable bandwidth, we therefore have that \\(f\_{\text{max,PZA}} \propto 1/\phi\\).
|
||||
|
||||
|
||||
### Lorentz force actuation {#lorentz-force-actuation}
|
||||
|
||||
Lorentz force actuated FSM are in general **low stiffness** systems, which typically have a control bandwidth beyond the suspension mode that is usually limited by the **internal modes of the moving part**.
|
||||
|
||||
The mover's mass is dominating the dynamics of low stiffness systems beyond the suspension mode.
|
||||
|
||||
<a id="figure--fig:csencsics20-typical-lorentz-fsm"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/csencsics20_typical_lorentz_fsm.png" caption="<span class='figure-number'>Figure 4: </span>Lorentz force actuator designs." >}}
|
||||
|
||||
\begin{equation}
|
||||
f\_\text{max,LFA} = \frac{1}{2\pi} k\_\text{LFA} \sqrt{\frac{1}{\phi J\_\text{init} + \Delta\_J + 2 d \phi^2}}
|
||||
\end{equation}
|
||||
|
||||
|
||||
### Hybrid reluctance force actuation {#hybrid-reluctance-force-actuation}
|
||||
|
||||
<a id="figure--fig:csencsics20-typical-hybrid-reluctance-fsm"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/csencsics20_typical_hybrid_reluctance_fsm.png" caption="<span class='figure-number'>Figure 5: </span>Hybrid reluctance actuator designs" >}}
|
||||
|
||||
|
||||
## Pareto front estimates for FSM systems {#pareto-front-estimates-for-fsm-systems}
|
||||
|
||||
<a id="figure--fig:csencsics20-pareto-estimate"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/csencsics20_pareto_estimate.png" caption="<span class='figure-number'>Figure 6: </span>Two dimensional performance space for FSM systems showing the tradeoff between range and bandwidth. Commercially available (symbols) as well as academically reported systems (dots) actuated by piezo (blue), Lorentz force (red) and reluctance actuators (green) are depicted." >}}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Csencsics, Ernst, and Georg Schitter. 2020. “Exploring the Pareto Fronts of Actuation Technologies for High Performance Mechatronic Systems.” <i>IEEE/ASME Transactions on Mechatronics</i>. IEEE.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,41 @@
|
||||
+++
|
||||
title = "The stewart platform manipulator: a review"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Dasgupta and Mruthyunjaya 2000</a>)
|
||||
|
||||
Author(s)
|
||||
: Dasgupta, B., & Mruthyunjaya, T.
|
||||
|
||||
Year
|
||||
: 2000
|
||||
|
||||
<a id="table--tab:parallel-vs-serial-manipulators"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:parallel-vs-serial-manipulators">Table 1</a>:</span>
|
||||
Parallel VS serial manipulators
|
||||
</div>
|
||||
|
||||
| | **Advantages** | **Disadvantages** |
|
||||
|--------------|---------------------------|-----------------------|
|
||||
| **Serial** | Maneuverability | Poor precision |
|
||||
| | Large workspace | Bends under high load |
|
||||
| | | Vibrate at high speed |
|
||||
| **Parallel** | High stiffness | Small workspace |
|
||||
| | Good dynamic performances | |
|
||||
| | Precise positioning | |
|
||||
|
||||
The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platform) connected through six extensible legs, each with spherical joints at both ends.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Dasgupta, B., and T. S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” <i>Mechanism and Machine Theory</i> 35 (1): 15–40. doi:<a href="https://doi.org/10.1016/s0094-114x(99)00006-3">10.1016/s0094-114x(99)00006-3</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,34 @@
|
||||
+++
|
||||
title = "A survey of control issues in nanopositioning"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Devasia, Eleftheriou, and Moheimani 2007</a>)
|
||||
|
||||
Author(s)
|
||||
: Devasia, S., Eleftheriou, E., & Moheimani, S. R.
|
||||
|
||||
Year
|
||||
: 2007
|
||||
|
||||
- Talks about Scanning Tunneling Microscope (STM) and Scanning Probe Microscope (SPM)
|
||||
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}): Creep, Hysteresis, Vibrations, Modeling errors
|
||||
- Interesting analysis about Bandwidth-Precision-Range tradeoffs
|
||||
- Control approaches for piezoelectric actuators: feedforward, Feedback, Iterative, Sensorless controls
|
||||
|
||||
<a id="figure--fig:devasia07-piezoelectric-tradeoff"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="<span class='figure-number'>Figure 1: </span>Tradeoffs between bandwidth, precision and range" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” <i>IEEE Transactions on Control Systems Technology</i> 15 (5). IEEE: 802–23.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,128 @@
|
||||
+++
|
||||
title = "Nanopositioning system with force feedback for high-performance tracking and vibration control"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Force Sensors]({{< relref "force_sensors.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Fleming 2010</a>)
|
||||
|
||||
Author(s)
|
||||
: Fleming, A.
|
||||
|
||||
Year
|
||||
: 2010
|
||||
|
||||
|
||||
## Summary {#summary}
|
||||
|
||||
- The noise generated by a piezoelectric force sensor is much less than a capacitive sensor
|
||||
- Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor
|
||||
- Noise of a piezoelectric force sensor
|
||||
- IFF with a piezoelectric stack actuator and piezoelectric force sensor
|
||||
- A force sensor is used as a displacement sensor below the frequency of the first zero
|
||||
- Sensor fusion architecture with a capacitive sensor and a force sensor and using complementary filters
|
||||
- Virtual sensor fusion architecture (called low-frequency bypass)
|
||||
- Analog implementation of the control strategies to avoid quantization noise, finite resolution and sampling delay
|
||||
|
||||
|
||||
## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
|
||||
|
||||
<a id="figure--fig:fleming10-piezo-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="<span class='figure-number'>Figure 1: </span>Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
|
||||
|
||||
The actuator experiences an internal stress in response to an applied voltage.
|
||||
This stress is represented by the voltage dependent force \\(F\_a\\) and is related to free displacement by
|
||||
\\[ \Delta L = \frac{F\_a}{k\_a} \\]
|
||||
|
||||
- \\(\Delta L\\) is the change in actuator length in [m]
|
||||
- \\(k\_a\\) is the actuator stiffness in [N/m]
|
||||
|
||||
The developed force \\(F\_a\\) is related to the applied voltage by:
|
||||
\\[ \Delta L = d\_{33} n V\_a \\]
|
||||
|
||||
- \\(d\_{33}\\) is the piezoelectric strain constant in [m/V]
|
||||
- \\(n\\) is the number of layers
|
||||
- \\(V\_a\\) is the applied voltage in [V]
|
||||
|
||||
Combining the two equations, we obtain:
|
||||
\\[ F\_a = d\_{33} n k\_a V\_a \\]
|
||||
|
||||
The ratio of the developed force to applied voltage is \\(d\_{33} n k\_a\\) in [N/V].
|
||||
We denote this constant by \\(g\_a\\) and:
|
||||
\\[ F\_a = g\_a V\_a, \quad g\_a = d\_{33} n k\_a \\]
|
||||
|
||||
|
||||
## Dynamics of a piezoelectric force sensor {#dynamics-of-a-piezoelectric-force-sensor}
|
||||
|
||||
Piezoelectric force sensors provide a high sensitivity and bandwidth with low noise at high frequencies.
|
||||
|
||||
If a **single wafer** of piezoelectric material is sandwiched between the actuator and platform:
|
||||
\\[ D = d\_{33} T \\]
|
||||
|
||||
- \\(D\\) is the amount of generated charge per unit area in \\([C/m^2]\\)
|
||||
- \\(T\\) is the stress in \\([N/m^2]\\)
|
||||
- \\(d\_{33}\\) is the piezoelectric strain constant in \\([m/V] = [C/N]\\)
|
||||
|
||||
The generated charge is then
|
||||
\\[ q = d\_{33} F\_s \\]
|
||||
|
||||
If an **n-layer** piezoelectric transducer is used as a force sensor, the generated charge is then:
|
||||
\\[ q = n d\_{33} F\_s \\]
|
||||
|
||||
---
|
||||
|
||||
We can use a **charge amplifier** to measure the force \\(F\_s\\).
|
||||
|
||||
{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="<span class='figure-number'>Figure 2: </span>Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \\(q\\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
|
||||
|
||||
The output voltage \\(V\_s\\) is equal to
|
||||
\\[ V\_s = -\frac{q}{C\_s} = -\frac{n d\_{33}F\_s}{C\_s} \\]
|
||||
that is, the scaling between the force and voltage is \\(-\frac{n d\_{33}F\_s}{C\_s}\ [V/N]\\) .
|
||||
|
||||
---
|
||||
|
||||
We can also use a voltage amplifier.
|
||||
In that case, the generated charge is deposited on the transducer's internal capacitance.
|
||||
|
||||
The open-circuit voltage of a piezoelectric force sensor is:
|
||||
\\[ V\_s = \frac{n d\_{33} F\_s}{C} \\]
|
||||
|
||||
- \\(C\\) is the transducer capacitance defined by \\(C = n \epsilon\_T A / h\\) in [F]
|
||||
- \\(A\\) is the area in \\([m^2]\\)
|
||||
- \\(h\\) is the layer thickness in [m]
|
||||
- \\(\epsilon\_T\\) is the dielectric permittivity under a constant stress in \\([F/m]\\)
|
||||
|
||||
We obtain
|
||||
\\[ V\_s = g\_s F\_s, \quad g\_s = \frac{n d\_{33}}{C} \\]
|
||||
|
||||
|
||||
## Noise of a piezoelectric force sensor {#noise-of-a-piezoelectric-force-sensor}
|
||||
|
||||
As piezoelectric sensors have a capacitive source impedance, the sensor noise density \\(N\_{V\_s}(\omega)\\) is primarily due to current noise \\(i\_n\\) reacting the capacitive source impedance:
|
||||
\\[ N\_{V\_s}(\omega) = i\_n \frac{1}{C \omega} \\]
|
||||
|
||||
- \\(N\_{V\_s}\\) is the measured noise in \\(V/\sqrt{\text{Hz}}\\)
|
||||
- \\(i\_n\\) is the current noise in \\(A/\sqrt{\text{Hz}}\\)
|
||||
- \\(C\\) is the capacitance of the piezoelectric in \\(F\\)
|
||||
|
||||
The current noise density of a general purpose LM833 FET-input op-amp is \\(0.5\ pA/\sqrt{\text{Hz}}\\).
|
||||
The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and \\(100 \mu F\\).
|
||||
|
||||
|
||||
## Tested feedback control strategies {#tested-feedback-control-strategies}
|
||||
|
||||
<a id="figure--fig:fleming10-fb-control-strats"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming10_fb_control_strats.png" caption="<span class='figure-number'>Figure 3: </span>Comparison of: (a) basic integral control. (b) direct tracking control. (c) dual-sensor feedback. (d) low frequency bypass" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Estimating the resolution of nanopositioning systems from frequency domain data"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Fleming 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Fleming, A. J.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In <i>2012 IEEE International Conference on Robotics and Automation</i>. doi:<a href="https://doi.org/10.1109/icra.2012.6224850">10.1109/icra.2012.6224850</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,189 @@
|
||||
+++
|
||||
title = "A review of nanometer resolution position sensors: operation and performance"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Position Sensors]({{< relref "position_sensors.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Fleming 2013</a>)
|
||||
|
||||
Author(s)
|
||||
: Fleming, A. J.
|
||||
|
||||
Year
|
||||
: 2013
|
||||
|
||||
- Define concise performance metric and provide expressions for errors sources (non-linearity, drift, noise)
|
||||
- Review current position sensor technologies and compare their performance
|
||||
|
||||
|
||||
## Sensor Characteristics {#sensor-characteristics}
|
||||
|
||||
|
||||
### Calibration and nonlinearity {#calibration-and-nonlinearity}
|
||||
|
||||
Usually quoted as a percentage of the fill-scale range (FSR):
|
||||
|
||||
\begin{equation}
|
||||
\text{mapping error (\\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
|
||||
\end{equation}
|
||||
|
||||
With \\(e\_m(v)\\) is the mapping error.
|
||||
|
||||
<a id="figure--fig:mapping-error"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="<span class='figure-number'>Figure 1: </span>The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
|
||||
|
||||
|
||||
### Drift and Stability {#drift-and-stability}
|
||||
|
||||
If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.
|
||||
|
||||
<a id="figure--fig:drift-stability"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="<span class='figure-number'>Figure 2: </span>The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
|
||||
|
||||
|
||||
### Bandwidth {#bandwidth}
|
||||
|
||||
The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\\,dB\\).
|
||||
|
||||
Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.
|
||||
|
||||
The frequency domain position error is
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
e\_{bw}(s) &= x(s) - v(s) \\\\
|
||||
&= x(s) (1 - P(s))
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
If the actual position is a sinewave of peak amplitude \\(A = \text{FSR}/2\\):
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\
|
||||
&\approx \pm A n \frac{f}{f\_c}
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
with \\(n\\) is the low pass filter order corresponding to the sensor dynamics and \\(f\_c\\) is the measurement bandwidth.
|
||||
|
||||
Thus, the sensor bandwidth must be significantly higher than the operating frequency if dynamic errors are to be avoided.
|
||||
|
||||
|
||||
### Noise {#noise}
|
||||
|
||||
In addition to the actual position signal, all sensors produce some additive measurement noise.
|
||||
In many types of sensor, the majority of noise arises from the thermal noise in resistors and the voltage and current noise in conditioning circuit transistors.
|
||||
These noise processes can usually be approximated by a Gaussian random process.<br />
|
||||
|
||||
A Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.
|
||||
|
||||
The autocorrelation function of a random process \\(\mathcal{X}\\) is
|
||||
|
||||
\begin{equation}
|
||||
R\_{\mathcal{X}}(\tau) = E[\mathcal{X}(t)\mathcal{X}(t + \tau)]
|
||||
\end{equation}
|
||||
|
||||
where \\(E\\) is the expected value operator.
|
||||
|
||||
The variance of the process is equal to \\(R\_\mathcal{X}(0)\\) and is the expected value of the varying part squared:
|
||||
|
||||
\begin{equation}
|
||||
\text{Var} \mathcal{X} = E \left[ (\mathcal{X} - E[\mathcal{X}])^2 \right]
|
||||
\end{equation}
|
||||
|
||||
The standard deviation \\(\sigma\\) is the square root of the variance:
|
||||
|
||||
\begin{equation}
|
||||
\sigma\_\mathcal{X} = \sqrt{\text{Var} \mathcal{X}}
|
||||
\end{equation}
|
||||
|
||||
The standard deviation is also the Root Mean Square (RMS) value of a zero-mean random process.
|
||||
|
||||
The Power Spectral Density \\(S\_\mathcal{X}(f)\\) of a random process represents the distribution of power (or variance) across frequency \\(f\\).
|
||||
|
||||
For example, if the random process under consideration was measured in volts, the power spectral density would have the units of \\(V^2/\text{Hz}\\).
|
||||
|
||||
The Power Spectral Density can be obtained from the autocorrelation function from the Wiener-Khinchin relation:
|
||||
|
||||
\begin{equation}
|
||||
S\_{\mathcal{X}} = 2 \mathcal{F}\\{ R\_\mathcal{X}(\tau) \\} = 2 \int\_{-\infty}^{\infty} R\_\mathcal{X}(\tau) e^{-2j\pi f \tau} d\tau
|
||||
\end{equation}
|
||||
|
||||
If the power Spectral Density is known, the variance of the generating process can be found from the area under the curve:
|
||||
|
||||
\begin{equation}
|
||||
\sigma\_\mathcal{X}^2 = E[\mathcal{X}^2(t)] = R\_\mathcal{X}(0) = \int\_0^\infty S\_\mathcal{X}(f) df
|
||||
\end{equation}
|
||||
|
||||
Rather than plotting the frequency distribution of power, it is often convenient to plot the frequency distribution of the standard deviation, which is referred to as the spectral density.
|
||||
It is related to the power spectral density by a square root:
|
||||
|
||||
\begin{equation}
|
||||
\text{spectral density} = \sqrt{S\_\mathcal{X}(f)}
|
||||
\end{equation}
|
||||
|
||||
The units of \\(\sqrt{S\_\mathcal{X}(f)}\\) are \\(\text{units}/\sqrt{Hz}\\).
|
||||
|
||||
The spectral density if preferred in the electronics literature as the RMS value of a noise process can be determined directly from the noise density and effective bandwidth.
|
||||
|
||||
|
||||
### Resolution {#resolution}
|
||||
|
||||
The random noise of a position sensor causes an uncertainty in the measured position.
|
||||
If the distance between two measured locations is smaller than the uncertainty, it is possible to mistake one point for the other.
|
||||
|
||||
To characterize the resolution, we use the probability that the measured value is within a certain error bound.
|
||||
|
||||
If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \\(\sigma\\) (RMS value).
|
||||
|
||||
The empirical rule states that there is a \\(99.7\\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
|
||||
This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
|
||||
|
||||
Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
|
||||
There is usually a trade-off between bandwidth and resolution ([Figure 3](#figure--fig:tradeoff-res-bandwidth)).
|
||||
|
||||
<a id="figure--fig:tradeoff-res-bandwidth"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="<span class='figure-number'>Figure 3: </span>The resolution versus banwidth of a position sensor." >}}
|
||||
|
||||
Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
|
||||
As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
|
||||
A convenient method for reporting this ratio is in parts-per-million (ppm):
|
||||
|
||||
\begin{equation}
|
||||
\text{DNR}\_{\text{ppm}} = 10^6 \frac{\text{full scale range}}{6\sigma\text{-resolution}}
|
||||
\end{equation}
|
||||
|
||||
|
||||
## Comparison and summary {#comparison-and-summary}
|
||||
|
||||
<a id="table--tab:summary-position-sensors"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a>:</span>
|
||||
Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
|
||||
</div>
|
||||
|
||||
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|
||||
|----------------|------------------------------------|---------|------------|-------------|-----------|
|
||||
| Metal foil | \\(10-500\\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
|
||||
| Piezoresistive | \\(1-500\\,\mu m\\) | 5 ppm | 0.5 nm | >100 kHz | 1% FSR |
|
||||
| Capacitive | \\(10\\,\mu m\\) to \\(10\\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
|
||||
| Electrothermal | \\(10\\,\mu m\\) to \\(1\\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
|
||||
| Eddy current | \\(100\\,\mu m\\) to \\(80\\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
|
||||
| LVDT | \\(0.5-500\\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
|
||||
| Interferometer | Meters | | 0.5 nm | >100kHz | 1 ppm FSR |
|
||||
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A. J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” <i>Sensors and Actuators a: Physical</i> 190: 106–26. doi:<a href="https://doi.org/10.1016/j.sna.2012.10.016">10.1016/j.sna.2012.10.016</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Low-order damping and tracking control for scanning probe systems"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Fleming, Teo, and Leang 2015</a>)
|
||||
|
||||
Author(s)
|
||||
: Fleming, A. J., Teo, Y. R., & Leang, K. K.
|
||||
|
||||
Year
|
||||
: 2015
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” <i>Frontiers in Mechanical Engineering</i> 1. doi:<a href="https://doi.org/10.3389/fmech.2015.00014">10.3389/fmech.2015.00014</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,42 @@
|
||||
+++
|
||||
title = "Nanometre-cutting machine using a stewart-platform parallel mechanism"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Furutani, Suzuki, and Kudoh 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Furutani, K., Suzuki, M., & Kudoh, R.
|
||||
|
||||
Year
|
||||
: 2004
|
||||
|
||||
- Lever mechanism to amplify the motion of piezoelectric stack actuators
|
||||
- Use of flexure joints
|
||||
- Eddy current displacement sensors for control (decentralized)
|
||||
|
||||
{{< figure src="/ox-hugo/furutani04_ctrl_arch.png" >}}
|
||||
|
||||
- Isotropic performance (cubic configuration even if not said so)
|
||||
|
||||
Possible sources of error:
|
||||
|
||||
- position error of the link ends in assembly => simulation of position error and it is not significant
|
||||
- Inaccurate modelling of the links
|
||||
- insufficient generative force
|
||||
- unwanted deformation of the links
|
||||
|
||||
To minimize the errors, a calibration is done between the required leg length and the wanted platform pose.
|
||||
Then, it is fitted with 4th order polynomial and included in the control architecture.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Furutani, K., M. Suzuki, and R. Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” <i>Measurement Science and Technology</i> 15 (2): 467–74. doi:<a href="https://doi.org/10.1088/0957-0233/15/2/022">10.1088/0957-0233/15/2/022</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Measurement technologies for precision positioning"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Position Sensors]({{< relref "position_sensors.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Gao et al. 2015</a>)
|
||||
|
||||
Author(s)
|
||||
: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
|
||||
|
||||
Year
|
||||
: 2015
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” <i>CIRP Annals</i> 64 (2): 773–96. doi:<a href="https://doi.org/10.1016/j.cirp.2015.05.009">10.1016/j.cirp.2015.05.009</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,42 @@
|
||||
+++
|
||||
title = "Implementation challenges for multivariable control: what you did not learn in school!"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Multivariable Control]({{< relref "multivariable_control.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Garg 2007</a>)
|
||||
|
||||
Author(s)
|
||||
: Garg, S.
|
||||
|
||||
Year
|
||||
: 2007
|
||||
|
||||
Discusses:
|
||||
|
||||
- When to use multivariable control and when not to?
|
||||
- Two major issues with implementing multivariable control: **gain scheduling** and **integrator wind up protection**
|
||||
|
||||
> Inline simple gain and phase margin measured for SISO, "robustness" determination of multivariable control requires complex analyses using **singular value techniques** and **Monte Carlo** simulations.
|
||||
|
||||
**When to use multivariable control**:
|
||||
|
||||
- System has high input/output coupling and not much separation between loop bandwidth
|
||||
- System is complex with large number of states
|
||||
- When sequential SISO loop closure will not meet performance requirements
|
||||
|
||||
Importance of having a mechanism to limit the control rate in the synthesis process.
|
||||
The control rate should be weighted appropriately in order to not saturate the system and stay in the linearity regime.
|
||||
|
||||
- importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In <i>AIAA Guidance, Navigation and Control Conference and Exhibit</i>. doi:<a href="https://doi.org/10.2514/6.2007-6334">10.2514/6.2007-6334</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,120 @@
|
||||
+++
|
||||
title = "Centralized Multivariable Control By Simplified Decoupling"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Decoupled Control]({{< relref "decoupled_control.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Garrido, Vázquez, and Morilla 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Garrido, J., Francisco V\\'azquez, & Morilla, F.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
Most decoupling approaches use the conventional decoupling scheme in [Figure 1](#figure--fig:garrido12-decoupling-control-system) with:
|
||||
|
||||
- \\(G(s)\\) the process matrix
|
||||
- \\(D(s)\\) the decoupler matrix
|
||||
- \\(C(s)\\) the diagonal control matrix
|
||||
|
||||
The design of the decoupler is obtained from:
|
||||
|
||||
\begin{equation}
|
||||
D(s) = G^{-1} (s) \cdot Q(s)
|
||||
\end{equation}
|
||||
|
||||
where \\(Q(s)\\) is the desired apparent process which is a diagonal matrix.
|
||||
|
||||
The main problem of this methodology is the fact that the complexity of the decoupler elements increases for high dimensional MIMO processes, which may require model reductions.
|
||||
|
||||
An alternative decoupling methods, called _inverted decoupling_, maintains very simple apparent processes and decoupler element independently of the system size.
|
||||
However, inverted decoupling cannot be applied to processes with multivariable [Right Half Plane Zeros]({{< relref "right_half_plane_zeros.md" >}}).
|
||||
|
||||
<a id="figure--fig:garrido12-decoupling-control-system"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/garrido12_decoupling_control_system.png" caption="<span class='figure-number'>Figure 1: </span>Block diagram of a decoupling control system" >}}
|
||||
|
||||
This work focuses on one of the most extended forms of conventional decoupling called simplified decoupling, in which \\(n\\) elements of the decoupler are set to unity.
|
||||
When the system has two inputs and two outputs (TITO), the simplified decoupling \\(G(s)\\) is given by:
|
||||
|
||||
\begin{equation}
|
||||
D(s) = \begin{bmatrix}
|
||||
1 & -g\_{12}(s)/g\_{11}(s) \\\\
|
||||
-g\_{21}(s)/g\_{22}(s) & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
And the decoupled apparent process \\(Q(s)\\) is given by:
|
||||
|
||||
\begin{equation}
|
||||
Q(s) = G(s) \cdot D(s) = \begin{bmatrix}
|
||||
g\_{11}(s) - \frac{g\_{21}(s g\_{12}(s))}{g\_{22}(s)} & 0 \\\\
|
||||
0 & g\_{22}(s) - \frac{g\_{21}(s)g\_{12}(s)}{g\_{11}(s)}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
In cases where the system is larger than 2x2, the decoupler elements set to unity are always the diagonal ones as found using:
|
||||
|
||||
\begin{equation}
|
||||
D(s) = G(s)^{-1} (\text{diag}(G(s)^{-1}))^{-1}
|
||||
\end{equation}
|
||||
|
||||
In this work, a simplified decoupling strategy is proposed for stable processes with possibly RHP zeros and time delays.
|
||||
|
||||
|
||||
## Methodology {#methodology}
|
||||
|
||||
Assuming that the process \\(G(s)\\) may have RHP zeros and time delays, but does not have any unstable poles, the decoupler matrix \\(D(s)\\) is obtained as follows (one of many possible configurations):
|
||||
|
||||
\begin{equation}
|
||||
D(s) = \begin{bmatrix}
|
||||
1 & \frac{\text{adj}G\_{12}}{\text{adj}G\_{22}} & \dots & \frac{\text{adj}G\_{1n}}{\text{adj}\_{nn}} \\\\
|
||||
\frac{\text{adj}G\_{21}}{\text{adj}G\_{11}} & 1 & \dots & \frac{\text{adj}G\_{2n}}{\text{adj}\_{nn}} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
\frac{\text{adj}G\_{n1}}{\text{adj}G\_{11}} & \frac{\text{adj}G\_{n2}}{\text{adj}G\_{22}} & \dots & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
And the decoupled apparent plant is:
|
||||
|
||||
\begin{equation}
|
||||
A(s) = \begin{bmatrix}
|
||||
\frac{|G|}{\text{adj}G\_{11}} & 0 & \dots & 0 \\\\
|
||||
0 & \frac{|G|}{\text{adj}G\_{22}} & \dots & 0 \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
0 & 0 & \dots & \frac{|G|}{\text{adj}G\_{nn}}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
where \\(|G(s)|\\) is the determinant of \\(G(s)\\), \\(\text{adj}G(s)\\) is the adjugate matrix of \\(G(s)\\), that is, the transpose of the cofactor matrix of \\(G(s)\\).
|
||||
|
||||
The proposed general simplified decoupling control is performed in three steps:
|
||||
|
||||
1. select a configuration: select the \\(n\\) elements of \\(D(s)\\) to be set to unity, one for each column
|
||||
2. Compose the decoupler elements of \\(D(s)\\)
|
||||
3. Design the \\(n\\) controllers of the diagonal control \\(C(s)\\) for the decoupled processes
|
||||
|
||||
The realizability requirement for the decoupler is that all of its elements must be proper, causal and stable.
|
||||
For processes with time delays, non-minimum phase zeros or different relative degrees, direct calculation of the decoupler element can lead to elements with RHP poles or negative relative degrees.
|
||||
|
||||
Several advice for the proper chose of the configuration are given in the paper.
|
||||
|
||||
|
||||
## Design and practical considerations {#design-and-practical-considerations}
|
||||
|
||||
It is usually necessary to approximate the expressions of \\(|G(s)|\\) and \\(\text{adj}G(s)\\) as it usually give non-rational expressions.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garrido, Juan, Francisco Vázquez, and Fernando Morilla. 2012. “Centralized Multivariable Control by Simplified Decoupling.” <i>Journal of Process Control</i> 22 (6): 1044–62. doi:<a href="https://doi.org/10.1016/j.jprocont.2012.04.008">10.1016/j.jprocont.2012.04.008</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,28 @@
|
||||
+++
|
||||
title = "An intelligent control system for multiple degree-of-freedom vibration isolation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Geng et al. 1995</a>)
|
||||
|
||||
Author(s)
|
||||
: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A.
|
||||
|
||||
Year
|
||||
: 1995
|
||||
|
||||
<a id="figure--fig:geng95-control-structure"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/geng95_control_structure.png" caption="<span class='figure-number'>Figure 1: </span>Local force feedback and adaptive acceleration feedback for active isolation" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Geng, Z. J., G. G. Pan, L. S. Haynes, B. K. Wada, and J. A. Garba. 1995. “An Intelligent Control System for Multiple Degree-of-Freedom Vibration Isolation.” <i>Journal of Intelligent Material Systems and Structures</i> 6 (6): 787–800. doi:<a href="https://doi.org/10.1177/1045389x9500600607">10.1177/1045389x9500600607</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,84 @@
|
||||
+++
|
||||
title = "Fly-scan-oriented motion analyses and upgraded beamline integration architecture for the high-dynamic double-crystal monochromator at sirius/lnls"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Geraldes et al. 2023</a>)
|
||||
|
||||
Author(s)
|
||||
: Geraldes, R. R., Luiz, S. A. L., Neto, J. L. d. B., Telles Ren\\'e Silva Soares, Reis, R. D. d., Calligaris, G. A., Witvoet, G., …
|
||||
|
||||
Year
|
||||
: 2023
|
||||
|
||||
|
||||
## Effect of different d spacing {#effect-of-different-d-spacing}
|
||||
|
||||
> Thus, if different d-spacings are found in the two crystals, an ideal energy matching for maximum flux would be related to slightly different \\(\theta\_B\\) in the crystals, such that the monochromatic beam would no longer be exactly parallel to the incoming beam, and **the magnitude of the deviation would be variable over the operational energy range**.
|
||||
|
||||
|
||||
## Effect of pitch error on source motion {#effect-of-pitch-error-on-source-motion}
|
||||
|
||||
> Then, considering that variations of the virtual source are often proportionally related to shifts of the beam at the sample through the beamline optics, **a common requirement is having them small compared with the source size**.
|
||||
> With **X-ray source sizes of about 5 um** and **L commonly of the order of 30m** for modern beamlines, a typical budget of 10% pushes **pitch errors to the range of 10 nrad** only.
|
||||
|
||||
|
||||
## Correct pitch errors with gap adjustments {#correct-pitch-errors-with-gap-adjustments}
|
||||
|
||||
> It can be seen that displacements in the virtual source related to pitch errors may be at least partly compensated by energy-dependent beam offset corrections via gap adjustments.
|
||||
|
||||
|
||||
## Allow some flux loss in order to have a more stable beam {#allow-some-flux-loss-in-order-to-have-a-more-stable-beam}
|
||||
|
||||
> The angular boundaries for pitch around an ideal energy tuning, which might be already out or perfect parallelism due to d-spacing variations, can be derived as a fraction of the angular bandwidth of the Darwin width of the crystals.
|
||||
> This can be used, for example, to **evaluate acceptable flux losses in trying to keep the incoming and outgoing beam parallel despite thermal effects**.
|
||||
|
||||
The pitch bandwidth for typical Si111 and Si311 can vary from 100urad at low energy to <1urad at high energy.
|
||||
|
||||
|
||||
## Analytical effect of miss-cut on the change of beam height {#analytical-effect-of-miss-cut-on-the-change-of-beam-height}
|
||||
|
||||
> This indicates that in reality the **gap motion range may need to be larger by a few percent than nominally expected**, that sensitivities at low angles may vary by more than one order of magnitude, that **calibrations for fixed exit may require more than the simpler trigonometric relation** of (2), and that the required velocities and accelerations related to the fly scan are in practice different from nominal ones.
|
||||
|
||||
|
||||
### Estimate the effect of the miss-cut on the beam error for our values of angles and miss-cut {#estimate-the-effect-of-the-miss-cut-on-the-beam-error-for-our-values-of-angles-and-miss-cut}
|
||||
|
||||
|
||||
## High dynamic range: low energy and high energy issues {#high-dynamic-range-low-energy-and-high-energy-issues}
|
||||
|
||||
> Hence, **differences of three to four orders of magnitude occur for the gap velocity for a given energy variation rate** within the operational range of the HD-DCM.
|
||||
>
|
||||
> For a control-based instrument like the HD-DCM, these aspects place demanding specifications on metrology and acquisition hardware, since very high resolution and low noise are required for the lower angular (higher energy) range, whereas high rates are necessary at the opposite limit.
|
||||
>
|
||||
> For example, while the angular resolution in the Bragg angle quadrature encoder is 50nrad for high angular resolution and small control errors, for an energy scan of 1keV/s, the crystal angular speed requirements would be around 0.1deg/s at the high energy range and as much as 40deg/s at the low energy limit.
|
||||
> In the latter case, the counting rates would have to be higher than the current electronics capacity of 10 MHz.
|
||||
>
|
||||
> Similarly for the gap, with a resolution of 0.1 nm from the quadrature laser interferometers for the nanometre-level control performance, an equivalent energy rate scan speed with Si(111) crystals without a miscut would translate to about 0.8 mm/s and 20 mm/s at the high and low energy limits, respectively.
|
||||
> In the latter case, counting rates would need to reach 200 MHz.
|
||||
|
||||
|
||||
## Bragg control has a bandwidth of 20Hz {#bragg-control-has-a-bandwidth-of-20hz}
|
||||
|
||||
|
||||
## Crystal control has a bandwidth between 150Hz and 250Hz {#crystal-control-has-a-bandwidth-between-150hz-and-250hz}
|
||||
|
||||
|
||||
## They are using the Bragg angle reference signal to measure the wanted crystal distance {#they-are-using-the-bragg-angle-reference-signal-to-measure-the-wanted-crystal-distance}
|
||||
|
||||
They are not using the encoder signal as we are doing.
|
||||
|
||||
|
||||
## Modes of operation {#modes-of-operation}
|
||||
|
||||
1. Standalone (similar as what we are using).
|
||||
2. Follower: follows an encoder signal from the ID
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Geraldes, Renan Ramalho, Sergio Augusto Lordano Luiz, João Leandro de Brito Neto, Telles René Silva Soares, Ricardo Donizeth dos Reis, Guilherme A. Calligaris, Gert Witvoet, and J. P. M. B. Vermeulen. 2023. “Fly-Scan-Oriented Motion Analyses and Upgraded Beamline Integration Architecture for the High-Dynamic Double-Crystal Monochromator at Sirius/Lnls.” <i>Journal of Synchrotron Radiation</i> 30 (1): 90–110. doi:<a href="https://doi.org/10.1107/s1600577522010724">10.1107/s1600577522010724</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,148 @@
|
||||
+++
|
||||
title = "Sensors and control of a space-based six-axis vibration isolation system"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Hauge and Campbell 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Hauge, G., & Campbell, M.
|
||||
|
||||
Year
|
||||
: 2004
|
||||
|
||||
**Discusses**:
|
||||
|
||||
- Choice of sensors and control architecture
|
||||
- Predictability and limitations of the system dynamics
|
||||
- Two-Sensor control architecture
|
||||
- Vibration isolation using a Stewart platform
|
||||
- Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
|
||||
|
||||
<a id="figure--fig:hauge04-stewart-platform"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Hexapod for active vibration isolation" >}}
|
||||
|
||||
**Stewart platform** ([Figure 1](#figure--fig:hauge04-stewart-platform)):
|
||||
|
||||
- Low corner frequency
|
||||
- Large actuator stroke (\\(\pm5mm\\))
|
||||
- Sensors in each strut ([Figure 2](#figure--fig:hauge05-struts)):
|
||||
- three-axis load cell
|
||||
- base and payload geophone in parallel with the struts
|
||||
- LVDT
|
||||
|
||||
<a id="figure--fig:hauge05-struts"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hauge05_struts.png" caption="<span class='figure-number'>Figure 2: </span>Strut" >}}
|
||||
|
||||
> Force sensors typically work well because they are not as sensitive to payload and base dynamics, but are limited in performance by a low-frequency zero pair resulting from the cross-axial stiffness.
|
||||
|
||||
**Performance Objective** (frequency domain metric):
|
||||
|
||||
- The transmissibility should be close to 1 between 0-1.5Hz
|
||||
\\(-3dB < |T(\omega)| < 3db\\)
|
||||
- The transmissibility should be below -20dB in the 5-20Hz range
|
||||
\\(|T(\omega)| < -20db\\)
|
||||
|
||||
With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and is used to obtain a scalar performance metric.
|
||||
|
||||
**Challenge**:
|
||||
|
||||
- small frequency separation between the two requirements
|
||||
|
||||
**Robustness**:
|
||||
|
||||
- minimization of the transmissibility amplification (Bode's "pop") outside the performance region
|
||||
|
||||
**Model**:
|
||||
|
||||
- single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems
|
||||
|
||||
<a id="figure--fig:hauge05-strut-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="<span class='figure-number'>Figure 3: </span>Strut model" >}}
|
||||
|
||||
**Zero Pair when using a Force Sensor**:
|
||||
|
||||
- The frequency of the zero pair corresponds to the resonance frequency of the payload mass and the "parasitic" stiffness (sum of the cross-axial, suspension, wiring stiffnesses)
|
||||
- This zero pair is usually not predictable nor repeatable
|
||||
- In this Stewart platform, this zero pair uncertainty is due to the internal wiring of the struts
|
||||
|
||||
**Control**:
|
||||
|
||||
- Single-axis controllers => combine them into a full six-axis controller => evaluate the full controller in terms of stability and robustness
|
||||
- Sensitivity weighted LQG controller (SWLQG) => address robustness in flexible dynamic systems
|
||||
- Three type of controller:
|
||||
- Force feedback (cell-based)
|
||||
- Inertial feedback (geophone-based)
|
||||
- Combined force/velocity feedback (load cell/geophone based)
|
||||
|
||||
> The use of multivariable and robust control on the full 6x6 hexapod does not improve performance over single-axis designs.
|
||||
|
||||
<a id="table--tab:hauge05-comp-load-cell-geophone"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a>:</span>
|
||||
Typical characteristics of sensors used for isolation in hexapod systems
|
||||
</div>
|
||||
|
||||
| | **Load cell** | **Geophone** |
|
||||
|-----------------------------------------|---------------------------------|-------------------------------------|
|
||||
| Type | Relative | Inertial |
|
||||
| Relationship with voice coil | Collocated and Dual | Non-Collocated and non-Dual |
|
||||
| Open loop transfer function | (+) Alternating poles/zeros | (-) Large phase drop |
|
||||
| Limitation from low-frequency zero pair | (-) Yes | (+) No |
|
||||
| Sensitive to payload/base dynamics | (+) No | (-) Yes |
|
||||
| Best frequency range | High (low-freq zero limitation) | Low (high-freq toll-off limitation) |
|
||||
|
||||
**Ability of a sensor-actuator pair to improve performance**:
|
||||
General system with input \\(u\\), performance \\(z\\), output \\(y\\) disturbance \\(u\\).
|
||||
|
||||
Given a sensor \\(u\\) and actuator \\(y\\) and a controller \\(u = -K(s) y\\), the closed loop disturbance to performance transfer function can be written as:
|
||||
|
||||
\\[ \left[ \frac{z}{w} \right]\_\text{CL} = \frac{G(s)\_{zw} + K(G(s)\_{zw} G(s)\_{yu} - G(s)\_{zu} G(s)\_{yw})}{1 + K G(s)\_{yu}} \\]
|
||||
|
||||
In order to obtain a significant performance improvement is to use a high gain controller, _provided_ the term \\(G(s)\_{zw} + K(G(s)\_{zw} G(s)\_{yu} - G(s)\_{zu} G(s)\_{yw})\\) is small.
|
||||
|
||||
We can compare the transfer function from \\(w\\) to \\(z\\) with and without a high gain controller.
|
||||
And we find that for \\(u\\) and \\(y\\) to be an acceptable pair for high gain control:
|
||||
\\[ \left| \frac{G(j\omega)\_{zw} G(j\omega)\_{yu} - G(j\omega)\_{zu} G(j\omega)\_{yw}}{K G(j\omega)\_{yu}} \right| \ll |G\_{zw}(j\omega)| \\]
|
||||
|
||||
**Controllers**:
|
||||
|
||||
**Force feedback**:
|
||||
|
||||
- Performance limited by the low frequency zero-pair
|
||||
- It is desirable to separate the zero-pair and first most are separated by at least a decade in frequency
|
||||
- This can be achieve by reducing the cross-axis stiffness
|
||||
- If the low frequency zero pair is inverted, robustness is lost
|
||||
- Thus, the force feedback controller should be designed to have combined performance and robustness at frequencies at least a decade above the zero pair
|
||||
- The presented controller as a high pass filter at to reduce the gain below the zero-pair, a lag at low frequency to improve phase margin, and a low pass filter for roll off
|
||||
|
||||
**Inertial feedback**:
|
||||
|
||||
- Non-Collocated => multiple phase drops that limit the bandwidth of the controller
|
||||
- Good performance, but the transmissibility "pops" due to low phase margin and thus this indicates robustness problems
|
||||
|
||||
**Combined force/velocity feedback**:
|
||||
|
||||
- Use the low frequency performance advantages of geophone sensor with the high robustness advantages of the load cell sensor
|
||||
- A Single-Input-Multiple-Outputs (SIMO) controller is found using LQG
|
||||
- The performance requirements are met
|
||||
- Good robustness
|
||||
|
||||
<a id="figure--fig:hauge04-obtained-transmissibility"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hauge04_obtained_transmissibility.png" caption="<span class='figure-number'>Figure 4: </span>Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hauge, G. S., and M. E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” <i>Journal of Sound and Vibration</i> 269 (3-5): 913–31. doi:<a href="https://doi.org/10.1016/s0022-460x(03)00206-2">10.1016/s0022-460x(03)00206-2</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,36 @@
|
||||
+++
|
||||
title = "Minimizing cross-talk in high-precision motion systems using data-based dynamic decoupling"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Decoupled Control]({{< relref "decoupled_control.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Heertjes and van Engelen 2011</a>)
|
||||
|
||||
Author(s)
|
||||
: Heertjes, M., & Engelen, A. v.
|
||||
|
||||
Year
|
||||
: 2011
|
||||
|
||||
> In the field of high-precision motion control, a static decoupling control design is generally used to command motion in the directions of an orthogonal basis.
|
||||
> Around the center-of-gravity of the system it then usually suffices to apply single-input single-output control in each of these directions separately.
|
||||
> Among the advantages are robust stability and performance through straightforward control designs and loop shaping techniques.
|
||||
>
|
||||
> If the static decoupling part does not fully achieve desired decoupling of the underlying MIMO motion system, a multi-variable controller can be sought to replace the SISO controller part.
|
||||
> A more natural approach would therefore be to replace the MIMO static decoupling part by a dynamic part and leave the SISO controller part intact.
|
||||
|
||||
<!--quoteend-->
|
||||
|
||||
> The aim of the paper is to minimize directly the cross-talk outputs via data-based optimization.
|
||||
> The criterion to be optimized consists solely of time-domain signals taken from a performance-relevant time interval.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Heertjes, Marcel, and Arjan van Engelen. 2011. “Minimizing Cross-Talk in High-Precision Motion Systems Using Data-Based Dynamic Decoupling.” <i>Control Engineering Practice</i> 19 (12): 1423–32. doi:<a href="https://doi.org/10.1016/j.conengprac.2011.07.016">10.1016/j.conengprac.2011.07.016</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Exploiting additional actuators and sensors for nano-positioning robust motion control"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Van Herpen et al. 2014</a>)
|
||||
|
||||
Author(s)
|
||||
: Herpen, R. v., Oomen, T., Kikken, E., Wal, M. v. d., Aangenent, W., & Steinbuch, M.
|
||||
|
||||
Year
|
||||
: 2014
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Herpen, Robbert van, Tom Oomen, Edward Kikken, Marc van de Wal, Wouter Aangenent, and Maarten Steinbuch. 2014. “Exploiting Additional Actuators and Sensors for Nano-Positioning Robust Motion Control.” <i>Mechatronics</i> 24 (6): 619–31. doi:<a href="https://doi.org/10.1016/j.mechatronics.2014.03.008">10.1016/j.mechatronics.2014.03.008</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,46 @@
|
||||
+++
|
||||
title = "An instrument for 3d x-ray nano-imaging"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Holler et al. 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
Instrument similar to the NASS.
|
||||
Obtain position stability of 10nm (standard deviation).
|
||||
|
||||
<a id="figure--fig:holler12-station"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/holler12_station.png" caption="<span class='figure-number'>Figure 1: </span>Schematic of the tomography setup" >}}
|
||||
|
||||
- **Limited resolution due to instrumentation**:
|
||||
The resolution of ptychographic tomography remains above 100nm due to instabilities and drifts of the scanning systems.
|
||||
- **Need of a Metrology System**:
|
||||
|
||||
> To achieve positioning accuracy and stability in the nanometer range, one cannot rely on the position encoders built into individual positioning stages.
|
||||
> A precise exteroceptive measurement of the relative position of the optical elements with respect to the sample is mandatory.
|
||||
> Thus, thermal drifts and parasitic motions can be measured and compensated for.
|
||||
- **Interferometer System Concept**:
|
||||
The sample is aligned with the X-ray with the XYZ piezo stage.
|
||||
As a result, the metrology sphere will be usually off center with respect to the rotation axis of the spindle.
|
||||
That implies that the laser will not propagate back to the interferometer at all rotation angles.
|
||||
A position sensitive detector (PSD) is used, it provides a measurement of the position of the sphere in the plane perpendicular to the laser.
|
||||
The interferometer is positionned on top of a translation stage. The PSD information is used to close the loop so that the interferometer follows the displacement of the metrology sphere.
|
||||
- **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” <i>Review of Scientific Instruments</i> 83 (7): 073703. doi:<a href="https://doi.org/10.1063/1.4737624">10.1063/1.4737624</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Active damping based on decoupled collocated control"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Active Damping]({{< relref "active_damping.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Holterman and de Vries 2005</a>)
|
||||
|
||||
Author(s)
|
||||
: Holterman, J., & deVries, T.
|
||||
|
||||
Year
|
||||
: 2005
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holterman, J., and T. J. A. de Vries. 2005. “Active Damping Based on Decoupled Collocated Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 10 (2): 135–45. doi:<a href="https://doi.org/10.1109/tmech.2005.844702">10.1109/tmech.2005.844702</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,74 @@
|
||||
+++
|
||||
title = "Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Actuators]({{< relref "actuators.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>)
|
||||
|
||||
Author(s)
|
||||
: Ito, S., & Schitter, G.
|
||||
|
||||
Year
|
||||
: 2016
|
||||
|
||||
|
||||
## Classification of high-precision actuators {#classification-of-high-precision-actuators}
|
||||
|
||||
<div class="table-caption">
|
||||
<span class="table-number">Table 1:</span>
|
||||
Zero/Low and High stiffness actuators
|
||||
</div>
|
||||
|
||||
| **Categories** | **Pros** | **Cons** |
|
||||
|----------------|---------------------------|-----------------------------|
|
||||
| Zero stiffness | No vibration transmission | Large and Heavy |
|
||||
| Low stiffness | High vibration isolation | Typically for low load |
|
||||
| High Stiffness | High control bandwidth | High vibration transmission |
|
||||
|
||||
|
||||
## Time Delay of Piezoelectric Electronics {#time-delay-of-piezoelectric-electronics}
|
||||
|
||||
In this paper, the piezoelectric actuator/electronics adds a time delay which is much higher than the time delay added by the voice coil/electronics.
|
||||
|
||||
|
||||
## Definition of low-stiffness and high-stiffness actuator {#definition-of-low-stiffness-and-high-stiffness-actuator}
|
||||
|
||||
- **Low Stiffness** actuator is defined as the ones where the transmissibility stays below 0dB at all frequency
|
||||
- **High Stiffness** actuator is defined as the ones where the transmissibility goes above 0dB at some frequency
|
||||
|
||||
<a id="figure--fig:ito16-low-high-stiffness-actuators"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="<span class='figure-number'>Figure 1: </span>Definition of low-stiffness and high-stiffness actuator" >}}
|
||||
|
||||
|
||||
## Low-Stiffness / High-Stiffness characteristics {#low-stiffness-high-stiffness-characteristics}
|
||||
|
||||
- The low stiffness actuators achieve smooth transition from active isolation to passive isolation.
|
||||
- The high stiffness actuators can have a gap between the passive and active isolation vibration where the vibrations are amplified in a certain frequency band.
|
||||
|
||||
|
||||
## Controller Design {#controller-design}
|
||||
|
||||
<a id="figure--fig:ito16-transmissibility"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="<span class='figure-number'>Figure 2: </span>Obtained transmissibility" >}}
|
||||
|
||||
|
||||
## Discussion {#discussion}
|
||||
|
||||
The stiffness requirement for low-stiffness actuators can be rephrased in the frequency domain as: "the cross-over frequency of the sensitivity function of the feedback system must be larger than \\(\sqrt{2} \omega\_r\\) with \\(\omega\_r\\) is the resonant frequency of the uncontrolled system".
|
||||
|
||||
In practice, this is difficult to achieve with piezoelectric actuators as their first resonant frequency \\(\omega\_r\\) is **too close to other resonant frequencies to ensure close-loop stability**.
|
||||
In contrast, the frequency band between the first and the other resonances of Lorentz actuators can be broad by design making them more suitable to construct a low-stiffness actuators.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>IEEE/ASME Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Flexure design for precision positioning using low-stiffness actuators"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Ito et al. 2016</a>)
|
||||
|
||||
Author(s)
|
||||
: Ito, S., Cigarini, F., Unger, S., & Schitter, G.
|
||||
|
||||
Year
|
||||
: 2016
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, Francesco Cigarini, Severin Unger, and Georg Schitter. 2016. “Flexure Design for Precision Positioning Using Low-Stiffness Actuators.” <i>IFAC-PapersOnLine</i> 49 (21): 200–205. doi:<a href="https://doi.org/10.1016/j.ifacol.2016.10.548">10.1016/j.ifacol.2016.10.548</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Jiao et al. 2018</a>)
|
||||
|
||||
Author(s)
|
||||
: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
|
||||
|
||||
Year
|
||||
: 2018
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Jiao, J., Y. Wu, K. Yu, and R. Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” <i>Journal of Vibration and Control</i> 25 (1): 151–71. doi:<a href="https://doi.org/10.1177/1077546318772474">10.1177/1077546318772474</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Robust control and H-Infinity optimization - Tutorial paper"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [H Infinity Control]({{< relref "h_infinity_control.md" >}}), [Weighting Functions]({{< relref "weighting_functions.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Kwakernaak 1993</a>)
|
||||
|
||||
Author(s)
|
||||
: Kwakernaak, H.
|
||||
|
||||
Year
|
||||
: 1993
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Kwakernaak, Huibert. 1993. “Robust Control and H$\Infty$-Optimization - Tutorial Paper.” <i>Automatica</i> 29 (2): 255–73. doi:<a href="https://doi.org/10.1016/0005-1098(93)90122-a">10.1016/0005-1098(93)90122-a</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,38 @@
|
||||
+++
|
||||
title = "A new isotropic and decoupled 6-dof parallel manipulator"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Legnani et al. 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
- Concepts of isotropy and decoupling for parallel manipulators
|
||||
- **isotropy**: the kinetostatic properties (same applicable force, same possible velocity, same stiffness) are identical in all directions (e.g. cubic configuration for Stewart platform)
|
||||
- **decoupling**: each DoF of the end effector can be controlled by a **single** actuator (not the case for the Stewart platform)
|
||||
|
||||
Example of generated isotropic manipulator (not decoupled).
|
||||
|
||||
<a id="figure--fig:legnani12-isotropy-gen"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="<span class='figure-number'>Figure 1: </span>Location of the leg axes using an isotropy generator" >}}
|
||||
|
||||
<a id="figure--fig:legnani12-generated-isotropy"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="<span class='figure-number'>Figure 2: </span>Isotropic configuration" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58: 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,26 @@
|
||||
+++
|
||||
title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Li, Hamann, and McInroy 2001</a>)
|
||||
|
||||
Author(s)
|
||||
: Li, X., Hamann, J. C., & McInroy, J. E.
|
||||
|
||||
Year
|
||||
: 2001
|
||||
|
||||
- if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In <i>Smart Structures and Materials 2001: Smart Structures and Integrated Systems</i>. doi:<a href="https://doi.org/10.1117/12.436521">10.1117/12.436521</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Disturbance attenuation in precise hexapod pointing using positive force feedback"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Lin and McInroy 2006</a>)
|
||||
|
||||
Author(s)
|
||||
: Lin, H., & McInroy, J. E.
|
||||
|
||||
Year
|
||||
: 2006
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lin, H., and J. E. McInroy. 2006. “Disturbance Attenuation in Precise Hexapod Pointing Using Positive Force Feedback.” <i>Control Engineering Practice</i> 14 (11): 1377–86. doi:<a href="https://doi.org/10.1016/j.conengprac.2005.10.002">10.1016/j.conengprac.2005.10.002</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Design and control of flexure jointed hexapods"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">McInroy and Hamann 2000</a>)
|
||||
|
||||
Author(s)
|
||||
: McInroy, J., & Hamann, J.
|
||||
|
||||
Year
|
||||
: 2000
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J. E., and J. C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” <i>IEEE Transactions on Robotics and Automation</i> 16 (4): 372–81. doi:<a href="https://doi.org/10.1109/70.864229">10.1109/70.864229</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,274 @@
|
||||
+++
|
||||
title = "Modeling and design of flexure jointed stewart platforms for control purposes"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_2">McInroy 2002</a>)
|
||||
|
||||
Author(s)
|
||||
: McInroy, J.
|
||||
|
||||
Year
|
||||
: 2002
|
||||
|
||||
This short paper is very similar to (<a href="#citeproc_bib_item_1">McInroy 1999</a>).
|
||||
|
||||
> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
> When pursuing micro-meter/micro-radian scale motion, two new phenomena become important:
|
||||
>
|
||||
> 1. joint friction and backlash can cause extremely nonlinear micro-dynamics
|
||||
> 2. base and/or payload vibrations become significant contributor to the motion
|
||||
|
||||
<!--quoteend-->
|
||||
|
||||
> If the spherical flexure is not properly matched to the particular application, it is shown that the complexity of the dynamics can greatly increase, thus limiting the control performance.
|
||||
|
||||
|
||||
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
|
||||
|
||||
<a id="figure--fig:mcinroy02-leg-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="<span class='figure-number'>Figure 1: </span>The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
|
||||
|
||||
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass ([Figure 1](#figure--fig:mcinroy02-leg-model)).
|
||||
|
||||
Thus, **the strut does not output force directly, but rather outputs a mechanically filtered force**.
|
||||
|
||||
The model of the strut are shown in [Figure 1](#figure--fig:mcinroy02-leg-model) with:
|
||||
|
||||
- \\(m\_{s\_i}\\) moving strut mass
|
||||
- \\(k\_i\\) spring constant
|
||||
- \\(b\_i\\) damping constant
|
||||
- \\(f\_{m\_i}\\) force applied by the actuator
|
||||
- \\(f\_{p\_i}\\) force exerted by the payload
|
||||
- \\(p\_i\\) three dimensional position of the top
|
||||
- \\(q\_i\\) three dimensional position of the bottom
|
||||
- \\(l\_i\\) strut length
|
||||
- \\(l\_{r\_i}\\) relaxed strut length
|
||||
|
||||
In general, **the strut mass and spherical flexure stiffness will cause payload forces that are not perfectly aligned with the strut**.
|
||||
|
||||
Applying Newton's second law and stacking the equations into a vector form gives:
|
||||
|
||||
\begin{equation}
|
||||
f\_p = f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2 \label{eq:strut\_dynamics\_vec}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\ddot{q}\_u = \left[ \hat{u}\_1^T \ddot{q}\_1 \ \dots \ \hat{u}\_6^T \ddot{q}\_6 \right]^T\\) notes the vector of base accelerations in the strut directions
|
||||
- \\(g\_u\\) denotes the vector of gravity accelerations in the strut directions
|
||||
- \\(Ms = \text{diag}([m\_1\ \dots \ m\_6])\\), \\(f\_p = [f\_{p\_1}\ \dots \ f\_{p\_6}]^T\\)
|
||||
- \\(v\_2 = [ \dot{\hat{u}}\_1^T \dot{v}\_1 \ \dots \ \dot{\hat{u}}\_6^T \dot{v}\_6 ]^T\\) contains nonlinear Coriolis and centripetal accelerations
|
||||
|
||||
|
||||
### Payload Dynamics {#payload-dynamics}
|
||||
|
||||
The payload is modeled as a rigid body:
|
||||
|
||||
\begin{equation}
|
||||
\underbrace{\begin{bmatrix}
|
||||
m I\_3 & 0\_{3\times 3} \\\\
|
||||
0\_{3\times 3} & {}^cI
|
||||
\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
|
||||
0\_{3 \times 1} \\\ \omega \times {}^cI\omega
|
||||
\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\ddot{\mathcal{X}}\\) is the \\(6 \times 1\\) generalized acceleration of the payload's center of mass
|
||||
- \\(\omega\\) is the \\(3 \times 1\\) payload's angular velocity vector
|
||||
- \\(\mathcal{F}\\) is the \\(6 \times 1\\) generalized force exerted on the payload
|
||||
- \\(M\_x\\) is the combined mass/inertia matrix of the payload, written in the payload frame {P}
|
||||
- \\(c(\omega)\\) represents the shown vector of Coriolis and centripetal terms
|
||||
|
||||
Note \\(\dot{\mathcal{X}} = [\dot{p}^T\ \omega^T]^T\\) denotes the time derivative of the payload's combined position and orientation (or pose) with respect to a universal frame of reference {U}.
|
||||
|
||||
First, consider the **generalized force due to struts**.
|
||||
Denoting this force as \\(\mathcal{F}\_s\\), it can be calculated form the strut forces as:
|
||||
|
||||
\begin{equation}
|
||||
\mathcal{F}\_s = {}^UJ^T f\_p = {}^U\_BR J^T f\_p
|
||||
\end{equation}
|
||||
|
||||
where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation matrix from {B} to {U}.
|
||||
|
||||
The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
|
||||
|
||||
\begin{equation}
|
||||
\mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
|
||||
- \\(g\\) is the gravity vector
|
||||
|
||||
By combining \ref{eq:strut\_dynamics\_vec}, \ref{eq:payload\_dynamics} and \ref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
|
||||
|
||||
\begin{equation}
|
||||
{}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh}
|
||||
\end{equation}
|
||||
|
||||
Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
|
||||
In the next section, a connection between the two will be found to complete the formulation
|
||||
|
||||
|
||||
## Direction of Payload Force {#direction-of-payload-force}
|
||||
|
||||
Many prior hexapod dynamic formulations assume that the strut exerts force only along its direction of motion.
|
||||
|
||||
The flexure joints Hexapods transmit forces (or torques) proportional to the deflection of the joints.
|
||||
This section establishes design guidelines for the spherical flexure joint to guarantee that the dynamics remain tractable for control.
|
||||
|
||||
<a id="figure--fig:mcinroy02-model-strut-joint"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="<span class='figure-number'>Figure 2: </span>A simplified dynamic model of a strut and its joint" >}}
|
||||
|
||||
[Figure 2](#figure--fig:mcinroy02-model-strut-joint) depicts a strut, along with the corresponding force diagram.
|
||||
The force diagram is obtained using standard finite element assumptions (\\(\sin \theta \approx \theta\\)).
|
||||
Damping terms are neglected.
|
||||
\\(k\_r\\) denotes the rotational stiffness of the spherical joint.
|
||||
|
||||
From [Figure 2](#figure--fig:mcinroy02-model-strut-joint) (b), Newton's second law yields:
|
||||
|
||||
\begin{equation}
|
||||
f\_p = \begin{bmatrix}
|
||||
-f\_m + m\_s \Delta \ddot{x} + k\Delta x \\\\
|
||||
m\_s \Delta \ddot{y} + \frac{k\_r}{l^2} \Delta y \\\\
|
||||
m\_s \Delta \ddot{z} + \frac{k\_r}{l^2} \Delta z
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
Note that the payload force is **not** in general aligned with the strut.
|
||||
The force is aligned perfectly with the strut only if \\(m\_s = 0\\) and \\(k\_r = 0\\) (i.e. the struts have negligible mass and the spherical joints have negligible rotational stiffness).
|
||||
|
||||
To examine the passive behavior, let \\(f\_m = 0\\) and consider a sinusoidal motion:
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} \Delta x \\\ \Delta y \\\ \Delta z \end{bmatrix} =
|
||||
\begin{bmatrix} A\_x \cos \omega t \\\ A\_y \cos \omega t \\\ A\_z \cos \omega t \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
This yields:
|
||||
|
||||
\begin{equation}
|
||||
f\_p = \begin{bmatrix}
|
||||
\Big( -m\_s \omega^2 + k \Big) A\_x \cos \omega t \\\\
|
||||
\Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_y \cos \omega t \\\\
|
||||
\Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_z \cos \omega t
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
The direction of \\(f\_p\\) depends upon to motion specifications, leg inertia and control algorithm.
|
||||
|
||||
The hypothesis that it is mostly along the strut direction can be tested by dividing the magnitude of the \\(x\\) component by the magnitude of the combined \\(y\\) and \\(z\\) components:
|
||||
|
||||
\begin{equation}
|
||||
x\_\text{gain} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|} \frac{|A\_x|}{\sqrt{A\_y^2 + A\_z^2}}
|
||||
\end{equation}
|
||||
|
||||
Note that large \\(x\_\text{gain}\\) indicates \\(x\\) direction dominance.
|
||||
|
||||
\\(x\_\text{gain}\\) is divided into two parts.
|
||||
The first part depends on the mechanical terms and the frequency of the movement:
|
||||
|
||||
\begin{equation}
|
||||
x\_{\text{gain}\_\omega} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|}
|
||||
\end{equation}
|
||||
|
||||
<div class="important">
|
||||
|
||||
In order to get dominance at low frequencies, the hexapod must be designed so that:
|
||||
|
||||
\begin{equation}
|
||||
\frac{k\_r}{l^2} \ll k \label{eq:cond\_stiff}
|
||||
\end{equation}
|
||||
|
||||
</div>
|
||||
|
||||
This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer.
|
||||
|
||||
By satisfying \ref{eq:cond\_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
|
||||
\\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\]
|
||||
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 \\]
|
||||
|
||||
<div class="important">
|
||||
|
||||
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:
|
||||
|
||||
\begin{equation}
|
||||
\text{control bandwidth} \ll \sqrt{\frac{k\_r}{m\_s l^2}} \label{eq:cond\_bandwidth}
|
||||
\end{equation}
|
||||
|
||||
</div>
|
||||
|
||||
The control bandwidth can be increase for hexapods that are designed so that \\(x\_{\text{gain}\_\omega} \gg 1\\) for \\(\omega \ll \sqrt{k/m\_s}\\).
|
||||
This can be achieve, for instance, by adding damping.
|
||||
In this case, it is reasonable to use:
|
||||
|
||||
\begin{equation}
|
||||
\text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
|
||||
\end{equation}
|
||||
|
||||
<div class="important">
|
||||
|
||||
By designing the flexure jointed hexapod and its controller so that both \ref{eq:cond\_stiff} and \ref{eq:cond\_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
|
||||
|
||||
Equation \ref{eq:eom\_fjh} is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
|
||||
|
||||
This section will derive this implicit relationship.
|
||||
Let denote:
|
||||
|
||||
- \\(\mathcal{X}\_B\\) the pose of {B} with respect to {U}
|
||||
- \\({}^B\mathcal{X}\_P\\) the pose of {P} with respect to {B}
|
||||
- \\({}^Uq\_i = {}^U\_BR {}^Bq\_i + {}^UP\_{BORG}\\) the position of the ith base attachment point, expressed in the universal frame {U}
|
||||
- \\(P\_{BORG}\\) the position of the origin of frame {B}
|
||||
|
||||
Note that although \\({}^Bq\_i\\) is fixed, \\({}^Uq\_i\\) varies due to base motion.
|
||||
|
||||
Differentiating twice and converting derivatives of rotation matrices into angular velocity cross products yields:
|
||||
|
||||
\begin{equation}
|
||||
{}^U\dot{q}\_i = \omega\_B \times {}^U\_BR {}^Bq\_i + \underbrace{{}^U\_BR {}^B\dot{q}\_i}\_{= 0} + v\_B
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
{}^U\ddot{q}\_i = \dot{\omega}\_B \times {}^U\_BR {}^Bq\_i + \omega\_B \times \omega\_B \times {}^U\_BR {}^Bq\_i + \dot{v}\_B
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\omega\_B\\) denotes the angular velocity of {B} with respect to {U}
|
||||
- \\(v\_B = {}^U\dot{P}\_{BORG}\\) denotes the linear velocity of the origin of {B} with respect to {U}
|
||||
|
||||
By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times a)\\) and putting the equation in a matrix form:
|
||||
|
||||
\begin{equation}
|
||||
{}^U \hat{u}\_i^T {}^U\ddot{q}\_i = \left[ {}^U\hat{u}\_i^T \left( {}^U\_BR {}^Bq\_i \times {}^U\hat{u}\_i \right)^T \right] \ddot{\mathcal{X}}\_B + {}^U\hat{u}\_i^T \left( \omega\_B \times \left[ \omega\_B \times {}^U\_BR {}^Bq\_i \right] \right)
|
||||
\end{equation}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J. E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>IEEE/ASME Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,170 @@
|
||||
+++
|
||||
title = "Dynamic modeling of flexure jointed hexapods for control purposes"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">McInroy 1999</a>)
|
||||
|
||||
Author(s)
|
||||
: McInroy, J.
|
||||
|
||||
Year
|
||||
: 1999
|
||||
|
||||
This conference paper has been further published in a journal as a short note (<a href="#citeproc_bib_item_2">McInroy 2002</a>).
|
||||
|
||||
|
||||
## Abstract {#abstract}
|
||||
|
||||
> This paper presents a new dynamic model suitable for control of flexure jointed hexapods (FJH).
|
||||
>
|
||||
> Novel contributions include:
|
||||
>
|
||||
> 1. Base acceleration inputs are included
|
||||
> 2. The dynamic model is experimentally verified
|
||||
> 3. The model is developed so that it is suitable for control
|
||||
> 4. A decoupled force control is derived
|
||||
|
||||
|
||||
## Strut Dynamics {#strut-dynamics}
|
||||
|
||||
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.
|
||||
|
||||
<a id="figure--fig:mcinroy99-general-hexapod"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="<span class='figure-number'>Figure 1: </span>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](#figure--fig:mcinroy99-strut-model).
|
||||
|
||||
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
|
||||
|
||||
<a id="figure--fig:mcinroy99-strut-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="<span class='figure-number'>Figure 2: </span>The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
|
||||
|
||||
<a id="table--tab:mcinroy99-strut-model"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a>:</span>
|
||||
Definition of quantities on <a href="#orgef559a8">2</a>
|
||||
</div>
|
||||
|
||||
| **Symbol** | **Meaning** |
|
||||
|------------------------------|--------------------------------------------|
|
||||
| \\(m\_i\\) | moving strut mass |
|
||||
| \\(k\_i\\) | spring constant |
|
||||
| \\(b\_i\\) | damping constant |
|
||||
| \\(f\_m\\) | force the actuator applies |
|
||||
| \\(f\_{p\_i}\\) | forced exerted by the payload |
|
||||
| \\(p\_i\\) | three dimensional position of the top |
|
||||
| \\(q\_i\\) | three dimensional position of the bottom |
|
||||
| \\(l\_i\\) | strut length |
|
||||
| \\(l\_{r\_i}\\) | relaxed strut length |
|
||||
| \\(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 (<a href="#citeproc_bib_item_2">McInroy 2002</a>)).
|
||||
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](#figure--fig:mcinroy99-strut-model) (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
|
||||
\end{equation}
|
||||
|
||||
The acceleration \\(\hat{u}\_i^T \ddot{p}\_i\\) can be written as:
|
||||
\\[ \hat{u}\_i^T \ddot{p}\_i = \ddot{l}\_i + \hat{u}\_i^T \ddot{q}\_i - \dot{\hat{u}}\_i^T \dot{v}\_i \\]
|
||||
|
||||
- [ ] Not sure how the last term is obtained
|
||||
|
||||
Separating strut and base accelerations, and putting all six strut equations in a single vector yields:
|
||||
|
||||
\begin{equation}
|
||||
f\_p = f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2 \label{eq:strut\_dynamics\_vec}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\ddot{q}\_u = \left[ \hat{u}\_1^T \ddot{q}\_1 \ \dots \ \hat{u}\_6^T \ddot{q}\_6 \right]^T\\) notes the vector of base accelerations in the strut directions
|
||||
- \\(g\_u\\) denotes the vector of gravity accelerations in the strut directions
|
||||
- \\(Ms = \diag([m\_1\ \dots \ m\_6])\\), \\(f\_p = [f\_{p\_1}\ \dots \ f\_{p\_6}]^T\\)
|
||||
- \\(v\_2 = [ \dot{\hat{u}}\_1^T \dot{v}\_1 \ \dots \ \dot{\hat{u}}\_6^T \dot{v}\_6 ]^T\\)
|
||||
|
||||
|
||||
## Payload Dynamics {#payload-dynamics}
|
||||
|
||||
The payload is modeled as a rigid body:
|
||||
|
||||
\begin{equation}
|
||||
\underbrace{\begin{bmatrix}
|
||||
m I\_3 & 0\_{3\times 3} \\\\
|
||||
0\_{3\times 3} & {}^cI
|
||||
\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
|
||||
0\_{3 \times 1} \\\ \omega \times {}^cI\omega
|
||||
\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\ddot{\mathcal{X}}\\) is the \\(6 \times 1\\) generalized acceleration of the payload's center of mass
|
||||
- \\(\omega\\) is the \\(3 \times 1\\) payload's angular velocity vector
|
||||
- \\(\mathcal{F}\\) is the \\(6 \times 1\\) generalized force exerted on the payload
|
||||
- \\(M\_x\\) is the combined mass/inertia matrix of the payload, written in the payload frame {P}
|
||||
- \\(c(\omega)\\) represents the shown vector of Coriolis and centripetal terms
|
||||
|
||||
Note \\(\dot{\mathcal{X}} = [\dot{p}^T\ \omega^T]^T\\) denotes the time derivative of the payload's combined position and orientation (or pose) with respect to a universal frame of reference {U}.
|
||||
|
||||
First, consider the **generalized force due to struts**.
|
||||
Denoting this force as \\(\mathcal{F}\_s\\), it can be calculated form the strut forces as:
|
||||
|
||||
\begin{equation}
|
||||
\mathcal{F}\_s = {}^UJ^T f\_p = {}^U\_BR J^T f\_p
|
||||
\end{equation}
|
||||
|
||||
where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation matrix from {B} to {U}.
|
||||
|
||||
The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
|
||||
|
||||
\begin{equation}
|
||||
\mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
|
||||
- \\(g\\) is the gravity vector
|
||||
|
||||
By combining \ref{eq:strut\_dynamics\_vec}, \ref{eq:payload\_dynamics} and \ref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
|
||||
|
||||
\begin{aligned}
|
||||
& {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\
|
||||
& - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega)
|
||||
\end{aligned}
|
||||
|
||||
Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
|
||||
In the next section, a connection between the two will be found to complete the formulation
|
||||
|
||||
|
||||
## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
|
||||
|
||||
|
||||
## Joint Space Dynamics {#joint-space-dynamics}
|
||||
|
||||
|
||||
## Control Example {#control-example}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J. E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>IEEE/ASME Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,41 @@
|
||||
+++
|
||||
title = "A review of the parallel structure mechanisms with kinematic decoupling"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Parallel Manipulators]({{< relref "parallel_manipulators.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Nosova 2020</a>)
|
||||
|
||||
Author(s)
|
||||
: Nosova, N. Y.
|
||||
|
||||
Year
|
||||
: 2020
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
Parallel mechanisms can be characterized by high speeds, since the engines are mounted on the base and the links have a relatively small mass.
|
||||
The disadvantages are: limited working space, the presence of singularities in the immediate vicinity of the workspace.
|
||||
|
||||
The kinematic decoupling for a parallel structure manipulator consists in that one movement of the output platform is provided by only one input link or group of links of the kinematic chain.
|
||||
|
||||
|
||||
## Types of Kinematic Decoupling {#types-of-kinematic-decoupling}
|
||||
|
||||
There are three different types of decoupling:
|
||||
|
||||
1. **strong coupling**: where each configuration parameter is a function of all joint variable (e.g. Stewart platform)
|
||||
2. **complete decoupling**: each configuration parameter is a function of only one joint variable (e.g. Ortoglide)
|
||||
3. **partial decoupling**: some configuration parameters are in function of only some joint variables
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Nosova, N. Yu. 2020. “A Review of the Parallel Structure Mechanisms with Kinematic Decoupling.” <i>Advanced Technologies in Robotics and Intelligent Systems</i>. Springer International Publishing, 247–55. doi:<a href="https://doi.org/10.1007/978-3-030-33491-8_30">10.1007/978-3-030-33491-8_30</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,185 @@
|
||||
+++
|
||||
title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Motion Control]({{< relref "motion_control.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Oomen 2018</a>)
|
||||
|
||||
Author(s)
|
||||
: Oomen, T.
|
||||
|
||||
Year
|
||||
: 2018
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
Control of positioning systems is traditionally simplified by an excellent mechanical design.
|
||||
In particular, the mechanical design is such that the system is stiff and highly reproducible.
|
||||
In conjunction with moderate performance requirements, the control bandwidth is well-below the resonance frequency of the flexible mechanics as is shown in [Figure 1](#figure--fig:oomen18-next-gen-loop-gain) (a).
|
||||
As a result, the system can often be completely **decoupled** in the frequency range relevant for control.
|
||||
Consequently, the control design is divided into well-manageable SISO control loops.
|
||||
|
||||
Although motion control design is well developed, presently available techniques mainly apply to positioning systems that behave as a rigid body in the relevant frequency range.
|
||||
On one hand, increasing performance requirements hamper the validity of this assumption, since the bandwidth has to increase, leading to flexible dynamics in the cross-over region, see [Figure 1](#figure--fig:oomen18-next-gen-loop-gain) (b).
|
||||
|
||||
<a id="figure--fig:oomen18-next-gen-loop-gain"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="<span class='figure-number'>Figure 1: </span>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." >}}
|
||||
|
||||
|
||||
## Traditional motion control {#traditional-motion-control}
|
||||
|
||||
In the frequency range that is relevant for control, the dynamical behavior is mainly determined by the mechanics.
|
||||
In particular, the mechanics can typically be described as:
|
||||
|
||||
\begin{equation}
|
||||
G\_m = \sum\_{i=1}^{n\_{RB}} \frac{c\_i b\_i^T}{s^2} + \sum\_{n\_{RB} + 1}^{n\_s} \frac{c\_i b\_i^T}{s^2 + 2\xi \omega\_i s + \omega\_i^2}
|
||||
\end{equation}
|
||||
|
||||
where the first term refers to rigid body modes and the second term to flexible modes.
|
||||
|
||||
- \\(n\_{RB}\\) is the number of rigid body modes
|
||||
- \\(c\_i \in \mathbb{R}^{n\_y}\\) and \\(b\_i \in \mathbb{R}^{n\_u}\\) are associated with the mode shapes
|
||||
- \\(\xi\_i, \omega\_i \in \mathbb{R}\_+\\)
|
||||
|
||||
In traditional positioning systems, the number of actuators \\(n\_u\\) and sensors \\(n\_y\\) equals the number of rigid body modes \\(n\_{RB}\\) and are positioned such that the matrix \\(\sum\_{i=1}^{n\_{RB}} c\_i b\_i^T\\) is invertible.
|
||||
In this case, matrices \\(T\_u\\) and \\(T\_y\\) can be selected such that:
|
||||
|
||||
\begin{equation}
|
||||
G = T\_y G\_m T\_u = \frac{1}{s^2} I\_{n\_{RB}} + G\_{\text{flex}}
|
||||
\end{equation}
|
||||
|
||||
A tradition motion control architecture is shown in [Figure 2](#figure--fig:oomen18-control-architecture).
|
||||
|
||||
<a id="figure--fig:oomen18-control-architecture"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/oomen18_control_architecture.png" caption="<span class='figure-number'>Figure 2: </span>Traditional motion control architecture" >}}
|
||||
|
||||
|
||||
### Traditional feedforward design {#traditional-feedforward-design}
|
||||
|
||||
[Feedforward Control]({{< relref "feedforward_control.md" >}}) can effectively compensate for reference induced error signals.
|
||||
In particular, \\(f\\) should be selected such that \\(r - G f\\) is minimized.
|
||||
In the low frequency range, the system is decoupled and \\(G\_{\text{flex}}\\) can be ignored, in which case \\(f = G^{-1} r\\).
|
||||
In practice, the feedforward signal is selected as \\(f = ms^2 r\\).
|
||||
|
||||
|
||||
### Traditional feedback design {#traditional-feedback-design}
|
||||
|
||||
The [Feedback Controller]({{< relref "feedback_control.md" >}}) has to minimize \\((1 + GK)^{-1}(\delta - v)\\).
|
||||
The main idea is that rigid body decoupling of \\(G\\) enables the shaping of the diagonal elements of \\(K\\) through a decentralized feedback controller.
|
||||
As a result, each diagonal element of \\(K\\) may be tuned independently.
|
||||
Typically, a PID controller is tuned through manual loop-shaping, followed by notch filters to account the the flexible modes that hamper stability and/or performance.
|
||||
|
||||
|
||||
### Traditional design procedure {#traditional-design-procedure}
|
||||
|
||||
Traditional motion control design divides the multi-variable control design problems into sub-problems that are manageable by manual control design.
|
||||
The traditional procedure consists of the following steps:
|
||||
|
||||
- identify an FRF of \\(G\_m\\)
|
||||
- decouple the plant to obtain an FRF of \\(G\\)
|
||||
- design \\(K\\) using manual loop-shaping, consisting of PID with notches
|
||||
- tune a feedforward controller, e.g. \\(f = m s^2 r\\)
|
||||
|
||||
|
||||
## Precision motion control developments {#precision-motion-control-developments}
|
||||
|
||||
|
||||
### Challenges {#challenges}
|
||||
|
||||
High performance mechatronic systems are becoming lighter and lighter.
|
||||
Such lightweight systems exhibit predominant flexible dynamical behavior, as well as an increased susceptibility to disturbances.
|
||||
|
||||
This leads to several challenges for motion control design:
|
||||
|
||||
- **Unmeasured performance variables** due to spatio-temporal deformations.
|
||||
In particular, the location where the performance is desired may not be directly measured.
|
||||
- **Many additional inputs and outputs** can be exploited to actively control the flexible dynamical behavior.
|
||||
Spatially distributed actuators can actively provide stiffness and damping to the mechanical deformations.
|
||||
- **Position dependent behavior** is almost unavoidable.
|
||||
For instance in gantry stage designs, mass distribution change due to motion, leading to additional position-dependent behavior.
|
||||
A key challenge lies in handling the position dependence of future systems
|
||||
- A **system-of-systems perspective** on motion control design provides a strong potential for performance enhancement of the overall system.
|
||||
In particular, typical manufacturing machines and scientific instruments involves multiple controlled subsystems where the two subsystems have to move relative to each other.
|
||||
Performance limitations in each subsystem will negatively impact the overall performance.
|
||||
A joint design enables that individual subsystems will be able to compensate each other's limitations.
|
||||
A main challenge lies in an increase of the complexity of the control problem.
|
||||
- **Thermal dynamics**, in addition to mechanical deformations are expected to become substantially more important due to increasing performance specifications.
|
||||
- **Vibrations**, such as flow induced vibrations of cooling liquids and floor vibrations, have to be attenuated.
|
||||
|
||||
|
||||
### Generalized plant approach {#generalized-plant-approach}
|
||||
|
||||
A generalized plant framework allows for a systematic way to address the future challenges in advanced motion control.
|
||||
|
||||
The generalized plant is depicted in [Figure 3](#figure--fig:oomen18-generalized-plant):
|
||||
|
||||
- \\(z\\) are the performance variables
|
||||
- \\(y\\) and \\(u\\) are the measured variables and measured variables, respectively
|
||||
- \\(w\\) contains the exogenous inputs, typically including both reference signals and disturbances.
|
||||
|
||||
<a id="figure--fig:oomen18-generalized-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/oomen18_generalized_plant.png" caption="<span class='figure-number'>Figure 3: </span>Generalized plant setup" >}}
|
||||
|
||||
|
||||
## Feedback and Identification for Control {#feedback-and-identification-for-control}
|
||||
|
||||
Feedback control is essential to deal with uncertainty in the system dynamics \\(G\\) and disturbances \\(v\\).
|
||||
Indeed, the main goal of feedback si to render the system insensitive to such uncertainties.
|
||||
|
||||
|
||||
### Norm-based control {#norm-based-control}
|
||||
|
||||
A model-based design is foreseen to be able to systematically address the above mentioned challenges.
|
||||
|
||||
To specify the control goal, the criterion:
|
||||
|
||||
\begin{equation}
|
||||
J(G, K) = \\| \mathcal{F}\_l(P(G), K) \\|
|
||||
\end{equation}
|
||||
|
||||
is posed, where the goal is to compute:
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{opt}} = \text{arg} \text{min}\_{K} J(G\_0, K)
|
||||
\end{equation}
|
||||
|
||||
Where \\(\\| \cdot \\|\\) denotes a suitable norm, e.g. \\(\mathcal{H}\_2\\) or \\(\mathcal{H}\_\infty\\), and \\(\mathcal{F}\_l\\) denotes a lower linear fractional transformation.
|
||||
|
||||
\\(G\_0\\) denotes the true system, which is generally unknown and represented by a model \\(\hat{G}\\).
|
||||
|
||||
|
||||
### Nominal modeling for control {#nominal-modeling-for-control}
|
||||
|
||||
To arrive at a mathematically tractable optimization problem, knowledge of the true system is represented through a model \\(\hat{G}\\).
|
||||
The central question is how to obtain such a model that is suitable for controller design.
|
||||
[System Identification]({{< relref "system_identification.md" >}}) as opposed to first principles modeling, is an inexpensive, fast and accurate approach to obtain such a model.
|
||||
Indeed, the machine is often already built, enabling direct experimentation.
|
||||
|
||||
The model \\(\hat{G}\\) that results from system identification is an approximation of the true system \\(G\_0\\) for several reasons:
|
||||
|
||||
- motion systems often contains an infinite number of modes \\(n\_s\\), while a model of limited complexity may be desirable from a control perspective
|
||||
- parasitic non-linearities are present, including nonlinear damping
|
||||
- identification experiments are based on finite time disturbed observations, leading to uncertainties on estimated parameters
|
||||
|
||||
|
||||
### Toward robust motion control {#toward-robust-motion-control}
|
||||
|
||||
Doing a model based control design using an identified model may not work well due to a lack of robustness.
|
||||
Indeed, if \\(K(\hat{G})\\) is designed solely based on \\(\hat{G}\\), there is no reason to assume that it achieves a suitable level of performance on \\(G\_0\\).
|
||||
This motivates a robust control design, where the **model quality is explicitly addressed during controller synthesis**.
|
||||
|
||||
|
||||
## Feedforward and learning {#feedforward-and-learning}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” <i>IEEJ Journal of Industry Applications</i> 7 (2): 127–40. doi:<a href="https://doi.org/10.1541/ieejjia.7.127">10.1541/ieejjia.7.127</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,53 @@
|
||||
+++
|
||||
title = "Force feedback versus acceleration feedback in active vibration isolation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Preumont et al. 2002</a>)
|
||||
|
||||
Author(s)
|
||||
: Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
|
||||
|
||||
Year
|
||||
: 2002
|
||||
|
||||
Summary:
|
||||
|
||||
- Compares the force feedback and acceleration feedback for active damping
|
||||
- The use of a force sensor always give alternating poles and zeros in the open-loop transfer function between for force actuator and the force sensor which **guarantees the stability of the closed loop**
|
||||
- Acceleration feedback produces alternating poles and zeros only when the flexible structure is stiff compared to the isolation system
|
||||
|
||||
The force applied to a **rigid body** is proportional to its acceleration, thus sensing the total interface force gives a measured of the absolute acceleration of the solid body.
|
||||
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](#figure--fig:preumont02-force-acc-fb-light)), the acceleration feedback gives larger damping on the higher mode.
|
||||
- For heavy payload ([Figure 2](#figure--fig:preumont02-force-acc-fb-heavy)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
|
||||
|
||||
<a id="figure--fig:preumont02-force-acc-fb-light"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="<span class='figure-number'>Figure 1: </span>Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
|
||||
|
||||
<a id="figure--fig:preumont02-force-acc-fb-heavy"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="<span class='figure-number'>Figure 2: </span>Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
|
||||
|
||||
Guaranteed stability of the force feedback:
|
||||
|
||||
> If two arbitrary flexible, undamped structures are connected with a single-axis soft isolator with force feedback, the poles and zeros of the open-loop transfer function from the force actuator to the force sensor alternate on the imaginary axis.
|
||||
|
||||
The same is true for the transfer function from the force actuator to the relative displacement of the actuator.
|
||||
|
||||
> According to physical interpretation of the zeros, they represent the resonances of the subsystem constrained by the sensor and the actuator.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback versus Acceleration Feedback in Active Vibration Isolation.” <i>Journal of Sound and Vibration</i> 257 (4): 605–13. doi:<a href="https://doi.org/10.1006/jsvi.2002.5047">10.1006/jsvi.2002.5047</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,53 @@
|
||||
+++
|
||||
title = "A six-axis single-stage active vibration isolator based on stewart platform"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Preumont et al. 2007</a>)
|
||||
|
||||
Author(s)
|
||||
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
|
||||
|
||||
Year
|
||||
: 2007
|
||||
|
||||
Summary:
|
||||
|
||||
- **Cubic** Stewart platform ([Figure 3](#figure--fig:preumont07-stewart-platform))
|
||||
- Provides uniform control capability
|
||||
- Uniform stiffness in all directions
|
||||
- minimizes the cross-coupling among actuators and sensors of different legs
|
||||
- Flexible joints ([Figure 2](#figure--fig:preumont07-flexible-joints))
|
||||
- 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](#figure--fig:preumont07-iff-effect-stiffness))
|
||||
- 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
|
||||
|
||||
<a id="figure--fig:preumont07-iff-effect-stiffness"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="<span class='figure-number'>Figure 1: </span>Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
|
||||
|
||||
<a id="figure--fig:preumont07-flexible-joints"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="<span class='figure-number'>Figure 2: </span>Flexible joints used for the Stewart platform" >}}
|
||||
|
||||
<a id="figure--fig:preumont07-stewart-platform"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="<span class='figure-number'>Figure 3: </span>Stewart platform" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>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.” <i>Journal of Sound and Vibration</i> 300 (3-5): 644–61. doi:<a href="https://doi.org/10.1016/j.jsv.2006.07.050">10.1016/j.jsv.2006.07.050</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,92 @@
|
||||
+++
|
||||
title = "Advances in internal model control technique: a review and future prospects"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Complementary Filters]({{< relref "complementary_filters.md" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Saxena and Hote 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Saxena, S., & Hote, Y.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
|
||||
## Proposed Filter \\(F(s)\\) {#proposed-filter-f--s}
|
||||
|
||||
\begin{align\*}
|
||||
F(s) &= \frac{1}{(\lambda s + 1)^n} \\\\
|
||||
F(s) &= \frac{n \lambda + 1}{(\lambda s + 1)^n}
|
||||
\end{align\*}
|
||||
|
||||
|
||||
## Internal Model Control {#internal-model-control}
|
||||
|
||||
Central concept in IMC: control can be acheive only if the control system involves, either implicitly or explicitly, some representation of the process to be controlled.
|
||||
|
||||
|
||||
### Basic IMC structure {#basic-imc-structure}
|
||||
|
||||
IMC can be considered as a special case of classical feedback structure with plant \\(G(s)\\) and controller \\(C(s)\\).
|
||||
|
||||
The plan model \\(G\_M(s)\\) is added and substracted into the feedback path of feedback controller.
|
||||
|
||||
The structure can then be modified and we obtain a new controller \\(Q(s)\\).
|
||||
|
||||
IMC is related to the classical controller through:
|
||||
|
||||
\begin{align\*}
|
||||
Q(s) = \frac{C(s)}{1+G\_M(s)C(s)} \\\\
|
||||
C(s) = \frac{Q(s)}{1-G\_M(s)Q(s)}
|
||||
\end{align\*}
|
||||
|
||||
Internal model control system is characterized by a control device consisting of the controller \\(Q(s)\\) and a predictive model \\(G\_M(s)\\) of the process (internal model).
|
||||
The internal model loop uses the difference between the outputs of the process \\(G(s)\\) to be controlled and the internal model.
|
||||
This difference \\(E(s)\\) represents the effect of disturbance and mismatch of the model.
|
||||
|
||||
|
||||
### Features of IMC Structure {#features-of-imc-structure}
|
||||
|
||||
Three properties:
|
||||
|
||||
- **Dual stability**: assume that, if the plant model is perfect (\\(G\_M(s) = G(s)\\)) and disturbance is absent, the system becomes open-loop and the closed-loop stability is characterized by the stability of \\(G(s)\\) and \\(Q(s)\\)
|
||||
- **Perfect control**: assume that, if the controller is equal to the model inverse (\\(Q(s) = G\_M^{-1}\\)) and \\(G(s) = G\_M(s)\\) with \\(G(s)\\) stable, then the system is perfectly controlled.
|
||||
- **Zero Offset**: assume that, if the steady state gain of the controller is equal to the inverse of model gain, then offset free control is obtained for constant step of ramp type inputs and disturbances. As expected, the equivalent classical controller leads to integral action.
|
||||
|
||||
Issues:
|
||||
|
||||
- the plant model is never perfect
|
||||
- inverting the model can cause instability
|
||||
- control signal may have large magnitude
|
||||
|
||||
|
||||
## Design procedure for IMC Compensator {#design-procedure-for-imc-compensator}
|
||||
|
||||
1. factorize the plant model as \\(G\_M(s) = G\_{M-}(s)G\_{M+}(s)\\) where \\(G\_{M-}(s)\\) is invertible and minimum phase and \\(G\_{M+}(s)\\) is non-invertible and contains all non-minimum phase elements (delays, RHP zeros). Then, the controller is the inverse of the invertible portion of the plant model: \\(Q\_1(s) = G\_{M-}^{-1}(s)\\).
|
||||
2. Filter selection: to make the controller proper and robust against the plant-model mismatch, a low pass filter of the form \\(F(s) = \frac{n \lambda}{(\lambda s + 1)^n}\\) is augmented with the inverted model \\(Q\_1(s)\\): \\(Q(s) = Q\_1(s) F(s)\\). \\(\lambda\\) is a tuning parameter which has an inverse relationship with the speed of closed loop response, \\(n\\) is selected such that \\(Q(s)\\) becomes proper.
|
||||
|
||||
|
||||
## Issues in IMC {#issues-in-imc}
|
||||
|
||||
|
||||
### Filter selection and tuning guidelines {#filter-selection-and-tuning-guidelines}
|
||||
|
||||
|
||||
## Some advantages and future prospects {#some-advantages-and-future-prospects}
|
||||
|
||||
|
||||
## Conclusion {#conclusion}
|
||||
|
||||
The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Saxena, S., and Y. V. Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” <i>IETE Technical Review</i> 29 (6): 461. doi:<a href="https://doi.org/10.4103/0256-4602.105001">10.4103/0256-4602.105001</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Design for precision: current status and trends"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Precision Engineering]({{< relref "precision_engineering.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Schellekens et al. 1998</a>)
|
||||
|
||||
Author(s)
|
||||
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
|
||||
|
||||
Year
|
||||
: 1998
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” <i>Cirp Annals</i>, no. 2: 557–86. doi:<a href="https://doi.org/10.1016/s0007-8506(07)63243-0">10.1016/s0007-8506(07)63243-0</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "On compensator design for linear time-invariant dual-input single-output systems"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Schroeck, Messner, and McNab 2001</a>)
|
||||
|
||||
Author(s)
|
||||
: Schroeck, S., Messner, W., & McNab, R.
|
||||
|
||||
Year
|
||||
: 2001
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” <i>IEEE/ASME Transactions on Mechatronics</i> 6 (1): 50–57. doi:<a href="https://doi.org/10.1109/3516.914391">10.1109/3516.914391</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Nanopositioning with multiple sensors: a case study in data storage"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Sebastian and Pantazi 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Sebastian, A., & Pantazi, A.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” <i>IEEE Transactions on Control Systems Technology</i> 20 (2): 382–94. doi:<a href="https://doi.org/10.1109/tcst.2011.2177982">10.1109/tcst.2011.2177982</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,100 @@
|
||||
+++
|
||||
title = "A concept of active mount for space applications"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Active Damping]({{< relref "active_damping.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Souleille et al. 2018</a>)
|
||||
|
||||
Author(s)
|
||||
: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C.
|
||||
|
||||
Year
|
||||
: 2018
|
||||
|
||||
This article discusses the use of Integral Force Feedback with amplified piezoelectric stack actuators.
|
||||
|
||||
> In the proposed configuration, it can also be noticed by the softening effect inherent to force control is limited by the metallic suspension.
|
||||
|
||||
|
||||
## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
|
||||
|
||||
[Figure 1](#figure--fig:souleille18-model-piezo) 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.
|
||||
|
||||
<a id="figure--fig:souleille18-model-piezo"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="<span class='figure-number'>Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
|
||||
|
||||
<div class="table-caption">
|
||||
<span class="table-number">Table 1:</span>
|
||||
Parameters used for the model of the APA 100M
|
||||
</div>
|
||||
|
||||
| | Value | Meaning |
|
||||
|------------|------------------------|----------------------------------------------------------------|
|
||||
| \\(m\\) | \\(1\\,[kg]\\) | Payload mass |
|
||||
| \\(k\_e\\) | \\(4.8\\,[N/\mu m]\\) | Stiffness used to adjust the pole of the isolator |
|
||||
| \\(k\_1\\) | \\(0.96\\,[N/\mu m]\\) | Stiffness of the metallic suspension when the stack is removed |
|
||||
| \\(k\_a\\) | \\(65\\,[N/\mu m]\\) | Stiffness of the actuator |
|
||||
| \\(c\_1\\) | \\(10\\,[N/(m/s)]\\) | Added viscous damping |
|
||||
|
||||
The dynamic equation of the system is:
|
||||
|
||||
\begin{equation}
|
||||
m \ddot{x}\_1 = \left( k\_1 + \frac{k\_ek\_a}{k\_e + k\_a} \right) ( w - x\_1) + c\_1 (\dot{w} - \dot{x}\_1) + F + \left( \frac{k\_e}{k\_e + k\_a} \right)f
|
||||
\end{equation}
|
||||
|
||||
The expression of the force measured by the force sensor is:
|
||||
|
||||
\begin{equation}
|
||||
F\_s = \left( -\frac{k\_e k\_a}{k\_e + k\_a} \right) x\_1 + \left( \frac{k\_e k\_a}{k\_e + k\_a} \right) w + \left( \frac{k\_e}{k\_e + k\_a} \right) f
|
||||
\end{equation}
|
||||
|
||||
and the control force is given by:
|
||||
|
||||
\begin{equation}
|
||||
f = F\_s G(s) = F\_s \frac{g}{s}
|
||||
\end{equation}
|
||||
|
||||
The effect of the controller are shown in [Figure 2](#figure--fig:souleille18-tf-iff-result):
|
||||
|
||||
- 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
|
||||
|
||||
<a id="figure--fig:souleille18-tf-iff-result"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="<span class='figure-number'>Figure 2: </span>Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}}
|
||||
|
||||
<a id="figure--fig:souleille18-root-locus"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/souleille18_root_locus.png" caption="<span class='figure-number'>Figure 3: </span>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](#figure--fig:souleille18-setup-flexible-payload)).
|
||||
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.
|
||||
|
||||
<a id="figure--fig:souleille18-setup-flexible-payload"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="<span class='figure-number'>Figure 4: </span>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](#figure--fig:souleille18-result-damping-transmissibility), both the suspension modes and the flexible modes of the payload can be critically damped.
|
||||
|
||||
<a id="figure--fig:souleille18-result-damping-transmissibility"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="<span class='figure-number'>Figure 5: </span>Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Souleille, A., T. Lampert, V. Lafarga, S. Hellegouarch, A. Rondineau, G. Rodrigues, and C. Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>CEAS Space Journal</i> 10 (2). Springer: 157–65.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,66 @@
|
||||
+++
|
||||
title = "A soft 6-axis active vibration isolator"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Spanos, Rahman, and Blackwood 1995</a>)
|
||||
|
||||
Author(s)
|
||||
: Spanos, J., Rahman, Z., & Blackwood, G.
|
||||
|
||||
Year
|
||||
: 1995
|
||||
|
||||
**Stewart Platform** ([Figure 1](#figure--fig:spanos95-stewart-platform)):
|
||||
|
||||
- Voice Coil
|
||||
- Flexible joints (cross-blades)
|
||||
- Force Sensors
|
||||
- Cubic Configuration
|
||||
|
||||
<a id="figure--fig:spanos95-stewart-platform"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Stewart Platform" >}}
|
||||
|
||||
Total mass of the paylaod: 30kg
|
||||
Center of gravity is 9cm above the geometry center of the mount (cube's center?).
|
||||
|
||||
Limitation of the **Decentralized Force Feedback**:
|
||||
|
||||
- high frequency pole due to internal resonances of the struts
|
||||
- low frequency zero due to the rotational stiffness of the flexible joints
|
||||
|
||||
After redesign of the struts:
|
||||
|
||||
- high frequency pole at 4.7kHz
|
||||
- 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.
|
||||
|
||||
<a id="figure--fig:spanos95-iff-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="<span class='figure-number'>Figure 2: </span>Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
|
||||
|
||||
The controller used consisted of:
|
||||
|
||||
- second order low pass filter to gain stabilize the plant at high frequencies and provide steep roll-off
|
||||
- first order lead filter to provide adequate phase margin at the high frequency crossover
|
||||
- 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](#figure--fig:spanos95-results).
|
||||
|
||||
<a id="figure--fig:spanos95-results"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/spanos95_results.png" caption="<span class='figure-number'>Figure 3: </span>Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In <i>Proceedings of 1995 American Control Conference - ACC’95</i>. doi:<a href="https://doi.org/10.1109/acc.1995.529280">10.1109/acc.1995.529280</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,37 @@
|
||||
+++
|
||||
title = "Interferometric characterization of rotation stages for x-ray nanotomography"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Stankevic et al. 2017</a>)
|
||||
|
||||
Author(s)
|
||||
: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
|
||||
|
||||
Year
|
||||
: 2017
|
||||
|
||||
- Similar Station than the NASS
|
||||
- Similar Metrology with fiber based interferometers and cylindrical reference mirror
|
||||
|
||||
<a id="figure--fig:stankevic17-station"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stankevic17_station.png" caption="<span class='figure-number'>Figure 1: </span>Positioning Station" >}}
|
||||
|
||||
- **Thermal expansion**: Stabilized down to \\(5mK/h\\) using passive water flow through the baseplate below the sample stage and in the interferometry reference frame.
|
||||
- **Controller**: Two Independant PID loops
|
||||
- Repeatable errors => feedforward (Look Up Table)
|
||||
- Non-repeatable errors => feedback
|
||||
- Result: 40nm runout error
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stankevic, T., C. Engblom, F. Langlois, F. Alves, A. Lestrade, N. Jobert, G. Cauchon, U. Vogt, and S. Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” <i>Review of Scientific Instruments</i> 88 (5): 053703. doi:<a href="https://doi.org/10.1063/1.4983405">10.1063/1.4983405</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,125 @@
|
||||
+++
|
||||
title = "Respect the unstable"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Stein 2003</a>)
|
||||
|
||||
Author(s)
|
||||
: Stein, G.
|
||||
|
||||
Year
|
||||
: 2003
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
> The second trend has been evident at our conferences, and certainly in our journal, over the years.
|
||||
> This trend is the increasing worship of abstract mathematical results in control at the expense of more specific examinations of their practical, physical consequences.
|
||||
|
||||
<div class="important">
|
||||
|
||||
**Basic facts about unstable plants**:
|
||||
|
||||
- Unstable systems are fundamentally, and quantifiably more difficult to control than stable ones
|
||||
- Controllers for unstable systems are operationally critical
|
||||
- Closed-loop systems with unstable components are only locally stable
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## The Bode Integrals {#the-bode-integrals}
|
||||
|
||||
<div class="important">
|
||||
|
||||
**Bode Integrals**:
|
||||
|
||||
The first integral applies to stable plants and the second to unstable plants.
|
||||
They are valid for every stabilizing controller, assuming only that both plan and controller have finite bandwidths.
|
||||
In words, the integrals state that the log of magnitude of sensitivity function of a SISO feedback system, integrated over frequency, is constant.
|
||||
The constant is zero for stable plants, and it is positive for unstable ones.
|
||||
It becomes larger as the number of unstable poles increases and/or as the poles more farther into the right-half plane.
|
||||
|
||||
\begin{align}
|
||||
\int\_0^\infty \ln |S(j\omega)| d \omega & = 0 \label{eq:bode\_integral\_stable} \\\\
|
||||
\int\_0^\infty \ln |S(j\omega)| d \omega & = \pi \sum\_{p \in P} \text{Re}(p) \label{eq:bode\_integral\_unstable}
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## A Bode Integral Interpretation {#a-bode-integral-interpretation}
|
||||
|
||||
Bode integral can be thought as **conservation laws**.
|
||||
They state that a certain quantity, the integrated value of the log of the magnitude of the sensitivity function, is conserved under the action of feedback.
|
||||
The total amount of this quantity is always the same.
|
||||
It is equal to zero for stable plant/compensator pairs, and it is equal to some fixed positive amount for unstable ones.
|
||||
|
||||
Since we are talking about the log of sensitivity magnitude, it follows that negative values are good, and positive values are bad.
|
||||
|
||||
<div class="definition">
|
||||
|
||||
It is curious, somehow, that our field has not adopted a name for this quantity being conserved (i.e. the integrated log of sensitivity magnitude).
|
||||
It is here proposed to call it **dirt**
|
||||
|
||||
</div>
|
||||
|
||||
The job of a serious control designer is then to more dirt from one place to another, using appropriate tools, without being able to get rid of any of it (illustrated in [Figure 1](#figure--fig:stein03-serious-design)).
|
||||
|
||||
<a id="figure--fig:stein03-serious-design"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stein03_serious_design.png" caption="<span class='figure-number'>Figure 1: </span>Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies" >}}
|
||||
|
||||
In the same spirit, the job of a more academic control designer with more abstract tools such as LQG, \\(\mathcal{H}\_\infty\\), is to set parameters (weights) of a synthesis machine to adjust the contours of the machine's digging blades to get just the right shape for the sensitivity function ([Figure 2](#figure--fig:stein03-formal-design)).
|
||||
|
||||
<a id="figure--fig:stein03-formal-design"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stein03_formal_design.png" caption="<span class='figure-number'>Figure 2: </span>Sensitivity shaping automated by modern control tools" >}}
|
||||
|
||||
|
||||
## Available bandwidth {#available-bandwidth}
|
||||
|
||||
An argument is sometimes made that the Bode integrals are not really restrictive because we only seek to dig holes over finite frequency bands.
|
||||
We then have an infinite frequency range left over into which to dump the dirt, so we can make the layer arbitrarily thin ([Figure 3](#figure--fig:stein03-spreading-it-thin)).
|
||||
|
||||
<a id="figure--fig:stein03-spreading-it-thin"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stein03_spreading_it_thin.png" caption="<span class='figure-number'>Figure 3: </span>It is possible to spead the increase of the sensitivity function over a larger frequency band" >}}
|
||||
|
||||
The weakness of this argument is evident from standard classical theory.
|
||||
A thin layer, say with \\(\ln|S| = \epsilon\\) requires a loop transfer function whose Nyquist diagram falls on a near-unit circle, centered at \\((-1 + j 0)\\) with a radius \\(\approx (1-\epsilon)\\), over a wide frequency range.
|
||||
This means that the loop cannot simply attenuate at high frequencies but must attenuate in a very precise way.
|
||||
The loop must maintain very good frequency response fidelity over wide frequency ranges.
|
||||
|
||||
But a key fact about physical systems is that they do not exhibit good frequency response fidelity beyond a certain bandwidth.
|
||||
This is due to uncertain or unmodeled dynamics in the plant, to digital control implementations, to power limits, to nonlinearities, and to many other factors.
|
||||
Let us call that bandwidth the available bandwidth" \\(\Omega\_a\\), to distinguish it from other bandwidths such as crossover or \\(3-dB\\) magnitude loss.
|
||||
The available bandwidth is the frequency up to which we can keep \\(G(j\omega) K(j\omega)\\) close to a nominal design and beyond which we can only guarantee that the actual loop magnitude will attenuate rapidly enough (e.g. \\(|G(j\omega) K(j\omeg\\))| < δ/ω^2$).
|
||||
In today's popular robust control jargon, the available bandwidth is the frequency range over which the unstructured multiplicative perturbations are substantially less than unity.
|
||||
|
||||
Note that the available bandwidth is not a function of the compensator or of the control design process.
|
||||
Rather, it is an a priori constraint imposed by the physical hardware we use in the control loop.
|
||||
Most importantly, the available bandwidth is always finite.
|
||||
|
||||
Given all this, Bode's integrals really reduce to finite integrals over the range \\(0 \ge \omega \ge \Omega\_a\\):
|
||||
|
||||
\begin{align}
|
||||
\int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \delta \\\\
|
||||
\int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \pi \sum\_{p \in P} \text{Re}(p) + \delta
|
||||
\end{align}
|
||||
|
||||
All the action of the feedback design, the sensitivity improvements as well as the sensitivity deterioration, must occur within \\(0 \ge \omega \ge \Omega\_a\\).
|
||||
Only a small error \\(\delta\\) occurs outside that range, associated with the tail of the complete integrals.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stein, Gunter. 2003. “Respect the Unstable.” <i>IEEE Control Systems Magazine</i> 23 (4). IEEE: 12–25.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Motion control, mechatronics design, and moore's law"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Steinbuch, Oomen, and Vermeulen 2021</a>)
|
||||
|
||||
Author(s)
|
||||
: Steinbuch, M., Oomen, T., & Vermeulen, H.
|
||||
|
||||
Year
|
||||
: 2021
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Steinbuch, Maarten, Tom Oomen, and Hans Vermeulen. 2021. “Motion Control, Mechatronics Design, and Moore’s Law.” <i>IEEJ Journal of Industry Applications</i>. The Institute of Electrical Engineers of Japan, 21006010.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Advanced motion control: an industrial perspective"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Steinbuch and Norg 1998</a>)
|
||||
|
||||
Author(s)
|
||||
: Steinbuch, M., & Norg, M.
|
||||
|
||||
Year
|
||||
: 1998
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Steinbuch, M., and M.L. Norg. 1998. “Advanced Motion Control: An Industrial Perspective.” <i>European Journal of Control</i> 4 (4): 278–93. doi:<a href="https://doi.org/10.1016/s0947-3580(98)70121-9">10.1016/s0947-3580(98)70121-9</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,202 @@
|
||||
+++
|
||||
title = "Tensor methods for mimo decoupling and control design using frequency response functions"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Decoupled Control]({{< relref "decoupled_control.md" >}}), [Multivariable Control]({{< relref "multivariable_control.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Stoev et al. 2017</a>)
|
||||
|
||||
Author(s)
|
||||
: Stoev, J., Ertveldt, J., Oomen, T., & Schoukens, J.
|
||||
|
||||
Year
|
||||
: 2017
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
By appropriate system design, most systems are either decoupled or can be decoupled using static input-output transformations.
|
||||
Hence, most motion system and their motion software architecture use SISO control design method and solutions.
|
||||
|
||||
The first step typically involves a FRF identification using specific excitation signals.
|
||||
Once the FRF is available, the controller \\(K\\) can be designed directly based on the FRF data.
|
||||
Many classical MIMO control design methods aim at decoupling the open loop function at some location in the feedback loop.
|
||||
Because their are strong non-intuitive aspect of MIMO loop-shaping, the following step-by-step approach is proposed, in which the design complexity is only increased if justified by the problem at hand:
|
||||
|
||||
- **[Interaction Analysis]({{< relref "interaction_analysis.md" >}})**.
|
||||
The goal is to identify two sided interactions in the plant dynamics.
|
||||
If there is no two sided interaction, then feedback design becomes a standard multi-loop SISO design problem.
|
||||
Two measured of the plant interaction are [Relative Gain Array]({{< relref "relative_gain_array.md" >}}) and [Structured Singular Value]({{< relref "structured_singular_value.md" >}}).
|
||||
- **Decoupling transformations**.
|
||||
To reduce interaction, one may redefine the input and output of the plant using a decoupling transformation.
|
||||
For motion systems, most transformations are found on the basis of **kinematic model**.
|
||||
Herein, combinations of the actuators are defined so that actuator variables act in independent (orthogonal) directions at the center of gravity.
|
||||
Similarly, combinations of the sensors are defined so that each translation and rotation of the center of gravity can be measured independently.
|
||||
This, this basically amounts to the **inversion of a kinematic model** of the plant.
|
||||
- Independent feedback control design
|
||||
- Sequential feedback control design
|
||||
- Norm based control design
|
||||
|
||||
All steps, except for the last, can be performed with a non-parametric model of the plant (i.e. an identified FRF).
|
||||
|
||||
|
||||
## MIMO frequency response decomposition {#mimo-frequency-response-decomposition}
|
||||
|
||||
The problem addressed in this paper is to decouple a given set of MIMO FRF.
|
||||
Such decoupled representation, if existing, would permit the MIMO FRF to be written as a linear combination of parallel SISO FRFs.
|
||||
The existing methods to convert the MIMO FRF into equivalent combination of SISO FRF fall into two groups:
|
||||
|
||||
- **matrix decomposition methods** use linear algebra based on eigen-value, or singular value decomposition which are able to diagonalize the FRF at a single frequency.
|
||||
- **optimization methods** formulate the problem of simultaneous diagonalization of the FRF at multiple frequencies as an optimization problem.
|
||||
|
||||
At each frequency \\(\omega\_i, i = 1 \dots N\_f\\), we have a square matrix \\(H(\omega\_i) \in \mathbb{C}^{N \times N}\\) with the complex response of the system relating the inputs and outputs.
|
||||
|
||||
**MIMO decoupling of dyadic system**:
|
||||
|
||||
\begin{align}
|
||||
H(\omega\_i) &= T\_y S(\omega\_i) T\_u + E(\omega\_i), \ i = 1 \dots N\_f \label{eq:decomposition} \\\\
|
||||
S(\omega\_i) &= \begin{bmatrix}
|
||||
S\_1(\omega\_i) & 0 & 0 \\\\
|
||||
0 & \ddots & 0 \\\\
|
||||
0 & 0 & S\_N(\omega\_i)
|
||||
\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
where \\(S(\omega\_i)\\) is a diagonal matrix containing SISO FRFs \\(S\_k(\omega\_i) \in \mathbb{C}\\) on the main diagonal, \\(T\_y \in \mathbb{R}^{N \times N}\\), \\(T\_u \in \mathbb{R}^{N \times N}\\), \\(E(\omega\_i)\\) is the error.
|
||||
|
||||
The approximate MIMO system decoupling is shown in [Figure 1](#figure--fig:stoev17-decoupled-system-schematic).
|
||||
|
||||
In practical cases, the matrix \\(\hat{S}(\omega\_i) = T\_y^{-1} H(\omega\_i) T\_u^{-1}\\) will not be purely diagonal, but rather diagonally dominated.
|
||||
|
||||
<a id="figure--fig:stoev17-decoupled-system-schematic"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stoev17_decoupled_system_schematic.png" caption="<span class='figure-number'>Figure 1: </span>MIMO FRF decomposition in parallel branches" >}}
|
||||
|
||||
The array \\(H(\omega\_i), i = 1 \dots N\_f\\) of complex matrices can be represented as a 3-dimensional sensor \\(\underline{H}\\).
|
||||
|
||||
<div class="important">
|
||||
|
||||
The core result of this paper is that the decomposition can be found by rephrasing \ref{eq:decomposition} as a "Canonical Polyadic Decomposition" (CPD).
|
||||
This is shown in [Figure 2](#figure--fig:stoev17-decompos-3d-tensor), where \\(T\_y,T\_u,S\_d\\) can be directly computed using a single Matlab function.
|
||||
|
||||
</div>
|
||||
|
||||
Mathematically equivalent form of CPD is shown in the lower part of [Figure 2](#figure--fig:stoev17-decompos-3d-tensor), where the tensor \\(\underline{S}\\) contains the rows of the matrix \\(S\_d\\) on each of its diagonals in the third dimension, which is exactly the problem of simultaneous diagonalization.
|
||||
|
||||
The transformation effectively diagonalises the original frequency response tensor \\(\underline{H}\\) using two transformation matrices \\(T\_y, T\_u\\).
|
||||
This operation is closely related to the SVD on a single matrix, however in this case the diagonalisation occurs for a set of matrices, each describing the MIMO FRF at different frequency.
|
||||
|
||||
<a id="figure--fig:stoev17-decompos-3d-tensor"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stoev17_decompos_3d_tensor.png" caption="<span class='figure-number'>Figure 2: </span>Decomposition of 3D tensor" >}}
|
||||
|
||||
The direct application of a CPD procedure on the above complex data tensor would result in complex solutions, including complex matrices \\(T\_y \in \mathbb{C}^{N \times N}\\), \\(T\_u \in \mathbb{C}^{N \times N}\\).
|
||||
This is not useful for a practical decoupling of physical systems as we require real solutions for \\(T\_y,T\_u\\).
|
||||
The direct solution we use for this is to take the imaginary and real part of the complex tensor \\(\underline{H} \in \mathbb{C}^{N \times N \times N\_f}\\), each of them a real tensor by itself, and stack them one behind the other in the dimension of the frequencies, thus getting an augmented real-valued tensor \\(\underline{\breve{H}} \in \mathbb{R}^{N \times N \times 2N\_f}\\).
|
||||
|
||||
|
||||
## Numerical Example {#numerical-example}
|
||||
|
||||
Let's now make a Matlab example using the [Tensorlab](https://www.tensorlab.net/) toolbox.
|
||||
|
||||
Let's define a 2x2 diagonal system:
|
||||
|
||||
```matlab
|
||||
S = [4e3/(s^2 + 25*s + 4e3) 0
|
||||
0 4e5/(s^2 + 250*s + 4e5)];
|
||||
```
|
||||
|
||||
And coupled this system with two random matrices:
|
||||
|
||||
```matlab
|
||||
Ty = [0.13 0.003
|
||||
0.43 0.51];
|
||||
|
||||
Tu = [0.32 0.67
|
||||
0.95 0.006];
|
||||
```
|
||||
|
||||
The couple system is defined:
|
||||
|
||||
```matlab
|
||||
H = Ty * S * Tu;
|
||||
```
|
||||
|
||||
Then, suppose with have the frequency response function of the coupled plant:
|
||||
|
||||
```matlab
|
||||
freqs = logspace(0,3,1000);
|
||||
H_frf = freqresp(H, freqs, 'Hz');
|
||||
```
|
||||
|
||||
<a id="figure--fig:stoev17-coupled-diagonal-plants"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stoev17_coupled_diagonal_plants.png" caption="<span class='figure-number'>Figure 3: </span>Diagonal and coupled plants" >}}
|
||||
|
||||
We take the real and imaginary part of the FRF and concatenate the two along the frequency dimension.
|
||||
|
||||
```matlab
|
||||
H_frf_real = cat(3, real(H_frf), imag(H_frf));
|
||||
```
|
||||
|
||||
Then random matrices are initialize the the CPD.
|
||||
|
||||
```matlab
|
||||
U = cpd_rnd(size(H_frf_real), size(H_frf_real,1));
|
||||
```
|
||||
|
||||
And the CPD is performed.
|
||||
|
||||
```matlab
|
||||
[T, ~] = cpd3_sd(H_frf_real, U);
|
||||
```
|
||||
|
||||
The obtained decoupling matrices are:
|
||||
|
||||
```matlab
|
||||
Ty_est = T{1};
|
||||
```
|
||||
|
||||
```text
|
||||
Ty_est =
|
||||
-0.289402459385387 -0.00647742171539879
|
||||
-0.957207509228524 -1.10111369041218
|
||||
```
|
||||
|
||||
```matlab
|
||||
Tu_est = T{2};
|
||||
```
|
||||
|
||||
```text
|
||||
Tu_est =
|
||||
0.430893809258741 0.999980044721872
|
||||
0.902402640256826 0.00631744869723862
|
||||
```
|
||||
|
||||
And the decoupled plant using the estimated optimal decoupling matrices is:
|
||||
|
||||
```matlab
|
||||
H_dec = inv(Ty_est) * H * inv(Tu_est);
|
||||
```
|
||||
|
||||
<a id="figure--fig:stoev17-results-decoupling-example"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/stoev17_results_decoupling_example.png" caption="<span class='figure-number'>Figure 4: </span>Diagonal, coupled and decoupled plants" >}}
|
||||
|
||||
|
||||
## Conclusion {#conclusion}
|
||||
|
||||
The paper presents an application for the tensor decomposition for the design of a static decoupling of a MIMO system.
|
||||
The results in this paper are obtained on a _non-parametric_ frequency domain model of the plant and indicate that the procedure is more robust that the eigen-value based decoupling.
|
||||
The advantages of this method with respect to some of th existing methods can be found when the FRF data available is disturbed by noise.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stoev, Julian, Julien Ertveldt, Tom Oomen, and Johan Schoukens. 2017. “Tensor Methods for Mimo Decoupling and Control Design Using Frequency Response Functions.” <i>Mechatronics</i> 45: 71–81. doi:<a href="https://doi.org/10.1016/j.mechatronics.2017.05.009">10.1016/j.mechatronics.2017.05.009</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Tang, Cao, and Yu 2018</a>)
|
||||
|
||||
Author(s)
|
||||
: Tang, J., Cao, D., & Yu, T.
|
||||
|
||||
Year
|
||||
: 2018
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tang, J., D. Cao, and T. Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” <i>Proceedings of the Institution of Mechanical Engineers, Part c: Journal of Mechanical Engineering Science</i> 233 (1): 132–45. doi:<a href="https://doi.org/10.1177/0954406218756941">10.1177/0954406218756941</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Six-axis vibration isolation system using soft actuators and multiple sensors"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Thayer et al. 2002</a>)
|
||||
|
||||
Author(s)
|
||||
: Thayer, D., Campbell, M., Vagners, J., & Flotow, A. v.
|
||||
|
||||
Year
|
||||
: 2002
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Thayer, D., M. Campbell, J. Vagners, and A. von Flotow. 2002. “Six-Axis Vibration Isolation System Using Soft Actuators and Multiple Sensors.” <i>Journal of Spacecraft and Rockets</i> 39 (2): 206–12. doi:<a href="https://doi.org/10.2514/2.3821">10.2514/2.3821</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,24 @@
|
||||
+++
|
||||
title = "Fiber-Based Distance Sensing Interferometry"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Interferometers]({{< relref "interferometers.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Thurner et al. 2015</a>)
|
||||
|
||||
Author(s)
|
||||
: Thurner, K., Quacquarelli, F. P., Braun, Pierre-Francois, Dal Savio, C., & Karrai, K.
|
||||
|
||||
Year
|
||||
: 2015
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” <i>Applied Optics</i> 54 (10). Optical Society of America: 3051–63.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,54 @@
|
||||
+++
|
||||
title = "Sensor fusion for active vibration isolation in precision equipment"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Tjepkema, van Dijk, and Soemers 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Tjepkema, D., Dijk, J. v., & Soemers, H.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
|
||||
## Relative motion Control {#relative-motion-control}
|
||||
|
||||
Control law: \\(f = -G(x-w)\\)
|
||||
|
||||
\\[ \frac{x}{w} = \frac{k+G}{ms^2 + k+G} \\]
|
||||
\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
|
||||
|
||||
|
||||
## Force Control {#force-control}
|
||||
|
||||
Control law: \\(f = -G F\_a = -G \left(f-k(x-w)\right)\\)
|
||||
|
||||
\\[ \frac{x}{w} = \frac{k}{(1+G)ms^2 + k} \\]
|
||||
\\[ \frac{x}{F} = \frac{1+G}{(1+G)ms^2 + k} \\]
|
||||
|
||||
|
||||
## Inertial Control {#inertial-control}
|
||||
|
||||
Control law: \\(f = -Gx\\)
|
||||
|
||||
\\[ \frac{x}{w} = \frac{k}{ms^2 + k+G} \\]
|
||||
\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
|
||||
|
||||
|
||||
## Design constraints and control bandwidth {#design-constraints-and-control-bandwidth}
|
||||
|
||||
Heavier sensor => lower noise but it is harder to maintain collocation with the actuator => that limits the bandwidth.
|
||||
There is a compromise between sensor noise and the influence of the sensor size on the system's design and on the control bandwidth.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tjepkema, D., J. van Dijk, and H. M. J. R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” <i>Journal of Sound and Vibration</i> 331 (4): 735–49. doi:<a href="https://doi.org/10.1016/j.jsv.2011.09.022">10.1016/j.jsv.2011.09.022</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Essential challenges in motion control education"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Ech et al. 2019</a>)
|
||||
|
||||
Author(s)
|
||||
: M. \VCech, J. K\\"onigsmarkov\\'a, Goubej, M., Oomen, T., & Visioli, A.
|
||||
|
||||
Year
|
||||
: 2019
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>ech, M., J. Königsmarková, M. Goubej, T. Oomen, and A. Visioli. 2019. “Essential Challenges in Motion Control Education.” <i>IFAC-PapersOnLine</i> 52 (9): 200–205. doi:<a href="https://doi.org/10.1016/j.ifacol.2019.08.196">10.1016/j.ifacol.2019.08.196</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,33 @@
|
||||
+++
|
||||
title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Wang et al. 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
**Introduction of Markers**:
|
||||
That limits the type of samples that is studied
|
||||
|
||||
There is a need for markerless nano-tomography
|
||||
=> the key requirement is the precision and stability of the positioning stages.
|
||||
|
||||
**Passive rotational run-out error system**:
|
||||
It uses calibrated metrology disc and capacitive sensors
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wang, J., Y.-c. K. Chen, Q. Yuan, A. Tkachuk, C. Erdonmez, B. Hornberger, and M. Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” <i>Applied Physics Letters</i> 100 (14): 143107. doi:<a href="https://doi.org/10.1063/1.3701579">10.1063/1.3701579</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,61 @@
|
||||
+++
|
||||
title = "Investigation on active vibration isolation of a stewart platform with piezoelectric actuators"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Wang et al. 2016</a>)
|
||||
|
||||
Author(s)
|
||||
: Wang, C., Xie, X., Chen, Y., & Zhang, Z.
|
||||
|
||||
Year
|
||||
: 2016
|
||||
|
||||
**Model of the Stewart platform**:
|
||||
|
||||
- Struts are treated as flexible beams
|
||||
- Payload and the base are treated as flexible plates
|
||||
- The FRF synthesis method permits to derive FRFs of the Stewart platform
|
||||
|
||||
The model is compared with a Finite Element model and is shown to give the same results.
|
||||
The proposed model is thus effective.
|
||||
|
||||
<a id="figure--fig:wang16-stewart-platform"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Stewart Platform" >}}
|
||||
|
||||
**Control**:
|
||||
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](#figure--fig:wang16-force-feedback)).
|
||||
|
||||
- the force sensor is mounted **between the base and the strut**
|
||||
|
||||
<a id="figure--fig:wang16-force-feedback"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="<span class='figure-number'>Figure 2: </span>Feedback of integrated forces in the platform" >}}
|
||||
|
||||
Sorts of HAC-LAC control:
|
||||
|
||||
- LAC: Decentralized integral force feedback
|
||||
- HAC: Inertial control using accelerometers. Use of the Jacobian to decouple the motion and then Fx-LMS based adaptive control is used
|
||||
|
||||
**Experimental validation**:
|
||||
|
||||
- All 6 transfer function from actuator force to force sensors are almost the same (gain offset)
|
||||
- Effectiveness of control methods are shown
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wang, C., X. Xie, Y. Chen, and Z. Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” <i>Journal of Sound and Vibration</i> 383. Elsevier BV: 1–19. doi:<a href="https://doi.org/10.1016/j.jsv.2016.07.021">10.1016/j.jsv.2016.07.021</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,141 @@
|
||||
+++
|
||||
title = "Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Yang et al. 2019</a>)
|
||||
|
||||
Author(s)
|
||||
: Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
|
||||
|
||||
Year
|
||||
: 2019
|
||||
|
||||
**Discusses**:
|
||||
|
||||
- flexible-rigid model of Stewart platform
|
||||
- the impact of joint stiffness is compensated using a displacement sensor and a force sensor
|
||||
- then the MIMO system is decoupled in modal space and 6 SISO controllers are applied for vibration isolation using force sensors
|
||||
|
||||
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](#figure--fig:yang19-stewart-platform)):
|
||||
|
||||
- piezoelectric actuators
|
||||
- flexible joints ([Figure 2](#figure--fig:yang19-flexible-joints))
|
||||
- force sensors (used for vibration isolation)
|
||||
- displacement sensors (used to decouple the dynamics)
|
||||
- cubic (even though not said explicitly)
|
||||
|
||||
<a id="figure--fig:yang19-stewart-platform"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Stewart Platform" >}}
|
||||
|
||||
<a id="figure--fig:yang19-flexible-joints"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="<span class='figure-number'>Figure 2: </span>Flexible Joints" >}}
|
||||
|
||||
The stiffness of the flexible joints ([Figure 2](#figure--fig:yang19-flexible-joints)) are computed with an FEM model and shown in [Table 1](#table--tab:yang19-stiffness-flexible-joints).
|
||||
|
||||
<a id="table--tab:yang19-stiffness-flexible-joints"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:yang19-stiffness-flexible-joints">Table 1</a>:</span>
|
||||
Stiffness of flexible joints obtained by FEM
|
||||
</div>
|
||||
|
||||
| \\(k\_{\theta u},\ k\_{\psi u}\\) | \\(72 Nm/rad\\) |
|
||||
|-----------------------------------|-----------------|
|
||||
| \\(k\_{\theta s}\\) | \\(51 Nm/rad\\) |
|
||||
| \\(k\_{\psi s}\\) | \\(62 Nm/rad\\) |
|
||||
| \\(k\_{\gamma s}\\) | \\(64 Nm/rad\\) |
|
||||
|
||||
**Dynamics**:
|
||||
If the bending and torsional stiffness of the flexible joints are neglected:
|
||||
\\[ M \ddot{x} + C \dot{x} + K x = J^T f \\]
|
||||
|
||||
- \\(M\\) is the mass matrix
|
||||
- \\(C\\) is the damping matrix
|
||||
- \\(K\\) is the stiffness matrix
|
||||
- \\(x\\) is the generalized coordinates, representing the displacement and orientation of the payload plate
|
||||
- \\(f\\) is the actuator forces
|
||||
- \\(J\\) is the Jacobian matrix
|
||||
|
||||
In this paper, the parasitic bending stiffness of the flexible joints are considered:
|
||||
\\[ M \ddot{x} + C \dot{x} + (K + K\_e) x = J^T f \\]
|
||||
where \\(K\_e\\) is the stiffness matrix induced by the parasitic stiffness of the flexible joints.
|
||||
|
||||
Analytical expression for \\(K\_e\\) are derived in the paper.
|
||||
|
||||
**Controller Design**:
|
||||
There is a strong coupling between the input forces and the state variables in the task space.
|
||||
The traditional modal decoupled control strategy cannot work with the flexible Stewart platform because it is impossible to achieve simultaneous diagonalization of the mass, damped and stiffness matrices.
|
||||
|
||||
To make the six-dof system decoupled into six single-dof isolators, a controller based on the leg's force and position feedback is designed.
|
||||
|
||||
> The idea is to synthesize the control force that can compensate the parasitic bending and torsional torques of the flexible joints and simultaneously achieve diagonalization of the matrices \\(M\\), \\(C\\) and \\(K\\)
|
||||
|
||||
The force measured by the force sensors are:
|
||||
\\[ y = f - k J x - c J \dot{x} \\]
|
||||
The displacements measured by the position sensors are:
|
||||
\\[ z = [\Delta l\_1\ \dots\ \Delta l\_6]^T \\]
|
||||
|
||||
Let's apply the feedback control based on both the force sensor and the position sensor:
|
||||
\\[ f = -H(s) y + (1 + H(s)) K\_{el} z \\]
|
||||
where \\(K\_{el} = J^{-T} K\_e J^T\\) is the stiffness matrix of the flexible joints expressed in joint space.
|
||||
|
||||
We thus obtain:
|
||||
\\[ f = \frac{H(s)}{1 + H(s)} (k J x + c J \dot{x}) + J^{-T} K\_e x \\]
|
||||
|
||||
If we substitute \\(f\\) in the dynamic equation, we obtain that the parasitic stiffness effect of the flexible joints has been compensated by the actuation forces and the system can now be decoupled in modal space \\(x = \Phi u\\).
|
||||
\\(\Phi\\) is the modal matrix selected such that \\(\Phi^T M \Phi = I\_6\\) and \\(k \Phi^T J^T J \Phi = \text{diag}(\omega\_1^2\ \dots\ \omega\_6^2)\\):
|
||||
\\[ s^2 + \frac{1}{1 + H(s)} \frac{c \omega\_i^2}{k} s + \frac{1}{1 + H(s)} \omega\_i^2 = 0, \quad i = 1,\ \dots,\ 6 \\]
|
||||
|
||||
The six-dof system is now transformed into a six one-dof system where \\(H(s)\\) can be designed for control purpose.
|
||||
|
||||
In order to apply this control strategy:
|
||||
|
||||
- A force sensor and displacement sensor are need in each strut
|
||||
- The joint stiffness has to be known
|
||||
- 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](#figure--fig:yang19-control-arch).
|
||||
|
||||
<a id="figure--fig:yang19-control-arch"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/yang19_control_arch.png" caption="<span class='figure-number'>Figure 3: </span>Control Architecture used" >}}
|
||||
|
||||
\\(H(s)\\) is designed as a proportional plus integral compensator:
|
||||
\\[ H(s) = k\_p + k\_i/s \\]
|
||||
|
||||
Substituting \\(H(s)\\) in the equation of motion gives that:
|
||||
|
||||
- an increase of \\(k\_i\\) increase the damping and thus suppress the resonance peaks
|
||||
- an increase of \\(k\_p\\) lowers the resonance frequency and thus the bandwidth of vibration isolation is examped
|
||||
|
||||
**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](#figure--fig:yang19-results).
|
||||
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.
|
||||
|
||||
<a id="figure--fig:yang19-results"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/yang19_results.png" caption="<span class='figure-number'>Figure 4: </span>Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
|
||||
|
||||
> A model-based controller is then designed based on the leg’s force and position feedback.
|
||||
> The position feedback compensates the effect of parasitic bending and torsional stiffness of the flexible joints.
|
||||
> The force feedback makes the six-DOF MIMO system decoupled into six SISO subsystems in modal space, where the control gains can be designed and analyzed more effectively and conveniently.
|
||||
> The proportional and integral gains in the sub-controller are used to separately regulate the vibration isolation bandwidth and active damping simultaneously for the six vibration modes.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yang, X., H. Wu, B. Chen, S. Kang, and S. Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” <i>Journal of Sound and Vibration</i> 439. Elsevier BV: 398–412. doi:<a href="https://doi.org/10.1016/j.jsv.2018.10.007">10.1016/j.jsv.2018.10.007</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,22 @@
|
||||
+++
|
||||
title = "Invited review article: high-speed flexure-guided nanopositioning: mechanical design and control issues"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Yong et al. 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Yong, Y. K., Moheimani, S. O. R., Kenton, B. J., & Leang, K. K.
|
||||
|
||||
Year
|
||||
: 2012
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yong, Y. K., S. O. R. Moheimani, B. J. Kenton, and K. K. Leang. 2012. “Invited Review Article: High-Speed Flexure-Guided Nanopositioning: Mechanical Design and Control Issues.” <i>Review of Scientific Instruments</i> 83 (12): 121101. doi:<a href="https://doi.org/10.1063/1.4765048">10.1063/1.4765048</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "Investigation on two-stage vibration suppression and precision pointing for space optical payloads"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Yun et al. 2020</a>)
|
||||
|
||||
Author(s)
|
||||
: Yun, H., Liu, L., Li, Q., & Yang, H.
|
||||
|
||||
Year
|
||||
: 2020
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” <i>Aerospace Science and Technology</i> 96: 105543. doi:<a href="https://doi.org/10.1016/j.ast.2019.105543">10.1016/j.ast.2019.105543</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,37 @@
|
||||
+++
|
||||
title = "Six dof active vibration control using stewart platform with non-cubic configuration"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Zhang et al. 2011</a>)
|
||||
|
||||
Author(s)
|
||||
: Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y.
|
||||
|
||||
Year
|
||||
: 2011
|
||||
|
||||
- **Non-cubic** stewart platform
|
||||
- **Flexible** joints
|
||||
- Magnetostrictive actuators
|
||||
- Strong coupled motions along different axes
|
||||
- Non-cubic architecture => permits to have larger workspace which was required
|
||||
- Structure parameters (radius of plates, length of struts) are determined by optimization of the condition number of the Jacobian matrix
|
||||
- **Accelerometers** for active isolation
|
||||
- Adaptive FIR filters for active isolation control
|
||||
|
||||
<a id="figure--fig:zhang11-platform"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zhang11_platform.png" caption="<span class='figure-number'>Figure 1: </span>Prototype of the non-cubic stewart platform" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zhang, Z., J. Liu, J. Mao, Y. Guo, and Y. Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In <i>2011 6th IEEE Conference on Industrial Electronics and Applications</i>. doi:<a href="https://doi.org/10.1109/iciea.2011.5975679">10.1109/iciea.2011.5975679</a>.</div>
|
||||
</div>
|
||||
Reference in New Issue
Block a user