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title = "Singular Value Decomposition"
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author = ["Thomas Dehaeze"]
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## SVD of a MIMO system {#svd-of-a-mimo-system}
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We are interested by the physical interpretation of the SVD when applied to the frequency response of a MIMO system \\(G(s)\\) with \\(m\\) inputs and \\(l\\) outputs.
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\begin{equation}
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G = U \Sigma V^H
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\end{equation}
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\\(\Sigma\\)
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: is an \\(l \times m\\) matrix with \\(k = \min\\{l, m\\}\\) non-negative **singular values** \\(\sigma\_i\\), arranged in descending order along its main diagonal, the other entries are zero.
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\\(U\\)
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: is an \\(l \times l\\) unitary matrix. The columns of \\(U\\), denoted \\(u\_i\\), represent the **output directions** of the plant. They are orthonormal.
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\\(V\\)
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: is an \\(m \times m\\) unitary matrix. The columns of \\(V\\), denoted \\(v\_i\\), represent the **input directions** of the plant. They are orthonormal.
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The input and output directions are related through the singular values:
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\begin{equation}
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G v\_i = \sigma\_i u\_i
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\end{equation}
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So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\).
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Furthermore, since \\(\normtwo{v\_i}=1\\) and \\(\normtwo{u\_i}=1\\), we see that **the singular value \\(\sigma\_i\\) directly gives the gain of the matrix \\(G\\) in this direction**.
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The **largest gain** for any input is equal to the **maximum singular value**:
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\\[\maxsv(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_1}}{\normtwo{v\_1}} \\]
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The **smallest gain** for any input direction is equal to the **minimum singular value**:
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\\[\minsv(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_k}}{\normtwo{v\_k}} \\]
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We define \\(u\_1 = \bar{u}\\), \\(v\_1 = \bar{v}\\), \\(u\_k=\ubar{u}\\) and \\(v\_k = \ubar{v}\\). Then is follows that:
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\\[ G\bar{v} = \maxsv \bar{u} ; \quad G\ubar{v} = \minsv \ubar{u} \\]
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## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
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This is taken from [Preumont's book](preumont18_vibrat_contr_activ_struc_fourt_edition.md).
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The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix:
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\\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\]
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With:
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- \\(U\\) and \\(V\\) orthogonal matrices. The columns \\(u\_i\\) and \\(v\_i\\) of \\(U\\) and \\(V\\) are the eigenvectors of the square matrices \\(JJ^T\\) and \\(J^TJ\\) respectively
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- \\(\Sigma\\) a rectangular diagonal matrix of dimension \\(m \times n\\) containing the square root of the common non-zero eigenvalues of \\(JJ^T\\) and \\(J^TJ\\)
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- \\(r\\) is the number of non-zero singular values of \\(J\\)
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The pseudo-inverse of \\(J\\) is:
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\\[ J^+ = V\Sigma^+U^T = \sum\_{i=1}^r \frac{1}{\sigma\_i} v\_i u\_i^T \\]
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The conditioning of the Jacobian is measured by the **condition number**:
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\\[ c(J) = \frac{\sigma\_{max}}{\sigma\_{min}} \\]
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When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum.
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This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\).
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<./biblio/references.bib>
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@@ -40,6 +40,9 @@ Tags
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| Newport | [link](https://www.newport.com/search/?q1=hexapod%3Arelevance%3Acompatibility%3AMETRIC%3AisObsolete%3Afalse%3A-excludeCountries%3AFR%3AnpCategory%3Ahexapods&ajax&text=hexapod) | USA |
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| Symetrie | [link](https://symetrie.fr/en/hexapods-en/positioning-hexapods/) | France |
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| CSA Engineering | [link](https://www.csaengineering.com/products-services/hexapod-positioning-systems/hexapod-models.html) | USA |
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| Aerotech | [link](https://www.aerotech.com/product-catalog/hexapods.aspx) | USA |
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| SmarAct | [link](https://www.smaract.com/smarpod) | Germany |
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| Gridbots | [link](https://www.gridbots.com/hexamove.html) | India |
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## Stewart Platforms at ESRF {#stewart-platforms-at-esrf}
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Papers by J.E. McInroy:
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- ([O’Brien et al. 1998](#org6990e82))
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- ([McInroy, O’Brien, and Neat 1999](#org2eaf165))
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- ([McInroy 1999](#org19d227f))
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- ([McInroy and Hamann 2000](#org0de51ce))
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- ([Chen and McInroy 2000](#orgfa121ee))
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- ([McInroy 2002](#org1ad3b68))
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- ([Li, Hamann, and McInroy 2001](#orga0bd81f))
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- ([Lin and McInroy 2003](#org01eae42))
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- ([Jafari and McInroy 2003](#org9490dc9))
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- ([Chen and McInroy 2004](#org3bd3a36))
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- ([O’Brien et al. 1998](#orgb1b8013))
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- ([McInroy, O’Brien, and Neat 1999](#org06dafcc))
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- ([McInroy 1999](#orga04f28d))
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- ([McInroy and Hamann 2000](#org61fde10))
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- ([Chen and McInroy 2000](#org4b76086))
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- ([McInroy 2002](#orgc3f19a5))
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- ([Li, Hamann, and McInroy 2001](#org469fdd4))
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- ([Lin and McInroy 2003](#org52d73bd))
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- ([Jafari and McInroy 2003](#orga8e7146))
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- ([Chen and McInroy 2004](#org97f0e30))
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## Bibliography {#bibliography}
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<a id="org3bd3a36"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
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<a id="org97f0e30"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
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<a id="orgfa121ee"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
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<a id="org4b76086"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
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<a id="org9490dc9"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
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<a id="orga8e7146"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
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<a id="org01eae42"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
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<a id="org52d73bd"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
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<a id="orga0bd81f"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
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<a id="org469fdd4"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
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<a id="org19d227f"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
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<a id="orga04f28d"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
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<a id="org1ad3b68"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
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<a id="orgc3f19a5"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
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<a id="org0de51ce"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
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<a id="org61fde10"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
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<a id="org2eaf165"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
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<a id="org06dafcc"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
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<a id="org6990e82"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.
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<a id="orgb1b8013"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.
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