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title = "Identification and decoupling control of flexure jointed hexapods"
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author = ["Dehaeze Thomas"]
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draft = false
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Tags
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: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
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Reference
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: (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>)
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Author(s)
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: Chen, Y., & McInroy, J.
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Year
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: 2000
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## Abstract {#abstract}
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> By exploiting properties of the joint space mass-inertia matrix of flexure jointed hexapods, a new **decoupling method** is proposed.
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> 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.
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> Prior decoupling control algorithms imposed severe constraints on the allowable geometry, workspace and payload.
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> This paper derives a new algorithm which removes these constraints, thus greatly expanding the applications.
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> 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.
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> This algorithm can be used for precision payload calibration, thus improving performance and removing the labor required to design the control for different payloads.
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> The new decoupling algorithm is experimentally compared to earlier techniques.
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> These experimental results indicate that the new approach is practical, and improves performance.
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## Introduction {#introduction}
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Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control.md" >}})) impose two constraints:
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- the payload mass/inertia matrix is diagonal
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- the geometry of the platform and attachment of the payload must be carefully chosen
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This limits the applications significantly.
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The algorithm derived herein removes these constraints, thus greatly expanding the potential applications.
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## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
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The derivation of the dynamic model is done in (<a href="#citeproc_bib_item_2">McInroy 1999</a>) ([Notes]({{< relref "mcinroy99_dynam.md" >}})).
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<a id="figure--fig:chen00-flexure-hexapod"></a>
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{{< 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" >}}
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In the joint space, the dynamics of a flexure jointed hexapod are written as:
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\begin{equation}
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\vec{f}\_b = \vec{f}\_m - \bm{K}(\vec{l} - \vec{l}\_r) - \bm{B} \dot{\vec{l}}
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\end{equation}
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\begin{aligned}
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& \left( {}^U\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} \right) \ddot{\vec{l}} + \\\\
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& {}^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) = \\\\
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& {}^U\_B\bm{R} \bm{J}^T \vec{f}\_m + \vec{\mathcal{F}}\_e + \vec{\mathcal{F}} + \vec{\mathcal{C}} - \\\\
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& \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
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\end{aligned}
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where:
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- \\(\bm{J}\\) is the \\(6 \times 6\\) hexapod Jacobian relating payload Cartesian movements, expressed in {P}, to strut length changes in the joint space
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- \\({}^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)
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- \\(\bm{J}\_c\\) and \\(\bm{J}\_B\\) are \\(6 \times 6\\) Jacobian matrices capturing base motion
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- \\({}^P\bm{M}\_x\\) is the \\(6 \times 6\\) mass-inertia matrix of the payload found with respect to the payload frame {P}
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- \\(\bm{M}\_s\\) is a diagonal \\(6 \times 6\\) matrix containing the moving mass of each strut
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- \\(\bm{B}\\) and \\(\bm{K}\\) are \\(6 \times 6\\) diagonal matrices containing the damping of stiffness, respectively, of each strut
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- \\(\vec{l}\\) is the \\(6 \times 1\\) vector of strut lengths, and \\(\vec{l}\_r\\) is the constant vector of relaxed strut length
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- \\(\vec{f}\_b\\) is the vector of forces exerted at the bottom of the strut
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- \\(\vec{f}\_m\\) is the vector of strut motor forces
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- \\(\ddot{\vec{q}}\_s\\) is a \\(6 \times 1\\) vector of base accelerations along each strut plus some Coriolis terms
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- \\(\vec{\mathcal{F}}\_e\\) is a vector of payload exogenous generalized forces
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- \\(\vec{\mathcal{C}}\\) is a vector containing all the Coriolis and centripetal terms except the Coriolis terms in \\(\ddot{\vec{q}}\_s\\)
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- \\(\vec{\mathcal{G}}\\) is a vector containing all gravity terms
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\begin{aligned}
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\bm{M}\_p & \ddot{\vec{p}}\_s + \bm{B} \dot{\vec{p}}\_s + \bm{K} \vec{p}\_s = \vec{f}\_m + \\\\
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& \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
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\end{aligned}
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where
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- \\(\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\\)
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- \\(\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}\\)
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\\(\bm{M}\_p\\) and \\(\bm{M}\_q\\) are joint space mass-inertia matrices.
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## Decoupling the Dynamics of Flexure Jointed Hexapods {#decoupling-the-dynamics-of-flexure-jointed-hexapods}
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## Identification of Joint Space Mass-Inertia Matrix {#identification-of-joint-space-mass-inertia-matrix}
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## Experimental Results {#experimental-results}
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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>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>
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<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>
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</div>
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