The stiffness of the flexible joints (Figure [2](#org8725dbf)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
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)\\):
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.
{{<figuresrc="/ox-hugo/yang19_results.png"caption="Figure 4: 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.
<aid="org8fbcee2"></a>Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” _Journal of Sound and Vibration_ 439 (January). Elsevier BV:398–412. <https://doi.org/10.1016/j.jsv.2018.10.007>.
