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Rework flexible joint section

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@ -106,6 +106,39 @@ Check the two reports:
For instance show that from FEM (/super element/, simscape), we get the characteristics found in the datasheet: stroke, stiffness, resonance frequency, etc...
- [X] Look here: https://gitlab.esrf.fr/dehaeze/nass-fem/-/tree/master?ref_type=heads
==============
Flexible joints:
*Here we just look at the wanted stiffness in different directions?*
*Or also design of the flexible joint*?
Because we talk about detail design in last section
*In this section, we talk only about important part of the flexible joint, and not connection to the struts of plates*
Prefix is =fem_joints=
- [X] *Should I consider rigid micro-station for simplified analysis?*
Yes
- [X] https://research.tdehaeze.xyz/nass-simscape/docs/flexible_joints_study.html
- [X] look at section 5.4 file:/home/thomas/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org
- [X] Show that flexible joints' stiffnesses has little impact on the Stewart platform behavior
Find paper of MrInroy that talks about that with associated conditions: [[cite:&mcinroy02_model_desig_flexur_joint_stewar]]
- [X] Look here: https://gitlab.esrf.fr/dehaeze/nass-fem/-/tree/master?ref_type=heads
- [X] Also check start of this report: file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org to show the Simscape model of the Flexible joint
*No this will be for the detail design phase*
Outline:
- Review of flexible joints ?
- Imperfection of the flexible joint: Model
- Study of the effect of limited stiffness in constrain directions and non-null stiffness in other directions
Compare with perfect flexible joint case
- Obtained Specification
- Design optimisation (FEM)
- Implementation of flexible elements in the Simscape model: close to simplified model
** Not used
*** Complete Strut with Encoder
:PROPERTIES:
@ -2387,57 +2420,43 @@ exportFig('figs/detail_fem_actuator_fem_vs_perfect_iff_plant.pdf', 'width', 'hal
* Flexible Joint Design
<<sec:detail_fem_joint>>
** Notes :noexport:
*Here we just look at the wanted stiffness in different directions?*
*Or also design of the flexible joint*?
Because we talk about detail design in last section
*In this section, we talk only about important part of the flexible joint, and not connection to the struts of plates*
Prefix is =fem_joints=
- [ ] *Should I consider rigid micro-station for simplified analysis?*
- [ ] https://research.tdehaeze.xyz/nass-simscape/docs/flexible_joints_study.html
- [X] look at section 5.4 file:/home/thomas/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org
- [X] Show that flexible joints' stiffnesses has little impact on the Stewart platform behavior
Find paper of MrInroy that talks about that with associated conditions: [[cite:&mcinroy02_model_desig_flexur_joint_stewar]]
- [X] Look here: https://gitlab.esrf.fr/dehaeze/nass-fem/-/tree/master?ref_type=heads
- [X] Also check start of this report: file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org to show the Simscape model of the Flexible joint
*No this will be for the detail design phase*
Outline:
- Review of flexible joints ?
- Imperfection of the flexible joint: Model
- Study of the effect of limited stiffness in constrain directions and non-null stiffness in other directions
Compare with perfect flexible joint case
- Obtained Specification
- Design optimisation (FEM)
- Implementation of flexible elements in the Simscape model: close to simplified model
** Introduction :ignore:
The flexible joints have few advantages compared to conventional joints such as the *absence of wear, friction and backlash* which allows extremely high-precision (predictable) motion.
The parasitic bending and torsional stiffness of these joints usually induce some *limitation on the control performance*. [[cite:&mcinroy02_model_desig_flexur_joint_stewar]]
High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy.
This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation.
For Stewart platforms requiring nanometric precision, numerous flexible joint designs have been developed and successfully implemented, as illustrated in Figure ref:fig:detail_fem_joints_examples.
For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts.
In this document is studied the effect of the mechanical behavior of the flexible joints that are located the extremities of each nano-hexapod's legs.
#+name: fig:detail_fem_joints_examples
#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_yang}) [[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_preumont}) [[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_wire}) [[cite:&du14_piezo_actuat_high_precis_flexib]].
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_yang}}
#+attr_latex: :options {0.35\textwidth}
#+begin_subfigure
#+attr_latex: :height 5cm
[[file:figs/detail_fem_joints_yang.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_preumont}}
#+attr_latex: :options {0.3\textwidth}
#+begin_subfigure
#+attr_latex: :height 5cm
[[file:figs/detail_fem_joints_preumont.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_wire}}
#+attr_latex: :options {0.3\textwidth}
#+begin_subfigure
#+attr_latex: :height 5cm
[[file:figs/detail_fem_joints_wire.png]]
#+end_subfigure
#+end_figure
Ideally, we want the x and y rotations to be free and all the translations to be blocked.
However, this is never the case and be have to consider:
- Non-null bending stiffnesses
- Non-null radial compliance
- Axial stiffness in the direction of the legs
While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance [[cite:&mcinroy02_model_desig_flexur_joint_stewar]].
This section examines how these non-ideal characteristics affect system behavior, focusing particularly on bending/torsional stiffness (Section ref:ssec:detail_fem_joint_bending) and axial compliance (Section ref:ssec:detail_fem_joint_axial).
This may impose some limitations, also, the goal is to specify the required joints stiffnesses.
Say that for simplicity (reduced number of parts, etc.), we consider the same joints for the fixed based and the top platform.
*Outline*:
- Perfect flexible joint
- Imperfection of the flexible joint: Model
- Study of the effect of limited stiffness in constrain directions and non-null stiffness in other directions
- Obtained Specification
- Design optimisation (FEM)
- Implementation of flexible elements in the Simscape model: close to simplified model
The analysis of bending and axial stiffness effects enables the establishment of comprehensive specifications for the flexible joints.
These specifications guide the development and optimization of a flexible joint design through finite element analysis (Section ref:ssec:detail_fem_joint_specs).
The validation process, detailed in Section ref:ssec:detail_fem_joint_validation, begins with the integration of the joints as "reduced order flexible bodies" in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -2470,55 +2489,29 @@ open(mdl); % Open Simscape Model
<<m-init-other>>
#+end_src
** Flexible joints for Stewart platforms
Review of different types of flexible joints for Stewart plaftorms (see Figure ref:fig:detail_fem_joints_examples).
Typical specifications:
- Bending stroke (i.e. long life time by staying away from yield stress, even at maximum deflection/load)
- Axial stiffness
- Bending stiffness
- Maximum axial load
- Well defined rotational axes
Typical values?
- $K_{\theta, \phi} = 15\,[Nm/rad]$ stiffness in flexion
- $K_{\psi} = 20\,[Nm/rad]$ stiffness in torsion
- \[ K_a = 60\,[N/\mu m] \] axial stiffness
#+name: fig:detail_fem_joints_examples
#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_yang}) [[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_preumont}) [[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_wire}) [[cite:&du14_piezo_actuat_high_precis_flexib]].
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_yang}}
#+attr_latex: :options {0.35\textwidth}
#+begin_subfigure
#+attr_latex: :height 5cm
[[file:figs/detail_fem_joints_yang.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_preumont}}
#+attr_latex: :options {0.3\textwidth}
#+begin_subfigure
#+attr_latex: :height 5cm
[[file:figs/detail_fem_joints_preumont.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_wire}}
#+attr_latex: :options {0.3\textwidth}
#+begin_subfigure
#+attr_latex: :height 5cm
[[file:figs/detail_fem_joints_wire.png]]
#+end_subfigure
#+end_figure
** Bending and Torsional Stiffness
<<sec:joints_rot_stiffness>>
<<ssec:detail_fem_joint_bending>>
Because of bending stiffness of the flexible joints, the forces applied by the struts are no longer aligned with the struts (additional forces applied by the "spring force" of the flexible joints).
The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction.
This additional spring forces can affect system dynamics.
In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.
- To simplify the analysis, the micro-station is considered rigid, and only the nano-hexapod is considered with:
- 1dof actuators, k=1N/um, without parallel stiffness to the force sensors
- The bending stiffness of all joints are varied and the dynamics is identified
To isolate and quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified single-degree-of-freedom actuators (stiffness $1\,N/\mu m$) without parallel force sensor stiffness.
Joint bending stiffness was varied from 0 (ideal case) to 500 Nm/rad.
Analysis of the plant dynamics reveals two significant effects.
For the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$, bending stiffness increases low-frequency coupling, though this remains small for realistic stiffness values (Figure ref:fig:detail_fem_joints_bending_stiffness_hac_plant).
In [[cite:&mcinroy02_model_desig_flexur_joint_stewar]], it is established that forces remain effectively aligned with the struts when the flexible joint bending stiffness is much small than the actuator stiffness multiplied by the square of the strut length.
For the nano-hexapod, this corresponds to having the bending stiffness much lower than 9000 Nm/rad.
This condition is more readily satisfied with the relatively stiff actuators selected, and could be problematic for softer Stewart platforms.
For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure ref:fig:detail_fem_joints_bending_stiffness_iff_plant).
This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant).
However, this time the parallel stiffness does not comes from the considered strut, but from the bending stiffness of the flexible joints of the other five struts.
This characteristic impacts the achievable damping using decentralized Integral Force Feedback [[cite:&preumont07_six_axis_singl_stage_activ]].
This is confirmed by the Root Locus plot in Figure ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof.
This effect becomes less significant when using the selected APA300ML actuators (Figure ref:fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness.
A parallel analysis of torsional stiffness revealed similar dynamic effects, though these proved less critical for system performance.
#+begin_src matlab
%% Identify the dynamics for several considered bending stiffnesses
@ -2567,23 +2560,6 @@ for i = 1:length(Kf)
end
#+end_src
HAC plant (transfer function from f to dL, as measured by the external metrology):
- It increase the coupling at low frequency, but is kept to small values for realistic values of the bending stiffness (Figure ref:fig:detail_fem_joints_bending_stiffness_hac_plant)
- Bending stiffness does not impact significantly the HAC plant.
The added stiffness increases the frequency of the suspension modes
Condition in [[cite:&mcinroy02_model_desig_flexur_joint_stewar]] to have forces aligned with the struts when considering rotational stiffness: kr << k*l^2
For the current nano hexapod configuration, it correspond to << 9000 Nm/rad.
This may be an issue for soft nano-hexapod (for instance k = 1e4 => << 90) => have to design very soft flexible joints.
Here, having relatively stiff actuators render this condition easier to achieve.
IFF Plant:
- Having bending stiffness adds complex conjugate zero at low frequency (Figure ref:fig:detail_fem_joints_bending_stiffness_iff_plant)
- Similar to having a stiffness in parallel to the struts (i.e., to the force sensor).
This can be explained since even if the force sensor is removed (i.e. zero axial stiffness of the strut), the strut will still act as a spring between the mobile and fixed plates because of the bending stiffness of the flexible joints.
The frequency of the zero gives an idea of the stiffness contribution of the flexible joint bending stiffness
- They therefore impose limitation for decentralized IFF, as discussed in [[cite:&preumont07_six_axis_singl_stage_activ]]
- This can be seen in the root locus plot of Figure ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof
#+begin_src matlab :exports none :results none
freqs = logspace(0, 3, 1000);
@ -2741,11 +2717,6 @@ leg.ItemTokenSize(1) = 15;
exportFig('figs/detail_fem_joints_bending_stiffness_iff_locus_1dof.pdf', 'width', 'half', 'height', 500);
#+end_src
However, as the APA300ML was chosen for the actuator, stiffness are already present in parallel to the force sensors:
- The dynamics is computed again for all considered values of the bending stiffnesses with the 2DoF model of the APA300ML
- Root locus for decentralized IFF are shown in Figure ref:fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml.
Now the effect of bending stiffness has little effect on the attainable damping, as its contribution as "parallel stiffness" is small compared to the parallel stiffness already present in the APA300ML.
#+begin_src matlab
%% Identify the dynamics for several considered bending stiffnesses - APA300ML
G_Kf_apa300ml = {zeros(length(Kf), 1)};
@ -2830,25 +2801,28 @@ exportFig('figs/detail_fem_joints_bending_stiffness_iff_locus_apa300ml.pdf', 'wi
#+end_subfigure
#+end_figure
Conclusion:
- Similar results for torsional stiffness, but less important
- thanks to the use of the APA, the requirements in terms of bending stiffness are less stringent
** Axial Stiffness
<<sec:joints_trans_stiffness>>
<<ssec:detail_fem_joint_axial>>
- Adding flexibility between the actuation point and the measurement point / point of interest is always detrimental for the control performances.
This is verified, and the goal is to estimate the minimum axial stiffness that the flexible joints should have
- Here, the mass of the strut should be considered.
It is set to 112g as specified in the APA300ML specification sheet.
The limited axial stiffness ($k_a$) of flexible joints introduces an additional compliance between the actuation point and the measurement point.
As explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapter 6]] and in [[cite:&rankers98_machin]] (effect called "actuator flexibility"), such intermediate flexibility invariably degrades control performance.
Therefore, determining the minimum acceptable axial stiffness that maintains nano-hexapod performance becomes crucial.
- Transfer functions are estimated for several axial stiffnesses (Figure ref:fig:detail_fem_joints_axial_stiffness_plants)
- IFF plant is not much affected (Figure ref:fig:detail_fem_joints_axial_stiffness_iff_plant).
Confirmed by the root locus plot of Figure ref:fig:detail_fem_joints_axial_stiffness_iff_locus
- "HAC" plant:
- Additional modes at high frequency corresponding to internal modes of the struts.
It adds coupling to the plant.
This is confirmed by computed the RGA-number for the damped plant (i.e. after applying decentralized IFF) in Figure ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant
The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects.
A parametric study was conducted by varying the axial stiffness from $1\,N/\mu m$ (matching actuator stiffness) to $1000\,N/\mu m$ (approximating rigid behavior).
The resulting frequency responses (Figure ref:fig:detail_fem_joints_axial_stiffness_plants) reveal distinct effects on system dynamics.
The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both frequency response data (Figure ref:fig:detail_fem_joints_axial_stiffness_iff_plant) and root locus analysis (Figure ref:fig:detail_fem_joints_axial_stiffness_iff_locus).
However, the externally measured (HAC) plant demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes.
This coupling is quantified through RGA analysis of the damped system (Figure ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance.
Above this resonance frequency, two critical limitations emerge.
First, the system exhibits strong coupling between control channels, making decentralized control strategies ineffective.
Second, control authority diminishes significantly near the resonant frequencies.
These effects fundamentally limit achievable control bandwidth, making high axial stiffness essential for system performance.
Based on this analysis, an axial stiffness specification of $100\,N/\mu m$ was established for the nano-hexapod joints.
#+begin_src matlab
%% Identify the dynamics for several considered axial stiffnesses
@ -3011,10 +2985,6 @@ exportFig('figs/detail_fem_joints_axial_stiffness_iff_plant.pdf', 'width', 'half
#+end_subfigure
#+end_figure
Integral force feedback
- [ ] Maybe show the damped plants instead?
- [ ] Root Locus: not a lot of effect
#+begin_src matlab :exports none :results none
%% Decentalized IFF
Kiff = -200 * ... % Gain
@ -3132,29 +3102,12 @@ exportFig('figs/detail_fem_joints_axial_stiffness_rga_hac_plant.pdf', 'width', '
#+end_subfigure
#+end_figure
Conclusion:
- The axial stiffness of the flexible joints should be maximized to limit additional coupling at high frequency that may negatively impact the achievable bandwidth
- It should be much higher than the stiffness of the actuator
- For the nano-hexapod 100N/um is a reasonable axial stiffness specification
- Above the resonance frequency linked to the limited axial stiffness of the flexible joint, the system becomes coupled and impossible to control
- Also, loose control authority at the frequency of the zero
** Specifications and Design flexible joints
<<ssec:detail_fem_joint_specs>>
** Obtained design / Specifications
- Summary of specifications (Table ref:tab:detail_fem_joints_specs)
- Explain choice of geometry:
- x and y rotations are coincident
- stiffness can be easily tuned
- high axial stiffness
- Explain how it is optimized:
- Extract stiffnesses from FEM
- Parameterized model in the FE software
- Quick optimization: (few iterations, could probably increase more the axial stiffness)
- There is a trade off between high axial stiffness and low bending/torsion stiffness
- Also check the yield strength
- Show obtained geometry Figure ref:fig:detail_fem_joints_design:
- "neck" size: 0.25mm
- Characteristics of the flexible joints obtained from FEA are summarized in Table ref:tab:detail_fem_joints_specs
The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics.
Critical specifications include sufficient bending stroke to ensure long-term operation below yield stress, high axial stiffness for precise positioning, low bending and torsional stiffnesses to minimize parasitic forces, adequate load capacity, and well-defined rotational axes.
Based on the dynamic analysis presented in previous sections, quantitative specifications were established and are summarized in Table ref:tab:detail_fem_joints_specs.
#+name: tab:detail_fem_joints_specs
#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model
@ -3168,6 +3121,15 @@ Conclusion:
| Torsion Stiffness $k_t$ | $< 500\,Nm/\text{rad}$ | 260 |
| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 |
Among various possible flexible joint architectures, the design shown in Figure ref:fig:detail_fem_joints_design was selected for three key advantages.
First, the geometry creates coincident x and y rotation axes, ensuring well-defined kinematic behavior through precise definition of the system's Jacobian matrix.
Second, the design allows easy tuning of different directional stiffnesses through a limited number of geometric parameters.
Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational degrees of freedom.
The joint geometry was optimized through parametric finite element analysis.
The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits.
The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through finite element analysis and documented in Table ref:tab:detail_fem_joints_specs.
#+name: fig:detail_fem_joints_design
#+caption: Designed flexible joints.
#+attr_latex: :options [htbp]
@ -3187,14 +3149,11 @@ Conclusion:
#+end_figure
** Validation with the Nano-Hexapod
<<ssec:detail_fem_joint_validation>>
To validate the designed flexible joint:
- FEM: modal reduction
two interface frames are defined (Figure ref:fig:detail_fem_joints_frames)
- additional 6 modes are extracted: size of reduced order mass and stiffness matrices: $18 \times 18$
- Imported in the multi-body model
- The transfer functions from forces and torques applied between frames $\{F\}$ and $\{M\}$ to the relative displacement/rotations of the two frames is extracted.
- The stiffness characteristics of the flexible joint is estimated from the low frequency gain of the obtained transfer functions. Same values are obtained with the reduced order model and the FEM.
The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis.
This high-fidelity representation was created by defining two interface frames (Figure ref:fig:detail_fem_joints_frames) and extracting six additional modes, resulting in reduced-order mass and stiffness matrices of dimension $18 \times 18$.
The computed transfer functions from actuator forces to both force sensor measurements ($\bm{f}$ to $\bm{f}_m$) and external metrology ($\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$) demonstrate dynamics consistent with predictions from earlier analyses (Figure ref:fig:detail_fem_joints_fem_vs_perfect_plants), thereby validating the joint design.
#+name: fig:detail_fem_joints_frames
#+caption: Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames $\{F\}$ and $\{M\}$ both located at the center of the rotation.
@ -3235,32 +3194,13 @@ k_f = K(4,4); % bending [Nm/rad]
k_t = K(6,6); % torsion [Nm/rad]
#+end_src
Depending on which characteristic of the flexible joint is to be modelled, several DoFs can be taken into account:
- 2DoF (universal joint) $k_f$
- 3DoF (spherical joint) taking into account torsion $k_f$, $k_t$
- 2DoF + axial stiffness $k_f$, $k_a$
- 3DoF + axial stiffness $k_f$, $k_t$, $k_a$
- 6DoF ("bushing joint") $k_f$, $k_t$, $k_a$, $k_s$
While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order.
The complete nano-hexapod model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints).
To improve computational efficiency, a low order representation was developed using simplified joint elements with selective compliance DoF.
Adding more degrees of freedom:
- can represent important features
- adds model states that may not be relevant for the dynamics, and may complexity the simulations without adding much information
After testing different configurations, a good compromise was found for the modelling of the nano-hexapod flexible joints:
- bottom joints: $k_f$ and $k_a$
- top joints: $k_f$, $k_t$ and $k_a$
Talk about model order:
- with flexible joints: 252 states:
- 12 for the payload (6 dof)
- 12 for the 2DoF struts
- 216 DoF for the flexible joints (18*6*2)
- 12 states for?
- with 3dof and 4dof: 48 states
- 12 for the payload (6 dof)
- 12 for the 2DoF struts
- 12 states for the bottom joints
- 12 states for the top joints
After evaluating various configurations, a compromise was achieved by modeling bottom joints with bending and axial stiffness ($k_f$ and $k_a$), and top joints with bending, torsional, and axial stiffness ($k_f$, $k_t$ and $k_a$).
This simplification reduces the total model order to 48 states: 12 for the payload, 12 for the struts, and 24 for the joints (12 each for bottom and top joints).
While additional degrees of freedom could potentially capture more dynamic features, the selected configuration preserves essential behavioral characteristics while minimizing computational complexity.
#+begin_src matlab
%% Compare Dynamics between "Reduced Order" flexible joints and "2-dof and 3-dof" joints