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Thomas Dehaeze
2025-12-02 15:12:57 +01:00
parent 9231f43cbb
commit 74b6c5fe92
2 changed files with 136 additions and 128 deletions
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@@ -120,7 +120,7 @@
The Delta Robot geometry is defined as shown in Figure [[fig:delta_robot_schematic]].
The geometry is fully defined by three parameters:
- =d=: Cube's size (i.e., the length of the cube edge) eqref:eq:detail_kinematics_cubic_s
- =d=: Cube's size (i.e., the length of the cube edge)
- =a=: Distance from cube's vertex to top flexible joint
- =L=: Distance between two flexible joints (i.e., the length of the struts)
@@ -155,7 +155,7 @@ Dynamical properties:
It has a mass of ~300g
- Payloads: payloads can weight up to 1kg
Let's initialize a Delta Robot architecture, and plot the obtained geometry (Figures ref:fig:delta_robot_architecture and ref:fig:delta_robot_architecture_top).
Let's initialize a Delta Robot architecture, and plot the obtained geometry (Figures [[fig:delta_robot_architecture]] and [[fig:delta_robot_architecture_top]]).
#+begin_src matlab
%% Geometry
@@ -202,17 +202,18 @@ exportFig('figs/delta_robot_architecture_top.pdf', 'width', 'wide', 'height', 't
** Kinematics: Jacobian Matrix and Mobility
Jacobian matrix between actuator displacement and top platform displacement.
There are three actuators in the following directions $\hat{s}_1$, $\hat{s}_2$ and $\hat{s}_3$;
\begin{equation}\label{eq:detail_kinematics_cubic_s}
\begin{equation}\label{eq:delta_robot_unit_vectors}
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}\quad
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix}\quad
\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\end{equation}
\begin{equation}
The Jacobian matrix is defined as shown in eqref:eq:delta_robot_jacobian.
\begin{equation}\label{eq:delta_robot_jacobian}
\bm{J} = \begin{bmatrix}
\hat{\bm{s}}_1^T \\ \hat{\bm{s}}_2^T \\ \hat{\bm{s}}_3^T
\end{bmatrix}
@@ -226,11 +227,13 @@ s3 = delta_robot.geometry.As(:,5);
J = [s1' ; s2' ; s3']
#+end_src
\begin{equation}
It links the small actuator displacement to the top platform displacement eqref:eq:delta_robot_inverse_kinematics.
\begin{equation}\label{eq:delta_robot_inverse_kinematics}
d\mathcal{L} = J d\mathcal{L}
\end{equation}
\begin{equation}
\begin{equation}\label{eq:delta_robot_forward_kinematics}
d\mathcal{X} = J^{-1} d\mathcal{L}
\end{equation}
@@ -648,7 +651,7 @@ Then, the impact of the flexible joint's imperfections will be studied.
The cube's edge length is equal to 50mm, the distance between cube's vertices and top joints is 20mm and the length of the struts (i.e. the distance between the two flexible joints of the same strut) is 50mm.
The actuator stiffness is $1\,N/\mu m$.
The obtained geometry is shown in Figure ref:fig:delta_robot_studied_geometry.
The obtained geometry is shown in Figure [[fig:delta_robot_studied_geometry]].
#+begin_src matlab
%% Geometry
@@ -706,7 +709,7 @@ G.OutputName = {'D1', 'D2', 'D3'};
The dynamics is first identified in perfect conditions (infinite axial stiffness of the joints, zero bending stiffness).
We get =State-space model with 3 outputs, 3 inputs, and 6 states.=
We get a perfectly decoupled system, with three identical modes in the X, Y and Z directions.
The dynamics is shown in Figure ref:fig:delta_robot_dynamics_perfect.
The dynamics is shown in Figure [[fig:delta_robot_dynamics_perfect]].
#+begin_src matlab :exports none :results none
%% Dynamics of the delta robot with perfect joints
@@ -753,7 +756,7 @@ Because the flexible joints will have some bending stiffness, the actuator in on
This will limit its effective stroke.
We want this parallel stiffness to be much smaller than the stiffness of the actuator.
The parallel stiffness seen by the actuator as a function of the bending stiffness of the flexible joints is computed and shown in Figure ref:fig:delta_robot_bending_stiffness_parallel_k.
The parallel stiffness seen by the actuator as a function of the bending stiffness of the flexible joints is computed and shown in Figure [[fig:delta_robot_bending_stiffness_parallel_k]].
#+begin_src matlab
%% Bending Stiffness
@@ -799,7 +802,7 @@ This should be validated with the final geometry.
** Bending Stiffness
Then, the dynamics is identified for a bending Stiffness of $50\,Nm/\text{rad}$ and compared with a Delta robot with no bending stiffness in Figure ref:fig:delta_robot_bending_stiffness_dynamics.
Then, the dynamics is identified for a bending Stiffness of $50\,Nm/\text{rad}$ and compared with a Delta robot with no bending stiffness in Figure [[fig:delta_robot_bending_stiffness_dynamics]].
It can be seen that the DC gain is a bit lower when the bending stiffness is considered and the resonance frequency is increased.
This simply means that the system stiffness is increased.
@@ -854,7 +857,7 @@ exportFig('figs/delta_robot_bending_stiffness_dynamics.pdf', 'width', 'wide', 'h
** Axial Stiffness
Now, the effect of the axial stiffness on the dynamics is studied (Figure ref:fig:delta_robot_axial_stiffness_dynamics).
Now, the effect of the axial stiffness on the dynamics is studied (Figure [[fig:delta_robot_axial_stiffness_dynamics]]).
Additional modes can be observed on the plant dynamics, which could limit the achievable bandwidth.
Therefore the axial stiffness should be maximized.
Having the axial stiffness 100 times stiffer than the actuator stiffness seems reasonable.
@@ -909,7 +912,7 @@ exportFig('figs/delta_robot_axial_stiffness_dynamics.pdf', 'width', 'wide', 'hei
** Torsional Stiffness
Now the compliance in torsion of the flexible joints is considered.
If we look at the compliance of the delta robot in rotation as a function of the torsional stiffness of the flexible joints (Figure ref:fig:delta_robot_kt_compliance), we see almost no effect: the system is not made more stiff by increasing the torsional stiffness of the joints.
If we look at the compliance of the delta robot in rotation as a function of the torsional stiffness of the flexible joints (Figure [[fig:delta_robot_kt_compliance]]), we see almost no effect: the system is not made more stiff by increasing the torsional stiffness of the joints.
#+begin_src matlab
%% Effect of torsional stiffness on the system compliance
@@ -965,7 +968,7 @@ exportFig('figs/delta_robot_kt_compliance.pdf', 'width', 'wide', 'height', 'norm
#+RESULTS:
[[file:figs/delta_robot_kt_compliance.png]]
If we have a look at the effect of the torsional stiffness on the plant dynamics (Figure ref:fig:delta_robot_kt_dynamics), we see almost no effect, except when super high values are reached ($10^6\,Nm/\text{rad}$), which are unrealistic.
If we have a look at the effect of the torsional stiffness on the plant dynamics (Figure [[fig:delta_robot_kt_dynamics]]), we see almost no effect, except when super high values are reached ($10^6\,Nm/\text{rad}$), which are unrealistic.
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
@@ -1031,7 +1034,7 @@ Therefore, the torsional stiffness is not a super important metric for the desig
** Shear Stiffness
As shown in Figure ref:fig:delta_robot_shear_stiffness_compliance, the shear stiffness of the flexible joints has some effect on the compliance in translation and almost no effect on the compliance in rotation.
As shown in Figure [[fig:delta_robot_shear_stiffness_compliance]], the shear stiffness of the flexible joints has some effect on the compliance in translation and almost no effect on the compliance in rotation.
This is quite logical, and so the shear stiffness should be maximized.
A value of $100\,N/\mu m$ seems reasonable.
@@ -1121,7 +1124,7 @@ And we see the effect of changing the cube's size.
- [ ] *Understand why such different dynamics between 3dof_a joints and 6dof joints with very high shear stiffnesses*
The effect of the cube's size on the plant dynamics is shown in Figure ref:fig:delta_robot_cube_size_plant_dynamics:
The effect of the cube's size on the plant dynamics is shown in Figure [[fig:delta_robot_cube_size_plant_dynamics]]:
- coupling decreases with the cube's size
- one resonance frequency increases with the cube's size (resonances in rotation), which may be beneficial from a control point of view
- coupling at the main resonance varies with the cube's size, but it may also depend on the relative position between the CoM and the cube's center
@@ -1200,7 +1203,7 @@ exportFig('figs/delta_robot_cube_size_plant_dynamics.pdf', 'width', 'wide', 'hei
*** Effect on the compliance
As shown in Figure ref:fig:delta_robot_cube_size_compliance_rotation, the stiffness of the delta robot in rotation increases with the cube's size.
As shown in Figure [[fig:delta_robot_cube_size_compliance_rotation]], the stiffness of the delta robot in rotation increases with the cube's size.
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
@@ -1277,7 +1280,7 @@ And we see the effect of changing the strut length.
*** Effect on the plant dynamics
As shown in Figure ref:fig:delta_robot_strut_length_plant_dynamics, having longer struts:
As shown in Figure [[fig:delta_robot_strut_length_plant_dynamics]], having longer struts:
- decreases the main resonance frequency: this means that the stiffness in the X,Y and Z directions is decreased when the length of the strut is longer.
This is reasonable as the "lever" arm is getting larger, so the bending stiffness and compression of the flexible joints have a larger effect on the top platform compliance.
- decreases the low frequency coupling: this effect is more difficult to physically understand
@@ -1361,7 +1364,7 @@ exportFig('figs/delta_robot_strut_length_plant_dynamics.pdf', 'width', 'wide', '
*** Effect on the compliance
As shown in Figure ref:fig:delta_robot_strut_length_compliance_rotation, the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot).
As shown in Figure [[fig:delta_robot_strut_length_compliance_rotation]], the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot).
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
@@ -1447,7 +1450,7 @@ exportFig('figs/delta_robot_strut_length_compliance_rotation.pdf', 'width', 'ful
To make things easier, we take a top platform with no mass, mass-less struts, and we put a payload on top of the platform.
As shown in Figure ref:fig:delta_robot_CoM_pos_effect_plant, having the CoM of the payload at the cube's center allow to have better decoupling properties above the suspension mode of the system (i.e. above the first mode).
As shown in Figure [[fig:delta_robot_CoM_pos_effect_plant]], having the CoM of the payload at the cube's center allow to have better decoupling properties above the suspension mode of the system (i.e. above the first mode).
This could allow to have a bandwidth exceeding the frequency of the first mode.
But how sensitive this decoupling is to the exact position of the CoM still need to be studied.