From 10bd39e90f5ee3981352dbb85fed48f0ab07fa48 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 2 Dec 2025 15:31:07 +0100 Subject: [PATCH] Mathjax test --- delta-robot.html | 165 +++++++++++++++++++++++++---------------------- 1 file changed, 87 insertions(+), 78 deletions(-) diff --git a/delta-robot.html b/delta-robot.html index 6a3fe36..3615ae1 100644 --- a/delta-robot.html +++ b/delta-robot.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Delta Robot @@ -11,22 +11,31 @@ - +MathJax = { + loader: {load: ['[tex]/tagformat']}, + tex: { + packages: {'[+]': ['tagformat']}, + + tags: "ams", + multlineWidth: "85%", + tagSide: "right", + tagIndent: ".8em", + + macros: { + bm: ["\\boldsymbol{#1}", 1] + }, + + tagformat: { + tag: (tag) => '(' + tag + ')', + ref: '', // use same format for qref and ef + number: (n) => n.toString(), + id: (id) => 'mjx-eqn:' + id.replace(/\s/g, '_'), + url: (id, base) => base + '#' + encodeURIComponent(id), + } + } +}; + +
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Table of Contents

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1. Geometry

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1. Geometry

The Delta Robot geometry is defined as shown in Figure 1. @@ -163,8 +172,8 @@ Let’s initialize a Delta Robot architecture, and plot the obtained geometr

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2. Kinematics: Jacobian Matrix and Mobility

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2. Kinematics: Jacobian Matrix and Mobility

There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\); @@ -237,8 +246,8 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um

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3. Kinematics: Degrees of Freedom

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3. Kinematics: Degrees of Freedom

In the perfect case (flexible joints having no stiffness in bending, and infinite stiffness in torsion and in the axial direction), the top platform is allowed to move only in the X, Y and Z directions while the three rotations are fixed. @@ -528,8 +537,8 @@ Therefore, to model some compliance of the top platform in rotation, both the ax

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4. Kinematics: Number of modes

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4. Kinematics: Number of modes

In the perfect condition (i.e. infinite stiffness in torsion and in compression of the flexible joints), the system has 6 states (i.e. 3 modes, one for each DoF: X, Y and Z). @@ -545,8 +554,8 @@ State-space model with 3 outputs, 3 inputs, and 6 states.

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5. Flexible Joint Design

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5. Flexible Joint Design

@@ -575,8 +584,8 @@ First, the dynamics of a “perfect” Delta-Robot is identified (i.e. w Then, the impact of the flexible joint’s imperfections will be studied.

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5.1. Studied Geometry

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5.1. Studied Geometry

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. @@ -609,8 +618,8 @@ The dynamics is shown in Figure 8

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5.2. Stiffness seen by the actuator

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5.2. Stiffness seen by the actuator

Because the flexible joints will have some bending stiffness, the actuator in one direction will “see” some stiffness due to the struts in the other directions. @@ -644,8 +653,8 @@ This should be validated with the final geometry.

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5.3. Bending Stiffness

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5.3. 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 10. @@ -665,8 +674,8 @@ It is not critical from a dynamical point of view, it just decreases the achieva

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5.4. Axial Stiffness

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5.4. Axial Stiffness

Now, the effect of the axial stiffness on the dynamics is studied (Figure 11). @@ -684,8 +693,8 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).

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5.5. Torsional Stiffness

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5.5. Torsional Stiffness

Now the compliance in torsion of the flexible joints is considered. @@ -718,8 +727,8 @@ Therefore, the torsional stiffness is not a super important metric for the desig

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5.6. Shear Stiffness

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5.6. Shear Stiffness

As shown in Figure 14, the shear stiffness of the flexible joints has some effect on the compliance in translation and almost no effect on the compliance in rotation. @@ -738,8 +747,8 @@ A value of \(100\,N/\mu m\) seems reasonable.

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5.7. Effect of cube’s size

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5.7. Effect of cube’s size

Let’s choose reasonable values for the flexible joints: @@ -755,8 +764,8 @@ Let’s choose reasonable values for the flexible joints: And we see the effect of changing the cube’s size.

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5.7.1. Effect on the plant dynamics

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5.7.1. Effect on the plant dynamics

  • [ ] Understand why such different dynamics between 3dof_a joints and 6dof joints with very high shear stiffnesses
    • @@ -779,8 +788,8 @@ The effect of the cube’s size on the plant dynamics is shown in Figure
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5.7.2. Effect on the compliance

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5.7.2. Effect on the compliance

As shown in Figure 16, the stiffness of the delta robot in rotation increases with the cube’s size. @@ -799,8 +808,8 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see

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5.8. Effect of the strut length ?

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5.8. Effect of the strut length ?

Let’s choose reasonable values for the flexible joints: @@ -815,8 +824,8 @@ Let’s choose reasonable values for the flexible joints: And we see the effect of changing the strut length.

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5.8.1. Effect on the plant dynamics

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5.8.1. Effect on the plant dynamics

As shown in Figure 17, having longer struts: @@ -840,8 +849,8 @@ So, the struts length can be optimized to not decrease too much the stiffness of

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5.8.2. Effect on the compliance

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5.8.2. Effect on the compliance

As shown in Figure 18, the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot). @@ -856,8 +865,8 @@ As shown in Figure 1

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5.9. Having the Center of Mass at the cube’s center

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5.9. Having the Center of Mass at the cube’s center

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. @@ -877,17 +886,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need

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5.10. Conclusion

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5.10. Conclusion

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6. Conclusion

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6. Conclusion

Author: Dehaeze Thomas

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Created: 2025-12-02 Tue 15:22

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Created: 2025-12-02 Tue 15:31