diff --git a/delta-robot.html b/delta-robot.html index a02ce5c..a1656f0 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 @@ -38,53 +38,61 @@

Table of Contents

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1. The Delta Robot Kinematics

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1. Studied Geometry

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1.1. Studied Geometry

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The Delta Robot geometry is defined as shown in Figure 1.

@@ -179,9 +187,9 @@ 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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1.2. Kinematics: Jacobian Matrix and Mobility

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There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);

@@ -253,9 +261,9 @@ 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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1.3. Kinematics: Degrees of Freedom

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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.

@@ -324,9 +332,9 @@ 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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1.4. Kinematics: Number of modes

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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).

@@ -341,14 +349,15 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
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5. Flexible Joint Design

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2. Flexible Joint Design

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-First, in Section 5.1, the dynamics of a “perfect” Delta-Robot is identified (i.e. with perfect 2DoF rotational joints). +First, in Section 2.1, the dynamics of a “perfect” Delta-Robot is identified (i.e. with perfect 2DoF rotational joints).

@@ -356,15 +365,15 @@ Then, the impact of the flexible joint’s imperfections will be studied. The goal is to extract specifications for the flexible joints of the six struts, in terms of:

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  • bending stiffness (Section 5.2)
    • -
  • axial stiffness (Section 5.3)
    • -
  • torsional stiffness (Section 5.4)
    • -
  • shear stiffness (Section 5.5)
    • +
  • bending stiffness (Section 2.2)
    • +
  • axial stiffness (Section 2.3)
    • +
  • torsional stiffness (Section 2.4)
    • +
  • shear stiffness (Section 2.5)
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5.1. Studied Geometry

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2.1. Studied Geometry

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@@ -400,16 +409,16 @@ The dynamics is shown in Figure 8
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5.2. Bending Stiffness

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2.2. Bending Stiffness

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5.2.1. Stiffness seen by the actuator, and decrease of the achievable stroke

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2.2.1. Stiffness seen by the actuator, and decrease of the achievable stroke

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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. This will limit its effective stroke. @@ -460,9 +469,9 @@ It is not critical from a dynamical point of view, it just decreases the achieva

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5.2.2. Effect on the coupling

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2.2.2. Effect on the coupling

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Here, reasonable values for the flexible joints (modelled as a 6DoF joint) stiffness are taken:

@@ -491,9 +500,9 @@ Therefore, the bending stiffness of the flexible joints should be minimized (10N
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5.3. Axial Stiffness

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2.3. Axial Stiffness

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@@ -514,9 +523,9 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
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5.4. Torsional Stiffness

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2.4. Torsional Stiffness

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@@ -552,9 +561,9 @@ Therefore, the torsional stiffness is not a super important metric for the desig

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5.5. Shear Stiffness

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2.5. Shear Stiffness

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@@ -576,9 +585,9 @@ A value of \(100\,N/\mu m\) seems reasonable.
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5.6. Conclusion

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2.6. Conclusion

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@@ -625,16 +634,20 @@ A value of \(100\,N/\mu m\) seems reasonable. -
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6. Effect of the Geometry

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3. Effect of the Geometry

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+In this section, the effect of the geometry on the system properties are studied. +The goal is to better understand the different trade-offs, and to extract specifications in terms of the Delta Robot geometry. +

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6.1. Effect of cube’s size

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3.1. Effect of cube’s size

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Let’s choose reasonable values for the flexible joints:

@@ -649,9 +662,9 @@ Let’s choose reasonable values for the flexible joints: And we see the effect of changing the cube’s size.

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6.1.1. Effect on the plant dynamics

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3.1.1. Effect on the plant dynamics

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  • [ ] Understand why such different dynamics between 3dof_a joints and 6dof joints with very high shear stiffnesses
@@ -673,9 +686,9 @@ The effect of the cube’s size on the plant dynamics is shown in Figure
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6.1.2. Effect on the compliance

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3.1.2. Effect on the compliance

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As shown in Figure 17, the stiffness of the delta robot in rotation increases with the cube’s size.

@@ -693,9 +706,9 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see
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6.2. Effect of the strut length

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3.2. Effect of the strut length

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Let’s choose reasonable values for the flexible joints:

@@ -709,9 +722,9 @@ Let’s choose reasonable values for the flexible joints: And we see the effect of changing the strut length.

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6.2.1. Effect on the compliance

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3.2.1. Effect on the compliance

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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).

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6.2.2. Effect on the plant dynamics

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3.2.2. Effect on the plant dynamics

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As shown in Figure 19, having longer struts:

@@ -764,9 +777,9 @@ So, the struts length can be optimized to not decrease too much the stiffness of
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6.3. Having the Center of Mass at the cube’s center

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

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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.

@@ -785,17 +798,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need
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6.4. Conclusion

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3.4. Conclusion

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7. Conclusion

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4. Conclusion

Author: Dehaeze Thomas

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Created: 2025-12-02 Tue 16:08

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Created: 2025-12-02 Tue 16:10

diff --git a/delta-robot.org b/delta-robot.org index 07fd858..8d2213d 100644 --- a/delta-robot.org +++ b/delta-robot.org @@ -88,7 +88,7 @@ org-ref-acronyms-before-parsing)) #+END_SRC -* The Delta Robot Kinematics :ignore: +* The Delta Robot Kinematics <> ** Introduction :ignore: ** Matlab Init :noexport:ignore: @@ -1189,6 +1189,9 @@ exportFig('figs/delta_robot_shear_stiffness_compliance.pdf', 'width', 'full', 'h * Effect of the Geometry <> ** Introduction :ignore: +In this section, the effect of the geometry on the system properties are studied. +The goal is to better understand the different trade-offs, and to extract specifications in terms of the Delta Robot geometry. + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <>
Table 1: Recommendations for the flexible joints