diff --git a/delta-robot.html b/delta-robot.html index d0a63ab..4a3ae3f 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 @@ -11,19 +11,19 @@ @@ -39,43 +39,43 @@

Table of Contents

- +

-
-

1. Geometry

+
+

1. Geometry

The Delta Robot geometry is defined as shown in Figure ref:fig:delta_robot_schematic. @@ -91,7 +91,7 @@ The geometry is fully defined by three parameters: -

+

delta_robot_schematic.png

Figure 1: Schematic of the Delta Robot

@@ -149,22 +149,22 @@ Let’s initialize a Delta Robot architecture, and plot the obtained geometr

-
+

delta_robot_architecture.png

Figure 2: Delta Robot Architecture

-
+

delta_robot_architecture_top.png

Figure 3: Delta Robot Architecture - Top View

-
-

2. Kinematics: Jacobian Matrix and Mobility

+
+

2. Kinematics: Jacobian Matrix and Mobility

Jacobian matrix between actuator displacement and top platform displacement. @@ -199,7 +199,7 @@ The achievable workspace is a cube whose edge length is equal to the actuator st

-
+

delta_robot_3d_workspace.png

Figure 4: 3D workspace

@@ -214,7 +214,7 @@ Depending on how the YZ plane is oriented (i.e., depending on the Rz angle of th

-
+

delta_robot_2d_workspace.png

Figure 5: 2D mobility for different orientations

@@ -226,15 +226,15 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um -
+

delta_robot_2d_workspace_optimal.png

Figure 6: 2D mobility for the optimal Rz angle

-
-

3. Kinematics: Degrees of Freedom

+
+

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. @@ -524,8 +524,8 @@ Therefore, to model some compliance of the top platform in rotation, both the ax

-
-

4. Kinematics: Number of modes

+
+

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). @@ -541,11 +541,11 @@ State-space model with 3 outputs, 3 inputs, and 6 states.

-
-

5. Flexible Joint Design

+
+

5. Flexible Joint Design

- +

The goal is to extract specifications for the flexible joints of the six struts. @@ -571,8 +571,8 @@ 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.

-
-

5.1. Studied Geometry

+
+

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. @@ -584,7 +584,7 @@ The obtained geometry is shown in Figure +

-
-

5.2. Stiffness seen by the actuator

+
+

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. @@ -619,7 +619,7 @@ The parallel stiffness seen by the actuator as a function of the bending stiffne

-
+

delta_robot_bending_stiffness_parallel_k.png

Figure 9: Effect of the bending stiffness of the flexible joints on the stiffness seen by the actuators

@@ -640,8 +640,8 @@ This should be validated with the final geometry.

-
-

5.3. Bending Stiffness

+
+

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 ref:fig:delta_robot_bending_stiffness_dynamics. @@ -654,15 +654,15 @@ It is not critical from a dynamical point of view, it just decreases the achieva

-
+

delta_robot_bending_stiffness_dynamics.png

Figure 10: Effect of the bending stiffness on the dynamics

-
-

5.4. Axial Stiffness

+
+

5.4. Axial Stiffness

Now, the effect of the axial stiffness on the dynamics is studied (Figure ref:fig:delta_robot_axial_stiffness_dynamics). @@ -673,15 +673,15 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).

-
+

delta_robot_axial_stiffness_dynamics.png

Figure 11: Effect of the joint’s axial stiffness on the plant dynamics

-
-

5.5. Torsional Stiffness

+
+

5.5. Torsional Stiffness

Now the compliance in torsion of the flexible joints is considered. @@ -692,7 +692,7 @@ If we look at the compliance of the delta robot in rotation as a function of the

-
+

delta_robot_kt_compliance.png

Figure 12: Effect of the joint’s torsional stiffness on the Delta Robot compliance

@@ -703,7 +703,7 @@ If we have a look at the effect of the torsional stiffness on the plant dynamics

-
+

delta_robot_kt_dynamics.png

Figure 13: Effect of the joint’s torsional stiffness on the Delta Robot plant dynamics

@@ -714,8 +714,8 @@ Therefore, the torsional stiffness is not a super important metric for the desig

-
-

5.6. Shear Stiffness

+
+

5.6. 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. @@ -727,15 +727,15 @@ A value of \(100\,N/\mu m\) seems reasonable.

-
+

delta_robot_shear_stiffness_compliance.png

Figure 14: Effect of the shear stiffness of the flexible joints on the Delta Robot compliance

-
-

5.7. Effect of cube’s size

+
+

5.7. Effect of cube’s size

Let’s choose reasonable values for the flexible joints: @@ -751,8 +751,8 @@ Let’s choose reasonable values for the flexible joints: And we see the effect of changing the cube’s size.

-
-

5.7.1. Effect on the plant dynamics

+
+

5.7.1. Effect on the plant dynamics

-
-

5.7.2. Effect on the compliance

+
+

5.7.2. 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.

-
+

delta_robot_cube_size_compliance_rotation.png

Figure 16: Effect of the cube’s size on the rotational compliance of the top platform

@@ -795,8 +795,8 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see
-
-

5.8. Effect of the strut length ?

+
+

5.8. Effect of the strut length ?

Let’s choose reasonable values for the flexible joints: @@ -811,8 +811,8 @@ Let’s choose reasonable values for the flexible joints: And we see the effect of changing the strut length.

-
-

5.8.1. Effect on the plant dynamics

+
+

5.8.1. Effect on the plant dynamics

As shown in Figure ref:fig:delta_robot_strut_length_plant_dynamics, having longer struts: @@ -829,22 +829,22 @@ So, the struts length can be optimized to not decrease too much the stiffness of

-
+

delta_robot_strut_length_plant_dynamics.png

Figure 17: Effect of the cube’s size on the plant dynamics

-