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
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
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
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.
@@ -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.
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
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.
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
Now, the effect of the axial stiffness on the dynamics is studied (Figure 11). @@ -684,8 +693,8 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
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
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.
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.
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
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.
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
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
Created: 2025-12-02 Tue 15:22
+Created: 2025-12-02 Tue 15:31