Figure 17: Bosch rexroth 30x60 extruded profile
Once this section was selected, we had to perform an additional iteration in order to update the parameters of the simulations: weight requirement will decrease from 2 kg/m to ∼1.5 kg/m. Since we have the moment of inertia of the section, we solve equation Eq. 59for the maximum displacement. This yields:
vmaxtip = 3.63 mm (63)
Which satisfies the requirement set at the beginning. With this initial draft of the robot structure, we proceed to the following stage of the design, in-troduced in the following pages, in which we add detail to the mechanical model and we perform a differentiated selection of the links for each of the arm sections.
3 .3 stiffness-based positioning precision map
The static analysis presented in the previous section provided an engineer-ing model to define the section properties for the links. The problem can be restated if we assume that the geometrical properties are known: it is interesting to note, in fact, that once the design is completed, it is possible
Figure 18: Complete diagram of efforts acting on the manipulator.
to create a 3D map of the effects of the assembly’s flexibility in terms of end-effector displacement.
In applications in which high positioning precision is of outmost impor-tance, a model that takes into account the intrinsic mechanical limitation due to the non-infinite rigidity could be applied to the control system to adaptively correct the commanding angles of the joints. In order to design such a model, we generalize the problem by parametrizing the characteris-tics of the system.
In addition, we now take into account the first link flexibility and the horizontal displacement it generates, which was ignored in the first level computations of the previous section.
Supposing that the last three links can be accounted for as a concen-trated force, we obtain the diagram presented in Fig. 18.
Once the moment diagram is computed in the general case, we can use beam deflection theory to obtain the displacements. Since we suppose to have symmetric sections:
v′′(x) =−Mz(x)
EIxx (64)
In order to find the displacement at the end effector, we need to extract the
3.3 stiffness-based positioning precision map 45
Figure 19: Total diagrams for a) normal force, shear and b) bending mo-ment.
position and attitude in an sequential fashion, and use the superposition of effects to compute the end result. The iterative algorithm can be described as follows (with x being the coordinate along the direction of the beam):
Where the rotation matrix R ensures the correctness of the sum in the case of non parallel links.
In this case, the moments to which the links are subject, expressed in the local reference frame of the i−th link, are:
M1(x) = M1= F2l2+ F3(l2+ l3) + q2· l22
The angular deflection and y-displacement caused by these bending
In terms of tip deflection, if we assume the base of the first link to be the origin of a 2D cartesian reference frame, the composite equation to compute it is:
3.3 stiffness-based positioning precision map 47
This approach can be extended to the computation of the end effector dis-placement for each joint configuration.
The equations need to take into account the general coordinatesθ2 and θ3and are independent fromθ1. A way to approach the problem would be to scale the distributed loads and the concentrated forces according to the configuration under analysis, that is:
q†2 = q2cos(θ2) q†3 = q3cos(θ3) (77a) F2† = F2cos(θ2) F3†= F3cos(θ3) (77b) Naturally, the rotation matrices will contain also the rotation due to the configuration. This, ultimately, allows to obtain a map of the entire manip-ulator’s workspace which describes the positioning error due to the link flexibility and payload loading.
The complete expressions in this case are the same of Eq. 71, 73, 74, with the only caveat of the modified loading. The visualization of this per-formance parameter is shown for different joint configurations in Fig. 20, where the norm of the displacement error is represented for the angular set [θ2; θ3] spanning from −π to +π.
In more detail, the same plot is separated in Fig. 21-25 into the x, y components of the resulting end effector displacement. In these plots, the parameters being used are:
Figure 20: End effector deflection, norm
F2= 50 N (79b)
F3= 30 N (79c)
q = 1.50 kg/m (79d)
Which are derived from the previous section. From the plot, it is pos-sible to start a refinement process of the current architecture: first of all, it can be seen by comparing the x and y displacement, that most of the displacement happens in the x-axis; this can be greatly improved by in-creasing the stiffness of the first link. Since the weight of link 1 will be unloaded by a proper bearing structure, the added weight will not influ-ence the sizing of the motors (apart from the added rotational inertia). To
3.3 stiffness-based positioning precision map 49
Figure 21: End effector deflection, x-component
Figure 22: End effector deflection, y-component
3.3 stiffness-based positioning precision map 51
Figure 23: End effector deflection, norm. Contour plot
Figure 24: End effector deflection, x-component. Contour plot
3.3 stiffness-based positioning precision map 53
Figure 25: End effector deflection, y-component. Contour plot
this extent, we modify the choice of link 1 with a more suitable profile: in this case, the Bosch 90x90L was selected:
Ix= 211.1 cm4 (was 19.6 cm4) (80a) Iy= 211.1 cm4 (was 5.1 cm4) (80b) q = 6.5 kg/m (was 1.5 kg/m) (80c) Then, for the selection of the next two links, we started from the baseline Bosch 30x60 configuration selected in the previous chapter and applied a tapered approach: we increased progressively the second link section until we reached a reasonable tradeoff between the increased stiffness and the increased weight (which will naturally influence the selection of motor 2).
Finally, we selected the Bosch 45x90SL profile, where SL stands for super−light. These are the specifications:
Ix = 73.40 cm4 (was 19.6 cm4) (81a) Iy = 9.1 cm4 (was 5.1 cm4) (81b) q = 2.4 kg/m (was 1.5 kg/m) (81c) The last link is kept unchanged and is a Bosch 30x60 profile. Before pro-ceeding with FEM analysis, the new design was tested for tip displacement.
The results are shown in Fig. 27-32: as expected, this new configuration drastically reduced the norm of the displacement, with a mean reduction of 54%. The highest displacement, happening when the arm is fully ex-tended, is reduced to:
vmaxtip = 1.41 mm (82)
The data used in the plots is summarized in the following:
E = 70·109Pa (83a)
I1= 211.1·10−8m4 (83b) I2= 73.40·10−8m4 (83c) I3= 19.60·10−8m4 (83d)
3.3 stiffness-based positioning precision map 55
Figure 26: Bosch profiles chosen for link 1, 2 and 3 respectively.
l1= 1 m (83e)
l2= 0.7 m (83f)
l3= 0.7 m (83g)
With the updated loads being:
F1= 100 N (84a)
F2= 50 N (84b)
F3= 30 N (84c)
q1= 6.5 kg/m (84d)
q2= 2.4 kg/m (84e)
q3= 1.50 kg/m (84f)
Reassuming the results obtained in this section, the links’ sizing param-eters are:
length [m] area [cm2] Ix[cm4] Iy[cm4] q [kg/m] mass [kg]
link 1 1 24.1 211.1 211.1 6.5 6.5
link 2 0.7 9.04 73.4 18.1 2.44 1.71
link 3 0.7 5.5 19.6 5.1 1.5 1.05
Figure 27: End effector deflection, modified designed, norm.
3.3 stiffness-based positioning precision map 57
Figure 28: End effector deflection, modified designed, x-component.
Figure 29: End effector deflection, modified designed, y-component.
3.3 stiffness-based positioning precision map 59
Figure 30: End effector deflection, modified designed, norm. Contour plot
Figure 31: End effector deflection, modified designed, x-component. Con-tour plot
3.3 stiffness-based positioning precision map 61
Figure 32: End effector deflection, modified designed, y-component. Con-tour plot
Figure 33: Torque profiles for different arm configurations, 3D plot.