2. Implementation of the Kinematic Coupling in the Laser-Cutting Machine Design

This work is a case-study type. In particular, it has been developed as part of a project run by a company that ranks among the world’s leaders in laser-cutting and welding fields [17]. Aim of the study is to evaluate the feasibility of kinematic coupling for the precise positioning of workpieces in a robotized laser-cutting machine designed by the company. Objectives of this work are the mechanical design of the coupling and the assessment of its accuracy, given the target precision of 0.02 mm required by the application. Figure 1a shows the CAD assembly of the machine.

Figure 1. (a) The CAD model of the laser-cutting machine; (b) one of the modules of the machine.

The system is made up of two separate modules that are placed next to one another. Each module is positioned on a cart (see Fig. 1b) that is pulled by a gear motor mounted on a linear guide. This translation occurs to move the module between the loading station, where workpieces are positioned on the cart and the actual machining station. Height-adjustable wheels are installed at the four corners of the structure to provide mobility within the working space. In the original configuration, the accuracy of the positioning of the cart in the working area depended on a traditional “mushroom-socket” type of coupling, which limits the repeatability of the connection as well as its precision, due to the effects of wear on the coupling elements. This phenomenon depends on the number of cycles, and it is difficult to predict its effect on the accuracy of the system. Moreover, any error in the positioning of the cart directly affected the precision of the manufacturing on the workpiece. For this reason, a solution based on a kinematic coupling has been investigated.

Among the several designs available, the one considering three V-shaped grooves and three spheres/balls, in a triangular configuration, has been adopted (see Fig. 2a).

Figure 2. The implementation of the three-groove kinematic coupling: (a) the model of the cart, sketched without springs and linear guides, along with coupling elements; sketches of disengaged (b) and engaged (c) configurations [Reference De Benedictis and Ferraresi16].

Other shapes as the Kelvin Clamp coupling have been preliminary considered but discarded in favor of the three-groove coupling solution from James Clerk Maxwell due to the symmetric configuration, which also reduces the costs of manufacturing. Both designs have been formerly developed in the early 1800 [Reference Slocum10]. The introduction of a dedicated positioning system based on kinematic coupling allows for improved precision with respect to the original configuration. Indeed, once in the machining station, the cart can be decoupled from the linear guide and the vertical motion required to achieve workpiece’s ultimate positioning is permitted. To provide the necessary level of accuracy, the final placement occurs only on the coupling parts. Two industrial air springs placed between the rails and the plane below are inflated and deflated to regulate the vertical movement (as shown in Fig. 2b–c). Side linear guides (not shown in Fig. 2a) are necessary since the air spring’s elasticity does not ensure a straight vertical motion.

3. Design of the Three-Groove Kinematic Coupling

Three balls and three V-shaped grooves positioned at the vertices of an equilateral triangle serve as the final connection structure [Reference Slocum18, Reference Hale19]. The ball element is a 100 mm diameter spherical shell with a base at 3/4 of the height overall. A threaded hole is bored into its flat surface to attach it to the base. This configuration requires that the spheres are stationary while the grooves are part of the cart that translates vertically (see Fig. 2b–c). Each groove is a cylinder with a 200 mm diameter and a 127.5 mm height. For each groove-ball couple, the contact surfaces are cut with an angle of 90° between them on one side of the groove.

The dimensioning of the components requires an appropriate choice of the materials. In particular, the behavior of a hardened stainless steel (440C SST, 200 GPa elastic modulus, 0.28 Poisson ratio), tungsten carbide (WC, 683 GPa elastic modulus, 0.24 Poisson ratio), Teflon (PTFE, 1.32 GPa elastic modulus, 0.46 Poisson ratio), and a ceramic (Si3N4, 240 GPa elastic modulus, 0.25 Poisson ratio) have been compared. In this preliminary analysis, the following data were considered to characterize the system:

• diameter of the coupling, depending on the available surface below the cart: 0.8 m;

• weight of the cart: 3000 N;

• weight of the workpiece: 4500 N.

The static analysis yielded the results shown in Fig. 3.

Figure 3. Comparison between the different materials tested for several ball diameters.

For this preliminary evaluation, the point of application of the load was considered coincident to the centroid of the coupling, hence all the elements of the coupling were subjected to the same stress. For each material and ball diameter, the ratio between the contact pressure and the allowable Hertzian stress $\sigma$ Hz, namely the contact (Hertzian) stress ratio $\sigma$ / $\sigma$ Hz, has been estimated. For metals, the allowable Hertzian stress has been set to 1.5 times the ultimate tensile strength [Reference Roark and Young20], whereas for the other materials, the ultimate compressive strength has been considered. In order to achieve low-stress ratio (below 50%), as well as to have smaller dimensions and lower production costs, 440C stainless steel was chosen. The final diameter of 100 mm led to 45% of stress ratio, as shown by data presented in Fig. 3.

Calculating the stresses and deflections at the contact, which have an impact on the positioning error permitted by the kinematic coupling, is necessary for the coupling’s design. The key phases of this technique are described below, and more details can be found in the literature [Reference Slocum18]. It begins with knowledge of the following information:

• the size of the ball elements and radii of curvature of the grooves;

• the locations of the contact points between the spheres and the grooves;

• the contact forces’ orientation;

• the coordinates of the points at which the preload forces are applied and their magnitude;

• the external load magnitude and its point of application;

• the elastic modulus and Poisson coefficient of each material considered.

The normals to the planes holding the contact force vectors for each ball must bisect the internal angles between two consecutive sides of the coupling triangle in order to satisfy the theoretical stability criterion for exact three-groove kinematic couplings [Reference Slocum10]. In the illustration of Fig. 4, for example, the normal to the plane π should split in half the angle between the sides joining the couples 12 and 23.

Figure 4. Equilateral triangle shape of the three-groove kinematic coupling.

Being defined by the geometry of the system, as well as the physical parameters listed above, the calculation of the six contact forces F _bi is obtained by Eq. (1), hence by performing the static analysis of the system. This calculation requires the knowledge of the external force F _e and moments M _e vectors acting on the coupling:

(1) \begin{equation} \sum {\boldsymbol{F}}_{e}=\mathbf{0},\,\sum {\boldsymbol{M}}_{e}=\mathbf{0}\quad \longrightarrow \quad F_{{b_{i}}},\quad (i=1\ldots 6) \end{equation}

The estimation of stresses and deflections depends on the geometry of the contact area. For this particular kinematic coupling design, the contact region is shaped as an ellipse (see Fig. 5) with semi-major axis R _max and semi-minor axis R _max. To perform their calculation, the equivalent elastic modulus E _e and the equivalent radius R _e must be known. They are given by Eq. (2):

(2) \begin{equation} E_{e}=\frac{1}{\frac{\left(1-{\nu _{g}}^{2}\right)}{E_{g}}+\frac{\left(1-{\nu _{b}}^{2}\right)}{E_{b}}},\quad R_{e}=\frac{1}{\frac{1}{R_{{g_{\max }}}}+\frac{1}{R_{{g_{\min }}}}+\frac{1}{R_{{b_{\max }}}}+\frac{1}{R_{{b_{\min }}}}} \end{equation}

where R _bmax and R _bmin have the same value, R _gmax is infinite (i.e., 1/ R _gmax = 0), and R _gmin is negative. As shown in Eq. (2), elastic modulus and Poisson coefficients are assigned for both coupling elements. Finally, the semi-major axis and semi-minor axis can be estimated as shown in Eq. (3), given the force F acting on the ball-groove coupling:

(3) \begin{equation} R_{\max }=\alpha \left(\frac{3FR_{e}}{2E_{e}}\right)^{1/3},\quad R_{\min }=\beta \left(\frac{3FR_{e}}{2E_{e}}\right)^{1/3} \end{equation}

where $\alpha$ and $\beta$ are functions of the geometric factor cos $\theta$ and can be both approximated as polynomial expressions. For this problem, cos $\theta$ reduces to R _e /|R _gmin| [Reference Slocum18]. Finally, the Hertzian contact stress $\sigma$ and deflection $\delta$ are calculated for each ball-groove couple with Eq. (4):

(4) \begin{equation} \sigma =\frac{3F}{2\pi R_{\max }R_{\min }},\quad \delta =\lambda \left(\frac{2F^{2}}{3R_{e}{E_{e}}^{2}}\right)^{1/3} \end{equation}

where $\lambda$ is a function with similar meaning to $\alpha$ and $\beta$ in Eq. (3).

Figure 5. Detail of ball-groove contact for a generic geometry of the groove [Reference De Benedictis and Ferraresi16].

The positioning errors in the kinematic coupling may be estimated once the contact stresses and deflections have been found. The weighted average of ball deflections obtained by using Eq. (4) may be used to determine the deviations $\delta_{\xi{c}}$ ( $\xi$ =x, y, z) of the centroid of the coupling triangle in the scenario where the spheres’ relative positioning is not changing noticeably. Rotation errors $\varepsilon$ _x and $\varepsilon$ _y are then calculated, starting from the segments’ lengths L and angles $\theta$ , see Fig. 4 for reference. This methodology yields the results shown in Eqs. (5) and (6), under small rotation angles assumption:

(5) \begin{equation} \varepsilon _{x}=\frac{\delta _{z1}}{L_{1,23}}\cos \theta _{23}+\frac{\delta _{z2}}{L_{2,31}}\cos \theta _{31}+\frac{\delta _{z2}}{L_{3,12}}\cos \theta _{12} \end{equation}

(6) \begin{equation} \varepsilon _{y}=\frac{\delta _{z1}}{L_{1,23}}\sin \theta _{23}+\frac{\delta _{z2}}{L_{2,31}}\sin \theta _{31}+\frac{\delta _{z2}}{L_{3,12}}\sin \theta _{12} \end{equation}

The average of the rotations about z, with regard to the coupling’s centroid, computed for each ball, is used to obtain the rotation error around the z-axis as shown in Eq. (7):

(7) \begin{equation} \varepsilon _{z}=\frac{\varepsilon _{z1}+\varepsilon _{z2}+\varepsilon _{z3}}{3} \end{equation}

The relation shown in Eq. (8) may be used to calculate the translation errors $\delta_{\xi}$ ( $\xi$ =x, y, z) at any location around the coupling once the rotation errors $\varepsilon$ _x , $\varepsilon$ _y , and $\varepsilon$ _z (in radians) have been calculated.

(8) \begin{equation} \left[\begin{array}{c} \delta _{x}\\[5pt] \delta _{y}\\[5pt] \delta _{z}\\[5pt] 1 \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & -\varepsilon _{z} & \varepsilon _{y} & \delta _{xc}\\[5pt] \varepsilon _{z} & 1 & -\varepsilon _{x} & \delta _{yc}\\[5pt] -\varepsilon _{y} & \varepsilon _{x} & 1 & \delta _{zc}\\[5pt] 0 & 0 & 0 & 1 \end{array}\right]\cdot \left[\begin{array}{c} x-x_{c}\\[5pt] y-y_{c}\\[5pt] z-z_{c}\\[5pt] 1 \end{array}\right]-\left[\begin{array}{c} x-x_{c}\\[5pt] y-y_{c}\\[5pt] z-z_{c}\\[5pt] 0 \end{array}\right] \end{equation}

4. Estimation of Positioning Accuracy Allowed by the Kinematic Coupling

In this section, the analyses carried out to assess the positioning accuracy of the coupling are presented, and their results are discussed. In particular, the methodology shown in the Section 3 was applied to estimate position and orientation errors for different loading conditions. The study represents a preliminary investigation aimed at implementing such a coupling system in a practical scenario, that is the laser-cutting machine presented in Section 2; therefore, all the analyses have been performed only at simulation level. To achieve realistic results, however, parameters from the actual machine have been considered when available, otherwise they have been estimated (as the overall expected mass of the coupling elements). The parameters considered for the simulations are listed below:

• cart’s payload: 450 kg;

• overall mass of the cart, coupling elements included: 300 kg;

• loading area on the cart: 1.1 m × 1.1 m;

• target positioning accuracy: 0.02 mm.

For each simulation, errors $\delta_{\xi}$ and $\varepsilon$ $_{\xi}$ ( $\xi$ =x, y, z) were estimated, as well as the Hertzian stress ratio $\sigma$ / $\sigma$ Hz.

The initial analysis looked at system deformations by taking into account objects of various weights put on the cart and placed at its center of gravity. This investigation was required to determine if the target repeatability could be obtained even under the most extreme load. Figure 6 shows the findings of this investigation. The displacement at the center of gravity is entirely vertical, since the cart and coupling centroids are coincident, and equivalent to about 0.006 mm when using a weight of 4500 N, which represents the anticipated payload. The stress ratio $\sigma$ / $\sigma$ Hz for this condition results equal to about 45%.

Figure 6. Vertical deflection (a) and Hertzian stress ratio (b) obtained when increasing external load is applied at the center of gravity of the cart. Arrows directed to the point corresponding to cart’s payload.

To emphasize the impact of load eccentricity on the overall accuracy, the location of the external load was then altered. In this study, it was shifted along the y axis (see Fig. 4 for reference), as allowed by the symmetry of the kinematic coupling triangle. Figure 7 shows the results of this analysis. Ball element 1 has a larger maximum $\sigma$ / $\sigma$ Hz than that found for balls 2 and 3. The higher stress ratio is attained when the weight is placed directly on the ball 1 (58%), but when the midpoint between balls 2 and 3 is taken into account, stress ratio is lowered to 34%. The trends for 2 and 3 are overlapping, as seen in Fig. 7b, due to the symmetry.

Figure 7. Rotation and vertical deflections (a) and Hertzian stress ratios (b) for several locations of the external load on the cart. Δy _L represents the deviation from the centroid of the coupling and of the cart.

According to Fig. 7a, the total deformation along the z-axis varies by the order of one thousandth of a millimeter. However, it should be noted that the deformation varies locally on the spheres. By examining each ball separately, the greatest vertical deformation at ball 1 is calculated to be 0.0114 mm, which is in any case less than the limit of 0.02 mm. It is also confirmed the impact of rotation about the x-axis. Indeed, the rotation error $\varepsilon$ _x decreases as the external load approaches the coupling’s center of gravity. When the force is applied directly to ball 1, the maximum $\varepsilon$ _x of 1.9e−5 is used, yielding an overall displacement of 0.0143 mm, hence still acceptable for this application.

Although being still lower than 0.02 mm, positioning error results significantly affected by eccentric location of the external load. This aspect is critical for the robustness and reliability of the solution, hence the effect of load balancing on the system has been evaluated in the final analysis. At the risk of increased bulkiness and weight of the system, it is expected to improve the precision of the coupling for most of the loading conditions. By balancing the moments exerted on the system made up of the cart and the coupling elements, it was possible to estimate the counterweight’s field of action, leading to the outcome shown in Fig. 8. The counterweight, which has plant view measurements of 0.34 m × 0.2 m, is a 50 kg cast iron industrial weight.

Figure 8. (a) Space of action of the counterweight: thick line denotes the cart, and dashed line denotes the area within the counterweight has to be contained in order to avoid obstructions with surrounding elements. The green-dot area represents the space available to the external load’s point of application that still results in perfect balancing. (b) The relative positioning of external load, cart, and counterweight centroids in an explanatory configuration.

The size and weight of the chosen counterweight determine how large is the green region visible in Fig. 8a. By making the deformations on the spheres consistent, balancing compensates the rotational errors that would otherwise be worsened by shifting away from the coupling’s centroid. Figure 9 shows the resulting deflections and rotation errors obtained for the unbalanced and balanced conditions. The deformations $\delta$ _z and $\varepsilon$ _x are always lower than that obtained for the unbalanced case: even by considering the largest errors shown in Fig. 9b, the maximum accuracy error results equal to 0.0138 mm.

Figure 9. Vertical deflection (a) and rotation error about the x axis (b) for unbalanced and balanced conditions. Δy _L represents the deviation from the centroid of the coupling and of the cart.