**1. Introduction**

The deburring process is mainly performed by using manual labor or computerized numerical control (CNC) machines, but in the research field also robotic solutions have been considered [1]. Workers can easily adapt to the unpredictability of the burr, but the dangerous working conditions and the difficulty of the task sugges<sup>t</sup> using dedicated machines. To do so, special CNC machines have been developed, even if a comparable robotic solution costs less than 1/3 than the cost of a CNC machine [2]; moreover, when compared to a CNC machine, an industrial robot can provide a wider workspace, is usually more flexible (since it has a higher number of degrees of freedom), and can provide a higher dexterity, adapting its movement to complex geometries. The drawback of robotic solutions relies on the fact that robots are generally less stiff and accurate than CNC machines [3]: this affects the finishing of the workpiece, can cause chatter [4], and increases the programming and setup time.

The literature related to robotic deburring is limited [1], when compared to the literature related to robotic machining. This can be related to the high variability of the burr profile, which may behave differently depending on the workpiece material [5,6] and may influence burr removal time [7]. In other fields, the effect of burr properties and grinding tool on the final result has been studied. In particular, it has been of grea<sup>t</sup> interest in the biomedical field, where the bone grinding is crucial for surgical operations [8]. In fact, the shape of the tool itself provides different deburring results both in terms of temperature and tool wear [9], which may be fundamental for industrial applications, in which the burr

**Citation:** Bottin, M.; Cocuzza, S.; Massaro, M. Variable Stiffness Mechanism for the Reduction of Cutting Forces in Robotic Deburring.*Appl. Sci.* **2021**, *11*, 2883. https:// doi.org/10.3390/app11062883

Academic Editor: Giuseppe Carbone

Received: 18 February 2021 Accepted: 18 March 2021 Published: 23 March 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

to be removed is made of a material sensible to temperature (e.g., plastics or aluminum) or in intensive and repetitive tasks (e.g., robotic deburring). In depth analysis of the grinding process has been developed for yttrium aluminum garne<sup>t</sup> (YAG) single crystals, in which it is shown that strain rate plays an important role in the final deformation [10]. This knowledge can be integrated in robotic deburring for specific applications.

In industrial robotics, most of the research effort has focused on obtaining a precise control of the robot motion: Tao et al. [11] developed a sliding mode control method based on radial basis function (RBF) neural network, which is able to learn uncertain control actions. Quian et al. [12] proposed a sensorless force controller that uses the information retrieved only from collaborative robot internal motor sensors to estimate the burr dimensions. Other methods used to handle the high variability of the burr profile include the use of sensors [13,14], but these approaches are not suitable in many deburring applications since the working environment is usually dirty and dusty. If the ambient is clean, however, sensors can be used effectively [15]. El Naser et al. [16] and Gou et al. [17] have developed a robotic arm designed purposely for deburring. This approach is similar to the development of a special CNC machine. On the other hand, Lai et al. [18] have presented a novel design that includes an additional manipulator to be linked to an existing industrial robot. In this way, the kinematic chain becomes similar to the one of a parallel robot, thus increasing stiffness. These last two approaches, however, requires a complex external equipment, which may be expensive and time-consuming for industrial applications.

To increase or decrease the stiffness of the mechanical system during specific tasks, variable stiffness actuators (VSA) and mechanisms have been developed. They can be useful in many fields of application, such as in machining operations, since the stiffness can be changed to improve the surface finishing or reduce vibrations. During the years different solutions have been proposed. Vuong et al. [19] proposed a mechanism that can be tuned to change its stiffness in a wide range, from zero to (theoretically) infinite. It uses multiple linear guides in combination with a moving pivot and linear springs to achieve this functionality. Another example of VSA can be found in [20], in which the authors focused on the compactness of the system and obtained a stiffness inversely proportional to the gear displacement. A variable stiffness robotic arm has also been designed [21]: in this case a 7-degrees-of-freedom manipulator takes advantage of VSA to perform different tasks in the SHERPA mission. Moreover, Petit et al. [22] developed an ad-hoc controller to take advantage of VSA capabilities.

Deburring is a redundant task: the workpiece can be machined on any point of the grinding surface of the grinding wheel. Therefore, it is possible to achieve functional redundancy [23], and the robot can perform the same task using different configurations. For example, if the stiffness and inertial properties of the robot are known or experimentally identified [24,25] it is possible to choose a proper configuration so that the kinematic chain of the robot is stiff enough to perform the grinding operation. However, this field of research has only been partially explored. To overcome the functional redundancy, Nemec et al. [26] proposed to create robot recipes by means of programming by demonstration: by analyzing the movements of an expert human demonstrator, the robot can perform the same task (in this case polishing) exploiting redundancy. Moreover, as can be seen in [27], the stiffness map of a robot can be used to improve the accuracy of the assigned task. In particular, in this work, the authors introduced a compensation strategy to improve a roll hemming process. A stiffness model of a parallel haptic mechanism suitable for real-time computation has been proposed in [28].

When traditional industrial robots are concerned, it is usually difficult to implement force feedback controllers for deburring control purposes. Moreover, industrial applications require high flexibility and reduced implementation costs.

In sum, the literature review suggests that novel deburring techniques are needed to improve flexibility and precision and reduce costs. Indeed, CNC machines provide the best performance, but are generally more expensive and less flexible when compared to robotic solutions. One of the main issues related to robotic deburring is that the tool can ge<sup>t</sup> damaged or stopped when the burr thickness exceeds a certain threshold. A simple system suitable for industrial application is thus required to automatically compensate cutting force peaks. A passive mechanical system is a possible solution: the robot programming remains unaffected, and, if unexpected deburring forces appear, the mechanical system will automatically adapt to reduce the forces applied to the robot.

To achieve this objective, in this paper a simple mechanism is presented that can be used to reduce the peaks of deburring forces that could harm the robot structure or deburring tool. The mechanism will reduce cutting forces automatically if the burr is too high thanks to its compliance, while it will return to the baseline configuration when the burr thickness is acceptable again. The proposed system has also the advantage not to need a high setup time, when compared to a force feedback controller implemented in an industrial robot.

The work is organized as follows: in Section 2 the formulation of deburring forces is covered; in Section 3 the proposed mechanism and its mathematical model are presented; in Section 4 the design of the mechanism is addressed; Section 5 presents the dynamic analysis and simulation of the system in selected test cases; finally, Section 6 concludes the paper.

#### **2. Deburring Forces**

Limited information is available about the estimation of deburring forces. However, a simple model has been developed starting from the grinding process [6,29,30]. In the deburring configuration considered (Figure 1), the material is removed by the cylindrical surface of the deburring wheel, with the workpiece moving from the right to the left (along the *y* axis) and the deburring wheel rotating counterclockwise.

**Figure 1.** Scheme of the workpiece position in relation to the mechanism axis of movement (**a**); deburring forces (**b**).

The material removing rate (MRR) of a single chunk of material can be calculated from the burr dimensions and workpiece movement speed:

$$MRR = b\_w \cdot h \cdot v\_a \quad \left[\frac{m^3}{s}\right] \tag{1}$$

where *bw* and *h* are the width and height of the material chunk, and *va* is the linear workpiece movement speed. If the material and its specific removal energy *u* are known and the rotational speed *ω* of the grinding wheel is defined, it is possible to calculate the tangential cutting force needed to remove the slice of material as follows:

$$F\_l = \frac{b\_w \cdot h \cdot v\_a \cdot u}{\omega \cdot R} \quad \text{[N]} \tag{2}$$

where *R* is the radius of the grinding wheel.

In practice, (2) depends on the position of the mechanism since the burr height that the grinding wheel removes varies with its *x* position. As a result, *Ft* can be expressed as:

$$F\_t = \frac{b\_w \cdot \upsilon\_a \cdot u}{\omega} \cdot \frac{(h-x)}{R} \tag{3}$$

It is worthy to point out that the other main force component involved in the cutting process is the force *Fn* normal to the burr surface. The ratio (*μ* = *Ft*/*Fn*) between *Ft* and *Fn* depends on the material, but its value is usually between 0.2 (ceramics and high strength steel) and 0.8 (mild steel) [31]

If the material is removed by a small portion of the grinding wheel surface (Figure 1):

$$h - x = R - R\cos(\theta) \quad \rightarrow \quad \theta = \arccos\left(1 - \frac{h - x}{R}\right) \tag{4}$$

Since *Ft* and *Fn* are applied evenly on the surface of the material being removed, it is possible to consider them applied in the middle of the surface, so they are inclined of *θ*/2 with respect to the *y* and *x* axis, respectively. The resulting force along the mechanism *x* axis is: 

$$\begin{aligned} F\_{tx} &= F\_t \sin\left(\frac{\theta}{2}\right), & F\_{tx} &= F\_t \cos\left(\frac{\theta}{2}\right) = \frac{F\_t}{\mu} \cos\left(\frac{\theta}{2}\right) \\\ F\_{tot} &= F\_{nx} - F\_{tx} = \frac{b\_w \cdot v\_d \cdot u}{\omega} \cdot \frac{(h-x)}{R} \left(\frac{1}{\mu} \cos\left(\frac{\theta}{2}\right) - \sin\left(\frac{\theta}{2}\right)\right) \end{aligned} \tag{5}$$

It is worth to notice that in practice *R h*, so usually *θ* → 0. Equation (5) is still valid for a generic deburring process, but in this case it is possible to simplify it to calculate the actual normal component of the deburring force applied to the mechanism *Fb*:

$$F\_b \approx \frac{b\_{\text{uv}} \cdot v\_a \cdot u}{\mu \text{ }\omega} \cdot \frac{(h-x)}{R} \tag{6}$$

#### **3. Proposed Mechanism**

It is assumed that the robot arm is holding the workpiece, while the grinding wheel is attached to the ground through the proposed mechanism (Figure 2). The robot endpoint trajectory assures the desired depth of cut and feed rate of the piece with respect to the grinding wheel. To develop a mechanism that is as simple as possible, the main compliance direction, assumed normal to the burr (i.e., parallel to the direction of the normal component of the deburring force (6)), should be fixed. The cutting forces that arise during the deburring can be projected along the tangent and normal directions to the burr profile [32], as discussed in Section 2, and the tangential component of the deburring force is balanced by the manipulator.

**Figure 2.** System configuration: the robot holds the workpiece whilst the compliant mechanism holds the grinding wheel.

The main objective is to obtain a mechanism that does not change configuration in normal conditions (assuring a precision cut thanks to its high stiffness), while it changes configuration and reduces its stiffness (and thus the forces applied to the grinding wheel) when the burr exceeds a certain threshold value. In this case, a second cutting cycle is necessary to completely remove the burr.

The proposed mechanism is a slider-crank, where the crank is attached to the ground and the grinding wheel is fixed to the slider. To introduce some forces that allow the system to return to the baseline configuration, some springs are connected to different parts of the mechanism. To obtain a suitable mechanism behavior, it is chosen to place the mechanism close to a singularity, namely close to the top-dead-center (TDC) point. As a result, when the mechanism is close to the TDC, the slider-crank mechanism can react with a maximum force value (see Section 3.1), below which the slider (which supports the grinding wheel) does not move. Of course, if the mechanism is exactly at the TDC, the force value becomes the one giving the structural break of the structure. If this threshold is exceeded, the grinding wheel moves away from the workpiece, thus removing less material, and consequently reducing the cutting force. The slider-crank is ideal for the task at hand since it is easy to build and is suitable for working conditions near the TDC. To avoid to reach the singular configuration, a mechanical stop is applied to the slider (described in more detail in Section 5).

#### *3.1. First Temptative Mechanism*

The possibility to add three translational springs to the classic slider-crank mechanism is explored (Figure 3): one for the *y*-translation of the crank-end (*k*1); one for the *x*-translation of the crank-end (*k*2); one for the *x*-translation of the slider (*k*3). In this first stage the springs are connected to sliders, so that their elongations remain parallel to the main axes *x* and *y*. The free lengths of the springs are such that they do not generate any forces when the mechanism is in the TDC (angle of the crank *θ* = 0, and angle of the rocker arm link *α* = 0), i.e., the free length of *k*1 is zero, the free length of *k*2 is *l*1, while the free length of *k*3 depends on the anchor point.

**Figure 3.** Slider-crank mechanism with three translational springs, two of which are connected to sliders so that their movement is purely along one axis. *l1* is the crank length, *l2* is the rocker arm link length.

When neglecting friction forces, the principle of virtual work gives:

$$
\delta L = 0 \to \sum\_{i} \delta L\_{i} = \sum\_{i} F\_{i} \cdot \delta \mathbf{x}\_{i} = 0 \tag{7}
$$

where *Fi* is the *i*-th force applied to the mechanism, *δxi* is the associated virtual displacement, and *δLi* the related virtual work. From the mechanism configuration of Figure 3 and (7) it is found that:

$$F \cdot \delta \mathbf{x} - k\_3 \mathbf{x} \cdot \delta \mathbf{x} - k\_1 s\_1 \cdot \delta s\_1 - k\_2 s\_2 \cdot \delta s\_2 = 0 \tag{8}$$

where:

$$\begin{array}{cccc} s\_1 = l\_1 \sin \theta & \rightarrow & \delta s\_1 = l\_1 \cos \theta \cdot \delta \theta\\ s\_2 = l\_1(1 - \cos \theta) & \rightarrow & \delta s\_2 = l\_1 \sin \theta \cdot \delta \theta \end{array} \tag{9}$$

Substituting *s*1, *s*2, and their differentials in (8) results in:

$$\left( (F - k\_3 \mathbf{x}) \delta \mathbf{x} - \left( k\_1 l\_1^2 \sin \theta \cos \theta + k\_2 l\_1^2 (1 - \cos \theta) \sin \theta \right) \delta \theta = 0 \tag{10}$$

It is possible to calculate the magnitude of the force *F* that is balanced by the springs in each configuration of the mechanism starting from the speed ratios of the slider-crank mechanism:

$$\pi\_{\theta x} = \frac{\theta}{\dot{\bar{x}}} = \frac{\cos(a)}{l\_1 \sin(a + \theta)} = \frac{\delta \theta}{\delta \mathbf{x}} \; \; \; \; \pi\_{\theta \mathbf{x}} = \frac{\dot{a}}{\dot{\bar{x}}} = \frac{\cos(\theta)}{l\_1 \sin(a + \theta)} = \frac{\delta a}{\delta \mathbf{x}} \tag{11}$$

and using them in (10), which results in:

.

$$F = k\_3 \text{x} + \frac{k\_1 l\_1 \sin \theta \cos \theta \cos a + k\_2 l\_1 (1 - \cos \theta) \sin \theta \cos a}{\sin(a + \theta)} \tag{12}$$

This equation is only a function of the variable *x*, since *α* and *θ* can be written as:

$$\begin{aligned} \theta &= \arccos\left(\frac{l\_1^2 + (l\_1 + l\_2 - x)^2 - l\_2^2}{2l\_1(l\_1 + l\_2 - x)}\right) \\ \kappa &= \arcsin\left(\frac{l\_1}{l\_2}\sin\theta\right) \end{aligned} \tag{13}$$

Starting from (12) it is possible to obtain different behaviors of the mechanism by changing the stiffnesses of springs and link lengths. As an example, in Figure 4 two configurations are shown: to the left only the springs *k*1 and *k*2 are used, while to the right only *k*1 and *k*3 are used. In the first case the mechanism can withstand higher forces with the increase of *x*, whereas in the second case the opposite happens.

**Figure 4.** Variation of the force necessary to maintain the mechanism in static equilibrium as a function of *x*. (**a**) Only *k*1 and *k*2 are used (*k*1 =1500 N/m, *k*2 = 3000 N/m). (**b**) Only *k*1 and *k*3 are used (*k*1 = 1500 N/m, *k*3 = 150 N/m).

It is important to notice that the force *F* results in a finite value when the mechanism is close to the TDC configuration, and this force is a function of *k*1 only, i.e., (12) when *x* → 0 and using (13) becomes:

$$\lim\_{x \to 0} F = k\_1 \frac{l\_1 l\_2}{l\_1 + l\_2} \tag{14}$$

In sum, both progressive and regressive forces as a function of the slider displacement can be obtained with the proposed mechanism and layout, together with a finite value force for configurations close to the TDC. These last two are precisely the ingredients necessary to build a system capable of reducing the cutting forces when the deburring height exceeds a given threshold. Indeed, during deburring the grinding wheel has to remove a certain

amount of material, while preserving the structural integrity of the robot. The mechanism is able to withstand a certain deburring force (finite value around the TDC). When this threshold is exceeded, the mechanism becomes compliant (with regressive curve), thus reducing the cutting forces.

#### *3.2. Final Mechanism*

The mechanism proposed in Section 3.1, although theoretically suitable for the target application, is difficult to build. In particular, it would be quite impractical to include sliders at the at the anchor points with the frame, in order to keep the springs aligned with the reference axes. Therefore, a new mechanism is proposed with the springs fixed at some distance from the crank fixed hinge (Figure 5). Even with this modification, the basic principles described in Section 3.1 hold valid.

**Figure 5.** Slider-crank mechanism with translational springs connected to fixed ends (**a**). Two particulars of the mechanism are shown in the right figure (**b**).

Additionally, in this modified design, the springs are unloaded when the mechanism is close to the TDC ( *x* → 0). The principle of virtual work is to be applied in the same way as in (7), but the virtual displacements of *k*1 and *k*2 (*<sup>s</sup>*1 and *s*2, respectively) have to be modified according to the new mechanism. Starting from the dimensions of Figure 5, the spring displacements are:

$$\begin{array}{c} s\_1 = \sqrt{l\_1^2 + d^2 - 2l\_1 d \cos(\gamma\_1 + \theta)} - \sqrt{\left(a - l\_1\right)^2 + c^2} \\ s\_2 = \sqrt{l\_1^2 + b^2} - \sqrt{l\_1^2 + b^2 - 2l\_1 b \sin(\theta)} \end{array} \tag{15}$$

where *d* is the distance between the anchor point of *k*1 and the crank hinge, and *γ*1 is the angle of the vector crank hinge-anchor point with respect to the *x*-axis:

$$d = \sqrt{a^2 + c^2}, \quad \gamma\_1 = \operatorname{atan}\left(\frac{c}{a}\right) \tag{16}$$

The differentials of *s*1 and *s*2 can be calculated as:

$$\begin{array}{l} \delta \mathbf{s}\_{1} = \frac{\partial \mathbf{s}\_{1}}{\partial \theta} \cdot \delta \theta = \frac{l\_{1}d \sin(\gamma\_{1} + \theta)}{\sqrt{l\_{1}^{2} + d^{2} - 2l\_{1}d \cos(\gamma\_{1} + \theta)}} \cdot \delta \theta\\ \delta \mathbf{s}\_{2} = \frac{\partial \mathbf{s}\_{2}}{\partial \theta} \cdot \delta \theta = \frac{l\_{1}b \cos(\theta)}{\sqrt{l\_{1}^{2} + b^{2} - 2l\_{1}b \sin(\theta)}} \cdot \delta \theta \end{array} \tag{17}$$

As a result, the force *F* at the slider becomes:

$$F = k\_3 \mathbf{x} + \left( k\_1 s\_1 \frac{l\_1 d \sin(\gamma\_1 + \theta)}{\sqrt{l\_1^2 + d^2 - 2l\_1 d \cos(\gamma\_1 + \theta)}} + k\_2 s\_2 \frac{l\_1 b \cos(\theta)}{\sqrt{l\_1^2 + b^2 - 2l\_1 b \sin(\theta)}} \right) \tau\_{\theta x} \tag{18}$$

The expression (18) for *x* → 0 simplifies to:

$$\lim\_{x \to 0} F = \frac{k\_1 l\_1 l\_2 c^2 \left(b^2 + l\_1^2\right) + k\_2 l\_1 l\_2 b^2 \left(\left(a - l\_1\right)^2 + c^2\right)}{\left(l\_1 + l\_2\right)\left(\left(a - l\_1\right)^2 + c^2\right)\left(b^2 + l\_1^2\right)}\tag{19}$$

Differently from (14), which depends on *k*1 only, (19) depends both on *k*1 and *k*2. Similarly to the mechanism of Figure 3, the new mechanism behaviour depends on its main dimensions. Nevertheless, in this case also the dimensions *a*, *b*, and *c* have a grea<sup>t</sup> impact on *F*. As an example, two different mechanisms are shown in Figure 6, whose main parameters are listed in Table 1. Similarly to the mechanism presented in Section 3.1., both progressive and regressive forces as a function of the *x* displacement can be obtained, with a finite value of the force around the TDC. The regressive curve is preferred for the application at hand since it is related to lower cutting forces.

**Figure 6.** Variation of the force necessary to maintain the mechanism in static equilibrium as a function of *x*. ((**a**) Mechanism #1, (**b**) Mechanism #2).


