## Abstract

The application of servocontrolled mechanical-bearing-based precision motion stages (MBMS) is well-established in advanced manufacturing, semiconductor industries, and metrological applications. Nevertheless, the performance of the motion stage is plagued by self-excited friction-induced vibrations. Recently, a passive mechanical friction isolator (FI) has been introduced to reduce the adverse impact of friction in MBMS, and accordingly, the dynamics of MBMS with FI were analyzed in the previous works. However, in the previous works, the nonlinear dynamics components of FI were not considered for the dynamical analysis of MBMS. This work presents a comprehensive, thorough analysis of an MBMS with a nonlinear FI. A servocontrolled MBMS with a nonlinear FI is modeled as a two DOF spring-mass-damper lumped parameter system. The linear stability analysis in the parametric space of reference velocity signal and differential gain reveals that including nonlinearity in FI significantly increases the local stability of the system's fixed-points. This further allows the implementation of larger differential gains in the servocontrolled motion stage. Furthermore, we perform a nonlinear analysis of the system and observe the existence of sub and supercritical Hopf bifurcation with or without any nonlinearity in the friction isolator. However, the region of sub and supercritical Hopf bifurcation on stability curves depends on the nonlinearity in FI. These observations are further verified by a detailed numerical bifurcation, which reveals the existence of nonlinear attractors in the system.

## 1 Introduction

Motion stages provide high-speed and high-precision positioning in manufacturing and metrology-related processes, which include machining, additive manufacturing, and semiconductor fabrications [1–4]. Among magnetic-based, flexural-based, fluidic-based, and mechanical-bearing-based motion stages (MBMS), MBMS are more prominent in the industries due to their cost-effectiveness, high off-axis stiffness, wide ranges of motions, and easy-to-install feature [5]. The motion control of MBMS in different applications is commonly realized by using different servocontrollers [6–8]. However, it can lead to the problem of self-excited limit cycles in the tracking error caused by the sliding friction between contact surfaces, which is also known as friction-induced vibrations (FIV) [6,9–14]. The adverse effects of friction on the control performance include oscillations of stick-slip phenomena tracking errors and long settling times. Therefore, understanding the dynamics of self-excited friction-induced vibrations under different conditions is essential to mitigate tracking error oscillation, which leads to better motion stage performances.

Different controllers have been proposed to eliminate friction-induced vibrations. The fundamental concept of these controllers to suppress FIV is to counteract the friction force by providing an equal and opposite force and suppress these FIV. These controllers can be divided into three categories as (1) high-gain controllers, (2) model-based controllers, and (3) advanced controllers. The use of these controllers in compensating for the effect of friction is well-established in the literature [11,13,15–18]. However, the performance of these controllers can be limited by environmental noise in the case of high-gain controllers, model inaccuracy in the case of model-based controllers, and low-performance computational/actuator hardware in the case of advanced controllers.

Recently, a mechanical device known as the friction isolator (FI) was proposed as a more robust and efficient approach to mitigating self-excited friction-induced vibrations in the MBMS system [19,20]. Unlike conventional motion stages in which the bearings are rigidly installed to the table, the compliant motion stages adopt FI as motion-compliant joints between the bearings and the table, isolating the table from nonlinear frictional effects. For more design details of FI, readers are referred to Refs. [19,20]. Furthermore, parametric analyses on the motion stage with FI showed that with optimum design parameter selections, FI could increase the fixed-point stability region to allow the use of higher control gains and reduce the amplitudes of the control error limit cycles [21,22]. However, the above-mentioned studies analyzed the dynamics of an MBMS with linear FI only. Nevertheless, the mechanical design of FI can introduce nonlinear elements, which are significantly compared to the linear elements [19] and hence, need to be considered in the dynamics of compliant motion stages for optimum selection of control parameters. The present paper examines this problem and provides a framework for further analysis. For this, we extend the model proposed in Refs. [22] by introducing a nonlinear element in the system to take into account nonlinear FI. The preliminary linear stability analysis reveals that linear FI underestimates the stability regime, and hence, it is desirable to consider nonlinearity in FI for a better selection of control parameters for steady operations. Furthermore, the nonlinear analysis using the analytical method, more specifically the method of multiple scales, reveals the existence of super and subcritical Hopf bifurcation in the system regardless of the presence of nonlinearity in FI. However, for the given value of system parameters, the region corresponding to supercritical bifurcation can be increased by properly selecting the nonlinear parameters.

The paper is arranged as follows. We outline the extended model of MBMS with nonlinear FI in Sec. 2. Later on, the linear stability analysis of fixed-points is investigated in Sec. 3. This is followed by a detailed analysis of the current system in Sec. 4. Results and detailed discussions from the previous sections are presented in Sec. 5. Finally, some conclusions are drawn in Sec. 6.

## 2 Mathematical Formulation

To establish the effect of nonlinearity in the friction isolator and the contribution of various system parameters to linear and nonlinear instability of PD-controlled MBMS, a nonlinear two DOF model, illustrated in Fig. 1, is obtained by extending the linear model discussed in Ref. [22]. In this model, the precision motion stage is modeled as a rigid mass $mt$, whereas the combined mass of the friction isolator and bearing is modeled as $mb$. The nonlinear interactionsbetween $mb$ and $mt$ are represented by a nonlinear spring with a stiffness function $K($.$),\u2009\u2009and\u2009\u2009a\u2009\u2009linear\u2009\u2009damper\u2009\u2009with\u2009\u2009damping\u2009coefficient\u2009\u2009cfi$. Also, in this model, $u1$ and $r(t)$ represent the output feedback control force from the PD controller and input reference/set-point signal to the PD controller, respectively. Therefore, if $X1(t)$ and $X2(t)$ represent the position of MBMS and FI, respectively, then the equations of motion for the system are given by

This solvability condition further puts the restriction on the ranges of $krq,krc$ to get unique real fixed-points for the given value of $kr$. Therefore, we have ensured that the numerical values of $krq$ and $krc$ are chosen such that we get unique real fixed-points of the system.

where $h0=1g0,h1=h0+xs3+\sigma 2vrv,h2=\u22123krch1+krq,h3=h1\u2212xs3\u2212\sigma 2vrv,h4=\u2212\sigma 2+\sigma 1g1h3vrv,h5=\u2009g0+vrvg1,h6=\alpha \sigma 1(h1\u2212xs3),h7=vrvg2+g1$, and $h8=g2+g3vrv$. Since nonlinearities in these equations appear as coefficients of higher orders of $\u03f5(>0)$, the unperturbed system can be obtained by setting $\u03f5=0$ in Eq. (17) for the linear stability analysis.

## 3 Linear Stability of Fixed-Points and Existence of Critical Points

Since the expressions for $Rem$ and $Imm(m=1,2,3)$ are lengthy, and hence, not reported in the work for the sake of brevity. In the next step, we analyze the system using the perturbation method, more specifically, the method of multiple scales (MMS).

## 4 The Method of Multiple Scales

The linear stability analysis of our system in Sec. 3 determines the fixed-points' local stability for a given set of operational and system parameters. For a given set of parameters, if a small perturbation to a fixed-point dies out with time, then it is locally stable; however, if it increases with time, then it is globally unstable. The sensitivity of fixed-points toward initial perturbation in a locally stable point and its time evolution depends on the nature of the existing system's nonlinearity. If all perturbations die out with time, irrespective of their magnitude, then a locally stable point is considered as a globally stable point for the fixed-point. However, the small perturbation dies out, and the large perturbation settles down to a limit cycle close to the critical point in a locally stable point. A globally stable point is different from a locally stable point, which further leads to the existence of bistable regions. Since such a phenomenon relies on the system's nonlinearity, linear stability analysis is insufficient to analyze the bistable regime. Therefore, we perform a thorough nonlinear analysis of the system at Hopf points to establish the globally stable region of the fixed-points.

## 5 Results and Discussions

In this section, we examine the analytical results presented in Secs. 3 and 4 through numerical simulations. For our numerical simulation, we use the parameter values listed in Table 1 . We first analyze the linear stability of the system in the parametric space of $\zeta $ and $vrv$ followed by the validation of our analytical formulation. Later on, by utilizing our analytical findings, we present the different regions of sub and supercritical Hopf bifurcation on linear stability boundaries. Finally, a detailed bifurcation analysis is presented.

$mt(kg)$ | 1.5 | $kp$ | $2e4$ |

$mb(\u2009N\u2212s/m)$ | 0.75 | $X0(\u2009m)$ | 0.0007353 |

$\sigma 0*(\u2009N/m)$ | $2.2e6$ | $\sigma 1*,\sigma 2*(N\u2212s/m)$ | $237,14.25$ |

$fc*(\u2009N)$ | 5.1 | $fs*(N)$ | 6.5 |

$\omega 0(rad/s)$ | 115.5 | $\kappa $ | 0.001 |

$\sigma 0$ | 110 | $\sigma 1$ | 1.37 |

$\sigma 2$ | 0.0823 | $fs$ | 0.44 |

$fc$ | 0.35 | $a$ | 2.5 |

$mt(kg)$ | 1.5 | $kp$ | $2e4$ |

$mb(\u2009N\u2212s/m)$ | 0.75 | $X0(\u2009m)$ | 0.0007353 |

$\sigma 0*(\u2009N/m)$ | $2.2e6$ | $\sigma 1*,\sigma 2*(N\u2212s/m)$ | $237,14.25$ |

$fc*(\u2009N)$ | 5.1 | $fs*(N)$ | 6.5 |

$\omega 0(rad/s)$ | 115.5 | $\kappa $ | 0.001 |

$\sigma 0$ | 110 | $\sigma 1$ | 1.37 |

$\sigma 2$ | 0.0823 | $fs$ | 0.44 |

$fc$ | 0.35 | $a$ | 2.5 |

### 5.1 Linear Stability Curves.

In this section, we present the effect of the nonlinear components of friction isolator on the linear stability of the system. For this step, we plot the stability curves for different combinations of $krq$ and $krc$ on the operational parameter region of $\zeta \u2212vrv$ and are shown in Figs. 2–4. For ease of understanding, the unstable and stable regions are denoted by “$U$” and “$S$,” respectively.

As mentioned earlier, it is mathematically challenging to get analytical expressions for $vrv,cr$ and $\zeta cr$, hence, we obtain stability boundaries numerically by solving Eqs. (21) and (22) along with Eq. (13) for the varying values of frequency ω in a range ω∈(ω_{1}, ω_{2}). Since ω_{1} and ω_{2} are functions of system parameters, their numerical values vary from one case to another. On solving Eqs. (21) and (22) along with Eq. (13), for a given range of frequency, we get negative values of ζ _{} and vrv. However, as negative values of the control gain and the reference signal are not feasible, we plot the stability curves for the positive values of parameters.

From Figs. 2–4, we can easily observe that, compared to the case of linear FI, the inclusion of quadratic and cubic nonlinearities in the FI increases the fixed-point's stability significantly. This observation further implies that the nonlinearities in FI support a wider range of stable operating conditions. However, the relative effects of quadratic/cubic nonlinearity on the stability region for a given value of cubic/quadratic nonlinearity in various scenarios are different. For example, Fig. 2 shows the stability boundaries for different values of cubic/quadratic nonlinearity for a given nonzero value of quadratic/cubic and linear stiffness whereas Fig. 3 shows the stability boundaries with different values of cubic/quadratic nonlinearity in the absence of quadratic/cubic nonlinearities and nonzero linear stiffness, and Fig. 4 shows stability boundaries for different values of cubic/quadratic nonlinearity in the absence of quadratic/cubic and linear stiffness. From Figs. 2 and 3, it can be observed that the relative effects of cubic and quadratic nonlinearities on the fixed-point's stability are almost identical with or without the other component of nonlinearity. This further implies that the rate of increase in stability with the increase in cubic nonlinearity is approximately the same as with the increase in quadratic nonlinearity. However, we emphasize that the overall stability at the given value of cubic nonlinearity (Fig. 3(a)) is higher than that at quadratic nonlinearity (Fig. 3(b)). Furthermore, from Fig. 4, it can be observed that although the rate of increase in stability with quadratic nonlinearity (Fig. 4(b)) is much higher than cubic nonlinearity, the overall stability boundary for a given value of cubic nonlinearity is significantly larger than the case of quadratic nonlinearity. These observations further suggest that increasing the cubic stiffness of FI is more beneficial than increasing the quadratic stiffness. Having established the effect of nonlinear stiffness on the fixed-point's stability, we analyze the Hopf bifurcation on the stability curves using analytical results obtained by MMS. However, before this step, we must validate our analytical results, which can be done by comparing them with numerical simulations and presented next.

### 5.2 Validation of Method of Multiple Scales.

To evaluate the accuracy of the MMS, we compare the solution of the system obtained from the slow-flow equations to the one obtained from Eq. (11) using the matlabode solver “ode45.” We first present the time response of the motion stage with nonlinear FI for two sets of operational parameters close to the Hopf point. In particular, we respectively choose two nondimensional reference velocities with a smaller and a larger value, i.e., $vrv=0.0495<vrv,cr=0.05,\zeta cr=0.11873$, and $vrv=0.09<vrv,cr=0.1,\zeta cr=0.09502$. Since both sets of parameters are in the unstable regime, we obtain a gradually increasing periodic response of different amplitudes as shown in Fig. 5. We emphasize that these time responses are shifted to the origin set at the fixed-point (Eq. (12)). From Fig. 5, it can be easily observed that the analytical solution of the system from MMS exhibits an excellent match with the numerical solution of the system. We repeat the same steps for the motion stage with linear FI and two sets of operational parameters, viz., $vrv=0.0495<\u2009vrv,cr=0.05,\zeta cr=0.24562$ and $vrv=0.09<vrv,cr=0.1,\zeta cr=0.19082117$. The results are shown in Fig. 6 and we observe a good match between the two approaches for the motion stage with linear FI. Hence, our analytical solutions (Eq. (45)) are valid.

### 5.3 Hopf Bifurcation.

In this section, we analyze the different regions of super and subcritical Hopf bifurcation on the stability lobes. When the system changes its stability through the Hopf bifurcation, the fixed-points settle down to a limit cycle close to Hopf point. Furthermore, the location of a limit cycle with respect to the Hopf point decides the nature of Hopf bifurcation. More specifically, in the case of supercritical Hopf bifurcation, these limit cycles exist in the unstable region only, whereas the existence of limit cycles close to the Hopf point in the stable regime signifies subcritical Hopf bifurcation. We emphasize that the presence of supercritical Hopf bifurcation leads to the fixed-point's global stability of the stable region, whereas subcritical bifurcation leads to a bistable region in the system. Therefore, it is an essential step toward the understanding of the criticality of Hopf bifurcation on the stable curves.

*a*), i.e., the nontrivial solution of Eq. (46

*a*) and is given by

Equation (47) plays an essential role in determining the criticality of Hopf bifurcation. In Eq. (47), if $q11q12$ is negative then for $R$ to be a real quantity $k1$ should be positive. This further implies that limit cycles will exist in a linearly unstable region only, and the Hopf bifurcation will be supercritical. However, if for another set of critical parameter values $q11q12$ get positive then for a real value of $R,k1$ should be negative. Hence, limit cycles exist in the linear stable region, and the Hopf-bifurcation will be subcritical. After determining the criteria for subcritical and supercritical Hopf bifurcations, we evaluate $q11q12$ at every Hopf point on the stability curve and decide the characteristic of Hopf bifurcation. Figures 7(a) and 7(b) show the characteristic of Hopf bifurcation on the stability boundary for the MBMS with nonlinear and linear FI, respectively. From both figures, we can observe that supercritical Hopf bifurcations occur at low values of $vrv$. At the same time, fixed-points lose stability through subcritical Hopf bifurcations for high values of $vrv$.

At first glance on Figs. 7(a) and 7(b), it appears that there is no effect of nonlinearity of FI on the subcritical and supercritical Hopf bifurcation regions on the stability curve, and transition point remains the same with nonlinear and linear FI. Therefore, to demonstrate the effect of nonlinear parameters of FI $(krq,krc)$ on the criticality of Hopf bifurcation, we plot $q11q12$ with different values of $\zeta cr$ and $vrv,cr$ for different sets of $krq$ and $krc$, and the results are shown in Figs. 8(a) and 8(b), respectively.

From Fig. 8(a), it can be noted that the inclusion of nonlinearity in FI reduces the range of $\zeta cr$ corresponding to supercritical Hopf bifurcation. However, this can be further justified by the fact that the nonlinearity shrinks the unstable region by decreasing $\zeta cr$ values which lead to a decrease in the effective range of $\zeta cr$ for supercritical Hopf bifurcation. Instead, Figs. 8(a) and 8(b) provide more information about the effect of $krq$ and $krc$ on the criticality of Hopf bifurcation. From Figs. 8(a) and 8(b), we can easily observe that the inclusion of nonlinearity in FI can decrease or increase the region of supercritical Hopf bifurcation depending on the numerical values of $krq$ and $krc$. The optimization of these values for a larger region of supercritical Hopf bifurcation and globally stable region is left for future work.

We emphasize that these analytical findings only provide the amplitude of limit cycles close to the Hopf point and characteristics of Hopf bifurcation on the stability boundaries. Therefore, to observe the global behavior of the system in the unstable region, we employ the numerical bifurcation analysis and present in the Sec. 5.4. Note that this step not only provides information about the large amplitude response of the system but also further verifies our analytical findings.

### 5.4 Bifurcation Analysis.

To perform the numerical bifurcation analysis for the motion stage with nonlinear and linear FI, we solve the system of ODEs given by Eq. (10) using matlabode solver “ode45.” The bifurcation plots, showing the extreme points of $x1$, i.e., the error amplitude of the motion stage (corresponding to $x2=0$), for the motion stage with nonlinear and linear FI have been shown in Figs. 9 and 10, respectively. The numerical bifurcation analysis can be performed by making either of the operational parameters a constant and varying the another. However, to show the existence of super- and subcritical Hopf bifurcations for lower and higher values of $vrv$, respectively, as observed in Sec. 5.3, we have fixed $\zeta $ and varied $vrv$. To plot these numerical bifurcation diagrams, we vary $vrv$ in upward and downward directions so that the system loses and gains stability through Hopf bifurcation. For completeness, we have also plotted these numerical bifurcation diagrams for two different values of $\zeta $. To get a better picture of the dynamics of the motion stage with nonlinear and linear FI, the bifurcation diagrams close to the Hopf points are shown in the inset of Figs. 9 and 10. From these numerical bifurcation diagrams, we can easily observe the existence of stable limit cycles with fixed-point solutions at higher values of $vrv$ for a given value of $\zeta $, which implies Hopf bifurcation is subcritical by nature. However, for lower values of $vrv$, stable limit cycles exist in the unstable region only, which indicates supercritical bifurcation. Both of these observations are consistent with our analytical findings in Sec. 5.3. Furthermore, in the case of the motion stage with nonlinear

FI, the response amplitude for higher values of $vrv$ is relatively smaller than the ones corresponding to the motion stage with linear FI. For a better understanding of the dynamics of a PD-controlled motion stage with nonlinear and linear FI, the zoomed views of Figs. 9 and 10 have been shown in Figs. 11 and 12, respectively. For the sake of brevity, we only present these zoomed views for differential gains of higher values, i.e., $\zeta =0.1$. The corresponding representative phase portraits for different values of $vrv$ have been shown inside these zoomed figures. From Figs. 11 and 12, we can easily observe that in both cases, close to Hopf points, stable period-1 solutions lose stability through period-doubling bifurcation. This further leads to the appearance of period-2 solutions, which can also be observed from the phase portraits (Figs. 11(a) and 11(d) and 12(a) and 12(d)). Furthermore, in the case of nonlinear FI, the system exhibits only period-4 solutions away from the Hopf points, and there is no exchange in the stability of limit cycles away from the Hopf points (Figs. 11(b) and 11(c)). However, in the case of linear FI, apart from the coexistence of period-1 and period-2 solutions (as can be seen by phase portraits for $vrv=0.02$ in Fig. 12(a)), there is a continuous exchange of stability between period-1 and period-2 solutions as shown in Figs. 12(a),*–*12(d). Also, when comparing Figs. 11 and 12 in terms of subplots (i) and (ii), we observe that the branch of stable period1 solutions close to Hopf point is significantly smaller in case of linear FI when compared to the case of nonlinear FI. This observation further signifies the importance of nonlinear FI over linear FI.

We perform the quantitative match between MMS results and numerical simulations for completeness. For this step, we use the fixed-arc-length continuation scheme [33] to get the branch of limit cycles close to the Hopf point and later compare it with the branch of limit cycles obtained using the slow-flow equation emerging from the Hopf point. These results are shown in Figs. 13(a) and 13(b) for MBMS with nonlinear and linear FI, respectively. Since the analysis has been performed close to the Hopf points, we can observe an excellent match between MMS and numerical simulations close to the Hopf point, which further verifies our analytical approach.

## 6 Conclusion

In this study, we analyzed the linear and nonlinear dynamics of a PD-controlled MBMS with a FI using analytical and numerical methods. Contrary to earlier studies where the nonlinearity in the friction isolator was ignored, the effects of nonlinearity from the friction isolator on the dynamics of the motion stage have been explored in this work. The dynamical effect of friction was captured through the LuGre friction model. A parametric study on the linear stability analysis revealed that compared to the linear friction isolator, the nonlinearity in the friction isolator increases the fixed-points' local stability in the operating parameter space of reference signal and differential gain. This further implied that the system's stability is underestimated when using the linear FI model. The nonlinearity in the FI should be considered in the modeling for a better prediction of steady operating conditions. The nonlinear analysis of MBMS with FI was performed analytically using MMS and numerical simulations. The analytical findings were verified by comparing them against numerical solutions, and a good match was observed. Both analytical and numerical simulations revealed the existence of supercritical and subcritical Hopf bifurcation. Furthermore, the parametric analysis on the criticality of Hopf bifurcation revealed the sensitivities of subcritical and supercritical Hopf bifurcation regions in terms of the nonlinearities of the friction isolator. On exploring the dynamics of MBMS with nonlinear FI in the unstable regime, we observed period-doubling bifurcations, period-4 solutions, and quasi-periodic solutions.

Finally, these findings suggest that the consideration of nonlinearity in the FI model is an essential step to get an accurate picture of global dynamics of the motion stage with a FI.

## Acknowledgment

This work was funded by National Science Foundation (NSF) Award CMMI #1855390: Towards a Fundamental Understanding of a Simple, Effective, and Robust Approach for Mitigating Friction in Nanopositioning Stages, and CMMI #2000984: Nonlinear Dynamics of Pneumatic Isolators in Ultra-Precision Manufacturing Machines.

## Funding Data

National Science Foundation (NSF) (Award No. CMMI #1855390; doi: 10.13039/100000001).

## Conflict of Interest

The authors declare that they have no conflict of interest.

## Data Availability Statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.