A constrained linear quadratic optimization algorithm toward jerk-decoupling cartridge design

doi:10.1016/j.jfranklin.2016.10.020

Journal of the Franklin Institute

Volume 354, Issue 1, January 2017, Pages 479-500

https://doi.org/10.1016/j.jfranklin.2016.10.020 Get rights and content

Abstract

Linear direct feed drives are widely used in machine tools, but an abrupt counter force from the secondary part will induce the jerk to the metro frame contacted with the linear motor and cause the vibration of auxiliary devices on it. The jerk-decoupling cartridge (JDC) provides a buffer to reduce such an impact. Design target for such a system includes both the tracking error and the jerk induced to the metro frame. To achieve both targets systematically, this work presents an integrated approach to efficiently determine parameters in the JDC and the position controller of the feed drive. The problem is firstly formulated as a nonlinear constrained optimization problem, which is then converted to a series of projection gradient optimization problems and step searching problems, which are either convex or linear. Thus, fast convergence of parameters are achieved within first several iterations. Through a series of simulation, the effectiveness of proposed methodology is verified.

Introduction

In high-volume manufacturing industries such as electronics and semiconductor sectors, there is a growing need for high-speed precision machines to achieve high production throughput and high production quality [1], [2], [3], [4], [5]. The linear direct feed drive is an excellent choice to meet the requirement of high speed, ultra precision and improved reliability due to the simplicity of its mechanical construction [6]. As shown in Fig. 1, there are three configurations of feed drive design in machine tool branch from the existing literature [7]. The first is fixed-mounted feed drive as shown in Fig. 1(a). In this design, the impulsive high reaction force is directly transmitted to the machine frame and residual vibration is induced to auxiliary devices on the machine frame. The second design named active-mounted feed drive in Fig. 1(b) has an additional linear motor installed between the machine frame and the secondary part. This is to actively compensate the reaction force [8]. In [9], the principle of mechanically coupled and opposite driving linear motors is presented to solve the problem of excitation force to the machine base and the maximum power limitation. The third configuration named suspended feed drive as in Fig. 1(c) decouples the reaction force from the machine frame by introducing the jerk-decoupling cartridge (JDC) between the secondary part and the machine frame [7], [10], [11]. The JDC has been implemented either in a linear feed drive [12] or a pinion-shaft or a ball-screw driven system [13]. Remarkably, by adjusting coefficients of the JDC, resonant modes of the system are adjustable in the design phase rather than suppress them in the control phase [11]. In [14], the jerk-decoupling technique is applied to a high-speed linear direct drive stage, the jerk transmitted to the machine frame is verified to be reduced significantly. The jerk-decoupling technique is also presented in [15] and a comparison with conventional designed direct linear drives shows the effectiveness of the proposed design. Notice that in Fig. 1(c), the impulse to the machine frame is not only determined by mechanical parameters of JDC [11], but also linked to the motion trajectory and controller parameters of the primary part. Recently, passive-assist device (PAD) consisting of a torsional spring in parallel with a rotary motor has been used to reduce the residual vibration [16]. A novel PAD concept is introduced in [17], [18], which uses magnetic repulsion to provide assistive force to the linear motor during acceleration/deceleration to reduce actuation force, thus the vibration is reduced by transmitting the assistive force to the ground rather than to the machine frame.

In this work, we first formulate the above mechatronics design problem into a linear quadratic optimization problem with a series of defined constraints. Later, we convert such constrained nonlinear optimization problem to a convex optimization dual problem. To efficiently solve it, we propose an algorithm based on the direct computation of projected gradient matrix and linear searching of incremental steps, thus fast convergence of parameters is achieved. In addition, this optimization result is robust to the slow-varying disturbance like bearing friction.

The paper is organized as follows: Section 2 presents the modelling of the linear direct feed drive. A hybrid Linear Quadratic Tracking (LQT) and Linear Quadratic Regulator (LQR) optimization problem is formulated in Section 3. In order to solve the formulated optimization problem, a gradient-based algorithm is presented in Section 4. Lastly, Section 5 showcases an example with accompanying simulation results. Section 6 concludes the paper with salient points.

Section snippets

Modelling of the linear feed drive with JDC configuration

A mechanical linear direct drive mounted on a metro frame through a spring–damper system is shown in Fig. 2 and associated symbols for all parameters are shown in Table 1.

Assuming that the primary part, the secondary part and the metro frame are all subjected to constant disturbances due to friction forces, dynamics of the system can be expressed as follows: $m_{1} {\ddot{y}}_{1} = f_{1} - d_{1}, m_{2} {\ddot{y}}_{2} = - f_{1} - f_{2} + d_{2}, m_{3} {\ddot{y}}_{3} = f_{2} - f_{3} + d_{3}, f_{1} = K_{f} i (t), f_{3} = k_{3} y_{3} + b_{3} {\dot{y}}_{3}, i (t) = K_{m} u_{m} (t) .$

Notice that $f_{1} = K_{f} i (t)$ in Eq. (1) is obtained under the

Linear quadratic formulation

We convert the original system model (1) to the following state space form: $\dot{x} = Ax + Bu + Ed, y = Cx,$ where $x^{T} = [y_{1} {\dot{y}}_{1} y_{2} {\dot{y}}_{2} y_{3} {\dot{y}}_{3}]$ and $d^{T} = [d_{1} d_{2} d_{3}]$ . Matrices A, B, C and E are given in Appendix B.

To reject constant disturbances, we take the derivative of Eq. (3), thus the state space model is given by $\ddot{x} = A \dot{x} + B \dot{u}, \dot{y} = C \dot{x},$ Notice that ${\dot{y}}_{2}$ and ${\dot{y}}_{3}$ can be extracted from $\dot{y} = C \dot{x}$ , but not y₁. To include y₁, we augment y₁ as a state variable. Thus, the state vector is defined as ${\hat{x}}^{T} = [y_{1} {\dot{y}}_{1} {\ddot{y}}_{1} {\dot{y}}_{2} {\ddot{y}}_{2} {\dot{y}}_{3} {\ddot{y}}_{3}] \equiv [x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$

Gradient-based constrained optimization algorithm

Due to the constrained feedback structure of the decentralized form in Eqs. (12), (13), (14), there is no standard closed-form solution to this optimization problem [23], [24]. Thus, we need iterative approaches to find the optimal solution. Nature-inspired heuristic optimization algorithms [25], [26] are well known among iterative approaches, such as genetic algorithms (GA) [27], Particle Swarm Optimization (PSO) [28], Ant Colony Optimization (ACO) [29], and Artificial bee colony (ABC)

Simulation

From generation of speed of the response and maximum acceleration, all the eigenvalues of matrix A_d are set to $- 20$ , then a₁, a₂, a₃ and a₄ are obtained as $- 160 000$ , $- 32 000$ , $- 2400$ and $- 80$ respectively, motion profiles of the reference trajectory are shown in Fig. 4.

Weighting matrices can be chosen based on different requirements. In the simulation, two sets of weighting matrices Q₁, R₁ and Q₂ are chosen for illustration:

•
Set 1: $Q_{1} = diag {1, 1, 1}$ , $R = diag {1, 1}$ , $Q_{2} = 10^{- 5} I_{7 \times 7}$ .
•
Set 2: $Q_{1} = diag {10, 1, 0.1}$ , $R =$

Conclusion

This paper presents a linear quadratic decentralized optimization algorithm to integrated design the optimal JDC mechanism and the corresponding controller into a high-speed motion stage under various constraints. The mechatronics design problem is converted to a linear quadratic optimization problem, the performance index is initiated with consideration of various design factors including the tracking accuracy of the primary part, velocities of the secondary part and the metro frame, the force

Acknowledgment

The authors would like to thank SIMTech-NUS Joint Lab on Precision Motion Systems (U12-R-024JL) for funding the project U13-R-036SU. The authors would also like to thank to Prof José C. Geromel for his sharing and discussion.

References (39)

Y. Altintas et al.
Machine tool feed drives
CIRP Ann. – Manuf. Technol.
(2011)
M. Rehm et al.
Mechanically coupled high dynamic linear motors-a new design approach and its control strategy
CIRP Ann. – Manuf. Technol.
(2014)
B. Denkena et al.
Energy optimized jerk-decoupling technology for translatory feed axes
CIRP Ann. – Manuf. Technol.
(2009)
D. Yoon et al.
Magnet assisted stage for vibration and heat reduction in wafer scanning
CIRP Ann. – Manuf. Technol.
(2015)
A. Bemporad et al.
The explicit linear quadratic regulator for constrained systems
Automatica
(2002)
D. Karaboga et al.
A comparative study of artificial bee colony algorithm
Appl. Math. Comput.
(2009)
J.C. Geromel et al.
An algorithm for optimal decentralized regulation of linear quadratic interconnected systems
Automatica
(1979)
J.C. Geromel et al.
Optimal decentralized control of dynamic systems
Automatica
(1982)
C.J. Evans
Precision engineeringan evolutionary perspective
Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci.
(2012)
K.K. Tan et al.
Precision Motion Control: Design and Implementation
(2007)

K.L. Barton et al.

A cross-coupled iterative learning control design for precision motion control

IEEE Trans. Control Syst. Technol.

(2008)

C. Hu et al.

Adaptive robust repetitive control of an industrial biaxial precision gantry for contouring tasks

IEEE Trans. Control Syst. Technol.

(2011)

H. Zhu et al.

Integrated servo-mechanical design of a fine stage for a coarse/fine dual-stage positioning system

IEEE/ASME Trans. Mechatron.

(2016)

J. Švéda, M. Valášek, Z. Šika, New active machine tool drive mounting on the frame, Applied and Computational Mechanics...

J. Tang et al.

Vibration control of rotationally periodic structures using passive piezoelectric shunt networks and active compensation

J. Vib. Acoust.

(1999)

B. Denkena et al.

Energy flow in jerk-decoupled translatory feed axes

Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci.

(2007)

D. Stoiber, Impulsentkoppelter direktantrieb, wO Patent App. PCT/EP1999/001,482, September 23,...

M. Knorr, D. Stoiber, Impulsgekoppelter transmissionsantrieb, eP Patent App. EP20,020,006,808, October 9,...

W. Lin, H. Yuan, Z. Zhong, W.J. Lin, G. Yang, Vibration suppression for impulsive motion in direct drive linear stage,...

Cited by (20)

Enhanced sensitivity shaping by data-based tuning of disturbance observer with non-binomial filter
2019, ISA Transactions
Citation Excerpt :
The Disturbance Observer (DOB) [1] is a simple yet effective way to deal with low frequency disturbances and it has been widely adopted in the industrial applications [2–4]. Although the traditional feedback controller is able to provide a basic disturbance rejection capability [5–8], the DOB becomes necessary when higher performance is demanded for motion systems in the presence of severe disturbances. By estimating the actual disturbance using the inverse system model, the DOB loop can attenuate external disturbances as well as disturbances resulting from model mismatches.
This work presents a new method of sensitivity shaping for motion control systems with disturbance observer (DOB). Although the traditional DOB design has been proved to be an effective means of rejecting disturbances in many applications, shaping of the closed-loop sensitivity is constrained since the $Q$ -filter in DOB is usually restricted to a standard Butterworth or binomial form, where the cutoff frequency is the only tunable parameter. To overcome this limitation and achieve better performance for more demanding motion systems, a general form of the $Q$ -filter is employed in this work. An data-based iteratively tuning procedure is subsequently developed according to a pre-defined cost function, so that the disturbance rejection in the low-frequency band is improved while the high-frequency noise attenuation and robust stability are not compromised. This method is data-based in the sense that the sensitivity shaping process is conducted based on the input–output data collected from the operating motion system, instead of relying on the identified system model. Simulation and experimental results are presented to show the advantage of this data-based sensitivity shaping approach.
A novel constrained H<inf>2</inf> optimization algorithm for mechatronics design in flexure-linked biaxial gantry
2017, ISA Transactions
Citation Excerpt :
The drawback of such a design is that it will possibly introduce additional resonant modes to the end-effector, being sensitive to chattering control signals [26]. In the existing literatures, the effects of resonant modes are mostly minimized using frequency domain based techniques, like gain stabilization or phase stabilization [24,27], or time domain based techniques, such as optimal control [28–31]. For this flexure-linked gantry, the assurance of the biaxial synchronization and orthogonality in the flexure-linked gantry requires simultaneous determination of controller parameters and stiffness of the flexure joints.
The biaxial gantry is widely used in many industrial processes that require high precision Cartesian motion. The conventional rigid-link version suffers from breaking down of joints if any de-synchronization between the two carriages occurs. To prevent above potential risk, a flexure-linked biaxial gantry is designed to allow a small rotation angle of the cross-arm. Nevertheless, the chattering of control signals and inappropriate design of the flexure joint will possibly induce resonant modes of the end-effector. Thus, in this work, the design requirements in terms of tracking accuracy, biaxial synchronization, and resonant mode suppression are achieved by integrated optimization of the stiffness of flexures and PID controller parameters for a class of point-to-point reference trajectories with same dynamics but different steps. From here, an $H_{2}$ optimization problem with defined constraints is formulated, and an efficient iterative solver is proposed by hybridizing direct computation of constrained projection gradient and line search of optimal step. Comparative experimental results obtained on the testbed are presented to verify the effectiveness of the proposed method.
Methodology to assess quality of estimated disturbances in active disturbance rejection control structure for mechanical system
2017, ISA Transactions
Citation Excerpt :
In systems with passive compensation of disturbances the bandwidth is low, which does not allow to identify the perturbations in a precise form, in this way a part of the perturbations are introduced in the system and they degrade its performance. On the other hand, in systems with active compensation of disturbances the bandwidth is greater, which allows to estimate more accurately the perturbations and of this way to avoid that they produce a negative effect in the closed loop system [13–16]. The active disturbances rejection control structure (ADRC), shown in Fig. 1, is a typical scheme of a controller with active compensation of disturbances.
A methodology to assess the quality of estimation of disturbances in mechanical systems, by state observers, in the control structure with active compensation of disturbances (ADRC) is presented. Evaluation is carried out by four performance indices that depend on the steady-state error between reference signals and output of the plant. These indices are related with the accuracy and precision of the closed loop system in the sense of norms L₂ and $L_{\infty}$ , for a set of reference signals representing the typical operating conditions of the mechanism. The effectiveness of the methodology is illustrated with the quality assessment of the estimated disturbance of five state observers to control of a simple pendulum and validated on a SCARA robot arm.
New results on H<inf>∞</inf> filtering for Markov jump systems with uncertain transition rates
2017, ISA Transactions
Citation Excerpt :
This shows that there is still room for improvement in the optimization problem of matrix inequality (18). In recent literature [41], a gradient-based optimization algorithm is developed to convert the non-LMI constrained optimization problem to a projected gradient problem, and thus fast convergence of parameters is achieved. Feasibility and optimization problems of matrix inequality (18) may be solved using the gradient-based optimization algorithm, which deserves further investigation.
In this paper, we study the $H_{\infty}$ filtering problem for a class of continuous-time Markov jump systems with time-varying uncertainties in transition rates, in which the uncertain transition rates are assumed to be affine parameter-dependent uncertainty models. By converting the affine parameter-dependent uncertainty models for transition rates into time-varying polytopic ones and using the Lyapunov function approach, a sufficient condition on the existence of an $H_{\infty}$ filter is obtained in terms of a parameter-dependent matrix inequality. Also, the parameter-dependent matrix inequality is converted into a set of parameter-free linear matrix inequalities which can be solved numerically. Illustrative examples are given to demonstrate the effectiveness and advantages of the approach.
Topology and Size-Shape Optimization of an Adaptive Compliant Gripper with High Mechanical Advantage for Grasping Irregular Objects
2019, Robotica
Parameter Space Optimization for Robust Controller Synthesis With Structured Feedback Gain
2023, IEEE Transactions on Cybernetics

View all citing articles on Scopus

View full text

A constrained linear quadratic optimization algorithm toward jerk-decoupling cartridge design

Abstract

Introduction

Section snippets

Modelling of the linear feed drive with JDC configuration

Linear quadratic formulation

Gradient-based constrained optimization algorithm

Simulation

Conclusion

Acknowledgment

CIRP Ann. – Manuf. Technol.

CIRP Ann. – Manuf. Technol.

CIRP Ann. – Manuf. Technol.

CIRP Ann. – Manuf. Technol.

Automatica

Appl. Math. Comput.

Automatica

Automatica

Precision engineeringan evolutionary perspective

Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci.

Precision Motion Control: Design and Implementation

A cross-coupled iterative learning control design for precision motion control

IEEE Trans. Control Syst. Technol.

Adaptive robust repetitive control of an industrial biaxial precision gantry for contouring tasks

IEEE Trans. Control Syst. Technol.

Integrated servo-mechanical design of a fine stage for a coarse/fine dual-stage positioning system

IEEE/ASME Trans. Mechatron.

Vibration control of rotationally periodic structures using passive piezoelectric shunt networks and active compensation

J. Vib. Acoust.

Energy flow in jerk-decoupled translatory feed axes

Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci.