Accurate physical modeling and synchronization control of dual-linear-motor-driven gantry with dynamic load


 To achieve high-accuracy tracking of dual-linear-motor-driven (DLMD) gantry, high-level synchronization between redundant actuators is a nonnegligible factor and also a difficult issue to be solved prior. Especially, when both XY axes are simultaneously operating to accomplish complex tasks efficiently, additional coupling effects will be generated by the dynamic load presented on the crossbeam, which makes the synchronization issue more complicated compared to the case with static load. However, due to the absence of an accurate model to fully reveal the complete coupling characteristics, existing approaches to this issue still have inherent limitations. Therefore, this paper focuses on the systematic physical modeling and synchronization control of DLMD gantry with a dynamic load presented on the crossbeam. A complete coupling mathematical model is established firstly, by fully considering two linear motions (X-axis and Y-axis) and also including the additional rotational motion of the crossbeam. Built upon the effective model information, corresponding solutions by compensating the dynamic load effects and actively controlling the rotational dynamic to regulate the internal forces have been proposed, leading to a novel adaptive robust synchronization control method. Comparative experiments are carried out, and the results show the effectiveness and superiority of the proposed method in dealing with synchronization issue subjected to dynamic load effects.


Introduction
Among the various types of precision cartesian robotic systems [1], H-type gantry [2] with dual-driven structure and direct actuators [3][4][5][6] is widely used in industrial applications, in which, a payload platform is driven by one motor along the crossbeam, while two other parallel arranged motors are rigidly connected to drive the motion of the entire crossbeam. Although the dual-driven structure is used to provide larger thrust and achieve better control performance, more severe requirements of control also arise due to its mechanical coupling. Namely, the motions of two parallel arranged motors are naturally restricted by each other through the coupling bridge. Thus, except for the desired linear motion of this axis, the additional crossbeam rotation will be caused if two motors are not well synchronized. Meanwhile, because of the physical restriction in the linear guide pairs, excessive internal forces might be generated simultaneously. Then, adverse effects, such as additional wear in the guiding parts, deformations or even damage of the hardware, may also be caused. Importantly, the normal operation of such kind of systems may be affected, not to mention high-accuracy tracking. Especially, as a kind of typical multi-axis machines, both axes of the gantry system are usually commanded to operate together to accomplish complex task with high efficiency. However, the movement of the payload platform along the crossbeam will cause more complicated dynamic coupling effects between the motions of two other axes (i.e., the dynamic load effects to be mainly studied in the following), which makes the realization of high-level synchronization more difficult.
To our knowledge, the synchronization control design on DLMD gantry is still far from mature. Part of the reasons might be due to the absence of an accurate dynamical model, leading the uncomplete knowledge of the system. Commonly, the motion of each motor is modeled individually [7][8][9], while the mechanical coupling between Accurate physical modeling and synchronization control of dual-linear-motor-driven gantry with dynamic load ·3· two parallel motors is essentially ignored, which makes the resulting model actually have no guidance meaning on controller development. Other researches on the coupling effects can mainly be concluded into two categories. The first one is the coupling modeling of the entire system [10][11][12][13], in which a three-degree-of-freedom (3DOF) modeling is usually presented, i.e., the linear motions of X-axis and Y-axis and the rotational motion of the crossbeam. Although certain coupling effects can be explained by those models, inherent conflicts between the handing of physical constraints and the generation of rotational motion still exist, leading to the doubts about the correctness of those work. For example, as in [12][13], under the assumption of pure rotational flexible joints, the rotation of the rigid crossbeam essentially can not be generated. The second type is only focusing on the subsystem with dual-driven structure [14][15]. By recognizing the relatively flexible feature of the bearing balls in direct-driven systems and subsequently modeling them by a set of distributed horizontal two-way springs [16], the unbalanced deformations of those components along with additional internal forces can be clearly indicated. Then, the generation of additional rotational motion of the crossbeam and the aforementioned adverse effects are properly explained. However, the movement of the platform on the crossbeam has not been involved, which means the complex coupling between the motions of two axes and the dynamic load effects are not fully considered. Therefore, an accurate dynamic model to reveal the complete properties of such kind of systems is still of great concern.
Limited by the aforementioned situation of modeling, earlier synchronized control schemes are usually based on the pure motion synchronization, e.g., the synchronized master command approach [17][18], the master-slave control [7][8], and the well-known cross-coupled control [8][9], [19]- [21]. Due to the ignorance of the mechanical coupling in dual-driven systems and the following internal forces issue, the achievable synchronization performance is actually limited, which has been detailedly examined in [14]. Meanwhile, as in cross-coupled control, the design of the cross-coupled compensator and the set of the related controller gains have no systematic guidelines, especially for dynamic load condition. Opposite to those control schemes based on pure motion synchronization, Yao's group develops a kind of thrust allocation technique based synchronization method [14], [22], by using the idea of internal forces regulation. Comparative experiments with the previous methods indicate the superiority of the proposed method. Essentially, it provides a simplified static solution to balance the rotational moment of the crossbeam. Thus, the transient performance of internal forces regulation is still limited. To overcome this weakness, a novel synchronization method by directly controlling the rotational dynamics to regulate the internal forces more efficiently has been proposed in [15]. The synchronization performance is further improved, simultaneously indicating the relationship between internal forces and rotational angles. However, similar to most of the existing works, only the simple case with static load is studied. Actually, systematic work on dealing with synchronization issue subjected to dynamic load effects can rarely been seen yet, leading to the urgent demand of further work.
In this paper, to efficiently guide the development of synchronization control scheme, an accurate physical modeling is presented firstly to capture the essential characteristics of DLMD gantry systems. Taking into account two obvious linear motions (X-axis and Y-axis) and the usually ignored rotation of the crossbeam along with the deformations of bearing balls, an integrated 3DOF dynamic model is derived to clearly reveal the essence of dynamic load effects. Built upon those knowledge, following the idea by actively suppressing the rotational dynamics to effectively regulate the internal forces, an novel three-input three-output (TITO) synchronization control method is then proposed. The dynamic load effects and various uncertainties are effectively compensated and denominated by the ARC controller. The proposed method is compared to the existing thrust allocation technique based synchronization methods. Experimental results on a practical DLMD gantry show the significant effect of dynamic load on the synchronization performance, and also verify the performance improvement of our proposed synchronization strategy.

System Description
A general DLMD gantry with X-and Y-axis sub-assemblies usually can be shown in Fig.1. Commonly, to guarantee the repeated positioning accuracy through mechanical structure, the crossbeam of large span and mass is rigidly connected with two parallel motors and sliders. The linear motion of Y-axis is guided by two parallel pairs of ball linear guide rails. Thus, the motions of two parallel motors of Y-axis are not mutually independent, but naturally restricted by each other through the coupling bridge. Meanwhile, a payload platform, i.e., the working head, is driven by another linear motor along the crossbeam to provide the linear motion of X-axis. Despite the high-rigidity achieved by such kind of structures, certain components of relatively flexibility still exist due to various factors, e.g., for the successfully assembling in practice. According to the existing researches [14][15][16], the relatively elastic property of bearing balls is most significant comparing to the other rigid components and connections. Thus, they can be modeled as distributed horizontal two-way springs with large stiffness.  Within this structure, it is usually assumed that the parallelism of two linear guides can be sufficiently guaranteed owing to the high-accuracy assembling in practice. When the crossbeam is orthogonal to the parallel guides of Y-axis, the horizontal two-way springs can be taken as perfectly balanced shown in Fig.1, which is defined as the equilibrium state without additional internal forces in this paper. However, subjected to the physical restriction between the carriages and the guide rails, the bearing balls would be squeezed unevenly when the thrusts of two coupled actuators are not properly allocated, leading to the unbalanced lateral deformations shown in Fig.2. Because of the high stiffness of the bearing balls, excessive internal forces could exhibit, which will cause some adverse effects such as increase of mechanical wear, influence on normal operation and even hardware damage. Meanwhile, the contemporaneous rotation of the crossbeam (i.e., the "pull and drag" phenomena) will also decrease the feed-accuracy of the end-effector, which is usually attached to the working head in practice.
Further, complex coupling effects exist between the working head's linear motion and the crossbeam's rotation. Meanwhile, the physical property of Y-axis will be changed due to the dynamic load presented on the crossbeam. For a short explanation, denote the mass centers of the crossbeam, the working head, and the entire moving body of Y-axis as C1, C2, and C shown in Fig.1. Define the spans of the two motors, encoders, and linear guides in Fig.1  Thus, it is particularly difficult to achieve high-level synchronization of such kind of systems with dynamic load, not to mention high-accuracy tracking performance.
In general, high-accuracy tracking is usually the common requirement of precision systems. However, as analyzed previously, high-level synchronization is the basis to achieve the former goal for DLMD gantry systems. As a result, both high-level synchronization between redundant actuators and high-accuracy tracking should be concurrently considered. Therefore, both objectives are pursued in this paper, where, the synchronization issue is mainly discussed.

Rigid-flexible Modeling
To deal with the complicated synchronization issue with dynamic load, an accurate dynamical model exactly revealing the dynamical properties is a prior basis to which leads to the following velocity vector Thus, the entire kinetic energy can be calculated where, where K  is the equivalent rotational stiffness of the bearing balls modeled in [14], [16], indicating the direct relationship between additional internal forces and rotational angles. Combined (4) with (3), the total energy of the system can be obtained by is the set of generalized coordinates; Mq, Cq and Kq are the matrices of inertia, Coriolis and centrifugal forces, and stiffness; F represents the matrix of generalized forces to be calculated later. Seen from Fig.2, without direct analysis of the unknown disturbance forces, the thrust * m F of each motor and the friction force * r F of each guide pair are the main external forces and can be modeled as where, * u represents the control input of each motor, with * K being the corresponding force constant; * B denotes the coefficient of viscous;   ( ) where,  represents the lumped modeling errors, including the approximation error of Coulomb friction and the previously ignored unknown disturbances.
For simplicity, normalizing (9) with respect to the force constant Apparently, the following properties always hold: (P1) In any finite work space

Assumption and control objectives
where,   Thus, by making e small, the above objectives can be achieved.

Adaptive Robust Synchronization Control
In this section, a control algorithm is presented to achieve high-level synchronization and high-accuracy tracking performance of DLMD gantry systems. The desired compensation technique is applied to form desired compensation integrated direct/indirect adaptive robust control (DCDIARC). The overall block diagram of the control design is shown in Fig.3. With the use of the previously established dynamic model, the unknown parameters will be accurately estimated online to properly compensate the main coupling effects and the parametric uncertainties, while the remaining uncertainties will be dominated by the robust feedback in this framework. The detailed design procedure is as follows.

Rate Limited Projection Type Adaptation Law
To separate the designs of controller and estimator for different purposes, i.e., robust tracking and accurate parameter estimation, the following rate-limited projection structure is used for online adaptation Where  is the adaptation rate matrix; ˆM  is the upper bound for the adaptation rate;  is the adaptation function. The detailed forms of  and  will be introduced later;  Seen from (17) and (18), it is obvious that the adaptation law (16) have the following properties for any  [32]:

DCDIARC
Within the above structure, the controller is designed first. An equivalent switch-function-like quantity is defined to simplify the controller design s e e = +  (19) where  is a 33  positive definite diagonal matrix.
Then, 2 a v and 2 s v are designed to deal with the Accurate physical modeling and synchronization control of dual-linear-motor-driven gantry with dynamic load ·9· remaining lumped model uncertainties in (21) (24) and (25) with the projection adaptation law (16) can always guarantee the boundedness of all signals in the closed-loop system. Furthermore, the Lyapunov function ( ) Vt in (20) can be bounded by  . Thus, the tracking errors are guaranteed for both transient performance and steady-state accuracy.

Parameter Estimation Algorithm
This subsection focuses on the fast and accurate parameter estimation, which may lead to better model compensation in the control design to further improve the performance. Especially, asymptotic tracking in the presence of parametric uncertainties only can be pursued here. To this end, 0 k d = is assumed for the time being.
Thus, (10) can be written as a linear regression form, shown in the following where f • represents the filtered output of signal • .
Then, for the purpose of fast and accurate online adaptation, the recursive least square estimation algorithm [37] with co-variance re-setting and exponential forgetting is used to design the estimator. Within adaptation law (16)   Then, a similar lemma summarizing the properties of the estimator as stated in [38], and certain additional results including asymptotic tracking can be further clarified: Lemma 1. The rate-limited projection type adaptation law (16) with the recursive least square estimator given by (35) and (36) guarantees:   (37) then, in addition to the robust performance given in Theorem 1, the control law (24), (25) combined with the recursive least square estimator (35), (36) (38) which leads to an equivalent resolution of 0.476 rad  for α. A dSPACE controller board is connected to the DLMD gantry. The control algorithms are directly compiled in MATLAB/Simulink and downloaded to the dSPACE for implementation, and the sampling period is set as 0.1ms.

Performance Indexes Notations
The same as [16], the following indexes are used to qualify the performance:  To simulate the dynamic load condition in practice, both XY axes are commanded to moving together along a "V" shape contour as shown in Fig.5. The corresponding desired smooth trajectories of two running periods are given in Fig.6. Seen from Fig.6, yd(t) is a unidirectional point-to-point S-curve with a travel distance of 0.3m, a ·11· maximum velocity of 0.6m/s, and a maximum acceleration of 6m/s 2 ; while xd(t) is a bidirectional point-to-point S-curve with a travel distance of 0.25m, a maximum velocity of 0.5m/s and a maximum acceleration of 5m/s 2 .

Figure. 5 "V" shape contour
To clearly illustrate the coupling effects under dynamic load condition and simultaneously verify the effectiveness and performance improvement of our proposed design, the following three controllers are compared:  C2: The static thrust allocation based synchronization control method proposed in [14] and also designed via DCDIARC. In this controller, the motor force of X-axis, i.e., C3: The dynamic thrust allocation based synchronization control method proposed in [22] and also designed via DCDIARC. The controller structure is almost the same as C2. The only difference is the thrust allocation coefficients     Accurate physical modeling and synchronization control of dual-linear-motor-driven gantry with dynamic load ·13· 7- Fig.10. Their performance indexes are calculated in Table  2. First, the rotation angles in Fig.9 clearly shows that C2 achieves the worst synchronization performance, i.e., significant rotational oscillations appear during moments A and B. Seen from Table II, the maximum oscillated amplitude of rotational angle is 34.05 rad  .
Simultaneously, additional control efforts need to be applied to deal with the subsequent excessive internal forces. Thus, under the same condition in set1, corresponding oscillations occur on the control inputs of C2 among three controllers as shown in Fig.10, leading to the maximum control effort 2 y Lu   in Table II. Thus, the direct relationship between the rotational angles and internal forces is clearly indicated. Meanwhile, the dynamic load effects have also been verified. Due to the movement of the working head, high-level synchronization between two parallel motors of Y-axis obviously cannot be achieved by static thrust allocation. Especially, due to the complicated coupling, the tracking performance of C2 is also severely affected. Seen from Fig.8, corresponding oscillated tracking errors have been caused during moments A and B. Oppositely, C3 can achieve better synchronization performance than C2 due to the online computation of thrust allocation coefficients according to the feedback signal of x As seen from Fig.9, the rotational angles of C3 can be suppressed within a small level to regulate the internal forces efficiently during the whole running periods. Correspondingly, seen from Fig.7, Fig.8 and Fig.10, the oscillations appearing when controlled by C2 have been eliminated by C3 to achieve better tracking performance and smoother control inputs. Although the dynamic load effects have been partly compensated by C3 through dynamic thrust allocation technique, the transient performance of the rotational dynamics and internal forces regulation can not be fully guaranteed due to the open-loop treatment of rotational dynamics.
Based on the above analyses, it can be predicted that the proposed controller C1 could achieve best synchronization performance by its direct control of rotational dynamics. Seen from Fig.9, the rotational angles of C1 are highly suppressed during the whole history. As seen from Table II under Set1, which is much better than that of C2 and clearly reflects the importance of internal forces regulation to guarantee high-accuracy tracking of DLMD gantry systems.

Figure 11 Parameter estimation history
On the other hand, combined with the parameter estimation histories in Fig.11 with Fig.7-Fig.8, it is seen that the initial tracking error and rotational angle are always large due to the poor model compensation at the beginning. However, owing to the fast and accurate online adaptation ability of DCDIARC, even under the heavy load condition, i.e., Set2, the parameter estimates can converge quickly to provide better model compensation for effectively dealing with parametric uncertainties and dynamic coupling effects. Therefore, C1 can still maintain excellent control performance under Set2, i.e., , which is very close to the results of C1 under Set1. Thus, the superiority of our proposed design can be verified. By using accurate online adaptation and active control the rotation dynamics to deal with dynamic load effects, both high-accuracy tracking of two axes and high-level synchronization between two parallel motors, i.e., strong suppression of the rotational angles to avoid large internal forces, can be simultaneously achieved.

Conclusion
In this paper, an accurate 3DOF physical model is

·14·
presented for general DLMD gantry with dynamic load, where both the rigid and flexible dynamics are considered, i.e., two linear motions (X-axis and Y-axis) and the crossbeam's rotation. The presented model clearly shows the generation of the rotational motion along with excessive internal forces, the dynamic load effects and the complicated coupling properties in this system. Built upon those knowledge, an novel TITO synchronization control method is then proposed by directly controlling the linear dynamics for high-accuracy tracking of both axes, and actively suppressing the rotational dynamic to regulate the internal forces for high-level synchronization between two parallel motors. Meanwhile, the DCDIARC algorithm is applied to deal with the parametric uncertainties and uncertain nonlinearities and also compensate the dynamic load effects. Systematic design procedure of controller with detailed stability proof has been clearly given. Comparative experiments are carried out and the results show that both excellent tracking of the linear motions of two axes and high rotational suppression of the crossbeam are realized by our proposed method.