## 1 Introduction  
One of the key technical bottlenecks in achieving ultra-precision machining is how to achieve accurate, stable, and reliable micro displacement of the tool or workpiece during the machining process [1]. For most precision and ultra-precision machining machines, a high-performance linear motion system is necessary and critical [2]. The linear motor-drive system does not require any intermediate mechanical transmission mechanism, and the linear motor directly provides thrust to the table, eliminating the consumption caused by the transmission mechanism and the limitation of transmission clearance, achieving \"zero transmission\" from the motor to the table [3]. Moreover, it has the characteristics of fast response speed, small electrical time constant, high thrust, and low loss, which can provide high dynamic response speed and acceleration as well as extremely high stiffness, high positioning accuracy, smooth and error-free motion, but cannot eliminate nonlinear friction [4]. Therefore, the low-speed linear motion of the table relative to the guide rail has become the main factor restricting the improvement of feed accuracy in CNC machine tools. Based on the differential motion synthesis mechanism of dual linear motors and modern servo drive control technology, a micro-feed servo system with dual linear-motor differential drives is proposed. By superimposing two quasi-equal macroscopic movements, namely \"the upper linear motor drives the upper table to move in a straight line\" and \"the under linear motor drives the under table to move in a straight line\", which have almost equal instantaneous velocities and opposite directions, the inevitable low-speed nonlinear crawling phenomenon caused by the inherent properties of the traditional electromechanical servo system structure is avoided, and high-precision differential micro-feed control is achieved, fundamentally eliminate the creeping of linear motion in micro-feed mechanisms. Many scholars use the lumped parameter method to model electromechanical transmission systems [5; 6; 7; 8; 9; 10; 11; 12]. [PERSON], [PERSON] [5] established the mathematical model of a permanent magnet linear synchronous motor (PMLSM) with uncertainties, that is, a second-order complementary sliding mode control (SOCSMC) with fast convergence and global robustness, to conquer the uncertainties and reduce chattering. [PERSON] [6] presented a comprehensive thermal model based on the lumped parameter approach for STLSRM, which predicts temperature distribution in different parts of this motor, including slot winding, end-winding, stator pole, stator yoke, and the moving part. [PERSON] [7] aims to review analytical methodologies, i.e., lumped parameter magnetic equivalent circuit (LPMEC), magnetic co-energy (MCE), Laplace equations (LE), Maxwell stress tensor (MST) method, and sub-domain modeling for the design of a segmented PM (SPM) consequent pole flux switching machine (SPMCPFSM). [PERSON] [8] proposes a fast-dynamic model, based on a model order reduction method, to control a permanent magnet synchronous machine at a wide range of speeds. The robustness of the control is observed particularly when the linear lumped parameter-based models are employed to represent a machine composed of nonlinear magnetic materials. [PERSON] [9] combined the merits of PMLFSM and FELFSM by proposing a novel Hybrid Excited LFSM (HELFSM), the proposed machine is excited by PMs, Field Excitation Coils (FECs), and Armature Windings (AWs). [PERSON] [10] designed a two-pole, three-phase, 7.5-kW line-start permanent magnet (LSPM) synchronous motor, which is an analytical technique for rotor geometry optimization based on a lumped magnetic parametric approach. For the palletizing robot's operating characteristics of high speed, high acceleration, and heavy load, [PERSON] [11] carried out kinematics analysis by using the lumped parameter method, which obtained a positive kinematics solution and workspace.