This paper addresses the problem of synthesizing four-link initial kinematic chains with rotational pairs for designing flat lever mechanisms with specified motion laws of the input and output links. The proposed method formulates the synthesis problem as an optimization process involving three moving planes Q, Q1 and Q2. Each plane performs distinct motions: Q is fixed, Q1 rotates around a point A, and Q2 undergoes plane-parallel motion. The goal is to determine the optimal position of point A∈Q, B∈Q1 and C∈Q2 such that the distance between points B and C remains nearly constant across multiple configurations. The synthesis problem is reduced to a sequence of linear systems of equations corresponding to three iterative minimization stages: Determining the fixed-point A and the radius R for given points B and C; Determining the point B and radius R for a given point C; Determining the point C and radius R for a given point B. The algorithm uses a hierarchical optimization process to iteratively refine the positions of A, B, and C until convergence is achieved. The convergence criteria are defined by the small changes in the positions of these points and the radius R between iterations. The proposed method avoids solving a highly nonlinear system directly by decomposing the problem into simpler linear systems, ensuring computational efficiency and robustness. The solution provides the desired positions of the points, forming an open kinematic chain ABCD with three degrees of freedom. The approach is applicable to the synthesis of complex planar mechanisms with unrestricted parameters, making it a valuable tool for mechanism design.

Contemporary energy storage deployments increasingly incorporate heterogeneous cell chemistries within a single installation, yet most commercial battery management circuits remain optimized for a single electrochemical type. This paper describes a microprocessor-controlled platform capable of autonomous diagnostics and charge equalization across four battery technologies: lithium-ion NMC, lithium iron phosphate (LiFePO₄), nickel-metal hydride (NiMH), and lead-acid. The measurement front-end is a 16-bit delta-sigma converter delivering 46 μV resolution and ±1.4 mV per-cell accuracy, interfaced to an STM32F446RE ARM Cortex-M4 processor at 180 MHz. Equalization combines resistive dissipation for voltage spreads below 50 mV with a switched-capacitor network achieving above 90 % energy-transfer efficiency for larger disparities. State-of-charge tracking couples’ coulomb integration with an adaptive extended Kalman observer, maintaining accuracy within ±3 % even from a 40 % seed-error initialization. Benchtop trials on four-cell series packs of each chemistry recorded inter-cell convergence to ±5 mV within 22 min, SOC root-mean-square error below 3 %, and correct automatic chemistry classification across all laboratory sessions. The novelty of the proposed system is not in the individual use of battery diagnostics or cell balancing, which are well-established topics, but in their integration into a single microprocessor-based multi-chemistry platform capable of automatic electrochemical-type identification, chemistry-specific parameter loading, adaptive SOC/SOH estimation, and hybrid passive-active balancing. This distinguishes the proposed approach from conventional BMS solutions optimized for only one battery chemistry.