## 1 Introduction
By using techniques such as geometric and topological reasoning and even fibered logical spaces for logical reasoning, mathematical modeling used in robotics shares connections with artificial intelligence. These multidisciplinary techniques, in turn, extend the interaction between AI and symbolic mathematical computation while integrating with robotic systems [9]. The equations containing geometric and nonholonomic connections that describe the motion of coupled body systems, Lagrange multipliers offer a way to account for these relationships that helps with the development of robotic control systems for cyclic motions [10]. For robot manipulators, Bound Model Predictive Control (BoundMPC) is a path planning algorithm for robots that optimizes an objective over a planned-ahead horizon based on input/output constraints such as joint limits and collisions. BoundMPC is suitable for adaptive constraint-obeying motion planning based on state-space modeling and error decomposition of path following and is therefore appropriate for use in industries where efficiency and security are required [11]. Dynamic Movement Primitives (DMPs) accommodate point-to-point and cyclic movements and reconstruct complex biological and robot motions through the assistance of differential equations. Quaternion-based DMPs allow orientation changes without singularities. Movement adaptation is facilitated through force feedback and reinforcement learning-based DMPs, as well as through Riemannian extensions to geometry that facilitate object contact and movement into the complex geometric space [12]. An end-to-end control structure for underwater vehicles was presented to avoid interruptions while station-keeping in wave-dominated seas. The system forecasts wave-induced forces and adapts control actions via the combination of a deterministic sea wave predictor (DSWP) and a nonlinear model predictive controller (NMPC). Based on experimental studies, the method improves autonomous vehicle behavior in adverse sea conditions and is robust to noisy predictions and delays [13]. A new wheel motion generator to track the centroidal motion of a quadruped-on-wheel robot, which can cross various rough terrains using model-based whole-body torque control, was proposed. The wheel contact model and the whole-body inverse kinematics model are derived using spatial vectors. The wheel motion is extracted out mathematically depending on the base and the legged motions, which serve as the kinematics model. The wheel motion generator combines both the kinematics model and the robot centroidal momentum/dynamics model. The models are decomposed into three components related to the base motion, the legged motion, and the wheel motion [14]. A horizontal-stability control framework was designed for a wheel-legged hybrid robot to ensure the stability and horizontal orientation of the robot's trunk when encountering unknown and rough terrain conditions. This framework primarily comprises a compliance controller and a terrain adaptation controller. The compliance controller is geared towards establishing compliant interactions with the terrain and tracking the desired ground reaction forces. This is achieved through the implementation of a novel adaptive impedance control method to uphold torque equilibrium in the robot's trunk [15]. Deep learning improves robot arm motion planning for e-commerce picking by creating coarse motions as a warm start for an optimizing motion planner. Unlike existing motion planning approaches, this solution improves warehouse automation efficiency by lowering calculation time drastically without sacrificing kinematically and dynamically feasible motions [16]. A memristor-based hybrid analog-digital computation system improves the robot's energy consumption and speed as it processes input of high frequencies, while decision-making is taken care of by digital.