A powerful mathematical foundation for the study, modelling, and control of robotic systems is provided by differential geometry, which is a key component of the field of robotics and control systems. This study examines differential geometry's use in robotics and control systems, emphasising its value in solving a number of problems that arise in these fields. In the introduction, the basic ideas of differential geometry, such as manifolds, tangent spaces, and differential forms, are briefly discussed. In order to create kinematic models that accurately depict the relationship between the robot's joint angles and the pose of its end-effector, it then investigates how these notions are used to define the configuration space of robotic systems. Differential geometry offers a simple framework for comprehending the topology and geometry of the robot's configuration space, enabling motion planning, workspace characterisation, and singularity analysis. The paper investigates the use of differential geometry in sensor integration, perception, and estimation issues in robotics, in addition to kinematics and dynamics. Tools for analysing sensor measurements on curved manifolds, such as vision data on a sphere or range measurements on a non-Euclidean surface, are provided by differential geometry. Because of this, it is now possible to create reliable perception algorithms and state estimate methods for robotic systems that operate in complicated situations. The study also examines recent developments in differential-geometric control theory, including motion planning on Lie groups, optimal control, and geometric control. Due to these advancements, differential geometry is now more widely applicable in robotics, allowing for the creation of complex control schemes that take into account the system's fundamental geometric structure.
Key words: Geometric structure, control theory, robotic system, differentiation, mathematical foundation
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