# A Method of Estimating an Object's Parameters Based on Simplication with Momentum for a Manipulator
## 1 Introduction  
Numerous advanced control strategies are built upon a dynamic model as their foundation. In robotic object manipulation, accurately determining object-specific parameters--such as mass, center of mass position, and elements of the inertia tensor--enables the robot to perform more precise and efficient manipulation motions [1]. For consistent objects, these parameters can be obtained through iterative manipulation or predefined data provided by an operator. However, when dealing with diverse objects, the robot must estimate these parameters iteratively for each new object it encounters. The development of strategies for robots to estimate inertial parameters has been widely studied, as summarized in a comprehensive survey [2], which classifies existing approaches into three main categories. The first approach estimates parameters using visual information and potential correlations between visual and inertial properties, often assuming uniform object density [3, 4, 5, 6]. This method eliminates the risk of damaging the object during estimation. However, it has inherent limitations, as it relies on pre-established relationships between visual and inertial properties. This makes it unsuitable for estimating the parameters of objects with varying internal contents, such as a box containing different items. The second approach involves instructing the robot to gently prod or nudge the object, and then calculating its parameters based on the forces applied by the robot andthe resulting reactions from the object [7; 8; 9; 10]. However, this technique requires strong assumptions about the robot's environment and precise control over the interaction process. For example, when a robot pushes an object on a table and measures its motion, it must have accurate knowledge of the friction coefficients between the object and the table, as well as between the pushing mechanism and the object. Additionally, it must ensure no slippage occurs between the robot's end-effector and the object's surface. The third approach involves securely attaching the object to the robot, initiating motion, and subsequently estimating the object's parameters using end-effector wrenches and object motion based on its dynamics [11; 12]. This method is relatively straightforward, as it requires only a secure connection between the robot and the object, achieved through a firm grasp or other suitable means. Numerous studies have explored this approach using various techniques. For instance, the authors of [13] estimated object parameters attached to a robot's end-effector using an offline least squares method, similar to robot dynamics estimation [14; 15; 16; 17], since the number of equations describing the object's dynamics is smaller than the number of parameters to be estimated. The offline least squares method is simple and independent of initial conditions. However, it cannot adapt to parameter changes occurring during the process. To address this limitation, researchers have developed online estimation strategies that can accommodate parameter variations during tasks, utilizing recursive least squares [18; 19; 20; 21] or extended Kalman filters [22; 23]. While online methods can adjust to parameter changes in real time, improper selection of initial conditions may lead to inaccurate estimation results. Recent advancements have incorporated visual data [24], generalized momentum observers with filters [25], and machine learning techniques [26; 27; 28; 29] to improve estimation performance. Parameter estimation methods have also been applied in vehicle research to enhance electronic control systems, contributing to improved vehicle safety and passenger comfort [30; 31; 32; 33]. This highlights the broad applicability of parameter estimation, extending beyond robotics into various other domains. However, these approaches face specific challenges due to the underconstrained nature of the problem, where the number of parameters to be estimated exceeds the available dynamic equations.