# A Method of Estimating an Object's Parameters Based on Simplication with Momentum for a Manipulator
## 2 System Modeling  
Assuming the robot securely grasps an object with its gripper installed at its end-effector to facilitate smooth motions, as shown in Figure 1, coordinate frames \(\{O\}\), \(\{R\}\), and \(\{G\}\) represent the robot's base, the robot's end-effector, and the center of mass (COM) of the object, respectively. The force and torque applied at the robot's end-effector will affect the motion of both the gripper and the object. By utilizing the specifications or CAD data of the gripper provided by the manufacturer, the parameters of the gripper can be predetermined, thereby enabling calculation of the parameters of the object.  
With \(\{R\}\) and \(\{G\}\) fixed to the object and the gripper securely attached to the robot's end-effector, a constant relative kinematic relationship between \(\{R\}\) and \(\{G\}\) is maintained throughout the parameter estimation process, ensuring their parallel alignment. The object has a mass denoted by \(m\), and the robot exerts forces (\(\mathbf{f}_{R}\)) and torques (\(\boldsymbol{\tau}_{R}\)) on it. Given this grip, the positions of the end-effector and the object's center of mass (COM) are represented by \(\mathbf{r}_{R}\) and \(\mathbf{r}_{G}\), respectively, while the linear acceleration of the COM is denoted by \(\mathbf{a}_{G}\). The object's dynamics can then be described by  
\[\mathbf{f}_{R}=m(\mathbf{a}_{G}-\mathbf{g}) \tag{1}\] \[\boldsymbol{\tau}_{R,b}+\mathbf{r}_{G/R,b}\times\mathbf{f}_{R,b}= \mathbf{I}_{G,b}\boldsymbol{\alpha}_{b}+\boldsymbol{\omega}_{b}\times( \mathbf{I}_{G,b}\boldsymbol{\omega}_{b}), \tag{2}\]  
where subscript \(b\) denotes that the corresponding vector is expressed using the body coordinate frames attached to the object, \(\mathbf{g}=\begin{bmatrix}0&0&-1\end{bmatrix}^{T}g\) denotes the gravitational acceleration vector with gravitational acceleration \(g\), and \(\mathbf{r}_{G/R}=\mathbf{r}_{G}-\mathbf{r}_{R}\). \(\mathbf{I}_{G,b}\) represents the inertia tensor matrix measured with the coordinate frames \(\{G\}\), defined by  
\[\mathbf{I}_{G,b}=\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\ I_{xy}&I_{yy}&I_{yz}\\ I_{xz}&I_{yz}&I_{zz}\end{bmatrix}. \tag{3}\]  
Figure 1: Free-body diagram of a carried object.  
Furthermore, \(\mathbf{\alpha}\) and \(\mathbf{\omega}\) denote the angular acceleration and velocity of the object, respectively.  
The position of COM, \(\mathbf{r}_{G}\), can be expressed as  
\[\mathbf{r}_{G}=\mathbf{r}_{R}+\mathbf{R}\mathbf{r}_{G/R,b}, \tag{4}\]  
where \(\mathbf{R}\) represents the rotation matrix from \(\{G\}\) to \(\{O\}\). By differentiating Equation (4) with respect to time,  
\[\mathbf{v}_{G}\triangleq\dot{\mathbf{r}}_{G}=\mathbf{v}_{R}+\mathbf{\omega}\times(\mathbf{R}\mathbf{ r}_{G/R,b}). \tag{5}\]  
By further differentiation of Equation (5),  
\[\begin{split}\mathbf{a}_{G}&\triangleq\dot{\mathbf{v}}_{G} \\ &=\mathbf{a}_{R}+\mathbf{\alpha}\times(\mathbf{R}\mathbf{r}_{G/R,b})+\mathbf{\omega} \times\Big{(}\mathbf{\omega}\times(\mathbf{R}\mathbf{r}_{G/R,b})\Big{)},\end{split} \tag{6}\]  
where \(\mathbf{v}_{R}\triangleq\dot{\mathbf{r}}_{R}\) and \(\mathbf{a}_{R}\triangleq\dot{\mathbf{v}}_{R}\).