## 2 Dynamic Modeling and Linearization
### Construction of Minimal Parameter Set and Linear Regression Expression  
In robotic manipulator dynamic modeling, the original parameterized model often contains a large number of redundant parameters, such as masses, moments of inertia, and center-of-mass positions. This redundancy can lead to non-uniqueness of the solution or reduced identification accuracy. To improve the observability of the model and enhance  
Figure 1: Schematic diagram of the manipulator links. (**a**) SolidWorks model of the six-degree-of-freedom robotic manipulator. (**b**) Definition of dynamic and kinematic parameters on a single link.  
the numerical stability of the identification system, the dynamic model is typically reconstructed into a Minimal Parameter Set (MPS) form and transformed into a standard linear regression structure, thereby enabling structure simplification for identification purposes.  
In Equation (6), terms \(M(q)\), \(C(q,\dot{q})\), and \(G(q)\) are all nonlinear functions with parameters embedded within the expressions, making direct identification difficult. To facilitate a structure suitable for least squares estimation, the model must be linearized and expressed in the form shown in Equation (7):  
\[\tau=Y(x)\cdot P+\varepsilon \tag{7}\]  
Here, \(x=\left\{q,\dot{q},\ddot{q}\right\}\) represents the state vector of the manipulator at the identification instant, \(q\) denotes the joint position vector, \(\dot{q}\) denotes the joint velocity vector, and \(\ddot{q}\) denotes the joint acceleration vector. These state variables determine the instantaneous dynamic behavior of all physical quantities in the Newton-Euler formulation. Through algebraic transformation, the mass, inertia, and center-of-mass parameters are combined with structural constants into a minimal identifiable parameter set \(P\). A linear regression matrix \(Y(x)\) is constructed accordingly, and \(\varepsilon\) represents residual terms such as unmodeled disturbances or sensor noise.  
From Equation (7), the linear regression form can be derived. To extract an explicit linear expression from the nonlinear dynamic model, an accurate regression matrix \(Y\) is constructed, with its elements being functions of the state variables. These include joint angles, joint velocities, joint accelerations, and combination functions, which are related to the system's inertia, Coriolis forces, and gravity terms. Based on the rigid-body dynamic characteristics of the robotic manipulator links, the minimal parameter set \(P\) mainly includes:  
\[P=\left[m_{1},m_{1}r_{1x},m_{1}r_{1y},m_{1}r_{1z},I_{1,xx},I_{1,yy},I_{1,zz},f _{01},f_{c1},\ldots,m_{6},\ldots,f_{c6}\right]^{T} \tag{8}\]  
It mainly includes mass terms \(m_{i}\), center-of-mass position terms \(m_{1}r_{1y}\), moment of inertia terms \(I_{1,xx}\), and friction parameter terms \(f_{c1}\)[18]. To accommodate the structural identification requirements of a six-degree-of-freedom manipulator, the state variables \(x=\left\{q,\dot{q},\ddot{q}\right\}\) are mapped into multiple dimensions of the regression matrix, as expressed in Equation (9):  
\[Y=\begin{bmatrix}\ddot{q}_{1}&\dot{q}_{1}&g&\ddot{q}_{1}\dot{q}_{1}&\cos(q_{1} )&\ldots\\ \ddot{q}_{2}&\dot{q}_{2}&g&\ddot{q}_{2}\dot{q}_{2}&\cos(q_{2})&\ldots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \ddot{q}_{6}&\dot{q}_{6}&g&\ddot{q}_{6}\dot{q}_{6}&\cos(q_{6})&\ldots\\ \end{bmatrix}_{6\times m} \tag{9}\]  
Since \(Y\) contains certain columns that are linearly dependent, it is ill-conditioned and cannot be directly estimated using the least squares method. Therefore, QR decomposition is applied to transform it into a full-rank form [18], as shown in Equations (10) and (11):  
\[Y_{min}=Y\cdot P_{b} \tag{10}\]  
\[P_{b}=K\cdot P \tag{11}\]  
Here, \(K\) is the parameter transformation matrix, which extracts the unique inertial subspace. After transformation, \(Y_{min}\) becomes the identification matrix with maximal observability.