# Inertial Parameter Identification in Robotics: A Survey
## 2 Robot Modeling and Control in the Context of Parameter Identification  
### Inverse and Direct Dynamic Models  
The inverse dynamic model (IDM) of a rigid serial robot manipulator with \(n\)-degrees-of-freedom relates the joint-space motion quantities \(\mathbf{q},\mathbf{\dot{q}},\mathbf{\ddot{q}}\in\mathbb{R}^{n}\), to the generalized forces \(\mathbf{\tau}_{idm}\in\mathbb{R}^{n}\) applied to the system. The IDM can be derived using the Euler-Lagrange formalism (c.f. [36]), resulting in the following equation  
\[\mathbf{M}(\mathbf{\chi},\mathbf{q})\mathbf{\ddot{q}}+C(\mathbf{\chi},\mathbf{q},\mathbf{\dot{q}})\mathbf{\dot{q }}+g(\mathbf{\chi},\mathbf{q})+\mathbf{\zeta}(\mathbf{\chi},\mathbf{\dot{q}})=\mathbf{\tau}_{idm}\,. \tag{1}\]  
where \(\mathbf{M}(\mathbf{\chi},\mathbf{q})\in\mathbb{R}^{n\times n}\) denotes the generalized inertia matrix, \(C(\mathbf{\chi},\mathbf{\dot{q}},\mathbf{\dot{q}})\in\mathbb{R}^{n\times n}\) the Coriolis and centripetal effects matrix, \(g(\mathbf{\chi},\mathbf{q})\in\mathbb{R}^{n}\) the gravitational torque vector, and \(\mathbf{\zeta}(\mathbf{\dot{q}})\in\mathbb{R}^{n}\) the friction vector. This equation is parameterized by the vector \(\mathbf{\chi}=[\mathbf{\chi}_{1}^{\top}\mathbf{\chi}_{2}^{\top}\cdots\mathbf{\chi}_{n}^{\top} ]^{\top}\in\mathbb{R}^{p}\) concatenating the standard dynamic parameters of each robot link \(j\), expressed as  
\[\mathbf{\chi}_{j}= \begin{bmatrix}XX_{j},&[PERSON]_{j},&XZ_{j},&[PERSON]_{j},&YZ_{j},&ZZ_{j},&MX_{j}, &MY_{j},&MZ_{j},&Ia_{j},&Fv_{j},&Fc_{j}\end{bmatrix} ^{\top}\,, \tag{2}\]  
where \(XX_{j},XY_{j},[PERSON]_{j},YY_{j},YZ_{j},[PERSON]_{j}\) are the elements of the inertia tensor \(\mathbf{L}_{j}\) of link \(j\), expressed at the link origin \(\mathcal{L}_{j}\), \(Ia_{j}\) refers to inertia of the actuator and transmission system, \(M_{j}\) is the link mass, \(X_{j}\), \(Y_{j},Z_{j}\) are the coordinates of the link Center-of-Mass in \(\mathcal{L}_{j}\) and \(MX_{j},MY_{j},&MZ_{j}\) are the corresponding first moments (c.f. [37]). Let \(\mathbf{h}(\mathbf{\chi},\mathbf{q},\mathbf{\dot{q}})\in\mathbb{R}^{n}\) be the vector that regroups the Coriolis, centripetal, gravitational and frictional effects, namely \(\mathbf{h}(\mathbf{\chi},\mathbf{q},\mathbf{\dot{q}})=\mathbf{C}(\mathbf{\chi},\mathbf{q},\mathbf{\dot{q}})\mathbf{ \dot{q}}+\mathbf{\dot{g}}(\mathbf{\chi},\mathbf{q})+\mathbf{\dot{\zeta}}(\mathbf{\chi},\mathbf{\dot{q}})\). The direct dynamic model (DDM), allowing calculation of the joint accelerations \(\mathbf{\ddot{q}}\) as a function of the joint positions \(\mathbf{q}\), joint velocities \(\mathbf{\dot{q}}\), generalized forces \(\mathbf{\tau}_{idm}\) and of the vector \(\mathbf{\chi}\) of dynamic parameters, can then be written as  
\[\mathbf{\ddot{q}}=\mathbf{M}^{-1}(\mathbf{\chi},\mathbf{q})(\mathbf{\tau}_{idm}-\mathbf{h}(\mathbf{\chi}, \mathbf{q},\mathbf{\dot{q}}))\,. \tag{3}\]  
From Equation (3), is it clear that the DDM is nonlinear with respect to the dynamic parameters and the robot's state vector defined as \(\mathbf{x}=[\mathbf{\dot{q}}^{\top}\mathbf{q}^{\top}]^{\top}\in\mathbb{R}^{2n}\). By contrast, the IDM has the interesting property of being linear in \(\mathbf{\chi}\), and can thus be reformulated as  
\[\mathbf{\tau}_{idm}=\mathbf{Y}_{\mathbf{\chi}}(\mathbf{\ddot{q}},\mathbf{\dot{q}},\mathbf{q})\mathbf{\chi}\,, \tag{4}\]  
where \(\mathbf{Y}_{\mathbf{\chi}}(\mathbf{\ddot{q}},\mathbf{\dot{q}},\mathbf{q})=\partial\mathbf{\tau}_{idm}/ \partial\mathbf{\chi}\in\mathbb{R}^{n\times p}\) denotes the closed-form expression of the Jacobian matrix of \(\mathbf{\tau}_{idm}\) with respect to \(\mathbf{\chi}\), often referred to as the _model regressor_. It is worth noting that the equality in Equation (4) only holds provided that the friction term \(\mathbf{\zeta}(\mathbf{\chi},\mathbf{\dot{q}})\) is also linear with respect to the parameter vector (in case this is not verified (4) is only a first-order approximation of a potentially much more complex system.)