# Experimental Investigation of Free-Motion Task Implementation on a Serial Metamorphic Manipulator
## 2 Preliminaries and Methodology  
### Dynamic Modeling of SMMs  
In this work, the terminology established in previous studies was utilized [28; 29]. The robot assembly, comprising both active and passive modules interconnected according to specific interconnection rules, is referred to as the _metamorphic structure_. Passive modules employed for reconfiguration are designated as _pseudojoints_, and the process of altering the pseudojoints' configuration is termed _metamorphosis_. The resultant robot configuration following metamorphosis is known as _anatomy_. A metamorphic structure can encompass a large number of anatomies, contingent on the number of pseudojoints used to form the serial chain. The formulation of the dynamic model was implemented through screw theory tools. A simplified model was extracted, since it helped isolate core dynamics, making it easier to understand the fundamental behavior of the system before more detailed models were introduced for fine-tuning and precision. Dynamic equations were defined in a compact analytical form that was aligned with the physical structure assembly, allowing the mechanics of reconfiguration to be parameterized and incorporated into the dynamic matrices' expressions without additional complexity [30]. A brief presentation of the derivation of the analytical dynamic model for SMMs is provided, mainly focused on the Generalized Inertia Matrix (GIM). For a robot with \(n_{a}\) active joints, the GIM is an \(n_{a}\times n_{a}\) symmetric, positive definite matrix that relates to the current active joint configuration, \(\mathbf{q_{a}}\), anatomy, \(\mathbf{q_{p}}\), and metamorphic structure, \(\mathbf{s}\). The detailed presentation of the structural parameters for the SMM prototype under study, are thoroughly discussed in [30]. The GIM of an SMM has the form  
\[\mathcal{M}(\mathbf{q_{a}},\mathbf{q_{p}},\mathbf{s})=\sum_{n=1}^{n_{e}}\mathbf{ J_{n}^{\top}}\cdot\mathbf{M_{n}^{b}}\cdot\mathbf{J_{n}^{b}} \tag{1}\]  
where \(\mathbf{J_{n}^{b}}\) are the Link Body Jacobians (LBJs) and \(\mathbf{M_{n}^{b}}\) are the Link Inertia Matrices (LIMs) with respect to each link's center of mass (CM). Both LBJs and LIMs should be constructed for the current SMM metamorphic structure and anatomy. In Figure 1, a metamorphic link \(n\) is illustrated along with the main geometric components used for analytical calculations. The link consists of the intermediate bodies (_pseudojoints_) between two active modules. The inertia matrix \(\mathbf{M_{m}^{b}}\) of the \(m\) intermediate bodies expressed in the local reference frame is predetermined from Computer-Aided Design (CAD) software FreeCAD 1.0. In order to extract the LIM of a metamorphic link, the body inertia matrices are expressed in the base frame \(\{S\}\). \[\mathbf{M_{m}^{s}}=Ad_{g_{sl_{m}}^{-1}}^{\top}\cdot\mathbf{M_{m}^{b}}\cdot Ad_{ g_{sl_{m}}^{-1}} \tag{2}\]  
Then, the LIM of a metamorphic link \(n\) with \(\{N_{m}\}\) intermediate bodies, expressed in the base frame \(\{S\}\), can be extracted  
\[\mathbf{M_{n}^{s}}=\sum_{m=1}^{\mathbf{N_{m}}}\mathbf{M_{m}^{s}} \tag{3}\]  
and is expressed in the CM frame of each link \(\{l_{n}\}\) given its spatial transformation \(g_{sl_{n}}\)  
\[\mathbf{M_{n}^{b}}=Ad_{g_{sl_{n}}}^{\top}\cdot\mathbf{M_{n}^{s}}\cdot Ad_{g_{sl _{n}}} \tag{4}\]  
The LBJ of the \(n\)th link is given by  
\[ \begin{split}\mathbf{J_{n}^{b}}&=\left[\mathcal{J}_ {1}^{(n)}\ \dots\ \mathcal{J}_{n-1}{}^{(n)}\ 0\ \dots\ 0\right]\\ &=Ad_{g_{sl_{n}}^{-1}}\cdot\left[\mathcal{A}_{n1}\sharp_{a1}^{g_{ s1}}\ \dots\ \mathcal{A}_{nm}\sharp_{a1}^{g_{s}}\ 0\ \dots\ 0\right]\end{split}  \tag{5}\]  
where the adjoint transformation \(\mathcal{A}_{nc}\in\mathbb{R}^{6\times 6}\) is utilized to form the \(c\)th Jacobian column of the \(n\)th link. \[\mathcal{A}_{nc}= \begin{cases}Ad_{g_{sl_{n}}^{-1}}^{-1}\cdot g_{sl_{n}+1}\cdot g _{sl_{n}+2}^{g_{sl_{n}+2}}\dots\cdot e^{g_{sl_{n}}q_{cn}}&n>c\\ I&n=c\\ 0&n<c\end{cases}