## 3 Natural Coordinates Formulation (NCF)
### Modeling Rigid Body Dynamics with NCF  
The first step involves selecting specific points and vectors on the rigid body that uniquely define its position and orientation in space. For instance, two points could determine a line segment representing a rigid link's orientation and length, while an additional non-collinear point could establish the body's plane. Figure 2: Elephant trunk with longitudinal muscles [15]. An elephant head, which is rigidly connected to the trunk, is regarded as a rigid body that can be described and formulated via the Natural Coordinate Formulation (NCF). The NCF is a method used in the kinematic and dynamic analysis of rigid multibody systems; a special feature of the NCF is that it leads to the generation of a constant mass matrix, which simplifies the computational process, and the generalized inertia forces, such as centrifugal and Coriolis forces, are zero, which is advantageous for analysis [16]. The rigid body is described in the Natural Coordinates Formulation (NCF) by the Cartesian coordinates of the basic points distributed throughout the body and by the Cartesian components of several unit vectors attached to these points that describe the orientation of the rigid body. For a spatial rigid body which is described with two points and two non-coplanar unit vectors, shown in Figure 3, the vector of the generalized coordinates of a rigid body can be expressed as  
\[\mathbf{q}=\left[ \begin{array}{cccc}\mathbf{r}_{i}^{\mathrm{T}}&\mathbf{r}_{j}^ {\mathrm{T}}&\mathbf{u}^{\mathrm{T}}&\mathbf{v}^{\mathrm{T}}\end{array} \right]^{ \mathrm{T}} \tag{1}\]  
where \(\mathbf{r}_{\mathrm{i}}\) and \(\mathbf{r}_{j}\) are the position vectors of points \(i\) and \(j\). The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors at point \(i\) which are perpendicular to each other and to vector \(\mathbf{r}_{ij}\). A generic point, \(P\), of the rigid body is described by a position vector, \(\mathbf{r}^{P}\), in the inertial frame \((X,Y,Z)\) and \(\bar{\mathbf{u}}\) in the local or body coordinate frame \((\bar{x},\bar{y},\bar{z})\). The position of the point \(\mathrm{P}\) relative to point \(i\) can be defined as  
\[\bar{\mathbf{u}} = \mathbf{r}^{P}-\mathbf{r}_{i}=c_{1}\mathbf{r}_{ij}+c_{2}\mathbf{ u}+c_{3}\mathbf{v} \tag{2}\] \[\Updownarrow\] \[\mathbf{r}^{P} = \mathbf{r}_{i}+c_{1}\big{(}\mathbf{r}_{j}-\mathbf{r}_{i}\big{)}+ c_{2}\mathbf{u}+c_{3}\mathbf{v}\] (3) \[\Updownarrow\] \[\mathbf{r}^{P} = (1-c_{1})\mathbf{r}_{i}+c_{1}\mathbf{r}_{j}+c_{2}\mathbf{u}+c_{3} \mathbf{v} \tag{4}\]  
Figure 3: Kinematics of rigid body using Natural Coordinate Formulation. which can be expressed in matrix form as  
\[\mathbf{r}^{p}= \begin{bmatrix}(1-c_{1})\mathbf{I}&c_{1}\mathbf{I}&c_{2}\mathbf{I}&c _{3}\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{r}_{i}\\ \mathbf{r}_{j}\\ \mathbf{u}\\ \mathbf{v}\end{bmatrix} =C\;\boldsymbol{q} \tag{5}\]  
where \(c_{1},c_{2}\), and \(c_{3}\) are constant elements of the vector \(\left(\mathbf{r}^{p}-\mathbf{r}_{i}\right)\) in the basis formed by vectors \(\mathbf{r}_{ij}\), \(\mathbf{u}\), and \(\mathbf{v}\). Therefore, the matrix \(\mathbf{C}\) depends on the geometry of the rigid body and does not vary with the motion of the system and is hence constant with time. The selection of vectors \(\mathbf{r}_{i},\mathbf{r}_{j},\mathbf{u}\), and \(\mathbf{v}\) in the previous equations can be chosen to reflect the actual anatomical attachment of the trunk to the head, while ensuring (in general) an effective mathematical representation for the rigid-flexible coupling model. In our formulation, \(\mathbf{r}_{i}\) and \(\mathbf{r}_{j}\) are reference points on the rigid head (center of mass and boundary point) that define its spatial orientation and serve as attachment points for the trunk. These points were selected to ensure a stable and realistic representation of the head's motion constraints.