*3.2. Kinematics of the Design Track Centerline*

Track geometry is the superposition of the design geometry and the irregularities. The components of the absolute position vector of an arbitrary point on the design track

centerline with respect to an inertial and global frame is a function of the arc-length *s*, as follows:

$$\mathbf{R}^t(\mathbf{s}) = \begin{bmatrix} R\_x^t(\mathbf{s}) \\ R\_y^t(\mathbf{s}) \\ R\_z^t(\mathbf{s}) \end{bmatrix} \tag{1}$$

where **R***<sup>t</sup>* contains the components of vector - *R<sup>t</sup>* shown in Figure 2. The geometry of the track centerline 3D-curve is defined by the horizontal profile and the vertical profile. Both profiles are defined in the rail industry using sections of variable length. Points between two sections are called vertices. Horizontal profile vertices do not necessary coincide with vertical profile vertices. Horizontal profile includes three types of sections: tangent (straight), curve (circular) and transitions (clothoid). Vertical profile includes two types of sections: constant-slope (straight) and transitions (cubic).

At each track section, the track centerline geometry is characterized by the following geometric values:


Figure 2 shows the TF < *X<sup>t</sup>* ,*Y<sup>t</sup>* , *Z<sup>t</sup>* > associated with the track centerline at each value of *s*. The orientation of the TF with respect to a GF can be measured with the Euler angles *ψ<sup>t</sup>* (*azimuth* or *heading* angle), *θ <sup>t</sup>* (vertical slope, positive when downwards in the forward direction) and *ϕ <sup>t</sup>* (*cant* or *superelevation angle*). The rotation matrix from the TF to the GF is given by:

$$\mathbf{A}^{t}(\mathbf{s}) = \begin{bmatrix} \mathbf{c}\theta^{t}\mathbf{c}\psi^{t} & \mathbf{s}\mathbf{q}^{t}\mathbf{s}\theta^{t}\mathbf{c}\psi^{t} - \mathbf{c}\mathbf{q}^{t}\mathbf{s}\psi^{t} & \mathbf{s}\mathbf{q}^{t}\mathbf{s}\psi^{t} + \mathbf{c}\mathbf{q}^{t}\mathbf{s}\theta^{t}\mathbf{c}\psi^{t} \\ \mathbf{c}\theta^{t}\mathbf{s}\psi^{t} & \mathbf{c}\mathbf{q}^{t}\mathbf{c}\psi^{t} + \mathbf{s}\mathbf{q}^{t}\mathbf{s}\theta^{t}\mathbf{s}\psi^{t} & \mathbf{c}\mathbf{q}^{t}\mathbf{s}\theta^{t}\mathbf{s}\psi^{t} - \mathbf{s}\mathbf{q}^{t}\mathbf{c}\psi^{t} \\ -\mathbf{s}\theta^{t} & \mathbf{s}\mathbf{q}^{t}\mathbf{c}\theta^{t} & \mathbf{c}\mathbf{q}^{t}\mathbf{c}\theta^{t} \end{bmatrix} \tag{2}$$

The azimuth *ψ<sup>t</sup>* can have an arbitrary value, however, the slope *θ <sup>t</sup>* and cant *ϕ <sup>t</sup>* angles can be considered as small angles, such that the rotation matrix from the TF to the GF can be approximated to:

$$\mathbf{A}^{t}(\mathbf{s}) \simeq \begin{bmatrix} \mathbf{c}\boldsymbol{\upmu}^{t} & -\mathbf{s}\boldsymbol{\upmu}^{t} & \boldsymbol{\uprho}^{t}\mathbf{s}\boldsymbol{\upmu}^{t} + \boldsymbol{\uptheta}^{t}\mathbf{c}\boldsymbol{\upmu}^{t} \\ \mathbf{s}\boldsymbol{\upmu}^{t} & \mathbf{c}\boldsymbol{\upmu}^{t} & \boldsymbol{\uptheta}^{t}\mathbf{s}\boldsymbol{\upmu}^{t} - \boldsymbol{\uprho}^{t}\mathbf{c}\boldsymbol{\upmu}^{t} \\ -\boldsymbol{\uptheta}^{t} & \mathbf{q}^{t} & 1 \end{bmatrix} \tag{3}$$

An ideal body that moves along the track, taking the same orientation as the track frame with a forward velocity *V* and a forward acceleration *V*˙ , has the following absolute velocity and acceleration:

$$
\dot{\mathbf{R}}^t = \begin{bmatrix} V \\ 0 \\ 0 \end{bmatrix}, \quad \dot{\mathbf{R}}^t = \begin{bmatrix} \dot{V} \\ \rho\_h V^2 \\ -\rho\_v V^2 \end{bmatrix} \tag{4}
$$

Similarly, the absolute angular velocity and the absolute angular acceleration of that body are given by:

$$
\boldsymbol{\ddot{\sigma}}^{t} = \begin{bmatrix} \rho\_{tw} V \\ \rho\_{\overline{v}} V \\ \rho\_{\overline{h}} V \end{bmatrix}, \quad \dot{\mathbf{a}}^{t} = \begin{bmatrix} \rho\_{tw} \dot{V} \\ \rho\_{\overline{v}} \dot{V} \\ \rho\_{\overline{h}} \dot{V} + \rho\_{\overline{h}}^{\prime} V^{2} \end{bmatrix} \tag{5}
$$
