# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions
## 2 Vector Labeling Standard  
To ensure clarity and consistency in the mathematical formulation of the proposed method, a structured vector labeling standard is adopted throughout this work. This convention enables the explicit representation of positions, directions, and reference frames involved in the kinematic chains, which are essential for formulating and evaluating boundary functions under geometric and collision constraints. The notation employed here is aligned with the vectorial formalism introduced in [17], where directional vectors and reference frames are systematically labeled to support the construction of transformation matrices and constraint expressions. This standard has been particularly useful in the current study for expressing relative positions and orientations in a modular and traceable manner. A graphical representation of the labeling convention is provided in Figure 1.  
The following definitions summarize the adopted labeling scheme:  
1. A relative reference frame is denoted by \[o_{a,b}= \begin{bmatrix}i_{a,b}&j_{a,b}&k_{a,b}&\mathbf{r}_{a,b}\end{bmatrix} = \begin{bmatrix}R_{a,b}&\mathbf{r}_{a,b}\end{bmatrix} \in\mathbb{R}^{3\times 4}\] (1) where \(o_{a,b}\) represents the orientation and position of frame \(b\) as observed from frame \(a\). The matrix \(R_{a,b}\in\mathbb{R}^{3\times 3}\) contains the orthonormal unit vectors (directional cosines) \(i_{a,b},i_{a,b},k_{a,b}\), and \(\mathbf{r}_{a,b}\in\mathbb{R}^{3\times 1}\) denotes the position vector of the origin of frame \(b\) with respect to frame \(a\). When only one subscript is used, such as \(o_{a}\), the frame is considered to be self-referenced. In this case, the expression corresponds to \[o_{a}\equiv o_{a,a}= \begin{bmatrix}i_{a}&j_{a}&k_{a}&\mathbf{r}_{a}\end{bmatrix} \] (2) where \(\mathbf{r}_{a}=\mathbf{0}\), and \(R_{a}\) corresponds to the canonical orientation of frame \(a\) with respect to itself. 2. The vectors \(i_{a},j_{a},k_{a}\in\mathbb{R}^{3\times 1}\) represent the directional cosines of frame \(a\), forming an orthonormal basis. Orthonormality implies that the vectors are mutually perpendicular and of unit length. 3. The vectors \(i,j\), and \(k\) are visually distinguished in illustrations using the colors red, green, and blue, respectively, as shown in Figure 1.
4. A position vector is represented as \[\mathbf{r}_{a,b}^{(c)}\in\mathbb{R}^{3\times 1},\] (3) where \(a\) and \(b\) are the initial (origin) and the terminal (destination) points of the vector, respectively, while \(c\) is the observer frame from which the vector is expressed. The notation \(\mathbf{r}_{a,b}^{(c)}\) is interpreted as \"the vector that connects the frame \(a\) to \(b\), observed from frame \(c\)\". If the observer frame is omitted, it is assumed that all vectors are observed in a common reference frame. When only one subscript is provided, such as \(\mathbf{r}_{a}\), the vector is interpreted as being self-referenced, i.e., \(\mathbf{r}_{a,a}\), which by convention is equal to the zero vector: \[\mathbf{r}_{a}=\mathbf{r}_{a,a}= \begin{bmatrix}0\\ 0\\ 0\end{bmatrix} .\] (4)  
Figure 1: Graphical representation of the vector labeling standard. If all vectors are expressed with respect to a common observer, the superscript \(c\) may be omitted for simplicity. 6. A kinematic chain is defined as an ordered sequence of reference frames that are linked by physical or geometric constraints. The relation between frames is denoted by the precedence operator: \[a\prec b\prec c,\] (5) which indicates that frame \(a\) precedes frame \(b\), and frame \(b\) precedes frame \(c\) in the kinematic structure. 7. Subscripts \(a\), \(b\), and \(c\) may represent alphanumeric labels associated with a kinematic sequence and are not limited to numeric indices. 8. The Cartesian components of a vector are denoted as \(x_{a,b},y_{a,b},z_{a,b}\in\mathbb{R}\). 9. The canonical orientation of a frame with respect to itself is defined as \[\overset{(a)}{i_{a}}=i_{a}= \begin{bmatrix}1\\ 0\\ 0\end{bmatrix} ,\quad\overset{(a)}{j_{a}}=j_{a}=