## 2 Kinematic Modeling  
The kinematic modeling described here follows the procedure of [PERSON] et al. [7] and [PERSON] et al. [6]. For a general procedure of kinematic modeling of wheeled mobile manipulators, please refer to [PERSON] et al. [5]. Let \(r\in\mathbb{R}^{m}\) be the end-effector's position and/or pose in the task space. The configuration vector \(\mathbf{q}\), also known as the generalized coordinates of the mobile manipulator, is given by the combination of the platform configuration vector \(\mathbf{q_{p}}\) and the robot arm configuration vector \(\mathbf{q_{a}}\). Figure 2 illustrates these configuration vectors. A frame \(x^{\prime}y^{\prime}\) is attached to the mobile platform at the center of the wheels' axle \((x_{p},y_{p})\), with respect to the world reference frame \(xy\), with its \(x^{\prime}\) axis pointing in the forward direction and the \(y^{\prime}\) axis pointing in the direction parallel to the wheels' axle. The angle between the \(x\) axis of the world reference frame and \(x^{\prime}\) attached to the platform is denoted as \(\theta_{p}\). Then, the platform configuration is given by \(\mathbf{q_{p}}= \begin{bmatrix}x_{p}&y_{p}&\theta_{p}\end{bmatrix} ^{T}\in\mathbb{R }^{3}\). The robot arm configuration is given by the position of its joints as \(\mathbf{q_{a}}= \begin{bmatrix}q_{a_{1}}&q_{a_{2}}&\dots&q_{a_{n}}\end{bmatrix} ^{T} \in\mathbb{R}^{n_{a}}\). Finally, the configuration vector of the mobile manipulator is \(\mathbf{q}= \begin{bmatrix}\mathbf{q_{p}}^{T}&\mathbf{q_{a}}^{T}\end{bmatrix} ^{T}\in \mathbb{R}^{n}\) with \(n=3+n_{a}\). The end-effector's position/pose is a function of the configuration vector defined by the kinematic map:  
\[\mathbf{r}=f(\mathbf{q_{p}},\mathbf{q_{a}}), \tag{1}\]  
The wheeled platform movement is constrained under the rolling without a slipping assumption on both wheels, which can be expressed as the following nonholonomic constraint:  
\[\mathbf{\dot{q}_{p}}=G(\mathbf{q_{p}})\mathbf{u_{p}}, \tag{2}\]  
where \(\mathbf{u_{p}}= \begin{bmatrix}v_{p}&\omega_{p}\end{bmatrix} ^{T}\in\mathbb{R}^{2}\) is the velocity input of the platform, consisting of the linear and angular velocities of the platform, which are known as pseudo velocities or quasi-velocities. The columns of matrix \(G(\mathbf{q_{p}})\) span the admissible velocity space for a given platform configuration. The matrix \(G(\mathbf{q_{p}})\) for a differential-drive platform is defined as  
\[G(\mathbf{q_{p}})= \begin{bmatrix}\cos\theta_{p}&0\\ \sin\theta_{p}&0\\ 0&1\end{bmatrix} ,\]  
On the other hand, the robotic arm kinematics at the velocity level is not constrained for any configuration, and therefore, the generalized velocities of the arm are equal to the velocity inputs of the arm:  
\[\mathbf{\dot{q}_{a}}=\mathbf{u_{a}}. \tag{3}\]  
Figure 2: Nonholonomic mobile manipulator schematic. The velocity input vector for the entire NMM can be constructed as \(\mathbf{u}= \begin{bmatrix}\mathbf{u_{p}}^{T}&\mathbf{u_{a}}^{T}\end{bmatrix} ^{T}\in\mathbb{R}^ {\delta_{m}}\), with \(\delta_{m}=2+n_{a}\). The dimension \(\delta_{m}\) is called the mobility degree of the mobile manipulator and indicates the space dimension of the admissible generalized velocities [6]. Using (2) and (3), the map from the velocity input vector \(u\) to the generalized velocities \(\dot{\mathbf{q}}= \begin{bmatrix}\dot{\mathbf{q}}_{p}^{T}&\dot{\mathbf{q}_{a}}^{T}\end{bmatrix} ^ {T}\) can be written as  
\[\dot{\mathbf{q}}=S(\mathbf{q})\mathbf{u}, \tag{4}\]  
with:  
\[S(\mathbf{q})= \begin{bmatrix}G(\mathbf{q}_{p})&0\\ 0&I_{n_{a}}\end{bmatrix} ,\]  
where \(G(\mathbf{q}_{p})\) maps the platform's velocity input vector \(\mathbf{u}_{p}= \begin{bmatrix}v_{p}&\omega_{p}\end{bmatrix} ^{T}\) to the platform's generalized velocities \(\dot{\mathbf{q}}_{p}= \begin{bmatrix}\dot{x}_{p}&\dot{y}_{p}&\dot{\phi}_{p}\end{bmatrix} ^ {T}\), and matrix \(I_{n_{a}}\) is an identity matrix of size \(n_{a}\) that sets the arm's generalized velocities equal to its input velocities \(\dot{\mathbf{q}}_{a}=\mathbf{u}_{a}\).