## 4 Manipulability Maximization and Constraints Satisfaction
### Manipulability Maximization  
The term manipulability, introduced by [PERSON] [12], describes the ability of a robotic system to provide end-effector's velocities in any direction for a given configuration. The manipulability of wheeled mobile manipulators was studied in detail in [5]. There exist different algebraic measures to characterize the manipulability of a robotic system [5; 12; 13; 14]. The most widely used measure, known as the manipulability measure and noted here as \(\Omega\), is given by \(\Omega=\sigma_{1}\sigma_{2}\ldots\sigma_{m}\), where \(\sigma_{1},\sigma_{2},\ldots,\sigma_{m}\) are the singular values of the system Jacobian \(J(\mathbf{q})\). Therefore, \(\Omega\) is the manipulability measure for system configuration \(\mathbf{q}\). The measure \(\Omega\) is proportional to the volume of the manipulability ellipsoid [12]. Furthermore, it can be shown that \(\Omega\) has the following property:  
\[\Omega=\sqrt{\det(J(\mathbf{q})J^{\mathsf{T}}(\mathbf{q}))} \tag{13}\]  
By definition, \(\Omega\geq 0\), and \(\Omega=0\) if and only if \(\text{rank}(J(\mathbf{q}))<m\). The elements of \(\mathbf{\
abla_{q}}\mathbf{\Omega}\) are given mathematically by  
\[ \begin{split}&\frac{\partial\Omega}{\partial q_{i}}=-\frac{1}{2} \Omega\cdot\text{trace}\Bigg{(}(JJ^{\mathsf{T}})^{-1}\bigg{(}\frac{\partial J} {\partial q_{i}}J^{\mathsf{T}}+J\bigg{(}\frac{\partial J}{\partial q_{i}} \bigg{)}^{\mathsf{T}}\bigg{)}\Bigg{)},\\ &\text{with }i=1,2,\ldots,n.\end{split}  \tag{14}\]  
As mentioned before, we are not only concerned with improving the manipulability of the whole system but also that of the arm alone. In this work, we present a new manipulability measure that better expresses the manipulability of a mobile manipulator, because both the manipulability of the arm and the whole system are intrinsically considered. Let us define the manipulability of the whole system, denoted as \(\Omega_{p+a}\), and the manipulability of the robot arm, denoted as \(\Omega_{a}\), to be the measures obtained using Equation (13) with the Jacobians \(\bar{J}\) and \(J_{a}\) from Equation (5), respectively. Notice that by using the reduced Jacobian \(\bar{J}\), the platform's nonholonomic constraints are included in the manipulability measure of the whole system. Our proposed manipulability measure for mobile manipulators is defined as  
\[\Omega_{MM}=\hat{\Omega}_{p+a}\hat{\Omega}_{a}, \tag{15}\]  
where \(\hat{\Omega}_{p+a}\) and \(\hat{\Omega}_{a}\) are the normalized manipulabilities computed by dividing them by their respective maximum values over all the possible configurations of the system:  
\[\hat{\Omega}_{p+a}=\frac{\Omega_{p+a}}{\Omega_{p+a_{max}}}\ \ \text{and}\ \ \ \hat{\Omega}_{a}=\frac{\Omega_{a}}{\Omega_{a_{max}}}.\]  
The normalized values are used in our proposed measure to make sure that all its components are in the same scale. This normalization also makes the measures invariant to units and chosen reference frame [14; 15]. Notice that \(\bar{J}= \begin{bmatrix}J_{p}&J_{a}\end{bmatrix} S(\mathbf{q})\in\mathbb{R}^{m\times \delta_{m}}\) in (5) indicates the relation between \(\bar{J}\) and \(J_{a}\). Hence, the singular values of \(\bar{J}\) are not necessarily the same as the singular values of \(J_{a}\). There are cases where \(J_{a}\) is rank-deficient while \(\bar{J}\) is not. For these cases, the whole system manipulability \(\Omega_{p+a}\
eq 0\) even though the arm is in a singular configuration. Therefore, the measure \(\Omega_{p+a}\) fails to express the ability of the arm alone to provide end-effector's velocities in any direction. As a result, maximizing the measure \(\Omega_{p+a}\) might result in the poor manipulability of the arm, even though the manipulability of the whole system is preserved or improved. This is an issue that has been discussed in the literature [6; 10]. On the other hand, our proposed measure \(\Omega_{MM}\) is defined by the product of the singular values of \(\bar{J}\) and \(J_{a}\), i.e., the product of the manipulability ellipsoids of \(\bar{J}\) and \(J_{a}\).