# Self-Recognition Grasping Operation with a Vision-Based Redundant Manipulator System
## 4 Simulation and Experiments
### The Optimal Solution with ISPF  
During \(\varphi\in[0,2\pi]\), countless solutions exist which can achieve the same pose at the end-effector of the redundant manipulator. However, for grasping operation, only one group of solutions are  
Figure 6: Grasping operation under different phase angles of \(\varphi\). (**a**) \(\varphi=0^{\circ}\), (**b**) \(\varphi=90^{\circ}\), (**c**) \(\varphi=180^{\circ}\), (**d**) \(\varphi=260^{\circ}\).  
Figure 7: Simplified manipulator configurations of multiple groups of solution. (**a**) A general view. (**b**) View from Y-Z plane.  
needed. In order to find the optimal solution, an optimizing factor can be introduced. Manipulability is an important and common factor representing the dexterity of robots. The optimal solution, when determined by manipulability, has a better adaption for the unstructured environment, especially in terms of humanoid control and dexterous manipulation.  
\[\omega=\sqrt{\boldsymbol{J}(\theta)\cdot\boldsymbol{J}(\theta)^{\mathrm{T}}} \tag{17}\]  
In Equation (17), \(\boldsymbol{J}(\theta)\) represents the Jacobian matrix of the manipulator at joint angles \(\theta\), where \(\theta=[\theta_{1},\theta_{2},\ldots,\theta_{7}]^{\mathrm{T}}\in\boldsymbol{R }^{\mathcal{T}\times 1}\). It can be calculated by the DH parameters of the robot and joint angles \(\theta\). \(\omega\) represents the manipulability of manipulator. Since \(\theta\) has been derived by the ISPF method, corresponding manipulability can be obtained during \(\varphi\in[0,2\pi]\), according to Equation (17). The variation of manipulability related to \(\varphi\) is shown in Figure 8.  
From Figure 8, the maximum value of manipulability can be found at \(\varphi=260^{\circ}\), and the corresponding solution is \(\theta_{\mathrm{max}}=[-77.0459\ 92.1113-99.9610\ 46.2262-0.3588-33.1059\ 90.4780](^{\circ})\). The minimal value of manipulability is at \(\varphi=40^{\circ}\), where the corresponding solution is \(\theta_{\mathrm{min}}=[-114.9950\ 70.7922\ 42.8671\ 46.2262-21.7377-16.7663-19.3553](^{\circ})\). The configurations at the minimum and maximum of manipulability are shown in Figure 9. Although the numerical difference of manipulability is not obvious, the configurations have a wide difference at the minimum and maximum of manipulability. Through the introduction of manipulability, the most dexterous grasping configuration can be selected from all the feasible solutions, with which dexterous grasping manipulation can be achieved.  
Figure 8: Manipulability related to phase angle \(\varphi\).  
Figure 9: The configuration at the minimum and maximum of manipulability. (**a**) The configuration at the minimal of manipulability. (**b**) The configuration at the maximum of manipulability.  
Via the introduction of manipulability, manipulation can be expressed as a single object optimization problem of obtaining the optimal solution for self-recognized grasping operation.  
\[\begin{array}{ll}find:&\varphi\\ \max:&\omega=\sqrt{J(\theta)\cdot J(\theta)^{\mathrm{T}}}\\ &\theta=f_{inv}(\varphi)\end{array} \tag{18}\]  
With Equation (18), a unique \(\theta\) value can be obtained to achieve dexterous self-recognized grasping operation with a redundant manipulator. Additionally, other optimal factors, such as the optimal torque, least time cost, and so on, can be introduced to replace manipulability in Equation (18), which will greatly expand the ISPF method in achieving various optimizations for various grasping operation requirements.