## 3 The Joint Space Stiffness and the Cartesian Stiffness  
The relationship between the joint space stiffness and the Cartesian stiffness [29; 30] of the humanoid manipulator is as follows:  
\[\mathbf{K_{p}}=\mathbf{J}^{\mathrm{T}}\mathbf{K_{\theta}}\mathbf{J}^{\mathrm{1}} \tag{10}\]  
where \(\mathbf{K_{p}}\) is the Cartesian stiffness matrix of \(6\times 6\); \(\mathbf{K_{\theta}}\) is the joint space stiffness diagonal matrix of \(7\times 7\), and each diagonal term represents the stiffness of the corresponding joint; \(\mathbf{J}\) is the Jacobian matrix of \(6\times 7\), which is derived from the forward kinematics of the humanoid manipulator. Because the Cartesian stiffness matrix is not a diagonal matrix, it is difficult to directly estimate the Cartesian space stiffness of the humanoid manipulator through this matrix. Therefore, the generalized displacement of the end effector of the humanoid manipulator is used to evaluate the Cartesian space stiffness. \[\Delta\mathbf{p}=\mathbf{C}\mathbf{F} \tag{11}\]  
where \(\Delta\mathbf{p}\) is the generalized displacement of the end effector of the humanoid manipulator, \(\Delta\mathbf{p}=\left[\Delta x,\Delta y,\Delta z,\Delta\theta_{x},\Delta\theta_ {y},\Delta\theta_{z}\right]^{T}\); \(\mathbf{F}\) is the force/torque on the end effector of the humanoid manipulator, \(\mathbf{F}=\left[F_{x},F_{y},F_{z},T_{x},T_{y},T_{z}\right]^{T}\); \(\mathbf{C}\) is the compliance matrix of the humanoid manipulator. \[\mathbf{C}=\mathbf{K_{p}^{\mathrm{-1}}}=\mathbf{J}\mathbf{K_{\theta}^{\mathrm{-1}}}\mathbf{J}^{ \mathrm{T}} \tag{12}\]  
Assuming that the gravitational load is \(1\) kg, and the coordinate of the center of gravity of the load relative to the coordinate system of the end effector is \((30,50,-40)\), the unit is millimeter. The relevant calculation equations of the force/torque of gravitational load acting on the end effector are given in Appendix B. The joint angle of the series parallel  
Figure 3: The structural model of the humanoid manipulator. hybrid 7-DOF humanoid manipulator is \(\mathbf{q}=\left[-\frac{\pi}{3},-\frac{\pi}{6},\frac{\pi}{18},-\frac{7\pi}{18},\frac{ \pi}{36},-\frac{\pi}{36},-\frac{\pi}{9}\right]^{T}\). Three kinds of joint space stiffness of the humanoid manipulator are given as follows:  
\[\left\{ \begin{array}{l}K_{\mathbf{el}}=diag\left[10^{4},10^{4},10^{4},10^{4},10^{ 4},10^{4},10^{4}\right]\\ K_{\mathbf{\theta}m}=diag\left[5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4}\right]\\ K_{\mathbf{\theta}n}=diag\left[10^{5},10^{5},10^{5},10^{5},10^{5},10^{5}\right] \end{array} \right.,\]  
the unit is Nm/rad. Then the Cartesian space stiffness of the humanoid manipulator with different joint space stiffness is compared by comparing the generalized displacement of the end effector of the humanoid manipulator, as shown in Figure 4.  
It can be seen from Figure 4 that when the humanoid manipulator has the same posture and load, the change of the joint space stiffness will affect the generalized displacement of the end effector, that is, the Cartesian space stiffness of the humanoid manipulator is directly related to the joint space stiffness of the humanoid manipulator. The smaller the joint space stiffness is, the better the overall flexibility of the humanoid manipulator. In this case, the generalized displacement of the end effector is larger when the same force/torque is applied to the end effector, and vice versa. Therefore, in the human robot interaction, the safety of human robot interaction may be improved by adjusting the joint space stiffness of the humanoid manipulator in real-time. Assuming that the gravitational load on the end effector of the humanoid manipulator is the same as above, the joint space stiffness of the series parallel hybrid 7-DOF humanoid manipulator is \(\mathbf{K}_{\mathbf{\theta}}=diag\left[5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4},5\times 10^{4}\right]\), the unit is Nm/rad.