# A Hybrid Mapping Method with Position and Stiffness for Manipulator Teleoperation
## 4 Stiffness Mapping Method for Compliant Teleoperation
### Stiffness Identification Algorithm at the Human Arm Endpoint  
The force fluctuations and impedance adjustments at the human arm endpoint can be regarded as internal force regulations exerted by group of extensor and flexor muscles. Based on the high correlation between sEMG signals and generated muscle tension, there is an approximate linear relationship between the force/stiffness and the sEMG signals of the human arm, which can be described as follows:  
\[\left[\begin{array}{c}F_{end}\\ \mathbf{K}_{end}\right]=\left[\begin{array}{c}\mathbf{T}_{F}\\ \mathbf{T}_{K}\end{array}\right]\mathbf{P}_{se}+\left[\begin{array}{c}0\\ \mathbf{K}_{0}\end{array}\right] \tag{15}\]  
where \(\mathbf{F}_{end}\) and \(\mathbf{K}_{end}\) are the force vector and the stiffness vector at the human arm endpoint, \(\mathbf{P}_{se}\) is the processed sEMG signals, \(\mathbf{T}_{F}\) and \(\mathbf{T}_{K}\) are the mapping matrices from the sEMG signals \(\mathbf{P}_{se}\) to the force vector \(\mathbf{F}_{end}\) and the stiffness vector \(\mathbf{K}_{end}\), and \(\mathbf{K}_{0}\) is the stiffness offset at the human arm endpoint.  
The force mapping matrix \(\mathbf{T}_{F}\) can be obtained by the regression analysis model. However, the stiffness mapping matrix \(\mathbf{T}_{K}\) is difficult to identify accurately. Relevant research [8] shows that the force \(\mathbf{F}_{end}\) and the stiffness \(\mathbf{K}_{end}\) at the human arm endpoint are decoupled if the motion and force at the human arm endpoint change little. That is to say, the muscular activities can be decomposed and mapped into two subspaces. The first mapping subspace corresponds to the force fluctuations and the second will be defined as the kernel of the former which correlates of stiffness regulations without causing any effect on generated joint/endpoint forces. Therefore, the sEMG signals \(\mathbf{P}_{se}\) can be decomposed into the force mapping subspace \(\mathbf{P}_{F}\) and the force mapping null space \(\mathbf{P}_{K}\).  
\[\mathbf{P}_{se}=\mathbf{P}_{F}\oplus\mathbf{P}_{K} \tag{16}\]  
Then we can obtain:  
\[\mathbf{P}_{se}=\mathbf{T}_{\mathbf{F}}^{-1}\mathbf{T}_{\mathbf{F}}\mathbf{P}_{se}+\left(\mathbf{I}-\mathbf{T} _{\mathbf{F}}^{-1}\mathbf{T}_{\mathbf{F}}\right)\mathbf{P}_{se}=\mathbf{P}_{F}+\mathbf{P}_{K} \tag{17}\]  
In order to obtain the relationship between \(\mathbf{P}_{K}\) and \(\mathbf{P}_{se}\), we define \(\mathbf{N}_{F}^{+}\) as the kernel of the force mapping matrix \(\mathbf{T}_{F}\).  
\[\mathbf{N}_{F}^{+}=\mathbf{I}-\mathbf{T}_{\mathbf{F}}^{\mathrm{T}}(\mathbf{T}_{\mathbf{F}}\mathbf{T}^{ \mathrm{T}})^{-1}\mathbf{T}_{\mathbf{F}} \tag{18}\]  
Then \(\mathbf{P}_{K}\) can be expressed as:  
\[\mathbf{P}_{K}=\mathbf{N}_{F}^{+}\mathbf{P}_{se} \tag{19}\]  
The stiffness at the human arm endpoint can be rewritten as follows:  
\[\mathbf{K}_{end}=\mathbf{K}_{0}+\ \psi\mathbf{P}_{K}=\mathbf{K}_{0}+\ \psi\mathbf{N}_{F}^{+}\mathbf{P}_{se} \tag{20}\]  
where \(\psi\) is the mapping matrix from the force mapping null space \(\mathbf{P}_{K}\) to the stiffness at the human arm endpoint \(\mathbf{P}_{end}\).