# A Hybrid Mapping Method with Position and Stiffness for Manipulator Teleoperation
## 3 Motion Mapping Method for Position Teleoperation
### Virtual Impedance Mapping in Fine Space  
When the operator teleoperates the slave manipulator in fine space, he should guarantee the safety of whole system. That is to say, the slave manipulator should be controlled to avoid the collisions between the manipulator and the environment, the manipulated target, or the obstacles. In this case, since there is a possibility of collision, the CyberForce with force feedback is utilized in fine space to maintain the safety. To meet this requirement, the virtual impedance mapping method is proposed in this paper to obtain the safe motion in the fine space.  
If there is an obstacle which can be detected by the stereo vision or prior knowledge in the remote environment, a corresponding virtual obstacle expressed with a bounding sphere can be established at the master side by the inverse process of Equation (3), shown in Figure 3. Moreover, a safe boundary sphere is used to enlarge the virtual obstacle to improve the system safety.  
The Euclidean distance \(\boldsymbol{d}\) between the input position \(\boldsymbol{P}\) and the virtual contact boundary \(\boldsymbol{P_{Sb}}\) (Figure 4) at the master side is defined in Equation (4):  
\[\boldsymbol{d}=\parallel\boldsymbol{P}-\boldsymbol{P_{Sb}}\parallel \tag{4}\]  
To improve the motion safety in fine space, the Cartesian impedance controller is utilized for the master, which can be expressed as Equation (5):  
\[\mathbf{M_{d}}\ddot{\boldsymbol{\xi}}+\mathbf{B_{d}}\dot{\boldsymbol{\xi}}+ \mathbf{K_{d}}\boldsymbol{\xi}=\boldsymbol{F_{d}} \tag{5}\]  
\[\boldsymbol{F_{d}}=k\boldsymbol{d} \tag{6}\]  
Figure 2: Workspace redefinition of robot.  
where \(\mathbf{M_{d}}\), \(\mathbf{B_{d}}\), \(\mathbf{K_{d}}\) are the desired inertia, viscosity and stiffness matrices of the virtual impedance controller for the master, \(\mathbf{F_{d}}\) is the desired Cartesian force, \(\boldsymbol{\xi}\) is the correction term for virtual impedance controller and \(k\) is the force scale factor.  
Therefore, the virtual impedance mapping for robot position teleoperation in fine space can be expressed as:  
\[\mathbf{X_{s}}=\mathbf{K_{m}}(\mathbf{X_{m}}-\boldsymbol{\xi})+\mathbf{X_{0}} \tag{7}\]  
The switch between the fixed scale mapping method and the virtual impedance mapping method can be achieved automatically according to the real workspace of robot. The switch is stable because the trajectories of manipulator with different mapping methods are continuous when the minimum distance of robot is around \(D_{1}\). The derivation is as follows.  
When the minimum distance of robot \(D_{min}\) equals to \(D_{1}\), we can obtain \(\boldsymbol{d}=0\) according to Equation (4). Therefore, Equation (5) can be rewritten as:  
\[\mathbf{M_{d}}\tilde{\boldsymbol{\xi}}+\mathbf{B_{d}}\tilde{\boldsymbol{\xi} }+\mathbf{K_{d}}\boldsymbol{\xi}=\mathbf{F_{d}}=k\boldsymbol{d}=0 \tag{8}\]  
Then, we can calculate \(\boldsymbol{\xi}=0\). After substituting this into Equation (7), we can obtain:  
\[\mathbf{X_{s\_D_{1}}}=\mathbf{K_{m}}\Big{(}\mathbf{X_{m\_D_{1}}}-0\Big{)}+ \mathbf{X_{0}}=\mathbf{K_{m}}\mathbf{X_{m\_D_{1}}}+\mathbf{X_{0}} \tag{9}\]  
where \(\mathbf{X_{m\_D_{1}}}\) and \(\mathbf{X_{s\_D_{1}}}\) represent the Cartesian position vector of the master and slave manipulator when the minimum distance of robot \(D_{min}\) equals to \(D_{1}\).  
The expression of Equation (9) is the same as Equation (3) when the minimum distance of robot \(D_{min}\) equals to \(D_{1}\), which shows that the trajectories of manipulator are continuous and the switch between the fixed scale mapping method and the virtual impedance mapping method is stable.  
Figure 4: Distance of virtual obstacle.  
Figure 3: Diagram of virtual impedance mapping.