## 2 Materials and Methods
### Velocity-Centric Motion Mapping  
For telemanipulation applications, the remote robot should maintain smooth, accurate tracking of user hand movements. However, raw hand movement data provided by the Vive tracking system contain intended hand motion, as well as tremors and noise. Thus, direct velocity mapping causes aggressive control maneuvers and jittering robot motion. Single Exponential Smoothing (SES) is applied to remove unintended short-term fluctuations and reduce hand tremor and noise. The SES for filtering out the noise from the hand motion series can be calculated recursively:  
\[V_{t+1|t}=\alpha V_{t}+\alpha(1-\alpha)V_{t-1}+\alpha(1-\alpha)^{2}V_{t-2}\ldots +\alpha(1-\alpha)^{k}V_{t-k} \tag{1}\]  
where \(V_{t}\) are velocity measurements and \(0\leq\alpha\leq 1\) is the smoothing parameter. The decrease rate of the weighting terms is controlled by the smoothing parameter. If \(\alpha\) is large and close to 1, more weight is given to the more recent hand motion observations. If \(\alpha\) approaches 0, the output velocity signal tends to be the average of the historical input velocity data. A value of \(\alpha=0.9\) gives the optimal performance in motion smoothing applications, according to experiments. The approach for intuitively controlling the robot so that its end-effector smoothly follows a user's hand trajectory, \(T_{u}(t)\), is to calculate and control the required n-vector of joint velocities \(\dot{\theta}\) directly from the relationship \(J(\theta)\dot{\theta}=V\), where the desired robot end-effector twist \(V_{r}=\left[v_{r}\;\omega_{r}\;\right]^{T}\) and \(J(\theta)\in\;R^{6\times n}\) are expressed in the same frame. At time k, the motion tracking system measures and determines the configuration of the user's hand, converts this calculated spatial velocity \(V_{u}=\left[v_{u}\;\omega_{u}\;\right]^{T}\) to a Cartesian twist command \(V_{r}=\left[v_{r}\;\omega_{r}\;\right]^{T}\) to the robot in the ROS coordinate system, employs the inverse-Jacobian solver to determine the appropriate joint rate vector, \(\dot{\theta}\), according to the desired twist representation of end-effector motion, \(V_{r}\), and derives the joint configuration sent to the UR5 robot controller, as shown by:  
\[\dot{\theta}=J^{\dagger}(\theta)\;SV_{r} \tag{2}\]  
\[V_{u}=\left[v_{u}\;\omega_{u}\;\right]^{T}=kV_{r}=\left[v_{r}\;\omega_{r}\; \right]^{T}=J_{r}(\theta)\dot{\theta}=\left[J_{p}(\theta)\;J_{\omega}(\theta )\;\right]\dot{\theta} \tag{3}\]  
Figure 3: Overview of the MR subspace interface with the digital twin of the UR5 robot superimposed on the 3D Point Cloud of the physical UR5 robot in the MR environment. (**a**) The pick-and-place task of two cubical and cylindrical objects given a goal position with different heights; (**b**) the assembly task involved grabbing one LEGO subassembly from a predefined spot and stacking it onto a fixed LEGO base on the table. The use of the pseudo-inverse in Equation (2) implicitly weights the cost of each of the 6 joint velocities identically, returns the minimized two-norm of joint velocities, and reduces the energy consumption of the robot. \(S\) is a positive scaling factor, and \(S=I\) is chosen in the application for precisely mimicking the user's movement, where \(I\in R^{6\times 6}\) is an identity matrix. By applying inverse-Jacobian-based kinematic techniques, real-time control of the spatial velocity of the TCP is achieved, instead of creating discrete path plans. Thus, a smooth trajectory and quality manipulation behavior can be achieved. The proposed teleoperated robotic system provides the operator with a more natural and intuitive scheme for interacting with the remote robot in comparison to conventional robotic teleoperation systems in which the robot is manipulated in joint space or the end-effector is driven at a certain specified velocity [48].