## 3 Offline Optimization and Adaptation Under Nonlinear Task Constraints
### Optimization and Adaptation Procedure
#### 3.3.2 Adaptation Procedure
The modulated joint positions \(\boldsymbol{q}_{\text{mod}}\) are then used for real-time task execution. Furthermore, at each timestep, the initial velocity for the next step is estimated using trajectory generalization based on the current position \(\boldsymbol{x}_{d}\). This process repeats iteratively until all \(T\) timesteps are completed. ## 5 Experimental Validation

In this section, we validate the proposed human-robot skill transfer strategy within an industrial scenario where a robot learns different skills from the human to move around the obstacle and grasp the emergency stop. We first conduct human-robot skill transfer experiments. After the robot learns the skills, we evaluate our method in different static (offline) and dynamic (real-time) environments, and compare our method to other similar movement adaptation methods. For clarity, we summarize the definitions of all the parameters employed in the experiment section in Table 1. \begin{table}
\begin{tabular}{c l} \hline \hline
**Offline Adaptation** & **Description** \\ \hline \(f(\boldsymbol{p})\) & Trajectory shape modulation for 3D positions \\ \(R_{\text{obs}}\) & The obstacle radius \\ \(\chi,\text{Y},\text{Z}\) & The 3D position sets for via-points range \\ \(\tau\) & The threshold for fine-tuning via-points \\ \(T\) & The forward kinematics function \\ \(d_{k,t}\) & The distance threshold to represent task constraints probability \\ \(\boldsymbol{n}_{z,t},\boldsymbol{b}_{z,t}\) & The hyperplane normal and bias vectors for the maximum Z \\ \(R_{\text{obs}}\) & The obstacle safe radius \\ \(\alpha_{k,t}\) & The confidence level to represent the task constraints probability \\ \(\lambda_{k,t}\) & The initial Lagrange multiplier \\ \(p(\boldsymbol{w}^{0})=\mathcal{N}(\boldsymbol{w}^{0};\boldsymbol{\mu}^{0}_{ \boldsymbol{w}^{\prime}},\boldsymbol{\Sigma}^{0}_{w})\) & Initial Gaussian distributions of ProMPs \\ \(p(\boldsymbol{w}^{*})=\mathcal{N}(\boldsymbol{w}^{*};\boldsymbol{\mu}^{*}_{ \boldsymbol{w}^{\prime}},\boldsymbol{\Sigma}^{*}_{w})\) & Optimized Gaussian distributions of ProMPs \\ \(p(\boldsymbol{q}_{t})=\mathcal{N}(\boldsymbol{q}_{t};\boldsymbol{\mu}^{0}_{ \boldsymbol{v}},\boldsymbol{\sigma}^{0}_{t})\) & The initial mean and standard deviation of each timestep \\ \(p(\boldsymbol{q}_{t})=\mathcal{N}(\boldsymbol{q}_{t};\boldsymbol{\mu}^{*}_{t}, \boldsymbol{\sigma}^{*}_{t})\) & The optimized mean and standard deviation of each timestep \\ \(\boldsymbol{q}^{\text{safe}}_{t}\in(\boldsymbol{\mu}^{*}_{t}-2\boldsymbol{ \sigma}^{*}_{t},\boldsymbol{\mu}^{*}_{t}+2\boldsymbol{\sigma}^{*}_{t})\) & The optimized joint limit range \\ \(\boldsymbol{\Psi}^{0}_{t}=\boldsymbol{\Psi}^{T}_{t}w^{*}\) & The initial joint value for Jacobian IK \\ \(dt\) & The interval between adjacent timesteps \\ \(\left(\boldsymbol{\xi}_{k}\right)\) & The ellipsoid-shaped distance function \\ \(p^{k}_{j},\boldsymbol{\xi}^{c}_{k}\) & The curvature parameter and the center of the distance function \\ \(\xi^{k}_{k},\xi^{k}_{k}\) & The linear and angular velocities of dynamic obstacles \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of all parameters employed in the experiment section. ### Template Trajectory Generalization from Only One Human Demonstration

The Ufactory Xarm 7-DoF manipulator is selected to validate our method. A Kinect V2 camera is positioned on the side to track continuous human hand movements, while a Zed2i binocular camera is mounted overhead to capture task-related object poses. For a single obstacle scenario, the robot is taught three distinct skills: moving above, left, and right around the obstacle using the Kinect FORTH system. Each skill is demonstrated only once, with the human hand centroid trajectories recorded (see Figure 5a). The Kinect FORTH system captures real-time 6D poses of the human hand centroid and provides a colored model of the fingers, enabling precise trajectory acquisition. For each skill, the trajectory is smoothed using a Moving Average Filter (MAF), normalized to 30 timesteps, and transformed into the robot base frame.