## 1 Introduction
[17] combines ProMPs with Gaussian multiplications, introducing primitives to handle obstacles, but this requires anticipating all possible situations. Alternatively, ref. [18] optimizes ProMPs by excluding regions using the Kullback-Leibler (KL) divergence and reward functions, with policies derived via Relative Entropy Policy Search (REPS) [19], while ref. [20] minimizes the Mahalanobis distance to the original ProMPs. However, these methods focus solely on the end-effector, neglecting collisions for other links and constraints, like virtual walls or position/velocity limits. They are efficient but struggle to handle multiple combined constraints. To address these limitations, ref. [21] integrates multiple task constraints in Joint and Cartesian space into a unified probabilistic framework, optimizing ProMPs using the gradient ascent-descent algorithm. However, the covariance matrix of ProMPs basis functions can face symmetric positive-definite issues during iteration, and optimizing timesteps satisfying task constraints increases computational cost. KMPs replace explicit basis functions with a kernel matrix, extending to linear constraints (LC-KMPs) [22] via quadratic programming (QP). Further, ref. [23] handles nonlinear constraints (EKMPs) by approximating them with the Taylor expansion to solve with QP. While EKMPs improve efficiency over [21], they face numerical instability and high computational costs as timesteps grow, and their reliance on Taylor expansion for nonlinear constraints limits the adaptability due to inaccurate estimations. When a robot adapts learned skills to new task-constrained scenarios, unexpected obstacles or human partners may appear during execution. Due to the significant replanning time required for MPs' adaptation, real-time reactions are hindered, limiting applications to offline and static environments. Thus, real-time movement adaptation is crucial for dynamic environments, often categorized into global and local strategies. Global methods focus on convergence towards the goal. For instance, Rapidly Exploring Random Trees (RRT)* utilizes space-filling trees to find feasible paths with automatic sampling length [24], and Covariant Hamiltonian Optimization for Motion Planning (CHOMP) employs gradient optimization for motion planning [25]. These methods exhibit fast convergence but struggle in dynamic environments. Model Predictive Control (MPC) reduces adaptation to a finite time horizon [26], but defining cost functions is challenging. Reinforcement learning-based adaptation trains end-to-end strategies [27], but it lacks reliable convergence guarantees and requires extensive training. Local methods generate nonlinear control fields evaluated in real-time. Artificial Potential Fields (APF) [28] represent obstacles as repulsive fields, offering computational efficiency, but are prone to local minima. Harmonic Potential Functions (HPF) [29] ensure no critical points in free space but lack analytical solutions. Inspired by HPF, Dynamical System Modulation (DSM) [30] ensures convergence using fluid dynamics around obstacles, with stability proven via Lyapunov theory and extensions to handle multiple obstacles [31], real-time sensing [32], and rotation dynamics [33]. DSM demonstrates high efficiency for real-time obstacle avoidance in dynamic environments, providing closed-form solutions without local minima. However, DSM assumes no initial collisions and requires dynamic modulation in advance. If new positions from the last timestep encounter collisions during convergence to the goal, the robot may stop, limiting real-time performance. Additionally, DSM treats the robot as a zero-dimensional point, focusing only on the end-effector and ignoring collisions with other rigid links. In this context, null space velocity control [34] and task priority control [35] are proposed for non-end-effector real-time obstacle avoidance. However, they assume the obstacles to be stationary and only consider robot non-end-effector portions.