Kinematic Analysis of a 3-RPS Parallel Mechanism for Passive Rehabilitation of Sports-Related Ankle Sprains

Ankle sprains are common among athletes, especially in ball sports, and often require targeted rehabilitation to restore joint mobility. This article presents the design and evaluation of a 3-RPS type parallel mechanism developed for the passive rehabilitation of lateral ankle sprains. The system replicates natural ankle movements using inverse kinematic models based on geometric methods and screw theory. To solve the forward kinematics, three numerical strategies were tested: gradient descent, gradient descent with momentum, and Newton-Raphson, with the latter being the most efficient. Workspace analysis confirmed that the mechanism covers the physiological ranges for dorsiflexion-plantarflexion (30° to -20°) and eversion-inversion (20° to -10 °), with vertical displacements ranging from 0.2044 m to 0.3046 m. Both positional and orientational workspaces were analyzed to ensure complete therapeutic coverage. A singularity analysis identified twelve critical configurations, all of which are outside the effective operating range. Structurally, the device is compact and lightweight (≤ 3.5 kg), ensuring portability and ease of use in clinical and sports environments. Its precision, stability, and ability to generate controlled, repeatable ankle movements make it a valuable tool for early-stage rehabilitation, promoting functional recovery in injured athletes.