## 1 Introduction  
Serious games using virtual reality (VR) for rehabilitation purposes have gained popularity in recent years. The Leap Motion Controller(r) (Ultraeap, Bristol, UK) has become popular for such upper-extremity (UE) applications because of its affordability and ease of use compared to immersive VR headsets [1, 2]. This technology has been especially applied in stroke rehabilitation [3, 4, 5]. Studies evaluating effectiveness in stroke rehabilitation vary widely in pathological conditions, sample sizes, rehabilitation protocols, and methodologies. However, they consistently demonstrate positive outcomes, underscoring their value as a highly effective tool [1]. These studies commonly feature virtual games, developed as engaging therapeutic exercises. Patient progress is typically assessed after multiple sessions using qualitative measures such as the Action Research Arm Test, Wolf Motor Function Test, or QuickDASH. However, there is notable variability in the virtual tasks, number of sessions, and assessment methods employed. Some studies incorporate quantitative functional indicators, such as dexterity (measured with tests like the Box and Block Test (BBT)), range of motion, or strength, before and after rehabilitation sessions to evaluate improvement [4; 6; 7]. Beyond stroke, studies using the Leap Motion Controller (LMC) for rehabilitation are relatively limited but include burn patients [6], spinal cord injury (SCI) patients [8; 9; 10], Parkinson's disease (PD) patients [7], or children with cerebral palsy [11]. A standardized approach to assess VR's training effectiveness is to integrate a validated test directly into the game, enabling consistent tracking within the virtual environment itself [10; 12]. The most common approach is using the BBT, which evaluates functionality by assessing manual dexterity within the activity domain, as outlined by the International Classification of Functioning, Disability, and Health. In patients with Parkinson's disease, this virtual BBT (VBBT) has been implemented with immersive glasses, yielding excellent results in terms of rehabilitation effectiveness and patient acceptance [13]. VBBT has shown a strong correlation in test scores with real BBT (RBBT) in both stroke patients [14; 15], as well as in those with SCI [8]. These investigations compare the number of blocks transferred within a particular time, consistently showing that fewer blocks are transferred in the virtual environment compared to the physical version while still demonstrating a strong correlation with RBBT [13; 15]. This trend may be attributed to several factors, including the need for users to adapt to the virtual interface and the different feedback mechanisms present in virtual environments. A notable finding from these studies is that performance improves over time as participants adapt to the virtual interface, with the learning curve being comparable for both the affected and non-affected hands [4]. Few studies have leveraged the hand-tracking capability of the VR systems to analyse trajectory patterns and their smoothness. [PERSON] et al. [14] computed smoothness indexes from hand trajectories, finding significantly smoother movements in healthy individuals compared to stroke patients. [PERSON] et al. [16] quantified movement patterns in SCI patients after robotic rehabilitation training, assessing smoothness and movement efficiency through hand kinematics obtained by performing virtual trajectories with LMC. Although the training focused on the UE rather than the hand, they hypothesized improvements in hand movement and precision. Reduced trajectory length and velocity peaks supported their hypothesis of improved motor control, though BBT performance, measured by number of blocks transferred, remained unchanged, likely due to the short duration of the training. However, the VR systems have not been used to track the hand movement (i.e., detailed kinematics of the hand joints), probably because of limitations in accuracy and reliability [17].