## 1 Introduction  
At present, there are about 2,000,000 people suffering from stroke each year in China, with approximately 66 percent surviving, usually accompanied by motor function defects [1]. Due to the increasing number of stroke patients and the relatively small number of professional physicians, the task of upper limb rehabilitation training and assessment are arduous. For many patients, hospitals are relatively far away. Therapists can only provide one-to-one treatment. Home upper limb rehabilitation training is a more flexible method; it can provide patients with timely assistance, according to the needs of patients. Therefore, it is very meaningful to design an effective home rehabilitation training and evaluation method. Virtual game technology has developed rapidly and become an effective tool for rehabilitation training of post-stroke patients at present [2; 3; 4; 5]. Preliminary studies have shown that a virtual reality game system can provide subjects with a high level of enjoyment and increase motivation to engage in activity [6]. Virtual reality can make the measurement more immersive, and improve the enthusiasm and participation of the subjects. In addition, a virtual game system is very natural and active; it can provide some potential benefits for therapists and patients to supplement the lack of traditional methods [7]. [PERSON] et al. designed a ball game to measure the hand movements and assess the intra-session and inter-session reliability of upper limb performance indices in patients [8]. [PERSON] al. designed a Virtual Occupational Therapy Assistant (VOTA) for upper limb motor function measurement, and investigated performance indicators, which were generated during the practice of virtual daily activities [9]. A Kinect cardiovascular rehabilitation system was used to enable participants to perform exercises independently and individually at home [10]; the questionnaire survey showed that most of the subjects enjoyed this system. The methods used in the assessment of upper limb rehabilitation mainly include the Fugl-Meyer assessment (FMA) [11], the Wolf motor function test [12], range of motion (ROM) testing [13], the Stroke impairment assessment set (SIAS) [14], and others [15; 16]. These traditional assessment methods rely heavily on trained occupational therapists (OCT) and usually consume significantly more time [17]. Furthermore, because the rehabilitation assessment process is not quantitative, different therapists may make different assessment decisions. As an accurate markerless human motion capture tool, Microsoft's Kinect camera has been used in rehabilitation [18]. [PERSON] et al. [19] proposed a Kinect-enabled system for home-based rehabilitation, using dynamic time warping and fuzzy logic. [PERSON] et al. [20] developed a Kinect-based rehabilitation system for physical rehabilitation in public school settings. [PERSON] et al. demonstrated the clinical significance and effectiveness of the Kinect-based reachable workspace in Duchenne muscular dystrophy (DMD) and amyotrophic lateral sclerosis (ALS) [21; 22; 23]. Smoothness of hand movement is also a criterion to evaluate the rehabilitation status of upper limbs. [PERSON] et al. used the Fourier amplitude spectrum of the velocity profile to analyze the smoothness of the motion [24]. [PERSON] et al. quantified motion with respect to the three-dimensional curvature of the hand trajectory, aiming to establish a new measurement method independent of the duration of motion [25]. [PERSON] et al. designed the Armeo device task, and analyzed the smoothness indices' mean/maximum velocity, normalized jerk and number of peaks in velocity profiles [26]. However, these rehabilitation training games were designed with less consideration of the upper limb reachable workspace. Moreover, there are few methods of home rehabilitation assessment.