## 1 Introduction  
Neurological rehabilitation is the key to helping hemiparetic patients regain movement. Rehabilitation therapy usually consists of extensive task-specific and repetitive movements of the patient's limbs in order for them to regain lost mobility. Traditionally, these therapies are carried out by physiotherapists and nurses and they require intensive physical and occupational resources. However, the advent of robotic rehabilitation is enabling patients to perform these exercises with the sole assistance of a robotic device, with similar or improved effectiveness [1]. Moreover, IoT technology allows these therapies to be carried out, even at the patient's house or at nearby clinical centers, further developing the concept of tele-rehabilitation and showcasing its efficiency [2, 3]. Robotic rehabilitation is usually undertaken together with VR environments in order to engage and push patients to keep training, as patients' motivation and cognitive involvement have a great impact on rehabilitation outcomes [4, 5]. These platforms have also shown their ability for learning purposes [6].  
Over the last few years, many robotic systems have been developed in order to perform tailored rehabilitation exercises at home [7]. Usually, these platforms are connected to a computer, where serious games (mazes, puzzles, etc.) are shown in two-dimensional (2D) and three-dimensional (3D) scenes. The user is able to interact physically with these games through the robotic systems. Thus, it is desirable to properly recreate the sensation of handling objects in a virtual world [8, 9]. A haptic device or a collaborative robot with force feedback can trace the user's movements in a virtual world and restore appropriate reaction forces during contact with virtual objects. Furthermore, force feedback can also be used in order to help patients complete the exercise (i.e., assist-as-needed approaches [10]), or it can be established as resistive forces that change during the course of therapy to compel the user to exert a physical effort (in terms of strength level and duration), according to the patient's actual muscular capacity [11].  
Rehabilitation games are relevant scenarios in which to incorporate complex haptic effects, because virtual tasks are easier to understand if the haptic feedback is realistic [12]. These effects have several benefits, among them: the possibility to enhance the virtual scene and the ability to facilitate the rehabilitation, such as adding or subtracting the weight of the device by modifying the virtual mass or even making it weightless [13]. These virtual effects add immersiveness and realism to virtual interactions.  
A wide variety of realistic haptic effects can be simulated by combining the three main mechanical components: spring, damper, and mass. Or, in terms of their characteristic parameters, by combining stiffness, viscosity, and inertia. To use all of these effects, an elementary prerequisite is to ensure the stability of the human-robot interaction and, to this end, Section 2 provides a summary of stability conditions. Some are already known in scientific literature, but others are novel--mainly analytical stability conditions for virtual mass, virtual mass-damper, virtual spring-mass, and virtual spring-mass-damper systems. However, despite there being an awareness of stability limits, little or no research in haptics has focused on subjective discrimination between elastic, viscous, and inertial effects. In Section 3, we present a user study that was carried out to gain some insight in this area. Section 4 gathers the results and Section 5 discusses them before the conclusion.