## 4 Rigid Robotic Exoskeletons for Hand Rehabilitation  
Rigid robotic hand exoskeletons were the first solutions developed in the framework of robotic devices for rehabilitation and assistive applications from a chronological point of view due to their ability to transmit high forces with high precision through rigid mechanical transmission systems [21]. Rigid transmission mechanisms, i.e., systems composed of rigid links connected by joints are the earliest and simplest methods of transmitting actuator motions to the required human joints. The exoskeleton mechanisms were designed to establish a direct and rigid connection between the actuator motion space and the corresponding human joint motion space. In fully actuated systems, direct and inverse kinematic relationships can be established and, as a result, any measurements, such as encoder signals, current, or torque obtained from the actuator can be used to evaluate, with suitable accuracy, the corresponding state of the fingers. This inherent correspondence between the actuator and the hand joints allows for intuitive and easy hand control and no further tracking systems are needed in most cases.  
In this context, two main types of actuation can be distinguished: direct drive and geared systems. Direct drive refers to systems where the actuators are directly connected to the structure [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32], while geared systems use a gear train to connect the actuator to the structure [33; 34; 35; 36; 37]. In fewer applications, the two approaches are merged [38].  
The general principle for wearable robots and specifically for rigid hand exoskeletons is to position and connect the actuators in the proximal part of the actuated limb or finger to limit the inertia and avoid user fatigue. Usually, in hand exoskeletons, the motors are positioned on the back side of the hand to ensure portability, minimize the device's complexity, and reduce transmission losses during joint movements. However, this solution does increase the overall weight on the user's hand due to the actuator's mass and size. Rigid hand exoskeletons weigh, on average, 500 g and have a relatively large volume compared to the human hand size [39]. Keeping the device portable while employing an actuator for each hand DOF is nearly impossible. To limit the device weight and complexity, most rigid exoskeletons for the hand are characterized by few actuators, and joint motions are coupled by a mechanical transmission system [40]. Various types of mechanical transmission and coupling systems have been designed and implemented [5]. These structures can actuate multiple DOFs simultaneously, in a controlled manner, using a single actuator. Based on this concept, different transmission systems can be identified: four-bar linkage mechanisms [23; 31; 41; 42; 43; 44], remote center of rotation [22; 28; 45], serial links mechanisms (suitable if the device needs to fit various users) [24; 27; 34; 36], and matched-axis mechanical structures [46; 47; 48]. Each of these proposed solutions allows the actuation of the hand DOFs while maintaining the device's portability and functionality. However, notwithstanding design solutions, these devices are often bulky and heavy, and most of them do not adequately account for a misalignment between the robot's and the user's joints [49]. For example, in [32] a revised version of the four-bar linkage mechanisms is presented. When the exoskeleton is not worn on the finger, it has three DOFs and an actuator; therefore, it is underactuated, while when it is worn, the finger biomechanical structure closes the mechanism and reduces the number of DOFs to one. In this way, the remote center of motion for the metacarpophalangeal joint is no longer necessary, leading to a significant simplification of the mechanical structure.  
Some examples of rigid hand exoskeletons are shown in Figure 3. In the following, we present a more thorough analysis of rigid hand exoskeletons used for rehabilitation in terms of actuation and functional features.