# Asymmetric Constrained Control of a Cervical Orthotic Device Based on Barrier Sliding Modes
## 1 Introduction  
### Preliminaries  
Currently, emerging technologies provide feasible solutions for medical problems [1]. In terms of physical medicine and rehabilitation, the improvement of classical static orthosis and prosthesis has helped patients to recover their functional abilities in a shorter period of time [2]. Static orthosis usually immobilizes or restricts the motion of human limbs. They are made with diverse materials depending on their applications [3]. On the other hand, an active orthosis is a device that increases the ability of a person suffering from pathology or a fracture augmenting the power at one or more joints of the extremities by means of a mechanical actuated structure [4]. Depending on the physical affectation, the specialists should suggest the kind of orthosis to use. Cervical orthoses are a particular case of these devices which are designed to control head movement and neck function, cervical orthoses vary greatly in the degree to which they immobilize the neck and unload head weight [5, 6]. The most common cervical orthoses applications include the physiotherapy of head pain such as migraines [7], correction of spinal deformities, rehabilitation in postoperative procedures [8], whiplash-associated disorders, and non-operative treatment for fractures [9, 10]. Some examples are the halo vest, Philadelphia collars, Minerva collars, aspen, stiff-neck Miami collar, among others [3, 11]. Even though, several studies support the effectiveness of these orthoses immobilizing the fracture sites [10], current studies have demonstrated the advantages of using active orthoses, not only in cervical injuries but also in lower-limb rehabilitation [12]. The design of a cervical orthosis must satisfy the support, prevention, and correction of cervical injuries in patients [3]. Moreover, it must satisfy three-point pressure in parts of the body where nerves and tissues are not affected by the structure system [13]; the manufacturing of cervical orthosis devices shouldbe circumferential for supporting the neck and limiting the range of motion. In addition, the manufacturing device needs to be elaborated with material that does not irritate the skin. Indeed, some researchers evaluated different types of orthosis that have effects on the quality of treatment depending if the injury is a fracture, postoperative treatment, or deformity [14]. The main parameters to be evaluated in the design of an orthosis are the restricted motion, the correct size, comfort assessment control of snaking, and control of flexion/extension/rotation [8; 10]. Some evaluations compare the efficiency of common cervical orthosis finding the main complications of wearing those devices [10]. The main issues related to wearing a cervical orthosis are infections, pressure in plastic vests, injuries in nerves, occipital ulceration, stiffness of facial muscles, and control motion [8; 15]. Robotic cervical orthoses have to ensure a correct control range of motion during rehabilitation for either postoperative surgeries, fractures, or cervical injuries during rehabilitation sessions [16; 17]. Some of them use new mechanical technologies for their design, as well as control algorithms to ensure the safety and correct treatment of the patient. The robotic cervical systems which are bio-inspired designs usually incorporate the measurements of bio-potentials such as electromyography or electroencephalography for active control [17; 18; 19; 20]. Therefore, the control theory applied in any orthosis device must guarantee the security of the patient during the tracking trajectory of natural neck motion [17]. Among others, adaptive controllers, impedance adaptive controllers [21], and active disturbance theory [22] have been used in the control of active orthoses. To enhance the safety capacity of active cervical orthoses, the control development ought to consider movement and velocity restrictions. Barrier Lyapunov functions (BLF) have been implemented to deal with state and control restrictions [23].