# Modeling and Control of an Underactuated System for Dynamic Body Weight Support
## 1 Introduction  
The analysis of devices for rehabilitation purposes related to kinesitherapy designed during the last dozen years has indicated that body weight support systems are a crucial component. At present, there are many simple devices available on the market that are passive systems, as well as technologically advanced equipment fitted with an active, mechatronic body weight support system. Active systems enable a defined support load to be maintained, giving high accuracy, whereas passive systems (load value manually adjusted with a lever, counterweight, spring, etc.) often cause overload and a large difference between the adjusted load setting and recorded load setting during training. Therefore, devices fitted with an active body load support system are more favorable for the proper performance of exercises [1, 2, 3]. During the re-education of human gait, these systems are used to reduce body weight and reduce leg loading at the same time. A person performing exercises is clamped to the rope end of a winch with an orthopedic harness. When comparing rehabilitation devices fitted with an active load support system, the stationary element is mostly related to the fixed load bearing structure (it does not follow patient movements) and human gait is forced by mechanical orthosis or a treadmill [4, 5, 6, 7, 8, 9, 10, 11]. In devices with a treadmill, training is recommended for persons who have restored their gait to a degree, enabling unaided movement, but who still require assistance, or when gait manner, pace, and strength need improvement [12, 13]. Such devices provide technological advancements related to virtual reality and biofeedback [14, 15, 16, 17, 18]. One of the latest kinesitherapy devices is a mechatronic treadmill with a body weight support system developed by the Silesian University of Technology [19]. The active body weight support system is similar to other technologically advanced winches in rehabilitation devices for the re-education of human gait [1, 20, 21, 22]. It has two independent drives: a linear drum and series elastic actuator (SEA)-type drive. Because the range of vertical human body movements is rather small during gait, this system is only a body weight dynamic compensation system. Specifically, the SEA drive maintains the load according to the programmed settings, whereas the rotating drum operation is used when big vertical movements occur, e.g., upright standing, knee bends, walking up stairs. The SEA system features a specific design including the kinematic chain adaptability required due to interactions with humans, and it protects against overloading resulting from a delayed servoddrive response [23, 24, 25]. The SEA system includes one active part (controlled) and one passive part, which are connected by a connector with elastic and damping properties. Thanks to this solution, the device is an underactuated system, whereby the number of generalized variables is greater than the number of programmable variables [26, 27, 28, 29]. Note that the SEA drive minimizes offset-related support load errors in a reliable and quick manner to prevent overload during the performance of exercises. Moreover, any hazards to human health are prevented. The development of a control system that meets both requirements poses an interesting engineering problem. Considering the necessity to ensure the reliable operation of the device during any situation, control systems used for that purpose offer stability. An analysis of current papers on similar devices for the re-education of gait indicated that these systems most often employ a classic feedback loop and the settings of controllers are programmed by way of experiments. No previous paper focused on studies regarding stability. Therefore, the current paper is innovative as it looks at the development of a control algorithm for a body weight support system that takes into account stability analysis according to the Lyapunov theory of stability.