**2. Dry gangrene**

Dry gangrene represents coagulative necrosis of ischemic tissue, caused by inadequate blood supply due to peripheral artery disorders. The term dry gangrene is used only for necrosis of the acral limb [6, 7]. Patients with atherosclerosis, hypercholesterolemia, and diabetes mellitus are susceptible to dry gangrene, particularly when they smoke. The low local oxygen level provokes putrefaction without bacterial growth. The affected portions become dry, solidified, and reddish black (**Figure 1**). Once gangrene has developed, the affected tissue is no longer salvageable. The boundary of the dried lesion is sharply demarcated from the nonischemic skin so that autoamputation may follow [8]. Because of the lack of infection, dry gangrene is not so emergent as wet gangrene and gas gangrene. However, dry gangrene may develop to wet gangrene when the secondary infection happens. Diabetes mellitus is a serious and the most important risk factor for developing both dry and wet gangrenes.

#### **Figure 1.**

*Dry gangrene (gross appearance of two cases). Atherosclerosis-induced dry gangrene is seen in the foot (left). The border of necrotic lesion is relatively sharp. In the right panel, the toes of a diabetic patient are dry and blackcolored, and wet gangrene with red swelling and epidermal blister formation followed (the courtesy of Drs. Mitsuhiro Tachibana and Yasuhito Kaneko at Department of Diagnostic Pathology and Dermatology, Shimada Municipal Hospital, Shimada, Japan).*


**Abstract**

2D Ballbot is an actual under-actuated system with second-order nonholonomic velocity constraints and input coupling case where only control input is employed to control two outputs of the system. Controlling such a system is not easy because it faces many changelings including nonlinearities, external disturbances, and uncertainties. This study proposed a robust control system for a Ballbot mobile robot. The proposed control scheme is constructed using the hierarchical sliding mode control technique. The kinematics and dynamics of the Ballbot are derived. A Lyapunov function is used to prove the stability of the closed-loop control system. The stabilizing and transferring problems are investigated through several simulations and experiments by using the actual Ballbot platform.

**Keywords:** sliding mode control, Ballbot, under-actuated system, Lyapunov analysis, stabilizing and transferring