## 1 Introduction  
Backstepping design is a systematic recursive design procedure based on the choice of a Lyapunov function. This approach is suitable for the design of a class of nonlinear systems in the strict feedback form. In the design procedure, the system variable is treated as an independent (or a fictitious) input for subsystems, and consequently, each step results in an updated control design for the next step. The control law for each step is refined with the satisfaction of the Lyapunov function, such that the stability for each subsystem can be guaranteed as outlined in [1]. In the traditional backstepping design, system robustness can be maintained by using high gain control (under the condition that the system uncertainties are state dependent). However, in the case of a motion system, the system may be subject not only to uncertain parameters, but also to un-modeled system dynamics; which is state independent, such that the high gain backstepping design may not result in the asymptotic stability of such a system. In recent years, several control strategies, such as adaptive control [2; 3; 4], sliding mode control [5; 6] and neural networks [7; 8], have been integrated into the logical backstepping design procedure in order to enhance system robustness. An integral action has been widely used in adaptive backstepping control for eliminating steady state error, while a smooth sign function was adopted for avoiding chatter in sliding mode control. Some nonlinear functions can be estimated using neural networks with a backstepping design scheme. Other robust backstepping control approaches for specific nonlinearities, such as system time delays [9], mismatched uncertainties [10], backlash-like hysteresis [11] and input uncertainties [12], were also developed to make the resulting system insensitive to model uncertainties and external disturbances. Moreover, in recent studies [13; 14; 15; 16], it has been demonstrated that the backstepping controller design concept can be applied for different nonlinear systems. However, only numerical simulations were provided. In this paper, this design concept is applied for the control of a servo mechanical system. Not only the theoretical derivation is given, both the numerical simulations and experimental verifications are addressed, as well, to verify the feasibility of the proposed method. In regard to [15], the authors proposed an integral type robust controller based on the sliding control design concept. However, the controllers cannot attain the so-called approaching/sliding conditions, so the stability proof will be relatively complicated. One of the main difficulties in dealing with motion control is to suppress the effect of friction, which exists in the moving mechanism. Two main kinds of friction behavior have been observed experimentally in the literature, namely the Stribeck effect and hysteresis [17], occurring in sliding and pre-sliding regions, respectively. These nonlinear properties of friction induce undesirable behavior that includes steady state positioning errors, tracking lags, limit cycles and stick-slip motion. In the recent decade, friction model-based compensation approaches and the disturbance observer have been widely adopted for motion control problems [18; 19; 20; 21; 22]. Due to the complexity of friction, recently, a nonlinear damping-based backstepping design was proposed, where the friction effect is taken as a nonlinear function and is going to be measured by a time delay estimate (TDE) [23]. Then, the control performance is further enhanced by considering the TDE together with an internal model control [24]. However, the friction behavior is a highly nonlinear phenomenon, which is difficult to be well described, especially for the inter-medium region between static friction [25] and dynamic friction. Even though there were several friction models in the literature, they can only be applied for specific friction regions. It is difficult to obtain quantitative analysis of friction due to its time and position varying nature.