## 1 Introduction
By utilizing mechanical sensors or biological signals [20; 21; 22], finite state controllers decompose a gait into several different motion stages and tune a series of corresponding static parameters for each state of every subject. Each motion stage needs at least three parameters: stiffness, damping, and spring balance angle. Thus, the number of tunable parameters dramatically increases with the number of motion states, motion modes, driven joints, and quasi-controlled limbs [23; 24]. The tuning process of these parameters is very complicated and time-consuming for the prosthetic user. More recently, a robust adaptive impedance method based on the nonscalar boundary layer and sliding surface for tracking knee angle is proposed [4], which overcomes the influence of parametric and nonparametric uncertainties. However, to achieve adequate joint trajectory tracking control, a precise robot dynamic model is required. Moreover, the calculation of the APK dynamic model is very complex, and certain model parameters are hard to obtain. The time-delay estimation (TDE) technique presented in [25] is an effective method to reckon nonlinear uncertain systems with time-delayed information. Then, the time-delay control using TDE technique is developed in many robot systems [26; 27; 28; 29; 30]. TDC is generally recognized as a simple, efficient, effective, robust model-free scheme. It mainly includes two parts: the TDE part, which is used to reckon the uncertain and nonlinear system dynamics, and the robust control part, which is used to enhance the dynamical performance. Much research has been focused on the latter part of TDC [28; 29; 31]. Among the many existing robust control methods, sliding mode control has advantages in dealing with uncertain, nonlinear, and bounded external disturbances [32], and has been widely used in TDC. To assure finite-time convergence and avoid the singularity, a model-free nonsingular terminal sliding mode control based on TDE is applied to a robot manipulator in [28]. Recently, an adaptive fractional-order nonsingular terminal sliding mode control with TDE scheme applied in joint tracking of manipulator is discussed [33]. Although the proposed scheme is superior to many other nonsingular terminal sliding mode controls, fractional-order sliding manifold is too complex in the practical applications. The TDC schemes mentioned above make full use of the basic framework of the TDE technique and adopt the improved sliding mode control schemes to ensure the tracking accuracy. However, these schemes mainly focus on a small TDE error. With the external disturbance or parameter variations, the TDE-based controller cannot always guarantee accuracy. Thus, humans encounter many uncertain disturbances in the actual walking. In this paper, a novel APK with a variable stiffness actuator (VSA) is presented for more flexibility. A new robust TDC scheme for the flexible joint is proposed to achieve improved tracking accuracy based on adaptive nonsingular fast terminal sliding mode control (ANFTSMC) without a sophisticated physical model or multi-tuning parameters. The nonlinear dynamics equation of APK can be effectively eliminated by using TDE. ANFTSMC offers fast adaptation and chattering reduction. To effectively settle the issue of disturbances, an adaptive fuzzy neural network compensator can be added, which can guarantee better robustness in the complex environment. The primary contributions of this paper are described in the following three points:  
* A novel APK with a variable stiffness actuator (VSA), which can provide the ability to adjust joint stiffness depending on the different gaits, is proposed. * An adaptive robust force/position controller for flexible APK by using the TDE technique combined with adaptive nonsingular fast terminal sliding mode control and fuzzy neural network is proposed. * The stability analysis of the control system by Lyapunov stability theory is carried out, and some demonstrations using the virtual prototype illustrate the effectiveness of the proposed algorithm.