## 1 Introduction  
In the past decades, millions of people have had problems with the motion ability of their lower limbs due to wars, diseases, traffic accidents, and natural disasters. In particular, some of them have lost their ability to work and been prevented from engaging in normal social contact [1]. Physical disability significantly reduces their quality of life and puts them under high physical and mental stress. Because the current medical level cannot make the limb regenerate, the prostheses for these amputees have become an essential means to compensate for the loss of limbs. Traditional mechanical knee prostheses cannot bring about a significant change in the lives of amputees because of unnatural walking, great physical exertion, and poor wearing experience of patients. Thanks to advances in actuation, microembedded computing, miniaturized sensing, energy storage, and automatic pattern recognition, many rehabilitative robots have been developed to help and recover human movement [2; 3]. To solve these limitations, APK, as a lower limb prosthesis, has drawn increasing research interests to help people deal with walking disabilities in the past years. Compared to a conventional passive prosthesis that has the shortcomings of inability to generate mechanical power and the lack of sensory feedback, APK improves the similarity of able-bodied gait, decreases hip work production, and reduces metabolic expenditure [4]. The active prosthesis is different from the passive prosthesis basically by the use of active components for locomotion assistance. Several research works focused on these prostheses and their control [5; 6; 7; 8; 9]. [PERSON] et al. [10] designed a transfemoral prosthesis actuated by a pneumatic cylinder and proposed an impedance-based finite-state controller. Because of the limitations of pneumatic power, an electrically powered robotic prosthesis was developed. In [11], a powered prosthetic knee actuated by brushless dc motor and a hierarchical finite state machine control approach is presented. [PERSON] et al. [12] discussed the optimization problem for the degree-of-freedom active knee joint actuated by a flat brushless DC electric motor for overcoming complex walking environment. The first commercial active prosthetic knee was the Ossur power knee, which utilized an echo-control strategy to mimic the movements of the subject's strong side leg. However, these prostheses did not take into account inherent elasticity, and they were affected by the low loads during use. Recent research has been developing an energy-efficient, compliant powered prosthetic knee. [PERSON] et al. [13] designed dual series-elastic actuators (SEAs) for storing and releasing the energy, where the agonist-antagonist SEAs could lower energy consumption of the active prosthesis. A clutch cable SEA for minimizing electrical energy consumption and satisfying the torque-angle behavior is discussed in [14]. However, the addition of a clutch paralleled with the motor increases the complexity and uncertainty of control. At the same time, a semi-active prosthetic knee for lowering energy consumption by utilizing a lockable parallel spring is proposed in [15]. However, despite promising previous work, there is a need for developing a more flexible bionic, lightweight APK that can be more easily adapted to a variety of activities and different prosthetic users. In the prostheses mentioned above, the impedance-based finite-state control method is the most popular control scheme. Many teams have successfully implemented it [16; 17; 18]. For example, when it comes to the variable stiffness mechanisms, a new structure combined with the parallel redundant mechanisms and variable impedance actuator is presented in [19], where a stiffness and position control for two degrees of freedom (DOF) is achieved by three actuators with flexible elements.