## 1 Introduction  
Lower-limb losses have become a growing global problem. Traffic accidents, wars, sports injuries, burns, complications from diabetes or tumors, and forced amputation are the common reasons for lower-limb losses [1]. In order to help lower-limb amputees restore mobility and return to normal life, scientists have conducted a lot of studies, especially on lower-limb prostheses which are good replacements for amputated parts. At present, most commercially available lower-limb prostheses are passive, because they are relatively cheap, simple in structure and convenient for clinical application. However, walking with passive prostheses will consume 20% to 30% more energy than healthy individuals [2]; the asymmetries between the passive prosthesis and the sound leg will cause back pain, osteoarthritis and osteoporosis [3]; the walking speed may be 10% to 65% slower [1]; mobility in different terrains is limited. Therefore, an intelligent lower-limb prosthesis is worthy of study due to its powerful potential in walking speed selection and adaptation to different terrains. Scientific researchers have made great progress in intelligent lower-limb prostheses recently including gait planning and control strategies. The common control framework of intelligent lower-limb prostheses is a hierarchical framework. The framework proposed in [4] includes a high-level controller, mid-level controller and low-level controller. The high-level controller focuses on human intention recognition including locomotion mode classification, gait parameter determination, and gait phase determination. The mid-levelcontroller receives the locomotion mode, gait parameters and gait phase and then computes the desired joint torques. Therefore, the gait parameters and gait phase are always related to the joint trajectory generation and controllers. Ref. [5] computes the gait phase by thigh motion; Ref. [6] uses dynamic movement primitives (DMPs) to represent joint motion trajectories and the amplitude parameters of DMPs are the functions of walking speed; Ref. [7] introduces a predictive model that represents gait kinematics as a continuous function of gait cycle percentage, speed, and incline; Ref. [8] applies finite state impedance control (FSIC) which sets different impedance parameters during different gait phases to simulate human joints' impedance characteristic. To achieve better locomotion mode conversion and dynamic motion, the fast and accurate determination of gait parameters such as walking speed are necessary. To estimate walking speed, wearable sensors such as inertial measurement units (IMUs) are used in previous works. For now, the walking speed estimation methods can be classified into three kinds: Direct integration method, kinematic model building and machine learning methods. Ref. [9] installs the IMUs on the lower limb and estimates the walking speed by the direct integration method. The potential integral drift problem can be addressed by real-time velocity correction. Ref. [10] applies wearable accelerometers on the wrist and proposes a virtual inverted pendulum model to estimate the walking speed. Ref. [11] develops subject-dependent and independent machine-learning models to determine walking speed. However, methods based on direct integration or kinematic model building only estimate the walking speed once after an entire gait cycle. As for machine learning methods, collecting large amounts of data on lower-limb amputees is unfriendly and the trained regression models or neural networks are not adaptive to amputees' daily gait changes. This paper designs a phase-based step frequency estimator and a user-dependent stride estimator to estimate walking speed fast and accurately. Compared to direct integration methods, the proposed method can continuously estimate the walking speed in real-time. Compared to the machine learning methods, the proposed method has no need to collect large amounts of data under different speeds which are friendly to the lower-limb amputees.