Robotic Leg Prosthesis: A Survey From Dynamic Model to Adaptive Control for Gait Coordination

Gait coordination (GC), meaning that one leg moves in the same pattern but with a specific phase lag to the other, is a spontaneous behavior in the walking of a healthy person. It is also crucial for unilateral amputees with the robotic leg prosthesis to perform ambulation cooperatively in the real world. However, achieving the GC for amputees poses significant challenges to the prostheses’ dynamic modeling and control design. Still, there has not been a clear survey on the initiation and evolution of the detailed solutions, hindering the precise decision of future explorations. To this end, this paper comprehensively reviews GC-oriented dynamic modeling and adaptive control methods for robotic leg prostheses. Considering the two representative environments concerned with adaptive control, we first classify the dynamic models into the deterministic model for structured terrain and the constrained stochastic model for stochastically uneven terrain. Inspired by the concept of synchronization, we then emphasize three typical problems for the GC realization, i.e., complete coordination on structured terrain, stochastic coordination on stochastically uneven terrain, and finite-time delayed stochastic coordination. Finally, we conclude with a discussion on the remaining challenges and opportunities in controlling robotic leg prostheses.


I. INTRODUCTION
R OBOTIC leg prostheses that emulate the gaits of missing biological limbs first emerged in public life over a century ago [1], [2].Since then, a wide range of robotic devices have been developed in this field.Most available robotic leg prostheses are designed as passive devices, thus leading to a limited capability to replicate the natural gait of healthy individuals [3].In contrast, powered prostheses that utilize microcontrollers can perform various tasks [4].These advanced prostheses can ensure walking stability [5] and save metabolic costs [6], [7] during ambulation.Despite these advancements, individuals using robotic leg prostheses are less stable and have worse symmetry between lower limbs [8] [9] than healthy people, primarily due to system uncertainties [10] and environmental disturbances [11] within dynamics, significantly limiting the broad application of robotic leg prostheses.
In contrast, humans can achieve stable and efficient ambulation in complex environments subject to various disturbances.It mainly benefits from the collaborative behaviors of locomotion systems observed in the real world [12], [13].Therefore, gait coordination (GC) awareness of prosthesis-human systems (PHSs) has attracted much attention from the dynamics field over the past decades [14].To enhance GC performance, recent advancements in robotic leg prostheses primarily focus on two aspects: dynamic models [15] and adaptive control methods [16], [17], [18].Nevertheless, no survey paper has presented a comprehensive review of these advances in robotic leg prostheses from a dynamics perspective, unifying the dynamic models and the adaptive control methods for GC, thus inspiring the focus of this paper.
For the first aspect, the dynamics involved in PHSs are often characterized by uncertainties or partial knowledge, such as the unknown mass distribution [19] and the intricate footground interaction [20].In other words, system uncertainties can arise from various sources, including inaccurate modeling of dynamics [21], sensor noise [22], [23], and environmental disturbances [24].These dynamic uncertainties may lead to system instability and poor GC performance, thus posing significant challenges in designing controllers for robotic leg prostheses.Therefore, alternative and improved models are necessary to address the control design of robotic leg prostheses.Specifically, by classifying human-environment interactions, one can establish accurate representations of dynamics for PHSs, allowing for a taxonomy of control concerns.In response to these challenges, extensive ongoing research has been conducted in the dynamic modeling of PHSs [19], [25], and this paper first aims to provide a comprehensive review of dynamic models in two representative environments, ranging from walking on structured terrains to walking on stochastically uneven terrains.
The second aspect is controlling the collaborative behaviors of lower limbs, thus forming GC of PHSs within the dynamics.Correspondingly, the advanced control design of robotic leg prostheses is increasingly investigating GC from a dynamics perspective [26].Therefore, taking advantage of the new insights into synchronization from the dynamics theory, we emphasize the definitions of GC in unilateral amputees with robotic leg prostheses.The origin of the word synchronization is a Greek root (σ υγ χρoυoζ , which means "to share the common time"), as an agreement or correlation in time of different processes [27].Typically, the synchronizations contain complete/identical synchronization [28], [29], stochastic synchronization [30] [31], lag synchronization [32], [33], etc.By understanding these definitions, we can develop control methods to address the challenge of GC corresponding to different considerations [34], [35], such as structured terrains [36], [37], stochastically uneven terrains [38], [39], [40], and the presence of unknown stride frequency [41].In this regard, we expect to outline three fundamental solutions related to GC of PHSs: complete coordination for structured terrains, stochastic coordination for stochastically uneven terrains, and finite-time delayed stochastic coordination considering synchronization delay.
In this article, we present an up-to-date review of the recent advancements in robotic leg prostheses, focusing on dynamic models and adaptive control methods for GC.We propose the definitions of GC and then classify the control concerns correspondingly.Considering the two representative environments and unknown stride frequency, we review the adaptive control methods that have grown alongside these robotic devices, which are as follows: 1) complete coordination for structured terrains; 2) stochastic coordination for stochastically uneven terrains; 3) finite-time delayed stochastic coordination considering the synchronization delay.Moreover, considering the hierarchical framework of the control advancements, the control methods reviewed in this paper can be divided into two categories: high-level and low-level.This overview has the following contributions: 1) From the perspective of nonlinear dynamics, this review, for the first time, presents the most comprehensive survey on GC in unilateral amputees with robotic leg prostheses, providing valuable insights and further research in this field.2) With an understanding of two representative environments in real-world ambulation, one can discuss the dynamics from the deterministic to the stochastic model, thus allowing for a more accurate classification of control concerns for this review.3) Taking advantage of the insights into the synchronization in dynamics theory, we outline the conceptualization of GC, thereby integrating three fundamental control solutions related to achieving the GC performance of PHSs.The remainder of this paper is organized as follows.Section II first presents the dynamic models for two representative environments.Section III gives the necessary definitions of GC and the corresponding control concerns for PHSs.Section IV reviews the control methods for three problems: walking on structured terrains, stochastically uneven terrains, and unknown synchronization delay, respectively.Eventually, Section V concludes this paper, followed by future research directions.

II. PROSTHESIS-HUMAN DYNAMICS
In the context of the unilateral amputees with robotic leg prostheses, the walking environments, i.e., the structured terrains and stochastically uneven terrains, can refer to two types of bipedal dynamics, respectively.A summary of these two dynamics is shown in Fig. 1.
The first is the deterministic model, which uses the Lagrangian method of multi-body dynamics.This modeling approach considers the interactions and dependencies between the human, prosthesis, and ground, aiming to achieve heterogeneous coupling and nonideal contact [57], [58], [59].The second type is the stochastic model, which operates within a probabilistic framework.This stochastic model incorporates the uncertainties and variability associated with ground reaction forces (GRFs) on stochastically uneven terrain.
Unlike the bipedal robot models with symmetrical structures [60], PHSs typically consist of three main components: human body, healthy lower limb, and leg prosthesis.These components interact with each other dynamically, forming a heterogeneous coupling.Consider a PHS that consists of an n b degree-of-freedom (DoF) human body α n b ×1 , an n h DoF healthy lower limb α n h ×1 , and an n p DoF prosthesis α n p ×1 in a vertical XOY coordinate frame.The full state of PHS can be represented as where are generalized coordinate vectors of the human body, healthy limb, and leg prosthesis, respectively.In addition, n = n b + n h + n p , and n h = n p .

A. Deterministic Model With Uncertainty
Under a deterministic framework, the dynamic model of PHSs can be derived using the Lagrange method [19], [25].
First, the kinetic and potential energies of each link are computed.Then, the total kinetic and potential energies, i.e., T f (q, q) and V f (q, q), can be obtained by summing, respectively [61].Finally, the equation of motion (EoM) can be derived through the Lagrangian equation of the second kind, i.e., d dt with and Q ex being the generalized force vector applied to the system.In this dynamic model, the generalized force vector Q ex contains two parts, namely, where the designed control torque vector u is applied to the active joints of the prosthesis.In addition, F e is the generalized force vector corresponding to GRFs [37], [62], i.e., and J T ∈ R n×s is the Jacobian matrix that transforms GRFs, i.e., F c ∈ R s×1 , into generalized forces.Generally, some researchers have used holonomic constraints [63] to determine these GRFs.Besides, the spring-damper combination [21], [64] is also a solution for the GRFs, and the relationship between the GRF and the foot-ground penetration of the i-th dimensional space is given as where δ i denotes the foot-ground penetration generated by contacts, k 0,i is the stiffness of the contact model, and C 0,i is the damping of the contact model.Importantly, since the integration of weight and physical noise in the load cells, GRF estimation [65], [66], [67] is a better alternative than direct GRF measurement.Overall, by applying Lagrange's equation (3) and corresponding detailed forms of generalized forces given by ( 4)- (7), one can obtain a deterministic model of PHSs [37]: where M(q) ∈ R n×n is the symmetric positive definite inertia matrix, C(q, q) ∈ R n×1 represents the Coriolis/centrifugal matrix, and N(q) ∈ R n×1 denotes the gravitational force vector.It is seen from ( 8) that all dynamic terms, i.e., M(q), C(q, q), N(q), and F c , are represented as deterministic forms.

B. Stochastic Model With Constraints
Unlike walking on structured terrain, stochastically uneven terrain may suffer from the random effect of GRFs, i.e., F c [72], which introduces stochastic nonlinearities into the deterministic system (8).To this end, the standard Wiener process provides a solution for establishing the stochastic models, referring to several stochastic differential equations (SDEs) discussed in an introduction [73].In the context of PHSs, the stochastic model of PHSs can be effectively modeled using SDEs [40], [41]: where M ∈ R n×n , C ∈ R n×1 , and N ∈ R n×1 are defined in (8), w is a standard Wiener process, h T 2 is an unknown nonlinear function vector, and Fc denotes the state-dependent term of generalized force vector.Therefore, such a stochastic system provides a refined representation of the system dynamics, taking into account the stochastic factors of the systems.
In the second category, the stochastic nonlinearity [40], [41], [72], [73] involves the particularity of joint structure and walking gait, thus leading to a state boundary issue.Specifically, the time-varying state constraints can be given by [11], [40], and [41] q = q (t) |q inf (t) ≤ E q (t) ≤ q sup (t) , (10) where q sup (t) and q inf (t) denote the upper and lower boundaries of the states vector q (t).Therefore, the most challenging problem in controller design for the stochastic PHSs arises here when the system should ensure the state constraints.Remark 1: Establishing stochastic models is rather intricate for dynamic modeling.One can establish the deterministic Euler-Lagrange and stochastic models by classifying two representative environments, allowing for a more accurate representation of system uncertainty and constrained randomness, respectively, as shown in TABLE I. Regarding these two dynamics, the differences in control design stay in two aspects.First, compared to the deterministic model ( 8), the stochastic model may lead to a differential operator subject to the Wiener process [73].Second, different from the deterministic desired trajectory (8), the reference trajectories are stochastic or even bounded [11], [40], [41], simultaneously.
Remark 2: It follows from the stochastic model that the state constraint is a basic issue that needs to be solved.However, a limitation of the proposed stochastic model is that the walking experiment only tested two subjects in one stochastic condition, namely, sandy terrain.The constraint data collected from other conditions, such as terrains and subjects, are required for the constraint modeling.Moreover, stochastic testing for amputees has not yet been conducted, posing significant challenges for the modeling of prosthetics.Thus, the modeling complexity is improved.Researchers can further investigate this stochastic model and develop effective control strategies for robotic leg prostheses.

III. CONTROL CONCERNS FOR GAIT COORDINATION
Different metrics have been used to measure the performance of a prosthetic controller.These metrics were grouped into physiological, functional, and behavioral outcomes, forming hierarchical and correlative results.Specifically, physiological outcomes are typically performed with the amputee's own metabolic cost, muscle activity, and intent recognition, providing a more relevant evaluation for amputees.Furthermore, the physiological outcomes directly relate to functional outcomes.Regarding functional outcomes, gait segmentation, terrain recognition, and control precision are helpful to quantify the performance of novel control strategies objectively.Notably, both physiological and functional products are crucial for behavioral outcomes, such as GC performance, symmetry in joint power, etc.The results are reported in TABLE II.
In this survey, we emphasize the spatiotemporal gait symmetry during walking [74].Hence, this section first proposes three definitions of GC by taking advantage of the insights into the synchronization theory, as shown in Fig. 2.Then, they discuss the connections between definitions and the prosthesis control methods.
Consider the dynamic model in ( 8), a set of positions and angular displacements of the system can be given by  where denote the generalized coordinate vectors of the human body, the healthy lower limb, and the prosthesis, respectively, with n = n b + n l + n p and n l = n p .

A. Complete/Identical Coordination
Complete/identical coordination refers to the most general form of synchronization, referring to state variables' equality of time evolution.
Definition 1 [28], [29]: Regarding (8)(11)(12), complete coordination is defined as the identity between the states of the prosthesis q p and that of the phase-delayed healthy lower limb q L .Suppose the gait cycle of walking is T g , then the phase delay should be T g 2 according to the natural behavior of human walking.Hence, the complete coordination subject to asymptotically stable can be defined as ) where e (t) := q p (t) − q L (t − ) is the coordination error vector, ∥•∥ denotes the Euclidean norm, and = T g 2 is a determined phase delay.
Remark 3: The definition of complete coordination (13) gives two hierarchical control objectives for the structured terrains.First of all, a high-level frame estimating the state of the healthy lower limb, i.e., q L (t), is required, thus providing the adaptation to complex tasks.Estimating different locomotive tasks and gait phases is an important component of prosthesis control.In addition, the low-level control is the response to ensuring the coordination error, i.e., e (t), thus solving the stable adaptation to model uncertainty.Note that these unknown dynamics are difficult to model accurately, and adaptive control is highly desired.The adaptation to complex tasks and stable adaptation to model uncertainty are both essential to the complete coordination of PHSs.

B. Stochastic Coordination
Stochastic coordination is an extended form of complete coordination, referring to the equality of time evolution of state variables in probability.
Definition 2 [30], [31]: In the presented system (9) (11) (12), stochastic coordination is defined as the identity between the states of the prosthesis q p and of the healthy lower limb q L subject to random noise.The existence of stochastic coordination implies that the response system is asymptotically stable in probability, namely, (14) where e (t) = q p (t) − q L (t − ) is the coordination error vector, E (x) denotes the expectation of x.
Remark 4: Regarding stochastic coordination (14), there are also two hierarchical control objectives for ambulation on stochastically uneven terrains.First, due to the stochastic disturbances, the reference trajectory, i.e., q L (t), for the adaptation to stochastically uneven terrains, is highly desired.It motivates us to discuss gait planning in the stochastic condition.Moreover, the control error, i.e., E ∥e (t)∥, turns to a probability framework, and state constraints (10) are required.It makes sense to discuss the constrained adaptation to stochastic coordination.Therefore, the definition of stochastic coordination leads to two control issues: adaptation to stochastically uneven terrains and constrained adaptation to stochastic coordination.

C. Finite-Time Delayed Stochastic Coordination
Given the dynamic effects of the synchronization delay, the finite-time delayed stochastic coordination can be provided as follows.
Definition 3 [32], [33]: In the presented system (9)(11) (12), finite-time delayed stochastic coordination is defined as the finite-time identity between the states of the prosthesis q p and of the healthy lower limb q L with an undetermined phase delay γ , i.e., q L (t − γ ), in probability.The existence of finite-time delayed stochastic coordination implies that the response system is finite-time stable in probability, namely, when t > T with T being the reaching time.
Remark 5: The finite-time delayed stochastic coordination, given by ( 15) and ( 16), poses two hierarchical control objectives during HRI.At the high-level, the intentionally delayed states of the healthy lower limb, i.e., q L (t − γ ), are essential for adaptation to HRI.Hence, it is important to identify the intentional delay subject to different HRI.Furthermore, the above challenge greatly affects the transient performance of stochastic coordination.They also sparked our interest in discussing the finite-time adaptation to the transient performance, given by lim t→ T E ∥e (t)∥.Thus, the finite-time delayed stochastic coordination indicates two concerns of the prosthetic control: adaptation to HRI and finite-time adaptation to transient performance.

IV. ADAPTIVE CONTROL FOR ROBOTIC LEG
PROSTHESES This section first reviews annual publications on essential solutions for robotic leg prostheses.First, we initiated our research by conducting a search on the Web of Science using four distinct sets of keywords: "lower limb prosthesis" or "leg prosthesis" in combination with "passive", "impedance", "classification", and "position".Subsequently, we tabulated the publications associated with these four control methods over the past two decades.Finally, to visually represent the data, we created a graph where a circle represents each control method.The size of each circle corresponds to the number of publications for that particular method in each year, thus providing a proportional scaling, as shown in Fig. 3.

A. Complete Coordination on Structured Terrains
According to (13), we discuss prior advancements for the complete coordination of PHSs walking on structured terrains, including adaptation to complex tasks (high-level) and stable adaptation to model uncertainty (low-level), as shown in TABLE III.
In the first category, we cover the adaptation solutions to complex tasks at the high level, including finite-state machine (FSM) approaches and continuous approaches.These solutions give the research progress in environmental adaptation but lead to a new issue of model uncertainty, constituting another focus of this subsection.In the second category, we discuss two parallel solutions for the stable adaptation to model uncertainty: the trial-and-error and adaptive control approaches.

TABLE III CONTROL METHODS FOR COMPLETE COORDINATION OF PHSS
If the gait characteristics change, FSM might generate incorrect gait divisions, leading to gait disorder or instability [10], [37].More importantly, the environmental adaptability of the FSM-based approach is also limited because of the unknown impedance parameters under different walking tasks.It means that the difficulty of the FSM approach lies in identifying the user's gait phase and adapting the impedance parameters to the complex tasks.Gao et al. [113] introduced an LMI algorithm based on terrain geometry to improve recognition accuracy, achieving an impressive average accuracy of 98.5%.On the other hand, Young and Hargrove [79], [99] [100], [101] investigated an accurate intent recognition system to recognize subject-specific patterns, performing seamless transitions between locomotion modes by learning impedance solutions [115], [116].Considering the real challenges presented in a human-robot system, efficiency, stability, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
and optimality are required.In this regard, Wen et al. [26], [117], [118], [119], [120] demonstrated reinforcement learning (RL) with humans in the loop, which can seamlessly incorporate experience replay and supplemental values derived from previous experiences.Concerning the FSM-based approach with humans in the loop, the RL framework is used to turn the impedance parameters of the robotic knee prosthesis [84].However, there are some open challenges in these FSMbased controllers.Specifically, impedance-based controllers have limited adaptability, making it hard to adapt the real-time trajectory to the user's intention.Therefore, the algorithms in [26], [84], [117], [118], and [119] are primarily applied to FSM-based controllers for fixed, subjectively prescribed target tasks.Notably, as an amputee walking with a prosthesis, a fixed prosthetic trajectory potentially hinders the gait coordination between the healthy and the robotic legs.To this end, Wu et al. [96] developed a new RL solution for a robotic knee to mimic the states of the intact leg, thus performing gait coordination.
Despite the FSM-based solution, continuous approaches have emerged recently [122].Chen et al. [123] investigated an unsupervised adaptation method to assist the amputee's terrain-adaptive ambulation in various environments, i.e., level ground, stair ascent/descent, etc. Considering the learning burden of users, De Vree and Carloni [124] proposed a musculoskeletal model based on deep reinforcement learning to achieve a natural gait while reducing training time.Furthermore, Kim and Hargrove [125] mapped the autonomous motion of the residual limb (thigh) to the impedance parameters of the prosthesis controller, thus achieving level walking under three stride lengths.The extended work was proposed to achieve the trajectory planning of the knee joint [126].Also, nonlinear autoregressive networks have been proposed to avoid control switching at different walking speeds [127].Inspired by a pre-existing control strategy developed for bipedal robots, Sinnet et al. [128] first applied the virtual constraints to prostheses for extracting the continuous reference trajectories.Similar works were done for trajectory planning of prostheses at different tasks [129], [130], [131], [132], [133], [134], [135].Compared to the FSM-based control, this strategy unifies a gait cycle without switching controller parameters for different gait phases.Hence, the adaptive control of the leg prosthesis turns into a conventional tracking problem.
Furthermore, researchers have pointed out that gait asymmetry in lower limb amputees may lead to limited task variability and secondary diseases [8] [9].To this end, the next dominant goal for the adaptation to complex tasks is the GC of PHSs.In recent years, advances in neuro-synergic mechanisms have facilitated research into coordination-based controllers [174], [136].Zhou et al. [137] developed a human-robot cooperation control based on a trajectory deformation algorithm, thus improving robot compliance in HRIs.Hence, the GC performance of PHSs needs to be considered in the control design of the robotic leg prostheses [138].For example, a neuromusculoskeletal model [139] is controlled under the muscle synergy hypothesis, thus achieving variable walking speeds [26].Additionally, Lee and Goldfarb [140] investigated the symmetry of the stair gaits to mimic the missing biological limb with controllers.However, due to nonlinear dynamics, the GC of PHSs in complex tasks remains challenging, and therefore, control design needs to be carried out from a dynamic perspective.
Unapplied solutions and potential applications: Despite the applied solutions, RL methods may possess limited or no capacity to acquire the physics knowledge pertaining to objects.Hence, a hierarchical learning approach of manipulation is proposed, thus acquiring different skills [121].Moreover, the coordination of muscle activity has been investigated, but the principles governing functional integration remain unapplied to prosthetic control [141], [142].
2) Stable Adaptation to Model Uncertainty: Applied solutions: Regarding the nonlinear dynamics of PHSs, control approaches comprise several parameter and model uncertainties, thus leading to problems such as unacceptable tracking errors in time-varying tasks.Geyer and Herr [144], [145], [146], [147] investigated the adaptive control for powered prosthetic legs as well as neuromuscular model-based control.Besides, Aaron provided the model-based control for robotic walking [148], emphasizing the interaction force estimation [20], [149].Notably, researchers use the trial-and-error approach to tune PD control parameters [133], [143], but it is still challenging to guarantee robustness in the presence of system uncertainties.This problem was first considered by Kalanovic et al. [150] and investigated using feedbackerror learning.Then, Richter et al. [151] proposed a feedback control system to regulate the contact force between the prosthesis and the ground.To further consider the stability, recent adaptive controllers are primarily designed based on Lyapunov theory (see Theorem 1).
Theorem 1 [5]: Consider a general n-dimensional nonautonomous system: where and x = 0 is the isolated equilibrium point of the system, making If there is a function with the fixed sign V (t, x), its derivative, along with the system (17), is given by and, if V (t, x) is a signed function that is different from V (t, x), then the origin of the system is asymptotically stable.Regarding rehabilitation devices, the adaptive sliding mode schemes are designed to cope with system uncertainties [5], [131], [155].Typically, Azimi et al. [156] designed the adaptive impedance controllers using sliding surfaces, thus tracking hip displacement and knee and thigh angles under uncertainties.We have previously shown how adaptive controllers can be constructed by an NN-based framework [37].Regarding the coordination of PHSs, Hong et al. [10] proposed a vision-locomotion coordination control for a leg prosthesis, thus helping the amputee cross the obstacles.
Unapplied solutions and potential applications: Under the Lyapunov regard, uncertainty issues have been solved Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Theorem 2 [162], [163]: For an unknown nonlinear function U ( ) : R u → R, it can be approximated by the RBFNN: ∈ R u denotes the input of NN, W T is the weight vector with s (s > 1) nodes, and where c i ∈ R u is the width and b i indicates the center of the function.Moreover, it proves that RBFNN can approximate the continuous function with an approximation error ε, given by where ε is the positive constant, W * is the ideal constant weight vector determined by According to these solutions, RBFNN-based methods have been used to deal with uncertainty in robot dynamics [164], [165], [166], [167], output constraints [168], [169], etc., thus making the controllers noise-resistant.These considerations have led to an NN-based technique that can handle HRIs [170].Typically, Yu et al. [171] proposed an adaptive neural control approach to achieve stable and efficient HRIs and considered unknown human impedance and robot dynamics in human-robot cooperative tasks [172].However, these methods regard the stable adaptation of the HRI as deterministic frameworks, lacking specific analysis of PHS, let alone the stochastic characteristics.

B. Stochastic Coordination on Stochastically Uneven Terrains
All of the results discussed above allow researchers to prove that the controllers are sound in the case of structured terrains.However, these methods share a common limitation: no suitable coordination framework for stochastically uneven terrains.
According to (14), we provide advances on how to achieve stochastic coordination using adaptive control methods.We also consider two categories depending on the controller levels: adaptation to stochastically uneven terrains (high-level) and constrained adaptation to random uncertainty (low-level), as shown in TABLE IV.
The first category discusses how the gait planning methods adapt to stochastically uneven terrains.This awareness is practical when the reference trajectories mismatch for stochastic coordination.The second category presents the constrained adaptation to stochastic coordination in some applications.As far as we know, it is instructive for the control design of robotic leg prostheses in a probability framework.
1) Adaptation to Stochastically Uneven Terrains: Applied solutions: Considering a stochastically uneven terrain such as sandy ground, recent studies have concerned randomness as the worst factor that can destroy the symmetry of GRFs [11].Besides, Chen et al. [175] verified the effects of stochastic factors on foot muscle activity and center of pressure (CoP).In this regard, prior researchers began to focus on the mechanical design of prostheses [6] and the walking mechanism on stochastically uneven terrains [11], [176], [177], as shown in Fig. 5.However, the impact of GRFs has not been studied from a dynamics perspective, especially the gait planning for GC.
To adapt to stochastically uneven terrains, robotic leg prostheses employ a two-step process.Firstly, they update reference trajectories through gait planning, considering the contralateral intact leg on specific terrain conditions.Subsequently, the prostheses track these updated reference trajectories, ensuring that the movements of the prosthetic leg align with the intended trajectory of the contralateral intact leg.To this end, coordination strategies have been proposed to design real-time signals of the prosthesis, avoiding the pre-programmed reference trajectories in stochastic Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 5. Prior research for robotic leg prostheses walking on stochastically uneven terrains.a Mechanical design of robotic leg prostheses [6]; thus, the stochastic effects can be compensated by the designed foot structure.b Walking mechanism in stochastically uneven terrains [11], [176], [177].The states and GRFs can be obtained in this condition.
controllers [178], [179], [180].Moreover, the states of the prosthesis need to be guaranteed within a specific range [11], and their boundaries are called state constraints [185], [186].It has been found that constraint violation may lead to instability or even safety problems.Therefore, the stochastic coordination of PHSs on stochastically uneven terrains is another challenging problem.
To further consider the specific constraints, gait planning for leg prostheses must combine stochastic coordination strategies with corresponding state constraints to satisfy the asymmetric time-varying outputs of human gait [11], [132].In this regard, the Fourier series can be used to model the upper and lower bounds of the periodic states, given by where A i,sup,0 , A i,inf,0 , A i,sup,k , A i,inf,k , B i,sup,k , and B i,inf,k are Fourier coefficients, and f g denotes gait frequency [40].However, these state constraints within the gait planning method can provide complex reference trajectories, leading to a problem with the constrained adaptation to stochastic coordination at the low-level.Unapplied solutions and potential applications: Regarding existing research on stochastic control, reference trajectories are usually under deterministic assumptions, i.e., sine functions [181], [182] and triangle composite functions [183], [184], etc.However, these pre-determined reference trajectories do not match the time-varying gaits of the human body, thus may lead to the instability of systems.Applied solutions: Regarding stochastically uneven terrains, outstanding features, such as stochastic disturbances, made the deterministic model a mismatch assumption in control design [19].In particular, the traditional Kelvin-Voigt contact model [64] has inherent limitations in coping with stochastic disturbances of GRFs.Modeling PHSs based on first principles is challenging when there are random disturbances in GRF.Based on this representation, data-driven methods are also applied to cope with walking movements [194], GRFs [187], and daily living [188].
In the probability framework, Kalman filtering that follows statistical properties has emerged under practical problems, such as estimating the center-of-mass motion [202], the instabilities of humanoid bipedal robots [203], and the motion prediction [204].In addition, extensive studies on filter-based control of prosthetic legs [66], [67], i.e., extended Kalman filter (EKF) [205]..() [206], are carried out, thus providing the state estimation for prosthetic control.However, when the controlled systems are in high nonlinearity, the filtering approaches may cause a time lag, which poses new challenges for controller design under the stochastic framework.
Unapplied solutions and potential applications: Inspired by model-free ideas, the data-driven methods [189], [190], [191], [192], [193] provide some efficient solutions for the accurate modeling of the stochastic PHSs.Moreover, the stochastic differential equation (SDE) provides a suitable expression for stochastic control problems [195].The solutions of SDE and ODE at a given initial value can be obtained, respectively.Thus, there are two completely different solutions in these two cases.In this regard, researchers have investigated several typical features of stochastic systems, including continuous-time stochastic nonlinear systems [196], nonstrict feedback stochastic nonlinear time-delay systems [197], non-affine stochastic nonlinear systems [198] and switched stochastic nonlinear uncertain systems [199].Regarding Euler-Lagrange systems, stochastic nonlinearity is solved by adaptive neural networks [200], [201].Besides, according to learning-based control techniques such as NNs [208] and fuzzy logic systems (FLSs) [209], Lyapunov technology is employed to solve the control problem within a probabilistic framework [210].However, the convergence time is not investigated in this study, thus preventing this method from being used in highly real-time ambulation.
In recent years, researchers have extended the Lyapunov stability theory to a stochastic framework.It is worth noting that the Wiener process has a non-zero second variation [211].Therefore, it is necessary to investigate the stochastic process by introducing Ito's Lemma [212], the stochastic differential operator, and semi-globally uniformly ultimately bounded (SGUUB), seeing Definitions 4-5 and Theorem 3, respectively.
Employing this Lyapunov technology is necessary to address the inherent nonlinearity and uncertainty of dynamic systems and consider the particularity of system states [179].These considerations lead researchers to investigate the constrained adaptation issue by stochastic Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Definition 4 [211], [212]: Consider an n-dimensional stochastic system: where x n ] T is the state vector, and f (x) : R n → R n and g (x) : R n → R n×r are locally Lipschitz function vectors.
For the system (26) and a given Lyapunov function V (x) ∈ C 2 , the differentiation of V (x) is given by where Tr {•} is the trace of the matrix •.
Definition 5 [207]: Given any compact subset ∈ R n with x (t 0 , w) = x 0 ∈ , if there exist a constant δ > 0 and a time constant T (δ, x 0 , w) such that E ∥x (t, w)∥ < δ for all t ≥ t 0 + T , then the state trajectory of switched stochastic system ( 26) is said to be SGUUB.
Theorem 3 [211], [212]: Suppose that there exist two positive constants A and B, a Lyapunov function V (x) ∈ C 2 , and functions δ 1 (|x|) and δ 2 (|x|), such that for ∀x ∈ R n and t > 0. Hence, there is a strong solution of the system (26) for each x ∈ R n and the following inequality holds that is, the stochastic system ( 26) is SGUUB in probability.

C. Finite-Time Coordination Considering Synchronization Delay
In the previous subsections, we focused on the asymptotic stability of the robotic leg prostheses; however, considering practical requirements, we are more interested in transient performance.We expect to discuss a new specification for finite-time delayed stochastic coordination.
According to (15) (16), this part emphasizes the background of finite-time coordination by discussing the synchronization delay in response to stride frequencies and their effects on HRIs.Like previous subsections, we begin by adapting to HRI (high-level) and then discuss the finite-time adaptation to transient performance (low-level), as shown in TABLE V.
In the first category, we introduce the adaptation solutions to HRI, including fully synchronized, rhythmic HRI and FLS modeling.These solutions give the research progress in HRI adaptation but lead to a new issue of time delay, constituting another focus of this subsection.In the second category, we discuss four parallel solutions for the finite-time adaptation to transient performance: finite-time stable approach, LKF design, Pade approximation, and BLF design.6.The mechanism of synchronization delay between the bilateral lower limbs.a The walking process [225].Participants' kinematics, such as the GRFs and lower limb states, are recorded by the motion capture system and the separate load cells under each treadmill belt.b The presence of the stride frequency and step length [226].c The FLS-based model of synchronization delay [41].
1) Adaptation to Human-Robot Interaction: Applied solutions: The unilateral amputees with robotic leg prostheses are a class of HRIs in which humans and robots perform coordination [220].In this way, one of the main functions of HRIs is to share behaviors or ambulation during HRIs [221].To ensure adaptability and stability, such HRIs usually perform their behaviors fully collaboratively.In this regard, collaborative behaviors of HRIs have been investigated in recent research, including co-behaviors [220], [221], sit-to-stand [222], and human-following [223], etc.
Despite the fully synchronized HRIs, rhythmic HRI control in human life is equally important, such as walking with a given stride frequency [224].Fig. 6 a shows the experimental setup that uses an instrumented split-belt treadmill to monitor and control the speed of each belt [225].The symmetric gait of the subjects is simulated by fixing the speed of the right and left belts.Based on the experimental data, we can determine the stride frequency and step length by subtracting the foot positions with zero velocity during the experiment, as shown in Fig. 6 b [226].Furthermore, since the generation of intentional delay is in response to the stride frequency, we can further use stride frequency (see Fig. 6 b) to calculate the synchronization delay between right leg and left leg, which can be modeled by the fuzzy logic system (see Fig. 6 c) [41].For this concern, the synchronization delays of PHSs need to be estimated and introduced for gait planning of the robotic leg prostheses, especially for different walking speeds.Unfortunately, current prosthesis controllers rarely consider the unknown mechanism of synchronization delays, thus making ambulation uncontrollable.
Regarding PHSs, the walking experiments are conducted to identify the synchronization delays at different velocities using FLSs, as shown by the red dashed line in Fig. 6 c.Then, to continuously mimic the contralateral healthy limb, the FLS-based gait planning of prosthetic reference trajectory is redesigned, given by where q * p (t) denotes the vector of prosthetic reference trajectories, q L (t) indicates the healthy limb state vector, h T 0 ∈ R n×1 and h T 1 ∈ R n×1 are unknown but bounded constant vectors, W (t) means a standard normal random, and FLSs-based synchronization delay γ controls the stride frequency, given by: where |ε| ≤ ε with ε > 0 being a desired error accuracy, referring to [41].Based on reference trajectories (30), we can investigate adaptive control to address GC.Indeed, the synchronization delay between the leg prosthesis and the healthy lower limb determines the unknown stride frequency and destroys the transient performance.In other words, the coupled PHSs may lose adaptation to walking speeds and need finite-time performance.Considering the adverse effects of the synchronization delay in HRIs, the control methods reviewed in the rest of the paper are appraised against transient performance.
Unapplied solutions and potential applications: From a classic dynamics perspective, phase delay is a major cause of instability in HRI.Regarding this issue, the traditional proportional-integral-derivative (PID) controllers are adapted to mitigate the adverse effects of delay.Furthermore, based on the accuracy of the system model, the dead-time compensators (DTCs) are designed for time delays.These solutions prove the potential applications devoted to compensating for time delays in PHS systems.
2) Finite-Time Adaptation to Transient Performance: Applied solutions: The transient performance of the robotic leg prosthesis is seldom considered in the existing works.
Typically, a high gain observer is proposed to estimate the prosthetic velocities from measurable signals in the control effort, thus obtaining an acceptable transient performance [227].Moreover, Sun et al. [228] designed a fixed-time sliding-mode controller to ensure the transient performance of the wearable robot.However, the transient performance is important for PHSs, particularly for prostheses with time delay requirements.
Unapplied solutions and potential applications: Regarding the aforementioned adverse effects of synchronization delay, finite-time control methods are employed to provide a better transient performance [229], [230], [231].Therefore, realizing finite-time coordination is another key issue of prosthesis control.Current HRI concerns are usually focused on the control results of asymptotic stability [184], thus remaining the synchronization delay to be solved.Moreover, few previous finite-time control approaches can handle the synchronization delay of PHSs.To this end, we survey pertinent studies in a broader scope.
In terms of transient performance, Na et al. [232] proved finite-time convergence and stability feedback based on the Lyapunov stability theory.Moreover, Yin et al. [233] investigated the finite-time stability of stochastic systems via the stochastic Lyapunov stability theory (see Definition 6 and Theorem 4).To achieve synchronization, researchers have studied the finite-time and fixed-time synchronization of coupled network systems [234].Then, this approach was extended to more complex systems incorporating NNs or FLSs, including stochastic multi-intelligent systems [235], non-triangular stochastic systems [236], higher-order stochastic nonlinear systems [237], and Markov jump nonlinear systems [238].However, considering the synchronization above delay of PHSs, the existing finite-time controllers are difficult to apply directly.
Regarding the delay issue [239], previous studies have evaluated the practical stability of the system [240], [241], [242] via a class of Lyapunov-Krasovskii functional (LKF) and finite-time convergence.Yu et al. [243] proposed a motion prediction method to deal with an unintentional time lag in human-robot cooperative transportation.Furthermore, the input delay is investigated for the stochastic systems based on the Pade approximation [245].Unlike unintentional delays, synchronization delays are intentionally introduced to improve the synchronization performance of the system [244].To further consider state constraints, BLF and LKF are designed to deal with time delays and constraints simultaneously [246], [247].Hence, the abovementioned methods for delay issues are urgently applied to PHSs.In addition, the resource limitations remain open, thus promoting the discussion of event-triggered control.
V. SUMMARY AND FUTURE PERSPECTIVES This paper surveyed dynamic models and adaptive control methods with a novel insight into GC.The first contribution of this work is the conceptualization of GC by integrating synchronization from dynamic theory.It also contributed to existing research by extending the deterministic model to the stochastic model.Research advances have been discussed regarding synchronization theory, including complete/identical coordination, stochastic coordination, and finite-time delayed stochastic coordination.We hope this review provides an accessible starting point and a new dynamic perspective on this emerging field.However, several challenges in this field still warrant further investigation and research.

A. Extension of Modeling Methods
This article discusses two types of dynamic models for HRIs, including the deterministic model and the stochastic dynamic model.The identified promising prospects in modeling methods pave the way for future advancements in coordination control for complex terrains.However, establishing stochastic models is rather intricate, thus leading to an open problem for stochastic coordination.By further refining these methods and addressing the remaining challenges, we can unlock the full potential of human-robot systems in ambulation at complex terrains [248], [249], [250].

B. Theoretical Improvement to RL
The control approaches reviewed in this article are designed and implemented using the Lyapunov method, which forms a sufficient condition.To achieve better output and regulation in the cooperative ambulation, the control optimization based on the Reinforcement learning algorithm [251], namely, the design of the Bellman equation [119], is a significant improvement compared with the control methods in this review.

C. Expansion of Coordination Control
The control approaches of this review emphasize GC of PHSs as asymptotically or finite-time stable.However, considering HRIs in high real-time ambulation, the arrival time of controlled states is not investigated.Thus, the previous works must be extended to a fixed-time scheme [252], [253].The second issue is the event-triggered control since a controlled robotic system should ideally be efficient, thus reducing the communication burden.However, most prosthetic controllers focus on time-triggered sampling, and we may use pertinent studies in a broader scope.In the context of the event-triggered control, the main idea is that the control feedback signals are only executed under triggered conditions.Despite this improvement, the existing triggered strategies, such as fixed threshold [254] and relative threshold [255], make it hard to apply to cooperative walking for amputee individuals.Hence, open challenges remain in obtaining an event-triggered scheme that is conducive to ensuring gait coordination and reducing communication sources.

Fig. 1 .
Fig. 1.Overview of the prosthesis-human dynamics.a Deterministic model with uncertainty for the structured terrain.The ground reaction forces can be modeled as a spring-damper formulation.b Stochastic model for the stochastically uneven terrain.The stochastic uncertainties coupled with time-varying state constraints.

Fig. 2 .
Fig. 2. Three representations of GC. a Complete coordination for structured terrains is related to the deterministic model and control method discussed in Section IV-A.b Stochastic coordination for uneven terrains.c Finite-time delayed stochastic coordination.These two concerns are related to the stochastic model and the control methods discussed in Sections IV-B and IV-C.

Fig. 3 .
Fig. 3. Annual publications of robotic leg prostheses.a Bubble chart of four typical solutions by year.b A comparison of the total number of publications during 20 years.The bubble size in the Bubble chart shows the number of publications, i.e., each legend bubble indicates 60 publications.

TABLE IV CONTROL
METHODS FOR STOCHASTIC COORDINATION OF PHSS