An event-triggered linear time-varying model predictive control framework for physiologically meaningful bipedal walking

Abstract

Bipedal walking, as the most representative form of locomotion, is important for health and quality of human life. Model predictive control (MPC) has been widely employed for bipedal walking generation, but existing methods often struggle to simultaneously ensure stable tracking performance, physiological plausibility, and acceptable computational cost. To address this issue, an event-triggered linear time-varying model predictive control (ET-LTV-MPC) framework is proposed. Based on oscillator-generated desired trajectories, the control framework integrates a time-varying linearized model for future state prediction with a re-optimization mechanism triggered by foot contact, allowing timely re-optimization at hybrid state transitions while explicitly enforcing physiological torque constraints. Numerical simulations have been conducted using a compass model and compared with representative controllers. The results show that the ET-LTV-MPC achieves superior tracking accuracy while maintaining the joint torque within the physiological constraints. It also reduces the peak error caused by foot contact and provides a more favorable balance between tracking accuracy and computational cost than NMPC. Parametric analysis reveals that the optimal prediction window is approximately 0.2 s into the future, which is comparable to reported sensorimotor response latencies during human walking. These findings demonstrate that the proposed ET-LTV-MPC provides an accurate, computationally efficient and physiologically meaningful tool for generating bipedal walking.

Keywords

bipedal walking biomechanical modeling optimal control model predictive control trajectory tracking

Introduction

Locomotion is one of the most fundamental motor skills that enables humans to interact with and adapt to the environment (Severini and Zych, 2022). With global populations aging, the incidence of mobility impairments and gait disorders continues to rise, making the study of human locomotion more urgent (Wang et al., 2024). Bipedal walking, as the most representative form of human locomotion, is crucial to human health and quality of life and has become a focus of this field (Crompton et al., 2023). Understanding the mechanisms underlying human walking not only provides valuable physiological insights for health and rehabilitation (Das et al., 2022), but also guides the design and control of humanoid robots and lower-limb exoskeletons (Narayan and Dwivedy, 2025; Reher and Ames, 2021).

The origin of the research on bipedal walking can be traced back to the 1990s. McGeer (1990) used a rimless wheel model to describe passive walking and demonstrated that stable limit cycles could be achieved on an incline purely through gravity and collisions. However, the applicability of this model was limited by slope steepness. To overcome this limitation, the researchers shifted focus to more complex biomechanical models. The simplest biomechanical model is called the compass model, which introduces two rigid links and a rotating hip joint to approximate the behavior of a bipedal robot (Darici and Kuo, 2023; Garcia et al., 1998). Later, to capture more details, these foundational models were expanded to multi-link models. For example, the five-link model introduced the upper body and the knee joint, providing a more detailed representation of human-like gait dynamics (Borzova and Hurmuzlu, 2004; Castillo et al., 2023). To improve accuracy, the addition of muscle tissue further enhanced the simulation of muscle-joint interactions and led to the development of musculoskeletal models (Schumacher et al., 2025; Sylvester et al., 2021).

To bridge the gap between biomechanical models and practical walking systems, the most critical aspect is control. Initially, passive walking control can achieve natural periodic gaits, but showed limited adaptability to terrain, making it difficult to achieve stable walking on flat ground (Collins et al., 2005). This led to the introduction of proportional–derivative (PD) control strategies to achieve basic trajectory tracking and gait stabilization (Aoi and Tsuchiya, 2007). Inspired by biological neural networks, control strategies based on the central pattern generator (CPG) were introduced to generate rhythmic movement patterns by mimicking the coordination of neuron firing in the spinal cord (Minassian et al., 2017). In addition, researchers developed methods based on the zero-moment point (ZMP), which enhance stability by ensuring that the ZMP stays within the support polygon, preventing the human body from losing balance during movement (Park and Youm, 2007). Furthermore, the hybrid zero dynamics (HZD) method was developed to combine passive dynamics with active control, focusing on stable internal behaviors to achieve bipedal walking (Sharbafi and Seyfarth, 2015).

Recently, there has been growing evidence that human motor control operates according to optimal principles (Mathis and Schneider, 2021; Razavian et al., 2023). Therefore, model predictive control (MPC) has emerged as a promising framework to explain bipedal walking because of its ability to explicitly handle system dynamics, constraints, and optimization problems. Brasseur et al. (2015) introduced a linear MPC based on a center of mass (CoM) model and significantly improved walking performance on flat ground by using linear differential inclusion to constrain nonlinear dynamic feasibility. Vu Trieu Minh et al. (2020) designed a nonlinear MPC (NMPC) architecture that computes joint torques and trajectories for a multi-link walking model, reproducing human walking patterns measured by a Vicon motion-capture system. Ding et al. (2021) developed an NMPC framework based on an inverted-pendulum-plus-flywheel model that adjusts step location, CoM height, and angular momentum, achieving robust bipedal walking in multiple scenarios. Dallard et al. (2025) proposed a closed-loop MPC scheme that enhances the MPC based on the stable linear inverted pendulum model with an explicit model of ZMP dynamics and contact forces, enabling robust walking on uneven terrain.

However, existing MPC-based studies on bipedal walking still have several limitations. First, most MPC methods rely on highly simplified CoM-ZMP or linear inverted pendulum models, and constraints are usually imposed at the CoM/ZMP level rather than directly on joint torques. Second, most MPC methods update the control input at fixed sampling intervals. In hybrid walking systems, such fixed-period updates can lead to transient mismatches between the prediction model and the true dynamics at impact, which may degrade tracking accuracy. Third, NMPC methods can capture richer nonlinear dynamics, but they remain computationally demanding and often require specialized solvers, reduced update rates, or powerful hardware to maintain performance. Finally, the majority of MPC-based walking studies primarily evaluate stability, robustness, and disturbance rejection, while rarely discussing how controller parameters relate to physiological characteristics of human walking.

To address the above limitations, this paper makes the following contributions:

An event-triggered linear time-varying model predictive control (ET-LTV-MPC) framework was developed for bipedal walking, which directly applies physiological constraints to joint torques rather than CoM/ZMP level. To address the transient mismatch between the predictive model and the actual dynamics during impact, an event-triggered mechanism is designed that optimizes the control input upon detection of foot contact. To alleviate the computational burden of NMPC, a linear time-varying prediction model is introduced, which captures the essential behavior of the nonlinear dynamics and simplifies the optimization problem as a quadratic program.

A series of experiments were conducted to provide quantitative reference and address the lack of systematic comparisons among bipedal walking controllers. First, the proposed ET-LTV-MPC was compared with PD and linear quadratic regulator (LQR) controllers, demonstrating the ET-LTV-MPC achieves superior trajectory tracking performance while keeping the control torque within physiological constraints. Then, the ET-LTV-MPC was compared with the conventional LTV-MPC and NMPC, demonstrating the ET-LTV-MPC reduces the peak error caused by foot contact and has a much shorter solution time and a better trade-off than NMPC.

This study linked the key MPC design parameters to physiological characteristics of human walking. In particular, the optimized joint torque generated by the proposed ET-LTV-MPC is shown to remain consistently within the physiological constraints observed in human walking. In addition, the optimal prediction horizon corresponds to predicting approximately 0.2 second into the future, which is comparable to reported sensorimotor response latencies during human walking. These findings make the controller design more convincing and interpretable from a bio-inspired control perspective, addressing the lack of exploration on how MPC relates to human walking physiology in existing studies.

The rest of this paper proceeds as follows. Section “Bipedal walking model” establishes the biomechanical modeling of the bipedal walking system based on the compass model. Section “Control framework design” details the design methodology for the proposed ET-LTV-MPC framework, as well as the PD and LQR controllers for comparison. Section “Results and discussion” presents the results of the numerical simulations conducted to validate the performance of the proposed controller and analyze its physiological plausibility. Finally, Section “Conclusion” summarizes the contribution of this paper and outlines the directions of future research.

Bipedal walking model

Bipedal walking is a periodic motion, and each cycle consists of two primary phases: the swing phase and the foot contact. In this study, the compass model is adopted for biomechanical modeling and controller validation, as it provides a clear representation of bipedal walking dynamics and has been widely used in related research (Garcia et al., 1998). This model has a swing leg and a stance leg that are connected at the hip. The hip mass is M, the leg mass is m, and the leg length is l. The foot of the stance leg is constrained on the ground, and the stance leg can only rotate around the hip joint. $θ_{1}$ is the angle between the stance leg and the perpendicular line. $θ_{2}$ is the angle between the stance leg and the swing leg. The model has a control input u that acts directly on $θ_{2}$ . The acceleration due to gravity is g. The model moves only in the $x - y$ plane and walks to the x-direction. The foot of the stance leg at the initial moment is set as the origin of coordinates. Figure 1 illustrates the structure of the compass model and shows the two phases of the walking cycle.

Figure 1.

The compass gait model used for bipedal walking analysis in this study.

Swing phase

In the swing phase, only the stance leg is in contact with the ground. The dynamic equation of the model can be obtained using Lagrangian mechanics.

The kinetic energy of the hip mass is

\begin{matrix} T_{h} = \frac{1}{2} {Ml}^{2} {\overset{\cdot}{θ}}_{1}^{2} \end{matrix}

(1)

The potential energy of the hip mass is

\begin{matrix} U_{h} = Mgl \cos θ_{1} \end{matrix}

(2)

Similarly, the kinetic energy of the swing leg mass is

\begin{matrix} T_{l} = \frac{1}{2} m (l^{2} {\overset{\cdot}{θ}}_{1}^{2} + l^{2} {({\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{1})}^{2} + 2 l^{2} {\overset{\cdot}{θ}}_{1} ({\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{1}) \cos θ_{2}) \end{matrix}

(3)

The potential energy of the swing leg mass is

\begin{matrix} U_{l} = mg (l \cos θ_{1} - l \cos (θ_{1} - θ_{2})) \end{matrix}

(4)

The Lagrangian equation of the system, defined as the total kinetic energy minus the total potential energy, is given as follows

\begin{matrix} L = T_{h} + T_{l} - (U_{h} + U_{l}) \end{matrix}

(5)

The generalized torques acting on the two angles are derived as

\begin{matrix} {\begin{matrix} τ_{1} = \frac{d}{dt} \frac{\partial L}{\partial θ_{1}} - \frac{\partial L}{\partial θ_{1}} = 0 \\ τ_{2} = \frac{d}{dt} \frac{\partial L}{\partial θ_{2}} - \frac{\partial L}{\partial θ_{2}} = u \end{matrix} \end{matrix}

(6)

By rearranging the above equations, the dynamic equation of the compass model in the swing phase can be expressed in matrix form as

\begin{matrix} D (θ) [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{1} \\ {\overset{\cdot\cdot}{θ}}_{2} \end{matrix}] + C (θ, \overset{\cdot}{θ}) + G (θ) = [\begin{matrix} 0 \\ u \end{matrix}] \end{matrix}

(7)

where

\begin{matrix} D (θ) = [\begin{matrix} {Ml}^{2} + {ml}^{2} (2 - 2 \cos θ_{2}) & {ml}^{2} (- 1 + \cos θ_{2}) \\ {ml}^{2} (- 1 + \cos θ_{2}) & {ml}^{2} \end{matrix}] \\ C (θ, \overset{\cdot}{θ}) = [\begin{matrix} {ml}^{2} (- \sin θ_{2} ({\overset{\cdot}{θ}}_{2}^{2} - 2 {\overset{\cdot}{θ}}_{1} {\overset{\cdot}{θ}}_{2})) \\ - {ml}^{2} {\overset{\cdot}{θ}}_{1}^{2} \sin θ_{2} \end{matrix}] \\ G (θ) = [\begin{matrix} - Mgl \sin θ_{1} - mgl (\sin θ_{1} - \sin (θ_{1} - θ_{2})) \\ - mgl \sin (θ_{1} - θ_{2}) \end{matrix}] \end{matrix}

Foot contact

When the swing leg lands on the ground, the foot contact occurs and leads to a sudden change of state. The foot contact is assumed to occur only when all the following contact conditions are simultaneously satisfied (Gregg et al., 2011; Gregg and Spong, 2010)

\begin{matrix} {\begin{matrix} h = l \cos θ_{1} - l \cos (θ_{2} - θ_{1}) = 0 \\ \overset{\cdot}{h} = - l \sin θ_{1} {\overset{\cdot}{θ}}_{1} + l \sin (θ_{2} - θ_{1}) ({\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{1}) < 0 \end{matrix} \end{matrix}

(8)

where h is the height of the swing foot. The first condition ensures that the height of the swing foot is zero. The second condition ensures that the height of the swing foot is decreasing.

At the moment of foot contact, an impact may occur between the tip of the swing leg and the ground. To simplify the analysis, the following assumptions are made:

The swing leg does not slip or rebound upon contact;

The stance leg leaves the ground without generating any interaction force;

The control torque is assumed to be continuous and non-impulsive during the impact;

Immediately after contact, the swing leg becomes the new stance leg, while the previous stance leg becomes the new swing leg. Under these assumptions, when the swing leg strikes the ground, an inelastic collision occurs, resulting in an instantaneous change of the system state.

According to the law of conservation of angular momentum, the angular momentum before and after the impact remains constant. The angular momentum of the whole system around the contact foot is conserved, leading to

\begin{matrix} M \cos θ_{2} {\overset{\cdot}{θ}}_{1}^{-} & = (M {\overset{\cdot}{θ}}_{1}^{+} + 2 m (1 - \cos θ_{2}^{+}) {\overset{\cdot}{θ}}_{1}^{+}) \\ - m (1 - \cos θ_{2}^{+}) {\overset{\cdot}{θ}}_{2}^{+} \end{matrix}

(9)

where ${(\cdot)}^{-}$ and ${(\cdot)}^{+}$ denote the values immediately before and after impact, respectively.

Similarly, the angular momentum of the partial system around the hip joint is conserved, leading to

\begin{matrix} 0 = (1 - \cos θ_{2}) {\overset{\cdot}{θ}}_{1}^{+} - {\overset{\cdot}{θ}}_{2}^{+} \end{matrix}

(10)

At the instant of impact, due to the exchange of the stance and swing legs, the following geometric relationships hold

\begin{matrix} {\begin{matrix} θ_{1}^{+} = - θ_{1}^{-} \\ θ_{2}^{+} = - θ_{2}^{-} \end{matrix} \end{matrix}

(11)

Combining the above relationships, the state transition of the system before and after foot contact can be expressed as

\begin{matrix} [\begin{matrix} θ_{1}^{+} \\ θ_{2}^{+} \\ {\overset{\cdot}{θ}}_{1}^{+} \\ {\overset{\cdot}{θ}}_{2}^{+} \end{matrix}] = [\begin{matrix} - θ_{1}^{-} \\ - θ_{2}^{-} \\ \frac{2 M \cos 2 θ_{1}^{-}}{2 M + m (1 - \cos 4 θ_{1}^{-})} {\overset{\cdot}{θ}}_{1}^{-} \\ \frac{2 M \cos 2 θ_{1}^{-} (1 - \cos 2 θ_{1}^{-})}{2 M + m (1 - \cos 4 θ_{1}^{-})} {\overset{\cdot}{θ}}_{1}^{-} \end{matrix}] \end{matrix}

(12)

Control framework design

This section presents the design of three control methods for the bipedal walking model. In addition to the proposed ET-LTV-MPC framework, classical LQR controller and PD controller are also implemented for comparison.

ET-LTV-MPC framework

Previous research has demonstrated that oscillators can effectively generate rhythmic signals for bipedal walking control (Kuo, 2002). Therefore, this study introduced an oscillator to generate the desired angle between two legs

\begin{matrix} θ_{2 d} = A \cos φ = A \cos ω t \end{matrix}

(13)

where constant A is the amplitude that determines the stride, φ is the phase, and $ω = \overset{\cdot}{φ}$ .

It should be noted that foot contact will cause a state change: $θ_{2}^{+} = - θ_{2}^{-}$ . Therefore, φ will also reset at that time: $φ^{+} = φ^{-} - π$ .

The bipedal walking system is underactuated because only the swing leg joint is actuated, while the stance leg moves passively according to the system dynamics. Previous research has demonstrated that when $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ are able to closely follow the rhythmic reference generated by the oscillator, the hybrid system that includes the foot contact transition and phase reset is capable of producing self-stable periodic gaits within specific parameter ranges (Aoi and Tsuchiya, 2007). These findings highlight that accurate tracking $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ plays a crucial role in achieving stable walking behavior. Therefore, the control objective can be formulated as a trajectory tracking problem for $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ following the desired rhythmic signal $θ_{2 d}$ and ${\overset{\cdot}{θ}}_{2 d}$ .

To achieve this objective, an ET-LTV-MPC framework is developed, which consists of five steps: time-varying model establishment, future state prediction, solution optimization, repeated execution, and foot contact triggering. Figure 2 shows the whole process of the proposed ET-LTV-MPC framework.

Figure 2.

Overview of the proposed ET-LTV-MPC framework for bipedal walking, which consists of five steps: model establishment, future state prediction, solution optimization, repeated execution, and foot contact triggering.

The first step is to establish a linear time-varying model for future state prediction. The continuous dynamic equation introduced in equation (7) can be rewritten as

\begin{matrix} \overset{\cdot}{x} = f (x, u) \end{matrix}

(14)

where $x = {[θ_{2}, {\overset{\cdot}{θ}}_{2}]}^{T}$ .

By performing a first-order Taylor linearization around the current reference state $x_{r}$ and input $u_{r}$ , the nonlinear model can be approximated as

\begin{matrix} \overset{\cdot}{x} \approx f (x_{r}, u_{r}) & + {\frac{\partial f (x, u)}{\partial x} |}_{x = x_{r}, u = u_{r}} (x - x_{r}) \\ + {\frac{\partial f (x, u)}{\partial u} |}_{x = x_{r}, u = u_{r}} (u - u_{r}) \end{matrix}

(15)

This can be expressed in the form

\begin{matrix} \overset{\cdot}{x} = Ax + Bu + C \end{matrix}

(16)

where A and B are Jacobian matrices with respect to x and u, respectively, and $C = f (x_{r}, u_{r}) - {Ax}_{r} - {Bu}_{r}$ is the offset term. The Jacobian matrices A and B are recalculated at each sampling time by centered finite differences around $(x_{r}, u_{r})$ . Let $e_{i}$ denote the i th unit vector and $ϵ_{x}, ϵ_{u}$ small perturbations. In the subsequent simulations, $ϵ_{x} = ϵ_{u} = 10^{- 6}$ was used. The i th column of A and the j th column of B are approximated as

\begin{matrix} {\begin{matrix} A_{:, i} = \frac{f (x_{r} + ε_{x} e_{i}, u_{r}) - f (x_{r} - ε_{x} e_{i}, u_{r})}{2 ε_{x}} \\ B_{:, j} = \frac{f (x_{r}, u_{r} + ε_{u} e_{j}) - f (x_{r}, u_{r} - ε_{u} e_{j})}{2 ε_{u}} \end{matrix} \end{matrix}

(17)

Applying the forward Euler method for discretization yields

\begin{matrix} \frac{x (k + 1) - x (k)}{T} = Ax (k) + Bu (k) + C \end{matrix}

(18)

where k represents the current time step and T is the sampling period.

Therefore, the discrete linear model used for the prediction can be obtained

\begin{matrix} x (k + 1) = \tilde{A} x (k) + \tilde{B} u (k) + \tilde{C} \end{matrix}

(19)

where $\tilde{A} = AT + I$ , $\tilde{B} = BT$ , $\tilde{C} = CT$ , and $I \in R^{2 \times 2}$ is the identity matrix. The pseudo-code for model establishment is shown in Figure 3, which follows a standard strategy to obtain a linear time-varying model (Schei, 1997; Zhakatayev et al., 2017).

Figure 3.

Pseudo-code for the algorithm to establish the linear time-varying model.

The second step is to make predictions for the future state using the established model. Given the current state $x (k)$ , the next step state $x (k + 1)$ can be obtained using the discrete linear model introduced in equation (19). By recursively applying this process, the future states within a prediction horizon $N_{p}$ can be expressed in matrix form as

\begin{matrix} X_{all} = F + Φ U \end{matrix}

(20)

where $X_{all}$ is the sequence of predicted states over the horizon

\begin{matrix} X_{all} = [\begin{matrix} x (k + 1) \\ x (k + 2) \\ ⋮ \\ x (k + N_{p}) \end{matrix}] \end{matrix}

(21)

and U is the vector of possible future control inputs

\begin{matrix} U = [\begin{matrix} u (k) \\ u (k + 1) \\ ⋮ \\ u (k + N_{p} - 1) \end{matrix}] \end{matrix}

(22)

The matrices F and Φ are defined as

\begin{matrix} F = [\begin{matrix} \tilde{A} \\ {\tilde{A}}^{2} \\ ⋮ \\ {\tilde{A}}^{N_{p}} \end{matrix}] x (k) + [\begin{matrix} \tilde{C} \\ \tilde{A} \tilde{C} + \tilde{C} \\ ⋮ \\ {\tilde{A}}^{N_{p} - 1} \tilde{C} + \dots + \tilde{C} \end{matrix}] \end{matrix}

(23)

\begin{matrix} Φ = [\begin{matrix} \tilde{B} & 0 & 0 & \dots & 0 \\ A B & \tilde{B} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{A}}^{N_{p} - 1} \tilde{B} & {\tilde{A}}^{N_{p} - 2} \tilde{B} & {\tilde{A}}^{N_{p} - 3} \tilde{B} & \dots & \tilde{B} \end{matrix}] \end{matrix}

(24)

The third step is to design a cost function and solve the optimal control sequence. Based on the current state $x (k)$ and possible future control input sequences U, the corresponding future state sequence $X_{all}$ can be predicted. Therefore, $X_{all}$ is a function of the control sequence U. The reference trajectory $x_{ref}$ is generated by the oscillator, which provides the desired rhythmic signals for $θ_{2 d}$ and $θ_{2 d}$ . Over the prediction horizon $N_{p}$ , the reference sequence can be expressed as

\begin{matrix} X_{ref} = [\begin{matrix} x_{ref} (k + 1) \\ x_{ref} (k + 2) \\ ⋮ \\ x_{ref} (k + N_{p}) \end{matrix}] = [\begin{matrix} θ_{2 d} (k + 1) & {\overset{\cdot}{θ}}_{2 d} (k + 1) \\ θ_{2 d} (k + 2) & {\overset{\cdot}{θ}}_{2 d} (k + 2) \\ ⋮ & ⋮ \\ θ_{2 d} (k + N_{p}) & {\overset{\cdot}{θ}}_{2 d} (k + N_{p}) \end{matrix}] \end{matrix}

(25)

The tracking error between the predicted states and the oscillator-generated desired trajectory is given by

\begin{matrix} E_{all} = X_{all} - X_{ref} \end{matrix}

(26)

The control objective of the walking system is to minimize both the tracking error and the control energy. Therefore, the cost function is designed as

\begin{matrix} f (U) = E_{all}^{T} {QE}_{all} + U^{T} RU \end{matrix}

(27)

where Q and R are symmetric positive definite weighting matrices that determine the relative importance of tracking accuracy and control effort.

Substituting equation (20) into the above cost function yields

\begin{matrix} f (U) = {(F + Φ U - X_{ref})}^{T} Q (F + Φ U - X_{ref}) + U^{T} RU \end{matrix}

(28)

Expanding and rearranging the terms gives the standard quadratic form

\begin{matrix} f (U) = U^{T} Φ^{T} Q Φ U + 2 {(F - X_{ref})}^{T} Q Φ U + U^{T} RU + const \end{matrix}

(29)

Therefore, the cost function can be expressed as

\begin{matrix} J (U) = \frac{1}{2} U^{T} M_{1} U + U^{T} M_{2} \end{matrix}

(30)

where $M_{1} = 2 (Φ^{T} Q Φ + R)$ and $M_{2} = 2 Φ^{T} Q (F - X_{ref})$ .

In practical human bipedal walking systems, there exist physiological and actuator constraints. The control input u represents the torque of the hip joint, which is limited by a maximum admissible value. Therefore, control constraints are introduced as

\begin{matrix} | u (k + i) | \leq u_{\max}, i = 0, 1, \dots, N_{p} - 1 \end{matrix}

(31)

In matrix form, this can be expressed as

\begin{matrix} [\begin{matrix} | u (k) | \\ | u (k + 1) | \\ ⋮ \\ | u (k + N_{p} - 1) | \end{matrix}] \leq [\begin{matrix} u_{\max} \\ u_{\max} \\ ⋮ \\ u_{\max} \end{matrix}] = U_{\max} \end{matrix}

(32)

This transforms the optimal control problem into a quadratic programming (QP) problem, which minimizes a quadratic cost function subject to linear inequality constraints

\begin{matrix} \begin{matrix} \min_{U} J (U) = & \frac{1}{2} U^{T} M_{1} U + U^{T} M_{2} \\ s . t . & | U | \leq U_{\max} \end{matrix} \end{matrix}

(33)

This QP problem is convex and can be solved efficiently using standard and mature optimization solvers. At each control step, the optimal control sequence $U^{*} (k)$ is obtained as

\begin{matrix} U^{*} (k) = [\begin{matrix} u^{*} (k) \\ u^{*} (k + 1) \\ ⋮ \\ u^{*} (k + N_{p} - 1) \end{matrix}] \end{matrix}

(34)

The fourth step is to apply the optimal control input to the bipedal walking system and repeat the process. To ensure real-time control performance, only the first element of the optimal control sequence $u^{*} (k)$ is implemented in the walking system at each control step, while the remaining elements are discarded. The above process is repeatedly executed with a sampling period of T. After each control cycle, the system state is updated and the optimization problem is solved again based on the new measured state. Since the linearization and prediction are performed around the current operating point, the established prediction model is time-varying and is updated at every iteration.

The fifth step is foot contact triggering. Foot contact may occur at any time during walking and can cause a sudden change in the system state. Previous studies on human walking have shown that foot contact information provides a strong timing cue and can reset the ongoing locomotor phase, with basic muscle activation patterns tightly linked to the contact event (Tamura et al., 2020). Inspired by this evidence, the controller in this study adopts a contact-triggered re-optimization scheme. In the repeated execution process of the MPC, the currently computed optimal control input $u^{*} (k)$ is applied to the system for a duration of T. However, if a foot contact event occurs within this interval, the system state will abruptly change, and continuing to apply the previous control input may lead to a significant increase in tracking error. Therefore, when foot contact is detected, the controller must immediately perform resampling and state measurement, update the prediction model, and resolve the optimization problem to compute a new control input. This event-triggered mechanism enables the controller to adapt to sudden hybrid state transitions.

It should be noted that the event-triggered mechanism designed in this study is different from the conventional event-triggered control strategy that replaces periodic updates in order to reduce the average update frequency or computational load. The proposed ET-LTV-MPC framework still operates under the periodic time event mechanism, and the event-triggered update is introduced only as an additional re-optimization when foot contact occurs between two adjacent sampling instants. Therefore, the role of the event-triggered mechanism here is not to eliminate periodic updates, but to complement them by improving responsiveness to hybrid state transitions.

LQR controller

The LQR controller is another classical optimal controller that has been widely applied in system control (Barreda et al., 2025). In this study, an LQR controller is designed using the continuous linear model introduced in equation (16). The control objective is to regulate the state $x (t)$ to follow the oscillator-generated reference $x_{ref} (t) = {[θ_{2 d} (t), {\overset{\cdot}{θ}}_{2 d} (t)]}^{T}$ by minimizing the quadratic cost function

\begin{matrix} J = \int_{0}^{\infty} [{(x - x_{ref})}^{T} Q (x - x_{ref}) + u^{T} Ru] d t \end{matrix}

(35)

where Q and R are symmetric positive definite weighting matrices that determine the trade-off between state tracking accuracy and control effort.

By minimizing J, the optimal continuous-time state-feedback control law is obtained as

\begin{matrix} u = - K (x - x_{ref}) \end{matrix}

(36)

where $K \in R^{1 \times 2}$ is the feedback gain matrix and is derived from the solution P of the algebraic Riccati equation

\begin{matrix} A^{T} P + PA - {PBR}^{- 1} B^{T} P + Q = 0 \end{matrix}

(37)

The feedback gain is then given by

\begin{matrix} K = R^{- 1} B^{T} P \end{matrix}

(38)

PD controller

A PD controller is also designed for comparison using the same control variables $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ .

The PD control law is defined as

\begin{matrix} u = - K_{p} (θ_{2} - θ_{2 d}) - K_{d} ({\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{2 d}) \end{matrix}

(39)

where $K_{p}$ and $K_{d}$ are the proportional and derivative gains, respectively. This controller directly uses instantaneous state feedback to minimize the position and velocity errors without prediction or optimization.

Results and discussion

The research objective is to develop an ET-LTV-MPC framework capable of generating more stable and physiologically constrained bipedal walking behavior. To validate the proposed control framework, numerical simulations were conducted in the MATLAB/Simulink environment. The simulation parameters of the bipedal walking model and the ET-LTV-MPC framework are listed in Table 1. The parameters of the bipedal walking model were set in accordance with previous research (Aoi and Tsuchiya, 2007). The simulation duration was set to 20 seconds. The weighting factors $Q_{1}$ , $Q_{2}$ , and R correspond to the relative importance of minimizing the errors in $θ_{2}$ , ${\overset{\cdot}{θ}}_{2}$ , and the control input u, respectively. Wu et al. (2019) reported hip moment profiles during walking at a self-selected speed, providing a physiological reference for the torque scale used in this study. Accordingly, $u_{\max}$ was set to 30 N·m.

Table 1.

Parameters of the bipedal walking model and the proposed ET-LTV-MPC framework for simulation.

Parameters	Values	Units
M	50	kg
m	10	kg
l	1	m
g	9.81	m/s²
T	0.01	s
A	0.105	rad
ω	1.5	rad/s
$N_{p}$	20	–
$Q_{1}$	2	–
$Q_{2}$	5	–
R	0.1	–
$u_{\max}$	30	N·m

Generation of periodic gait

Figure 4 illustrates one walking cycle of the simulation based on the designed ET-LTV-MPC framework, including both the swing phase and the foot contact event. The upper plot shows the sequential motion of the stance and swing legs, where the green arrows indicate the corresponding time instants in the angle and angular velocity plots below, and the red arrow marks the moment of foot contact. The lower plots present the variations of $θ_{1}$ , $θ_{2}$ , ${\overset{\cdot}{θ}}_{1}$ , and ${\overset{\cdot}{θ}}_{2}$ within a single gait period. It can be observed that the angles and angular velocities change smoothly during the swing phase, and a sudden state transition occurs at the moment of foot contact due to the impact and leg exchange.

Figure 4.

One walking cycle of the compass model.

Figure 5 shows the simulation results of $θ_{1}$ , $θ_{2}$ , ${\overset{\cdot}{θ}}_{1},$ and ${\overset{\cdot}{θ}}_{2}$ over a total duration of 20 seconds. The results indicate that all state variables converge to a stable pattern with clear periodicity, exhibiting periodic and bounded oscillations. The abrupt sign changes at each foot contact reflect the phase reset mechanism embedded in the controller. During consecutive cycles, the waveforms remain almost identical in both shape and amplitude, suggesting that the gait rhythm is well maintained. These observations confirm that the proposed ET-LTV-MPC framework can effectively reproduce a bipedal walking pattern and maintain stable periodic behavior over the simulated duration.

Figure 5.

Simulation results of $θ_{1}$ , $θ_{2}$ , ${\overset{\cdot}{θ}}_{1},$ and ${\overset{\cdot}{θ}}_{2}$ over a total duration of 20 seconds.

Influence of prediction horizon

To investigate the trajectory tracking performance, simulations were conducted to evaluate how the ET-LTV-MPC framework performs when following the desired rhythmic trajectories generated by the oscillator. The objective of this experiment is to examine how the prediction horizon $N_{p}$ affects the control performance, that is, how far into the future the controller should predict to achieve optimal results. The tracking accuracy was quantified using the root mean square error (RMSE), which measures the average magnitude of the tracking error between the actual and reference trajectories. A smaller RMSE indicates a higher tracking accuracy.

Table 2 and Figure 6 show the variation of the RMSE for $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ with respect to the prediction horizon $N_{p}$ . The RMSE decreases rapidly as $N_{p}$ increases from small values and reaches its minimum around $N_{p} = 20$ , where the RMSE values are $4.37 \times 10^{- 4}$ rad for $θ_{2}$ and $1.4 \times 10^{- 3}$ rad/s for ${\overset{\cdot}{θ}}_{2}$ . Beyond this point, both errors gradually increase again. This result indicates that a moderate prediction horizon provides the best trade-off between prediction accuracy and computational efficiency. When $N_{p}$ is too small, the controller lacks sufficient future information to plan effective control actions, whereas excessively large $N_{p}$ values may accumulate model approximation errors and increase computational load. Therefore, $N_{p} = 20$ was selected as the optimal prediction horizon for subsequent simulations.

Table 2.

RMSE of $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ versus prediction horizon $N_{p}$ for the ET-LTV-MPC framework.

$N_{p}$	RMSE ( $θ_{2}$ ) (rad)	RMSE ( ${\overset{\cdot}{θ}}_{2}$ ) (rad/s)
1	$3.10 \times 10^{- 3}$	$4.60 \times 10^{- 3}$
5	$1.40 \times 10^{- 3}$	$1.90 \times 10^{- 3}$
10	$1.10 \times 10^{- 3}$	$1.60 \times 10^{- 3}$
15	$6.91 \times 10^{- 4}$	$1.40 \times 10^{- 3}$
20	$4.37 \times 10^{- 4}$	$1.40 \times 10^{- 3}$
25	$6.33 \times 10^{- 4}$	$1.70 \times 10^{- 3}$
30	$9.75 \times 10^{- 4}$	$2.20 \times 10^{- 3}$
35	$1.40 \times 10^{- 3}$	$2.70 \times 10^{- 3}$
40	$1.80 \times 10^{- 3}$	$3.20 \times 10^{- 3}$
45	$2.20 \times 10^{- 3}$	$3.80 \times 10^{- 3}$
50	$2.70 \times 10^{- 3}$	$4.40 \times 10^{- 3}$

Figure 6.

Variation of the RMSE of $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ with respect to the prediction horizon $N_{p}$ .

Since the sampling time of $T = 0.01$ second and the optimal prediction horizon $N_{p} = 20$ , the ET-LTV-MPC framework effectively predicts about 0.2 second into the future. This window is comparable to reported sensorimotor response latencies during human walking. For example, electromyography (EMG) responses to mediolateral balance perturbations include a medium latency R1 at 100–150 ms and a long latency R2 at 170–250 ms (Hof and Duysens, 2018). Visually guided foot-placement corrections to displaced stepping targets occur with a kinematic latency of about 155 ms and are preceded by EMG changes at 123 ms (Zhang et al., 2020). Therefore, the optimal prediction window identified in this study lies within a physiologically relevant time scale, suggesting that a short look-ahead horizon may be sufficient for accurate tracking in the present bipedal walking model. This provides a physiological interpretation for the observed performance peak at $N_{p} = 20$ and supports the physiological relevance of the proposed ET-LTV-MPC framework for bipedal walking control.

Comparison with PD and LQR controllers

To further evaluate the performance of the proposed ET-LTV-MPC framework, comparative simulations were conducted with the PD controller and the LQR controller. For the PD controller, increasing the proportional and derivative gains can effectively improve tracking accuracy and reduce error, but it also leads to a significant increase in control input amplitude. Similarly, for the LQR controller, enhancing the state-error weighting or reducing the input weighting in the cost function can also reduce tracking error, but again at the expense of higher control torque. Therefore, in order to keep the control inputs within the physiological torque limits, the PD and LQR gains were tuned conservatively. In contrast, the ET-LTV-MPC framework inherently satisfies the physiological constraint through optimization, maintaining the input torque within the predefined bound during the entire control process. When all three controllers were tuned so that their control inputs remained within the physiological constraint, the trajectory tracking results were obtained as shown in Figures 7 and 8. Figures 7 and 8 compare the trajectory tracking performance of the three controllers for $θ_{2}$ and ${\overset{\cdot}{θ}}_{2}$ , and their corresponding tracking errors e, $\overset{\cdot}{e}$ . The proposed ET-LTV-MPC framework achieves the closest tracking to the desired reference trajectory, exhibiting minimal phase lag and steady-state error, whereas both the PD and LQR controllers show larger deviations, particularly near the transition phases of the gait cycle.

Figure 7.

Tracking performance of $θ_{2}$ and the corresponding error e under PD controller, LQR controller, and ET-LTV-MPC framework.

Figure 8.

Tracking performance of ${\overset{\cdot}{θ}}_{2}$ and the corresponding error $\overset{\cdot}{e}$ under PD controller, LQR controller, and ET-LTV-MPC framework.

The inferior performance of the PD and LQR controllers can be attributed to their limited ability to account for future state evolution. Both controllers generate control actions based solely on the instantaneous state errors without explicitly considering the system’s future dynamics. Although the LQR controller is optimal in the sense of minimizing a quadratic cost function, it operates under a fixed linearized model and static weighting matrices, which restricts its ability to adapt to the time-varying and nonlinear nature of bipedal walking. Consequently, both PD and LQR controllers tend to exhibit phase lag and oscillatory behavior when tracking periodic trajectories. In contrast, the proposed ET-LTV-MPC framework continuously updates a linear time-varying model around the current operating point, enabling it to predict and optimize over future states within each control horizon. This predictive mechanism allows the controller to anticipate dynamic changes rather than merely react to them, leading to smoother transitions and enhanced stability. Furthermore, unlike conventional event-triggered strategies based on state-deviation conditions (Zhang et al., 2024), the event-triggered mechanism in this study is designed to respond specifically to foot contact events. When foot contact occurs, the system undergoes a sudden hybrid state transition, and maintaining the previous control input could induce large tracking errors. Therefore, upon detecting foot contact, the controller immediately resamples the system state, updates the prediction model, and resolves the optimization problem to compute a new control input. This mechanism enables the controller to remain adaptive and stable through the discrete–continuous transitions inherent in bipedal walking, ensuring both high tracking accuracy and physiological plausibility.

To further evaluate the practicality of the proposed controller under physiological constraints, the trade-off between tracking accuracy and control effort is examined. For the PD controller and the LQR controller, increasing the gains for the former and increasing the state error weight for the latter can not only improve tracking accuracy but also amplify the control torque. To ensure a fair comparison with the proposed method, the gains of the PD controller and the LQR controller were adjusted so that their tracking RMSE matched the level achieved by the ET-LTV-MPC framework, and then the required torques were compared.

Figure 9 shows the control torques for the three controllers under the same RMSE level. The PD controller produces relatively large torque amplitudes with abrupt jumps at phase transitions, while the LQR controller exhibits even greater torque peaks that frequently exceed the physiological limit. This excessive control effort is inconsistent with human motor behavior, which relies on a smooth and energetically efficient torque modulation. In contrast, the proposed ET-LTV-MPC framework maintains the torque strictly within the physiological constraint and generates rhythmic periodic control inputs that closely resemble the modulation patterns observed in human walking.

Figure 9.

Control torques of the PD controller, LQR controller, and ET-LTV-MPC framework.

The simulation results collectively demonstrate that the proposed ET-LTV-MPC framework achieves superior performance compared to conventional PD and LQR controllers. It provides accurate trajectory tracking, maintains control torque strictly within physiological limits, and generates smooth and rhythmic control patterns consistent with human walking. Parametric analysis shows that a moderate prediction horizon of $N_{p} = 20$ offers the best balance between prediction accuracy and computational efficiency, corresponding to an effective prediction of approximately 0.2 seconds into the future. This time scale is comparable to sensorimotor delays observed in human gait, further supporting the physiological relevance of the proposed approach. These findings indicate that the proposed control framework not only contributes to the understanding of the control mechanism in human walking but also holds practical implications for the control of humanoid robots and lower-limb exoskeletons. By reducing unnecessary torque peaks, the proposed controller may also be beneficial for actuator-friendly control in future robotic or assistive applications.

Comparison with LTV-MPC and NMPC

Furthermore, to clarify the contribution of the event-triggered mechanism and to clarify the rationale for using a linear time-varying model in the controller design, two additional comparative simulations were conducted. In the first experiment, the proposed ET-LTV-MPC was compared with a conventional LTV-MPC without event-triggered re-optimization. Both controllers use the same time-varying linearized model, sampling period $T = 0.01$ second and prediction horizon $N_{p} = 20$ . Under these conditions, the LTV-MPC yields RMSE values of $1.06 \times 10^{- 3}$ rad for $θ_{2}$ and $2.02 \times 10^{- 3}$ rad/s for ${\overset{\cdot}{θ}}_{2}$ , which are larger than the corresponding errors of $4.37 \times 10^{- 4}$ rad and $1.40 \times 10^{- 3}$ rad/s obtained with the ET-LTV-MPC. Although the control torque generated by the LTV-MPC also remains within the physiological constraint, its tracking error peaks are noticeably larger, especially near foot contact events, as illustrated in Figure 10. This degradation arises from the fixed 0.01 second sampling and control period: once a new control torque is computed, it is held constant until the next sampling instant, while foot contact may occur at any time within this interval. As shown in Figure 11(a), when contact occurs, the system dynamics change abruptly, but the LTV-MPC continues to apply a torque computed from the pre-contact model, which inevitably causes a transient model–plant mismatch and larger tracking errors. In contrast, the ET-LTV-MPC uses foot contact as an explicit event trigger: when contact is detected, the controller immediately resamples the state, updates the linear time-varying model to reflect the new support configuration, and re-solves the optimization problem to obtain a new control input that is applied without waiting for the next periodic sampling instant, as shown in Figure 11(b). By synchronizing re-optimization with the hybrid transitions of the walking, this event-triggered mechanism keeps the prediction model consistent with the actual dynamics and effectively suppresses error growth at impact, thereby enhancing the stability of the resulting periodic walking pattern.

Figure 10.

Enlarged tracking performance around a foot contact event under the conventional LTV-MPC and the proposed ET-LTV-MPC: (a) $θ_{2}$ , (b) ${\overset{\cdot}{θ}}_{2}$ .

Figure 11.

Control torque around a foot contact event: (a) conventional LTV-MPC with fixed 0.01 second sampling period, which continues to apply the torque computed from the pre-contact model; (b) proposed ET-LTV-MPC, where the detected foot contact triggers an immediate re-optimization based on the post-contact model.

The difference in update timing is further illustrated in Figure 12. As shown in Figure 12(a), the conventional LTV-MPC updates the control input only at the periodic sampling instants. In contrast, Figure 12(b) shows that the proposed ET-LTV-MPC retains the same periodic updates but introduces an additional immediate update when foot contact is detected. In the 20 second simulation, the periodic mechanism performs 2000 optimizations due to the fixed sampling period of 0.01 second, while the event-triggered mechanism introduces only 29 extra re-optimizations associated with foot contact events. Therefore, the additional computational burden caused by the event-triggered mechanism is only 1.45%, while the tracking performance is noticeably improved.

Figure 12.

Update instants around a foot contact event: (a) conventional LTV-MPC with periodic time event mechanism, (b) proposed ET-LTV-MPC with additional event-triggered update.

Second, to examine the real-time feasibility of the proposed controller and to clarify the trade-off between model fidelity and computational complexity, a set of computational experiments were carried out. In this study, the QP problems generated by ET-LTV-MPC are solved in MATLAB/Simulink using the QP solver of the Model Predictive Control Toolbox. The solver is set to active-set, so the QP problem is solved by the KWIK active-set algorithm provided in the block. The default solver options are adopted, where the maximum number of iterations is automatically computed but not smaller than 120, the inequality-constraint tolerance is $1 \times 10^{- 6}$ , and warm start is enabled. To evaluate the computational burden of the proposed ET-LTV-MPC, the Simulink Profiler is used to measure the execution time. On a laptop equipped with an Intel Core Ultra 5 125U processor running at 1.30 GHz and 16 GB of RAM, the measured average value of solver time is $1.01 \times 10^{- 4}$ second. This value is more than one order of magnitude smaller than the controller sampling period, which indicates that the QP problem generated by ET-LTV-MPC can be solved in real time.

To quantitatively evaluate the benefit of model linearization, an additional NMPC is implemented using the full nonlinear walking model as the prediction model. The NMPC uses the same parameters as the proposed ET-LTV-MPC. On the same hardware platform, the average solver time measured by the Simulink Profiler is $1.30 \times 10^{- 2}$ second, which means that the NMPC implementation is about 130 times slower than the ET-LTV-MPC implementation. The NMPC achieves an RMSE of $3.46 \times 10^{- 4}$ rad for $θ_{2}$ and $8.88 \times 10^{- 4}$ rad/s for ${\overset{\cdot}{θ}}_{2}$ , which is slightly smaller than the values of $4.37 \times 10^{- 4}$ rad and $1.40 \times 10^{- 3}$ rad/s obtained with the ET-LTV-MPC. These results show that the NMPC, while achieving slightly better tracking performance, suffers from a high computational cost due to the strongly nonlinear gait model. As a result, real-time implementation at the chosen sampling time is difficult and the controller is difficult to deploy on practical robotic hardware. In contrast, the proposed ET-LTV-MPC with a linear time-varying model offers comparable tracking accuracy with a drastically reduced computational burden, highlighting the necessity of model linearization.

Conclusion

In this paper, a novel ET-LTV-MPC framework was developed for physiologically constrained human bipedal walking. Based on an oscillator-generated desired trajectory, the controller combines a time-varying linearized model for future state prediction with a contact-triggered re-optimization mechanism. Numerical simulations demonstrated that the proposed controller successfully achieved periodic and stable gait patterns. The results of the comparative simulations show that the proposed framework maintains high tracking accuracy without violating physiological torque constraints, reduces the peak error caused by foot contact, and achieves a favorable trade-off between tracking performance and computational cost. Furthermore, the identified optimal prediction window is approximately 0.2 second into the future, which is comparable to reported sensorimotor response latencies during human walking. These findings demonstrate that the proposed framework provides a physiologically plausible and computationally efficient approach to bipedal walking control. However, this study still has limitations. First, the proposed control framework is currently only applicable to two-dimensional models and has not been validated on three-dimensional multi-joint models. Second, all simulations are conducted on flat, rigid ground and do not consider ground or environmental uncertainties such as uneven terrain and variations in friction. Future work will extend the proposed framework to more complex models and incorporate ground uncertainties to further assess its robustness in realistic locomotion scenarios.

Footnotes

Acknowledgements

The authors would like to thank editor and anonymous referees for their helpful and very delicate comments.

ORCID iDs

Mingquan Zhang

Masahiro Todoh

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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