Active Ankle Rehabilitation Training Based on Extended State Observer with Autotomy Mechanism to Enhance Interaction

Abstract

Stroke-related ankle dysfunction requires robust human–robot interaction, yet abrupt motion disturbances and measurement noise can degrade trajectory tracking and interaction quality. This study proposes a biologically inspired extended state observer with an autotomy mechanism (AESO). The design maps the finite state transition associated with biological autotomy to a continuous change between baseline and high observer bandwidths, thereby addressing the response–accuracy trade-off of conventional extended state observers (ESOs). An active rehabilitation training framework is then constructed by integrating AESO with a proportional–derivative controller and an admittance model for a parallel ankle rehabilitation robot. Boundedness during the bandwidth transition and terminal ultimate-error scaling are analyzed theoretically. Simulations and within-participant experiments involving eight healthy adults compare AESO with fixed-, bi-, and scalable-bandwidth ESOs during static and dynamic tracking tasks. The results show that AESO generally reduces tracking error, improves trajectory smoothness, and lowers normalized interaction torque relative to the baseline observers. These findings support the engineering feasibility of AESO for disturbance-robust ankle rehabilitation; its clinical benefits require confirmation in people after stroke and in longitudinal training studies.

Keywords

Ankle rehabilitation robot / Human–robot interaction / Extended state observer / Admittance model

1 Introduction

Stroke is a prevalent neurological disorder and frequently causes lower-limb dysfunction, including foot drop and inversion [1]. Because the ankle contributes to weight bearing, postural stability, and gait progression, targeted ankle rehabilitation is important for functional recovery and activities of daily living after stroke [2].

Admittance control is widely used in active ankle rehabilitation because it converts measured human–robot interaction (HRI) into compliant robot motion [3,4]. By responding to the participant’s motion and torque feedback, the controller can limit excessive mechanical loading and discomfort [5]. Its performance, however, can deteriorate when voluntary or involuntary motion, sensor noise, actuator interference, or other disturbances enter the interaction loop [6]. These disturbances may produce abrupt end-effector deviations and fluctuations in joint acceleration, thereby reducing the consistency of HRI [7].

Observers provide a practical way to estimate unmeasured states and disturbances from available measurements [8]. In particular, an extended state observer (ESO) augments the system state with lumped model uncertainty and external disturbance, reducing dependence on a perfectly known plant model [9]. A central design difficulty is the selection of observer bandwidth [10]. Increasing bandwidth improves the response to rapidly varying disturbances, but also amplifies measurement noise; decreasing bandwidth improves steady-state accuracy and noise attenuation, but slows disturbance estimation [9]. A fixed bandwidth therefore cannot provide the preferred response under all rehabilitation conditions.

Previous studies have expanded the estimation range of ESOs by incorporating disturbance derivatives [11], feedforward compensation [12,13], proportional error feedback and pole placement [14], and cascaded or parallel observer structures [15,16]. Low-bandwidth estimators tailored to periodic disturbances have also been reported [17]. These approaches improve disturbance rejection, but most retain a fixed bandwidth or optimize a particular disturbance class.

Adaptive-bandwidth ESOs attempt to use the available bandwidth more selectively [18]. Existing methods include discrete state switching [19], model-based adaptive laws [20,21], and feedback driven by tracking error or estimated disturbance intensity [22,23]. Two questions nevertheless remain coupled: when a substantial bandwidth change is warranted and how the observer should move between bandwidth states without introducing an additional switching transient.

Autotomy offers a decision-making analogy for these two questions. An organism remains in its normal state under a low perceived threat and initiates an emergency response only when the stimulus reaches a critical level; the physical transition then occurs over a finite interval [24]. We translate perceived threat and the finite transition into the estimated total disturbance and a continuous change from a baseline bandwidth to a high bandwidth. The biological threshold motivates the concept of a transition onset, while the present implementation uses prescribed transition timing rather than a real-time disturbance-dependent threshold law. This correspondence directly motivates the proposed ESO with an autotomy mechanism (AESO) and connects the biological inspiration to the bandwidth dilemma of conventional ESOs. Based on AESO, we develop an active HRI framework for a parallel ankle rehabilitation robot (PARR), establish boundedness during the bandwidth transition and characterize terminal ultimate estimation-error scaling, and evaluate the framework in simulation and human-participant experiments.

The remainder of this paper is organized as follows. Section 2 presents the AESO-based active rehabilitation framework and its estimation-error analysis. Section 3 describes the simulation study. Section 4 reports the experimental protocol, results, and rehabilitation relevance. Section 5 summarizes the conclusions and future work.

2 The Proposed Active Rehabilitation Training Control Framework Based on AESO

The proposed framework is implemented on our previously developed PARR [7,25]. It comprises an admittance model, a proportional–derivative (PD) controller, the proposed AESO, and a linear extended state observer (LESO). Section 2.1 describes the signal flow, Section 2.2 establishes the engineering correspondence with autotomy, and the subsequent sections formulate AESO and analyze estimation-error boundedness and terminal ultimate-error scaling.

2.1 Overview of the framework

Fig. 1 shows the signal flow in the order of propagation. The measured interaction torque T_q first enters the LESO, whose estimate D_t compensates the disturbance in the interaction-torque channel. The compensated torque is converted by the admittance model into the generated reference position {\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}. The PD controller combines this reference with the AESO state estimates z_1 and z_2 and produces the PARR control input u. The encoder measures the actual platform position {\theta }_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}; this measurement, denoted by y in the observer equations, is fed back to the AESO. The AESO estimates the system states and the lumped disturbance. Its disturbance estimate z_3 is available to the autotomy module as the disturbance-feedback signal. In the present study, however, the bandwidth-state transition is commanded according to a prescribed schedule rather than triggered automatically by a threshold on z_3. Thus, the z_3 path identifies the feedback input for a future disturbance-dependent trigger, whereas the implemented observer uses the prescribed transition \omega (t).

2.2 Autotomy mechanism

Autotomy is a defensive behavior in which certain reptiles and arthropods intentionally shed an appendage to escape predation [24,26]. Because the loss of an appendage carries an immediate biological cost, the response is not advantageous for every weak stimulus. Threat-sensitive predator-avoidance theory instead suggests that an organism evaluates risk and initiates the costly response only when the perceived threat warrants it [27,28]. Experimental observations further support risk-dependent behavior and the existence of a stimulus threshold for appendage detachment [29,30]. These features motivate two engineering requirements: retaining a low-cost normal state under weak disturbances and replacing an abrupt state switch with a finite transition when a stronger response is required.

For control design, we abstract autotomy as a continuous transition from a normal state s_1 to an emergency state s_2 over the finite interval [t_1,t_2], as illustrated in Fig. 2. The endpoint-normalized definition in Eq. 1 is equal to s_1 before the interval, changes continuously within the interval, and is equal to s_2 afterward:

(1)

Under this abstraction, AESO uses a prescribed finite-time transition between two bandwidth states. The biological stimulus-threshold concept motivates the transition onset but is not implemented as a real-time triggering law. The model does not represent biological tissue regeneration; returning the observer to its baseline bandwidth is solely an engineering state transition. Disturbance-dependent triggering and recovery laws remain topics for future work. Table 1 summarizes the resulting engineering correspondence.

2.3 AESO

Consider the nth-order system in Eq. 2:

(2)\begin{array}{l}\{\begin{array}{rl}{x}^{(n)}(t)& =f(t,x(t),\dot{x}(t),\cdots ,x^{(n-1)}(t))+\epsilon (t)+bu(t),\\ y(t)& =x(t),\end{array}\end{array}

Here, x(t),\dot{x}(t),\dots ,x^{(n-1)}(t) are the system states; f(\cdot ) represents known or unknown system dynamics; \epsilon (t) is the external disturbance; u(t) is the input; b is the scalar input gain; and y(t) is the measurable output.

The unknown dynamics and external disturbance are combined into the total disturbance:

(3)\begin{array}{l}{x}_{n+1}(t)=f(\cdot )+\epsilon (t)\end{array}

Define its derivative as

(4)\begin{array}{l}d(x,\epsilon )\triangleq \dot{f}(\cdot )+\dot{\epsilon }\end{array}

The augmented state equation is then given by Eq. 5.

(5)\begin{array}{l}\{\begin{array}{l}{\dot{x}}_1(t)=x_2(t)\\ ⋮\\ {\dot{x}}_n(t)=x_{n+1}(t)+bu(t)\\ {\dot{x}}_{n+1}(t)=d(x,\epsilon )\\ y(t)=x_1(t)\end{array}\end{array}

The proposed AESO is

(6)\begin{array}{l}\{\begin{array}{l}{\dot{\hat{x}}}_1(t)={\hat{x}}_2(t)+{\beta }_1(\omega (t))(x_1(t)-{\hat{x}}_1(t)),\\ ⋮\\ {\dot{\hat{x}}}_n(t)={\hat{x}}_{n+1}(t)+{\beta }_n(\omega (t))(x_1(t)-{\hat{x}}_1(t))+bu(t),\\ {\dot{\hat{x}}}_{n+1}(t)={\beta }_{n+1}(\omega (t))(x_1(t)-{\hat{x}}_1(t)).\end{array}\end{array}

Here, {\hat{x}}_i(t) is the estimate of the ith augmented state, and {\beta }_i(\omega (t)) is the corresponding AESO gain:

(7){\beta }_i(\omega (t))=\frac{(n+1)!}{i!\left(n+1-i\right)!}\omega (t{)}^i,

In Eq. 7, \omega (t) is the AESO bandwidth and n is the order of the plant in Eq. 2. Applying the transition model in Eq. 1, the bandwidth changes continuously from the baseline value {\omega }_1 to the high-response value {\omega }_2 over [t_1,t_2]:

(8)\begin{array}{l}\omega (t)=S({\omega }_1,{\omega }_2,t)\end{array}

Theorem 1. Assume that , , e(t_1) is finite, and for t\ge t_1, with D independent of {\omega }_2. Then the estimation error remains bounded during [t_1,t_2]. Moreover, for the ideal error model in Eq. 12, there is a constant C>0, independent of {\omega }_2, such that

(9)\begin{array}{l}\underset{t\to \mathrm{\infty }}{lim sup}|e_i(t)|\le \frac{CD}{{\omega }_2^{n+2-i}},i=1,2,\dots ,n+1.\end{array}

This bound describes the no-measurement-noise model used in the analysis.

2.4 Convergence analysis of the proposed AESO

This section analyzes estimation-error boundedness during the finite bandwidth transition and terminal ultimate-error scaling after the transition.

Define the componentwise estimation error as

(10)\begin{array}{l}{e}_i(t)=x_i(t)-{\hat{x}}_i(t),i=1,2,\cdots ,n+1\end{array}

and collect the components into

(11)\begin{array}{l}e(t)=[e_1(t)\cdots e_{n+1}(t){]}^{\mathrm{T}}\end{array}

The AESO estimation error satisfies

(12)\begin{array}{l}\{\begin{array}{l}{\dot{e}}_1(t)=e_2(t)-{\beta }_1(\omega (t))e_1(t),\\ ⋮\\ {\dot{e}}_n(t)=e_{n+1}(t)-{\beta }_n(\omega (t))e_1(t),\\ {\dot{e}}_{n+1}(t)=-{\beta }_{n+1}(\omega (t))e_1(t)-d(x,\epsilon )\end{array}\end{array}

or, in compact form,

(13)\begin{array}{l}\dot{e}(t)=A(t)e(t)+q(-d(x,\epsilon ))\end{array}

where

(14)\begin{array}{l}A(t)=\left[\begin{array}{cccc}-{\beta }_1(\omega (t))& 1& \cdots & 0\\ ⋮& \ddots & 1& ⋮\\ -{\beta }_n(\omega (t))& \cdots & 0& 1\\ -{\beta }_{n+1}(\omega (t))& \cdots & 0& 0\end{array}\right],q=\left[\begin{array}{c}0\\ ⋮\\ 0\\ 1\end{array}\right]\end{array}

Lemma 1. Consider the linear time-varying system

(15)\begin{array}{l}\dot{x}=A(t)x,x\in {\mathbb{R}}^n\end{array}

If A(t) can be decomposed into a time-invariant Hurwitz matrix A_H and a time-varying matrix A_V(t),

(16)\begin{array}{l}\dot{x}=(A_H+A_V(t))x\end{array}

If, in addition,

(17)\begin{array}{l}\underset{t\to \mathrm{\infty }}{lim}{A}_V(t)=\mathbf{0}\end{array}

and

(18)

then the origin of the homogeneous system in Eq. 15 is asymptotically stable.

For the proof of Theorem 1, write the global disturbance-derivative bound as

(19)

After the transition, \omega (t)={\omega }_2 and the error-system matrix is

(20)\begin{array}{l}{A}_H=\left[\begin{array}{cccc}-{\beta }_1({\omega }_2)& 1& \cdots & 0\\ ⋮& \ddots & 1& ⋮\\ -{\beta }_n({\omega }_2)& \cdots & 0& 1\\ -{\beta }_{n+1}({\omega }_2)& \cdots & 0& 0\end{array}\right].\end{array}

Using the binomial gains in Eq. 7, its characteristic polynomial is

(21)\begin{array}{l}det(\lambda I-A_H)=(\lambda +{\omega }_2{)}^{n+1},\end{array}

so A_H is Hurwitz for every {\omega }_2>0. For completeness, during the transition define

(22)\begin{array}{l}{A}_V(t)=\left[\begin{array}{cccc}{\beta }_1({\omega }_2)-{\beta }_1(\omega (t))& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮\\ {\beta }_n({\omega }_2)-{\beta }_n(\omega (t))& 0& \cdots & 0\\ {\beta }_{n+1}({\omega }_2)-{\beta }_{n+1}(\omega (t))& 0& \cdots & 0\end{array}\right].\end{array}

Because \omega (t) reaches {\omega }_2 after the finite transition, regardless of whether the transition approaches that endpoint from below or above,

(23)\begin{array}{l}\underset{t\to \mathrm{\infty }}{lim}\left({\beta }_i({\omega }_2)-{\beta }_i(\omega (t))\right)=0,i=1,\dots ,n+1,\end{array}

and hence

(24)\begin{array}{l}\underset{t\to \mathrm{\infty }}{lim}{A}_V(t)=\mathbf{0}.\end{array}

Moreover, because A_V(t)=0 after t_2,

(25)

where \Vert \cdot {\Vert }_1 is the induced matrix 1-norm. Lemma 1 therefore applies to the homogeneous error system.

Boundedness over the finite transition follows directly, without assuming a bounded Lyapunov gradient. Define

(26)

The variation-of-constants formula and Grönwall’s inequality give, for every t\in [t_1,t_2],

(27)\Vert e(t){\Vert }_2\le e^{{\overline{A}}_T(t-t_1)}\Vert e(t_1){\Vert }_2+{\int }_{t_1}^t{e}^{{\overline{A}}_T(t-s)}|d(x,\epsilon )|\mathrm{d}s.

Consequently,

(28)

It remains to establish the terminal high-gain scaling. Let

(29)\begin{array}{l}{a}_i=(\genfrac{}{}{0ex}{}{n+1}{i}),{\xi }_i={\omega }_2^{n+1-i}{e}_i,i=1,\dots ,n+1.\end{array}

For t\ge t_2, the scaled error satisfies

(30)\begin{array}{l}\dot{\xi }={\omega }_2{A}_0\xi -qd(x,\epsilon ),A_0=\left[\begin{array}{cccc}-a_1& 1& \cdots & 0\\ ⋮& \ddots & 1& ⋮\\ -a_n& \cdots & 0& 1\\ -a_{n+1}& \cdots & 0& 0\end{array}\right].\end{array}

The normalized matrix has characteristic polynomial

(31)\begin{array}{l}det(\lambda I-A_0)=(\lambda +1{)}^{n+1}.\end{array}

Therefore, constants K\ge 1 and \alpha >0, depending only on n, exist such that

(32)\begin{array}{l}\Vert e^{A_0\tau }{\Vert }_2\le K{e}^{-\alpha \tau },\tau \ge 0.\end{array}

Applying this bound to the scaled system yields

(33)\Vert \xi (t){\Vert }_2\le K{e}^{-\alpha {\omega }_2(t-t_2)}\Vert \xi (t_2){\Vert }_2+{\displaystyle \frac{KD}{\alpha {\omega }_2}},t\ge t_2.

Because e_i={\xi }_i/{\omega }_2^{n+1-i},

(34)\underset{t\to \mathrm{\infty }}{lim sup}|e_i(t)|\le \frac{KD}{\alpha {\omega }_2^{n+2-i}},i=1,\dots ,n+1.

Thus Eq. 9 holds with C=K/\alpha, and

(35)\begin{array}{l}\underset{{\omega }_2\to \mathrm{\infty }}{lim}\underset{t\to \mathrm{\infty }}{lim sup}|e_i(t)|=0.\end{array}

For a return transition S({\omega }_2,{\omega }_1,t), the finite-interval bound in Eq. 28 is unchanged, and the terminal analysis applies after replacing {\omega }_2 by the terminal bandwidth {\omega }_1. The asymptotic statement above applies to the ideal error model without measurement noise. It does not imply that the bandwidth should be increased without limit in practice, because sensor noise and unmodeled dynamics constrain the usable bandwidth.

3 Simulation of the Framework with AESO

AESO was compared with three baseline observers: a fixed-bandwidth ESO (FESO), a bi-bandwidth ESO (BESO), and a scalable-bandwidth ESO (SESO).

3.1 The baseline ESO algorithms

3.1.1 FESO

The FESO offers a simple and computationally efficient approach for estimating a system’s unknown states and disturbances. It achieves this by employing a fixed bandwidth in the observer design, as in Eq. 36.

(36)\begin{array}{l}\omega =300\end{array}

3.1.2 BESO

BESO switches between two bandwidth levels according to the sign of the error derivative [31]:

(37)\begin{array}{l}\omega (e)={\omega }_0\gamma ={\omega }_0{\eta }^{-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\dot{e})},\end{array}

where {\omega }_0 is the nominal bandwidth, \gamma is the directional switching factor, and \eta is the bi-bandwidth scaling factor. The sign function produces a step-type bandwidth change.

3.1.3 SESO

SESO continuously scales its bandwidth with the magnitude of the output-estimation error [23]:

(38)\begin{array}{l}\omega (e)={\omega }_0+\eta \left|e\right|,\end{array}

where {\omega }_0 is the nominal bandwidth and \eta >0 is the scaling factor. Unlike FESO and the step-type BESO, SESO produces a continuous error-dependent bandwidth.

3.2 Simulation protocol

The simulation compares the disturbance-rejection performance of AESO with FESO, BESO, and SESO under a common plant, command, and disturbance protocol.

Following our previous PARR model [7], the admittance transfer function G_A(s) is

(39)G_A(s)=\frac{1}{3.125s^2+2.5s}

The approximate transfer function of the PARR, G_R(s), is

(40)G_R(s)=\frac{1}{s^2+0.8s+0.8}

For AESO, {\omega }_1=300 and {\omega }_2=600. In the simulation, prescribed high-state commands were applied at the onset of the two D1 pulses (t=2 and 12 s), and return commands were applied at their offset (t=4 and 14 s); these commands were not generated by an online threshold on z_3. After each command, the local transition interval in Eq. 8 was set from t_1=0.001 s to t_2=0.05 s, giving a transition duration of 0.049 s. A return transition uses the same function with the two endpoint bandwidths interchanged. For comparable bandwidth ranges, BESO used {\omega }_0=450 and \eta =0.75 in Eq. 37, whereas SESO used {\omega }_0=80 and \eta =0.5 in Eq. 38. Fig. 3 distinguishes the green input-torque command (amplitude 10 in the simulation torque scale, period 1 s) from the four orange disturbance signals. D1 and D2 enter the reference-position channel, D3 enters the torque-measurement channel, and D4 enters the encoder channel.

3.2.1 Pulse disturbance (D1)

D1 was defined as a {10}^{\circ } pulse position disturbance with a 2 s pulse width and a 10 s period, applied over 2–4 s and 12–14 s. The amplitude was not derived from clinical tremor or involuntary-motion data. Accordingly, D1 represents a standardized severe stress test rather than a clinically representative involuntary motion. The amplitude was set to {10}^{\circ } because this value equals the maximum absolute target angle in both experimental tracking tasks. D1 therefore introduces an abrupt deviation on the scale of the task and enables a consistent comparison of transient recovery across the four observers.

3.2.2 Triangular wave disturbance (D2)

D2 was a triangular position disturbance with a 2 s period and an amplitude range of 0^{\circ }–1^{\circ }, representing a repeatable low-amplitude periodic perturbation.

3.2.3 Interference noise (D3)

White noise with a power spectral density of {10}^{-3} was added to the torque-measurement channel.

3.2.4 Sensor noise (D4)

White noise with a power spectral density of {10}^{-10} was added to the encoder output to represent position-measurement noise.

The simulation was performed in MATLAB R2022a with a fixed step of 0.001 s.

3.3 Simulation results

Fig. 4 presents position, tracking error, and observer bandwidth. All four observer conditions enabled the PD-controlled plant to follow the desired trajectory. The RMS tracking errors for AESO, BESO, SESO, and FESO were 2.03\times {10}^{-3\circ }, 1.97\times {10}^{-3\circ }, 2.97\times {10}^{-3\circ }, and 2.30\times {10}^{-3\circ }, respectively. The light-blue intervals identify the two D1 pulses. BESO produced the smallest overall RMS error, whereas SESO showed the largest. AESO did not yield the absolute minimum RMS value, but its continuous finite-time bandwidth changes avoided the repeated step switching of BESO and the prolonged error-sensitive variation of SESO. These results indicate a compromise between transient response and steady-state behavior.

4 Experimental Evaluation and Discussion of the Proposed AESO-Based Active Rehabilitation Training Framework

4.1 Experimental setup

The PARR is described in our previous studies [7,25]. A six-axis circular load cell (M3715C, SRI, China) measured the interaction torque between the participant and the robot, and encoders measured the pose of the moving platform. Both signals were transmitted to the control computer through serial communication.

Eight healthy adults (one female and seven males; age, 24.67\pm 2.33 years; body mass, 68.5\pm 16.5 kg) with no history of neurological disease or lower-limb musculoskeletal injury participated. The study involving human participants was approved by Ethical Committee of the Rehabilitation Hospital Affiliated to the National Research Center for Rehabilitation Technical Aids with the protocol No. S20220202. The procedure was conducted in accordance with the Declaration of Helsinki.

Each participant completed both tracking tasks under all four observer conditions (FESO, BESO, SESO, and AESO), so the experiment used a within-participant repeated-measures design and each participant served as their own control. Participants sat on an adjustable chair with the tested foot secured to the PARR platform and the knee maintained at approximately {90}^{\circ }. A monitor displayed the target and current platform positions in real time. Participants were instructed to interact with the PARR so that the platform followed the displayed target as accurately as possible.

The two protocol-defined tracking tasks are illustrated in Fig. 5.

4.1.1 Static tracking task

Participants tracked a repeated sequence of discrete targets: {10}^{\circ }, -{10}^{\circ }, 5^{\circ }, -5^{\circ }, and back to {10}^{\circ }. Each target was held for 10 s.

4.1.2 Dynamic tracking task

Participants tracked a sinusoidal target varying from -{10}^{\circ } to {10}^{\circ } with a 10 s period and controlled the PARR platform to match the displayed trajectory in real time.

To reduce expectation bias, participants were blinded to the observer condition used in each trial.

4.2 Performance metrics and statistical methods

Five performance metrics were used to compare AESO with the three baseline observers: generation error, response error, tracking error, trajectory jitter index (TJI), and interaction-torque magnitude. For TJI, the actual and generated-reference trajectories were treated as separate scalar outcomes. One condition-level value per participant, task, observer condition, and scalar outcome was obtained before group aggregation; repeated time samples were not treated as independent participants. Data are summarized as mean \pm standard deviation across the eight participants. A one-way repeated-measures analysis of variance (ANOVA) was used, with observer condition (FESO, BESO, SESO, and AESO) as the within-participant factor. The static and dynamic tasks were analyzed separately, and a separate ANOVA was performed for each scalar outcome, including each TJI trajectory. Statistical significance was set at . When an ANOVA indicated a significant observer-condition effect, post hoc within-participant pairwise comparisons were used to identify the specific differences shown by the brackets in the figures. Generation, response, and tracking errors were retained in degrees, whereas TJI is dimensionless. Interaction torque was normalized within participant as described below.

The desired trajectory is the task command shown on the display, the generated reference trajectory is the output of the admittance model, and the actual trajectory is the encoder-measured PARR position.

4.2.1 Generation error

Generation error is the difference between the generated reference and desired trajectories and quantifies reference-generation accuracy.

4.2.2 Response error

Response error is the difference between the actual and generated reference trajectories and quantifies closed-loop robot response.

4.2.3 Tracking error

Tracking error is the difference between the desired and actual trajectories and represents overall task performance.

4.2.4 Smoothness (TJI)

TJI is based on dimensionless squared jerk (DSJ), a dimensionless measure of motion smoothness [32]. Because jerk is the third time derivative of position, DSJ for a trajectory \mathrm{\Theta }(t) is defined as

(41)DSJ(\mathrm{\Theta })={\int }_{t_1}^{t_2}\stackrel{⃛}{\mathrm{\Theta }}(t{)}^2\mathrm{d}t\frac{(t_2-t_1{)}^5}{A_{\mathrm{\Theta }}^2},

where A_{\mathrm{\Theta }} is the total excursion of the trajectory being evaluated over [t_1,t_2].

Because the task command itself has nonzero DSJ, TJI normalizes the evaluated trajectory by the corresponding desired trajectory:

(42)TJI({\mathrm{\Theta }}_{\mathrm{e}\mathrm{v}\mathrm{a}})=\frac{DSJ({\mathrm{\Theta }}_{\mathrm{e}\mathrm{v}\mathrm{a}})}{DSJ({\mathrm{\Theta }}_d)}

Here, {\mathrm{\Theta }}_d is neither a theoretical minimum-jerk trajectory nor a trajectory measured from the robot. It is the protocol-defined command displayed to the participant: the recorded step-and-hold command sequence for the static task and the prescribed sinusoidal command for the dynamic task. DSJ was evaluated from the sampled command and evaluated-trajectory sequences using the same finite-difference operator, timestamps, and analysis interval. Thus, the static-task calculation uses the finite sample-to-sample changes in the recorded command rather than the distributional derivative of an ideal discontinuous step. Within each task, the same desired trajectory and timing were used for FESO, BESO, SESO, and AESO, providing a common denominator for comparison. Lower TJI indicates a smoother evaluated trajectory relative to that task command.

4.3 Results of trajectory tracking performance

Fig. 6 shows representative tracking data from Participant #1. Panels (A)–(D) correspond to the static task and panels (E)–(H) to the dynamic task. The enlarged regions show the overshoot, oscillation, and lag that distinguish the four observer conditions.

Fig. 7A summarizes the static task. BESO yielded the numerically smallest tracking error ({2.742}^{\circ }), followed closely by AESO ({2.785}^{\circ }). AESO had lower tracking error than FESO and SESO (), whereas no significant AESO–BESO difference is marked. AESO also had a lower response error than FESO (). For generation error, SESO was higher than BESO () and AESO ().

In the dynamic task (Fig. 7B), AESO produced the lowest mean tracking error ({1.047}^{\circ }), representing reductions of 45.67%, 24.89%, and 12.90% relative to FESO, BESO, and SESO, respectively. The marked AESO–FESO difference was significant (); no significant AESO–BESO or AESO–SESO tracking-error difference is marked. Response error did not differ significantly among the four conditions. AESO also produced the lowest mean generation error ({0.834}^{\circ }), with marked differences from FESO () and SESO (), but not from BESO.

The condition-specific patterns help explain these results. In the static task, BESO responded rapidly to each step but showed visible oscillation, whereas SESO had smaller steady-state fluctuations but greater lag. AESO combined a continuous bandwidth transition with a high-response state and therefore remained close to the best static tracking result without the repeated step switching of BESO. In the dynamic task, AESO had the smallest mean generation and tracking errors. Because not every comparison with an adaptive-bandwidth baseline was significant, the evidence indicates a balanced performance advantage across metrics and task conditions.

4.4 Results of smoothness

Fig. 8 presents the TJI results. In the static task, the AESO actual-trajectory TJI was 43.72% of the SESO value, and the AESO generated-reference TJI was 22.77% of the BESO value; both marked comparisons were significant. AESO also produced the smallest mean TJI for both trajectories in the dynamic task. These findings indicate that the continuous bandwidth transition was associated with less trajectory jitter, while the marked brackets in the figure specify which pairwise differences reached significance.

4.5 Results of interaction torque

Fig. 9 shows representative interaction torque for Participant #1. For this representative participant, the static-task BESO trace showed a longer high-torque interval than the SESO or AESO trace, and the dynamic-task BESO and SESO traces fluctuated more rapidly. These trace-level observations illustrate differing corrective-effort patterns but are not group-level statistical comparisons.

Interaction torque is sensitive to participant strength, body mass, and voluntary effort. We therefore normalized the mean interaction-torque magnitude within each participant and task before group aggregation. For participant i and observer condition c,

(43)T_{i,c}^{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=100\frac{\overline{|T_{i,c}|}}{\overline{|T_{i,\mathrm{F}\mathrm{E}\mathrm{S}\mathrm{O}}|}}\mathrm{\%},

so that FESO equals 100% for every participant and serves only as a within-participant reference. Fig. 10 shows the resulting group summaries. AESO had the lowest mean normalized torque in both tasks. In the dynamic task, its value was 44.34% lower than BESO and 17.84% lower than SESO.

4.6 Rehabilitation relevance and study limitations

The engineering outcomes may have practical relevance to rehabilitation. Smaller tracking errors may reduce the need for abrupt corrective motion, and lower TJI may correspond to more continuous velocity and acceleration profiles. Lower interaction torque may indicate that the user needs fewer or smaller compensatory efforts to correct the robot’s motion. Together, these effects may reduce unnecessary muscle activation and local fatigue, make the interaction more predictable, and support comfort and adherence during repeated training.

These implications are hypotheses rather than demonstrated clinical benefits. The present study included only eight healthy young adults and did not directly measure pain, muscle soreness, perceived comfort, fatigue, adherence, or long-term rehabilitation outcomes. Clinical efficacy therefore remains to be established in people after stroke and in longitudinal rehabilitation studies.

5 Conclusions and Future Work

This study developed an active rehabilitation training framework that integrates AESO with a PD controller and an admittance model for PARR. The endpoint-normalized autotomy transition maps a prescribed finite state change to a continuous bandwidth transition, and the convergence analysis establishes bounded estimation behavior during that transition. Simulation and within-participant experiments with eight healthy adults showed that AESO provides a useful balance among tracking accuracy, trajectory smoothness, and interaction-torque magnitude relative to fixed-, bi-, and scalable-bandwidth ESOs. BESO produced the smallest simulation root mean square error and the smallest mean static tracking error, whereas AESO achieved the smallest mean dynamic tracking error, the smallest mean TJI, and the lowest normalized interaction torque in both experimental tasks.

From a rehabilitation perspective, the combination of smaller tracking errors, smoother motion, and lower interaction torque may reduce abrupt corrections and unnecessary compensatory effort, potentially supporting comfort and adherence during repeated practice. Because the present participants were healthy adults and pain, soreness, fatigue, comfort, adherence, and clinical recovery were not measured, these potential benefits must be tested in people after stroke and in longitudinal rehabilitation studies.

Future work will combine surface electromyography (sEMG) with ankle kinematics to estimate actively exerted ankle torque and voluntary participation [33,34]. Recent related studies have demonstrated continuous lower-limb joint-moment and ground-reaction-force prediction from multimodal wearable signals [35], low-delay continuous multi-terrain motion recognition through fusion of sEMG and inertial measurement unit signals [36], and lightweight continuous prediction of lower-limb kinematics and dynamics from marker trajectories [37]. Although these models are not components of the present study, they provide concrete sensing and inference options for the proposed future extension. These physiological and biomechanical estimates will be investigated for adapting assistance level or admittance parameters, while AESO continues to estimate system states and unknown disturbances. Such a combination may help distinguish intentional effort from unintended perturbation and enable patient-specific adaptive training. sEMG was not used as a real-time AESO input or as an experimental outcome in the present study. Future work will also investigate disturbance-dependent bandwidth triggering and recovery laws; these functions are not part of the current prescribed-transition AESO.

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