Introduction. Restoring independent walking is one of the primary rehabilitation tasks for patients with musculoskeletal diseases and injuries, stroke sequelae, and other central nervous system disorders [1]. In 2019, there were 2.41 billion people worldwide with conditions that could benefit from rehabilitation, 1.71 billion of whom had musculoskeletal diseases [2]. These statistics take into account a wide range of disorders and reflect the global need for accessible technical means that allow for repeated and safe performance of motor exercises [3]. However, it should be noted, the data cited above allow only a very approximate assessment of the number of patients requiring gait restoration in particular [4].

Robotic systems, exoskeletons, and treadmill platforms equipped with body weight support and biofeedback systems are used for motor rehabilitation [5]. Traditional robotic systems and exoskeletons, which rigidly immobilize the patient's limbs, are capable of providing the required high-intensity training [6], but they limit the natural kinematic variability of gait. This significant drawback, in the long term, critically reduces the efficiency of transferring newly acquired motor skills to everyday independent walking [4]. For this reason, treadmills integrated into complex systems with weight support systems and virtual reality biofeedback modules are used in rehabilitation practice [4]. Most platforms operate at a fixed speed manually set by the operator, effectively imposing a strictly defined gait rhythm on the patient. Under these conditions, the treadmill distorts the natural linear and nonlinear dynamics of stride variability. The patient is forced to continuously adjust their biomechanics to the belt motion, instead of the belt adjusting to their current intentions, physical capacity, or fatigue level.

To overcome these limitations, adaptive treadmill platforms are being actively developed [7]. They can adjust the treadmill belt speed in real time with minimal latency, according to the user's kinematic and spatial characteristics. The central challenges in creating adaptive treadmill platforms are the synthesis and parametric optimization of control algorithms (controllers). They must, on the one hand, activate the energy system and an adequate response system to changes in the user's speed, and on the other, maintain stability, minimize jerking, and prevent the body from the loss of balance. Excessively high controller sensitivity will lead to high-frequency changes in treadmill speed and, consequently, increase spatial and temporal stride variability. Insufficient sensitivity and high algorithm inertia, on the other hand, will cause significant longitudinal spatial displacement of the user. The outcome in this case may be the patient exiting the safe working area of the treadmill.

The benefits of user-controlled belt speed are supported by experimental studies. The authors [8] compared fixed and adaptive modes in 18 patients with chronic stroke. With adaptive control, the step width was 13.7 cm, and with a fixed speed — 16.8 cm. The discrepancy turned out to be statistically significant (p < 0.0001). Step length and duration did not differ significantly between conditions. The results obtained indicate that the method of speed control can change individual biomechanical characteristics of walking even at a comparable average speed of movement [8].

The literature describes various methods of forming a control action to match the speed of the belt with a self-selected walking pace. The authors [9] assessed the user's speed based on the maximum speed of the foot transfer. After additional data processing, the average error in speed estimation was reduced to 0.0663 m/s, or approximately 51%. The limitation of this approach is associated with the need to register foot movement and pre-set the controller to the user’s characteristics. The authors [10] used force platform data and a Kalman filter to estimate human speed and position. The root-mean-square deviations relative to the optical motion capture system were 0.023 m/s for speed and 0.014 m for position, while the self‑selected walking speed measurements correlated at over 0.93 with the 10‑meter walk test. However, achieving this level of accuracy comes at the cost of a rather sophisticated and expensive treadmill platform.

Studies on speed control based on strain gauge sensors are published, but the implementation of such systems in mass clinical practice is extremely limited due to their high cost and technical complexity of calibration [11]. A more accessible alternative is the use of kinematic data (position and speed of the projection of the center of mass) obtained using optical or inertial tracking systems [12]. Computer vision systems require only a video camera; there is no need to attach sensors to the user. However, in this case, accuracy and stability depend on factors such as frame rate, lighting conditions, camera placement, occlusions, and the pose recognition algorithm [13]. The comparison of these approaches is especially important for the control of a treadmill platform, since differences in the discreteness, latency and noise level of the original coordinates are directly transmitted to the control system [14].

When developing treadmill platforms, linear, nonlinear, zonal, proportional-differential, and proportional-integral-differential (PID) control functions are used [15]. The selection of function and its parameters largely depends on the platform hardware, the length of the working area, the tracking system, and user safety requirements. The control system must not only keep the user within the working area but also smooth out transient processes, provide a smooth change in belt speed, and maintain balance.

In previous study [15], the authors compared six control functions using VR trackers and computer vision. This work was primarily a comparative selection and did not address the problem of sequential parametric optimization of the key functions according to a single criterion that takes into account the transient and stationary characteristics of the system.

The literature review reveals that most studies are conducted within a pre-selected measurement loop. In some studies, the primary metrics are the accuracy of speed and position estimation, while in others, the spatial-temporal characteristics of the stride, ground reaction forces, or the user's subjective assessment are considered. The differences in experimental protocols and criteria prevent a direct comparison of control functions and the reasonable selection of their coefficients. The problem of parametric optimization of linear, nonlinear, and PID functions using a single integral criterion, followed by comparison of optimized configurations for different user position tracking methods, has been insufficiently studied. Such a criterion must simultaneously consider position stability, the duration and amplitude of transient processes, tracking microdynamics, and subjective comfort.

The objective of this study was to experimentally compare and parametrically optimize the linear, nonlinear, and PID control functions of an adaptive treadmill platform using VR trackers and a CV system. To achieve this goal, five tasks were solved.

The scientific novelty of the study lies in the development of a procedure for parametric optimization of the control functions of an adaptive treadmill platform based on an integral criterion that combines indicators of position stability, transient processes, microdynamics of tracking, and subjective comfort, as well as in establishing the preferred parameters of the linear, nonlinear and PID functions for two measuring loops.

The practical significance of the work is related to the formation of initial parameters and recommendations for selecting a control function for systems based on VR trackers and computer vision.

Materials and Methods. The first stage of the study examined the hardware and overall architecture of the adaptive treadmill platform control system. The experiments utilized an adaptive treadmill platform developed by Tambov State Technical University (Tambov, Russian Federation). The control system integrated a treadmill platform, user position tracking tools, a coordinate preprocessing module, a control signal generation software module, a control microcontroller, and a belt drive. Based on the user's current coordinates, the software module calculated the required belt speed and transmitted the corresponding command to the drive controller. The servo drive had a power of 1.5 kW. The motor was controlled by a driver connected to an ESP32 microcontroller. Belt speed was determined based on encoder data. A key feature of the platform was the absence of handrails. This eliminated obstacles for computer vision and improved the quality of postural stability assessment, since human movements were analyzed without the need for postural compensation using hand support.

Key technical characteristics of the experimental platform:

Note that the platform is capable of speed of up to 5.5 m/s. However, as part of the experiment, the indicator was programmatically limited to 2.0 m/s. This is due to safety requirements and corresponds to a fast pace, that is, suitable for most musculoskeletal rehabilitation scenarios.

The study used two tracking systems [16]. The first (S1) was a set of HTC Vive Tracker 3.0 virtual reality sensors (HTC Corporation, Taiwan) with a recording frequency of 60 Hz and a positioning error of less than 0.01 m [17]. Two trackers of the system were symmetrically attached to the user’s waist: one anterior and one posterior [18]. This placement provided stable position tracking: even if one tracker temporarily lost connection or was blocked, the second continued to transmit coordinates.

The second system (S2) integrated computer vision and a camera with a resolution of 1920 × 1080 pixels and a recording rate of 30 frames per second [19]. The camera was installed at a height of 1.2 m and a distance of 1.5 m from the edge of the treadmill platform. The MediaPipe Pose model version 0.9 and OpenCV version 4.8.1.78 were used to recognize human body position [12].

The following applicability criteria were used when selecting the systems: positioning error of up to 5 cm, refresh rate of at least 30 Hz, and total latency of no more than 100 ms. Configuration S1 met these criteria. Configuration S2 met them with proper camera installation and controlled shooting conditions, including illumination and absence of occlusions [20].

The mathematical model of the controller involves dividing the treadmill belt into several functional areas, each of which determines the logic of the system behavior.

When recognizing a user, the computer vision system converts the current body position using the formula:

(1)

where LTR — platform length, m; ux — current position of the user in the frame along Х axis, received from MediaPipe Pose at two points of the belt; sx, ex — saved initial and final positions [21].

For a long treadmill platform, formula (1) can be transformed, and the metric position of a user relative to the middle of the platform, and not its beginning can be obtained:

(2)

For the CV system, formula (2) is used, that is, option 2. Virtual reality trackers can also be calibrated at the center of the treadmill. Therefore, the metric position (U) from both systems is determined in a single relative coordinate system.

Three basic control functions were implemented [15], each of which received as input the user current position U at time t and returned the target value of the belt speed.

The linear function FL determined the platform speed based on the user position U relative to the safety area boundary:

where LTRS — length of the safe area, m; kl — set size of the working area, m; VP — maximum platform speed, m/s.

The nonlinear function FN is based on a smoother change in speed. Its form depends on the value of coefficient kα:

In previous study [15], it was found that acceptable values of parameter kα were in the range 0.3 ≤ kα ≤ 0.7.

PID function FPID is based on the PID control law and uses information about the user offset relative to the previous measurement, as well as his speed (the derivative of this offset):

Here, Ut+Δt, Ut — the user current and previous position at time t + Δt and t, respectively. FPID (Ut+Δt, t + Δt) — new speed value for time t + Δt at position Ut+Δt. The correction factors kp, ki, and kd determine the effect of the proportional, integral, and differential components of function FPID.

A set of metrics has been developed to evaluate the performance of control functions.

User Position Stability Metrics

Transient Process Metrics (Acceleration Dynamics)

Microdynamics Metrics (Tracking Quality)

Subjective Metric

The metrics had different dimensions and different directions of improvement; therefore, normalization was performed before calculating the integral criterion. For each metric Ri, normalized value was calculated:

where — normalized value of the i-th metric; Ri — initial value of the metric obtained in the experiment; — maximum value of the metric in all experiments of the stage under consideration; — base value of the metric.

Zero is used as the base value, since for most metrics included in the integral criterion, the target value is to approach zero.

For metrics whose reduction corresponds to an improvement in control quality, the normalized value itself is included in the integral criterion: . For a subjective assessment of movement quality, a higher value corresponds to a better result, so the direction of the scale is inverted, and the transformation: is used.

All indicators converged toward a common optimization objective, and the integral control quality criterion was calculated as a weighted sum of normalized metrics. Weight coefficients wi were distributed using expert assessments, with the highest priority given to physical safety and stability parameters: RSTO (0.1684); RZTO (0.1579); RZFMEAN (0.1579); RSAO (0.1263); RZD (0.1263); RZAO (0.1263); RG (0.0737); RZFSMAE (0.0421); RZMD (0.0211). Thus, the optimization objective function was reduced to minimizing value RQ, which would serve as the objective function when selecting the parameters of the linear, nonlinear and PID control functions:

where wi — weight coefficient of the i-th metric; N — total number of metrics.

To test the robustness of the integral criterion to expert weight assignment, a sensitivity analysis was conducted. In each iteration, the weight coefficients were independently randomly changed within ±20% of the initial values, after which they were re-normalized so that their sum remained equal to one. For each set of weights, the integral estimates of the six combinations of control function and tracking system were recalculated, and a new ranking was formed. A total of 1000 iterations were performed. The similarity of each resulting ranking to the initial one was assessed using the Kendall rank correlation coefficient.

Thus, the calculation procedure was performed in four stages:

The obtained value RQ was used as the objective function when selecting the parameters of the linear, nonlinear and PID control functions.

Five healthy male subjects participated in the experiment to select the control function parameters. This sample was selected to ensure that the algorithms could be evaluated on stable and reproducible gait trajectories. The results reflect the system bench‑level tuning. Their applicability to patients with gait disorders will require separate validation.

During the final comparison of control functions, the separate sample was expanded to 10 healthy subjects, which made it possible to test the selected parameters in a single comparative experiment.

Three passes for each parametric configuration (5 options for linear, 12 for nonlinear, 16 for PID) were performed. The duration of one pass was 60 seconds. The first 10–15 seconds were occupied by the transition process (acceleration and synchronization), the remaining 45–50 seconds were used to evaluate the movement in a more stable mode. The selected duration avoided noticeable fatigue of the participants and at the same time obtained a sufficient time series for calculations.

The subject's task was to start walking from a complete rest, gradually reach a comfortable walking speed, and maintain it throughout the experiment. After 60 seconds, the controller automatically applied gradual braking. During all tests, an operator was present near the platform with an emergency stop button.

165 unique experiments were conducted, yielding 495 valid time series records of user position and walking speed.

The study was conducted in two stages. In the preliminary stage, parametric tuning of the linear, nonlinear and PID control functions was performed using five healthy male subjects. In the main stage, the selected configurations were tested on a separate group of ten healthy male subjects who were not involved in the parameter tuning. Participants were 22.5 ± 3.2 years old, with an average height of 179 ± 5.23 cm. This split sample allowed for the separation of the function tuning procedure from the subsequent comparative testing of the selected configurations.

Research Results. At the first stage, the selection of the working area parameter FL for the linear control function was performed. The results of calculating metrics for various parameter values in the range from 0.5 to 1.5 meters are presented in Table 1.

At low working-area settings, the highest surge amplitudes of belt speed and the greatest positional deviation of the user during initial acceleration are observed. Increasing the working area results in longer setting time for both speed and position. Moreover, at the maximum working-area value, a notable increase is seen in both the stationarity interval of the user position and the expected value of the high-frequency position oscillation amplitude.

The maximum subjective score RG was obtained at kl = 1.5 m, and the minimum value of the integral quality criterion for the linear function in the studied range was obtained with a working area of 1 m.

Next, parameters were selected for the nonlinear control function FN. In this case, two parameters were varied simultaneously: the length of the working area and the nonlinearity coefficient kα. The experimental results are presented in Table 2, where the values of the main normalized metrics and the integral quality criterion for control are listed for each configuration.

Analysis of the data obtained showed that for a nonlinear function, the minimum value of the integral criterion (1.24) was obtained with a working area of 0.75 m and a nonlinearity coefficient of 0.3.

Additionally, the individual stability of the selected parameters of the nonlinear function was assessed. For each of the five participants, the integral criterion was calculated separately for all combinations of working area length and nonlinearity coefficient. For two participants, the individual minimum coincided with the group minimum, while for the other two, the best values were located in close proximity to the group minimum. For one subject, the distribution of integral scores differed significantly from the group result. Consequently, the parameters selected based on averaged data can be considered as initial control settings.

At the next stage, the PID control function FPID was investigated. With fixed proportional coefficient kp = 2.0, the values of integral ki and differential kd coefficients were varied (Table 3).

The minimum value of the integral performance criterion for the PID function (1.15) was obtained with integral and differential coefficients of 0.5 and 0.5, respectively. Increasing the differential coefficient in the studied configurations almost always resulted in an increase in the amplitude of the belt speed surge. In each group, the lowest integral coefficient values maintained a higher user position stationarity interval.

As an additional configuration indicator, user exiting beyond the control function effective area was recorded. Out of 495 preliminary-stage records, such instances occurred in 7.7% of cases. Moreover, most exits beyond the control function effective area were recorded within a 0.5-meter working-area, which provided further justification for selecting longer working areas.

After selecting the parameters, three control functions were compared with the selected configurations. The impact of the user position tracking system, computer vision (CV) and virtual reality (VR) trackers, was also assessed. The comparison results for an expanded sample of participants (10 healthy male subjects) are presented in Figure 1 and Table 4. Note the difference in the color of the graphs in Figure 1 a and Figure 1 b–k. This difference in design emphasizes that the former visualizes metrics whose values need to be maximized, while the latter visualize metrics whose values need to be minimized.

A final comparison of all three control functions revealed that the integral criterion values depend on both the selected control function and the body position tracking system. When using a CV system, the weighted sum of the metrics is 2.874 for the linear function, 2.683 for the nonlinear function, and 2.694 for the PID function. The corresponding values when using VR trackers are: 2.002, 2.248 and 2.121.

Using a CV system, the minimum integral criterion value was obtained for the nonlinear control function. With VR trackers, the lowest integral criterion value for the linear control function provided high subjective user assessment results.

To assess the dependence of the results on the selected weight coefficients, the sensitivity of the integral criterion was analyzed. When each weight was randomly varied within ±20%, the average Kendall coefficient between the original and recalculated rankings was 0.983 ± 0.029, indicating a high stability of the overall sequence of configurations.

For all three VR tracking configurations, changing the weights did not change the ranking positions in any of the 1000 iterations. For the CV system, the ranking was less robust: the probability of deterioration in the nonlinear function position was 2.7%. For the PID function, the probability of improvement was 2.5%, while the probability of deterioration was 22.7%. The linear function improved the position in 22.7% of the iterations. Thus, the conclusion about the preference of the linear function for VR tracking turned out to be robust to moderate changes in the priorities of individual metrics. For the CV system, the differences between the functions were less clear-cut, especially between the nonlinear and PID.

The statistical significance of differences between control functions and tracking systems was tested using nonparametric analysis based on the Mann-Whitney test (Table 5). Pairwise comparison was performed for all control function combinations, separately for VR tracking systems and MediaPipe. The analysis was conducted across all key metrics.

Additionally, the combined normalized profiles formed from the nine metrics included in the integral criterion and their unweighted sum were compared. It was found that, when using VR trackers, statistically significant differences were observed between the linear and nonlinear functions (p < 0.001), as well as between the linear and PID functions (p = 0.0020). The difference between the nonlinear and PID functions did not reach the accepted level of statistical significance (p = 0.0574). For the CV system, no statistically significant differences in the combined profiles were found (p = 0.1407–0.5664), indicating less pronounced differences between the control functions. Values were obtained by pairwise comparison of the combined normalized indicators for each control function and tracking system. The integral criterion was not directly used in this analysis.

Discussion. Based on the results of the descriptive comparison and sensitivity analysis, the linear function with a 1.0 m working area can be considered the baseline configuration for the VR tracker loop. For computer vision, the lowest observed value of the integral criterion was obtained using the nonlinear function with a 0.75 m working area and a nonlinearity coefficient of 0.3. However, the close values of the PID function and the partial change in ranks resulting from weight variations preclude any definitive claim of superiority for either of the two options.

The selection of control function cannot be considered separately from the tracking system. When using VR trackers, a user position is determined by a metric system with greater stability and update rate, allowing for a simpler control function. Under these conditions, the linear law demonstrated the lowest integral criterion value and the highest subjective rating.

For the computer vision system, the differences between the nonlinear and PID functions were less pronounced. In the experiment, the minimum value of the integral criterion was obtained for the nonlinear function, but the PID function showed a similar result. This permits both configurations to be regarded as viable for the CV loop, but with consideration given to the position recognition quality, data update rate, and the need for additional coordinate filtering.

A more complex controller structure does not always improve the final control quality. The PID function demonstrated the lowest integral criterion value during the parametric tuning stage on a small sample, but in a general comparison with VR tracking, the best results were obtained for the linear law.

The system latency of the measurement loop was 31.3 ± 0.72 ms for the VR trackers and approximately 34.86 ms for the CV system. Both values are below the accepted threshold of 100 ms used as the acceptance criterion for real-time control. However, the computer vision system requires more stringent control of shooting conditions, as recognition quality depends on lighting, camera position, and possible occlusions.

The results obtained are consistent with the general logic of research into adaptive treadmill platforms, where control quality is determined not only by the control law but also by the stability of the user position measurement. Furthermore, the presented study refines the data of previous scientific studies through demonstrating a direct comparison of control functions in two measurement loops: VR tracking and computer vision.

We note the limitations of the study: the small sample size and the participation of only healthy volunteers. Therefore, the research results should be considered as a preliminary engineering validation stage, tuning the adaptive treadmill platforms, and selecting the initial parameters of the control functions. For patients with gait or balance disorders, or severe stride asymmetry, different values for the optimal coefficients and a different response to changes in belt speed are possible. Accordingly, in the future, it is planned to test the statistical significance of differences between individual control functions on a wider sample, calculate confidence intervals, analyze the effect of individual user characteristics, and test the clinical applicability of the proposed settings.

Conclusion. This paper addresses the problem of parametric optimization of the linear, nonlinear and PID control functions of an adaptive treadmill for musculoskeletal rehabilitation systems. An integrated control quality criterion is proposed and applied, combining metrically and physiologically interpretable parameters: average user displacement, duration and amplitude of transient processes, tracking microdynamics, and subjective assessment of movement quality. A procedure for selecting control function parameters based on the integrated quality criterion is tested.

Let us list three major results of the work.

The results obtained have practical significance for the development of adaptive treadmill platforms and the tuning of control algorithms in rehabilitation systems. Prospects for further research include testing the proposed parameters in experiments with a larger number of participants, including patients with gait disorders. Furthermore, plans are underway to develop a procedure for individually tuning control functions based on data from the first minutes of movement on the platform.