A Sensitive and Accurate Walking Speed Prediction Method Using Ankle Torque Estimation for a User-Driven Treadmill Interface

To achieve safe and immersive interface with a user-driven treadmill (UDT), robustness of the user position must be ensured by sensitively estimating and accurately converging to the intentional walking speed (IWS). The existing IWS estimation using a linear observer with the cart model (<inline-formula> <tex-math notation="LaTeX">$1^{\mathrm {st}}$ </tex-math></inline-formula> order dynamics) can exponentially converge to the true IWS. However, when the estimation sensitivity is increased by increasing the gain, this method causes severe postural instability due to the generation of excessive anomalous forces. Thus, the existing method has an implicit limitation with regards to increasing the position robustness because of the postural instability issues. In this paper, to simultaneously achieve sensitive and accurate IWS estimation while reducing postural instability on a UDT, in addition to the cart model, we have also utilized the inverted pendulum-based gait model (IPGM) as a <inline-formula> <tex-math notation="LaTeX">$2^{\mathrm {nd}}$ </tex-math></inline-formula> order dynamic to estimate the intentional walking acceleration (IWA) generated by the ankle torque. Thus, the proposed IWS prediction method uses the cart model for accurate convergence to IWS and the IPGM to follow sensitively the change in the IWS. In the proposed method, the internal states of the existing observers applied to the <inline-formula> <tex-math notation="LaTeX">$1^{\mathrm {st}}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$2^{\mathrm {nd}}$ </tex-math></inline-formula> order dynamics are shared recursively to estimate the ankle torque acting as a disturbance for the IPGM and to sensitively predict the change in the IWS. Experiments show that the proposed method can significantly facilitate the users in following a profile of desired walking speeds more accurately than the existing IWS estimation method under the same position robustness setup.


I. INTRODUCTION
18 Treadmills are widely utilized in virtual reality (VR) as a 19 representative device for locomotion interface (LI) to allow 20 users to participate actively in VR with realistic spatial sen-21 sations [1], [2], [3]. This functionality is achieved through a user-driven treadmill (UDT) that tracks the user's locomotion 23 intention and allows the generation of unlimited level-ground 24 conditions without limiting the user's motions. To ensure safe 25 and immersive gait interface with a UDT, the user's position 26 The associate editor coordinating the review of this manuscript and approving it for publication was Mark Kok Yew Ng . must be maintained in a reference position above the treadmill 27 belt even when their walking speed is changing arbitrarily, 28 and their spatial and temporal gait parameters should not 29 be significantly different from those during over-ground or 30 conventional treadmill walking [4], [5]. 31 If the belt motion of a UDT does not sensitively follow a 32 user's intentional walking speed (IWS), it causes a position 33 error, the excess of which can induce the user to fall down. 34 Thus, the main objectives of a UDT control scheme are to 35 ensure the position robustness by accurately estimating the 36 IWS, and to generate the appropriate control commands when 37 the user dynamically changes their walking speed [5], [6]. 38 gait pattern, it suffers from inaccurate prediction of the IWS 86 when the user is starting to walk from a standstill (i.e., from 87 zero to preferred speed). Thus, it is difficult to apply it to a 88 wide range of walking patterns and speeds. 89 To alleviate the limitations of the works presented in [5] 90 and [12], Kim. et. al. proposed a feedforward strategy called 91 the attenuator [13], which can keep a time required to con-92 verge to the IWS by attenuating from an exponential conver-93 gence rate (linear observer) to a proportional rate based on the 94 MSFV method. However, while using the attenuator concept, 95 it may still be difficult to guarantee the position robustness 96 due to the reduced convergence rate when the gait speed is 97 changing dynamically. 98 Therefore, in the presented work, we have aimed to 99 design a walking speed prediction method that can sensi-100 tively respond to changes in the IWS to increase position 101 robustness, while overcoming the postural instability by using 102 an appropriate gait dynamics model. The proposed method 103 involves simultaneous utilization of the existing 1 st order 104 dynamics to guarantee the accurate IWS convergence and 105 the 2 nd order dynamics of the inverted pendulum-based gait 106 model (IPGM) [14], [15] to represent the movement of the 107 lower extremities. While the conventional method involving 108 the 1 st order dynamics tracks the true IWS, another distur-109 bance observer working with the 2 nd order dynamics detects 110 the amount of change in the IWS through the estimation of 111 the ankle torque. 112 The effectiveness of the proposed method is validated 113 through experimentation with 10 subjects where it is com-114 pared to the existing controller reported in [12]. The experi-115 ment results revealed that use of the proposed method allowed 116 the subjects to change their gait speed more accurately 117 according to cue speeds given to them as reference com-118 mands, while maintaining the position robustness at the same 119 level as the conventional method. The existing IWS estimation, which is based on the cart 123 model shown in Figure 1 [12], has the problem of gait insta-124 bility that occurs when a user tries to change their walking 125 speed [13]. According to the research reported in [12], the 126 true IWS (v w ) acting as a disturbance in a UDT can be 127 estimated using the cart model, and it can be expressed as 128 follows 129 where, x 1 is the position of the COM, v w denotes the true 131 IWS, which is considered as the disturbance of where g is the gravitational acceleration defined as 9.81m/s 2 , 156 m is the user's mass, T ank represents the generated ankle (v c , a c ), a more accurate IWS estimation can be achieved. term (µ). This feedback control scheme helps to increase the 179 robustness of the position error by estimating the slowly time-180 varying uncertainty of the closed-loop system.

181
In the existing LI method without the additional distur-182 bance observer, the true IWS (v w ) is observed by the extended 183 state observer (ESO) [18] based on the cart model only, 184 as given by Eq. (1), and it directly feeds forward to the final 185 control command (v c ), shown using the white-dashed line 186 in Figure 3. Therefore, according to the existing method for 187 designing a LI controller, the conventional control command 188 can be given as follows wherev w_temp is the value observed by the ESO based on 191 Eq. (1) and µ is the feedback command from the RISE con-192 troller. The key feature ofv w_temp is exponential convergence 193 to v w according to the property of the 1 st order dynamics.

194
Meanwhile, to sensitively predict the change in the IWS 195 by estimating the generated ankle torque (T ank ) in the IPGM, 196 the additional disturbance observer is utilized as shown in 197 Figure 3 (Gray-line), and the conventional control command 198 represented by Eq. (4) is modified by utilizing the additional 199 disturbance observer as follows: wherex 2 is the observed amount of change in the IWS, which 202 is computed by the 2 nd order dynamics represented by Eq. (2) 203 and (3),v w is observed using the hyperbolic tangent tracking 204 differentiator-based nonlinear disturbance observer (HTDO) 205 [17] to accurately converge to the steady-state IWS with the 206 disturbance rejection property, andv w_total is the summation 207 of these observed values.
where, ρ, m 1 and m 2 are positive design parameters, z input is 229 the input value that requires signal processing, z 1 is the output 230 value after signal processing, and z 2 is a state representing the 231 differential value of z 1 . Moreover, the HTTD is also applied 232 to the motion capture system to reduce the random noise in where, β 1 , β 2 , β 3 , α 1 , α 2 , α 3 and ξ are the gain parameters value predicted by the other observer to compensate for the 255 uncertainties generated by the estimation error inx 3 .

256
In Eq. (7), f is a nonlinear gain function about ε, given as 257 If α is set to 1, this ESO becomes the Luenberger observer 260 (Linear observer) because the output of the nonlinear gain 261 function becomes identical to the input error (ε). On the other 262 hand, it takes the form of a sliding mode observer when α is 263 set to 0. Thus, if the value of α is increasing, the ESO is at a 264 sensitive setting. Under α = 0, the maximum value of f (ε, 265 0, ξ ) defined by the sliding mode observer mode is just 1. 266 When 0 < α < 1, the function f has the characteristic that the 267 smaller the error, the relatively greater is the output and the 268 larger the error, the smaller is the output.
(2) and (3) from Eq. (7) (i.e.,ê 1 =x 1 − 276 It should be noted that, 277 since f is working as only a nonlinear gain with respect to ε, 278 Eq. (9) is calculated by replacingê 1 with f (ε, α, ξ ) for the 279 simplified stability analysis [21]. Thus, the solution of Eq. (9) 280 is computed as To find the bounded condition of Eq. (10) for any time (t), A 283 in the error dynamics should be Hurwitz. Assuming that a comp 284 is bounded and z c is closed, the gain satisfying the Hurwitz 285 condition is given as To correspond to a height of the COM, which is different for 288 each person, the gain should be set as β 2 β 3 .

289
For immersive LI, the gain should be set to match the gen-290 erated ankle torque during real human walking by adjusting 291 the gain parameters of the ESO (x 1 → x 1 (t) andx 3 → 292 T ank /m). In research on gait analysis [22][23], the maximum 293 ankle torque (Nm) per body weight (kg) in the normal walk-294 ing speed range (1.25∼2 m/s) is known to be approximately 295 1.3 to 2 Nm/kg. Thus, the gains were set as; β 1 = 10, β 2 = 296 360, β 3 = 290, α 1 = 0.4, α 2 = 0.4, α 3 = 0.25 and ξ = 0.001, 297 so thatx 3 converges to the ankle joint torque trends suggested 298 in the existing gait research. Moreover, the stability is also 299 satisfied for the height of COM just above 0.14m 300 VOLUME 10, 2022 Through gain tuning,x 3 was suggested to estimate the ankle 303 torque, which causes the IWS change (x 2 ). However, it is 304 difficult to predictx 2 accurately, since the estimated ankle 305 torque (x 3 ) is highly gain-dependent, and the available infor-306 mation is too limited to determine it correctly. Therefore, 307 a comp , mentioned in the previous section, needs to be applied 308 to compensate for this inaccuracy.

340
For the stability analysis of HTDO, it satisfies the conver-341 gence condition as follows [24] When ρ z → ∞, the stability analysis can be obtained as 344 follows Which means that the variation in a comp is much faster than 348 -a c +â w . This can be clarified by the following equations.
Thus, when we regard ''−a c + â w_temp + a comp '' as ''v w '', 351 it is clear that Eq. (15) and Eq. (16) are established by 352 the theorem reported in [25]. It should be noted that, in an 353 actual system, −a c is bounded due to the physical limita-354 tions. Therefore, it is reasonable to assume that a comp is 355 estimated much faster thanâ w_temp − a c . Moreover, a comp 356 must be bounded becauseâ w_temp is bounded by the 1 st order 357 dynamics-based ESO and the HTTD. Finally, the parameters 358 of the HTDO are set as; ρ z = 10, m z1 = 1 and m z2 = 6.

359
To briefly explain the proposed IWS prediction method 360 shown in Figure 4, the HTDO calculates the compensated 361 IWS (v w ), and for an accurate IWA, it estimates the com-362 pensation value as a comp which is the convergence error in 363 the IWS calculated by the previous final control command 364 (a c ) and the predicted IWA (â w_temp ) based on the 1 st order 365 dynamics. Next, the ESO based on the IPGM computes the 366 amount of change in IWS (x 2 ), and suppliesx 1 to the ESO 367 based on the cart model with improved accuracy achieved by 368 using a comp . Again, in the cart model based ESO, the temporal 369 IWS (v w_temp ) is re-estimated by the predicted user location 370 update (x 1 ), and the temporal predicted IWS (v w_temp ) is 371 supplied to the HTDO to recursively improve the accuracy of 372 IWS estimation. Thus, it helps to converge to more accurate 373 IWS and T ank /m values.

375
The primary objective of the feedback controller is to gen-376 erate a control command to set the position of the user at 377 a desired location. For this purpose, the feedback controller 378 needs to be designed as shown in Figure 5 (See also Figure 3). 379 This controller utilizes the RISE controller given by [26] 380 where, k a is an adjustable control gain. The controller defined  where t is the simulation time. It should be noted that p x is 425 selected from p both as the ankle joint position of the lower 426 extremity that is in the stance phase. During the double 427 support phase of gait, this is calculated as the average of the 428 positions of the two ankle joints, and this value is almost 429 equal to the position of the COM. To perform the simulation 430 as close to the implemented system as possible, the random 431 noise of the motion capture camera was included in the 432 simulation in the range of ±1 mm. The loop frequency of the 433 motion capture was set to 100Hz, and that of the controller 434 was set to 1kHz.

435
In the simulation result shown in Figure 6(a), the given 436 IWS (v w ) is 1.4m/s and the applied time constant is 0.1s. 437 The 1 st order dynamics-based ESO converges quickly to the 438 IWS but is affected by the oscillatory movement of the ankle 439 position, whilev w stably converges to the steady state IWS 440 with noise rejection. 441 Meanwhile, By using the estimated T ank /m, as shown 442 in Figure 6(b),x 2 can sensitively respond to the amount 443 VOLUME 10, 2022 of change in the IWS and is reduced when the IWS goes 444 to a steady state. In Figure 6   the maximum belt speed of the developed UDT is 3.5 m/s. 478 The user's pelvic and foot motions during treadmill walking 479 were captured by the motion capture system (VICON) and 480 used to measure x 1 and p x with a sampling rate of 100 Hz. 481 The markers for tracking the position of COM were placed 482 on the posterior superior iliac spines. Considering the user's 483 convenience and safety, these spinal markers were attached 484 to the harness. The markers for tracking the ankles were 485 attached to the switch soles that are specially designed soles 486 installed in the user's ankle joint (see Figure 7). The switch 487 soles have a movable joint so that they do not interfere with 488 the movement of the user's ankle joint. The switches are 489 active when the lower extremity that they are attached to is in 490 the stance phase and deactivated during swing phase. During 491 double support phase, p x is computed as the average value of 492 both the ankle positions, same as the simulation performed 493 above.

494
If the user performs a run, the double support phase occurs 495 in the air. Thus, the switch sole signals of both the lower 496 extremities become temporarily off. At this time, since the 497 position of the ankle joint of the lower extremity where the 498 propulsive force is generated is located behind the ankle 499 joint where the braking force and switch sole signals will be 500 activated, p x is decided to be the ankle joint having the smaller 501 position value.

502
The hardware connection configuration is shown in 503 Figure 8. The software running on each PC are executed in 504 real-time with a 1kHz loop frequency, and real-time syn-505 chronization is realized by connecting the PCs through direct 506 connected TCP/IP communication. The motion capture PC 507 interfaces with the switch signals from the switch soles, and 508 it measures the user's COM (x 1 ) and ankle joint position (p x ) 509 via VICON. The measured information is transmitted to the 510 high-level PC, and the control command (v c ) is generated. 511 The low-level PC and the high-level PC transmit the current 512 treadmill belt speed (v) and control command (v c ) to each 513 other in real time. The PCI-type PLC installed in the low-514 level PC is connected to the servo amplifier via Mechatrolink, 515 which is an open field network used to simplify system 516 configuration while ensuring synchronization [27]. The loop 517 frequency of the PLC used is guaranteed to be 10kHz, there-518 fore, it can stably execute the calculated control command 519 (v c ) given by the high-level control PC.

521
To obtain objective performance results, all the participants 522 were given the same walking speed profile, and it was 523 observed that how closely they were able to follow the given 524 profile. The desired speed profile was provided visually to 525 the participants through the graphic shown in Figure 9 (left), 526 FIGURE 8. System configuration used for executing locomotion interface using the proposed IWS prediction method.  From the collected data, the root mean square (RMS) of the 563 error between the cue speed and the treadmill belt speed and 564 RMS of error in user position are computed, respectively. 565 Furthermore, to compare the gait pattern changes due to 566 the existing and the proposed controllers, an analysis of the 567 spatio-temporal gait parameters was performed on the tread-568 mill, which included the total number of steps, average step 569 length, cadence and walk ratio. The step length is measured 570 when double stance is determined by the switch sole, which 571 represents approximately the distance between the ipsilateral 572 and the contralateral heel at each heel contact. The walk 573 ratio represents the relationship between the amplitude and 574 frequency of rhythmic leg movements during walking and 575 was calculated as the average step length divided by the 576 cadence [28]. Although cadence usually uses number of steps 577 per minute, in this paper, it is defined as the number of steps 578 per second since the experiment time was only 30s.

579
For post-experimental data analysis, a one-way repeated 580 measures analysis of variance (RMANOVA) was performed 581 to study the effects of the proposed controller under the 582 various speed changes on the RMS of the error between the 583 cue speed and the treadmill belt speed, RMS of the error in 584 user position, total number of steps, average step length and 585 walk ratio. Mauchly's test of Sphericity was used to confirm 586 the validity of the RMANOVA results. Post hoc tests were 587 conducted using the Bonferroni correction method. Partial eta 588 squared (µ 2 p ) was calculated as a measure of the effect size 589 for one-way RMANOVA. All statistical analyses were carried 590 out using SPSS V20.0 (IBM Corp., USA).

593
The average position error for all the participants, and its 594 standard deviation (STD), in the time domain is shown in 595 Figure 10. For synchronizing the results of each participant, 596 post-processing was performed by referring to the cue speed 597 data included in each participant's results. The yellow-shaded 598 zone represents the accelerating and decelerating gait speeds. 599 Both the controllers have relatively large position errors when 600 the gait speed changes (yellow-shaded periods in Figure 10, 601 except at the start and end of the experiment). 602 Figure 11 shows the time domain experimental results in 603 the form of the average and STD of all the participants' 604 walking speeds compared to the given cue speed profile. 605 Analyzing both the LI controllers, the STD tends to increase 606 at the beginning and end of the experiment. The reason for 607 this large deviation in walking speeds of the participants at 608 the beginning and ending periods is that they experience 609 the incorrect speed convergence rate from the UDT, which 610 is exacerbated by the large gait speed changes (1.5 m/s) 611 during these periods as compared to the other experiment 612 VOLUME 10, 2022  However, when the user is trying to follow the desired 618 speed cue with the existing method, there is a large difference 619 in the gait speeds of the different participants. In the exper-620 iment period where the gait speed cue is maintained after 621 accelerating or decelerating, the existing controller has more 622 difficulty in following the cue speed. Thus, the trend of the 623 STD remains larger than the proposed method.

624
Combining the experimental results for the cue speed fol-625 lowing (command following) and the position error, the STD 626 of the position error of each participant in Figure 10 and 627 the gait speed error in Figure 11 show similar trends. That 628 is, both the STDs of the existing controller show relatively 629 larger errors and fluctuations. This is because the accuracy of 630 gait speed estimation was reduced due to the generation of 631 excessive control commands that did not match the intention 632 of the participants who were trying to follow the speed cues. 633 Thus, the current position robustness setting in the existing 634 method is relatively unsuitable in this situation. It means that 635 the IWS convergence rate should be lowered by a smaller 636 gain in the linear observer, which will also reduce the posi-637 tion robustness. However, in the case of the presented IWS 638 prediction, it can follow the IWS more precisely. 639 Figure 12 shows the estimated ankle torque per mass of 640 a male participant with the speed cue. In the section at 641 the start of the experiment, the ankle torque per mass (x 3 ) 642 showed a tendency to increase, reaching a maximum value 643 of 2.7 Nm/kg. In the section where the walking speed is 644 constant, the ankle torque shows a trend that is similar to the 645 simulation results (1.5 Nm/kg) and the gait analysis research 646 reported in [22] and [23]. When the cue for a reduction 647 in walking speed is given, the ankle torque also tends to 648 decrease, and reaches a minimum value of −2.3 Nm/kg. 649 Thus, the proposed IWS estimator can predict the ankle joint 650 torque well during the actual gait interface, similar to the 651 simulation result presented in Figure 6 Table 2, and results of the 656 one-way RMANOVA carried out to study the effects of the 657 proposed controller are presented in Table 3. The average 658      In case of the step length, the existing and the proposed 678 controller showed 0.3874±0.062 m and 0.3871±0.038 m, 679 respectively. While for the walking ratio, the outcomes were 680 0.231±0.053 m/step/s and 0.235±0.036 m/step/s, respec-681 tively. There was also no significant difference in any of 682 the spatio-temporal gait parameters. Thus, the overall results 683 suggest that the proposed controller has relatively better per-684 formance in following the IWS and it does not change the 685 user's gait pattern. Moreover, after the experiment, we asked 686 each participant about their feelings or opinions about both 687 the controllers. All the participants were of the opinion that it 688 was easier to follow the speed cues with the LI controller that 689 utilized the proposed IWS prediction method. This perception 690 is supported by the result obtained from the quantitative data. 691

692
In this paper, we proposed an IWS estimator that uses the 693 position information of the ankle joint and the COM of the 694 subject, which is more accurate and sensitive than the con-695 ventional method. The acceleration/deceleration generated 696 during walking is set as the disturbance of the UDT system, 697 and the torque generated at the ankle joint, which is the cause 698 of this disturbance, is accurately predicted by the proposed 699 observer scheme to quickly respond to the IWA by using the 700 1 st and 2 nd order dynamics that represent the UDT-human 701 dynamics. Results from the experiments with 10 participants 702 show that, as compared to the previously developed method, 703 the proposed controller has significantly better performance 704 in following the user's IWS, while maintaining the same level 705 of position robustness and having no significant effect on 706 their gait pattern.