ADJUSTABLE SURFACE STIFFNESS TREADMILL

Abstract

A treadmill with adjustable vertical surface stiffness is disclosed. The treadmill is used to study gait adaptation and develop interventions for gait rehabilitation for conditions such as stroke. The surface stiffness is set by controlling the length of a cantilever beam which acts as an adjustable stiffness spring suspension underneath the treadmill. The treadmill also contains an additional novel mechanism to reduce the amount by which it is allowed to sag under its own weight as the stiffness decreases.

Claims

1. An adjustable stiffness treadmill, comprising: a first treadmill motor; a first treadmill driven by the first treadmill motor, the first treadmill having a linear guide permitting vertical movement of the first treadmill and limiting lateral movement of the first treadmill; a cantilever beam supporting the first treadmill; and a support mechanism supporting the cantilever beam, the support mechanism positionable about the length of the cantilever beam whereby the spring tension of the cantilever beam may be selectively stiffened.

2. The device of claim 1, wherein the support mechanism comprises a rack and pinion mechanism configured and arranged to adjust the position of the support mechanism about the cantilever beam.

3. The device of claim 1, wherein the support mechanism comprises a plurality of cantilever constraint rollers contacting the cantilever beam.

4. The device of claim 1, wherein the cantilever beam comprises a cantilever spring.

5. The device of claim 1, further comprising: a preload mechanism configured and arranged to reduce sag in the first treadmill.

6. The device of claim 1, further comprising: a second treadmill motor; and a second treadmill driven by the second treadmill motor; the second treadmill positioned adjacent and parallel to the first treadmill, whereby a subject's left leg and right leg are on separate treadmills when in use.

7. The device of claim 1 further comprising a plurality of motion capture cameras positioned around the first treadmill.

8. The device of claim 6, further comprising a plurality of motion capture cameras positioned around the first treadmill and second treadmill.

9. An adjustable stiffness treadmill, comprising: a first treadmill motor; a first treadmill driven by the first treadmill motor, the first treadmill having a linear guide permitting vertical movement of the first treadmill and limiting lateral movement of the first treadmill; a cantilever beam supporting the first treadmill; a support mechanism supporting the cantilever beam, the support mechanism positionable about the length of the cantilever beam whereby the spring tension of the cantilever beam may be selectively stiffened; a second treadmill motor; and a second treadmill driven by the second treadmill motor; the second treadmill positioned adjacent and parallel to the first treadmill, whereby a subject's left leg and right leg are on separate treadmills when in use.

10. The device of claim 9, wherein the support mechanism comprises a rack and pinion mechanism configured and arranged to adjust the position of the support mechanism about the cantilever beam.

11. The device of claim 9, wherein the support mechanism comprises a plurality of cantilever constraint rollers contacting the cantilever beam.

12. The device of claim 9, wherein the cantilever beam comprises a cantilever spring.

13. The device of claim 9, further comprising: a preload mechanism configured and arranged to reduce sag in the first treadmill.

14. The device of claim 9, further comprising a plurality of motion capture cameras positioned around the first treadmill and second treadmill.

15. An adjustable stiffness treadmill, comprising: a first treadmill motor; a first treadmill driven by the first treadmill motor, the first treadmill having a linear guide permitting vertical movement of the first treadmill and limiting lateral movement of the first treadmill; a cantilever beam supporting the first treadmill; a support mechanism supporting the cantilever beam, the support mechanism positionable about the length of the cantilever beam whereby the spring tension of the cantilever beam may be selectively stiffened; and a preload mechanism configured and arranged to reduce sag in the first treadmill.

16. The device of claim 15, wherein the support mechanism comprises a rack and pinion mechanism configured and arranged to adjust the position of the support mechanism about the cantilever beam.

17. The device of claim 15, wherein the support mechanism comprises a plurality of cantilever constraint rollers contacting the cantilever beam.

18. The device of claim 15, wherein the cantilever beam comprises a cantilever spring.

19. The device of claim 15, further comprising: a second treadmill motor; and a second treadmill driven by the second treadmill motor; the second treadmill positioned adjacent and parallel to the first treadmill, whereby a subject's left leg and right leg are on separate treadmills when in use.

20. The device of claim 15, further comprising a plurality of motion capture cameras positioned around the first treadmill.

21. The device of claim 19, further comprising a plurality of motion capture cameras positioned around the first treadmill and second treadmill.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] These and other features, aspects, and advantages of the present invention will become better understood with reference to the following description, appended claims, and accompanying drawings where:

[0009] FIG. 1 is a partially exploded view of an embodiment of the adjustable stiffness treadmill;

[0010] FIG. 2A is a top, rear perspective of an embodiment of the adjustable stiffness treadmill;

[0011] FIG. 2B is a top, front perspective of an embodiment of the adjustable stiffness treadmill, with the platform over the cantilever removed;

[0012] FIG. 2C is a top, left, rear perspective of an embodiment of the adjustable stiffness treadmill, with the platform over the cantilever removed;

[0013] FIG. 3 is a right side view of the cantilever spring and support mechanism of an embodiment of the adjustable stiffness treadmill;

[0014] FIG. 4 is a side perspective view of the cantilever spring and support mechanism of an embodiment of the adjustable stiffness treadmill;

[0015] FIG. 5 is a side perspective view of a preload mechanism of an embodiment of the adjustable stiffness treadmill;

[0016] FIG. 6A shows an illustration of parameters and geometry defining the adjustable stiffness spring mechanism; and FIG. 6B shows an illustration of parameters and geometry for the unloaded deflection compensation mechanism;

[0017] FIG. 7A is an illustration of a chart showing linear stiffness of the adjustable stiffness mechanism was characterized at several position settings by recording treadmill displacements under loading with discretely applied weights; and FIG. 7B is an illustration of a chart showing surface stiffness vs effective spring length using the corresponding linear fits illustrated in FIG. A;

[0018] FIG. 8A is an illustration of a chart showing transient response to a step input of 676 N and the modeled response of a second-order system; and FIG. 8B is an illustration of a chart showing a Bode chart of the frequency response for the modeled second-order systems;

[0019] FIGS. 9A-9F are an illustration of a series of charts showing motor control performance increasing and decreasing stiffness between 300 kN/m and 15 kN/m with and without an applied load to the treadmill; where FIG. 9A shows effective spring length for decreasing stiffness and FIG. 9B shows increasing stiffness; FIG. 9C shows rate of change of effective spring length for decreasing stiffness and FIG. 9D shows increasing stiffness; FIG. 9E shows electric current drawn by the motor to track the trajectories in FIGS. 9A and 9C; and FIG. 9F shows electric current drawn by the motor to track the trajectories in FIGS. 9B and 9D;

[0020] FIG. 10 is an illustration of a chart showing vertical deflection of the treadmill during 1.25 m/s walking; and stiffness changes from 300 kN/m to 30 kN/m within one step at t=4.1s;

[0021] FIGS. 11A and 11B are illustrations of Table I and Table II, respectively; and

[0022] FIG. 12 is an illustration of Table III.

DESCRIPTION OF THE PREFERRED EMBODIMENT

[0023] Referring to FIGS. 1, 2A-2C, and 3-5, an embodiment of a design of an adjustable stiffness treadmill system is shown generally at 10. The treadmill 10 may comprise five subassemblies: a narrow, low-inertia treadmill 12; a leaf spring variable stiffness suspension 14; a preload linkage mechanism 16; a structural frame with vertical linear constraints 18; and an offboard motor assembly 20 coupled to the treadmill via universal joints 22. The treadmill 10 is designed to translate vertically up and down along its vertical rails 18, supported by the bending stiffness of a cantilevered sheet of spring steel 14, the effective length of which is modified via a servo-controlled rack and pinion mechanism on a cart 24. The treadmill geometry, offboard motor, and spring-based weight compensation (as opposed to an inertial counterweight) contribute toward minimizing the inertia of the treadmill, increasing the transparency of the device and increasing the usable range of walking speeds. This system is located inside a motion capture volume with twelve cameras (Qualisys, Gothenburg, Sweden) and mounted directly

[0024] adjacent and parallel to a separate, rigidly mounted treadmill [37]. The system is designed to function as a dual belt treadmill, with each belt controlled independently with a separate motor, to allow the future study of behavioral response to imposed ground stiffness asymmetry.

[0025] The geometric design targets of this system were to remain below the height of other adjustable stiffness treadmills (70 cm [22]) and exceed the effective walking surface length, ideally reaching a length achieved by treadmills used for adaptation studies (Bertec Fully Instrumented Treadmill, approx. 175 cm [38]). Based on previous asymmetric stiffness experiments [24], other device capabilities, and our own simulation work [39], we aimed to achieve a minimum stiffness of approximately 5 kN/m. Finally, we sought to design a suspension system with a minimum natural frequency of 1.5 Hz and a mechanical bandwidth of at least 2.0 Hz to ensure the system response would not be distorted for cyclical loads at walking frequency.

[0026] The key principle of the adjustable stiffness mechanism is illustrated in FIG. 6A. Vertical stiffness (dF/dd) is a function of the effective cantilever length L.sub.e of the leaf spring

[00001]$\begin{array}{cc}{L}_e=L_s-x& \left(1\right)\end{array}$

where L.sub.s is the unmodified length of the beam and x is the distance of the cantilever constraint from the base of the beam. Variable stiffness mechanisms with this operating principle have been developed and accurately characterized for rotary joints [31]-[33]. This section extends the prior work to characterize the length-dependent stiffness behavior of a vertically displacing load with a separate torsion spring-based preload mechanism.

[0027] Assuming a small deflection angle, the force-deflection relationship of the leaf spring can be modeled as a cantilever beam:

[00002]$\begin{array}{cc}d=\frac{{F\left(L_e\right)}^3}{3\mathrm{EI}}& \left(2\right)\end{array}$

where d and F are the deflection of and force applied to the beam tip normal to its longitudinal axis, respectively, E is the Young's modulus, and/is the area moment of inertia of the beam. Rearranging Eqn. 2 to solve for F and differentiating with respect to the deflection d, the stiffness of the beam

[00003]$\begin{array}{cc}K=\frac{\mathrm{dF}}{\mathrm{dd}}=\frac{3\mathrm{EI}}{{\left(L_e\right)}^3}& \left(3\right)\end{array}$

is proportional to the inverse cube of the effective beam length L.sub.e.

C. Preload Mechanism

[0028] Because our mechanism is loaded by the treadmill and the unsupported weight of the spring, we add a distributed load term to Eqn. 2 and a point load to the beam tip to account for the added downward force:

[00004]$\begin{array}{cc}d=\frac{\left(F+W_t\right)L_e^3}{3\mathrm{EI}}+\frac{L_{}^4}{8\mathrm{EI}}& \left(4\right)\end{array}$

where w is the load per unit length associated with the weight of the exposed length of the spring, and W, is the weight of the treadmill. After rearranging Eqn. 4 to solve for F

[00005]$\begin{array}{cc}F=\frac{3\mathrm{EI}}{L_e^3}d-\frac{3}{8}{L}_e-W_t& \left(5\right)\end{array}$

it becomes apparent that stiffness K remains defined by Eqn. 3 because the added weight terms are not dependent on d. However, a nonzero deflection is present when F=0 and L.sub.e is non-zero and may become substantial as L.sub.e becomes large.

[0029] To offset unloaded deflection from the variable spring weight and the fixed weight of the treadmill, a pretensioned torsion spring assembly was coupled between the end of the leaf spring and the ground. This assembly is designed to provide a pseudo-constant preload force as the device deflects under load. This was accomplished by preloading the springs such that a relatively small range of their full stroke is driven by the range of motion of the device. The vertical force applied to the leaf spring from each set of jaws of this preload device can be characterized as

[00006]$\begin{array}{cc}{F}_p={\mathrm{NK}}_t\frac{{}_0-}{L_p}\cos \left(\frac{}{2}\right)& \left(6\right)\end{array}$

where K.sub.t is the torsional spring stiffness, N is the number of torsion springs in parallel, .sub.0 is the rest angle of the torsion springs, and L.sub.p and are the link length and angle between the links (FIG. 6B).

[0030] The angle between the links can be defined with mechanism geometry parameters

[00007]$\begin{array}{cc}=2{\sin }^{-1}\left(\frac{h-d}{2L_p}\right)& \left(7\right)\end{array}$

where h is the vertical distance between the mechanism connections the treadmill and to the ground at d=0.

[0031] Substituting Eqn. 7 into Eqn. 6 and setting .sub.0=2 to indicate a maximum torsional deflection of one full revolution for the selected springs, the full equation for the preload force

[00008]$\begin{array}{cc}{F}_p=\frac{{\mathrm{NK}}_t}{L_p}\left[2-2{\sin }^{-1}\left(\frac{h-d}{2L_p}\right)\right]\cos \left[{\sin }^{-1}\left(\frac{h-d}{2L_p}\right)\right]& \left(8\right)\end{array}$

which can be simplified to

[00009]$\begin{array}{cc}{F}_p=K^{}\left[2-2{\sin }^{-1}\left(\frac{h-d}{2L_p}\right)\right]\sqrt{1-{\left(\frac{h-d}{2L_p}\right)}^2}& \left(9\right)\end{array}$

where K represents

[00010]$\frac{{\mathrm{NK}}_t}{L_p}$

as the combined linear stiffness term, the second term represents the spring deflection in radians, and the third term represents the alignment of the resultant force with the movement axis of the treadmill as a unitless scalar bounded between 0 and 1.

[0032] For small (hd)/2L.sub.p, the torsion spring deflection scales approximately linearly with the vertical deflection of the treadmill. The resultant force alignment scalar is more sensitive to this ratio, but nonlinear contributions from force alignment can be outweighed by large contributions of starting preload force.

[0033] In practice, the deflection compensation mechanism was implemented with two linkage sets nested in parallel to achieve sufficient upward force within the geometric constraints of our design. Eqn. 9 expresses F.sub.p as a function of the treadmill deflection, subject to static geometric parameters of each linkage. The resultant calculation for F.sub.p with multiple linkages is therefore the sum of Eqn. 9 calculated for each linkage for a given d. The implemented geometry can be competently described as a preloaded linear spring system, as confirmed by a linear approximation of preload force

[00011]$\begin{array}{cc}{F}_p=K_{\mathrm{pre}}d+F_{\mathrm{pre},0}& \left(10\right)\end{array}$

calculated via least squares regression. The linear approximation remains within 1% of the model defined by Eqn. 9 for the parameters of our mechanism. By adding Eqn. 10 to Eqn. 5, the required external force for a given deflection can be expressed as

[00012]$\begin{array}{cc}F=\left[\frac{3\mathrm{EI}}{{\left(L_e-x\right)}^3}+K_{\mathrm{pre}}\right]d-\frac{3}{8}{L}_e-W_t+F_{\mathrm{pre},0}& \left(11\right)\end{array}$

and the expected unloaded deflection is

[00013]$\begin{array}{cc}{d}_O=\frac{W_t+\frac{3}{8}{L}_e-F_{\mathrm{pre},0}}{\frac{3\mathrm{EI}}{{\left(L_s-x\right)}^3}+K_{\mathrm{pre}}}.& \left(12\right)\end{array}$

[0034] The vertical stiffness of the full assembly can be thus represented by

[00014]$\begin{array}{cc}\frac{3\mathrm{EI}}{{\left(L_e\right)}^3}+K_{\mathrm{pre}}.& \left(13\right)\end{array}$

[0035] Actual values for the design parameters defined above are reported in Table I for the adjustable stiffness mechanism, and in Table II for the preload mechanism. (See FIGS. 11A and 11B). Stiffness mechanism values were chosen to achieve our targeted stiffness range and allow for large vertical deflections of 10 cm while remaining below the material fatigue limit of the spring. Preload mechanism values were chosen to maximize the static weight compensation term within the geometric constraints of the treadmill structure. The vertical stiffness is modeled to range between 5.2-3100 kN/m for these parameters. Unloaded deflection is modeled to range from 0-39 mm for this range, with deflection exceeding 10 mm below a stiffness setting of 28 kN/m. Without the compensation mechanism, maximum unloaded deflection is modeled to reach 81 mm and exceed 10 mm below 56 kN/m.

[0036] The remainder of this section details the practical implementation of the design modeled above into a functional prototype.

D. Design Implementation

[0037] 1) Mechanical Construction: The cantilever beam 14 consists of three hardened 4140 alloy steel plates suspended parallel to the ground by an aluminum base 26 (FIGS. 1, 2A-2C, 3-5). Two pairs of galvanized steel rollers 28 mounted to a cart 24 function as a moving cantilever base by constraining the displacement and curvature of the spring behind the lead roller pair 28. Rollers 28 of a softer material than the leaf spring 14 were selected to prevent abrasion from changing the spring dimensions. The cart 24 rides on a linear track with ball bearings and is driven by a servo motor 34 actuated rack 36 and pinion 38. The rack 38 is 1 m long, allowing the effective length of the spring 14 to vary from 0.12 m to 1.12 m. The minimum effective spring length is determined by the clearance required between the cantilever rollers 28 and the attachments at the spring tip 32. The reaction forces from the spring 14 are transmitted to the treadmill 12 through a rigid link coupled to the tip 32 of the spring 14 and the base of the treadmill 12 via steel shafts 30 and needle bearings. These couplings function as revolute joints which allow the treadmill 12 to deflect vertically and the spring tip 32 to deflect through an arc without locking the mechanism.

[0038] The preload mechanism 16 beneath the treadmill 12 is constructed from a linkage machined from aluminum 6061 plates. The linkage 40 is coupled to the spring tip 32 attachment shaft with needle bearings and to the ground-fixed structural frame 42 via an identical shaft and set of needle bearings. A total of 21 torsion springs with 0.77 N-m/rad stiffness each are mounted in parallel within this mechanism at an average preload of 78% of their maximum allowable deflection (Table II), resulting in a vertical force of 266 N under no additional load, assuming no friction losses. The preload mechanism 16 is equipped with pins and hard stops to limit the maximum vertical deflection of the treadmill 12 to be flush with the height of the adjacent rigidly mounted treadmill 44.

[0039] The treadmill 12, which has a mass of 49.6 kg (55% of which is counteracted by the preload mechanism 16), is constructed from aluminum U-channels and sheets. The top surface is lined with low-friction phenolic wear plates. The belt tension can be adjusted via a lead screw mechanism which positions the front roller.

[0040] The treadmill 12 is constrained to only move in the vertical direction via a set of linear bearings and support-rail shafts 46. The linear motion system is designed with up to 10 cm of allowable vertical travel, the maximum downward limit of which may be reduced in 1.3 cm increments by a set of telescoping pylons 48 which function as hard stops. These pylons 48 also include hard stop upward travel limits, redundant with the mechanical limits included in the preload mechanism. The design of this subassembly was a critical part of the system design because the treadmill's 12 length makes it highly sensitive to angular misalignment. Furthermore, the smoothness of the system 10 response may be severely worsened by attempts to limit this misalignment by over constraining the linear motion system, causing it to bind. Therefore, the linear motion system was designed to maximize the length of bearing contact with the vertical guide rails within the overall height constraint, achieving 30.5 cm of effective bearing length for 10 cm of maximum travel. Two guide rails were spaced at 88.6 cm to remain below a 3:1 bearing width to length ratio, minimizing the possibility for binding interfering with the smoothness of the response. A wider spacing and additional guide rails may further reduce pitch angle but risk stick-slip behavior at low stiffness and energetic walking.

[0041] To avoid restricting a user's natural step length or other gait kinematics, the treadmill 12 was designed with an effective walking surface length of 1.68 m. This is significantly longer than other elastically-suspended treadmills; therefore, additional measures must be taken to prevent the inertia of the treadmill from reducing the natural frequency of the system below that required for walking. We used two approaches to limit the treadmill inertia. First, the drive motor for the treadmill belt is located offboard. This is critical because the treadmill drive motor 20 has a mass of 43 kg, which is nearly equal to the mass of treadmill 12 itself. This motor 20 drives the belt as it deflects vertically by coupling to the rear roller through universal joints 22. Second, unloaded deflection is counteracted by the spring powered preload mechanism 16 described previously rather than by an inertial counterweight.

[0042] The treadmill 12, 44 is surrounded by a wooden platform 50, 52 to provide a rigid surface to safely step onto and mitigate the perception of balancing on a moving surface high above the ground. This platform 50, 52 is located 13 cm below the treadmill 12, 44 top surface. The treadmill 12 may therefore sink nearly to the surrounding ground level at maximum deflection. Furthermore, the platform 50 encases the adjustable stiffness mechanism for added safety. [0043] 2) Actuator Configuration and Control: The position of the cantilever spring 14 which defines the surface stiffness is controlled by a servomotor 34 powering a rack and pinion 36, 38 actuator. While similar systems have been developed using a lead screw actuator [22], [23], [34], the screw length required for the target stiffness range would likely cause detrimental transverse vibration (i.e., screw whip) for the high rotation speed required to complete a maximal stiffness change within one step, except for very large screw diameters. The length of a rack and pinion system 36, 38 does not affect its performance if the rack 36 is stationary, with the tradeoff that the inertial load the motor 34 drives includes its own mass.

[0044] The servomotor 34 was selected by calculating the maximum horizontal force the motor must exert and the time in which the largest stiffness change must be achieved (<500 ms). The maximum force the motor 34 must overcome to move the cantilever 14 occurs at the stiffness setting where the vertical displacement caused by the applied downward force on the treadmill 12 is equal to the mechanical deflection limits. Under this condition, the spring 14 maximally deflects while transferring all of the load to the movable cantilever base cart 24. The horizontal force can be estimated using a formula that has been previously derived for cantilever beam deflection [36], [40]:

[00015]$\begin{array}{cc}{F}_m=\frac{F^2{L}_{}^2}{2\mathrm{EI}}& \left(14\right)\end{array}$

[0045] Assuming a maximum vertical load of 1.5 body weights of the heaviest allowable user (150 kg) and including a safety factor of 1.5, this force is estimated at 882 N. Therefore, we selected a 1 kW servomotor (SV2L-210B, AutomationDirect, Cumming, GA) and 1.5 kW drive (SV2A-2150, AutomationDirect, Cumming, GA) with an attached 5:1 ratio gearbox and 24-bit incremental encoder to drive the system. This actuation system is rated to move the cantilever position at a maximum speed of 2.35 m/s with a force of 1100 N intermittently, or 430 N continuously. Assuming that the motor 34 could reach maximum speed within 100 ms, we estimated that these performance specifications would allow a change from maximum to minimum stiffness (or vice versa) in under 500 ms with an applied load on the treadmill 12 up to our designed maximum. To further minimize demands on the motor 34, an integrated brake engages after the desired position is reached.

[0046] Stiffness control is performed on the servo drive via position and velocity trajectory tracking with a PI control architecture and 8 kHz sample frequency. Because stiffness can be mapped to effective spring length and does not require active control to maintain once set, static position targets were stored in the drive corresponding with the values characterized in the following section. The stiffness setting may be selected and triggered by controlling the state of corresponding digital inputs. Position targets and velocity profiles may be reprogrammed if modifications are made to the design or if more gradual change in stiffness is desired. Mechanical limit switches and hard stops mounted to a linear guide shaft prevent the cantilever cart from exceeding its designed range of motion and provide a reference position for homing the motor encoder should the zero position be lost. The system is further equipped with an emergency stop located at the operator workstation. System re-homing and stiffness control commands may be triggered by an operator using a custom pendant control panel.

[0047] A 3.7 kW AC motor (Leeson Electric Corporation, Grafton, WI) drives the treadmill belt via a 2.22:1 timing belt pulley, allowing a maintained maximum belt speed of 3.25 m/s with up to 1100 N of propulsive or braking force applied to the belt by the user. The belt speed is controlled with a 16-bit microprocessor based AC motor drive (Leeson Electric Corporation, Grafton, WI). An identical motor and drive power the belt for the adjacent rigid treadmill.

IV. Evaluation

[0048] To characterize the dynamics of the system and evaluate its ability to function as an experimental device, we performed a series of tests. The evaluation tests were designed to (1) quantify the actual stiffness behavior and range of the treadmill, (2) the natural oscillation frequency and damping ratio of the treadmill at different stiffnesses, and (3) the speed of the maximal stiffness change under load.

A. Stiffness Characterization

[0049] To quantify the relationship between surface stiffness and effective spring length, we measured vertical deflection under static loads at 6 cantilever positions across the available range of spring lengths. We sequentially stacked and removed exercise weights on the treadmill belt up to 99 kg (971 N). We recorded the vertical position of the treadmill for each change in weight so as to observe the linearity of the stiffness relationship and capture any hysteresis effects. We measured the vertical position of the treadmill surface by applying six reflective markers at the four corners and two long-edge midpoints of the walking surface and recorded their positions with the motion capture system mentioned in Section III recording at 100 Hz. Each deflection measurement represents the vertical height of the markers averaged across three seconds (300 frames of data), and averaged again across all six markers. We discarded measurements for which the treadmill stopped against the mechanical limits, which occurred only for the minimum stiffness condition.

[0050] For each spring length, we fit a linear regression model to applied force vs. displacement, as illustrated in FIG. 7A. Using the slope of each linear fit as the approximate linear stiffness, we calculated the relationship between the stiffness and effective spring length (FIG. 7B). We recorded a maximum stiffness of 280 kN/m and a minimum stiffness of 8.6 kN/m, with all linear approximations achieving R2 values exceeding 0.97 except for the minimum stiffness condition, in which the R2 was 0.75, due to hysteresis.

[0051] Deflection of the walking surface without added load remained below 20 mm for most conditions, but it reached 21 mm before loading and 39 mm after loading at minimum stiffness. In deriving an expression for treadmill stiffness with respect to cantilever position, the original model overestimated the stiffness at short spring lengths. However, after applying an offset of 16 cm to the effective spring length and maintaining all other parameters, the measured stiffness closely followed the relationship predicted by the model (R.sup.2=0.999) (FIG. 7B). Note that we proceeded to use the 15 kN/m setting as the minimum stiffness setting in all further tests due to the high degree of hysteresis, the magnitude of unloaded deflection, and the likelihood of the treadmill displacement being constrained under the test loads by the mechanical stops at 8.6 kN/m. We also approximate the maximum stiffness condition as 300 kN/m due to measurement precision, explained further in Section V. Qualitatively, these stiffnesses range from stepping on the surface of a plywood track (100-200 kN/m, [41]), to a sprung ballet dance floor (60 kN/m, [42]), to stepping on the surface of a trampoline (<20 kN/m, [43]).

B. System Identification

[0052] To evaluate the speed of the passive device dynamics relative to the frequency of human walking, we applied a step change in vertical load to the treadmill at 300, 30, and 15 kN/m and measured the transient response of the vertical displacement of the walking surface using the same marker and camera setup as from the stiffness characterization evaluation. We approximated a step change in downward force by having a volunteer (mass=68.9 kg, or 676 N) step onto the treadmill from a waiting position with one foot held over the walking surface. To account for variability in the load application, we repeated this procedure four times for each stiffness condition and averaged the treadmill response, synchronized to when the belt position deflected by 1 mm.

[0053] To quantify the system dynamics, we approximated the response as a second-order spring-mass-damper system. The second-order model was fit to the data via iterative error minimization using the Matlab system identification toolbox (Mathworks, Natick, MA). Model fit assessed from 0 to 100% as normalized root mean squared error (NRMSE) was 94.7%, 93.6%, and 94.4% for the 300 kN/m, 30 kN/m, and 15 kN/m conditions, respectively. The actual response and modeled second-order response are illustrated in FIG. 8A. From high to low stiffness, the natural frequency was 3.2, 2.4, and 2.1 Hz, all of which exceed the average stride frequency of walking (approx. 0.9-1.1 Hz, [44]). The system is underdamped, with damping ratios of 0.48, 0.30, and 0.27 from high to low stiffness. Extrapolating the frequency response of these second-order models (FIG. 8B), phase lag is less than 20 at typical walking cadence. Response magnitude falls below-3 dB at 4.1, 3.5, and 3.1 Hz from high to low stiffness.

C. Stiffness Control Performance

[0054] To evaluate the active control of the adjustable stiffness, we applied position control trajectories between the 15 kN/m to 300 kN/m settings with and without a load disturbance applied to the treadmill in the form of a person standing on the belt (80.7 kg, or 792 N). Rather than applying step changes to the reference position, we applied trapezoidal trajectories of 450 ms duration with smooth acceleration and deceleration for the initial and final 40 ms to minimize peak current demands on the motor. We recorded cantilever position and velocity from the embedded motor encoder and motor current as reported from the motor driver at a frequency of 8 kHz. Each position command was repeated 6 times.

[0055] We calculated the mean position, velocity, and motor current for increasing and decreasing stiffness change with and without added weight to the treadmill (FIGS. 9A-9F). Note that the 90% rise time is unaffected by the presence of a load disturbance and remains at 340 ms for all conditions.

D. Operation During Walking

[0056] Finally, to qualitatively demonstrate the behavior of the system in a practical human experiment scenario, we recorded the vertical treadmill position during a stiffness change from 300 to 30 kN/m while a person (74.4 kg) walked on the belts at 1.25 m/s (FIG. 10). The purpose of this demonstration is not to analyze the effect of the perturbation on gait, but to illustrate how the information from the previous tests translates to the intended use case.

V. Discussion

[0057] We developed a treadmill and suspension system capable of adjusting the surface stiffness within a range from 15 to 300 kN/m of well characterized linear stiffness during a single step, swing or stance phase. The AdjuSST meets these requirements while maintaining an effective walking surface length of 1.68 m, more than double the step length of a person of average height, and deflecting vertically downward up to 10 cm. The natural frequency of the AdjuSST exceeds the input frequency of walking at all available stiffnesses. Therefore, the AdjuSST successfully achieved our proposed contributions. While we did not conduct a formal analysis of gait in our tests for this manuscript, test users were able to walk in excess of 10 consecutive minutes on the treadmill and experience multiple stiffness perturbations which required visible adjustments to their gait mechanics. Test users reported that the sensation of the stiffness change was novel and surprising, but that they adjusted within a few minutes. [0058] 1) Stiffness characterization: Measuring the static displacement of the treadmill in response to fixed loads demonstrated that the effective vertical stiffness is linear across the available range. We observed increasing hysteresis as stiffness decreased. This is likely due to static friction in the mechanical suspension resisting rebound of the treadmill surface for incremental reductions in vertical load. Additionally, while the stiffness behavior is well characterized and offers a useful range, the cantilever constraints at both the moving cart and the fixed base are not ideal and result in a smaller upper bound on stiffness than we anticipated. This is most likely due to the relatively short length of the constraints and the presence of small clearances between cantilever rollers and the spring, which combine to allow the spring to bend upward behind the constraint.

[0059] We report the maximum stiffness as a rounded approximation because the calculation of high stiffness becomes sensitive to the resolution of the displacement measurement. The motion capture camera system was calibrated to within 0.5 mm of error, and the largest deflection measured at maximum stiffness was 3.4 mm, meaning that the maximum stiffness could plausibly be a value between 250 and 340 kN/m. [0060] 2) Passive suspension dynamics: Identification of the system dynamics revealed that the treadmill suspension has sufficient frequency bandwidth and resilience (i.e., not dominated by energy dissipation) to present a spring-like response to walking gait without encountering substantial phase lag or failure to return to the undeflected position before the next heel strike. Assuming running cadence is between 70 and 110 strides per minute [45], [46] and that both feet interact with the same belt, input frequency during running would vary between 2.3-3.7 Hz (FIG. 8B). Thus, the system response is likely to be distorted for low stiffness at high running cadences, but it may near resonance in the higher stiffness range at slow running cadences. This resonant frequency band may be useful in running contexts, where the objective of changing stiffness is more likely to align with improving energy economy by taking advantage of the foot-ground interaction dynamics [26], [41], [47] than with the asymmetric stiffness perturbation paradigm the treadmill was originally designed to enable.

[0061] Finally, we found that the treadmill response is underdamped, which provides a useful baseline for potentially studying the effects of added damping in the future. [0062] 3) System actuation: The actuation performance tests demonstrated that the AdjuSST can change between stiffness extremes within one step, irrespective of the walker's weight on the treadmill. For the tracking profile tested, motor current remained near the rated continuous operation limit during the stiffness change. The maximum rated intermittent current is briefly reached only when decelerating at the end of an increasing stiffness change with the disturbance load applied. Though the rated continuous current was exceeded, the system will not be continuously actuated, and will at most be actuated twice within one second with long idle periods in between when performing single step perturbations. Note that the system is also capable of following more gradually changing trajectories at reduced demand on the motor. [0063] 4) Limitations: Despite the successful contributions of this design, our approach has some limitations. Despite our countermeasures, some deflection at zero added load is present for lower stiffnesses because the preload mechanism does not compensate the full weight of the treadmill. While we have limited this deflection to under 1 cm across most of the stiffness range, it becomes noticeable at the extreme low end. This deflection remains for two main reasons: (1) The steel spring is substantially heavy that fully eliminating zero-load deflection will create a dead zone at higher stiffnesses, where a noticeable preload must be overcome before the belt will begin to deflect downward, and (2) the number of springs or jaws required to eliminate zero-load deflection is limited by the geometric constraints of our design. The design presented is a compromise between surface uniformity, treadmill size, and the ability to present an instantly reactive surface. The preload mechanism design is flexible enough to allow this operating point to be shifted toward increased preload if desired by the addition of higher stiffness torsion springs. This design also cannot reach the high range of stiffness achieved by the VST because the cantilever constraint cannot bypass the spring in either direction, as can be done by locating the fulcrum point of a rigid lever directly at one of the load insertion points. We chose to accept this limitation because our target range was achievable with this design, and it comes with other benefits, such as requiring low startup torque from the motor under heavy loads, enabling the performance to be minimally affected by load disturbances on the treadmill. [0064] 5) Comparison with other adjustable surface stiffness treadmills: As previously mentioned, two other adjustable surface stiffness treadmill designs have been presented [22], [23]. Notably, an updated design to the VST with a longer walking surface and vertical deflection was published during the review process of this manuscript, and we have included it in this comparison. We also have included a commercially available instrumented treadmill widely used for gait research, but which does not have adjustable stiffness. Evaluation metrics that can be directly compared between designs are included in Table III. (FIG. 12). Perhaps the clearest distinction between the AdjuSST and other designs is the physical dimensions: the effective walking surface is notably longer than the VST and TwAS. Both our design and the new VST 2.0 approach the walking length of the commercial treadmill. As stated in the above paragraph, the stiffness range of the AdjuSST is narrower than the VST, but AdjuSST achieves significantly higher stiffness than the TwaS, and marginally lower minimum stiffness than the VST 2.0. Like the TwaS and now the VST 2.0, the AdjuSST deflects linearly rather than angularly about a fixed axis like the VST. The belts of the AdjuSST are driven by motors rated for 5 more mechanical power than the VST and more than double the VST 2.0, approaching the power offered by the commercial treadmill. This power is important for potentially extending toward running applications, which require both higher speeds and larger torques to maintain constant belt speed against braking/propulsion forces. The drive motors of the TwaS are unspecified, though are reported to run the belts at up to 3.6 m/s, which is slightly faster than the AdjuSST at its current drive pulley ratio. All three designs are reported to be able to complete a maximum stiffness change in under 500 ms, though the VST is the fastest, with rise times under 250 ms. Unfortunately, a direct comparison of the system dynamics of the spring suspension is not possible, as this evaluation was not performed with the other designs. The Bertec treadmill provides a useful reference point in this regard, however-suspending a treadmill with an adjustable spring introduces the possibility of frequency-limited system dynamics, which are orders of magnitude away from being a concern for a rigidly mounted top-of-the-line commercial device. The open-loop response of the VST stiffness change was characterized at a similar bandwidth to the AdjuSST suspension, but these measures are not perfectly analogous. We can speculate that the VST 2.0 has more tightly constrained vertical motion than the AdjuSST owing to its redundant linear guides and compact design, but has slower system dynamics at a higher damping ratio owing to its greater than double sprung mass. It is possible that stiffness perturbations applied by these two devices will affect gait differently, but those differences remain to be seen.

V. CONCLUSION

[0065] This paper presented and evaluated a novel adjustable surface stiffness treadmill design. Overall, the evaluation of the AdjuSST has demonstrated that it is a capable platform for conducting experiments involving asymmetric perturbations of surface stiffness. In context with alternative designs, the AdjuSST contributes meaningful improvements in effective walking surface length while maintaining a useful stiffness range and responsive spring suspension. Future work with this system will focus on experiments which perturb human gait with prolonged asymmetric stiffness changes to investigate adaptation effects and aftereffects relevant to neuromotor rehabilitation.

[0066] It would be appreciated by those skilled in the art that various changes and modifications can be made to the illustrated embodiments without departing from the spirit of the present invention. All such modifications and changes are intended to be within the scope of the present invention except as limited by the scope of the appended claims.

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