Gait & Posture 41 (2015) 954–956
Contents lists available at ScienceDirect
Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost
Technical Note
A method for measuring instantaneous treadmill speed Nathan M. Olson a,b,1, Glenn K. Klute a,b,* a b
Department of Veterans Affairs, Puget Sound Health Care System, Seattle, WA 98108, USA Department of Mechanical Engineering, University of Washington, Seattle, WA 98105, USA
A R T I C L E I N F O
A B S T R A C T
Article history: Received 1 December 2014 Received in revised form 27 February 2015 Accepted 3 March 2015
This paper presents a new method for measuring instantaneous treadmill speed. The method consists of two parts: (1) a motion-capture system and reflective squares placed directly on the treadmill belt, and (2) an algorithm for estimating belt speed and distance traveled from the measured positions of the reflective squares. The method was tested with a single subject walking on an instrumented treadmill. The method is easy to set up and results in an accurate estimate of treadmill speed that is conveniently synchronized with the output of the motion-capture system. Published by Elsevier B.V.
Keywords: Treadmill instrumentation Walking speed Estimation algorithm
1. Introduction When performing analysis of treadmill walking, it is important to have accurate measurements of treadmill speed and distance traveled. Obtaining these measurements is not trivial, especially when the treadmill speed is varying, such as in self-paced mode where the speed is controlled by subject position feedback [1]. Many methods have been used to measure treadmill speed and distance traveled. Electronic treadmill interfaces have been used directly [2,3]. Speed has also been derived from stance leg kinematics recorded by motion-capture systems [4,5]. These methods do not work when the test subject is running, and require a scheme for switching between reflective markers on the two feet. Another method is to place a block with a reflective marker at front of the treadmill and measure its speed using a motion-capture system as it is carried to the back of the treadmill [6]. This method is only able to measure speed as long as the block is on the belt. Other researchers have used a rubber wheel connected to an electric generator and placed against the treadmill belt [7], but this method may be difficult to set up and is not easily integrated with motion-capture data. The method presented here uses a motion-capture system to track reflective squares placed directly on the treadmill belt. It
* Corresponding author at: VAPSHCS, 1660 S. Columbian Way, MS-151, Seattle, WA 98108, USA. Tel.: +1 206 764 2991. E-mail addresses: [email protected] (N.M. Olson), [email protected] (G.K. Klute). 1 Address: VAPSHCS, 1660 S. Columbian Way, MS-151, Seattle, WA 98108, USA. Tel.: +1 206 277 2991. http://dx.doi.org/10.1016/j.gaitpost.2015.03.003 0966-6362/Published by Elsevier B.V.
requires no additional equipment or measuring devices, can be applied to both single- and split-belt treadmills, and has the added benefit of being inherently synchronized with motion-capture data. Standard gait lab methods are used to capture the data, then an algorithm is used to estimate belt speed and distance traveled. 2. Methods Three 1 cm2 squares of reflective tape were placed on the top surface of the left treadmill belt at the outer edge of a dual-belt treadmill (Bertec; Columbus, OH). The reflective squares were approximately equally spaced, so that either one or two reflective squares were visible in all belt positions. The reflective squares were tracked by a 12-camera Vicon MX motion-capture system (Vicon; Oxford, UK) operating at 120 Hz. The motion-capture system is designed to track reflective spheres, but it successfully tracked the reflective squares without any changes. A subject, who provided informed consent to this Institutional Review Boardapproved protocol, walked at a constant self-selected speed of 1.1 m/s and also in self-paced mode for 60 s while motion-capture data were collected. A custom C++ application automatically extracted the positions of the three reflective squares from the unfiltered data in the C3D file (Coordinate 3D; www.c3d.org) exported by the motioncapture system. The reflective squares were detected in a 160 cm long cylinder with a 3 cm diameter, and labeled sequentially 1, 2, or 3 in order of appearance. If the first frame contained two reflective squares, the square closest to the back of the treadmill was labeled 1. The application source code is available in Supplementary material.
N.M. Olson, G.K. Klute / Gait & Posture 41 (2015) 954–956
yik ¼ ½ 1
0
d2i
d3i
Ni ½ x
v
d12
d13
T
l k þek
(2)
where yi is the measured position of reflective square i, d is the Kronecker delta, Ni is the number of laps completed by reflective square i, and the measurement noise e is a normally distributed random variable with zero mean and standard deviation R. The RTS algorithm behaves as a low-pass filter on the treadmill speed and the distance traveled. The ratio of process noise to measurement noise determines the cutoff frequency of the filter. For cutoff frequencies much lower than the sampling frequency, the cutoff frequency is given by rffiffiffiffi 1 Q (3) fc ¼ 2p R While only the ratio of Q to R has an effect on the calculated treadmill speed and distance traveled, the RTS algorithm also calculates uncertainties that are proportional to Q and R. In this application, zero, one, or two reflective squares might be available on a given motion-capture frame. If an otherwise measurable reflective square was blocked from observation (e.g., by a human subject), the algorithm propagates the system dynamics without correction, thus making this approach robust to gaps in reflective square measurement. For the examples presented here, two, one, and zero measurements were available in 28.4%, 67.8%, and 3.7% of frames, respectively.
Treadmill speed [m/s]
where k is the motion-capture frame number, x is the distance traveled by reflective square 1, v is the belt speed, dij is the distance between reflective squares i and j, l is the length of the belt, dt is the time step, and the process noise q is a normally distributed random variable with zero mean and standard deviation Q. The measurement equation for the position of reflective square i in the direction of belt motion is given by
1.15
1.10
1.05
1.00
RTS marker 1 marker 2 marker 3 FC
0.95
0.90
RTO
Table 1 Comparison between lengths measured with tape measure and estimated using RTS algorithm. Parameter
Estimated
Tape measure
Difference
Belt length [mm] d12 [mm] d13 [mm]
3871 973 2350
3858 969 2336
13 (0.3%) 4 (0.4%) 14 (0.6%)
LTO
0.5
0.0
LHS
RTO
1.0 Time [s]
RHS
1.5
2.0
1.35
1.30
1.25
1.20
RTS marker 1 marker 2 marker 3 FC
1.15
1.10
LHS
0.0
3. Results To check the accuracy of the distance estimation, the treadmill belt and the distances between reflective squares were measured with a tape measure (Table 1). The error in the belt length estimation when compared with the tape measure, using an initial uncertainty (error standard deviation) of 1637 mm, was 0.3%. This error in the length scale is also incorporated into the estimates of treadmill speed and distance traveled, and arises from limitations in calibrating the motion-capture system. Treadmill speed was also calculated by applying the Woltring routine [9] to each individual reflective square, and by applying the method proposed by Fusco and Cretual [5]. The treadmill speed computed using Fusco and Cretual’s method was also filtered using the Woltring routine. All methods used a 6 Hz cutoff frequency. Figs. 1 and 2 show the treadmill speed calculated using these three methods for constant speed and self-paced walking, respectively. When only one reflective square is visible, the RTS algorithm and the Woltring routine match within 0.01 m/s.
RHS
Fig. 1. Estimated treadmill speed of left belt during the first 2 s of a 60 s data collection with the treadmill commanded to a constant speed of 1.1 m/s. The black curve shows the results of the RTS algorithm, the gray curve shows the results of Fusco and Cretual’s (FC) method [5], and the three dashed curves show the speed of each of the three reflective squares (Marker 1, Marker 2, and Marker 3) calculated using the Woltring routine [9]. LHS = left heel strike; LTO = left toe-off; RHS = right heel strike; RTO = right toe-off.
Treadmill speed [m/s]
The position and velocity of the belt were estimated from the position of the reflective squares using the Rauch–Tung–Striebel (RTS) algorithm for maximum likelihood estimation [8] using the five-state discrete-time system model 2 3 2 3 3 2 32 x x 0 1 dt 0 0 0 6 q dt 7 6 v 7 6 0 1 0 0 0 76 v 7 6 7 6 7 7 6 76 6 7 6 d12 7 7 76 ¼6 (1) 6 7 6 0 0 1 0 0 76 d12 7 þ 6 0 7 4 0 5 4 d13 5 4 0 0 0 1 0 54 d13 5 0 l kþ1 l k 0 0 0 0 1 k
955
RTO RHS
0.5
LTO
1.0 Time [s]
RTO RHS
LHS
1.5
2.0
Fig. 2. Estimated treadmill speed of left belt during the first 2 s of a 60 s data collection with the treadmill in self-paced mode. The black curve shows the results of the RTS algorithm, the gray curve shows the results of Fusco and Cretual’s (FC) method [5], and the three dashed curves show the speed of each of the three reflective squares (Marker 1, Marker 2, and Marker 3) calculated using the Woltring routine [9]. LHS = left heel strike; LTO = left toe-off; RHS = right heel strike; RTO = right toe-off.
However, when two reflective squares are visible, the RTS algorithm provides a smooth transition, despite the difference between the speeds of the two visible reflective squares as estimated using the Woltring routine. The Fusco and Cretual’s method exhibits large dips in computed treadmill speed during the double support phase, which are not present in the reflective square data.
4. Discussion A limitation of this approach is that errors in the length scale in the position measurements are unobservable to the estimation algorithm. Thus, the errors in the position and velocity estimates is proportional to the error in the length scale. However, if the length of the treadmill belt is accurately known, this error can be removed
N.M. Olson, G.K. Klute / Gait & Posture 41 (2015) 954–956
956
by scaling the position and velocity estimates by the ratio of the true length of the belt to the estimated length of the belt. Although only one belt was tracked in this study, the method could be easily extended to both belts on a split-belt treadmill by applying the same algorithm to three reflective squares on each belt. This study demonstrates the effectiveness of using reflective squares attached to a treadmill belt in conjunction with a motioncapture system to estimate instantaneous treadmill speed. The method is easy to set up, inherently synchronized with motioncapture kinematics measurements, and does not require any additional instruments. The RTS algorithm is inherently robust to data gaps and provides accurate estimates of treadmill speed and distance traveled. Acknowledgements This research was supported by the Department of Veterans Affairs, Rehabilitation Research and Development Service (A9243C and A9248-S) and Department of Defense Deployment Related Medical Research Program of the Office of the Congressionally Directed Medical Research Programs (W81XWH-09-2-0144). Conflict of interest The authors have no conflicts of interest.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.gaitpost. 2015.03.003. References [1] Minetti AE, Boldrini L, Brusamolin L, Zamparo P, McKee T. A feedback-controlled treadmill (treadmill-on-demand) and the spontaneous speed of walking and running in humans. J. Appl. Physiol. 2003;95:838–43. [2] Segers V, Lenoir M, Aerts P, De Clercq D. Kinematics of the transition between walking and running when gradually changing speed. Gait Posture 2007;26:349–61. [3] Sloot LH, van der Krogt MM, Harlaar J. Energy exchange between subject and belt during treadmill walking. J. Biomech. 2014;47:1510–3. [4] Alton F, Baldey L, Caplan S, Morrissey MC. A kinematic comparison of overground and treadmill walking. Clin. Biomech. 1998;13:434–40. [5] Fusco N, Cretual A. Instantaneous treadmill speed determination using subject’s kinematic data. Gait Posture 2008;28:663–7. [6] Savelberg HH, Vorstenbosch MA, Kamman EH, van de Weijer JG, Schambardt HC. Intra-stride belt-speed variation affects treadmill locomotion. Gait Posture 1998;7:26–34. [7] Paolini G, Della Croce U, Riley PO, Newton FK, Kerrigan C. Testing of a triinstrumented-treadmill unit for kinetic analysis of locomotion tasks in static and dynamic loading conditions. Med. Eng. Phys. 2007;29:404–11. [8] Rauch HE, Striebel CT, Tung F. Maximum likelihood estimates of linear dynamic systems. AIAA J. 1965;3:1445–50. [9] Woltring HJ. A Fortran package for generalized, cross-validatory spline smoothing and differentiation. Adv. Eng. Softw. 1986;8:104–13.
Contents lists available at ScienceDirect
Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost
Technical Note
A method for measuring instantaneous treadmill speed Nathan M. Olson a,b,1, Glenn K. Klute a,b,* a b
Department of Veterans Affairs, Puget Sound Health Care System, Seattle, WA 98108, USA Department of Mechanical Engineering, University of Washington, Seattle, WA 98105, USA
A R T I C L E I N F O
A B S T R A C T
Article history: Received 1 December 2014 Received in revised form 27 February 2015 Accepted 3 March 2015
This paper presents a new method for measuring instantaneous treadmill speed. The method consists of two parts: (1) a motion-capture system and reflective squares placed directly on the treadmill belt, and (2) an algorithm for estimating belt speed and distance traveled from the measured positions of the reflective squares. The method was tested with a single subject walking on an instrumented treadmill. The method is easy to set up and results in an accurate estimate of treadmill speed that is conveniently synchronized with the output of the motion-capture system. Published by Elsevier B.V.
Keywords: Treadmill instrumentation Walking speed Estimation algorithm
1. Introduction When performing analysis of treadmill walking, it is important to have accurate measurements of treadmill speed and distance traveled. Obtaining these measurements is not trivial, especially when the treadmill speed is varying, such as in self-paced mode where the speed is controlled by subject position feedback [1]. Many methods have been used to measure treadmill speed and distance traveled. Electronic treadmill interfaces have been used directly [2,3]. Speed has also been derived from stance leg kinematics recorded by motion-capture systems [4,5]. These methods do not work when the test subject is running, and require a scheme for switching between reflective markers on the two feet. Another method is to place a block with a reflective marker at front of the treadmill and measure its speed using a motion-capture system as it is carried to the back of the treadmill [6]. This method is only able to measure speed as long as the block is on the belt. Other researchers have used a rubber wheel connected to an electric generator and placed against the treadmill belt [7], but this method may be difficult to set up and is not easily integrated with motion-capture data. The method presented here uses a motion-capture system to track reflective squares placed directly on the treadmill belt. It
* Corresponding author at: VAPSHCS, 1660 S. Columbian Way, MS-151, Seattle, WA 98108, USA. Tel.: +1 206 764 2991. E-mail addresses: [email protected] (N.M. Olson), [email protected] (G.K. Klute). 1 Address: VAPSHCS, 1660 S. Columbian Way, MS-151, Seattle, WA 98108, USA. Tel.: +1 206 277 2991. http://dx.doi.org/10.1016/j.gaitpost.2015.03.003 0966-6362/Published by Elsevier B.V.
requires no additional equipment or measuring devices, can be applied to both single- and split-belt treadmills, and has the added benefit of being inherently synchronized with motion-capture data. Standard gait lab methods are used to capture the data, then an algorithm is used to estimate belt speed and distance traveled. 2. Methods Three 1 cm2 squares of reflective tape were placed on the top surface of the left treadmill belt at the outer edge of a dual-belt treadmill (Bertec; Columbus, OH). The reflective squares were approximately equally spaced, so that either one or two reflective squares were visible in all belt positions. The reflective squares were tracked by a 12-camera Vicon MX motion-capture system (Vicon; Oxford, UK) operating at 120 Hz. The motion-capture system is designed to track reflective spheres, but it successfully tracked the reflective squares without any changes. A subject, who provided informed consent to this Institutional Review Boardapproved protocol, walked at a constant self-selected speed of 1.1 m/s and also in self-paced mode for 60 s while motion-capture data were collected. A custom C++ application automatically extracted the positions of the three reflective squares from the unfiltered data in the C3D file (Coordinate 3D; www.c3d.org) exported by the motioncapture system. The reflective squares were detected in a 160 cm long cylinder with a 3 cm diameter, and labeled sequentially 1, 2, or 3 in order of appearance. If the first frame contained two reflective squares, the square closest to the back of the treadmill was labeled 1. The application source code is available in Supplementary material.
N.M. Olson, G.K. Klute / Gait & Posture 41 (2015) 954–956
yik ¼ ½ 1
0
d2i
d3i
Ni ½ x
v
d12
d13
T
l k þek
(2)
where yi is the measured position of reflective square i, d is the Kronecker delta, Ni is the number of laps completed by reflective square i, and the measurement noise e is a normally distributed random variable with zero mean and standard deviation R. The RTS algorithm behaves as a low-pass filter on the treadmill speed and the distance traveled. The ratio of process noise to measurement noise determines the cutoff frequency of the filter. For cutoff frequencies much lower than the sampling frequency, the cutoff frequency is given by rffiffiffiffi 1 Q (3) fc ¼ 2p R While only the ratio of Q to R has an effect on the calculated treadmill speed and distance traveled, the RTS algorithm also calculates uncertainties that are proportional to Q and R. In this application, zero, one, or two reflective squares might be available on a given motion-capture frame. If an otherwise measurable reflective square was blocked from observation (e.g., by a human subject), the algorithm propagates the system dynamics without correction, thus making this approach robust to gaps in reflective square measurement. For the examples presented here, two, one, and zero measurements were available in 28.4%, 67.8%, and 3.7% of frames, respectively.
Treadmill speed [m/s]
where k is the motion-capture frame number, x is the distance traveled by reflective square 1, v is the belt speed, dij is the distance between reflective squares i and j, l is the length of the belt, dt is the time step, and the process noise q is a normally distributed random variable with zero mean and standard deviation Q. The measurement equation for the position of reflective square i in the direction of belt motion is given by
1.15
1.10
1.05
1.00
RTS marker 1 marker 2 marker 3 FC
0.95
0.90
RTO
Table 1 Comparison between lengths measured with tape measure and estimated using RTS algorithm. Parameter
Estimated
Tape measure
Difference
Belt length [mm] d12 [mm] d13 [mm]
3871 973 2350
3858 969 2336
13 (0.3%) 4 (0.4%) 14 (0.6%)
LTO
0.5
0.0
LHS
RTO
1.0 Time [s]
RHS
1.5
2.0
1.35
1.30
1.25
1.20
RTS marker 1 marker 2 marker 3 FC
1.15
1.10
LHS
0.0
3. Results To check the accuracy of the distance estimation, the treadmill belt and the distances between reflective squares were measured with a tape measure (Table 1). The error in the belt length estimation when compared with the tape measure, using an initial uncertainty (error standard deviation) of 1637 mm, was 0.3%. This error in the length scale is also incorporated into the estimates of treadmill speed and distance traveled, and arises from limitations in calibrating the motion-capture system. Treadmill speed was also calculated by applying the Woltring routine [9] to each individual reflective square, and by applying the method proposed by Fusco and Cretual [5]. The treadmill speed computed using Fusco and Cretual’s method was also filtered using the Woltring routine. All methods used a 6 Hz cutoff frequency. Figs. 1 and 2 show the treadmill speed calculated using these three methods for constant speed and self-paced walking, respectively. When only one reflective square is visible, the RTS algorithm and the Woltring routine match within 0.01 m/s.
RHS
Fig. 1. Estimated treadmill speed of left belt during the first 2 s of a 60 s data collection with the treadmill commanded to a constant speed of 1.1 m/s. The black curve shows the results of the RTS algorithm, the gray curve shows the results of Fusco and Cretual’s (FC) method [5], and the three dashed curves show the speed of each of the three reflective squares (Marker 1, Marker 2, and Marker 3) calculated using the Woltring routine [9]. LHS = left heel strike; LTO = left toe-off; RHS = right heel strike; RTO = right toe-off.
Treadmill speed [m/s]
The position and velocity of the belt were estimated from the position of the reflective squares using the Rauch–Tung–Striebel (RTS) algorithm for maximum likelihood estimation [8] using the five-state discrete-time system model 2 3 2 3 3 2 32 x x 0 1 dt 0 0 0 6 q dt 7 6 v 7 6 0 1 0 0 0 76 v 7 6 7 6 7 7 6 76 6 7 6 d12 7 7 76 ¼6 (1) 6 7 6 0 0 1 0 0 76 d12 7 þ 6 0 7 4 0 5 4 d13 5 4 0 0 0 1 0 54 d13 5 0 l kþ1 l k 0 0 0 0 1 k
955
RTO RHS
0.5
LTO
1.0 Time [s]
RTO RHS
LHS
1.5
2.0
Fig. 2. Estimated treadmill speed of left belt during the first 2 s of a 60 s data collection with the treadmill in self-paced mode. The black curve shows the results of the RTS algorithm, the gray curve shows the results of Fusco and Cretual’s (FC) method [5], and the three dashed curves show the speed of each of the three reflective squares (Marker 1, Marker 2, and Marker 3) calculated using the Woltring routine [9]. LHS = left heel strike; LTO = left toe-off; RHS = right heel strike; RTO = right toe-off.
However, when two reflective squares are visible, the RTS algorithm provides a smooth transition, despite the difference between the speeds of the two visible reflective squares as estimated using the Woltring routine. The Fusco and Cretual’s method exhibits large dips in computed treadmill speed during the double support phase, which are not present in the reflective square data.
4. Discussion A limitation of this approach is that errors in the length scale in the position measurements are unobservable to the estimation algorithm. Thus, the errors in the position and velocity estimates is proportional to the error in the length scale. However, if the length of the treadmill belt is accurately known, this error can be removed
N.M. Olson, G.K. Klute / Gait & Posture 41 (2015) 954–956
956
by scaling the position and velocity estimates by the ratio of the true length of the belt to the estimated length of the belt. Although only one belt was tracked in this study, the method could be easily extended to both belts on a split-belt treadmill by applying the same algorithm to three reflective squares on each belt. This study demonstrates the effectiveness of using reflective squares attached to a treadmill belt in conjunction with a motioncapture system to estimate instantaneous treadmill speed. The method is easy to set up, inherently synchronized with motioncapture kinematics measurements, and does not require any additional instruments. The RTS algorithm is inherently robust to data gaps and provides accurate estimates of treadmill speed and distance traveled. Acknowledgements This research was supported by the Department of Veterans Affairs, Rehabilitation Research and Development Service (A9243C and A9248-S) and Department of Defense Deployment Related Medical Research Program of the Office of the Congressionally Directed Medical Research Programs (W81XWH-09-2-0144). Conflict of interest The authors have no conflicts of interest.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.gaitpost. 2015.03.003. References [1] Minetti AE, Boldrini L, Brusamolin L, Zamparo P, McKee T. A feedback-controlled treadmill (treadmill-on-demand) and the spontaneous speed of walking and running in humans. J. Appl. Physiol. 2003;95:838–43. [2] Segers V, Lenoir M, Aerts P, De Clercq D. Kinematics of the transition between walking and running when gradually changing speed. Gait Posture 2007;26:349–61. [3] Sloot LH, van der Krogt MM, Harlaar J. Energy exchange between subject and belt during treadmill walking. J. Biomech. 2014;47:1510–3. [4] Alton F, Baldey L, Caplan S, Morrissey MC. A kinematic comparison of overground and treadmill walking. Clin. Biomech. 1998;13:434–40. [5] Fusco N, Cretual A. Instantaneous treadmill speed determination using subject’s kinematic data. Gait Posture 2008;28:663–7. [6] Savelberg HH, Vorstenbosch MA, Kamman EH, van de Weijer JG, Schambardt HC. Intra-stride belt-speed variation affects treadmill locomotion. Gait Posture 1998;7:26–34. [7] Paolini G, Della Croce U, Riley PO, Newton FK, Kerrigan C. Testing of a triinstrumented-treadmill unit for kinetic analysis of locomotion tasks in static and dynamic loading conditions. Med. Eng. Phys. 2007;29:404–11. [8] Rauch HE, Striebel CT, Tung F. Maximum likelihood estimates of linear dynamic systems. AIAA J. 1965;3:1445–50. [9] Woltring HJ. A Fortran package for generalized, cross-validatory spline smoothing and differentiation. Adv. Eng. Softw. 1986;8:104–13.
