Estimating Stair Running Performance Using Inertial Sensors

Stair running, both ascending and descending, is a challenging aerobic exercise that many athletes, recreational runners, and soldiers perform during training. Studying biomechanics of stair running over multiple steps has been limited by the practical challenges presented while using optical-based motion tracking systems. We propose using foot-mounted inertial measurement units (IMUs) as a solution as they enable unrestricted motion capture in any environment and without need for external references. In particular, this paper presents methods for estimating foot velocity and trajectory during stair running using foot-mounted IMUs. Computational methods leverage the stationary periods occurring during the stance phase and known stair geometry to estimate foot orientation and trajectory, ultimately used to calculate stride metrics. These calculations, applied to human participant stair running data, reveal performance trends through timing, trajectory, energy, and force stride metrics. We present the results of our analysis of experimental data collected on eleven subjects. Overall, we determine that for either ascending or descending, the stance time is the strongest predictor of speed as shown by its high correlation with stride time. Keywords: wearable sensors; inertial measurement units; motion tracking; human performance; stair running 1. Introduction We present a method for using inertial measurement units (IMUs) to measure the kinematics and performance of stair running. Running on stairs is a mechanically challenging task. Stair ascent (both walking and running) challenges the body to achieve center of mass translation forward and upward against gravity

wearable sensors; inertial measurement units; motion tracking; human performance; stair running 1. Introduction We present a method for using inertial measurement units (IMUs) to measure the kinematics and performance of stair running. Running on stairs is a mechanically challenging task. Stair ascent (both walking and running) challenges the body to achieve center of mass translation forward and upward against gravity (repeatedly generating upward ground reaction forces larger than the downward bodyweight force). Therefore, studying stair ascent can provide insights into an individual s aerobic conditioning [1], athletic strength and lower extremity power [2], and performance [3]. Stair descent, in contrast, challenges the body to achieve the desired forward and downward trajectory while controlling and leveraging the assistance of gravity. Therefore, stair descent performance is often studied in clinical populations to assess the level of lower extremity joint stability and control [4]. Furthermore, each footfall needs to land on the relatively small surface of each step, therefore, successful performance of both stair ascent and decent require body coordination across multiple body segments in order to avoid trips or falls. Overground running has been studied extensively from different points of view [5,6]; a detailed review of early research being provided by Novacheck [7]. Sensors2017,17, 2647; doi:10.3390/s17112647

Sensors2017,17, 2647 2 of 14 On the other hand, in-depth biomechanical analysis of stair running has been limited by inadequate biomechanical tracking tools. Optical-based motion capture systems and instrumented walkways, which are commonly used for studying gait, are limited by practical challenges in order to appropriately position cameras for the desired motion capture volume. Consequently, past studies of stair climbing focus on the functional walking pace [8–10] and have estimated overall energy expenditure [11], basic timing measures [12], and joint angles [6]. In contrast, we propose using foot-mounted IMUs as a motion capture instrument. Body-worn IMUs enable human motion analysis in outdoor and other contextually-relevant settings (e.g., training facilities, game settings, obstacle courses) and have been used in a wide array of biomechanics applications; see, for example, [13–20]. Our approach uses foot-mounted IMUs to measure the foot kinematic variables (acceleration and angular velocity) during stair running. Doing so enables one to track a large number of steps, to understand transient and steady state running on stairs, and to also deduce performance measures. IMUs are portable, unobtrusive, and unconstrained (e.g., they do not need external references) motion tracking devices. However, IMU data (and quantities computed therefrom) are affected by several sources of error (e.g., bias instability, scale factor errors, acceleration, and temperature sensitivity) that must be accounted for during motion tracking applications [21]. In this paper, we present specialized algorithms that address these sources of error to estimate the foot trajectory and velocity during stair running. In particular we extend the Zero velocity UPdaTe (ZUPT) algorithm [22], which has been validated to provide accurate foot motions [14,23], by adding additional drift corrections speci c to the constraints of stair running (known riser dimensions). We further employ those estimates to deduce metrics for evaluating stair running performance and explore the metrics utilizing experimental data collected on 11 subjects. We hypothesized that the metrics that could be de ned were related to the overall speed, thereby providing an ability to assess stair running techniques. 2. Materials and Methods We tested 11 healthy volunteer subjects (three female, eight male; age:25.6 3.7 years ; mean

for evaluating stair running performance and explore the metrics utilizing experimental data collected on 11 subjects. We hypothesized that the metrics that could be de ned were related to the overall speed, thereby providing an ability to assess stair running techniques. 2. Materials and Methods We tested 11 healthy volunteer subjects (three female, eight male; age:25.6 3.7 years ; mean SD ). The University of Michigan IRB approved the study and all subjects provided informed consent. Subjects were instructed to run up a long staircase at maximum speed, without skipping treads. After pausing for approximately ten seconds, the subjects ran down the same ight of stairs at maximum speed returning to the starting position, again without skipping treads. The staircase provided 16 strides total during the steady state (eight left and eight right). Subjects were not instructed which foot to begin stepping with for the task. The staircase rise height was 18 cm and the depth was 30 cm. The subjects wore two IMU data loggers (Opals, APDM, Portland OR, USA; 128 Hz sampling, 6gacceleration, 2000 deg/s angular rate), one mounted on each shoe af xed using athletic tape to the top of the laces (see Figure). The IMUs measure and store three components of linear acceleration(a f= [ax,ay,az]) from the on-board accelerometer and three components of angular velocity(w f= wx,wy,wz ) from the on-board angular rate gyro, both relative to the sensor- xed axes(x,y,z) . These sensor axes de ne the IMU frame of reference. We also de ne a navigation frame that overlaps with the IMU frame during initialization. The navigation frame remains af xed to the world during the experiment, while the IMU frame moves with the subject's foot. Since the IMU sensor measurements are relative, there is no need to follow a strict anatomical calibration. However, since the IMU reference frame determines the navigation frame during initialization, it is advisable to approximately align the IMU axes to the desired navigation frame (see Figure). In-depth explanations of how (strap-down) IMUs are used, particularly for navigation applications, are provided in [24,25]. Major results from this eld that we

is no need to follow a strict anatomical calibration. However, since the IMU reference frame determines the navigation frame during initialization, it is advisable to approximately align the IMU axes to the desired navigation frame (see Figure). In-depth explanations of how (strap-down) IMUs are used, particularly for navigation applications, are provided in [24,25]. Major results from this eld that we employ are summarized below.

Sensors2017,17, 2647 3 of 14Sensors 2017, 17, 2647 3 of 14 (a) ( b)( c) Figure 1. IMU data logger setup: (a) APDM IMU device showing the IMU sensor axes; (b) IMU attached to the shoe showing the IMU frame axes convention used in this paper; and (c) video stills showing the IMUs mounted on a subject's shoe climbing stairs ascent. 2.1. Orientation Estimation Estimating the foot trajectory from IMU data begins with first estimating the orientation of the foot-mounted IMU. For this purpose, we choose a quaternion ( —) representation of the IMU orientation. Unlike the more common Euler angle representation that suffers from gimbal-lock, the quaternion representation readily describes any arbitrary sequence of rotations [26]. Quaternions represent an orientation as a rotation angle about a rotation axis. Thus, quaternions are defined using four parameters, one defining the angle of rotation and three defining the axis of rotation (e.g., three direction cosines). The four quaternion parameters satisfy the differential equation: — 6= —∘ ¹ 2 (1) ¹=c0, ñ ë, ñ ì, ñ íg (2) in which the operator ∘ denotes quaternion multiplication [25,27] and ¹ is a four-element vector containing the aforementioned measured angular velocity components k ñ Ùo. Thus, the solution of (1) using the measured ¹ yields the gyro-estimated orientation of the IMU as a function of time. The gyro-estimated orientation will inevitably drift due to sensor errors, including bias drift, scale factor errors, and acceleration sensitivity. Our algorithm fuses the gyro-estimated orientation with accelerometer-estimated tilt angles from vertical (roll and pitch). This is achieved using a Kalman filter [28,29]. When the IMUs are mounted on the feet, the foot and the attached IMU are essentially stationary during specific time periods ( P æ) for the stance phase of each stride. The stationary periods are detected by observing the gyroscope and accelerometer measurements (see [14] Section 2.1 for more information about how P æ is determined). During stationary periods, the accelerometer measures the components of gravity ( )) along each sense axis. These measures are used to form accelerometer-estimated roll and pitch angles ( =[ ö Ô,

phase of each stride. The stationary periods are detected by observing the gyroscope and accelerometer measurements (see [14] Section 2.1 for more information about how P æ is determined). During stationary periods, the accelerometer measures the components of gravity ( )) along each sense axis. These measures are used to form accelerometer-estimated roll and pitch angles ( =[ ö Ô, à Ô]) per: ö Ô=sin ? 5 @ = ë ) A (3) à Ô=−sin ? 5 l = ì ) cos ö Ô p (4) Next, we use the gyroscope-estimated quaternion ( —) value to calculate the equivalent Euler angles k ž = c ö Ú, à Ú, ð Úgo, which also includes the estimated yaw angle k ð Úo (that is temporarily ignored as it cannot be detected from the accelerometers) . The Kalman filter states k žÝ=c öà, ààgo are estimated as a combination of the gyroscope-based and accelerometer-based tilt estimates. We assume that all gyroscope error contributions and accelerometer-based tilt errors can be modeled as zero mean Gaussian noise. Since the process and measurement covariance errors are sensor- dependent only, once the Kalman filter is tuned the parameters are valid for all participants. The Figure 1. IMU data logger setup: (a) APDM IMU device showing the IMU sensor axes; (b) IMU attached to the shoe showing the IMU frame axes convention used in this paper; and (c) video stills showing the IMUs mounted on a subject s shoe climbing stairs ascent. 2.1. Orientation Estimation Estimating the foot trajectory from IMU data begins with rst estimating the orientation of the foot-mounted IMU. For this purpose, we choose a quaternion(q)representation of the IMU orientation. Unlike the more common Euler angle representation that suffers from gimbal-lock, the quaternion representation readily describes any arbitrary sequence of rotations [26]. Quaternions represent an orientation as a rotation angle about a rotation axis. Thus, quaternions are de ned using four parameters, one de ning the angle of rotation and three de ning the axis of rotation (e.g., three direction cosines). The four quaternion parameters satisfy the differential equation: . q= q W 2

readily describes any arbitrary sequence of rotations [26]. Quaternions represent an orientation as a rotation angle about a rotation axis. Thus, quaternions are de ned using four parameters, one de ning the angle of rotation and three de ning the axis of rotation (e.g., three direction cosines). The four quaternion parameters satisfy the differential equation: . q= q W 2 (1) W= 0,wx,wy,wz (2) in which the operator denotes quaternion multiplication [25,27] andWis a four-element vector containing the aforementioned measured angular velocity components(w f). Thus, the solution of (1) using the measuredWyields the gyro-estimated orientation of the IMU as a function of time. The gyro-estimated orientation will inevitably drift due to sensor errors, including bias drift, scale factor errors, and acceleration sensitivity. Our algorithm fuses the gyro-estimated orientation with accelerometer-estimated tilt angles from vertical (roll and pitch). This is achieved using a Kalman lter [28,29] . When the IMUs are mounted on the feet, the foot and the attached IMU are essentially stationary during speci c time periods(ts)for the stance phase of each stride. The stationary periods are detected by observing the gyroscope and accelerometer measurements (see [14] Section for more information about howtsis determined). During stationary periods, the accelerometer measures the components of gravity(G)along each sense axis. These measures are used to form accelerometer-estimated roll and pitch angles(z= [fa,qa])per: fa=sin 1 ax G (3) qa= sin 1 ayG cosfa (4) Next, we use the gyroscope-estimated quaternion(q)value to calculate the equivalent Euler angles(x= [fg,qg,yg]) , which also includes the estimated yaw angle yg (that is temporarily ignored as it cannot be detected from the accelerometers). The Kalman lter states(ˆx= [ˆf,ˆq]) are estimated as a combination of the gyroscope-based and accelerometer-based tilt estimates. We assume that all gyroscope error contributions and accelerometer-based tilt errors can be modeled as zero mean Gaussian noise. Since the process and measurement covariance errors are sensor-dependent only,

Sensors2017,17, 2647 4 of 14 once the Kalman lter is tuned the parameters are valid for all participants. The updated state is then converted back to its corresponding quaternion value. Figure orientation estimation algorithm.Sensors 2017, 17, 2647 4 of 14 updated state is then converted back to its corresponding quaternion value. Figure 2 illustrates a block diagram of this orientation estimation algorithm. Figure 2. Angular velocity components measured by the gyroscope are integrated once to obtain orientation estimates ž. The accelerometer components are used to estimate tilt (roll and pitch) during stationary periods P æ. The Kalman filter bounds the tilt errors by fusing the gyro-based orientation and accelerometer-based tilt to establish the “corrected orientation” žÝ. 2.2. Foot Trajectory Estimation The resulting orientation estimates are used to resolve the foot IMU frame-acceleration components k ‡ Ùo into the navigation frame acceleration components ( ‡ á). The V-axis component of the resultant world-referenced acceleration ‡ á will be affected by gravity ): ‡ ê= ˜ Ù á ‡ Ù+ ) (5) in which ˜ Ù á is the rotation matrix from the foot IMU frame to the navigation frame as computed from the quaternion — [24,25]. Next, integrating ‡ á once and then twice yields the foot IMU velocity ( œ) and position ( –): œ= œ â+± ‡ á @ P ç ç Ú (6) –= – â+± œ @ P ç ç Ú (7) Since the experiment starts with a stationary phase, the initial velocity ( œ â) is zero and at a position ( – â) also designated as zero. However, this can be generalized to a non-zero initial velocity or position for applications that require such. Examples of software implementations of (1)–(7) are found in [30,31]. The velocity estimated from (6) is often polluted by residual drift error (deriving from both the gyro and the accelerometer) which leads to (often slowly varying) velocity errors. The velocity drift error can be estimated and (approximately) eliminated using the following procedure. During the stationary times ( P æ) any remaining estimated velocity during these times can be assumed to be

velocity estimated from (6) is often polluted by residual drift error (deriving from both the gyro and the accelerometer) which leads to (often slowly varying) velocity errors. The velocity drift error can be estimated and (approximately) eliminated using the following procedure. During the stationary times ( P æ) any remaining estimated velocity during these times can be assumed to be caused by drift error. These velocity errors are used to correct both the velocity (6) and position (7) estimates using an algorithm known as the Zero velocity UPdaTe (ZUPT). A block diagram for the ZUPT algorithm is illustrated in Figure 3 and further details of its implementation can be found in [14,22]. ž ± B( P) @ P ç ç Ú Kalman Filter ž Corrected Orientation žÝ Angular Velocity Tilt Estimation Linear Acceleration Noise + Orientation P æ ‡ Ù Ó Œ Figure 2. Angular velocity components measured by the gyroscope are integrated once to obtain orientation estimatesx. The accelerometer components are used to estimate tilt (roll and pitch) during stationary periodsts. The Kalman lter bounds the tilt errors by fusing the gyro-based orientation and accelerometer-based tilt to establish the “corrected orientation” ˆx. 2.2. Foot Trajectory Estimation The resulting orientation estimates are used to resolve the foot IMU frame-acceleration components(a f)into the navigation frame acceleration components(an). Thez-axis component of the resultant world-referenced accelerationanwill be affected by gravityG: aw=r n f a f+G (5) in whichr n fis the rotation matrix from the foot IMU frame to the navigation frame as computed from the quaternionq[24,25]. Next, integratinganonce and then twice yields the foot IMU velocity(v)and position(p): v=vo+ Z t to andt (6) p=p o + Z t to vdt (7) Since the experiment starts with a stationary phase, the initial velocity(vo)is zero and at a position(p o )also designated as zero. However, this can be generalized to a non-zero initial velocity or position for applications that require such. Examples of software implementations of (1)–(7) are found in [30,31]. The velocity estimated from (6) is often polluted by residual drift error (deriving from both the gyro and the accelerometer) which leads to

zero and at a position(p o )also designated as zero. However, this can be generalized to a non-zero initial velocity or position for applications that require such. Examples of software implementations of (1)–(7) are found in [30,31]. The velocity estimated from (6) is often polluted by residual drift error (deriving from both the gyro and the accelerometer) which leads to (often slowly varying) velocity errors. The velocity drift error can be estimated and (approximately) eliminated using the following procedure. During the stationary times(ts)any remaining estimated velocity during these times can be assumed to be caused by drift error. These velocity errors are used to correct both the velocity (6) and position (7) estimates using an algorithm known as the Zero velocity UPdaTe (ZUPT). A block diagram for the ZUPT algorithm is illustrated in Figure in [14,22].

Sensors2017,17, 2647 5 of 14Sensors 2017, 17, 2647 5 of 14 Figure 3. The accelerometer measurements are resolved in the world coordinate frame using the corrected orientation. The resultant accelerations are integrated twice to determine velocity and position. During stationary periods P æ, any remaining velocity is considered an error and its value is used to reset the position and acceleration errors. 2.3. Elevation Correction Since the riser (step height) and tread (step depth) dimensions of the stairs are known, we add an additional correction to the position estimate. In particular, we designed a single-state Kalman filter that makes corrections to the IMU-derived vertical foot position ( ž=[ L í]) knowing the riser height ( *) and the number of steps ( J) to yield an elevation observation per footfall ( =[ * J]) . The filter makes corrections to its state ( žÝ=[ L íÞ]) whenever the foot reaches a new tread during the stationary time ( P æ). The filter assumes that the state and observation are both affected by uncorrelated white noise. A block diagram showing this filter is illustrated in Figure 4. Finally, we apply a linear interpolation in order to provide backward corrections to obtain the complete foot trajectory for each stride. Figure 4. A Kalman filter makes foot elevation corrections using the known step height (riser), during each stationary time P æ. 2.4. Gait Timing Variables We used a wavelet analysis to establish the beginning (foot-strike) and end (toe-off) of each foot/ground contact period [32]. This approach is effective at identifying gait events because when the foot strikes or leaves the ground, the acceleration and angular velocity signals contain significantly more high-frequency content than at other times of the gait cycle. The wavelet analysis is used to identify time points when the measured signals contain significant content above 20 Hz, corresponding to either foot-strikes or toe-offs. Foot-strike time ( P æ ç å Ü Þ Ø) was defined as the time when the foot first contacts a tread. For running on stairs, the toe is more likely to contact the tread first (whereas, during

This paper presents methods for estimating foot velocity and trajectory during stair running using foot-mounted IMUs.