## 1 Introduction

Foot motion assessment plays a central role in diagnosis and treatment of walking disabilities. Functional electrical stimulation (FES) and active orthoses represent effective tools for online gait support. Both technologies rely on realtime foot motion assessment. The conventional method for motion assessment is optical motion capture (OMC), which has the decisive disadvantage that it is not suitable for realtime applications. Furthermore, it is an expensive technology and strictly limiting in space and time, as it restricts the analysis to a laboratory environment. Ambulatory realtime motion capture can be performed by the use of inertial measurement units (IMU), which typically comprise accelerometers, gyroscopes, and magnetometers. They represent an inexpensive and easy-to-handle technology without any of the mentioned limitations of OMC. There are a number of standard methods that estimate the orientation of the foot based on measurement data of an IMU attached to the foot, see for example [1, 3, 7]. Since, in indoor environments, the magnetic field is often heavily disturbed and far from homogeneous, we consider an IMU-based method that completely avoids to use magnetometer readings to calculate pitch and roll angles. Furthermore, we allow for almost arbitrary mounting orientation in the sense that we only assume one of the local IMU coordinate axes to lie in the sagittal plane of the foot.

In this contribution, we focus on the question how accurate the obtained orientation measurements are with respect to the conventional OMC method under different walking conditions. Unlike most IMU validation studies, e.g. [1], we place the optical markers on anatomical landmarks instead of attaching them to the IMUs. Thereby, we compare the entire gait analysis methods instead of comparing only the measurement systems. In the following, we briefly explain the method, before we validate it in trials with transfemoral amputees walking with shoes and in trials with healthy subjects walking barefoot.

## 2 Methods

The measured angular rates and accelerations of the paretic foot are used to detect the gait events*toe-off* and *initial contact*, which mark the beginning and the end of the *swing phase*, as well as *full contact* and *heel-rise*, which mark the beginning and the end of the*foot-flat phase*. In the following,*t*_{to},*t*_{ic}, *t*_{fc} and *t*_{hr} will denote the according time instants of the considered step, respectively. As depicted in Figure 1, the period of time between heel-rise and toe-off is referred to as*pre-swing*, and the phase between initial contact and full contact is called*loading response*. This is in accordance with standard literature, see e.g. [2]. The employed gait phase detection algorithm is described in [5] and, for the sake of brevity, is not further discussed here. Instead, we assume that we have realtime information of the current gait phase and consider the task of foot orientation angle measurement in the following.

The foot’s pitch angle *α* and roll angle *β* are defined in Figure 2. The local(By ”local” we denote the coordinates in the moving sensor coordinate system.) coordinates of *x*_{foot},*y*_{foot} ∈ ℝ^{3×1} are unknown because the sensor cannot be attached such that the local coordinate axes coincide with the anatomical axes of the foot, as Figure 3 illustrates. However, it is possible to assure that the local axis *x*_{local} = (1, 0, 0)^{T} of the IMU lies in the sagittal plane of the foot. We therefore assume that this is assured and determine the local coordinates of the foot’s posterior-anterior and mediolateral axis as follows: During every foot-flat phase, the local accelerometer readings *a*(*t*) are integrated over time, and the resulting vector is normalized to unit length:

where ‖·‖_{2} denotes the Euclidean norm. Since gravitational acceleration dominates when the foot is (almost) at rest, *z*_{ff} is (almost) vertical. Therefore, we calculate the local coordinates of the mediolateral axis *y*_{foot} as follows:

Likewise, we calculate the local coordinates of the posterior-anterior axis *x*_{foot} as follows:

Please note that, by construction, *x*_{foot} and *y*_{foot} are horizontal during stance and their local coordinates do not change with time, since the sensor moves along with the foot, see Figure 3 for illustration.

For each step, a strap-down integration [4] of the angular rates is started at heel-rise*t*_{hr} that yields the rotation matrix *R*_{ff}(*t*). This matrix transforms the local measurement vectors *a*(*t*) and *g*(*t*) of any time instant*t* between two foot-flat phases to the local coordinate system of the preceding foot-flat phase, which we refer to as the reference coordinate system of that step. By transforming *x*_{foot} to the reference coordinate system, in which the vertical axis *z*_{ff} is known, we calculate the time-dependent foot orientation angle in pitch direction:

Please note that the pitch angle is positive when the toes are above the heel and negative when vice versa. Likewise, we calculate the time-dependent foot orientation angle *β* in roll direction:

Due to the side-dependent axis definition (3), the roll angle *β* is always positive when the foot’s outer edge is above the inner edge, both for a right and a left foot.

However, it is important to note that orientation strap-down integration is always subject to drift, since, even with proper calibration, the gyroscopes have non-zero bias. Therefore, *α*(*t*) and *β*(*t*) also drift(For example, with the employed sensor hardware, we found that, at the end of a step, *α*(*t*_{fc}) is typically in the range of 2°.) between each two foot-flat phases. At every full contact *t*_{fc}, however, we can remove the drift from *α*(*t*), *β*(*t*) on the time interval *t* ∈ [*t*_{hr},*t*_{fc}] by assuming that neither the gyroscope bias nor the slope of the ground changed significantly during the swing phase:

In the following, we validate these IMU-based angles with respect to optical motion capture. Since the standard setup with only one optical marker on the forefoot and one on the ankle does not permit the calculation of roll orientation angles, we restrict the validation to foot pitch angles.

## 3 Experimental procedure and results of the validation

For evaluation of the proposed methods, the recorded data of two different experiments is processed. In both trial series, a wireless IMU was attached dorsally on the instep of each foot and optical markers of a visual motion capture system were attached to the lateral malleolus and the head of the fifth metatarsal (see Figure 3). In the first experiment, two healthy subjects walked barefoot at slow and fast pace, whereas in the second one, two transfemoral amputees walked with a leg prosthesis at self-assessed speed while wearing shoes on both the prosthetic and the contralateral foot. At least six trials were completed by each subject, for each of which the foot pitch angle is computed according to (6). For the sake of comparison, we also calculate the foot pitch from the three-dimensional optical marker positions.

Deviations between inertial and optical measurements. Columns show the walking velocity (vel.), the average root mean square error (RMSE) with its standard deviation with and without drift correction. Averages are taken over several steps. The rows refer to different subjects and their different footwear – barefoot (baref.), shoe and shoe on the prosthetic side (prosth.).

walking vel. | RMSE α [°] drift not removed | RMSEdrift removed | |

Subj.1-shoe | 0.81 | 2.66 ± 0.73 | 2.24 ± 0.41 |

Subj.1-prosth. | 0.81 | 1.40 ± 0.21 | 1.20 ± 0.26 |

Subj.2-shoe | 1.42 | 5.78 ± 1.49 | 3.97 ± 0.51 |

Subj.2-prosth. | 1.42 | 3.97 ± 1.31 | 2.93 ± 0.75 |

Subj.3-baref. | 1.62 | 3.61 ± 0.87 | 3.63 ± 0.83 |

Subj.3-baref. | 0.85 | 3.14 ± 0.39 | 3.09 ± 0.36 |

Subj.4-baref. | 1.52 | 3.90 ± 0.91 | 3.79 ± 0.73 |

Subj.4-baref. | 0.78 | 3.37 ± 0.37 | 3.41 ± 0.31 |

The root mean square error (RMSE) with and without drift correction is determined for each trial of each subject. We then calculate the average and standard deviations over all trials for each foot. Table 1 shows the results, as well as corresponding walking velocities. In Figure 4a, the pitch angles of Subject 2 (cf. Table 1) are plotted over one step for both the prosthetic and the contralateral side. Furthermore, pitch angles for different walking velocities of the barefoot walking Subject 3 are presented in Figure 4b.

## 4 Discussion

We first analyze the influence of the footwear. It is evident that deviations between the results of both measurement systems are larger in barefoot trials than in trials with shoes. This is most likely due to slight deformations of the foot during stance-phase and push-off, which lead to relative motion of the IMU and the optical markers with respect to each other. Since the IMU was attached directly to the skin in the barefoot trials, the effect is stronger therein than in the shoe-trials, in which the shoe dampens this effect. Likewise, the data in Table 1 demonstrates that the deviations are slightly smaller on the prosthetic side, which is more rigid and allows for less relative motion of the markers and IMU with respect to each other.

If we furthermore compare RMSEs between the slow and fast trials of Subject 3 and 4, it is noticeable that the deviations are significantly smaller in slower walking. The same observation is also made when comparing (the prosthetic or contralateral foot of) Subject 1 and Subject 2, the latter of which walked at a much faster pace.

By applying the periodic online drift correction proposed in (9), the RMSE was reduced by more than 15% in the shoe-walkers. In the barefoot walkers, this correction had no significant effect.

## 5 Conclusion

A method for realtime foot pitch and roll angle estimation was considered that avoids magnetometers and copes with almost arbitrary mounting orientation. It was validated with respect to a conventional optical motion capture method in subjects with different footwear. Results indicate that deformations of the foot and shoe during foot-flat and push-off lead to relative motion of the optical markers and the IMUs with respect to each other, which are strongest in barefoot walking and weakest in prosthetic feet. Nevertheless, the deviation between inertial and optical measurements are in the few-degrees range under all tested conditions. For slow walking with shoes, accuracies below 2° were obtained.

With respect to the intended application in FES-based gait neuroprostheses and active orthoses, we conclude that the obtained deviations are small enough to facilitate precise feedback control of the foot motion.

We would like to express our deep gratitude to Julius Thiele from TU Berlin as well as to Noelia Chia Bejarano and Simona Ferrante from Politecnico di Milano for providing the datasets that were analyzed in this study. Furthermore, we thank Sebastian Scheel and Dirk Reinhardt for their skillful support in automatic data evaluation.

**Funding**: Being conducted in the research project BeMobil, this work is funded by the German Federal Ministry of Research and Education (FKZ 16SV7069K).

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Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

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