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.

Figure 1 Employed finite state automaton model (phases and transitions) of the gait cycle of one side. During foot-flat phase, the foot rests on the ground, while it has no ground contact during swing phase.

Figure 2 The angle *α* that the foot’s posterior-anterior axis*x* _{foot} and its projection into the horizontal plane (dotted lines) confine is a proper measure of foot pitch, while foot roll can be quantified analogously using the foot’s mediolateral axis *y*_{foot}. The local coordinate axes of the IMU are *not* parallel to any anatomical axis, but the local x-axis lies in the sagittal plane of the foot.

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:

$${\widehat{z}}_{\text{ff}}:={\displaystyle \sum _{t\in \text{foot-flat}}a(t),\text{\hspace{1em}}{z}_{\text{ff}}}:={\widehat{z}}_{\text{ff}}/\left|\right|{\widehat{z}}_{\text{ff}}|{|}_{2,}$$(1)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:

$$\begin{array}{l}{\widehat{y}}_{\text{foot}}:={z}_{\text{ff}}\times {\chi}_{\text{local}},\\ {y}_{\text{foot}}:=+{\widehat{y}}_{\text{foot}}/\left|\right|{\widehat{y}}_{\text{foot}}|{|}_{2}\text{for}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{left}\text{\hspace{0.17em}}\text{foot},\end{array}$$(2)$$\text{and}\text{\hspace{0.17em}}{y}_{\text{foot}}:=-{\widehat{y}}_{\text{foot}}/\left|\right|{\widehat{y}}_{\text{foot}}\left|\right|2\text{for}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{right}\text{\hspace{0.17em}}\text{foot}.$$(3)Likewise, we calculate the local coordinates of the posterior-anterior axis *x*_{foot} as follows:

$$\begin{array}{l}{\widehat{x}}_{\text{foot}}:={z}_{\text{ff}}\times {x}_{\text{local}}\times {z}_{\text{ff}},\\ {\widehat{x}}_{\text{foot}}:={\widehat{x}}_{\text{foot}}/\left|\right|{\widehat{x}}_{\text{foot}}|{|}_{2}.\end{array}$$(4)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:

$$\alpha (t):=\frac{\pi}{2}-\sphericalangle ({z}_{\text{ff}},{R}_{\text{ff}}(t){x}_{\text{foot}}),$$(5)$$=\mathrm{arcsin}\left({z}_{\text{ff}}^{T}{R}_{\text{ff}}(t){x}_{\text{foot}}\right)\text{\hspace{1em}}\in [-\frac{\pi}{2},+\frac{\pi}{2}].$$(6)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:

$$\beta (t):=\frac{\pi}{2}-\sphericalangle ({z}_{\text{ff}},{R}_{\text{ff}}(t){y}_{\text{foot}}),$$(7)$$=\mathrm{arcsin}\left({z}_{\text{ff}}^{T}{R}_{\text{ff}}(t){y}_{\text{foot}}\right)\text{\hspace{1em}}\in [-\frac{\pi}{2},+\frac{\pi}{2}].$$(8)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:

$$\tilde{\alpha}(t):=\alpha (t)-\frac{t-{t}_{\text{hr}}}{{t}_{\text{fc}}-{t}_{\text{hr}}}(\alpha ({t}_{\text{fc}})-\alpha ({t}_{\text{hr}})),$$(9)$$\tilde{\beta}(t):=\beta (t)-\frac{t-{t}_{\text{hr}}}{{t}_{\text{fc}}-{t}_{\text{hr}}}(\beta ({t}_{\text{fc}})-\beta ({t}_{\text{hr}})).$$(10)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.

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