12 subjects (10 males/2 females, age 54 ± 14) with hemiplegia were included. Subject characteristics are given in Table  1. All subjects provided informed consent and the study was conducted in accordance with the principles of the Declaration of Helsinki and was approved by the local Ethics Committee (CPP Nîmes).

With: Stroke diagnosis: ischemic (isch)/hemorrhagic (hemo)/infra-tentional (IT)/supra-tentorial (ST). FAC: Functional ambulation Categories (Collen et al., 23 ). BI: Barthel index assessment of daily activity impairment (autonomy evaluations). OS: orthopaedic shoes. - SC: simple cane- CC: Canadian crutch- TC: tripod cane.

The algorithm was first introduced and described by Heliot and Espiau in 32 , where more technical details can be found. The method allows estimating the phase of the gait cycle of a healthy person using a micro-sensor, which associates 3 accelerometers and 3 magnetometers, placed on person’s thigh. The method was designed to allow humanoid robot to mimic the walk of a human demonstrator 32 . Ahn et al., based on a similar framework, have proposed a walking model, which can reproduce some behaviors of human walking 33 . But this approach is not intended to be used online for human motion tracking and has not been validated experimentally.

Our algorithm is based on defining OFFLINE a model of a system, which in our case is human gait. In the field of control of locomotion in humanoid robots, the bipedal gait has been modeled as a non-linear oscillator 32 34 . Two common oscillators can be used for this purpose: the van der Pol oscillator and the Rayleigh oscillator, which are very similar. The Van der Pol oscillator was used in 32 , and experimentally proved to be robust and suitable for human gait modeling. Therefore, the chosen model in this study is Van der Pol oscillator which details are given in 35 . Briefly, the Van der Pol oscillator is a self-oscillating system that has a stable periodic solution with a period of T₀ and a frequency of ω₀. The equation of Van der Pol oscillator used in this study is given by (1).

x ¨ 2 - μ 1 - bx - x 2 x ˙ 2 + ω 0 = 0

where μ, and ω₀ are positive constants. The choice of μ, ω₀, and b constants is explained bellow.In the phase plane (position versus velocity), periodic stable solution of the oscillator is a trajectory called a limit cycle (Figure  1). The phase φ is a coordinate along this limit cycle:

dϕ dt = ω 0

ω 0 = 2 π T 0

The phase grows uniformly in the direction of motion and gains 2π at each rotation. There is a need to define the φ variable in such a way that it rotates uniformly in the cycle of the oscillator. We therefore define OFFLINE an isochron matrix that allows us to associate a phase value to any point in the vicinity of the limit cycle. Each point in the vicinity of the limit cycle is associated with a point in the limit cycle (Figure  1). This makes it possible to compute a phase φ even in the vicinity of the limit cycle, for example when the observed behavior differs from that of the reference.

Based on control theory framework, it is theoretically possible to build an observer of this system that will be able to estimate the internal state of the system, in our case gait of a person with drop foot modeled as Van der Pol oscillator, from the system output measurement. In control theory, a state observer is defined as a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system 34 , and could be represented using the following set of equations:

x ˙ 1 = x 2 x ¨ 2 - μ 1 - x 2 x ˙ 2 + ω 0 = 0 y = x 1

where: y is the output of the system, and x1, and x2 are the states of the system which should be estimated. In our case, y corresponds to a measurement of shank angle of the unaffected leg. The shank angle was chosen because it is the easiest to be obtained by the sensor technology used in this study. Knowing the value of y, and solving the system of equations (3), we can obtain state variables x1 and x2. From the observer’s state variables we can compute the phase φ of the oscillator. We defined the gait cycle index GCI from φ normalization in order to get values ranging from 0 and 100% (the cycle starts and stops at unaffected leg heel strike):

GCI = 100 φ / max φ

The principle of described algorithm is given in Figure  2.

The last step is defending the values of μ, ω₀, and b constants in (1) that would fit the sensor measurement into this oscillator model. This problem can be solved by using dynamic optimization technique called Feasible Sequential Programming Technique (SQP) 36 . SQP technique finds the values of μ, ω₀, and b constants that minimize a quadratic function of the error between measurement and output of our model. The mathematical details about the calculations are given in 37 . Prior to the experiments the participant was asked to walk a couple of steps. The measurement of unaffected shank angle was used to identify oscillator parameters.We summarize the different steps of the method (see Figures  2 and 3).

1. The subject is asked to walk a few steps during which the shank angle is measured.

2. The oscillator parameters μ, ω₀, and b are identified through a dynamic optimization problem: the error between the measured shank angle and the oscillator output is minimized. The isochron matrix is computed.

3. The oscillator parameters are updated in the observer of the system.

• Oscillator design (OFFLINE)

Gait Cycle Index estimation (ONLINE) - the observer computes φ and GCI variables in real-time based on the measured shank angle y.

The developed system is based on a wireless architecture of sensors and actuators using WSN430 technology (https://www.iot-lab.info/). A WSN430 node is an electronic system which insures 3 functions: data acquisition using daughter board sensor-specific, data processing based on a micro-controller (Texas Instruments MSP430) and wireless radio- frequency communication (based on Texas Instruments CC1100). In our architecture, 3 types of WSN430 nodes are used: one sink node connected to a laptop via a serial port, one sensor node placed on the participants’ lower limbs, and one control node which triggers the stimulator as shown in Figure  4. This node allows communication with the network of sensor node and actuator node.

In the protocol, each subject walked along the GAITRite walkway. Each subject performed 3 successive trials for each of the 3 conditions: C1) No stimulation, C2) Stimulation triggered on the basis of heel switch information, C3) stimulation triggered on the basis of the GCI information. C2 and C3 were applied in a randomized order among the patients (see Table  1).

Condition C2 was used as a control condition for stimulation. In this preliminary validation work, the goal was to reproduce the behavior of heel switch triggering using our experimental observation algorithm. As the swing phase in normal human walking lasts 40% of the gait, the stimulation was triggered for 0 ≤ GCI ≤ 40% for almost all the subjects in C3 condition. The exceptions will be discussed here after.

Based on the GAITRite information, the data recorded during the trials were slip into strides.

• The validation of the GCI estimation algorithm was done on the C3 condition. To assess variability between strides, we computed the mean value of the cross-correlation between the GCI variable waveform over the strides of each patient. Similarly, we applied this to shank inclination waveform. To assess the validity of the GCI information, we computed the standard deviation of the GCI values corresponding to heel OFF and heel ON (heel strike) events extracted from GAITRite software.

• A t-test was carried out to compare the average walking speed in each condition (C1, C2, and C3).

In this study we used the observation of the unaffected leg shank angle to detect the heel strike moment instead of commonly used heel-floor contact. Strong positive correlation between unaffected leg shank inclinations demonstrates that the shank angle is a variable, which has similar repeatable shape during the gait of a person with drop foot, and could be potentially used to trigger the electrical stimulator.

In order to validate the relevance of the GCI information, we compared the GCI values computed in the C3 condition, corresponding to Heel OFF and Heel ON events extracted from GAITRite data for the affected leg (Figure  7). The aim was to determine the extent to which the GCI variable varies from one stride to the next (Table  3) and if a given GCI value could reliably represent a fixed gait event. Even though the oscillator parameters should ideally be identified on a reference trajectory from a gait event after the patient is stimulated and not on the C1 condition, as was done, the results show that the GCI variable was stable over the strides. The GCI information is more reliable when the standard deviation (Std) is low. The mean Std for all patients was 6.3% of the GCI for Heel OFF, which roughly corresponds (assuming that the GCI evolution is linear) to 0.13 s and 15% of GCI for Heel ON, which corresponds (assuming that the GCI evolution is linear) to 0.3 s. The maximum standard deviation was 22% (patient 11), which corresponds (assuming that the GCI evolution is linear) to 0.44 sec (half a step). From a mathematical point of view, the performances are good (<15%). From a functional point of view the corresponding error in seconds compared with the duration of gait cycle of a person with drop foot would need to be improved in order to allow for a fine triggering of the stimulation. These averaged performances will have to be improved in some patients in order to allow triggering a stimulator at any instant of the gait cycle. This should be possible by identifying the reference model parameters after an adaptation period of the patient to stimulation.

From the Table  3, we can observe low heel strike coefficient of variations for subject 7 and subject 8. Those subjects’ walk was more impaired compared to other subjects. This makes general comments difficult on the performance of GCI over all the subjects who have participated in this study.

Among all 12 subjects, 282 strides were analyzed for the trials in the C3 condition. In only 4% of the cases, the stimulation was not triggered in affected leg heel strike moment, as expected. That was due to problems with the wireless communication such as loss of wireless signal (Table  2). For patient 5 it was impossible to test the C2 condition. Indeed, we could not use a footswitch under the affected side heel neither the unaffected side toes. Indeed, the gait of this patient was too impaired and the foot strikes too lateral to make a direct link between the footswitch state and the stimulator ON/OFF status. However, it was possible to test the C3 condition for this same patient. In this case, stimulation was ON for GCI² ranging from 30% to 70%. Patients 1 to 4 and 7 to 12 were stimulated for GCI values ranging from 0% to 40%. Indeed, for these patients the swing phase duration was similar to normal gait. Patient 6 was stimulated for GCI values ranging from 0% to 50%.

The proposed approach allows selecting instants in between heel OFF and heel ON in order to appropriately trigger stimulation ON or OFF. This opens the possibility of exploring new strategies for stimulation like starting stimulation before heel OFF or applying biphasic stimulation in between heel OFF and heel ON. Compared to other approaches the optimization of oscillator parameters to define the reference model for each patient takes several minutes but can be an automatized procedure, which would be done once by the clinician. The use of this system by a patient is not more constraining than existing ones.