In the LoS test, as a dynamic reaching test used in this study, eight targets are located at different angles with combinations of A–P (anterior–posterior) and M–L (medio–lateral) directions. The subject observes the targets and his COP in real-time as small circles on a display monitor. When a target is illuminated, the subject leans toward it, moving his COP to reach and hit the target, and then returns to the upright stance (Fig. 1).

A sample of dynamic reaching test (LoS). a The active target is in pink, which is supposed to be hit by the COP (the small red circle) by voluntary lean. When the target is hit, the color changes to blue. The COP trajectory is in green. b The corresponding posture movement towards the target (the subject has to lean toward the target using ankle strategy); c The experimental setup

The experimental setup mainly consists of a force plate, with the sampling frequency of 80 Hz, recording the participant's COP and a monitor for displaying the targets and the COP position (Fig. 1). The force plate used in this study was custom made and had a fair concurrent validity (ICC = 0.818–0.989) compared with Kistler force plate as the reference (Kistler group, Switzerland) [17]. Also, the positions of all targets, with respect to the central point, are determined at the beginning of the test. The targets are located at 30% of the subjects' theoretical limits of stability found from Eq. (1):

$$d=0.3\times _\times \mathit\left(\theta \right),$$

(1)

where d is the distance of the target from the central point, \(_\) is the height of the subject’s center of mass, and \(\theta\) is the theoretical limit of the leaning angle in each different directions. In the AP direction, the values for \(\theta\) are 6.25° (target 1 (Fig. 1a), 4.5° (targets 2 and 3), 1.7° (targets 4 and 5), and 4.4° (targets 7, 8, and 9). In the ML direction, these values are 0° (targets 1 and 8), 6° (targets 2, 3, 6, and 7), and 8° (targets 4 and 5) [18, 19].

The COP trajectories of the subjects while performing the tests were recorded. Before the commencement of the data collection, the procedure was explained to each participant. Then, they did the test once to familiarize themselves with the setup (no data was collected). Next, they performed and repeated the tests four times. At the beginning of the test, the subjects had to stay in the upright stance position for 5 s. This was used to find their baseline upright posture (central position of the COP) and calibrate the positions of the targets accordingly.

The clinical assessments of the PD patients included Unified Parkinson's Disease Rating Scale (UPDRS) [20] and Functional Reach Test (FRT) [21] and were performed by a clinician. The clinical results were used to compare and validate the control parameters.

A sample of the COP displacement during a test is shown in Fig. 2. The displacement signal is divided into five sections: (1) Preparation, (2) Planning, (3) Anticipatory Postural Adjustments (APA), (4) Reaching, and (5) Returning [22, 23]. ‘Preparation’ refers to the duration before the target turns on. The subject is expecting the target, but there is no action. When a target turns on, the CNS requires a time, known as reaction time (\(_\)), to initiate internal planning. Before moving toward the target, anticipatory postural adjustment (APA) occurs, which is an undershoot in the opposite direction of the target position. APA is commonly observed in voluntary movements. The CNS considers the act of reaching as a perturbation to the existing upright posture and counteracts this perturbation before the reaching movement starts by performing the APA [24]. Also, since APA is not based on any sensory information, the CNS controls the APA phase using an internal model [24]. As recommended in [22], a threshold-based algorithm was used to detect the onset of APA, with the threshold set at twice the standard deviation of the initial signal.

COP displacement and its segments. Preparation: the time before the target turns on; Planning: the time required for the CNS to plan for the movement; APA: a COP movement for counteracting the expected mechanical effects of the perturbations due to the reaching movement; Reaching: from the end of the APA until reaching the targets; Returning: from the end of the reaching until returning to the initial level. \(_, _, _,\mathrm _\) are reaction time, APA time, reaching time, and returning time, respectively, ‘APA size’ is the maximum displacement of the COP during the APA, and ‘Leanmax’ is the distance between the COP signal in the upright position and its maximum value after hitting the target

Reaction time, \(_\), was considered as the time between the onset of the target illumination and the start of the APA. The end of the APA was determined as the time the COP returned to the baseline [22]. Thus, the APA duration (\(_\)) is the time between the onset and the end of APA, and the APA size is the peak-to-peak value of the COP displacement in the APA phase (Fig. 2). As shown in the figure, Reaching time (\(_\)) is from when the APA starts until the COP displacement reaches the maximum value. Finally, Returning time (\(_\)) is the duration of the COP returning back from its maximum peak to the next small peak that emerges after the COP is settled around (or passed) its initial value.

The proposed model is shown in Fig. 3. The model consists of two main sections; ‘motor planning,’ which mimics the CNS in planning the task, and the ‘postural control’ section, which models the neuromechanics of the motion assuming only the usage of ankle strategy. After receiving the visual input as a step function (i.e., a target turns on), the CNS performs motion planning (motor planning section); according to the pertinent literature [25,26,27], the CNS constructs and uses an internal model for optimal planning of the reaching task, which typically has a bell-shaped velocity profile; the created path is the input of the second section of the model, i.e. the postural control section [28]. In the postural control section, the task is performed based on the CNS commands and visual feedback. The feedforward controller was added to account for producing the anticipatory postural adjustment (APA) [29, 30].

The proposed model for the dynamic reaching test. 'Visual input' is a step function which indicates the target appearance; 'Motor planning' is the trajectory planning section of the task; 'Postural control' is a feedback/feedforward control model for performing the motion commanded as \(\theta _}\) from motor planning module, completing the whole task of reach (hitting the target).”

To the best of our knowledge, there is no investigation on motion planning in lower-limb dynamic reaching tests. In upper extremities reaching tests, however, there are several works which proposed a bell-shaped velocity profile based on the motion optimality hypothesis [31,32,33]. These studies used a polynomial expression as a model for the movement trajectory [25, 31, 34]. A similar formulation was used here for the body lean angle (Eq. 2):

$$_=_+\left(_-_\right)\left(_^-_^-_^\right), \tau =\frac_}_},$$

(2)

where \(_\) is the reference angle of the body to move toward the target, \(_\) and \(_\) are the angle of the body at the initial and target points, respectively; \(_\) and \(_\) are reaction time and reaching time, respectively; \(_, _, } _\) are constants that determine the geometry of the bell-shaped pattern taken from [25].

The postural control section, as shown in Fig. 3, is an extension of the models previously proposed for static tests [12, 14]. It consists of two major control paths, feedforward (FF) and a time-dependent summation block of feedback (FB) controllers.

Most of the studies on APA found that the CNS uses a feedforward controller for this phase of the movement [24, 30, 35]. Even the APA in the gait initiation is shown to be a feedforward control [29]. Therefore, the feedforward controller, in the proposed model, is considered for the APA section, and the remaining is feedback controlled. It is assumed that the CNS shifts smoothly from a feedforward to feedback control as the reaching task progresses. As a result, a time-dependent transition is proposed in this study. As shown in Fig. 3, a summation block, in which the weights of the three control signals are regulated through coefficient \(\alpha\), is added to the control model and defined as follows (Eq. 3):

$$\left\c} < t < t_ + t_} } & } \times \alpha + \tau _} \left( \right) + \tau _} } \\ + t_} } & } + \tau _} } \\ \end } \right.,$$

(3)

where, \(t\), \(_\), and \(_\) are time, reaction time, and the duration of the APA phase, respectively. Also \(_\),\(_\), and \(_\) are the output torques provided by feedback, feedforward, and ankle intrinsic biomechanics, respectively. The latter refers to the intrinsic torque provided by the stiffness and damping of the ankle joint, and their values are taken from the literature [12]. \(\tau\) is the total torque; \(\alpha\) is a time-varying parameter from 0 (at the beginning of the APA) to 1 (at the end of APA). The feedback controller is a PI controller (Eq. 4):

$$_=_\left(_-\theta \right)+_\left(}_-\dot\right),$$

(4)

with \(_\) is from Eq. (2), KI\(=5 N.m/deg/s\) [12, 14], and \(_\) was determined subject-specifically. The feedforward controller is an inverse model of the system (inverted pendulum) with \(_\) gain to be determined for each subject (Eq. 5):

$$_=_\left(_^-mg_\right),$$

(5)

where \(_\), m, and \(_\) are moment of inertia, mass, and height of COM of the subject, respectively, and \(g=9.81\frac^}\) is the gravitational acceleration.

There are two unknown parameters, \(_\) (proportional gain in the feedback controller) and \(_\) (feedforward gain), in the postural control section that were identified subject-specifically. These parameters were identified based on two COP parameters, Leanmax and APA size, extracted from the experimental data (Fig. 2). In other words, \(_ \, \text \, _\) values were identified such that the error between Leanmax and APA size of the model and experiment were minimized at the same time (Eq. 6).

$$}\left\\left|Lea_}(model)}-Lea_}(experiment)}\right|\\ \left|APA siz_-APA siz_\right|\end\right..$$

(6)

Also, as it is shown in Fig. 4, the variation of APA size and Leanmax values are smooth and there is no concern about local minima.

The effects of KFF and KP on the COP. a Leanmax, b APA size

Twenty-four PD patients (\(61.5\pm 9.6 })\), with level of 1 to 3 according to the Hoehn and Yahr scale (\(2.37 \pm 0.74)\) and disease duration of \(8.9 \pm 5.4 }\), and 24 healthy subjects (\(54.9\pm 7.4 })\) participated in this study (Table 1). The data for PD patients, diagnosed by a neurologist, was taken from a previous study in the same research group [18]. Healthy subjects had no previous balance disorders. Clinical assessments (UPDRS and FRT in Table 1) were done for PD patients to assess their balance function. The test protocol was approved by the ethics committee of the Iran University of Medical Science (IR.IUMS.REC.1400.826). All participants provided written confirmed consent according to the Declaration of Helsinki.

COP data collected from each subject was used to identify a subject-specific set of model parameters. For this purpose, each trial was first divided into eight sections (i.e. directions of targets). Then, Reaction time (\(_\)), Reaching time (\(_\)), Return time (\(_\)), Leanmax and APA size (Fig. 2), and APA duration (\(_\)) were extracted for each direction. The averages of these parameters for each direction over four trials were used to identify the model parameters. Reaction time, Reaching time, and Return time were used to identify the motor planning parameters. Leanmax and APA size were used for the identification of the control gains (\(_\) and \(_\)) as explained in "The proposed model for dynamic reaching". Finally, the independent t-test was used to compare the control parameters in each direction between the groups of healthy subjects and PD patients. The significance level was set to 0.05.