### Participants

Sixteen participants (7 females, 9 males, between 19 and 36 years, BMI: 24.58 ± 5.04 kg/m^{2}) with no history of neurological conditions and normal or corrected-to-normal vision took part in the experiments upon signing the informed consent form. The study was approved by the Institutional Review Board of Northeastern University in accordance with the Declaration of Helsinki (IRB# 18-01-19).

### Experimental apparatus

Participants stood on a narrow wooden beam (width 3.65 cm, height 7.62 cm) that was placed on the floor on top of a force plate (AMTI, Watertown, MA, USA, Fig. 1). They held two aluminum canes, one in each hand, to support themselves (length 117 cm, mass 680 g). The two canes were instrumented with a 6-DOF load cell at the bottom of each cane to measure the forces applied to the canes (MCW-500 Walker Sensors, AMTI Watertown, MA, USA). All force data were recorded at 500 Hz sampling rate. To record the participants’ movements in 3D, whole-body kinematics was recorded by 12 optical cameras at a sampling rate of 100 Hz (Qualisys, Göteborg, Sweden). Each participant was equipped with a standard biomechanical set of 43 reflective markers, following the C-Motion Plug-In Gait marker set. To track the orientation of the canes in 3D, 4 additional markers were attached to each cane.

### Experimental protocol

Participants were asked to stand barefoot in tandem stance on the narrow beam and maintain their gaze fixed to a point marked on the wall. They could choose which foot was at the front of their stance and they kept the same foot in front in all trials. For all experimental conditions, participants supported themselves with two canes, one held in each hand, their arms comfortably extended and their trunk kept upright (Fig. 1). Participants were asked to apply one of three levels of force on the canes: minimum (Min), i.e., as little as they could, preferred (Pref), i.e., as much as they liked, maximum (Max), i.e., as much as possible. They performed the same three force conditions in two arm configurations: their arms extended out horizontally in the frontal plane (Planar), and stretched out forward forming approximately a 45° angle at the shoulder with the frontal plane (Tripod), midway between the sagittal and the frontal planes. The planar configuration limited the base of support to a line orthogonal to the beam, while the tripod configuration enlarged the base of support to a triangular area. Standing on a narrow beam elicited instability around the vertical, mainly in the frontal plane. Therefore, we chose the planar configuration to counter such effects directly. The rationale for choosing straight arms was twofold: first, this simple joint configuration minimized differences across individuals and, second, it eliminated additional stiffness of the elbow and the wrist joints acting at the hand/cane junction.

In addition, two reference conditions were tested: in the first condition participants stood on the ground in the same tandem stance without holding canes (Off Beam–No Canes). This condition provided a reference to understand the instability induced by the beam. In the second control condition, participants stood on the beam without the cane support (On Beam–No Canes). In this difficult condition they were allowed to move their arms freely to help maintain balance on the beam. Comparisons with the main experimental conditions established a baseline for the mechanical challenges posed by standing on the beam and the supportive effects of the canes when standing on the beam (Hypothesis 0).

Each combination of force on the canes and arm configuration was repeated three times, grouped into two blocks. One block was performed with the planar cane placement, the second one with the tripod configuration. Each block presented 3 trials for each of the 3 force levels (Maximum, Minimum, Preferred). The arm configurations were grouped into two blocks to avoid too much disruption from changing cane placements. Prior to each block, participants performed the two reference conditions, first standing on the ground without canes (Off Beam–No Canes), then standing on the beam without canes (On Beam–No Canes). Each trial lasted 30 s; the entire recording session lasted approximately one hour, including the time to place the markers on the body.

### Data preprocessing

All analyses were carried out with custom software written in Matlab (The Mathworks Inc., Natick, MA, USA). All kinematic and kinetic data were filtered with a zero-lag, 3rd-order, low-pass Butterworth filter at 10 Hz (functions: *butter*, *filtfilt*). In order to exclude any familiarization or fatigue effects, data from the first 10% and last 10% of each trial were excluded from the analysis. The weight of the cane was subtracted from the vertical component of the force measured at the canes to estimate the effective force applied by participants. In the control condition where participants stood on the beam without cane support, they occasionally lost balance and stepped off the beam. These trials were excluded from the analysis.

As the feet of the participant were not in direct contact with the force plate but only with the beam, the center of pressure recorded by the force plate (Ground-CoP) was different from that resulting from the feet-beam interaction (Beam-CoP). The discrepancy was evaluated by the following equations

$${\text{Beam-CoP}}_{\mathrm{x}}={\text{Ground-CoP}}_{\mathrm{x}}+h\frac{{F}_{x}^{g}}{{F}_{z}^{g}},$$

(1)

$${\text{Beam-CoP}}_{y}={\text{Ground-CoP}}_{y}+h\frac{{F}_{y}^{g}}{{F}_{z}^{g}},$$

(2)

where *h* is the height of the beam and \({F}^{g}=\left[{F}_{x}^{g},{F}_{y}^{g},{F}_{z}^{g}\right]\) is the ground reaction force recorded by the force plate. The x-axis corresponded to the medio-lateral (ML) direction, the y-axis to the anterior–posterior (AP) direction, and the z-axis to the vertical direction, as illustrated in Fig. 1.

As \({F}_{z}^{g}>> {F}_{x,y}^{g}\), the additional terms on the right side of Eqs. (1) and (2) were negligible. Thus, in the following only the Ground-CoP was considered. For the sake of clarity, the CoP on the ground was referred to as the Feet-CoP.

When two canes touched the floor, the participant had three regions of contact with the ground: the feet on the beam, and the tips of the two canes. The feet were on the beam that was placed on the force platform, thus measuring the ground reaction force and the center of pressure. Information about the force applied on the canes was provided by the load cells at the tip of the canes. The center of pressure of each cane *cop* was computed by the ratio between the moments, \(m_{x}\) and \(m_{y}\), and the forces, \(f_{z}\), measured by the load cells

$$cop_{x} = \frac{{m_{y} }}{{f_{z} }},$$

(3)

$$cop_{y} = \frac{{m_{x} }}{{f_{z} }}.$$

(4)

The spatial positions of the tips of the canes were determined from the markers attached to the canes. With all variables in the laboratory coordinate frame, the total center of pressure (Total-CoP) was computed as the ratio of the total moments, \(M_{x,y}\), and the total force, \(F_{z}\). The total moments were defined as the sum of the product of the vertical force at each point of contact with the respective moment arm. The moment arm at each point was computed as the sum of the center of pressure with the relative position *a* \(=\left[{\mathrm{a}}_{\mathrm{x}},{\mathrm{a}}_{\mathrm{y}},0\right]\), which in turn is the vector from the origin of the coordinate frame to the point of contact. As it was desirable to compute the CoP in the medio-lateral (ML) and antero-posterior (AP) directions, the total moments in the AP and ML directions were determined, respectively, as shown in Eqs. (5) and (6)

$$M_{x} = \sum\limits_{i = 1}^{3} {(a_{y}^{i} + cop_{y}^{i} )} f_{z}^{i}$$

(5)

$$M_{y} = \sum\limits_{i = 1}^{3} {(a_{x}^{i} + cop_{x}^{i} )} f_{z}^{i} .$$

(6)

The index \(i\) indicates the current point of contact (i = 1: feet, i = 2: left cane, i = 3: right cane). Following the rule applied previously, the total CoP was determined as

$${\text{Total-CoP}}_{\mathrm{x}}=\frac{{M}_{y}}{{F}_{z}},$$

(7)

$${\text{Total-CoP}}_{\mathrm{y}}=\frac{{M}_{x}}{{F}_{z}},$$

(8)

where \(F_{z} = \sum\limits_{i = 1}^{3} {f_{z}^{i} }\).

For each participant a kinematic model of 15 rigid body segments (head, trunk, pelvis, left and right upper arms, forearms, hands, thighs, shanks and feet) was fit to the kinematic data using C-Motion Visual3D (Germantown, MD). The whole-body center of mass (CoM) was computed in Visual3D.

### Dependent measures

To obtain a metric for postural sway, the fluctuations of the CoP were summarized by the standard deviations of the CoP in two orthogonal directions, the anterior–posterior (AP) and the medio-lateral (ML) directions. These two directions were calculated separately because of the asymmetric constraints of the beam, i.e., the base of support in the AP direction was significantly larger than in the ML direction. Another measure of postural sway was defined as the area of the 95% tolerance ellipse. The same metrics were computed for both Feet-CoP and Total-CoP. Similarly, the fluctuations of the center of mass (CoM) were quantified by the area of the 95% tolerance ellipse. This area was calculated in the horizontal (x–y) plane to make it comparable to the areas of the CoPs. Note that movements in the vertical (z) direction were negligible. To quantify movements of the hand at the tip of the cane, the path length of the hand movement was calculated as the integral of the root mean squared sum of the derivatives of the x-, y- and z-components,

$${\text{path length}}={\int }_{start}^{end}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}+{\left(\frac{dz}{dt}\right)}^{2}}dt .$$

(9)

### Statistical analysis

A linear mixed model was used to evaluate the differences in the variability of the CoM and the CoP between the three levels of force (Min, Pref, Max) applied to the canes and the two cane placements (Planar, Tripod). The mixed model compared the experimental conditions (fixed effects), i.e., beam, force, and arm configuration conditions, which were consistent across participants, and accounted for the effects of normally distributed variability between participants (random effects). This linear model allowed the two control conditions (On Beam–No Canes and Off Beam–No Canes) to be included, even though they were not part of the balanced 3 (force levels) × 2 (cane placements) design. To identify the model that best fit each dependent variable, an iterative procedure was adopted to assess whether the inclusion of random effects was justified [18]. Then, according to the hypothesis testing method [19, 20], we iteratively compared different models that assessed whether it was necessary to include interaction terms or random-effect slopes. In Eq. (10), *B* is the beam condition (On Beam–No Canes or Off Beam–No Canes); *F* is the force condition (three levels: Min, Pref, Max), *C* is the arm configuration (two levels: Planar and Tripod), *Y* is the dependent variable for each participant *i* and each trial *j*. \(\beta\) are the fixed-effects coefficients, *S* are the random-effects coefficients from the participants,

$${Y}_{ij}={\beta }_{0}+{S}_{0i}+{\beta }_{b} {B}_{j} +\left({\beta }_{F}+{S}_{Fi}\right){F}_{j}+{\beta }_{c} {C}_{j} +{\epsilon }_{ij}.$$

(10)

To better compare the force conditions in which participants were standing on the beam with the canes on the ground, a second model (see Eq. 11) was tested on a subset of the data, excluding the trials of the two reference conditions,

$${Y}_{ij}={\beta }_{0}+{S}_{0i}+\left({\beta }_{F}+{S}_{Fi}\right){F}_{j}+{\beta }_{c} {C}_{j} +{\epsilon }_{ij}.$$

(11)

Additional multiple comparisons were conducted across experimental conditions by pairwise t-tests with Bonferroni corrections. The significance level was set to *p* = 0.05.

All statistical analyses were carried out in R, with packages *stats*, *lme4* and *lmerTest* [21].