Rolling balance board of adjustable geometry as a tool to assess balancing skill and to estimate reaction time delay

Abstract

The relation between balancing performance and reaction time is investigated for human subjects balancing on rolling balance board of adjustable physical parameters: adjustable rolling radius R and adjustable board elevation h. A well-defined measure of balancing performance is whether a subject can or cannot balance on balance board with a given geometry (R, h). The balancing ability is linked to the stabilizability of the underlying two-degree-of-freedom mechanical model subject to a delayed proportional–derivative feedback control. Although different sensory perceptions involve different reaction times at different hierarchical feedback loops, their effect is modelled as a single lumped reaction time delay. Stabilizability is investigated in terms of the time delay in the mechanical model: if the delay is larger than a critical value (critical delay), then no stabilizing feedback control exists. Series of balancing trials by 15 human subjects show that it is more difficult to balance on balance board configuration associated with smaller critical delay, than on balance boards associated with larger critical delay. Experiments verify the feature of the mechanical model that a change in the rolling radius R results in larger change in the difficulty of the task than the same change in the board elevation h does. The rolling balance board characterized by the two well-defined parameters R and h can therefore be a useful device to assess human balancing skill and to estimate the corresponding lumped reaction time delay.

1. Introduction

Research on human balancing is getting more and more focus in the last years due to the increasing number of accidents related to the loss of balance mainly among the elderly. Impaired medical conditions, inactive lifestyle, foot problems, attenuated vision and increased reaction time all increase the risk of falling [1–3]. One of the most common questions focuses on the mathematical modelling of the operation of the central nervous system (CNS) during different balancing tasks. Several perspectives exist in the available literature to model the CNS, such as delayed proportional–derivative (PD) feedback [4–6], delayed proportional–derivative–acceleration feedback [7–9], intermittent control [10–15] and predictor feedback [4,16], just to mention a few. Most of the above works analyse the operation of the CNS by comparing a simple balancing task performed by human subjects with dynamic properties of the underlying mechanical model.

Since the model of the human body and the operation of the CNS involve many uncertain parameters, it is advantageous to consider tasks associated with low-degree-of-freedom mechanical models with well-defined physical properties. For instance, stick balancing [4,6,11,16], ankle strategy during quiet standing [5,9,15,17,18] and ball and beam balancing [19] are often modelled as a single inverted pendulum system, while ankle–hip strategy during quiet standing [20,21] and standing on a balance board [22–24] are modelled as a generalized double inverted pendulum.

Balancing ability of human beings is related to the adaptability and flexibility of the muscles and the tendons, to the performance of the sensory systems (visual, vestibular and somatosensory system), and to the reaction time. These properties may be involved in the mechanical model as passive stiffness and/or damping at the joints [10,13], robustness of the control gains [4,5] and feedback delay in the control loop [6,16]. Comparison of experiments with the corresponding mathematical model and analysis of parameter changes may help to identify the type of the control concept applied during balancing and to estimate the associated feedback delay and the control gains.

A key limitation to balancing abilities is set by the reaction time delay. This limitation can well be demonstrated on the example of stick balancing on the fingertip [6,16]. Balancing a short stick is more challenging than balancing a longer one. This is because shorter sticks fall faster than the time needed to make a corrective motion. Even expert stick balancer subjects cannot balance a stick of length shorter than 20 cm [16]. In control system theory, it is known that a stick of length ℓ cannot be stabilized in its unstable upward position by delayed PD feedback if the feedback delay is larger than the critical delay

$${\tau }_{\mathrm{crit},\mathrm{PD}}=\sqrt{\frac{4\ell }{3g}}\hspace{0.17em},$$ 1.1

where g is the gravitational acceleration [6,16,25]. The delay-dependent stabilizability of feedback systems well demonstrates the negative effect of reaction delay on balancing skill. In the case of quiet standing, reaction time delay does not set such a strong stabilizability condition because the contribution by passive ankle stiffness strongly increases the time constant of the upright stance dynamics [26,27]. One technique to provoke loss of balance during upright stance is to ask subjects to stand on an unstable platform, e.g. on spring-supported moving platform [28,29], on skateboard [30] or on pinned [22,23,31] or rolling balance board [32,33].

Balance board experiments for pinned balance board with a fixed geometry have been performed in the literature [22–24,34]. The concept in our paper is that human subjects are asked to stand quietly on a balance board rolling in the sagittal plane. The geometry of the balance board (rolling radius, board elevation) can be adjusted such that the balancing task becomes more difficult or even impossible. The corresponding mechanical model involves the well-defined exact geometry and inertia of the balance board, a simplified single inverted pendulum model of the human subject and a delayed PD model of the operation of the CNS. The mechanical model allows to calculate the theoretical critical delay: if the feedback delay in the control loop is larger than the critical delay, then the system cannot be stabilized by any combination of the control gains. The main hypothesis of the paper is that loss of balance of human subjects during balancing trials on the balance board with different geometry can directly be linked to the critical time delay of the underlying mechanical model. This way the reaction delay of human subjects can be estimated based on their efficiency in balancing on different balance boards.

2. Methods

Rolling balance board with adjustable geometry is used to analyse human balancing ability under different levels of open-loop instability. Stabilizability is used as a measure to assess reaction time delay of human subjects based on a combined mechanical model of the human subject, the balance board and the control action performed by the CNS.

2.1. Balance board

The uniaxial rolling balance board used during the experiments consists of two rolling wheels, which are connected to a board by four fixing screws as shown in figure 1. The elements were made from plywood of thickness 21 mm and density 700 kg m⁻³. The geometry of the balance board was designed to be adjustable to realize dynamic balance conditions of different levels of difficulty. The wheels are available with different radii (R = 50, 75, 100, 125, 150, 200, 250 mm) and the elevation h of the board measured from the ground can be changed in steps of 25 mm in the case of each wheel. Preliminary measurements showed that changing the wheel radius R and/or the board elevation h has a great influence on the difficulty of the task: standing on the balance board is more challenging in the case of small wheel radius R and/or large board elevation h [32,33].

Figure 1.

Uniaxial rolling balance board with adjustable geometry. The adjustable parameters are the radius R of the wheels and elevation h between the ground and the board.

2.2. Mechanical model

Human balancing on a rolling balance board in the sagittal plane was modelled by a two-degree-of-freedom rigid body system as shown in figure 2. The human body was considered as a single inverted pendulum of mass m_h, height l and mass moment of inertia $I_{\mathrm{h}}=\frac{1}{12}{m}_{\mathrm{h}}{l}^2$ for the centre of gravity. The ankles were modelled as a joint between the single inverted pendulum and the balance board. The passive stiffness of the ankle joint was considered as a torsional spring of stiffness s_t according to [26] as

$$s_{\mathrm{t}}=0.91m_{\mathrm{h}}g\frac{l}{2}.$$ 2.1

The control torque T(t) acts at the ankle. Although there are evidences that passive damping contributes to the stabilization during balancing [35], experimental estimation of the damping parameter is more uncertain than that of the passive stiffness and it also changes with age [17,36]. Also, for a simple inverted pendulum model subject to delayed PD feedback, the effect of passive damping on the critical delay is negligible [37]. Therefore, in this model, passive damping at the ankle joint is neglected.

Figure 2.

Two-degree-of-freedom mechanical model of human balancing on uniaxial rolling balance board in the sagittal plane.

The generalized coordinates describing the motion of the two-degree-of-freedom model are the angle φ of the human body measured from the vertical and the angle $\vartheta$ of the balance board measured from the horizontal line as shown in figure 2. The position of the ankle joint is characterized by the horizontal distance e from the symmetry axis of the balance board and by the vertical distance f from the board. The centre of gravity l_b, mass m_b and mass moment of inertia I_b of the balance board were calculated based on the actual configuration of the balance board in terms of the two adjustable parameters R and h.

The governing equations were derived using Lagrange’s equation of the second kind and were linearized about the upper (unstable) equilibrium. The equation can be written as

$$\mathbf{M}\ddot{\mathbf{q}}(t)+\mathbf{S}\mathbf{q}(t)=\mathbf{Q}(t),$$ 2.2

where M, S and Q(t) are the mass, the stiffness matrices and the vector of generalized forces, respectively. The vector of generalized coordinates is

$$\mathbf{q}(t)=\left[\begin{array}{c}\phi (t)\\ \vartheta (t)\end{array}\right].$$ 2.3

2.3. Control concept and reaction time delay

Many different control concepts are used to model the behaviour of the CNS as was mentioned in the Introduction. They all have a common feature: time delay is involved in the feedback in order to model the reaction time of human beings. The position and orientation of the human body are perceived by the receptors of visual, vestibular and somatosensory systems. These signals are delivered to the CNS, which, after processing the information, determines the necessary control action to be transmitted to the musculature to maintain balance. The process requires certain amount of time (called reaction time) which presents a feedback delay in the control loop.

The operation of the CNS is modelled as delayed feedback of the available sensory information. It is assumed that the angles φ, $\vartheta$ and their derivatives $\dot{\phi }$, $\dot{\vartheta }$ (i.e. angular velocities) are used for full state feedback. The corresponding control torque reads

$$T(t)=\mathbf{P}\mathbf{q}(t-\tau )+\mathbf{D}\dot{\mathbf{q}}(t-\tau ),$$ 2.4

where

$$\mathbf{P}=[P_{\phi }{P}_{\vartheta }],\mathbf{D}=[D_{\phi }{D}_{\vartheta }]$$ 2.5

are the matrices of proportional and derivative control gains, respectively. This gives

$$T(t)=P_{\phi }\phi (t-\tau )+D_{\phi }\dot{\phi }(t-\tau )+P_{\vartheta }\vartheta (t-\tau )+D_{\vartheta }\dot{\vartheta }(t-\tau ).$$ 2.6

The vector of generalized forces reads

$$\mathbf{Q}(t)=\mathbf{B}T(t),$$ 2.7

where

$$\mathbf{B}=\left[\begin{array}{c}1\\ -1\end{array}\right].$$ 2.8

Thus, the governing equation of the closed-loop system can be written in the form

$$\mathbf{M}\ddot{\mathbf{q}}(t)+\mathbf{S}\mathbf{q}(t)=\mathbf{B}\mathbf{P}\mathbf{q}(t-\tau )+\mathbf{B}\mathbf{D}\dot{\mathbf{q}}(t-\tau ),$$ 2.9

which is a delay-differential equation.

Note that the control model in (2.9) can be considered as a kind of ‘simplified lumped model’. More sophisticated and more realistic control models exist in the literature involving hierarchical levels of control loops associated with different feedback delays [38,39]. Also, different sensory information requires different time duration to be delivered and to be processed, which supports the existence of multiple, combined or distributed time delays in the feedback loop. However, identification of different loop delays in a hierarchical control structure is circuitous. Therefore, the latency in the feedback loop is modelled as a single lumped delay denoted by τ in (2.9). In quiet standing, the lumped reaction time delay is typically in the range of 90–160 ms for healthy subjects [17,40–42] and may be increased up to 180–250 ms for the elderly [43]. In the case of balancing on a balance board, higher reaction times are expected due to the complexity of the task. For balancing on a pinned balance board, the delay range of 100–300 ms was analysed in [22] and in [44].

2.4. Stability analysis

The characteristic equation (see [45]) of (2.9) can be written in the form

$$D(s)=A(s)+B(s)\hspace{0.17em}{\mathrm{e}}^{-s\tau },$$ 2.10

where

$$A(s)=s^4+a_3{s}^3+a_2{s}^2+a_{1}s+a_0$$ 2.11

and

$$B(s)=b_3{s}^3+b_2{s}^2+b_{1}s+b_0.$$ 2.12

The coefficients of polynomial A(s) are determined by the physical parameters of the mechanical model and the control gains are involved in the coefficients of B(s). Since there is no passive damping in the system, a₁ = a₃ = 0. The system is stable if all the characteristic exponents of (2.10) are located on the left half of the complex plane [45].

Time delay in feedback systems typically has a destabilizing effect, which can be visualized by stability diagrams. For a fixed delay, the domain of stabilizing control gains can be determined by standard tools, such as the D-subdivision method [45] or by numerical techniques, e.g. the semi-discretization method [25]. Constructing series of stability diagrams in the space of the four control gains is more economical using numerical techniques than analytical methods. Figure 3 shows an example for a representation of the four-dimensional stability diagram such that two-dimensional diagrams in the plane $(P_{\vartheta },D_{\vartheta })$ are arranged in a table form for different P_φ and D_φ values. The individual subplots in the plane $(P_{\vartheta },D_{\vartheta })$ are actually two-dimensional projections of the four-dimensional stability diagram.

Figure 3.

Stability diagrams for m_h = 57 kg, l = 160 cm, e = 12 cm, f = 7 cm, R = 125 mm, h = 75 mm, τ = 150 ms. Stable domains are indicated by grey shading. The numerical values of $P_{\phi },D_{\phi },P_{\vartheta }$ and $D_{\vartheta }$ are given in SI base units.

2.5. Stabilizability and critical feedback delay

If the lumped delay τ in (2.9) is increased, then the size of the domain of stabilizing control gains typically decreases. If the delay exceeds a critical value, called critical time delay, τ_crit, then no stabilizing control gains can be found. The critical feedback delay can be related to the reaction time of human beings: if the reaction time is larger than the critical delay, then balancing is not possible due to the lack of stabilizing control gains. Conversely, should the reaction time be smaller than the critical delay, CNS might be able to find the proper control gains to stabilize the system around the unstable equilibrium. The critical time delay is affected by the mechanical properties of the system, thus by the actual configuration of the balance board, by the wheel radius R and by the elevation h of the board.

Stabilizability of the control system in terms of the feedback delay can be assessed by analytical techniques such as by the investigation of the D-curves [6,25], by using the multiplicity-induced dominance of the characteristic roots [33,46], by the Walton–Marshall method [47], or by numerical techniques such as the semi-discretization [25] combined with the multi-dimensional bisection method [48]. Here, the critical delay τ_crit and the associated critical control gains P_{φ ,cr}, $P_{\vartheta ,\mathrm{cr}}$, D_φ,cr and $D_{\vartheta ,\mathrm{cr}}$ are determined numerically using the semi-discretization method along a brute-force scheme. For this analysis, a more practical definition of critical delay is used, which takes into account the robustness of the controller to noise and to sensory uncertainties. Since the perceived sensory variables are multiplied with the control gains as shown by (2.6), sensory uncertainties can directly be linked to uncertainties in the control gains.

Here, we pose a practical robust stability concept. Namely, the system associated with the quadruplet ($P_{\phi },P_{\vartheta },D_{\phi },D_{\vartheta }$) is said to be robustly stable if the system remains stable even for $\pm 5\text{\%}$ perturbation of the control gains. This robust stability concept can easily be evaluated when numerical algorithms are used with ±5% relative resolution of the parameters space. Perturbation of all the four control gains gives 3⁴ − 1 = 80 different quadruplets of the control gains. These quadruplets can also be pictured as the points of a 3 × 3 × 3 × 3-sized cube in the four-dimensional parameter space. If all of these 80 quadruplets gives a stabilizing feedback control then the original quadruplet is said to be robust to ±5% perturbation of the control gains. However, due to the finite resolution of the parameter space, this concept may be too conservative. It can easily happen that only some portion of the 80 neighbouring quadruplets give a stable closed-loop system, but the neighbouring quadruplets of a slightly shifted parameter point are all stable. This phenomenon is demonstrated in figure 4 for a two-dimensional parameter space. Stable region in the plane (P, D) is indicated by grey shading. Red and green dots indicate unstable and stable control gain pairs according to the unequally spaced resolution. The investigated point is indicated by A. It can be seen that only 7 out of the 3² − 1 = 8 neighbouring points of A are stable (green) and one of them is out of the stable region (red). In this sense, parameter point A is not completely robust since there is a combination of perturbations that gives an unstable closed-loop system. However, if the original grid is shifted a bit and the investigated point is B, then it can be seen that all the 8 neighbouring points (blue dots) are within the stable region. In order to account for this effect caused by the discretization of the parameter space, the following condition of robust stability was specified. A quadruplet ($P_{\phi },P_{\vartheta },D_{\phi },D_{\vartheta }$) is called robust stable if 70 out of its 80 neighbouring quadruplets are stable. This robust stability condition can easily be checked by investigating the stability of all the neighbouring parameter quadruplets.

Figure 4.

Demonstration of the robustness concept based on the stability of the neighbouring parameter points.

The critical delay is specified as the delay for which there exist a robustly stabilizing quadruplet of control gains in the above sense, but for larger delay, no robustly stabilizing quadruplet exists.

The main steps of the algorithm are as follows.

• 1. The parameters R and h are fixed.
• 2. The feedback delay is set to a starting value τ = 20 ms.
• 3.
  The four-dimensional space of the control gains is discretized with an unequal spacing such that

  $$q_n={(1+p)}^n{q}_{\min },n=0,1,\dots ,89,$$ 2.13

  with $q_n=P_{\phi },P_{\vartheta },D_{\phi },D_{\vartheta }$, with starting values P_φ,min = 200, $P_{\vartheta ,\min }=400$, D_φ,min = 10, $D_{\vartheta ,\min }=80$. The case p = 0.05 gives the intended ±5% spacing of the resolution. This gives a 90 × 90 × 90 × 90 grid of the parameter space with 90⁴ = 65 610 000 parameter points. This parameter grid was found to be large enough to cover the domain of stabilizing control gains for different τ, R and h values in each step.
• 4. Stability for each parameter point was determined by the semi-discretization method with discretization time step Δt = 10 ms and the delay was written as τ = rΔt where the delay resolution r was varied according to the investigated delay. The set of stabilizing control gains was stored.
• 5. For each stabilizing quadruplets of gains, it was checked whether the neighbouring quadruplets obtained by $\pm 5\text{\%}$ perturbation are stable or not. If at least 70 out of the 80 neighbouring quadruplets were stable, then the original quadruplet was assessed as robustly stable.
• 6. If there was at least one robustly stable quadruplet for a given delay, then the system is assessed as robustly stabilizable and the delay was increased by Δt = 10 ms and the process was started from point 4 again.
• 7. The critical delay was the one for which there was a robustly stable quadruplet, but for a 10 ms larger delay, there was no such quadruplet.
• 8. The critical delay was determined according to points 4–7 for all the possible sets of the parameters R and h.

2.6. Experimental procedure

Experimental sessions consisted of a reaction time test and balancing trials with balance boards. Standing on the balance board strongly resorts to the muscles at the ankle and therefore results in fatigue and debility, which may cause increasing reaction time. Therefore, the first task of the subjects was to execute the reaction time test then they started the balancing trials on the balance board with different geometry.

2.6.1. Participants

Fifteen young healthy subjects were involved in the measurement (nine males, six females: 23.56 ± 1.26 years, 69.70 ± 14.28 kg, 174.46 ± 8.01 cm, BMI: 22.82 ± 4.16). Participants did not report any known visual or balance pathology and had not taken part in any experiments involving a balance board before. The research was performed following the principles of the Declaration of Helsinki. All participants provided informed consent for all research testing and were given the opportunity to withdraw from the study at any time.

2.6.2. Reaction time test

Since balancing on the balance board is performed by corrective actions at the ankle joints (subjects were asked to hold their hip and knee rigid), the reaction time was measured between a visual input and a mechanical action at the foot. For this purpose, a complex reaction time tester (CRTT) developed by [49] was used. CRTT consists of a programmable set of LED lights and speakers for visual and auditory inputs and a set of buttons and pedals to detect the subjects’ reactions. During the reaction time test, subjects were standing and leaning against a table, as shown in figure 5a. CRTT was set to provide 10 randomly timed light signals and subjects were asked to push a pedal with their dominant foot as fast as they can as response for each light flash. The reaction time τ_CRTT was estimated as the elapsed time between the light flash and the pedal push.

Figure 5.

(a) Reaction time test. (b) Experimental setup of the balancing trials. Reflective markers were placed on the body segments and the balance board. OptiTrack motion capture system can be seen in the background.

Note that reaction time is defined as the time between the onset of the stimulus (light flash) and the initiation of motor response (onset of pedal push) while CRTT measures the time between the light flash and the completion of pedal push. Hence, τ_CRTT is rather a response time, i.e. the reaction time plus the movement time [50]. In order to estimate the movement time, CRTT test was performed with markers placed on the foot that were recorded by the OptiTrack motion capture system as described in §2.6.3. This test was performed by subject S9. This way, the time duration of the pedal push was measured as the time between the onset of movement (change in the velocity of the foot) and the maximum displacement (≈7.6 mm). The movement time was measured to be on average ≈110 ms for subject S9.

2.6.3. Balancing trials

The second task was to perform balancing trials in the sagittal plane starting on the balance board with the greatest available radius R = 250 mm. During the balancing tests, the feet were placed on the board such that the middle of the feet was approximately aligned to the symmetry axis of the balance board. This way the control effort can be exerted in both $+\vartheta$ and $-\vartheta$ directions. First, the subject was asked to balance on the lowest board position h = 50 mm. If the trial was successful, the subject had to balance on the balance board having the same wheel radius, but the board was set to the highest position. Then the wheel radius was decreased by one size and the subject was asked to balance on the lowest, then on the highest board positions again. The process terminated when balancing was not successful neither on the lowest nor on the highest board position. In order to avoid the effect of multiple-try learning process, the subjects had only two attempts to perform a successful trial on each balance board setup.

A balancing trial was assessed to be successful if the subject were able to stand at least 60 s long on the balance board so that the edges of the board did not touch the ground. In order to standardize the trials, the following experimental protocol was introduced. Subjects were asked to stand on the balance board in shoulder width spread with open eyes, stretched legs, hands behind the back, straight-held trunk and head as shown in figure 5b. They were instructed to not raise their sole from the board, to not bend at the hip and to clasp the hands behind the back.

OptiTrack motion capture system and Motive software were used to record the trials with 8 Prime 13 cameras. The sampling frequency was 120 Hz satisfying the Shannon sampling theory. Reflective markers were placed on the shoulder, the hip, the knee and three markers on the balance board as shown in figure 5b. A wooden handrail was set next to the participants in order to provide a support while standing up on the balance board and also in order to prevent potential falls during balancing. Capturing of the trials started after the participants released the handrail and took on the requested posture. Motive software was used to acquire the spatial position of each marker in .csv file format.

Balancing trials were processed by a self-developed script in Matlab environment. The reference coordinate system of the Motive software was adjusted such that the sagittal plane of the human body was parallel to the x–y plane, therefore only x and y coordinates of the markers were needed to analyse the balancing trials. Time history of the human body tilt angle φ measured from the vertical was determined using the line connecting the markers placed on the knee (P3) and the shoulder (P1) by simple trigonometrical functions. Time history of the board tilt angle $\vartheta$ measured from the horizontal line was calculated based on the trajectory of markers B1 and B3, similarly. The measured time histories of φ and $\vartheta$ were shifted such that their means were zero.

3. Results

The result of the stabilizability analysis based on the mechanical model is evaluated first. Then, the experimental results are presented for the reaction time test and for the estimation of feedback delay associated with the balancing trials.

3.1. Stabilizability diagram

Preliminary calculations showed that changes in the human parameters have negligible effect on the critical delay. It was found that $\pm 20\text{\%}$ change in the human parameters resulted in less than ±10 ms change in the critical delay. Therefore, for the numerical calculations, averaged human parameters were used, which are listed in table 1. Parameters e and f related to the ankle position were chosen based on the data of the participants. The mechanical properties of the balance board (l_b, m_b, I_b) were calculated based on the actual set of R and h.

Table 1.

Parameters of the mechanical model used for the estimation of the critical delay.

parameter	notation	value
mass of human body	mh	70 kg
height of human body	l	170 cm
ankle position (vertical)	e	12 cm
ankle position (horizontal)	f	7 cm
gravitational acceleration	g	9.81 m s−2

In order to investigate the effect of both the wheel radius R and the board elevation h, the critical delay was determined for the lower (h = 50 mm) and the upper (h = R) board positions for a range of wheel radius between R_min = 50 mm and R_max = 250 mm and also for a middle board position (h = (50 + R)/2) for wheel radii larger than 100 mm (see the black dots in figure 6). Note that the parameter points h = (50 + R)/2 do not necessarily coincide with the actual board setting, where h was set in steps of 25 mm. Balancing trials were performed at the lower (h = 50 mm) and the upper (h = R) board positions (see §3.2).

Figure 6.

Stabilizability diagram. Investigated (R, h) pairs are indicated by black dots. The critical delay is shown by grey scale.

The corresponding stabilizability diagram is shown in figure 6, where the grey scale shows the critical feedback delay as function of the wheel radius R and the board elevation h. As can be seen, the largest critical delay is obtained for the lower board position (h = 50 mm) with the greatest wheel radius (R = 250 mm). The critical delay decreases with decreasing wheel radius and with increasing board elevation. The smallest critical time delay is obtained for the configuration with R = 50 mm and h = 50 mm. The main hypothesis in this paper is that the smaller the critical delay the more difficult the balancing task. Blue line indicates the contour line associated with τ_crit = 200 ms. According to the mechanical model, a subject with 200 ms reaction time delay cannot balance on a balance board with (R, h) pairs located to the left of the blue separation line.

3.2. Balancing trials

An example of experimental results for subject S7 can be seen in figure 7. Green and red markers indicate successful and unsuccessful balancing trials. Time histories of φ and $\vartheta$ are shown for balancing trials on different balance board configurations. The variation of the tilt angles is larger for more difficult balancing tasks. Amplitudes are the smallest in the case of the lowest elevation (h = 50 mm) with the largest wheel (R = 250 mm). Increasing the board elevation or decreasing the radius result in larger variation of φ and $\vartheta$. The subject was able to balance on the balance board with R = 100 mm and h = 100 mm only for 6 s, i.e. this balancing trial was unsuccessful.

Figure 7.

Experimental results for subject S7 with some sample time histories for different balance board configurations. Limit of balancing ability (stabilizability) is between the successful (green dots) and unsuccessful (red dots) trials indicated by blue line.

The reaction time delay is estimated based on the successful and unsuccessful balancing trials. Subject S7 was able to balance successfully on the balance board configurations (R, h) = (100, 50) and (125, 125) and could not balance on the balance board with (R, h) = (75, 50) and (100, 100) (see the green and red markers with black edge, respectively). Hence, the limit of stabilizability, indicated by blue line, is somewhere between these four points. The estimated reaction delay τ_e is assessed as the average of the critical delays associated with these four (R, h) pairs, which gives τ_{e, 7} = 138 ms.

The results of the balancing trials are shown in figure 8 for all the 15 participants. Again, green and red markers denote (R, h) pairs where the balancing trial was successful and unsuccessful. The smallest wheel radius associated with stable balancing is referred to as upper or lower critical radius, depending whether the board is in the upper or in the lower position. All subjects were able to balance successfully on the balance board of R = 150 mm with both lower and upper board elevation. One third of the subjects (S1, S4, S5, S13, S15) showed similar balancing performance independently of the board elevation: they balance successfully on the boards with R = 100 mm with both the lower and upper board elevation. For them, upper and lower critical radius are both 100 mm. However, for the majority of the participants, the lower and upper critical radii were different as can be seen in figure 8. For all of them, the lower critical radius was larger than the upper one. Thus, the experiments confirm the hypothesis that it is more difficult to balance on balance board configuration associated with smaller theoretical critical delay than on balance boards associated with larger critical delay. The tendency of the difficulty of the task follows that of the mechanical model. For a fixed h, the smaller R, the more difficult the balancing task. For a fixed R, the larger h, the more difficult the balancing task.

Figure 8.

Theoretical and experimental stabilizability diagrams. The estimated reaction time delays τ_e are assigned by the blue separation lines.

3.3. Estimation of the reaction time

The reaction time of the subjects while balancing on balance board is estimated by comparing the stabilizability diagram and the experimental results. The estimated reaction time τ_e is calculated as the average of the four critical delays associated with the parameter points (R, h) next to the line which separates successful and unsuccessful balancing trials (see markers with black edge next to the blue separation line in figure 7). The subjects can be divided into five groups based on the distribution pattern of the successful and unsuccessful trials, therefore the estimated reaction times take five different values.

• 1. The reaction time of the first group (S1, S4, S5, S13, S15) is estimated to be τ_e ≈ 115 ms. For them, the lower and the upper critical radius are the same.
• 2. The second group is formed by S3, S7 and S14. The critical upper and lower radius of these subjects are 100 and 125 mm, respectively. Hence, their reaction time is estimated to be τ_e ≈ 138 ms.
• 3. Subjects S2, S6, S8 and S9 have the same lower critical radius as the subjects in the second group; however, they were not able to balance successfully on the balance board with R = 125 mm with upper board elevation. The corresponding reaction time is estimated to be τ_e ≈ 150 ms.
• 4. The reaction time of the fourth group (S10, S11) is estimated by τ_e ≈ 163 ms in a similar way.
• 5. S12 was the only subject who was able to balance on a balance board with radius R = 75 mm. The corresponding reaction time was the smallest one, namely τ_e ≈ 103 ms.

Based on balancing performance with upper and lower board elevation, an upper and a lower estimation for the reaction time delay can be given. These limits are shown by the red error bars in figure 9a. The maximum of the error bar corresponds to the larger critical delay associated with the two neighbouring green markers and the minimum is given by the smaller critical delay associated with the red ones.

Figure 9.

(a) Response time τ_CRTT measured by CRTT (black) and the lumped reaction delay τ_e estimated indirectly based on the critical time delay τ_crit of the mechanical model (red). Error bars indicate minimum–maximum values. (b) The ratio of the average values of τ_CRTT and τ_e.

3.4. Reaction time test

The directly measured response time τ_CRTT and indirectly estimated reaction times τ_e of the 15 subjects are presented by black and red colour in figure 9a, respectively. As can be seen, both the response times and the reaction times range over a wide scale. The average value for τ_CRTT for the 15 subjects is 248 ms (range between 169 and 314 ms), while the average value of τ_e is 134 ms (range between 103 and 163 ms). Nevertheless, it is a remarkable observation that τ_CRTT is about twice as large as τ_e for all subjects. This is because τ_CRTT involves also the movement time, while τ_e is a pure lumped reaction delay in the mechanical model. The difference between the two delays is the movement time (≈110 ms for subject S9). In spite of the difference in the magnitude of the two delays, there is a strong correlation between them. The correlation coefficient of the measured and the estimated reaction time is 0.55. If the outlier subject S8 is excluded, then the correlation coefficient is 0.84. Figure 9b shows the ratio of τ_CRTT and τ_e. The average is τ_CRTT/τ_e = 1.86 (range between 1.13 and 2.21).

4. Discussion

A natural question might be whether the mechanical model is detailed enough to capture the main points of balancing on a balance board. For instance, single inverted pendulum (ankle strategy) can be replaced by double inverted pendulum (ankle–hip strategy). An obvious advantage of the single inverted pendulum model is that it involves only a minimum number of parameters related to the human body. Another claim related to the model might be the concept of the feedback mechanism. Delayed PD feedback accounts for the two most important features of neural motor control: (i) actuation is performed based on perceived sensory signals and (ii) there is a reaction time delay. Of course, other types of control concepts may also apply, for instance, intermittent feedback [10,13,20], acceleration feedback [7,8], predictor feedback [16] or hierarchical control [38,39]. An advantage of delayed PD feedback model is that while it is widely used in the literature [5,6,9,17,18,27] it can be described with a small number of parameters. In the case of the two-degree-of-freedom model of balancing on a balance board, the control model involves only five parameters: the four control gains and the feedback delay. It should also be mentioned that actual balancing tasks are strongly affected by the imperfections of the sensory perception and the motor actuation. More sophisticated models exist in the literature that involve for instance sensory dead zones and/or control torque saturation [10,27]. However, the size of the dead zone and the level of saturation of the torque are extremely uncertain parameters, whose estimation is practically impossible for all the four feedback signals (φ, $\dot{\phi }$, $\vartheta$, $\dot{\vartheta }$) and for the control torque. Overall, we believe that the presented two-degree-of-freedom mechanical model combined with a delayed PD feedback provides a good estimation of the global dynamics of balancing on a balance board.

An important and often debated question is whether prediction is involved in the feedback mechanism or not. Clearly, prediction cannot be made in the case of random/unexpected perturbations that are independent of the subjects’ movement [28,51–53]. In the case of balancing on a balance board, the control mechanism is purely reactive (the movement depends on subjects’ movement only) and prediction is certainly involved in the feedback mechanism to some extent since subjects know that they are standing on a balance board under the effect of the gravitational field (similarly as in quiet standing). However, prediction requires the knowledge of the state at all time instants in order to provide initial condition for the prediction. This is not the case in human balancing due to the presence of sensory dead zones. If the controlled variable is below the sensory threshold then no precise prediction can be made due to the lack of precise initial conditions. In this case, sensory threshold crossing can be perceived only with a reaction delay and corrective movement can be made based on delayed state feedback only. In this sense, sensory dead zone presents a kind of random perturbation throughout the control process. This implies that delayed PD feedback is an essential element of the feedback mechanism during human motor control tasks.

The original hypothesis that there is a relation between the level of difficulty of balancing on balance board and the critical time delay of the corresponding mechanical model was verified by the measurements. Subjects could not balance themselves on balance board configurations associated with small theoretical critical delay and could balance successfully on configurations with large critical delays. The relation between the difficulty of balancing and the rolling radius R and board elevation h also followed the expectations. It was more difficult to balance on balance boards with small radius (small R) and with high elevation of the board (large h). Since R and h are two well-defined and adjustable parameters of the system, investigation of their effect on the balancing performance can easily be evaluated based on the mechanical model. This underlines that rolling balance board, while being a relatively simple equipment, can effectively be used to estimate the balancing skill of subjects. Further question can be whether practice on rolling balance boards of different levels of difficulty in a systematic manner can help to improve balancing skills. For a multi-session training period with limited parameter variation (only R was changed), skill development was observed to be more pronounced for more difficult configurations (radius R = 75 mm) than for less difficult ones (radii R = 100, 125 mm) [54].

An important observation is that although both R and h affect stabilizability of the system, their contributions are not the same. The slope of the contour curves τ_crit in the plane (R, h) is around 3. This means that a change in R results in about 3-times larger change in τ_crit than the same change in h. Thus, in practical implementation, changing the radius can be used to adjust the critical delay in larger steps, while changing the board elevation h can be used for fine tuning.

The relation between the difficulty of the task and the critical delay of the corresponding mechanical model allows a kind of indirect estimation of the reaction delay of human subjects while standing on a balance board. It should be however emphasized that this reaction delay is different from the response time τ_CRTT measured directly by CRTT. Namely, the response time is the reaction time plus the movement time, which in general is about the same as the reaction time. Hence, the response time is about twice the reaction time [50]. This was confirmed by our measurement: the ratio of the response time and the estimated reaction time was on average 1.86.

The estimated reaction time of the 15 subjects was in the range of 100–160 ms which is slightly larger than the reaction times 90–125 ms used in the literature for quiet standing [17,27,40,41]. This can be explained by the fact that balancing on a balance board is a more complex task than standing quietly on a fixed platform, therefore the CNS requires more time to process the information and then to actuate on it.

Overall, the relation between different combinations of R, h and the critical delay of the mechanical model establish a well-defined measure of balancing skill. The two parameters R and h allow a more sophisticated differentiation of balancing subjects as opposed to stick balancing investigations, where the length of the balanced stick is the only measure of skill. Although the mechanical model and the corresponding control concept was relatively simple, it still involves the main component of balancing on a balance board: a two-degree-of-freedom model subjected to a feedback of the state variables with a reaction delay. The model was able to give an estimation of the combined visual/vestibular/somatosensory reaction time delay as a lumped delay during balancing on balance board.

Ethics

The research was performed following the principles of the Declaration of Helsinki.

Data accessibility

Additional data are available in the electronic supplementary material.

Authors' contributions

C.A.M. conducted the balancing trials, developed the mechanical model, performed numerical calculations and evaluated the results. A.Z. designed and manufactured the balance boards and provided support during the measurements and the evaluation. T.I. conceived and supervised the research project. All authors contributed to the writing of the paper and gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

The research reported in this paper and carried out at BME has been supported by the NRDI Fund (TKP2020 IES, grant no. BME-IE-BIO and TKP2020 NC, grant no. BME-NC) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology, by the Hungarian-Chinese Bilateral Scientific and Technological Cooperation Fund under grant no. 2018-2.1.14-TÉT-CN-2018-00008, by the Hungarian National Research, Development and Innovation Office (Project id.: NKFI-FK18 128636) and by the UNKP-20-3 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.