Stability Criteria and Motor Performance in Delayed Haptic Dyadic Interactions Mediated by Robots

TL;DR

The paper tackles stability in robot-mediated delayed dyadic haptic interactions by deriving analytical stability criteria using a zero-crossing approach on a delay-augmented two-mass–spring–damper model. It identifies delay-independent stability for $k\le k_m$ with $k_m=(b_1^2+b_2^2)/[2(m_1+m_2)]$ and a delay-dependent margin $\delta_m$ for $k>k_m$, with identical-parameter simplifications yielding $k_m=b^2/(2m)$ and explicit expressions for $\delta_m$. The work combines frequency-domain analysis, DDE simulations in MATLAB, dynamic parameter identification, and H-MAN/Hebi experiments to validate how stiffness, inertia, damping, and delay shape stability and motor performance in a dyadic setting. These results offer design guidelines and motivate delay-buffer strategies to enable robust remote dyadic interactions in rehabilitation and teleoperation contexts.

Abstract

This paper establishes analytical stability criteria for robot-mediated human-human (dyadic) interaction systems, focusing on haptic communication under network-induced time delays. Through frequency-domain analysis supported by numerical simulations, we identify both delay-independent and delay-dependent stability criteria. The delay-independent criterion guarantees stability irrespective of the delay, whereas the delay-dependent criterion is characterised by a maximum tolerable delay before instability occurs. The criteria demonstrate dependence on controller and robot dynamic parameters, where increasing stiffness reduces the maximum tolerable delay in a non-linear manner, thereby heightening system vulnerability. The proposed criteria can be generalised to a wide range of robot-mediated interactions and serve as design guidelines for stable remote dyadic systems. Experiments with robots performing human-like movements further illustrate the correlation between stability and motor performance. The findings of this paper suggest the prerequisites for effective delay-compensation strategies.

Figures (10)

• Figure 1: Free body diagram of robot-mediated dyadic interactions, modelled as a dyadic mass–spring–damper system. $m_1$ and $m_2$ in kg denote the mass (inertia) of each robot. $b_1$ and $b_2$ in Ns/m denote the damping (friction) of each robot. $k$ (N/m) represents the virtual spring connection. $x_1(t)$ and $x_2(t)$ define the continuous movement in metres. $f_1(t)$ and $f_2(t)$ define the interactive force in N exerted by human operators on the robots.
• Figure 2: Control diagram of dyadic interaction with round-trip time delay. The time delay $\delta$ is expressed in seconds. The dashed box highlights the dynamic system of each robotic mediator, as illustrated in Fig. \ref{['fig:fbd']}.
• Figure 3: Open-loop transfer function obtained from Equation \ref{['eq:dynamic_equation']}, where $L(s) = \frac{-\left(k e^{-\delta s}\right)^2}{(m_1 s^2 + b_1 s + k)(m_2 s^2 + b_2 s + k)}$. The poles of this open-loop transfer function are strictly negative. The Nyquist criterion is applied to assess system stability by examining the encirclement of the point $(-1,0)$. To cover a sufficient range of controllable parameters, stiffness is varied from $0.5\mathrm{K}$, $\mathrm{K}$, to $2\mathrm{K}$, where $\mathrm{K} = \mathcal{S}(M,B,M,B)$, and the time delay is varied from $0.5\Delta$, $\Delta$, to $2\Delta$, where $\Delta = \mathcal{D}\left(M,B,M,B,2K\right)$. The nominal values of $M$, $B$, $K$, and $\Delta$ are summarised in Table \ref{['tab:notation']}.
• Figure 4: Effect of modelled stiffness, delay, inertia, and damping on the magnitude ratio and phase shift, shown as the frequency response of the delayed dyadic mass–spring–damper system. Magnitude ratio has been used to indicate correlations among different parameters of a multi-DOF mass–spring–damper system [12]. Phase (in degrees) indicates how the frequency response is shifted or delayed relative to the input. (a) Increasing stiffness alone increases the magnitude ratio at higher input frequencies. Stiffness greater than $\mathrm{K}$ ($2\mathrm{K}$ or $4\mathrm{K}$) can increase the magnitude ratio above $1$, which may cause the response to diverge depending on the frequency. (b) Increasing delay alone shifts the phase to the left, delaying the response without affecting the magnitude ratio. (c) Increasing inertia usually increases the magnitude ratio, thereby reducing stability under frequencies near $-180$ deg phase shift. (d) Increasing damping enhances stability by reducing the magnitude ratio; the system becomes unstable when damping is less than $B$ (e.g., $0.5B$).
• Figure 5: MATLAB dde23 can solve delay differential equations with constant delays and was used to simulate the dynamic response under unit force inputs applied in opposite directions ($f_{1,x} = 1,\, f_{2,x} = -1,\, f_{1,y} = 1,\, f_{2,y} = -1$ N). (a–g) The system is strictly stable because the response asymptotically converges to equilibrium along both x and y axes. (h) The system is strictly stable along the y axis but marginally stable along the x axis, as the response along x neither diverges nor converges. (i) The system is unstable, with the response asymptotically diverging along both axes. (j–l) The system exhibits instability across all selected delay conditions.

Theorems & Definitions (1)

• proof