The Effect of Gait Stability Based on Two Types of Impact Strategies for Two-Link Walking and Brachiating Robots

1 Introduction

• Walking research reports passive periodic gaits, period-doubling, strange attractors, and chaos; brachiation findings are sparser and include stable and unstable gaits and fold and Neimark-Sacker bifurcations.The walking models also inspired variations incorporating springs, dampers, limbs, and forcing functions.Source: §1 · p. 1
• The stated gap is the absence of a unified bifurcation analysis of two-link walking and brachiating gaits, whose dynamics are studied through step-to-step maps.With bilateral symmetry and coordinate relabeling, a gait taking n steps is a period-n fixed point.Source: §1 · p. 1
• State-based switching triggers collision at zero surface-to-end-effector distance, while time-based switching triggers it after a preset duration.The two strategies produce the same period-one gaits, yet the paper reports different long-term gait behavior across impacts.Source: §1 · pp. 1–2

1.1 Statement of Contributions

• The study compares gait stability under both switching strategies and finds that near fold bifurcations, unstable gaits under time-based switching can converge to stable gaits in mirrored families.The stated bifurcation types are period-doubling, Neimark-Sacker, and fold under state-based switching, and Neimark-Sacker and fold under time-based switching.Source: §1.1 · p. 2, Figure 2 · p. 3

2 The Two-Link Model

• The discrete step-to-step map carries one post-impact state through continuous motion to the next impact, and a gait is a fixed point of these dynamics.The switching time is the first time the switching function reaches zero.Source: §2 · p. 4
• Numerical continuation traces period-one fixed points, whose stability properties are the main focus.The periodicity map is used to identify these fixed points.Source: §2 · p. 4

2.1 The Continuous Dynamics

• During swing, the two-link robot follows double-pendulum dynamics expressed using the inertia, Coriolis, and gravitational terms.The model’s equations are nondimensionalized, with time, length, and torque scaled as described in the section.Source: §2.1 · p. 4

2.2 The Jump Map

• At impact, an instantaneous plastic collision for equal link lengths applies a jump map that relabels coordinates and maps pre-impact to post-impact velocity.The impact map relates the pre-impact velocity to the post-impact velocity.Source: §2.2 · pp. 4–5

2.3 The Switching Function

• The switching function defines instantaneous slope from the robot’s configuration, which coincides with the preset surface incline at step boundaries and removes slope as a free parameter without loss of generality.The instantaneous slope is the angle between the line connecting the hands or feet and the horizontal.Source: Defining the Walking Surface · p. 5

3 Methodology and Results

• The study computes gait families by continuation, assesses period-one stability from eigenvalues, and simulates unstable gaits for 1,000 impacts.The physical parameters match, and the continuation algorithm uses a fixed step size of 0.001.Source: §3 · p. 5

3.1 The Big Picture

• Under SBS, three gait families have stable regions and the three simple bifurcation types, while TBS has two smaller stable regions with NS or FD bifurcations.The families are shown in switching-time–joint-angle plots; colors distinguish stable and unstable gaits.Source: §3.1 · p. 6, Figure 3 · p. 6

3.2 Exploring a Brachiating Family of Gaits

• On the brachiating branch, SBS gaits are stable for slopes −2.0° to −24.5° before a PD bifurcation at σcrit ≈ −24.57° produces stable period-two motion.The transition is marked by an eigenvalue crossing the unit circle at −1; SBS also has a marginally stable eigenvalue at +1.Source: §3.2 · p. 7
• On the same brachiating branch under TBS, period-one gaits are stable from τ = 1.90 to 2.44, with an NS bifurcation at τcrit ≈ 2.14 and closed invariant curves beyond it.The bifurcation is marked by complex conjugate eigenvalues λ1,2 ≈ −0.88 ± 0.47i crossing the unit circle; trajectories spiral from the unstable fixed point toward a stable invariant curve.Source: §3.2 · pp. 7–8

3.3 Fold Bifurcations and Symmetry in Time-Based Switching

• Under TBS, the brachiating branch also has an FD bifurcation at τcrit ≈ 1.92, where an eigenvalue crosses the unit circle at +1.Bilateral symmetry pairs each gait (x0, τ, σ) with its reflected counterpart (−x0, τ, −σ); unstable trajectories can converge to either family’s stable region.Source: §3.3 · p. 8

3.4 Exploring a Walking Family of Gaits After a Fall

• Under SBS, unstable walking gaits near stable walking enter period-doubling orbits, then generally converge to stable period-one brachiating gaits before reaching period-doubling brachiating orbits.The transitions occur over σ ∈ [3.29°, 3.86°], σ ∈ [4.06°, 24.47°], and after the brachiating-family PD point at σ ≈ 24.57°, respectively.Source: §3.4 · p. 9, Figure 6 · p. 9
• Most unstable period-one walking gaits converge to stable period-one brachiating gaits, though some near τ ≈ 2.7 converge to brachiating motions above the unstable curve.Other outcomes include stable walking or brachiating gaits of period 2k, and PD orbits of brachiating gaits beyond the stable period-one region.Source: §3.4 · p. 9

• State-based switching yields fold, period-doubling, and Neimark–Sacker bifurcations, while time-based switching yields Neimark–Sacker and fold bifurcations.Source: §4 · p. 10
• Under time-based switching, mirrored gait families in bilaterally symmetric models have intertwined basins of attraction.Source: §4 · p. 10
• Under state-based switching, basins after a robot falls can carry unstable period-one walking gaits toward stable brachiating gaits.Source: §4 · p. 10
• The authors report a first combined analysis connecting walking and brachiating gaits in the space of passive dynamic walking gaits.Source: §4 · p. 10
• The study does not investigate chaotic regimes; future work is planned to analyze routes to chaos and potential strange attractors.Source: §4 · p. 10