A compliant ankle-actuated compass walker with triggering timing control

Passive dynamic walkers are widely adopted as a mathematical model to represent biped walking. The stable locomotion of these models is limited to tilted surfaces, requiring gravitational energy. Various techniques, such as actuation through the ankle and hip joints, have been proposed to extend the applicability of these models to level ground and rough terrain with improved locomotion efficiency. However, most of these techniques rely on impulsive energy injection schemes and torsional springs, which are quite challenging to implement in a physical platform. Here, a new model is proposed, named triggering controlled ankle actuated compass gait (TC-AACG), which allows non-instantaneous compliant ankle pushoff. The proposed technique can be implemented in physical platforms via series elastic actuators (SEAs). Our systematic examination shows that the proposed approach extends the locomotion capabilities of a biped model compared to impulsive ankle pushoff approach. We provide extensive simulation analysis investigating the locomotion speed, mechanical cost of transport, and basin of attraction of the proposed model.

💡 Research Summary

The paper introduces a novel under‑actuated biped model called the Triggering Controlled Ankle Actuated Compass Gait (TC‑AACG), which integrates a series‑elastic actuator (SEA)–like linear spring at the ankle joint. Unlike traditional passive compass gait (PCG) walkers that require a downhill slope for energy replenishment, TC‑AACG can generate forward propulsion on level ground by exploiting a compliant ankle push‑off that is not instantaneous. The ankle spring is characterized by stiffness k and a pre‑compression length r₀. Its activation is governed by a “trigger angle” θ_trig_s: when the supporting leg angle θ_s exceeds this threshold, the spring is released. However, actual extension of the spring also demands that the propulsive force generated by the spring and the rotating body (F_p = k·r₀ + m_b·l·θ̇_s²) exceeds the impeding gravitational component (F_i = m_b·g·cosθ_s).

The gait cycle is divided into three phases: single‑support, single‑support push‑off, and double‑support. The dynamics of each phase are derived using Lagrangian mechanics. In the push‑off phase the configuration vector expands to include the spring length r, yielding a three‑degree‑of‑freedom system; during double‑support only r remains as a single degree of freedom. Transition events (ankle push‑off, collision, liftoff) are precisely defined, and the collision map accounts for both pre‑collision and post‑collision push‑off scenarios.

Stability is examined through a Poincaré section taken at the start of the single‑support phase (the liftoff state). The Poincaré map Ḡ_p maps the state from one stride to the next. Fixed points are identified, and their linear stability is assessed via the Jacobian D Ḡ_p; eigenvalues with magnitude less than one define the set of stable fixed points Γ_s_j. By sweeping the parameter space (r₀, θ_trig_s) the authors construct stability regions S_s_j for period‑1, period‑2, period‑4, and period‑8 gaits. Three distinct regions emerge: R₁ where the spring never triggers, R₂ where it triggers but does not extend, and R₃ where it both triggers and extends during the single‑support phase.

Simulation results reveal that when θ_trig_s ≤ –0.4 rad the ankle spring is triggered just before impact, producing a pre‑collision push‑off that resembles impulsive strategies but without the need for high‑power, short‑duration actuation. For –0.4 rad < θ_trig_s < 0 rad the spring is triggered earlier (pre‑apex), leading to a post‑collision push‑off that directs more energy vertically, thereby increasing the mechanical cost of transport (CoT). The optimal region for efficient locomotion is identified around θ_trig_s ≈ –0.19 rad with r₀ between 0.10 m and 0.144 m. In this regime the model attains forward speeds up to ~1.2 m s⁻¹ while maintaining a low CoT (~0.2), outperforming traditional impulsive ankle actuation.

The authors also examine the basin of attraction for various fixed points, showing that the compliant, time‑controlled ankle actuation considerably enlarges the set of initial conditions that converge to stable walking compared with the classic PCG. Moreover, the ability to vary the trigger timing provides a flexible tool to adapt the gait to different terrains or speed requirements without redesigning the mechanical hardware.

In conclusion, TC‑AACG demonstrates that a non‑instantaneous, SEA‑based ankle push‑off, governed by a simple trigger angle, can extend the applicability of compass‑gait walkers to level ground, improve energy efficiency, and increase robustness. The paper provides a thorough theoretical framework, parametric stability analysis, and performance metrics, laying the groundwork for future experimental validation on physical robots.