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Spiking Neuromorphic Control for Periodic Orbits Stabilization in Underactuated Robots: A Greedy Lyapunov-Based Approach

Maksim Surov, Oleg Sumenkov

arXiv:2610.37310v1cs.ROcs.SYeess.SY

TL;DR

Underactuated-robot orbit tracking with fixed-amplitude spikes raises questions about achievable stabilization. The paper constructs a greedy Lyapunov-based controller from transverse hybrid LTV dynamics, then demonstrates practical orbit stabilization in butterfly-robot experiments.

  • Problem

    For underactuated robots, it is unclear whether tracking a periodic orbit is achievable with control signals restricted to sums of constant-amplitude spikes.

    Source: §I · p. 1

  • Method

    The authors use transverse coordinates to form an auxiliary hybrid LTV system and apply a greedy Lyapunov-based design to construct a spiking controller.

    Source: §I · p. 1

  • Results

    Experiments show that transverse variables and the robot's phase trajectory converge to neighborhoods of the origin and reference orbit, respectively, for spike amplitudes including a = 0.02.

    Source: B. Neuromorphic Controller Design · p. 5

  • Takeaways & Limitations

    Theoretical analysis and physical-robot experiments support practical feasibility, including real-time implementation and effectiveness under disturbances, measurement noise, and delays.

    Source: §V · p. 6

Abstract

from arXiv · show

This work investigates the use of event-based spiking controllers, implemented as spiking neural networks, for trajectory tracking in underactuated robots. Specifically, we address the problem of stabilizing a prescribed periodic orbit. It is well known that, by applying the method of transverse linearization to the robot dynamics in a neighborhood of a periodic trajectory, the dynamics can be approximately described by a linear time-varying system with periodic coefficients (a periodic LTV system). The problem of orbital stabilization can therefore be reduced to stabilizing the trivial solution of this LTV system. We show that a previously proposed Lyapunov-based greedy design, originally developed for LTI systems, can also be extended to the corresponding linearized transverse dynamics, which take the form of a periodic LTV system. This enables the construction of a feedback controller for orbital stabilization of the robot. The effectiveness of the proposed method is demonstrated on the trajectory stabilization problem for the butterfly robot.

I. INTRODUCTION

The paper extends greedy Lyapunov-based spiking control from LTI systems to practical stabilization of periodic orbits in underactuated robots, with experimental validation on a butterfly robot.

I. INTRODUCTION

  • The work addresses a gap between neuromorphic control implementations and their theoretical justification, despite recent stability-analysis tools for event-driven systems.Spike-based control has also motivated interest through its potential for energy-efficient, robust, and computationally simple control,.Source: §I · p. 1
  • For underactuated robots, fewer control signals than degrees of freedom make orbit stabilization challenging, and fixed-amplitude spike inputs leave achievable tracking and stability unclear.The paper situates this challenge among applications such as walking bipeds, pendulumlike systems, and spacecraft,,.Source: §I · p. 1
  • A fixed-amplitude spiking controller achieves practical stabilization of a reference periodic orbit, with the approach experimentally validated on a butterfly robot.The controller design is constructive and establishes practical stability of the reference orbit.Source: §I · p. 1
  • Transverse coordinates reduce the robot's spiking-input dynamics near the orbit to an auxiliary hybrid LTV system stabilized using the greedy Lyapunov-based approach.This extends a design originally developed for LTI systems.Source: §I · p. 1

II. PROBLEM STATEMENT

The paper formulates practical orbital stabilization for an underactuated robot as a state-feedback design problem, using a local notion of stability around a prescribed trajectory.

II. PROBLEM STATEMENT

  • The robot's dynamics can be expressed through Euler–Lagrange equations, with a non-degenerate kinetic-energy matrix, generalized forces, and an input-to-force mapping.The desired trajectory and corresponding input are T-periodic, and the trajectory's phase-space points form the orbit.Source: §II · p. 2
  • Practical orbital stability requires solutions to remain bounded near the orbit and eventually enter and stay within a neighborhood of it.The neighborhood radius ρ is the practical accuracy; unlike asymptotic orbital stability, convergence to the orbit itself is not required.Source: §II · p. 2
  • The stability requirement is local: the definition is assumed to hold only for initial conditions in a subset containing the orbit.For initial conditions far from the orbit, the requirements may fail.Source: §II · p. 2
  • The design problem is to find state feedback u = u(q, ˙q) that ensures practical orbital stability of a desired trajectory.The robot dynamics are considered in control-affine form with a scalar control input and continuously differentiable functions f and g.Source: §II · pp. 1–2

A. Robot Dynamics Linearization Near a Reference Orbit

A periodic transverse-coordinate transformation separates phase along the reference orbit from deviations across it, enabling orbital stabilization to be treated as stabilization of transverse dynamics.

A. Robot Dynamics Linearization Near a Reference Orbit

  • The coordinate transformation x = α(τ, ξ) uses τ as a projection variable and ξ as the transverse variables, with the reference orbit represented by ξ = 0.The transformation is periodic in τ and locally diffeomorphic near the orbit.Source: A. Robot Dynamics Linearization Near a… · p. 2
  • The transverse dynamics have periodic coefficients, and linearization near ξ = 0 yields a periodic LTV system with matrices A(τ) and B(τ).Controllability over one period guarantees periodic stabilizing gains; typical numerical gain computation uses an LQR problem,.Source: A. Robot Dynamics Linearization Near a… · pp. 2–3
  • In these coordinates, orbital stabilization reduces to stabilizing the trivial solution of the transverse system.A continuous feedback that asymptotically stabilizes the transverse dynamics induces a control law that asymptotically stabilizes the orbit locally.Source: A. Robot Dynamics Linearization Near a… · p. 3

III. TRIGGER-EVENT CONTROL FOR ORBITAL

The section formulates orbital stabilization as event-triggered impulsive control of transverse dynamics and identifies obstacles to directly applying existing LTI event-triggered methods.

III. TRIGGER-EVENT CONTROL FOR ORBITAL

  • The proposed event-triggered neuromorphic approach applies greedy Lyapunov-based control to transverse dynamics for orbital stabilization.The feedback is in impulse form, with state-dependent event times computed from firing conditions.Source: §III · p. 3
  • Existing LTI event-triggered methods do not directly extend because impulsive feedback makes the closed-loop vector field discontinuous, complicating transverse linearization.The projection variable τ may jump abruptly, making dynamics expressed in terms of d/dτ problematic.Source: §III · p. 3
  • Even assuming the linear approximation remains valid, event-triggered algorithms for periodic LTV systems are not readily available.These are the second obstacle identified for extending existing approaches.Source: §III · p. 3

A. Hybrid transverse dynamics

The projection is chosen to keep the phase variable τ continuous through impulses, yielding a two-phase hybrid description of the transverse dynamics. The resulting flow and jump maps support event-triggered stabilization of the orbit.

A. Hybrid transverse dynamics

  • Choosing τ = π(q) makes τ unaffected by the velocity jump, so it remains continuous during impulses.The projection depends only on generalized coordinates, which gives a well-defined two-phase evolution of the transverse variables.Source: A. Hybrid transverse dynamics · p. 3
  • The impulse-control dynamics split into a flow phase when τ ≠ τ0 and a jump phase when τ reaches τ0.The jump map leaves τ unchanged and updates the transverse coordinates ξ.Source: A. Hybrid transverse dynamics · p. 3
  • Because τ is strictly increasing near the orbit, the transverse dynamics can be written as a hybrid system with flow and jump phases.The jump map follows by integrating the transverse dynamics across the impulse.Source: A. Hybrid transverse dynamics · p. 3
  • The transverse hybrid system retains higher-order terms in ξ, and its spike times are computed using the greedy Lyapunov-based approach.The supplied passage identifies r1 as quadratic and higher-order, and r2 as linear and higher-order in ξ.Source: A. Hybrid transverse dynamics · p. 4

B. Lyapunov-based greedy approach for periodic LTV systems

The greedy controller uses a quadratic Lyapunov function to trigger spikes that decrease its value in a periodic LTV system. The analysis establishes practical stability and connects transverse stability to practical orbital stability, while leaving nonlinear robustness analysis for future work.

B. Lyapunov-based greedy approach for periodic LTV systems

  • The controller uses V(τ,ξ) = ξ⊤P(τ)ξ and sets firing conditions so that a spike instantaneously decreases V.P(τ) is a positive-definite periodic solution of a matrix Riccati differential equation.Source: B. Lyapunov-based greedy approach for… · p. 4
  • The trivial solution of the periodic LTV system is practically stable under the stated event-trigger conditions.The result assumes continuous T-periodic A(τ) and B(τ), positive spike intensity a, and design parameters Q > 0 and r > 0.Source: B. Lyapunov-based greedy approach for… · p. 4
  • During flow, the Lyapunov analysis bounds V, and V does not increase at spike events.The proof concludes that V converges exponentially to an interval; ξ then enters a ball of radius r in finite time and remains there.Source: B. Lyapunov-based greedy approach for… · p. 4
  • A rigorous analysis of the nonlinear hybrid system and its robustness to disturbances and unmodeled dynamics is left for future work.The stated stability result concerns the closed-loop linearized system and only provides a basis for expecting local practical stability of the nonlinear transverse dynamics.Source: B. Lyapunov-based greedy approach for… · p. 4
  • Practical stability of the transverse dynamics implies practical orbital stability: the trajectory remains within a tube around the target orbit after finite time.The stated connection assumes typical continuity requirements for α.Source: B. Lyapunov-based greedy approach for… · p. 4

IV. EXPERIMENTAL RESULTS

The controller is evaluated on a physical Butterfly robot with two degrees of freedom and one control input. The experiment uses a periodic trajectory that advances both angular coordinates by π over a 2.38-second period.

IV. EXPERIMENTAL RESULTS

  • The experiment assumes ideal rolling, perfect spherical symmetry of the ball, and rotational symmetry of the mechanism.Under these assumptions, the configuration space is topologically a torus, parameterized by the link angle ϑ and ball angle ϕ.Source: §IV · p. 5
  • The Butterfly robot has two degrees of freedom but one control input, with a motor-driven link and a ball that rolls passively along its surface.The link is mounted on a fixed base and rotates in the vertical plane; its angle is measured by an encoder and the ball position is estimated using vision.Source: §IV · p. 5
  • The reference trajectory advances ϑ and ϕ by π per period, with T = 2.38 sec.Rotational symmetry returns the system to an identical configuration at the end of each period.Source: §IV · p. 5

A. Periodic LTV system: Transverse linearization

The Butterfly robot’s transverse coordinates are formed relative to the desired trajectory using τ as a projection variable. Since ϕ⋆(t) is strictly increasing, it can serve as the projection coordinate for representing the robot dynamics transversely.

A. Periodic LTV system: Transverse linearization

  • The desired ball-position trajectory ϕ⋆(t) is strictly increasing, allowing it to be chosen as the projection variable τ.This choice is used to represent the robot dynamics in transverse coordinates.Source: A. Periodic LTV system: Transverse… · p. 5
  • The transverse state ξ records deviations in link angle and both angular velocities from their desired trajectory values.Its components are ξ1 = ϑ − ϑ⋆(τ), ξ2 = ϑ̇ − ϑ̇⋆(τ), and ξ3 = ϕ̇ − ϕ̇⋆(τ).Source: A. Periodic LTV system: Transverse… · p. 5
  • The inverse coordinate transformation reconstructs the robot state from τ and ξ, and the resulting dynamics take the transverse form established earlier.The reconstruction uses ϕ = ϕ⋆(τ), while the other state components add their ξ deviations to the reference values.Source: A. Periodic LTV system: Transverse… · p. 5

B. Neuromorphic Controller Design

The controller uses a Riccati-based Lyapunov function for the hybrid transverse dynamics. Physical-robot experiments show practical orbital tracking and robustness to disturbances, noise, and delays.

B. Neuromorphic Controller Design

  • The design uses V(τ,ξ) = ξ ⊤P(τ)ξ, with P(τ) obtained as the stabilizing solution to a matrix Riccati differential equation.The design uses Q = diag{20, 1, 1} and R = 2 · 104; P(τ), B(τ), and u⋆(τ) are interpolated with a B-spline for real-time computation.Source: B. Neuromorphic Controller Design · p. 5
  • The physical-robot results demonstrate practical feasibility and robustness to disturbances, measurement noise, and delays.Frequent spike triggering is attributed to these nonidealities.Source: B. Neuromorphic Controller Design · pp. 5–6
  • At a = 0.02, transverse coordinates converge near the origin and the phase trajectory near the reference orbit.The experiments also reported good tracking for spike amplitudes a ∈ {5,6,10,12,20} × 10−3.Source: B. Neuromorphic Controller Design · p. 5, Figure 3 · p. 6, Figure 4 · p. 6

V. CONCLUSIONS

The paper represents event-based spiking control of an underactuated robot as a linear hybrid system with periodic coefficients, enabling firing conditions and practical orbital stability. Physical-robot experiments support feasibility, while a complete nonlinear-system stability and robustness proof remains future work.

V. CONCLUSIONS

  • Event-based spiking control yields a linear hybrid system with periodic coefficients, supporting firing conditions and practical orbital stability of the reference trajectory.The conclusion presents this as a theoretical justification for periodic-orbit tracking.Source: §V · p. 6
  • Physical-robot experiments confirm the controller can be implemented in practice, computed in real time, and remain effective amid disturbances, measurement noise, and delays.These nonidealities are not accounted for in the analysis.Source: §V · p. 6
  • A complete theoretical proof of nonlinear-system stability and robustness, and investigation of different firing strategies, are left for future work.The presented results are also described as readily extendable to systems with m > 1 control signals.Source: §V · p. 6
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