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Robust Semi-passive Velocity Field Control with Boundedness Guarantees for Safe Interaction between Mechanical Systems and Physical Environment

Van Trong Dang, Sumitaka Honji, Takahiro Wada

arXiv:2608.30193v1eess.SYcs.RO

TL;DR

Energetic passivity can constrain task performance, while disturbances may push system energy and states beyond safe operating regions. The paper proposes robust time-varying SPVFC that selectively preserves passivity, regulates energy and power flow, and proves bounded convergence under disturbances. Simulations demonstrate the method’s effectiveness, while the authors identify broad power-flow bounds and future input-saturation work as limitations.

  • Problem

    Fully passive interaction control can be conservative for task execution, and external disturbances can drive energy and states beyond operational regions.

  • Method

    The paper develops robust time-varying SPVFC that is passive above an energy threshold, permits non-passive behavior otherwise, and uses energy compensation to regulate energy and power flow.

  • Results

    Theoretical results establish bounded convergence of energy and system states under external disturbances, with simulations demonstrating the proposed method’s effectiveness.

  • Takeaways & Limitations

    The approach relaxes conventional passivity conservatism while maintaining bounded energy, tracking errors, and instantaneous power flow within the supported interaction setting.

  • Takeaways & Limitations

    The theoretically bounded power-flow domain remains relatively broad, and input saturation is left for future investigation.

Abstract

from arXiv · show

Controllers that guarantee energetic passivity with respect to the pair of external force and velocity realize safe interaction between the mechanical system and its physical environment. However, solely adhering to energetic passivity constraints may impose fundamental limitations on control performance and, in some cases, prevent the successful execution of controlled tasks. In addition, external disturbances from the physical environment can drive the system energy level and states beyond operational regions, thereby undermining task performance and safety. In this paper, we study a robust time-varying semi-passive velocity field control to aim to relax the inherently conservative nature of fully passive control methods in a controlled manner. Specifically, the proposed control method guarantees passivity of the closed-loop system with respect to the force-velocity input-output pair when the energy level exceeds a predefined level, while permitting non-passive behaviors to preserve task performance otherwise. Furthermore, the energy level and the states of the closed-loop system are proved to converge to bounded domains even in the presence of unpredicted disturbances. Additionally, the proposed method also enables constraining power flow between the closed-loop system and its physical environment to enhance safety in the interaction process. Numerical simulation examples demonstrate the effectiveness of the proposed method.

I. INTRODUCTION

Physical interaction control must balance energetic safety with task performance, while disturbances can drive energy and states outside operational regions. The paper motivates robust semi-passive control to address these challenges under bounded-disturbance assumptions.

  • Motivation: Energetic passivity supports safe force–velocity interaction but can restrict control performance and cause sluggish or stalled responses.Passive systems dissipate externally supplied energy without internally generating energy.
  • Motivation: Energy upper bounds alone do not constrain transfer rate, so instantaneous power flow may remain hazardous during interaction.The motivation distinguishes stored-energy limits from power-flow limits.
  • Related work: Prior passivity-based methods address robustness and safety, while energy-compensation approaches relax conservatism through adaptive injection or dissipation.The cited approaches include robust PBC, barrier-function combinations, and velocity-field control with energy compensation.
  • Contribution: The proposed robust time-varying SPVFC is passive above a predefined energy level and permits non-passive behavior otherwise to preserve task performance.It is designed for nonlinear mechanical systems and provides a smooth transition between passive and non-passive regimes.
  • Problem formulation: The paper targets bounded energy and motion under external disturbances while enforcing power-flow constraints for safer physical interaction.The problem setting assumes bounded external torque and models nonlinear mechanical dynamics with control and environmental inputs.

III. MAIN RESULTS

The paper develops a robust time-varying SPVFC using an augmented mechanical system with a fictitious flywheel and energy-compensation terms. The design regulates energy and power flow while supporting trajectory tracking through boundedness and finite-time compensation properties.

  • Robust time-varying SPVFC: The controller augments the mechanical system with a fictitious flywheel that stores and supplements kinetic energy during physical interaction.The augmented state combines the mechanical system and flywheel dynamics.
  • Robust time-varying SPVFC: A time-varying velocity field encodes desired trajectories for timed tracking tasks, unlike prior time-invariant fields mainly used for contour following.The augmented field includes separate mechanical-system and flywheel velocity components linked through kinetic-energy conservation.
  • Robust time-varying SPVFC: The improved SPVFC adds integer- and fractional-order energy-compensation terms to regulate the passivity domain, stability, energy level, and power flow.The control law includes matrices S1 and S2 governed by control parameters and a smooth saturation function.
  • Robust time-varying SPVFC: When energy error is high, the controller dissipates energy; when energy is depleted, it injects energy to maintain operation and support task completion.The design aims to keep energy error within the interval [−δ2, δ3].
  • Robust time-varying SPVFC: The linear compensation term improves compensation rate, while the nonlinear fractional-power term provides finite-time energy compensation.Both terms contribute to regulation of the energy level and power flow.

B. Passivity and its conditions

The robust time-varying SPVFC guarantees passivity above a configured energy threshold while permitting non-passive behavior below it, enabling a smooth relaxation of fully passive control.

  • The closed-loop system is passive with respect to the force–velocity pair when its energy satisfies the specified constraint.
  • The controller’s smooth saturation function creates a continuous transition between passive and non-passive behaviors.
  • Non-passive behavior is permitted below the threshold to inject energy and maintain task performance.
  • The passive and non-passive domains can be configured through the parameters k_d and δ_2.
  • The system energy tends toward the designed region through the compensation terms D_1 and D_2, with external-force effects analyzed separately.

C. Energy Boundedness Analysis

The boundedness analysis proves that the closed-loop energy converges to a bounded region after finite settling time despite external disturbances, with convergence rates configured by control parameters.

  • The closed-loop kinetic energy converges to the bounded region B_1 under external disturbances.
  • When the energy exceeds the designed upper interval, the derivative condition makes the energy converge to k_d + δ_3 in finite time.
  • When the energy falls below the designed lower interval, its derivative is nonnegative and the energy converges toward the lower boundary.
  • The energy remains bounded after the finite settling interval T_E ≤ max {T_1, T_2}, while compensation terms remain inactive inside the dead-zone.
  • The convergence rate can be configured for each application through γ_1, γ_4, ζ_1, and ζ_2.

D. Uniform Ultimate Boundedness Analysis

The analysis proves uniform ultimate boundedness of the tracking errors using Lyapunov arguments, with parameter choices controlling both the bounded region and convergence behavior.

  • D. Uniform Ultimate Boundedness Analysis: The derivation addresses a potential gap by further analyzing β(t), because an intermediate inequality alone cannot directly establish UUB of ev.The additional lemma ensures the constrained region for V3 is invariant under the stated design conditions.
  • D. Uniform Ultimate Boundedness Analysis: The auxiliary error function V3(ev) remains bounded by ¯V3 for all t ≥ 0 when its initial value satisfies V3(ev(0)) ≤ ¯V3.The control parameter κ and the initial condition are selected to keep trajectories inside the constrained region.
  • D. Uniform Ultimate Boundedness Analysis: The tracking errors of the closed-loop mechanical system converge to a uniform ultimate bounded region B2 after a finite settling time Ts.The proof combines the robust SPVFC, a time-varying velocity field, and Lyapunov analysis under external disturbances.
  • D. Uniform Ultimate Boundedness Analysis: For V4 above its disturbance-dependent threshold, its derivative satisfies ˙V4 ≤ −ϑ2ΞV4, driving the system errors into a bounded region Bu.The threshold depends on Λ, ϑ2, and Ξ, while Ξ is selected from the design parameters.
  • D. Uniform Ultimate Boundedness Analysis: Larger κ, Ea, and ψ produce a tighter bounded region B2 but can increase the settling time Ts.Parameter selection therefore trades bounded-error size against convergence speed.

E. Power Flow Constraint

The power-flow analysis bounds energy transfer between the closed-loop system and its environment, addressing safety concerns that energy-level bounds alone do not capture.

  • E. Power Flow Constraint: The power flow between the closed-loop mechanical system and its physical environment is bounded under the conditions of Theorems 2, 3, and 4.The result is stated as a bound on |P(t)|.
  • E. Power Flow Constraint: During t < TE, energy-level regulation gradually mitigates power flow while preventing excessive compensation.The compensation terms are regulated by the energy level through the control law.
  • E. Power Flow Constraint: P(t) > 0 denotes power entering the mechanical system, whereas P(t) < 0 denotes power flowing from the system to the environment.The sign convention distinguishes the two directions of environmental energy transfer.
  • E. Power Flow Constraint: The bounded power-flow domain is relatively large because it is derived using the augmented system’s energy-bounded region.A subsequent bound is obtained using Theorem 4 and the reverse triangle inequality.
  • E. Power Flow Constraint: The method explicitly addresses passivity conservatism while maintaining a smooth transition between passive and non-passive behaviors.This contrasts with switching passivity approaches that may introduce discontinuities and oscillations.

IV. SIMULATION EXAMPLE

The paper uses numerical simulation examples involving a robot manipulator to validate the effectiveness of the proposed method.

  • IV. SIMULATION EXAMPLE: Numerical simulation examples of a robot manipulator are provided to validate the effectiveness of the proposed method.

A. Simulation Settings

The simulation considers a fully actuated two-link robot manipulator tracking a circular end-effector trajectory on a horizontal surface.

  • A. Simulation Settings: The simulated plant is a fully actuated two-link robot manipulator operating on a horizontal surface.Its kinematics and dynamic matrices are specified for the simulation model.
  • A. Simulation Settings: The model includes a fictitious flywheel with mass mf = 10kg and specified initial conditions for the augmented system.
  • A. Simulation Settings: The desired end-effector path is circular with radius R = 0.3 m and center xR = yR = 0.35 m.
  • A. Simulation Settings: The control configuration sets ψ = 30I2, Ea = 10 J, ζ1 = 3, ζ2 = 5, and ka = 10J, alongside the remaining listed gains and bounds.The additional settings include K1 = K2 = 2I3, κ = 0.5, δ1 = δ4 = 0.01, δ2 = δ3 = 1, and ηmin = ηmax = 1.

B. Numerical Simulation under External Disturbances

Simulations evaluate the proposed controller under three external-disturbance types, showing robust trajectory tracking, bounded energy and tracking errors, and bounded power flow.

  • Three disturbance types model oscillatory interactions, resistive contact forces, and disturbances aligned with robot motion.
  • The robot maintains robust desired-trajectory tracking under all three external disturbances.
  • [9, 11] J: the closed-loop energy converges to the prescribed bounded region within approximately 0.1s in every disturbance scenario.The energy varies within the domain for Disturbance 1, stays at the lower bound for Disturbance 2, and approaches the upper bound for Disturbance 3.
  • 0.82, 1.25, 1.50 and 0.24s, 0.14s, 0.05s: disturbance-specific upper bounds and settling times differ with disturbance magnitude.
  • Power flows for all three disturbance scenarios satisfy the bounded region derived from Corollary 1.

C. Numerical Simulation of Control Parameter Effect

The simulations show that increasing the control parameter improves tracking-error bounds and convergence speed while preserving the prescribed energy region under parameter and disturbance changes.

  • The three conditions compare nominal control, κ = 1.0, and κ = 1.0 with 1.5-times amplified disturbance amplitude.
  • Increasing κ improves tracking accuracy because it directly influences the error bound and settling time.
  • 0.56: Condition 2 bounds the tracking-error norm more tightly than Condition 1's 0.82 bound.
  • [9, 11] J: all three conditions remain within the prescribed energy working region regardless of κ and disturbance magnitude.

D. Comparative Simulations

Comparative simulations examine the proposed continuous SPVFC against original PVFC and a switching semi-passive controller under combined disturbances. The proposed method preserves task performance and energy bounds while avoiding observed depletion and transient peaking behaviors.

  • The comparison uses continuous SPVFC, original PVFC, and a semi-passive framework switching between conservative and nominal controllers.
  • Under doubled Disturbances 1 and 2, original PVFC exhibits energy depletion near t ≈7.5s, degrading task performance and terminating operation.
  • The switching comparison method compensates energy but exhibits transient peaking that degrades tracking performance and appears in its control torques.
  • Under doubled Disturbances 1 and 3, both compared methods achieve task performance, but accumulate injected energy and violate the upper energy bound.
  • The proposed method preserves prescribed energy bounds while maintaining bounded tracking behavior across the comparative disturbance scenarios.
  • The study validates the method through numerical simulations of a two-link robot manipulator subjected to multiple external disturbances.
  • A remaining limitation is that the power-flow bounded domain is relatively broad, motivating future power-flow adjustment mechanisms for tighter bounds.

APPENDIX

The appendix reports results for Conditions 1–3 under Disturbances 2 and 3.

  • Conditions 1, 2, and 3 under Disturbances 2 and 3 are presented in Figs. A.1 and A.2, respectively.
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