State-Dependent Torque Perturbations

• State-dependent torque perturbations are modifications to applied torque that vary with the system’s instantaneous state, enabling advanced adaptive control.
• These perturbations are applied in robotics, frictional mechanics, and astrophysics to improve robustness, facilitate bifurcation analysis, and handle nonstationary disturbances.
• Simulation-based injections and adaptive compensation techniques using state features yield measurable improvements in tracking accuracy and overall dynamic performance.

State-dependent torque perturbations are explicit or implicit modifications to the torque input acting on a mechanical, robotic, or astrophysical system, where the perturbation is a deterministic or stochastic function of the system’s instantaneous state. Such perturbations arise in diverse settings including adaptive control under exogenous disturbances, the injection of dynamical discrepancy models in sim-to-real robotics, frictional phenomena governed by rate-and-state laws, nonsmooth or hybrid balancing dynamics, and linearized responses in many-body astrophysical systems. State-dependent torque perturbations are structurally more expressive than parameter-randomized or static-force models, underpinning both advanced robustness techniques and the emergence of novel bifurcation behaviors.

1. Foundations and Definitions

State-dependent torque perturbations occur when the applied or experienced torque at any instant depends explicitly on the mechanical state—joint positions $q$, velocities $\dot q$, actuator commands, or exogenous signals. The general form is:

$\tau(t) = \tau_\mathrm{ctrl}(t) + \delta \tau(s_t)$

where $s_t$ denotes the state at time $t$ and $\delta \tau(s_t)$ the perturbation, which may be learned, adaptive, nonsmooth, or physically motivated.

In robotic systems, these perturbations serve either as disturbance models for robustness training (Cha et al., 9 Apr 2025), as feedback for dissipative convergence (Materassi et al., 2018), as adaptive compensation for unknown inputs (Stewart et al., 2024), or as ON/OFF-switching in nonsmooth control (Simpson et al., 2011). In frictional and astrophysical applications, the torque’s state dependence may result from microscopic processes or distribution function gradients (Singh et al., 2015, Dootson et al., 2022).

2. Techniques for State-Dependent Torque Injection and Modeling

2.1. Simulation-Based Perturbation Injection

In sim-to-real transfer for locomotion, explicit state-dependent joint torque perturbations are injected during simulation rollouts:

$$\tau_{\mathrm{sim}}(s_t) = \tau_\pi(o_t) + \delta \tau(s_t)$$

Here, $\tau_\pi$ is the nominal policy output and $\delta \tau(s_t)$ is parameterized by a neural network $\tau_\phi$, which receives a privileged observation vector including body pose, velocities, commanded torque, and foot forces. $\phi$ is sampled per rollout for expressive stochastic coverage of unmodeled realities, directly in torque space, transcending classical domain randomization which only perturbs a limited parameter set $p_{DR}$ (Cha et al., 9 Apr 2025).

2.2. Control Law and Adaptive Compensation

For bipedal walking on uncertain surfaces, torque perturbations induced by moving supports are modeled within the dynamics, e.g.,

$$d(t) = -\frac{1}{k_p} \left( \ddot x_{ws}(t) - \frac{x^c_{sc}(t)}{z_{sc}(t)}\ddot z_{ws}(t) \right)$$

and rejected via adaptive estimation and feedforward control, ensuring bounded tracking error even under time-varying exogenous torque disturbances whose structure depends on the instantaneous robot and environment state (Stewart et al., 2024).

2.3. Nonsmooth and Hybrid Feedback

Under switching control schemes (e.g., PD-torque with ON/OFF regions in phase space), the effective applied torque is a state-dependent function which may be discontinuous, yielding piecewise-smooth or hybrid dynamics. Structure and bifurcation characteristics are determined by switching surfaces in the state space, time delays, and gain parameters (Simpson et al., 2011).

2.4. Physics- and Distribution-Based Frameworks

• In frictional rotary systems, the local torque is state-dependent through the slip velocity and internal state variable $\theta(r,t)$,

$$\tau(r,t) = \sigma\left[\mu_0 + a \ln\frac{V(r,t)}{V_0} + b \ln\frac{V_0 \theta(r,t)}{D_c}\right]$$

and the total torque is an integral over radius—hence a nonlinear, state-dependent functional (Singh et al., 2015).

• In self-gravitating stellar discs, the torque response to external perturbation is a functional of the phase-space distribution function’s gradient,

$$T(t) \propto \int d\mathbf{J} \sum_{\mathbf{m}} \left[ m_\phi \, \mathbf{m} \cdot \frac{\partial f_0}{\partial \mathbf{J}} \right]$$

such that only states with nonzero gradients support finite torque transfer (Dootson et al., 2022).

3. Analytical Structures and Bifurcations

State dependence transforms the dynamical portrait in several ways:

• In piecewise-smooth or hybrid systems (e.g., balancing with ON/OFF torque), border-collision and grazing bifurcations generate attracting or repelling limit cycles (“zigzag” or “spiral” orbits), homoclinic connections, or complex bursting. Critical values of switching parameters, delay, or gains lead to abrupt transitions in stability structure; Filippov flows and discontinuity-induced bifurcations dominate phase diagrams (Simpson et al., 2011).
• In metriplectic control of rigid bodies, state-dependent dissipative torques,

$$\tau(L) = 2 k C'(L^2) \left[ \omega^2 L - (\omega \cdot L)\omega \right]$$

asymptotically align angular momentum with inertia axes without energy loss, exploiting negative-semidefinite metrics in the state variables (Materassi et al., 2018).

• In frictional and astrophysical settings, state-dependent torques lead to nontrivial relaxation and response phenomena: monotonic decay or possible acceleration of angular velocity per the sign of state evolution coefficients, and resonance/amplification or suppression depending on system distribution functions and their gradients (Singh et al., 2015, Dootson et al., 2022).

4. Empirical Performance and Robustness

Empirical studies across distinct domains highlight the practical significance of state-dependent torque perturbations:

• In humanoid locomotion, injection of neural-network-parametrized state-dependent torque noise yields robust motor policies that generalize to complex “reality gaps” (e.g., unmodeled compliance, soft ground) where standard domain-randomized counterparts catastrophically fail. Key metrics such as mean absolute velocity error and episode failure rate show strict superiority of state-dependent perturbation methods on both simulation benchmarks and real-robot trials (Cha et al., 9 Apr 2025).
• In adaptive ankle torque control for bipedal robots on dynamic surfaces, the explicit treatment and online estimation of state-dependent perturbations enables tracking error and actuator limits to remain bounded under strong, time-varying disturbances. Adaptive compensation outperforms classical PD+feedforward architectures, maintaining constraint compliance and achieving substantially reduced RMS errors in all test scenarios (Stewart et al., 2024).
• In the rotary friction context, analytical and numerical results confirm that under high stiffness and both velocity-strengthening and -weakening regimes, the state-dependent component is crucial to explaining torque relaxation, velocity response, and the detailed dependence of dynamic torque on velocity and prior loading (Singh et al., 2015).

5. Theoretical Generalizations and Control Synthesis

Several theoretical frameworks support systematic construction of state-dependent torque control laws:

• Metriplectic extensions combine antisymmetric (Hamiltonian) and symmetric (dissipative, state-dependent) brackets to achieve energy preserving yet dissipative trajectories, generalizable to broad classes of mechanical and thermodynamical systems (Materassi et al., 2018).
• In hybrid and adaptive control, embedding continuous error systems within hybrid dynamics allows for adaptive estimation and compensation of state-dependent inputs, effectively mapping structurally hybrid problems into a space where Lyapunov techniques guarantee boundedness and convergence (Stewart et al., 2024).
• For simulation-based policy learning, explicit parameterization of the perturbation (e.g., as an MLP with zero bias and privileged inputs, randomized per rollout) enables practitioners to match or overcomplete the space of naturally occurring reality gaps, reducing the need for detailed simulator parameterizations and alleviating the tuning burden (Cha et al., 9 Apr 2025).

6. Practical Recommendations and Limitations

The successful deployment of state-dependent torque perturbations is contingent on the following:

• Choice of input state features for the perturbation model must encompass the full range of physically relevant state variables; in simulation-driven applications, privileged information is often available but not required at deployment (Cha et al., 9 Apr 2025).
• Perturbation amplitude should be scaled to produce challenging yet learnable disturbances; gradually increasing noise limits until mild performance degradation ensures optimal training coverage (Cha et al., 9 Apr 2025).
• Episodic rather than stepwise parameter randomization preserves rollout consistency and prevents instability in policy learning (Cha et al., 9 Apr 2025).
• For rigorous stability and performance guarantees in adaptive schemes, projection and forgetting mechanisms are used in online parameter estimation (Stewart et al., 2024).
• While self-gravity amplifies state-dependent torques in many-body astrophysics, the linear response approach becomes invalid if nonlinear orbit trapping dominates, highlighting current theoretical boundaries (Dootson et al., 2022).

7. Context and Domains of Application

State-dependent torque perturbations constitute a unifying mathematical and algorithmic structure with broad relevance in:

Domain	Role of State-Dependent Torque	Reference
Sim-to-real robotics	Robustness to unknown reality gaps, policy generalization	(Cha et al., 9 Apr 2025)
Adaptive robot control	Compensation for time-varying, hybrid disturbances	(Stewart et al., 2024)
Mechanical systems	Nonsmooth hybrid stabilization via switching	(Simpson et al., 2011)
Rigid body dynamics	Metriplectic dissipative convergence to target manifolds	(Materassi et al., 2018)
Frictional mechanics	Rate-and-state models of torque with internal variables	(Singh et al., 2015)
Astrophysical systems	Linear response and resonance transfer via DF gradients	(Dootson et al., 2022)

Significance arises from both practical improvements—higher robustness, reduced tuning burden, new capabilities under strong nonstationary disturbances—and theoretical depth, with novel bifurcation structures, attractor formation, and systematic dissipative control. The framework is extensible to any domain where torque or force inputs can be explicitly state-parameterized and highlights the advantages of direct joint-space intervention over restricted parameter-space perturbations.