Internal Moving Mass Actuators

• Internal moving mass actuators are devices that generate motion by shifting internal masses, leveraging Newtonian dynamics through inertial, gravitational, and capacitive forces.
• They are integrated in applications such as precision optomechanics, robotics, and underwater vehicles, where compact design and direct control of motion are critical.
• Key challenges include calibration, periodic error management, and handling complex dynamical couplings, which are addressed with advanced modeling and iterative waveform correction.

An internal moving mass actuator is a device or system in which actuation is achieved by the controlled motion of a mass or masses inside the structure or platform itself, as opposed to external mechanisms or direct-force actuators. This actuation principle leverages the exchange between the internal mass and the host body to produce motion, torque, or dynamical effects by means of inertial, gravitational, or reaction forces. Internal moving mass actuators are critical in fields such as precision optomechanics, robotics, underwater vehicles, and nonholonomic locomotion, where compactness, decoupling from the environment, or direct control of internal degrees of freedom are essential.

1. Physical Principles and Core Mechanisms

Internal moving mass actuators exploit Newtonian and Lagrangian dynamics. When an internal mass $m_i$ is displaced or accelerated within a host system (e.g., membrane, robot body, or vehicle hull), the resulting change in momentum/torque is imparted onto the host, generating net actuation. The underlying mechanisms are:

• Inertial actuation: Movement of an internal mass causes reaction forces on the host, driving its motion or vibration.
• Gravitational actuation: Shifting the internal mass redistributes the center of mass, enabling gravitational torques (e.g., for rolling balls or underwater pitch control).
• Capacitive/electrostatic actuation: For mass-loaded membranes, external voltages across an electrode gap generate controlled forces on the embedded mass via the electric field (Depellette et al., 22 Dec 2025).

The general dynamical model for internal moving mass actuation is derived using Newton–Euler or Lagrangian frameworks, incorporating constraints, inertial couplings, and, where relevant, nonholonomic or fluid interactions (Rambech et al., 29 Oct 2025, Putkaradze et al., 2018).

2. Design Architectures and Representative Implementations

Contemporary implementations span a diverse set of platforms:

• Mass-loaded membrane optomechanics: A microfabricated high-stress silicon nitride membrane (2 mm × 2 mm × 100 nm, $\sigma \approx 0.9$ GPa) is loaded with a central 1.3 mg gold sphere. Capacitive electrodes actuate the membrane for strokes $>700$ nm, while the scheme also integrates with a 3D microwave cavity for optomechanical readout (Depellette et al., 22 Dec 2025).
• Rolling balls and disks: Internal point masses traverse fixed trajectories (e.g., circular, elliptical, figure-eight rails) within a spherical shell. Modulating their position and acceleration enables control of rolling, precession, and net locomotion (Putkaradze et al., 2018). The nonholonomic nature necessitates explicit handling of rolling constraints.
• Nonholonomic crawling and sleighs: Internal masses in Chaplygin sleighs or other platforms are periodically moved to drive net translation or chaotic stochastic motion. Distinct actuation regimes include direct-averaging (slow oscillations) and parametric/mode-coupled excitation (Bizyaev et al., 2018).
• Underwater vehicles: Point masses movable along internal axes (e.g., surge axis) adjust the vehicle's center of gravity, inducing pitch, roll, or even surge depending on the dynamic coupling. The Newton–Euler formalism captures the coupled rigid-body and actuator dynamics, including added mass and restoring forces (Rambech et al., 29 Oct 2025).
• Piezo motors with internal periodic processes: Stacked piezoelectric architectures displace internal masses or legs in periodic cycles. Waveform design and harmonics control minimize periodic nonlinearities (Hasselmann et al., 2022).

3. Mathematical Modeling and Control Formalisms

Internal moving mass systems are typically modeled as coupled rigid bodies with extended state spaces. The governing equations take the form:

• Membrane-mass systems: The loaded oscillator dynamics (mass $m_\mathrm{eff}$, frequency $\omega_0$), with actuation force

$$F_\mathrm{elec}(x) = \frac{1}{2} \epsilon_0 S_gm V^2 / (d-x)^2$$

and a frequency shift under DC bias (Depellette et al., 22 Dec 2025):

$$\Delta \omega \approx - \frac{V_{DC}^2}{4 m_\mathrm{eff} \omega_0} \left. \frac{d^2C}{dx^2} \right|_{x=0}$$

• Nonholonomic robots: The evolution of body-fixed angular velocity $\Omega$, internal mass coordinates $\chi_i(\theta_i)$, and external pose is set by

$$\dot\Omega = [\sum m_i \widehat{s_i}^2 - I]^{-1} \left[ \Omega \times I\Omega + r\,\tilde{F}\times\Gamma + \sum m_i s_i \times \left\{ g\Gamma + \Omega \times (\Omega \times s_i + 2\dot{\chi}_i) + \ddot{\chi}_i \right\} \right]$$

with appropriate kinematic constraints (Putkaradze et al., 2018).

• Underwater vehicles: The actuator-augmented Newton–Euler model comprises the extended state $x = [\eta, r_p]^\top$ and the coupled equations

$$(M_S + M_A + M_P(r_p))\,\dot\nu' + [C(\nu) + C_P(\nu, r_p)]\,\nu' + g(\eta) + g_P(\eta, r_p) = \tau'$$

with explicit expressions for Coriolis, added mass, and restoring terms (Rambech et al., 29 Oct 2025).

Control design often leverages feedback linearization, Pontryagin’s minimum principle, or Fourier-based iterative correction (for cyclic actuators) (Hasselmann et al., 2022).

4. Performance Metrics and Experimental Observations

Quantitative metrics for internal moving mass actuators include:

System	Amplitude	Q Factor	Tunability	Bandwidth
Mass-loaded SiN membrane (Depellette et al., 22 Dec 2025)	$>700$ nm	$\sim2$ kHz/5 Hz (room), $10^5$ (cryo)	$>10\%$	DC-30 V
Rolling ball (Putkaradze et al., 2018)	Rails set scale	Set by geometry	Depends on rail/CM	$\sim$ kHz
Piezo motor (LT40) (Hasselmann et al., 2022)	Step $1.7-2.1$ μm	Error $<0.012$ μm	User-scaled	$100-200$ Hz

Displacement sensitivities down to femtometer/$\sqrt{\text{Hz}}$ are achieved in optomechanical membrane systems (Depellette et al., 22 Dec 2025). For piezo-motor systems, periodic subdivision error is systematically reduced by up to $10\times$ using iterative phase-correction waveforms (Hasselmann et al., 2022).

In nonholonomic and underwater systems, actuation performance depends on geometric parameters (mass amplitude, rail offset), control waveforms, and, where relevant, environmental coupling (e.g., friction, fluid dynamics).

5. Error Sources and Calibration Approaches

Periodic nonlinearities and residual errors arise from phase mismatches, friction, hysteresis, and component tolerances—particularly in actuators with internal periodic processes (Hasselmann et al., 2022). These are modeled via harmonic expansions:

$$a(\phi) = \frac{c_0}{2\pi}\phi + \sum_{k=1}^K A_k \sin(k\phi+\theta_k)$$

where the $A_k$ quantify subdivisional error.

The phase-optimization algorithm achieves error suppression by:

1. Measuring actuator output over many cycles.
2. Fourier filtering the harmonics to isolate and invert the strictly periodic error.
3. Remapping command waveforms via a corrected phase $\psi(\phi)$.
4. Iterating waveform construction until sub-micrometer residuals are attained.

Sensor resolution, lookup-table granularity, and waveform amplitude scaling are all critical for minimizing nonlinearity and maximizing force/efficiency (Hasselmann et al., 2022).

6. Applications and Integration in Advanced Systems

Internal moving mass actuators are foundational in:

• Quantum optomechanics and gravity experiments: The actuation and readout of a mass-loaded SiN membrane combine mass actuation, frequency tuning, and integrated cavity coupling for high-sensitivity and quantum-state control (Depellette et al., 22 Dec 2025). The scheme enables, for example, studies of gravitational coupling from milligram-scale source masses at cryogenic temperatures.
• Robotics and rolling-ball robots: Multitrajectory designs (elliptic, figure-eight, circular) extend the actuation space and control versatility for omnidirectional and energy-efficient gaits in nonholonomic robots (Putkaradze et al., 2018).
• Nonholonomic vehicles: Oscillating internal masses in systems like the Chaplygin sleigh generate robust mean locomotion (including chaotic regimes), with parameter tunability for thrust or diffusion scaling (Bizyaev et al., 2018).
• Underwater vehicles: Internal moving mass mechanisms augment standard maneuvering models, enabling fine control of pitch/roll and transition between operational modes without compromising exterior hull integrity, as validated in Remus 100 AUV simulations (Rambech et al., 29 Oct 2025).
• Precision positioners: Advanced piezo-motor actuators benefit from error-canceling waveform algorithms to achieve high step-size repeatability and force maximization under a wide range of loads (Hasselmann et al., 2022).

7. Advantages, Limitations, and Design Considerations

Internal moving mass actuation provides several advantages:

• Direct, contactless actuation with minimal external disturbance.
• High actuation amplitude per unit voltage (in capacitive systems).
• Electrostatic/electromagnetic tunability for resonance matching and adaptive control.
• Seamless integration with readout systems—especially in quantum and precision applications.
• Energy efficiency via gravitational/inertial exploitation.

Limitations include:

• Sensitivity to rail geometry, component tolerances, and possible dynamical singularities (e.g., $\sum m_i \|s_{i\perp}\|^2 \approx I$ in rolling robots), which can lead to loss of control authority or system singularities (Putkaradze et al., 2018).
• Residual periodic errors and hysteresis in cyclic actuators, requiring iterative calibration (Hasselmann et al., 2022).
• Increased dynamical coupling and complex Coriolis/mass matrices in multibody or underwater systems (Rambech et al., 29 Oct 2025).

Design optimization requires:

• Careful mass, lever arm, and waveform selection to maximize actuation and minimize nonlinearity.
• Tuning of oscillation amplitude and direction for desired response regimes, as outlined in random-walk and parametric actuation analyses (Bizyaev et al., 2018).
• Ensuring mechanical and electrical interfaces do not degrade system $Q$ or introduce loss channels (Depellette et al., 22 Dec 2025).

Internal moving mass actuators thus enable a class of systems characterized by compactness, scalability, and the ability to realize sophisticated motion, control, and sensing tasks across a diverse range of modern engineering and physics research domains.