Conjugate Momentum Observer in Robotics & Quantum Sensing

• Conjugate momentum observer is a framework using generalized momentum (derived from Lagrangian dynamics) to reconstruct unknown external forces or conjugate variables.
• In robotics, it employs a residual filter based on momentum mismatch, achieving high accuracy in contact and disturbance estimation under varying model constraints.
• In quantum measurement and other fields, the concept extends to sequential and variational techniques that address noncommutativity and model dependency challenges.

Searching arXiv for the papers and adjacent terminology to ground the article. arXiv Search Query: "conjugate momentum observer robotics arXiv" The available literature suggests that “Conjugate Momentum Observer” is not a single standardized term, but a family of closely related constructions whose meaning depends on context. In robotics and control, it most often denotes a generalized-momentum residual observer built from the Lagrangian momentum $p=M(q)\dot q$, used to estimate contact forces, external disturbances, or actuator-side forces without explicit acceleration estimation (Payne et al., 2024, Gupta et al., 2024, Pitroda et al., 2024, Johnston et al., 5 May 2025). In quantum measurement and sensing, the phrase is better interpreted as an observer of conjugate variables—typically position and momentum—implemented through sequential measurements, classically correlated probe-and-measurement schedules, or phase-space positive operator-valued measures rather than a control-theoretic state observer (Carmeli et al., 2011, Park et al., 2022, Cooper et al., 31 Mar 2026). In other areas, the phrase is potentially misleading: in general relativity, the relevant construction concerns observer-defined linear and angular momentum $(P^a,J^{ab})$, not canonical conjugate momentum (Flanagan et al., 2014); in nuclear many-body theory, the conjugate momentum appears as a second generator coordinate in the dynamical GCM rather than as an engineering observer (Hizawa et al., 2020).

1. Terminological scope

A useful way to organize the term is by the object being reconstructed or operationally assigned.

Domain	Meaning of “momentum”	Representative papers
Robotics and control	Generalized or conjugate momentum $p=M(q)\dot q$	(Payne et al., 2024, Gupta et al., 2024, Pitroda et al., 2024, Johnston et al., 5 May 2025, Singh et al., 21 Sep 2025)
Quantum measurement and sensing	Conjugate observables such as $X$ and $P$	(Carmeli et al., 2011, Park et al., 2022, Cooper et al., 31 Mar 2026)
General relativity	Observer-defined $(P^a,J^{ab})$, not canonical momentum	(Flanagan et al., 2014)
Nuclear many-body theory	Conjugate momentum as a generator coordinate paired with $q$	(Hizawa et al., 2020)

In the robotics literature, the central variable is usually the generalized momentum

$p=M\dot q,$

or its coordinate-dependent variant in reduced or modal coordinates, and the observer estimates an external generalized force from discrepancies in the momentum balance (Payne et al., 2024, Johnston et al., 5 May 2025). This usage is explicitly close to generalized or canonical momentum in the Lagrangian sense, although several papers note that they do not introduce a Hamiltonian-state observer in canonical phase-space form (Payne et al., 2024, Singh et al., 21 Sep 2025).

The quantum-measurement literature uses a different notion of “observer.” There the issue is not estimation from a plant model but operational access to noncommuting variables. Sequential measurement theory shows that every Weyl-Heisenberg covariant phase-space observable can be implemented as a sequential measurement of two conjugate observables (Carmeli et al., 2011), while multi-parameter sensing can estimate small conjugate shifts in position and momentum through classically correlated squeezed probes and matched homodyne measurements (Park et al., 2022). A later cold-atom formulation extends this logic to quantum gas microscopes by measuring either a Husimi-$Q$ phase-space distribution or spatially resolved momentum-weighted densities (Cooper et al., 31 Mar 2026).

2. Generalized-momentum observers in robotics

The classical robotics form begins from Euler–Lagrange or floating-base rigid-body dynamics and uses generalized momentum to eliminate explicit acceleration dependence. In the bipedal contact-estimation formulation of “Multi-Momentum Observer Contact Estimation for Bipedal Robots,” the starting point is the floating-base constrained dynamics

$M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$

with generalized momentum

$(P^a,J^{ab})$0

Using $(P^a,J^{ab})$1, the momentum rate is rewritten as

$(P^a,J^{ab})$2

so the unknown term is inferred from a momentum mismatch rather than from $(P^a,J^{ab})$3 (Payne et al., 2024). The baseline observer is written as

$(P^a,J^{ab})$4

and the actual proposal is to run multiple such observers in parallel under different contact constraints (Payne et al., 2024).

For continuum robots, the same principle is transferred to modal generalized coordinates $(P^a,J^{ab})$5. The paper “A Modal-Space Formulation for Momentum Observer Contact Estimation and Effects of Uncertainty for Continuum Robots” defines the conjugate or generalized momentum as

$(P^a,J^{ab})$6

with dynamics

$(P^a,J^{ab})$7

After rewriting the momentum balance, the unknown contact generalized force appears additively, and the residual satisfies

$(P^a,J^{ab})$8

so each component is a first-order low-pass filtered estimate of the true contact generalized force (Johnston et al., 5 May 2025). The paper explicitly states that, in the Lagrangian setting, this “generalized momentum” is exactly the canonical or conjugate momentum associated with the modal generalized coordinates (Johnston et al., 5 May 2025).

A related but distinct development appears in “Generalized Momenta-Based Koopman Formalism for Robust Control of Euler-Lagrangian Systems,” where the state is constructed directly in momentum coordinates,

$(P^a,J^{ab})$9

The paper does not propose a standalone momentum observer; instead it uses a generalized-momentum-based state representation to decouple the actuation channel and then adds a linear Generalized Extended State Observer for lumped disturbances (Singh et al., 21 Sep 2025). This suggests a neighboring usage in which conjugate momentum is the preferred state coordinate rather than the primary observer output.

3. Contact, disturbance, and actuator-force estimation

A major robotics use of conjugate-momentum observers is contact or external-force estimation. In the bipedal multi-model architecture, one observer is built for each contact hypothesis, with reduced dynamics

$p=M(q)\dot q$0

and momentum observer

$p=M(q)\dot q$1

The active contact mode is inferred from which constrained model produces the smallest residual, augmented by a Markov-style transition model (Payne et al., 2024). Reported contact-mode accuracy is 98.44% for a single 5 s planar simulation trial at 1 kHz with 70 dB signal-to-noise ratio, and 77.12% on Sarcos Guardian XO hardware; the paper also reports 76.10% for direct most-likely mode choice and states that the transition model reduces spurious switching (Payne et al., 2024).

The observer has also been adapted to systems in which the unknown is not foot contact but an aerodynamic or propulsion-related generalized force. In “Conjugate Momentum-Based Estimation of External Forces for Bio-Inspired Morphing Wing Flight,” the Aerobat dynamics are written as

$p=M(q)\dot q$2

with conjugate momentum

$p=M(q)\dot q$3

The paper gives the force-estimation equation

$p=M(q)\dot q$4

where $p=M(q)\dot q$5 is the unknown generalized external force and $p=M(q)\dot q$6 is a fixed diagonal gain matrix (Gupta et al., 2024). The reported $p=M(q)\dot q$7 values are $p=M(q)\dot q$8 for $p=M(q)\dot q$9, $X$0 for $X$1, and $X$2 for $X$3 under the stated simulation disturbance profile (Gupta et al., 2024).

A closely related estimator appears in “Conjugate momentum based thruster force estimate in dynamic multimodal robot,” where Harpy’s dynamics are written as

$X$4

the generalized momentum is

$X$5

and the residual is defined so that

$X$6

The estimated body-frame thruster force is then reconstructed by

$X$7

The paper reports normalized RMSE values $X$8, $X$9, and $P$0 for $P$1, $P$2, and $P$3, respectively, in the constraint-model case (Pitroda et al., 2024).

For multimodal legged-aerial locomotion, “Optimization free control and ground force estimation with momentum observer for a multimodal legged aerial robot” uses a Conjugate Momentum Observer to estimate the net generalized ground-contact wrench from the reduced-order body dynamics

$P$4

With generalized momentum $P$5, the observer residual is designed so that

$P$6

and the paper states that the momentum observer tracks the aggregate GRF wrench better than the constrained model, especially in the normal direction and during impulsive contact events (Krishnamurthy et al., 2024).

4. Observer-defined momentum in general relativity

In general relativity, the phrase can be misleading. “Observer dependence of angular momentum in general relativity and its relationship to the gravitational-wave memory effect” is not about conjugate momentum in the Hamiltonian sense. It defines a procedure by which observers can locally assign

$P$7

at a spacetime point using only nearby geometry, with $P$8 regarded as an element of the dual of the Poincaré Lie algebra acting on the tangent space (Flanagan et al., 2014). The pair is represented by

$P$9

and under an origin shift $(P^a,J^{ab})$0,

$(P^a,J^{ab})$1

The operational construction uses local curvature data. Observers measure $(P^a,J^{ab})$2 and $(P^a,J^{ab})$3, form curvature invariants

$(P^a,J^{ab})$4

define effective mass and radius,

$(P^a,J^{ab})$5

construct a source displacement $(P^a,J^{ab})$6, a 4-velocity $(P^a,J^{ab})$7, linear momentum $(P^a,J^{ab})$8, a spin vector $(P^a,J^{ab})$9, and finally

$q$0

These definitions are chosen so that in stationary vacuum regions near future null infinity they reproduce the expected source charges (Flanagan et al., 2014).

The paper then introduces transport equations for comparing these quantities between observers,

$q$1

and shows that curvature induces observer dependence through generalized holonomy (Flanagan et al., 2014). Bursts of gravitational waves with memory give nontrivial generalized holonomy, so disagreement in angular momentum assignment is closely related to memory. This literature therefore concerns observer-defined momentum, not canonical conjugate variables.

5. Quantum observers of conjugate variables

In quantum measurement theory, the most direct analogue is a measurement-theoretic observer for a conjugate pair such as position and momentum. “Sequential Measurements of Conjugate Observables” provides a structure theorem for covariant instruments and proves that every Weyl-Heisenberg covariant phase-space observable can be implemented as a sequential measurement of two conjugate observables (Carmeli et al., 2011). For a first instrument $q$2 followed by a measurement of $q$3, the sequential observable is

$q$4

and the marginals are necessarily unsharp, of the form

$q$5

In the standard position-momentum realization on $q$6, the coupling is the generalized von Neumann interaction $q$7, and the resulting joint observable is a covariant phase-space POVM rather than a sharp simultaneous $q$8-$q$9 measurement (Carmeli et al., 2011).

“Optimal estimation of conjugate shifts in position and momentum by classically correlated probes and measurements” studies small displacements of a harmonic oscillator with

$p=M\dot q,$0

The paper shows that independent sets of differently squeezed Gaussian probes, classically correlated with matched quadrature measurements, attain the same asymptotic precision as the best known entangled Gaussian and non-Gaussian schemes for estimating the two displacement parameters (Park et al., 2022). The effective classically correlated probe ensemble is

$p=M\dot q,$1

with total classical Fisher information

$p=M\dot q,$2

For the equal-weight squeezed-probe strategy,

$p=M\dot q,$3

matching the entanglement benchmark in the asymptotic regime (Park et al., 2022).

“Phase-space microscopes for quantum gases” recasts the issue operationally for quantum gas microscopes. In the Husimi-$p=M\dot q,$4 mode, the apparatus implements a coherent-state POVM with outcome distribution

$p=M\dot q,$5

which is exactly the Husimi-$p=M\dot q,$6 representation (Cooper et al., 31 Mar 2026). The uncertainty tradeoff is explicit: $p=M\dot q,$7 so $p=M\dot q,$8 up to factors of order one (Cooper et al., 31 Mar 2026). In the averaged mode, the microscope instead measures spatially resolved momentum-weighted densities such as

$p=M\dot q,$9

and

$Q$0

These protocols are observers of conjugate-variable structure, but not momentum observers in the robotics sense (Cooper et al., 31 Mar 2026).

6. Conjugate momentum as an auxiliary collective variable

In nuclear many-body theory, “Generator coordinate method with a conjugate momentum” extends the generator coordinate method by including both a collective coordinate $Q$1 and its conjugate momentum $Q$2 as generator coordinates (Hizawa et al., 2020). The DGCM ansatz is

$Q$3

with conjugacy condition

$Q$4

If $Q$5 is generated by a constrained mean-field operator $Q$6, one defines

$Q$7

which leads, after Fourier transformation in $Q$8, to the projector

$Q$9

The conjugate momentum here is therefore an auxiliary variable whose integration reconstructs projected collective content, not an externally measured observer signal (Hizawa et al., 2020).

The application to particle-number projection makes the canonical pairing especially explicit. With $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$0 and gauge angle $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$1, the DGCM state is

$M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$2

and the number projector is

$M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$3

The method significantly lowers the ground-state energy, especially for magic nuclei where the BCS pairing gap vanishes, and the paper suggests it as a simpler alternative to VAP (Hizawa et al., 2020). This usage is structurally “observer-like” only in the broad sense that the conjugate variable restores missing dynamical information.

7. Conceptual synthesis and recurring limitations

Across the cited literature, a common structural pattern emerges: conjugate or generalized momentum is chosen because it exposes unknown forces or conjugate-variable information through a balance law or through a covariant measurement construction. In robotics, the generic form is a momentum balance in which the unknown generalized force enters additively and can be estimated by a stable residual filter (Payne et al., 2024, Johnston et al., 5 May 2025). In quantum measurement, the analogous object is a covariant phase-space POVM or a classically scheduled multi-run estimator that preserves the operational meaning of simultaneous access to conjugate variables while respecting noncommutativity (Carmeli et al., 2011, Park et al., 2022, Cooper et al., 31 Mar 2026). In general relativity and DGCM, the term “observer” points instead to local assignment or variational reconstruction rather than to a disturbance observer (Flanagan et al., 2014, Hizawa et al., 2020).

Several limitations recur. Robotics formulations are model dependent and often sensitive to velocity estimation, contact-mode ambiguity, frictional uncertainty, or poor dynamic-state estimates (Payne et al., 2024, Johnston et al., 5 May 2025, Gupta et al., 2024, Pitroda et al., 2024, Krishnamurthy et al., 2024). The continuum-robot formulation cannot localize contact from the momentum observer alone and identifies unobservable wrench directions through the rank properties of the contact Jacobian (Johnston et al., 5 May 2025). The multimodal legged-aerial and thruster-estimation papers rely heavily on accurate terrain or contact-force knowledge to separate contact from thrust effects (Pitroda et al., 2024, Krishnamurthy et al., 2024). Quantum constructions do not evade noncommutativity; they replace sharp simultaneous measurement by unsharp joint measurement, sequential measurement, or many-run asymptotic estimation (Carmeli et al., 2011, Park et al., 2022, Cooper et al., 31 Mar 2026). The general-relativistic construction is explicitly not about canonical conjugate momentum at all (Flanagan et al., 2014).

Taken together, the literature supports a narrow and a broad reading. In the narrow sense, a conjugate momentum observer is a Lagrangian residual observer built from $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$4, used to estimate external generalized forces. In the broader sense, it denotes any operational scheme that reconstructs information encoded in a conjugate pair—whether through generalized momentum in robot dynamics, sequential measurement of $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$5 and $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$6, classically correlated squeezed-probe estimation of $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$7, phase-space microscopy, or DGCM augmentation by $M\ddot{q} + C\dot{q} + G + A^{\top}\lambda = B^{\top}\left(\tau_{\text{mot} + \tau_{\text{ext}\right),$8.