Submanifold-Constrained Passivity

• Submanifold-constrained passivity is a dissipativity framework where storage functions vanish on designated subspaces, ensuring stability and convergence within safe operational regions.
• The approach leverages off-manifold projections, polynomial approximations, and SOS programming to reformulate dissipation inequalities under practical constraints.
• Applications include consensus in multi-agent systems, manipulator control under collision and singularity constraints, and intrinsic passivity on manifolds like SO(3) for rigid-body dynamics.

Search arXiv for papers on "submanifold-constrained passivity" and the listed arXiv ids. Submanifold-constrained passivity denotes a class of passivity and dissipativity formulations in which storage functions, supply rates, or convergence criteria are defined relative to a submanifold, feasible set, or safe set rather than the full ambient state-output space. In the cited literature, this viewpoint appears in consensus as convergence to the agreement submanifold, in rigid-body attitude dynamics constrained to $SO(3)$, in manipulator control under singularity and collision constraints, and in local passivity analysis under operational limitations (Yue et al., 29 Aug 2025, Terunuma et al., 23 May 2026, Zhang et al., 2024, Zakeri et al., 2019). The unifying idea is that the relevant notion of “distance” is measured from a constraint-compatible set—such as $S=\operatorname{span}(\mathbb{1})$, a safe set $\mathcal{C}$, or the manifold $SO(3)$—and the passivity inequality is reformulated so that the closed-loop analysis is aligned with that geometry.

1. Definition and geometric formulation

A precise formulation appears in the setting of output agreement on digraphs, where the target set is the agreement submanifold $S=\operatorname{span}(\mathbb{1}_n)$. There, disagreement is represented by the projection

$\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$

and output agreement is equivalent to convergence of this projected quantity to zero (Yue et al., 29 Aug 2025).

Within that framework, an $M$-constrained storage function is a differentiable function $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$ such that $Q(x)=0$ whenever the relevant output lies in $M$, and $S=\operatorname{span}(\mathbb{1})$0 otherwise. The associated notion of submanifold-constrained passivity, or $S=\operatorname{span}(\mathbb{1})$1-passivity, is defined by the existence of an $S=\operatorname{span}(\mathbb{1})$2-constrained storage function $S=\operatorname{span}(\mathbb{1})$3 and real parameters $S=\operatorname{span}(\mathbb{1})$4 satisfying

$S=\operatorname{span}(\mathbb{1})$5

This shifts the dissipativity inequality from the full output $S=\operatorname{span}(\mathbb{1})$6 to the off-manifold component $S=\operatorname{span}(\mathbb{1})$7, and from storage minimized at the origin to storage minimized on the target submanifold (Yue et al., 29 Aug 2025).

A related formulation on balanced digraphs introduces the signal space $S=\operatorname{span}(\mathbb{1})$8, with

$S=\operatorname{span}(\mathbb{1})$9

so that passivity analysis is carried out directly in terms of distance to the consensus submanifold. In that setting, the projected-output viewpoint is the core analytic device for treating nonlinear consensus under directed coupling (Yue et al., 2024).

2. Constrained dissipativity and local passivity on admissible sets

A broader antecedent is local dissipativity under operational limitations, where admissible behavior is restricted by state and input sets $\mathcal{C}$0 and $\mathcal{C}$1. Dissipativity is required only for trajectories satisfying $\mathcal{C}$2 and $\mathcal{C}$3, with

$\mathcal{C}$4

Local passivity is the special case $\mathcal{C}$5, and local OFP and IFP indices are defined on the same constrained region through the supply rates $\mathcal{C}$6 and $\mathcal{C}$7, respectively (Zakeri et al., 2019).

The computational treatment in that work proceeds by polynomial approximation of the nonlinear dynamics via Taylor’s theorem and a multivariate generalisation of Bernstein polynomials. The dissipation inequality is then converted into polynomial nonnegativity conditions over $\mathcal{C}$8, relaxed to SOS programming, and used both to certify local passivity and to maximize local passivity indices (Zakeri et al., 2019). A central implication stated there is that passivity and passivity indices are local or submanifold-dependent properties for nonlinear systems.

Submanifold-constrained passivity also appears in continuous-time constrained optimization. For affine equality constraints, the primal-dual gradient dynamics admit a Brayton-Moser formulation

$\mathcal{C}$9

with a Krasovskii-type storage function

$SO(3)$0

Its derivative satisfies

$SO(3)$1

and with external input gives passivity from $SO(3)$2 to $SO(3)$3. In that analysis, the state space is effectively constrained to the submanifold defined by the affine equality constraints, while inequality constraints are modeled as a state dependent switching system whose passivity is preserved under arbitrary switching; the two passive subsystems are then interconnected in a power conserving way (Kosaraju et al., 2017).

3. Agreement submanifolds in nonlinear network systems

In nonlinear multi-agent systems, submanifold-constrained passivity is used to recast consensus as convergence to a manifold rather than to a point. On balanced digraphs, the problem is formulated as convergence to $SO(3)$4, and the forward and feedback parts of the network are analyzed with respect to $SO(3)$5. The resulting inequalities take the form

$SO(3)$6

$SO(3)$7

with $SO(3)$8 and $SO(3)$9. A sufficient condition for asymptotic output agreement is

$S=\operatorname{span}(\mathbb{1}_n)$0

where $S=\operatorname{span}(\mathbb{1}_n)$1 is the maximum out-degree (Yue et al., 2024).

That result is explicitly sufficient rather than necessary, and it guarantees output agreement rather than average consensus in the nonlinear case. The numerical example is a continuous neural network with nonlinear nodes and edge controllers, used to illustrate both the applicability of the criterion and its non-necessity in some cases (Yue et al., 2024).

A later generalization treats arbitrary digraphs and any passive agents. There, the decisive step is a compensation theorem formulated directly in terms of submanifold-constrained passivity. If the agent relation admits an $S=\operatorname{span}(\mathbb{1}_n)$2-constrained storage function $S=\operatorname{span}(\mathbb{1}_n)$3 and the controller relation admits a storage function $S=\operatorname{span}(\mathbb{1}_n)$4, and if for some positive $S=\operatorname{span}(\mathbb{1}_n)$5

$S=\operatorname{span}(\mathbb{1}_n)$6

then the network achieves output agreement: $S=\operatorname{span}(\mathbb{1}_n)$7 The stated graph condition is the existence of a globally reachable node, and the framework is presented as removing the requirement for output-strictly passive agents that limited earlier passivity-based consensus results (Yue et al., 29 Aug 2025).

4. Safety-constrained manipulation and “passive when feasible”

In manipulator control, the term is used for architectures that preserve passivity only on the feasible or safe submanifold determined by barrier constraints. One formulation begins from task-space Passivity-Based Control with storage

$S=\operatorname{span}(\mathbb{1}_n)$8

for which passivity holds when

$S=\operatorname{span}(\mathbb{1}_n)$9

The difficulty is that standard constrained task-space PBC guarantees passivity only when constraints are not active, and singularities may require infinite or infeasibly large torques (Kurtz et al., 2021).

To address this, singularity avoidance is encoded through a Control Barrier Function built from the manipulability index

$\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$0

with the safe set

$\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$1

Because $\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$2 has relative degree $\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$3, an Exponential CBF is imposed: $\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$4 The key step is to make the reference-system input $\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$5 an optimization variable and add $\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$6 as a convex constraint in the QP. The resulting formulation is stated to ensure both forward invariance of the constraint set and passivity, even when the barrier is active (Kurtz et al., 2021).

A more expansive manipulator architecture constrains a dynamical-system-based impedance control law with a relaxed hierarchical CBF-QP subject to multiple concurrent, possibly contradicting, constraints: joint limits, self-collisions, external collisions, and singularities. The storage function is

$\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$7

and relative-degree-two constraints are enforced through ECBFs of the form

$\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$8

Hard constraints are assigned to joint limits and self-collision; soft constraints, with slack variables, to external collision and singularity (Zhang et al., 2024).

The central property is stated in that work as follows: the robot is passive if an optimal $\proj_{S^\perp}(y)=\left(I-\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top\right)y,$9 can be found in the feasible set, i.e., when $M$0. When safety constraints are active and the QP departs from the passive reference, unconstrained passivity need not hold, but invariance of the safety sets is preserved. This is the sense in which the paper uses the phrase “submanifold-constrained passivity” or “passive when feasible” (Zhang et al., 2024).

5. Intrinsic passivity on $M$1 and rigid-body networks

In rigid-body attitude control, the constraint manifold is $M$2 itself. Each rigid body is modeled by

$M$3

with $M$4. A passivity-based control law is written intrinsically as

$M$5

and a synchronization potential can be chosen as

$M$6

The storage function is

$M$7

with dissipation relation

$M$8

which yields strict passivity with respect to the port $M$9 (Terunuma et al., 23 May 2026).

The manifold constraint is not treated as an auxiliary condition but as an intrinsic part of the control design. The closed-loop vector field is tangent to $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$0, and the Nagumo theorem is invoked to show that the closed-loop system never leaves the submanifold. Lyapunov theory together with LaSalle’s invariance principle adapted to manifolds is then used to establish asymptotic convergence to $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$1 for initial conditions on $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$2 (Terunuma et al., 23 May 2026).

The human-in-the-loop extension adds a semi-autonomous architecture in which a multi-robot system preserves invariance of the average information fed back to the human operator through stealthy control, while human intervention is mediated through a virtual leader coupled to the robots via a passivity-based attitude synchronization law. Closed-loop stability is rigorously proved under the assumption that the human behaves as a passive system, and simulation studies are used to identify the human operator as a dynamical system and to examine passivity properties of the identified model (Terunuma et al., 23 May 2026).

6. Passivization, structured synthesis, and scope conditions

Submanifold-constrained passivity also intersects with passivization and controller synthesis when passivity must hold over subsets, families of equilibria, or structurally restricted controller classes. A general characterization of passivizing input-output transformations starts from

$Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$3

and classifies all linear transformations that map a system with given shortage of passivity to a system with prescribed excess of passivity. For SISO systems, every such transformation is parameterized as

$Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$4

with $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$5 invertible and all entries non-negative; for MIMO systems, the corresponding condition is expressed through a generalized Riccati inequality

$Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$6

That paper explicitly extends the same framework to simultaneous passivation over multiple equilibria and to passivation over subsets or submanifolds (Sharf et al., 2019).

For fixed-structured controllers, the structural restriction itself is represented by a linearly parameterized family

$Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$7

with $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$8. Necessary and sufficient conditions are then given for existence of a controller in that family that passivates the closed-loop system, and the maximal IFP or OFP index can be obtained through convex optimization with SOS constraints (Su et al., 2019). The description in that work states that passivity is enforced on the intersection of the set of stabilizing controllers and the submanifold specified by the controller structure.

Several recurring scope conditions follow from these formulations. Submanifold-constrained passivity does not, in general, imply unconstrained passivity: in constrained manipulators, safety invariance may take precedence when constraints are active (Zhang et al., 2024). Consensus conditions such as $Q:\mathbb{R}^n\to\mathbb{R}_{\ge 0}$9 are sufficient but not necessary (Yue et al., 2024). In nonlinear consensus on balanced digraphs, output agreement rather than average consensus is the guaranteed property (Yue et al., 2024). In local passivity analysis, the certified property depends on the admissible region $Q(x)=0$0, and enlarging that region can reduce the maximal provable passivity index (Zakeri et al., 2019).

Taken together, these works define submanifold-constrained passivity as a geometric specialization of dissipativity theory: storage vanishes on the target manifold or feasible set, dissipation is measured in off-manifold directions, and control synthesis is organized so that stability, safety, synchronization, or agreement are established relative to the constraint geometry rather than the unconstrained ambient dynamics (Yue et al., 29 Aug 2025).