Discrete Virtual Holonomic Constraints

• DVHC are algebraic constraints enforced at discrete instants in hybrid systems, defining the geometric relationship between system states such as the center-of-mass position and orientation.
• They induce low-dimensional discrete zero dynamics that govern orbital behavior, enabling the use of return-map methods and impulsive control strategies.
• Control synthesis for DVHC employs layered feedback schemes to enforce constraints and stabilize desired periodic motions in both underactuated and high-dimensional cooperative systems.

Searching arXiv for papers on Discrete Virtual Holonomic Constraints to ground the article in current literature. Discrete Virtual Holonomic Constraints (DVHC) are algebraic constraints enforced only at discrete instants of a hybrid system, rather than continuously in time. In the devil-stick problems studied in 2025, a DVHC specifies the center-of-mass (CoM) location as a function of the stick orientation at the instants when impulsive inputs are applied, and this construction induces a discrete zero dynamics (DZD) that governs orbital behavior. In cooperative quadrupedal locomotion, closely related discrete virtual constraints appear as modified local outputs designed so that an augmented periodic orbit remains invariant under both continuous and discrete dynamics. Across these settings, DVHC provides a framework for geometric constraint specification, hybrid-order reduction, and orbit stabilization through return-map methods (Khandelwal et al., 20 Aug 2025, Khandelwal et al., 9 Sep 2025, Hamed et al., 2019).

1. Formal definition and basic interpretation

A continuous virtual holonomic constraint is an algebraic relation $h(q)=0$ in the generalized coordinates $q$, enforced for all $t$ by continuous-time feedback. By contrast, a DVHC is enforced only at the discrete instants $t_k$ when impulses are applied (Khandelwal et al., 20 Aug 2025). In the propeller-motion formulation for a devil-stick, the generalized coordinates immediately before the $k$-th impulse are

$q_k^- := [h_k^-,\theta_k]^T,$

with $h_k^- \in \mathbb{R}^2$ the CoM position and $\theta_k$ the orientation. The DVHC is

$\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$

where $\Phi:\mathbb{R}\to\mathbb{R}^2$ is a prescribed virtual trajectory, for example

$q$0

Thus, at each impulse instant the condition $q$1 is re-imposed, but between impulses $q$2 need not lie on the circle (Khandelwal et al., 20 Aug 2025).

The planar-juggling formulation uses the same principle with a different geometry. If the pre-impact state is

$q$3

then the DVHC is

$q$4

Its stated interpretation is that, at each impact, the CoM is “snapped” to the point $q$5 via an impulsive control (Khandelwal et al., 9 Sep 2025).

In cooperative locomotion, the discrete-constraint idea is embedded in distributed virtual-output design. Each agent $q$6 implements modified virtual outputs

$q$7

and these are required to vanish on the augmented orbit $q$8. In that setting, DVHC are enforced through local distributed feedback laws rather than through a single impulsive law (Hamed et al., 2019).

A recurring point in the literature is that DVHC are geometric rather than time-parametrized prescriptions. One stated consequence is that they obviate the need for a time-varying reference trajectory in the devil-stick setting (Khandelwal et al., 9 Sep 2025).

2. Hybrid formulation and discrete zero dynamics

The devil-stick problems are modeled as hybrid systems with alternating impulse and flight phases. In the propeller-motion formulation, an impulse $q$9 is applied normal to the stick at signed distance $t$0, and the state then evolves under free fall for duration $t$1 until the next impulse (Khandelwal et al., 20 Aug 2025). The impulsive update is

$t$2

$t$3

The subsequent flight dynamics are

$t$4

$t$5

Because the next impulse is applied after a prescribed rotation increment $t$6, one enforces

$t$7

which yields $t$8 and $t$9 (Khandelwal et al., 20 Aug 2025).

Under the DVHC, the pre-impact position and velocity are constrained by

$t_k$0

where

$t_k$1

Substitution into the hybrid update yields a two-dimensional DZD in $t_k$2 alone (Khandelwal et al., 20 Aug 2025).

For propeller motion, the implicit return equations are

$t_k$3

$t_k$4

with

$t_k$5

An explicit quadratic solution for $t_k$6 is also given in the source (Khandelwal et al., 20 Aug 2025).

The planar-juggling formulation arrives at the same dimensional reduction in a different coordinate chart. Enforcing the DVHC exactly, with $t_k$7 in positions and $t_k$8 in velocities, yields a reduced map on

$t_k$9

so that the 4-dimensional hybrid underactuated system reduces to a 2-dimensional discrete dynamics when the DVHC is enforced exactly (Khandelwal et al., 9 Sep 2025).

In cooperative locomotion, the hybrid setting is substantially larger. The single-agent model is

$k$0

with $k$1 an 8-vertex directed cycle for Vision 60, and the two-agent model $k$2 is constructed on the strong-product graph $k$3. Eliminating multipliers yields impact maps of the form

$k$4

and the design objective is that the augmented orbit

$k$5

remain invariant under both continuous and discrete dynamics (Hamed et al., 2019).

3. Periodic motions and stability criteria

In the propeller-motion problem, a periodic orbit occurs when

$k$6

which simplifies the DZD to

$k$7

$k$8

Around a nominal periodic solution $k$9 of period $q_k^- := [h_k^-,\theta_k]^T,$0, stability is assessed באמצעות the Floquet map

$q_k^- := [h_k^-,\theta_k]^T,$1

Asymptotic attraction requires all eigenvalues $q_k^- := [h_k^-,\theta_k]^T,$2 to satisfy $q_k^- := [h_k^-,\theta_k]^T,$3, whereas $q_k^- := [h_k^-,\theta_k]^T,$4 gives only Lyapunov, or neutral, stability (Khandelwal et al., 20 Aug 2025).

For planar juggling, the target motions are period-2 juggling orbits. The stated symmetry condition is

$q_k^- := [h_k^-,\theta_k]^T,$5

which implies $q_k^- := [h_k^-,\theta_k]^T,$6. Under this symmetry, the DZD admits a one-parameter family of fixed points with

$q_k^- := [h_k^-,\theta_k]^T,$7

Orbital stability is then determined by the linearization of $q_k^- := [h_k^-,\theta_k]^T,$8 about the 2-cycle, and the criterion reported is

$q_k^- := [h_k^-,\theta_k]^T,$9

as necessary and sufficient. The source further states that the family of orbits is non-degenerate and that appropriately chosen DVHC parameters $h_k^- \in \mathbb{R}^2$0 yield locally attracting 2-cycles (Khandelwal et al., 9 Sep 2025).

A central point in both devil-stick papers is that exact DVHC enforcement alone does not automatically produce a unique, asymptotically selected orbit. In the propeller study, if $h_k^- \in \mathbb{R}^2$1 is chosen so that no periodic zero-dynamics orbit exists, the DVHC is still enforced but $h_k^- \in \mathbb{R}^2$2 and the approximate invariant $h_k^- \in \mathbb{R}^2$3 drift slowly over many rotations; with $h_k^- \in \mathbb{R}^2$4, no periodic orbits exist and $h_k^- \in \mathbb{R}^2$5 are aperiodic while $h_k^- \in \mathbb{R}^2$6 drifts monotonically (Khandelwal et al., 20 Aug 2025). This separates manifold enforcement from orbit selection.

In cooperative locomotion, stability is evaluated by a Poincaré return map $h_k^- \in \mathbb{R}^2$7 on a switching manifold $h_k^- \in \mathbb{R}^2$8, linearized about a fixed point $h_k^- \in \mathbb{R}^2$9. The design parameters $\theta_k$0 are then adjusted so that the eigenvalues of $\theta_k$1 lie strictly inside the unit circle (Hamed et al., 2019).

4. Control synthesis

The devil-stick controllers are layered. The first layer enforces the DVHC; the second stabilizes a chosen orbit on the induced DZD (Khandelwal et al., 20 Aug 2025, Khandelwal et al., 9 Sep 2025).

For propeller motion, the DVHC-enforcing impulsive controller defines the position and velocity errors at $\theta_k$2 as

$\theta_k$3

It imposes the contraction law

$\theta_k$4

From the hybrid update, one obtains two scalar equations in the unknowns $\theta_k$5 and $\theta_k$6; after eliminating $\theta_k$7, a quadratic in $\theta_k$8 is solved for $\theta_k$9, then $\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$0 is recovered, and finally

$\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$1

The source states that as $\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$2, the velocity error $\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$3, so the DVHC manifold is reached and maintained at the impulse instants (Khandelwal et al., 20 Aug 2025).

The planar-juggling controller uses the same structure, with

$\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$4

and the exponential law

$\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$5

On the zero dynamics, closed-form expressions are reported: $\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$6

$\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$7

$\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$8

These expressions are specific to the planar-juggling geometry $\rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0,$9 (Khandelwal et al., 9 Sep 2025).

The orbit-stabilizing layer in both devil-stick papers is formulated through an Impulse-Controlled Poincaré Map. A section $\Phi:\mathbb{R}\to\mathbb{R}^2$0 is chosen, the return state $\Phi:\mathbb{R}\to\mathbb{R}^2$1 is defined on successive intersections, and the map

$\Phi:\mathbb{R}\to\mathbb{R}^2$2

is linearized about a fixed point: $\Phi:\mathbb{R}\to\mathbb{R}^2$3 If $\Phi:\mathbb{R}\to\mathbb{R}^2$4 is stabilizable or controllable, a discrete feedback

$\Phi:\mathbb{R}\to\mathbb{R}^2$5

is selected so that the eigenvalues of $\Phi:\mathbb{R}\to\mathbb{R}^2$6 lie inside the unit circle. In both sources, one stated choice is LQR with $\Phi:\mathbb{R}\to\mathbb{R}^2$7 and $\Phi:\mathbb{R}\to\mathbb{R}^2$8 (Khandelwal et al., 20 Aug 2025, Khandelwal et al., 9 Sep 2025).

In cooperative locomotion, control synthesis is distributed and continuous between impacts. Each agent solves, at each 1 kHz sample, a local quadratic program

$\Phi:\mathbb{R}\to\mathbb{R}^2$9

subject to

$q$00

The cost keeps $q$01 close to the nominal HZD law $q$02, while $q$03 penalizes input-output linearization defect (Hamed et al., 2019).

5. Demonstrated systems and simulation evidence

The propeller-motion study reports four classes of simulation evidence (Khandelwal et al., 20 Aug 2025). For DVHC enforcement without orbit stabilization, the parameters are

$q$04

From arbitrary $q$05, $q$06, and $q$07, the reported behavior is $q$08 exponentially and $q$09; $q$10 and $q$11 converge to periodic sequences; $q$12 settles into a one-parameter family of orbits; and an approximate invariant $q$13 flattens out. On the Poincaré section $q$14, a desired orbit with $q$15 and LQR weights $q$16, $q$17 yields convergence to a single orbit, with impulse and lever-arm corrections vanishing after $q$18. Additional simulations show aperiodic DVHC motion when $q$19, exact $q$20-periodicity for certain rational choices of $q$21, and equidistribution of $q$22 on $q$23 for irrational $q$24 (Khandelwal et al., 20 Aug 2025).

The planar-juggling study reports a separate parameter set: $q$25

$q$26

From arbitrary $q$27 initial conditions, the errors $q$28 and $q$29 decay $q$30 per impact. The resulting 2-periodic orbit has

$q$31

$q$32

For orbital stabilization, the study chooses $q$33 in the planar-symmetric case, with $q$34, and reports convergence of $q$35 and $q$36 in $q$37 impacts (Khandelwal et al., 9 Sep 2025).

The cooperative-locomotion study addresses a much larger hybrid model. Its numerical model has 64 continuous-time domains, 192 discrete-time transitions, 96 state variables, and 36 control inputs. The implementation is on two Vision 60 quadrupeds with Kinova arms, and the reported outcome is that the distributed QP-based controllers validate the effectiveness of DVHC and the associated distributed feedback design (Hamed et al., 2019).

Taken together, these examples show that DVHC have been used in both low-dimensional underactuated juggling and high-dimensional cooperative legged locomotion. A plausible implication is that the framework is not tied to a single morphology, but to hybrid systems in which actuation, switching, or impact structure makes event-based manifold enforcement natural.

6. Relation to continuous VHCs, HZD, and conceptual scope

The relation between DVHC and continuous VHCs is explicit in the propeller-motion paper. As $q$38, one has $q$39 and

$q$40

Expanding the discrete dynamics in this limit recovers the zero dynamics of continuous propeller motion under a VHC: $q$41 For $q$42, this reduces to

$q$43

the planar pendulum-type zero dynamics whose periodic orbits govern continuous propeller maneuvers (Khandelwal et al., 20 Aug 2025). This establishes DVHC as a discrete analogue of continuous VHCs in a precise limiting sense.

The cooperative-locomotion construction connects DVHC to hybrid zero dynamics (HZD). For a single agent, the nominal outputs

$q$44

are designed so that

$q$45

is hybrid-invariant and yields a stable periodic orbit $q$46. The distributed extension modifies these outputs through measurable global variables $q$47, so that on the augmented cooperative orbit $q$48, each agent applies exactly the nominal single-agent controller. By Theorem 2 and Corollary 5 in the source, $q$49 is then a periodic orbit of the complex hybrid model $q$50 (Hamed et al., 2019).

Two misconceptions are directly addressed by the reported results. First, DVHC are not continuous path constraints: in the devil-stick setting, the CoM is constrained only at impulse instants, and between impulses it may depart from the prescribed geometric locus (Khandelwal et al., 20 Aug 2025). Second, exact constraint enforcement is not identical to orbital stabilization: additional return-map feedback is used precisely because the induced DZD can admit families of periodic motions, slow drift, or aperiodic behavior, depending on parameter choices (Khandelwal et al., 20 Aug 2025, Khandelwal et al., 9 Sep 2025).

The literature therefore presents DVHC as a hybrid-systems methodology in which one prescribes geometry at discrete events, derives a low-dimensional DZD, and then designs event-based or distributed feedback to stabilize desired orbits. This suggests a unifying interpretation of DVHC as event-centered virtual-constraint design for underactuated and contact-rich robotic systems.