Distributed Nonlinear Dynamic Inversion (DNDI)

• DNDI is a distributed control method that combines local nonlinear dynamic inversion with consensus tracking over communication graphs.
• It employs input–output linearization at the agent level and graph-based protocols to coordinate missile guidance and aerial robotics applications.
• The approach addresses practical challenges including robustness and realization constraints by integrating data-driven inversion and incremental control techniques.

Distributed Nonlinear Dynamic Inversion (DNDI) denotes a class of nonlinear control designs in which local nonlinear dynamics are inverted at the agent level while coordination objectives are imposed through distributed interaction laws over a communication graph. In the material presently available, the clearest explicit formulation appears in cooperative missile guidance, where DNDI is defined as the combination of nonlinear dynamic inversion or input–output linearization for each missile and a distributed consensus-tracking protocol that uses local and neighbor information to enforce coordinated terminal behavior (Mondal et al., 22 Sep 2025). Closely related work extends the surrounding landscape in three directions: robust incremental inversion for aerial robots (Hachem et al., 13 Jan 2025), data-driven online inversion of identified nonlinear predictors (Novara et al., 2014), and distributed nonlinear control frameworks that are inverse-like but not based on exact cancellation, such as separable control contraction metrics (Shiromoto et al., 2018) and distributed inverse dynamics for floating-base legged systems (Khandelwal et al., 2024). Taken together, these works place DNDI at the intersection of nonlinear inversion, distributed coordination, and practical constraint handling.

1. Definition, scope, and conceptual position

In the cooperative missile-guidance formulation, DNDI extends nonlinear dynamic inversion to a distributed multi-agent setting: each missile inverts its own nonlinear guidance dynamics, while the desired closed-loop behavior is expressed not as purely local tracking but as distributed consensus tracking with respect to neighbors and a virtual leader (Mondal et al., 22 Sep 2025). In that formulation, DNDI is therefore the synthesis of two ingredients: input–output linearization at the agent level and graph-based leader-following coordination at the network level.

This usage is narrower than a generic phrase such as “distributed inversion-based control.” The missile-guidance paper is explicitly about multiple follower missiles attacking a stationary ground target, with cooperative variables chosen as time-to-go and line-of-sight (LOS) rate (Mondal et al., 22 Sep 2025). By contrast, the other supplied papers do not formulate DNDI under that name. The robust aerial-robotics work develops a cascaded single-agent INDI architecture with robust outer-loop synthesis rather than distributed coordination (Hachem et al., 13 Jan 2025). The nonlinear inversion control paper presents a centralized, data-driven SISO inversion method for unknown discrete-time plants (Novara et al., 2014). The separable-CCM paper provides a distributed nonlinear tracking framework without exact plant inversion (Shiromoto et al., 2018). The quadruped paper uses distributed inverse dynamics and constrained contact-force allocation rather than graph-based multi-agent coordination (Khandelwal et al., 2024).

Accordingly, DNDI should be understood in two layers. In the strict sense represented directly in the literature excerpt, it is a distributed input–output linearization method with consensus-tracking error dynamics (Mondal et al., 22 Sep 2025). In a broader research sense, the surrounding papers suggest a family resemblance among methods that separate local nonlinear inversion or inverse-dynamics realization from higher-level distributed objectives, but that broader interpretation is an inference rather than a formal shared definition.

2. Canonical mathematical structure

The generic nonlinear agent used in the DNDI missile-guidance formulation is written as

$\dot X_i = f(X_i)+g(X_i)U_i, \qquad Y_i = h(X_i).$

Differentiating the output yields

$$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$

with

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$

This is the standard dynamic-inversion step at the output level (Mondal et al., 22 Sep 2025).

The distributed extension is introduced through the consensus-tracking error. For scalar output,

$$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$

and for vector output $Y_i\in\mathbb{R}^p$,

$$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$

equivalently

$$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$

The desired error dynamics are chosen as

$\dot E_i + K_i E_i = 0.$

Substituting the output dynamics and solving algebraically for the control gives the distributed inversion law

$$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$

This equation is the central DNDI law in the missile-guidance paper (Mondal et al., 22 Sep 2025).

The communication model is a weighted undirected connected graph

$G=\{V,E\},$

with adjacency matrix $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$0, degree matrix

$$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$1

and Laplacian

$$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$2

Leader access is encoded by $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$3 (Mondal et al., 22 Sep 2025). The resulting architecture is distributed in the sense that each agent uses its own output, neighboring outputs, and, if applicable, leader information. The missile-guidance formulation further requires neighbor output derivatives $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$4, or quantities from which they can be obtained (Mondal et al., 22 Sep 2025).

This structure differs sharply from centralized nonlinear inversion of a single plant. In the data-driven nonlinear inversion control method, for example, the controller is a centralized one-step right-inversion of an identified predictor

$$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$5

implemented by

$$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$6

with the inversion performed over a scalar constrained optimization rather than over a graph-coupled consensus law (Novara et al., 2014). That contrast clarifies what is specifically “distributed” in DNDI: not merely local inversion, but inversion embedded inside networked cooperative error dynamics.

3. Missile-guidance formulation and cooperative objectives

The most explicit DNDI application in the supplied material concerns a salvo of follower missiles engaging a stationary ground target through a virtual-leader architecture (Mondal et al., 22 Sep 2025). The purpose is simultaneous target interception rather than isolated individual interception. The paper states two control objectives: the follower missiles are required to track a virtual time-to-go profile, and the tracking of zero LOS rate assures interception (Mondal et al., 22 Sep 2025).

For the $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$7-th missile, the planar motion model is

$$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$8

With $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$9 the relative distance, $$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$0 the LOS angle, and $$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$1, the relative kinematics are

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$2

Differentiation yields

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$3

with transformed inputs

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$4

This orthogonal transformation maps tangential and normal acceleration commands into range-aligned and LOS-aligned virtual inputs (Mondal et al., 22 Sep 2025).

The state variables are

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$5

and the chosen outputs are

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$6

Their first derivatives are

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$7

and

$$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$8

Hence the output vector has relative degree $$f_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]f(X_i), \qquad g_Y(X_i)=\left[\frac{\partial h}{\partial X_i}\right]g(X_i).$$9 with respect to

$$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$0

and

$$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$1

The inversion matrix is diagonal and nonsingular provided $$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$2 and $$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$3 (Mondal et al., 22 Sep 2025).

The virtual leader output is defined as

$$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$4

with $$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$5 described as an average initial or final interception time reference (Mondal et al., 22 Sep 2025). Distributed coordination is then imposed on the vector

$$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$6

so that time-to-go consensus implies simultaneous impact and LOS-rate regulation implies a collision course.

The same paper states that all missiles hit the target simultaneously in simulation, that $$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$7 reaches consensus and tracks the leader time-to-go profile, that relative ranges converge to zero simultaneously, and that LOS rates converge to zero (Mondal et al., 22 Sep 2025). At the same time, its validation remains qualitative: no exact impact-time errors, miss distances, convergence times, peak acceleration commands, or robustness margins are reported in the provided text (Mondal et al., 22 Sep 2025). The paper also does not provide a new Lyapunov proof for the missile-specific closed loop; the exponential decay of $$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$8 follows directly only from the imposed first-order error model, contingent on valid inversion and persistence in the nonsingular operating region (Mondal et al., 22 Sep 2025).

4. Local inversion mechanisms, robustness, and realizability

The broader DNDI problem is not only one of distributed coordination; it is also one of ensuring that each agent can realize the commanded virtual controls robustly under modeling error, disturbances, and sensing limitations. The aerial-robotics INDI work is especially relevant on this point (Hachem et al., 13 Jan 2025).

That paper contrasts classical nonlinear dynamic inversion for

$$e_i = \sum_{j\in N_i} a_{ij}(y_i-y_j)+\beta_i(y_i-y_L),$$9

with incremental nonlinear dynamic inversion. Classical NDI depends on explicit cancellation or inversion of the nonlinear model, so performance can degrade when $Y_i\in\mathbb{R}^p$0, $Y_i\in\mathbb{R}^p$1, actuator effectiveness, or disturbances are uncertain (Hachem et al., 13 Jan 2025). INDI instead uses a first-order expansion around the previous sample $Y_i\in\mathbb{R}^p$2,

$Y_i\in\mathbb{R}^p$3

neglects the state increment term at high sampling rate, and obtains

$Y_i\in\mathbb{R}^p$4

The control depends on control effectiveness and measured state-derivative data, but not directly on the uncertain drift term $Y_i\in\mathbb{R}^p$5 (Hachem et al., 13 Jan 2025). This measured-increment structure is the source of INDI robustness to model mismatch.

The same paper develops a full cascaded architecture for a multirotor drone. In the translational loop, a position controller generates commanded translational accelerations $Y_i\in\mathbb{R}^p$6, and an outer INDI law converts these into commanded thrust magnitude and Euler angles. In the rotational loop, an attitude controller generates commanded angular accelerations $Y_i\in\mathbb{R}^p$7, and an inner INDI law converts these into motor angular-velocity commands (Hachem et al., 13 Jan 2025). The relevance to DNDI lies in the modular decomposition: local low-level inversion can remain onboard each vehicle, while an outermost command generator could, in principle, be replaced by distributed coordination laws (Hachem et al., 13 Jan 2025).

For the outer guidance loop, the incremental translational relation is

$Y_i\in\mathbb{R}^p$8

and for the inner stabilization loop the commanded motor angular velocity is

$Y_i\in\mathbb{R}^p$9

Disturbance compensation is implicit rather than observer-based: disturbance forces and torques are reconstructed through measured-filtered accelerations and rates (Hachem et al., 13 Jan 2025). The paper stresses that the same second-order filter must be used consistently for all incremented and measured signals to keep timing aligned (Hachem et al., 13 Jan 2025).

The robust outer-loop augmentation uses low-order structured $$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$0 synthesis. The generalized plant is designed so that the controller $$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$1 minimizes the closed-loop induced norm from

$$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$2

to weighted performance outputs

$$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$3

with inequalities such as

$$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$4

The structured low-order controller is synthesized via nonsmooth optimization using MATLAB’s systune (Hachem et al., 13 Jan 2025).

Quantitatively, the reduced-order INDI/$$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$5 controller improves disturbance attenuation by nearly $$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$6 relative to INDI/PD in the attitude loop, while the full-order version improves it by more than $$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$7; in the position loop under force disturbances of amplitude between $$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$8 N and $$E_i = (\bar d_i+\bar\beta_i)Y_i-\bar a_i \mathbf Y-\bar\beta_i Y_L,$$9 N, the reduced-order INDI/$$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$0 controller improves disturbance attenuation by about $$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$1 (Hachem et al., 13 Jan 2025). The abstract and conclusion state that disturbance rejection improves by more than $$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$2 overall for both rotational and translational dynamics (Hachem et al., 13 Jan 2025). This is not distributed control, but it identifies a local-agent robustness problem that any practical DNDI system must solve.

The data-driven inversion paper adds a different realizability perspective. There the plant is an unknown nonlinear discrete-time SISO system

$$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$3

approximated by a polynomial predictor

$$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$4

with online inversion

$$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$5

Because $$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$6 is polynomial in a scalar decision variable, the candidate minimizers are the real roots of $$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$7 in the admissible set together with the saturation endpoints (Novara et al., 2014). This is not DNDI, but it shows a distinct route to nonlinear inversion when analytical models are unavailable: identify a local predictor from data, then perform constrained right-inversion online. A plausible implication is that future DNDI systems could combine distributed coordination with local identification-and-inversion primitives rather than exact first-principles models.

5. Related distributed nonlinear control frameworks

DNDI exists alongside distributed nonlinear control frameworks that achieve trajectory tracking or stabilization without exact input–output cancellation. The clearest example in the supplied material is the separable-CCM approach (Shiromoto et al., 2018).

That paper considers networked nonlinear systems on a physical interaction graph $$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$8 and a communication graph $$E_i = \sum_{j\in N_i} a_{ij}(Y_i-Y_j) + \beta_i(Y_i-Y_L).$$9, with local subsystem dynamics

$\dot E_i + K_i E_i = 0.$0

and stacked form

$\dot E_i + K_i E_i = 0.$1

The goal is universal exponential stabilizability of any forward-complete target trajectory $\dot E_i + K_i E_i = 0.$2, under controllers that are $\dot E_i + K_i E_i = 0.$3-admissable in the sense that local input $\dot E_i + K_i E_i = 0.$4 depends only on local and communicated neighbor states and targets (Shiromoto et al., 2018).

The core condition is the existence of a dual metric $\dot E_i + K_i E_i = 0.$5 and structured matrix function $\dot E_i + K_i E_i = 0.$6 satisfying the pointwise LMI

$\dot E_i + K_i E_i = 0.$7

With $\dot E_i + K_i E_i = 0.$8, the differential controller is

$\dot E_i + K_i E_i = 0.$9

and the actual nonlinear feedback is reconstructed along a minimizing geodesic $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$0 by

$$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$1

Distribution is enforced structurally: $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$2 is block diagonal with local dependence, and $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$3 respects the communication sparsity pattern (Shiromoto et al., 2018). The method is therefore distributed, nonlinear, and tracking-oriented, but it is not exact dynamic inversion.

Its significance for DNDI is comparative. The paper is explicit that the framework is useful for systems that are not feedback linearizable and even for a network with over $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$4 states whose linearization is uncontrollable (Shiromoto et al., 2018). This stands in contrast to exact DNDI formulations that rely on the invertibility of $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$5 and well-defined relative degree. The CCM result therefore functions as an alternative design doctrine: instead of canceling nonlinearities, impose contraction through convex synthesis and geodesic reconstruction.

The quadruped paper offers another adjacent architecture. The full floating-base robot dynamics are

$$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$6

and the controller first computes a virtual generalized force as if the $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$7-DoF system were fully actuated: $$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$8 Under perfect modeling,

$$U_i = (g_Y(X_i))^{-1} \Bigg[ -f_Y(X_i) + (\bar d_i+\bar\beta_i)^{-1} \Big( \bar a_i \dot{\mathbf Y} +\bar\beta_i \dot Y_L - K_i\big( (\bar d_i+\bar\beta_i)Y_i-\bar a_i\mathbf Y-\bar\beta_i Y_L \big) \Big) \Bigg].$$9

which is a computed-torque or dynamic-inversion relation (Khandelwal et al., 2024). The generalized force is then partitioned into base and joint components, with the base wrench realized through stance-contact forces by solving

$G=\{V,E\},$0

subject to exact friction-cone and unilateral constraints, and the final joint torque chosen as

$G=\{V,E\},$1

The orthogonality condition

$G=\{V,E\},$2

ensures that the projected joint-tracking torque does not alter the effective base wrench (Khandelwal et al., 2024).

This is not DNDI in the missile-guidance sense, because the “distribution” is across contact channels and task subspaces rather than across communicating agents. Yet it illustrates how inversion-based nonlinear control becomes constrained force allocation in underactuated systems. The paper reports that its geometric projected-gradient solver is about $G=\{V,E\},$3 faster than qpOASES, with about $G=\{V,E\},$4 lower residual and about $G=\{V,E\},$5 lower constraint violation, while the overall controller reduces foot slip, improves orientation tracking, and uses about $G=\{V,E\},$6 less power than a QP-based balance controller in the reported tests (Khandelwal et al., 2024). For DNDI, the broader lesson is that exact inversion alone is often insufficient; realization constraints determine the actual control architecture.

6. Limitations, misconceptions, and likely research directions

A common misconception is that DNDI is simply distributed consensus appended to any nonlinear controller. The missile-guidance formulation is more specific: it requires a well-defined input–output map

$G=\{V,E\},$7

with invertible $G=\{V,E\},$8, and it embeds graph-based consensus directly at the output-derivative level (Mondal et al., 22 Sep 2025). Another misconception is that any distributed nonlinear controller is a DNDI controller. The separable-CCM method is distributed and nonlinear but avoids exact inversion altogether (Shiromoto et al., 2018); the quadruped method uses inverse dynamics but not multi-agent consensus (Khandelwal et al., 2024). Terminological precision is therefore important.

The missile-guidance DNDI paper also has explicit limitations. It does not treat impact-angle control, target assignment among multiple targets, inter-missile spacing or collision constraints, actuator-constrained optimality, or finite-time proofs (Mondal et al., 22 Sep 2025). It does not explicitly model communication delay, packet loss, asynchronous update, bandwidth limits, quantization, or exchanged-data noise (Mondal et al., 22 Sep 2025). It does not provide a missile-specific Lyapunov proof, a boundedness proof for all internal states, or a singularity-avoidance argument for $G=\{V,E\},$9 or $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$00 (Mondal et al., 22 Sep 2025). These omissions are not peripheral: they identify the gap between a clean DNDI derivation and a fully fieldable distributed guidance law.

The aerial-robotics INDI work exposes a different set of limitations that carry over naturally to DNDI-style systems. Incremental inversion depends on high controller update rate, negligible state change over one sample, slowly varying nonlinear terms relative to the sample period, and sufficiently accurate local control-effectiveness estimates (Hachem et al., 13 Jan 2025). Differentiation of noisy gyro signals requires careful filtering, which introduces phase lag (Hachem et al., 13 Jan 2025). The approach also relies on reasonably accurate local matrices such as $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$01, $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$02, and $$\dot Y_i = \left[\frac{\partial h}{\partial X_i}\right]\dot X_i = \left[\frac{\partial h}{\partial X_i}\right]\big(f(X_i)+g(X_i)U_i\big) = f_Y(X_i)+g_Y(X_i)U_i,$$03; severe mismatch can degrade inversion (Hachem et al., 13 Jan 2025). These are local-agent constraints, but distributed coordination cannot compensate for them.

The data-driven inversion paper highlights a further issue: scalability. Its polynomial root-finding inversion is computationally attractive because it is one-dimensional and SISO (Novara et al., 2014). Extending that exact mechanism to MIMO or networked settings would no longer reduce to a scalar polynomial stationary condition (Novara et al., 2014). This suggests that data-driven DNDI would require new decomposition or approximation strategies rather than a direct reuse of the SISO method.

A plausible research trajectory, suggested by the supplied materials but not proved in any single paper, is a layered architecture in which distributed coordination generates local virtual commands, while each agent realizes those commands through a robust local inversion layer. The robust INDI cascaded architecture already exhibits the modular separation needed for such a design (Hachem et al., 13 Jan 2025). The missile-guidance DNDI law provides the graph-coupled cooperative layer (Mondal et al., 22 Sep 2025). The data-driven inversion paper suggests how local inversion might be learned when accurate models are unavailable (Novara et al., 2014). The CCM framework suggests a fallback when exact inversion is structurally impossible or too brittle (Shiromoto et al., 2018). The quadruped work suggests how realization constraints can be absorbed into optimization and projection layers rather than analytic inversion alone (Khandelwal et al., 2024).

In that sense, DNDI is best regarded not as a single finished methodology but as a design pattern whose strictest current form is distributed input–output linearization with consensus tracking, and whose practical development depends on robust local inversion, communication-aware coordination, and feasibility-preserving realization layers.