AugMPC: Augmented Model Predictive Control

• AugMPC is a family of strategies that integrates an MPC core with supplementary mechanisms to address model mismatch, contact scheduling, and computational constraints.
• In satellite applications, AugMPC embeds an incremental integrator into a constrained linear MPC, significantly reducing tracking errors while preserving QP structure.
• For legged locomotion, AugMPC combines reinforcement learning for scheduling with a low-level contact-explicit MPC to manage non-periodic gaiting and real-time dynamics.

Searching arXiv for papers using the term “AugMPC” and closely related formulations to ground the article. AugMPC is a non-uniform label in contemporary model predictive control literature. In explicit usage, it denotes at least two distinct architectures: an augmented constrained linear MPC for agile earth observation satellites, where an incremental augmented state embeds integral action while preserving a quadratic-program structure, and an RL-augmented MPC for non-gaited legged and hybrid locomotion, where reinforcement learning supplies contact-schedule and navigation commands to a low-level contact-explicit MPC (Wang et al., 9 Mar 2026, Patrizi et al., 11 Mar 2026). In a broader technical sense, several adjacent works are “AugMPC-style” because they augment nominal MPC with sensitivity corrections, uncertainty-set dynamics, invariant error sets, value-aware losses, or solver-level augmented-Lagrangian structure rather than replacing MPC outright (Hose et al., 2024, Parsi et al., 2022).

1. Terminology and scope

The term is best understood as a family of augmentation strategies rather than a single canonical controller. The common design pattern is to retain a recognizable MPC core while adding an auxiliary mechanism that compensates for a specific deficiency: linearization error, contact-schedule combinatorics, parameter variation, observer mismatch, or online computational burden.

Usage of “AugMPC”	Core augmentation	Representative paper
Satellite attitude control	Incremental augmented state with embedded integrator	"Augmented Model Predictive Control: A Balance between Satellite Agility and Computation Complexity" (Wang et al., 9 Mar 2026)
Legged and hybrid locomotion	RL policy for gait/contact scheduling above contact-explicit MPC	"RL-Augmented MPC for Non-Gaited Legged and Hybrid Locomotion" (Patrizi et al., 11 Mar 2026)
AugMPC-style adaptive/robust approximation	Sensitivities, uncertainty sets, invariant sets, or value functions augment nominal MPC	(Hose et al., 2024, Parsi et al., 2022, Dey et al., 2022, Milios et al., 12 Nov 2025)

This plurality of meanings is not merely terminological. It reflects different loci of augmentation: inside the prediction model, outside the optimizer as a supervisory policy, within the uncertainty description, or within the learning objective. A plausible implication is that “AugMPC” is most useful as a structural descriptor—MPC plus a targeted augmentation—rather than as a uniquely defined algorithmic class.

2. Augmented constrained linear MPC for agile satellites

In the satellite-control usage, AugMPC refers specifically to Augmented-CLMPC, proposed for agile earth observation satellites as a middle ground between standard linear MPC and nonlinear MPC (Wang et al., 9 Mar 2026). The motivating tradeoff is explicit: LMPC is computationally light but suffers from model mismatch and steady-state tracking error, especially for moving targets, whereas NMPC uses the full nonlinear quaternion and rigid-body dynamics but is computationally expensive and can exceed the sampling period on hardware.

The nonlinear plant model is quaternion-based. With attitude quaternion $\boldsymbol{q}=[q_0,q_1,q_2,q_3]^T$, angular velocity $\boldsymbol{\omega}=[P,Q,R]^T$, control torque $\boldsymbol{T}$, reaction-wheel momentum $\boldsymbol{h}_{rw}$, and inertia matrix $\boldsymbol{J}$, the kinematics and dynamics are given by

$$\dot{\boldsymbol{q}}=\frac{1}{2} \left[\begin{array}{rrrr} q_0&-q_1&-q_2&-q_3\ q_1&q_0&-q_3&q_2\ q_2&q_3&q_0&-q_1\ q_3&-q_2&q_1&q_0 \end{array}\right] \left[\begin{array}{c} 0\ \boldsymbol{\omega} \end{array}\right],$$

$$\dot{\boldsymbol{\omega}}=\boldsymbol{J}^{-1}\left(\boldsymbol{T}-\boldsymbol{\omega}\times(\boldsymbol{J}\boldsymbol{\omega}+\boldsymbol{h}_{rw})\right),\qquad \dot{\boldsymbol{h}_{rw}}=-\boldsymbol{T}.$$

The target attitude quaternion is $\boldsymbol{q}_t$, with error quaternion

$$\boldsymbol{q}_e=\boldsymbol{q}\otimes\boldsymbol{q}_t^{-1}=[\eta;\boldsymbol{\xi}],$$

and near $\boldsymbol{\xi}=\boldsymbol{0}$ the paper linearizes to

$\boldsymbol{\omega}=[P,Q,R]^T$0

with the actual torque command

$\boldsymbol{\omega}=[P,Q,R]^T$1

The augmentation is introduced by moving from the discrete linear model

$\boldsymbol{\omega}=[P,Q,R]^T$2

to the incremental augmented model

$\boldsymbol{\omega}=[P,Q,R]^T$3

with

$\boldsymbol{\omega}=[P,Q,R]^T$4

and

$\boldsymbol{\omega}=[P,Q,R]^T$5

The lower-right identity block in $\boldsymbol{\omega}=[P,Q,R]^T$6 is the essential mechanism: it embeds an integrator into the prediction model and compensates for model mismatch caused by system linearization and feedforward control.

The output prediction over horizon $\boldsymbol{\omega}=[P,Q,R]^T$7 is written as

$\boldsymbol{\omega}=[P,Q,R]^T$8

and the AugMPC cost is

$\boldsymbol{\omega}=[P,Q,R]^T$9

The online optimization remains a quadratic programming problem:

$\boldsymbol{T}$0

subject to reaction-wheel torque and momentum limits,

$\boldsymbol{T}$1

$\boldsymbol{T}$2

The solver is qpOASES.

The central claim is empirical rather than theorem-driven. The paper does not provide a formal proof of closed-loop stability or recursive feasibility for AugMPC; instead it argues structurally that the controller remains a standard constrained linear MPC/QP and shows experimentally that the embedded integrator eliminates the moving-target steady-state error observed in ordinary LMPC (Wang et al., 9 Mar 2026).

3. RL-augmented MPC for non-gaited locomotion

A second explicit use of the label appears in legged and hybrid locomotion, where AugMPC denotes a contact-explicit hierarchical architecture coupling Reinforcement Learning (RL) and Model Predictive Control (MPC) (Patrizi et al., 11 Mar 2026). The problem is the combinatorial difficulty of deciding contact timing online. If contact timing is optimized directly inside MPC, the resulting problem becomes mixed-integer or otherwise difficult to solve in real time. AugMPC moves that burden upward: RL decides navigation-level commands and contact schedule modifications, while the low-level MPC executes a full rigid-body trajectory optimization under a predefined schedule.

The architecture has three layers. A high-level stochastic policy trained with Soft Actor-Critic outputs a base twist reference and contact-injection commands. A low-level receding-horizon locomotion MPC solves a nonlinear trajectory optimization with explicit contact constraints and sends references to a joint impedance controller. RL therefore augments MPC rather than replacing it; the same MPC remains active during both training and deployment (Patrizi et al., 11 Mar 2026).

For the flat-terrain formulation, the policy action is

$\boldsymbol{T}$3

where $\boldsymbol{T}$4 is the MPC twist reference and $\boldsymbol{T}$5 contains one contact-injection scalar per foot. A new flight phase for foot $\boldsymbol{T}$6 is injected when $\boldsymbol{T}$7. The policy does not output an entire gait cycle or a clock variable. Instead it requests immediate insertion of a flight phase at a fixed injection node $\boldsymbol{T}$8 in the horizon. This is the key mechanism through which acyclic gaits emerge.

The low-level MPC solves

$\boldsymbol{T}$9

subject to

$\boldsymbol{h}_{rw}$0

with

$\boldsymbol{h}_{rw}$1

The formulation includes integrator and inverse-dynamics equalities, no-slip point-contact or rolling-contact constraints, friction-cone and unilaterality inequalities, joint-velocity limits, and costs for velocity regularization, acceleration regularization, force regularization, base-twist tracking, capture terms, and vertical foot tracking. The NLP is solved using an equality-constrained multiple-shooting variant of iLQR, derived from DDP, in the Horizon framework; inequality constraints are approximated using quadratic barriers; and the controller is implemented in a real-time iteration scheme.

The contact schedule is parameterized as alternating flight phases and contact phases for each foot. Flight phases activate force-vanishing and vertical-foot-tracking terms, whereas contact phases activate point-contact or rolling-contact constraints, friction, unilaterality, and force-regularization terms. The injection node trades off latency and feasibility; the experiments use $\boldsymbol{h}_{rw}$2. This makes the RL action space compact while still permitting non-periodic, task-dependent contact timing.

The empirical scope is unusually broad. The paper reports validation across platforms spanning 50 kg to 120 kg, including a simplified quadruped, a Unitree B2-W wheeled quadruped, and Centauro, a 120 kg wheeled-legged humanoid robot. Training uses Isaac Sim, evaluation includes MuJoCo for sim-to-sim transfer, and real deployment uses XBot2. The reported training setup uses 800 parallel environments, action dimension 10, observation dimension up to 250 for Centauro, and policies obtained within 4--10\times 10^6 environment steps, corresponding to roughly 9--29 simulated days; 12 simulated days correspond to about 5.8 hours of wall-clock time. The framework reaches real-time factors up to 50, and onboard policy inference on Centauro takes 0.334 ms on an Intel Core i7-11800H CPU (Patrizi et al., 11 Mar 2026).

The principal result is that the architecture achieves zero-shot sim-to-sim transfer without domain randomization across all tested platforms and zero-shot sim-to-real transfer without domain randomization on Centauro. In the reported energy comparison, hybrid locomotion on Centauro converges to average CoT $\boldsymbol{h}_{rw}$3, versus $\boldsymbol{h}_{rw}$4 for legged locomotion. The authors interpret the resulting behaviors—alternating single and double flight phases, symmetric and asymmetric trotting, and timing adaptation near targets—as evidence that non-periodic gaiting can be delegated to RL while preserving the structure and constraint handling of contact-explicit MPC.

4. AugMPC-style augmentations beyond the explicit label

Outside the two papers that explicitly use the term, a wider literature develops controllers that are AugMPC-like because they augment MPC with additional structure targeted at adaptation, robustness, or approximation.

A first class augments approximate MPC with local sensitivity information. "Parameter-Adaptive Approximate MPC: Tuning Neural-Network Controllers without Retraining" introduces a neural AMPC architecture in which one network approximates the nominal policy and a second network approximates the parameter Jacobian, yielding

$\boldsymbol{h}_{rw}$5

The method reuses the same trained networks while adapting online to known changes in plant or MPC parameters. The paper frames this as a parameter-adaptive approximate MPC or sensitivity-corrected neural MPC, and proves a local stability-transfer result under bounded approximation error, unchanged active set, and input-robust stability of the underlying MPC (Hose et al., 2024).

A second class augments MPC with uncertainty-set dynamics. "Dual adaptive MPC using an exact set-membership reformulation" embeds the evolution of a set-membership parameter set directly into the MPC problem. Strong duality converts the predicted set-update LP into equality and nonnegativity constraints on dual variables,

$\boldsymbol{h}_{rw}$6

This makes future uncertainty reduction decision-dependent, thereby inducing performance-oriented exploration while maintaining robust constraint satisfaction and recursive feasibility (Parsi et al., 2022).

A closely related adaptive-robust line addresses time-varying additive uncertainty through an online Feasible Parameter Set. In "Adaptive MPC under Time Varying Uncertainty: Robust and Stochastic," the plant is

$\boldsymbol{h}_{rw}$7

with bounded rate of change $\boldsymbol{h}_{rw}$8. The FPS $\boldsymbol{h}_{rw}$9 is updated recursively as a polyhedron and used inside both robust and stochastic MPC formulations. The resulting algorithms prove recursive feasibility and show that the optimal robust value function is an ISS Lyapunov function for the closed loop (Bujarbaruah et al., 2019).

A third class augments output-feedback MPC with invariant observer-error geometry. "Adaptive Output Feedback Model Predictive Control" combines an adaptive observer with a two-tube robust MPC. The estimated-state tube is

$\boldsymbol{J}$0

while the true-state tube is the Minkowski augmentation

$\boldsymbol{J}$1

where $\boldsymbol{J}$2 is an invariant estimation-error set. The corresponding tightened constraint set is

$\boldsymbol{J}$3

This yields recursive feasibility and hard-constraint satisfaction for output-feedback adaptive MPC, with boundedness of the closed loop (Dey et al., 2022).

A fourth class augments MPC through uncertainty envelopes and semidefinite programming. "A Semi-Definite Programming Approach to Robust Adaptive MPC under State Dependent Uncertainty" models nonlinear additive uncertainty $\boldsymbol{J}$4 as globally Lipschitz, constructs sample-induced quadratic envelopes of its graph, intersects them online, and computes horizon-wise ellipsoidal outer approximations by SDP via the S-procedure. The robust MPC then enforces constraints for all disturbances consistent with the learned envelope (Bujarbaruah et al., 2019).

A fifth class augments the imitation-learning pipeline for AMPC with value-function structure. "Statistically Consistent Approximate Model Predictive Control" replaces behavioral cloning with the one-step look-ahead loss

$\boldsymbol{J}$5

where $\boldsymbol{J}$6 approximates the optimal value of a stabilizing soft-constrained MPC. The paper proves statistical consistency of exact minimizers and ISS guarantees for approximate minimizers, explicitly targeting the set-valued nature of nonlinear MPC solution maps (Milios et al., 12 Nov 2025).

Finally, some work uses augmentation at the solver level. "A construction-free coordinate-descent augmented-Lagrangian method for embedded linear MPC based on ARX models" specializes an augmented-Lagrangian and cyclic coordinate-descent scheme to ARX-based linear MPC. The method is construction-free, matrix-free, and library-free, avoids explicit QP construction when the ARX model changes online, and targets adaptive or LPV embedded settings (Wu et al., 2022). In a different robustification direction, "A Simple Robust MPC for Linear Systems with Parametric and Additive Uncertainty" compresses multiplicative and additive uncertainty into an effective additive term

$\boldsymbol{J}$7

then combines exact horizon-one treatment, net-additive multi-step tightening, and adaptive horizon selection to prove recursive feasibility and ISS while reducing online burden relative to tube MPC (Bujarbaruah et al., 2021).

5. Guarantees, computational structure, and performance

Across these usages, AugMPC is less a single theorem than a spectrum of tradeoffs between fidelity, adaptation, and computational tractability.

In the satellite formulation, the central advantage is that augmentation preserves the QP structure and runtime of LMPC while recovering much of the tracking quality associated with NMPC (Wang et al., 9 Mar 2026). On the reported experimental platform, Augmented-CLMPC runs in about 0.3 s per control update, similar to the other LMPC methods, whereas NMPC requires about 2.2 s. With sampling period $\boldsymbol{J}$8, the paper states that NMPC exceeds the control sampling period and is therefore not practically real-time on the target hardware. In moving-target simulations, Augmented-CLMPC reduces steady-state error from about 1.99--6.04 deg for ULMPC/CLMPC down to about 0.37 deg, essentially matching NMPC’s 0.34--0.35 deg; in the hardest phase it keeps mean squared steady-state error at 0.14 deg$\boldsymbol{J}$9 versus 36.63 and 37.48 deg$$\dot{\boldsymbol{q}}=\frac{1}{2} \left[\begin{array}{rrrr} q_0&-q_1&-q_2&-q_3\ q_1&q_0&-q_3&q_2\ q_2&q_3&q_0&-q_1\ q_3&-q_2&q_1&q_0 \end{array}\right] \left[\begin{array}{c} 0\ \boldsymbol{\omega} \end{array}\right],$$0 for ULMPC and CLMPC. The paper also evaluates observability using a commercial camera swath angle of 0.8022 deg and reports that Augmented-CLMPC and NMPC achieve nearly continuous observability across all phases (Wang et al., 9 Mar 2026).

In the RL-augmented locomotion formulation, the central computational move is hierarchical decomposition rather than model simplification. Contact scheduling is exported to the policy, while MPC remains contact-explicit and full rigid-body (Patrizi et al., 11 Mar 2026). This does not yield a general proof of recursive feasibility or stability, but it does produce a tractable real-time pipeline with explicit dynamics, friction, and contact constraints. The strongest evidence is transfer: successful zero-shot sim-to-sim transfer across all tested platforms and zero-shot sim-to-real transfer on Centauro, both without domain randomization. This suggests that the MPC layer acts as a strong structural prior and regularizer for RL, although that causal interpretation remains an inference rather than an explicit theorem.

In the adaptive and robust AugMPC-style papers, guarantees are typically stronger but more local or more conservative. Parameter-adaptive AMPC proves a local stability neighborhood

$$\dot{\boldsymbol{q}}=\frac{1}{2} \left[\begin{array}{rrrr} q_0&-q_1&-q_2&-q_3\ q_1&q_0&-q_3&q_2\ q_2&q_3&q_0&-q_1\ q_3&-q_2&q_1&q_0 \end{array}\right] \left[\begin{array}{c} 0\ \boldsymbol{\omega} \end{array}\right],$$1

under bounded nominal approximation error, bounded sensitivity error, fixed active set, and input-robust stability of the underlying MPC (Hose et al., 2024). Dual adaptive MPC proves robust constraint satisfaction and recursive feasibility but yields a nonconvex online problem because exact dual reformulation introduces bilinear constraints (Parsi et al., 2022). The time-varying-offset adaptive MPC proves recursive feasibility for both robust and stochastic variants and shows that the optimal robust value function is an ISS Lyapunov function (Bujarbaruah et al., 2019). Adaptive output-feedback MPC proves recursive feasibility and hard-constraint satisfaction through its homothetic-and-invariant two-tube construction, but leaves full asymptotic stability analysis open (Dey et al., 2022). Value-aware AMPC proves statistical consistency to the pointwise minimizer set of the augmented loss and ISS of approximate minimizers, thereby addressing a failure mode of behavioral cloning when the MPC solution is set-valued (Milios et al., 12 Nov 2025).

Solver-level augmentations reveal a different axis of performance. The ARX-based augmented-Lagrangian solver remains linear-MPC-specific, but its construction-free design is tailored to adaptive settings where model coefficients change every sample. In the reported examples it is always faster than total construction plus solve time of qpOASES and OSQP, and in several cases faster than their pure solve time alone (Wu et al., 2022). This suggests that, in some AugMPC contexts, augmentation is best interpreted as a computational architecture rather than a control law modification.

6. Limitations and research directions

Despite the breadth of results, no single AugMPC formulation dominates across domains. The satellite controller remains tied to a linearized prediction model, so sufficiently severe nonlinear regimes should still favor NMPC in principle, and the paper provides no formal stability or recursive-feasibility proof (Wang et al., 9 Mar 2026). The locomotion architecture depends on a high-quality rigid-body MPC and on training a high-level policy whose action frequency can be critical, especially in legged mode; poor values of $$\dot{\boldsymbol{q}}=\frac{1}{2} \left[\begin{array}{rrrr} q_0&-q_1&-q_2&-q_3\ q_1&q_0&-q_3&q_2\ q_2&q_3&q_0&-q_1\ q_3&-q_2&q_1&q_0 \end{array}\right] \left[\begin{array}{c} 0\ \boldsymbol{\omega} \end{array}\right],$$2 can lead to local minima where stepping fails to emerge (Patrizi et al., 11 Mar 2026).

The adaptive and AMPC-like variants expose complementary limitations. Sensitivity-corrected neural AMPC is only first-order in $$\dot{\boldsymbol{q}}=\frac{1}{2} \left[\begin{array}{rrrr} q_0&-q_1&-q_2&-q_3\ q_1&q_0&-q_3&q_2\ q_2&q_3&q_0&-q_1\ q_3&-q_2&q_1&q_0 \end{array}\right] \left[\begin{array}{c} 0\ \boldsymbol{\omega} \end{array}\right],$$3, requires the changed parameter value to be known online, and relies on differentiable NLP sensitivities; the paper explicitly notes that active-set changes can degrade the linear predictor (Hose et al., 2024). Exact dual adaptive MPC is nonconvex online, and its exact embedding of future set-membership evolution increases computational burden (Parsi et al., 2022). Output-feedback two-tube MPC is presently specialized to SISO LTI systems in observable canonical form (Dey et al., 2022). Value-aware AMPC requires learning a high-quality approximation of the soft-constrained MPC value function and assumes existence of a continuous optimal selection over the support of interest (Milios et al., 12 Nov 2025).

Taken together, these works indicate that AugMPC is best viewed as a design principle: retain MPC where its structure, constraints, and interpretability matter most, and augment only the part that blocks deployment. In satellite control, that augmentation is an embedded integrator. In legged locomotion, it is a learned contact-schedule supervisor. In adaptive and approximate control, it may be a sensitivity map, a shrinking parameter set, an invariant error set, or a value function. The unifying theme is selective augmentation rather than wholesale replacement of MPC.