MUJICA: Unified Wheeled-Legged Control

• MUJICA is a unified control framework for wheeled-legged robots that integrates omnidirectional moving, platform climbing, and fall recovery under a single proprioceptive policy.
• It employs a two-stage architecture where a shared low-level policy uses a dedicated skill indicator and a high-level selector autonomously switches modes based on proprioceptive history.
• The system achieves robust sim-to-real performance by incorporating a state estimator and enforcing hardware-aware DC-motor constraints to ensure safe, adaptive control.

Searching arXiv for the specified MUJICA paper and related entries to ground the article in current arXiv records. MUJICA, short for Multi-skill Unified Joint Integration of Control Architecture, is a unified control framework for wheeled-legged robots that integrates diverse low-level skills—including omnidirectional moving, high platform climbing, and fall recovery—within a single fully proprioceptive policy. The framework is designed for the specific difficulties of wheeled-legged locomotion: the need to balance wheeled driving with legged control, to remain robust under noisy proprioceptive sensing, and to respect real-world motor constraints that are often oversimplified in reinforcement learning. In the reported implementation, MUJICA is validated in simulation and on the Unitree Go2-W robot, with an emphasis on sim-to-real robustness, seamless transitions across locomotion modes, and adaptive behavior based solely on proprioceptions (Li et al., 13 May 2026).

1. Scope and problem formulation

MUJICA is motivated by three obstacles identified for wheeled-legged robots in unstructured environments. First, existing multi-skill blind controllers usually handle only a narrow family of tasks, often with similar dynamics, such as flat-ground walking or stair/slope traversal, rather than unifying strongly dissimilar skills such as omnidirectional moving, high platform climbing, and fall recovery in one policy. Second, many multitask methods rely on manual switching or multiple separate experts, which increases system complexity and can create interference between behaviors. Third, real robots exhibit actuator constraints that are frequently modeled too coarsely in RL; for DC motors, the torque limit depends not only on magnitude but also on joint velocity, and for some joints also on joint position (Li et al., 13 May 2026).

The control problem is modeled as a Constrained Partially Observable Markov Decision Process (C-POMDP):

$$\max \mathbb{E}_{\pi}\left[\sum_{t=0}^\infty \gamma^t R\left(\boldsymbol{s}_{t},\bm{a}_t,\boldsymbol{s}_{t+1}\right)\right] \ \text{s.t.}\ \ \mathbb{E}_{\pi}\left[\sum_{t=0}^\infty \gamma^t C_i\left(\boldsymbol{s}_{t},\boldsymbol{a}_t,\boldsymbol{s}_{t+1}\right)\right] \leq \delta_i,\forall i \in \left\{1,...,k\right\}$$

A defining design choice is that the deployment policy is blind: it uses only proprioceptive observations, while privileged information is used during training for critics and auxiliary supervision. This places MUJICA in the class of blind locomotion systems, but with an explicit focus on multi-skill wheeled-legged control under hardware-feasible actuation (Li et al., 13 May 2026).

The proprioceptive observation is

$\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$

and the privileged state is

$$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$

Here, the observation includes base angular velocity, gravity vector in the base frame, command input, joint positions, joint velocities, previous action, and a skill indicator $\zeta_t$; the privileged state augments this with base linear velocity, collision-related privileged signals, wheel-ground distances, and a latent state or hidden embedding (Li et al., 13 May 2026).

2. Unified multi-skill architecture

MUJICA is organized into two stages. In Stage S1, a single low-level policy is trained jointly on three skills: omnidirectional moving on ordinary terrains, high platform climbing, and fall recovery. These skills are unified by conditioning the shared policy on a skill indicator $\zeta_t$, with a unique indicator variable for each skill. The paper states that the indicator does not merely act as a tag; it separates the policy’s behavior space along a dedicated dimension, allowing one network to represent distinct low-level behaviors without collapsing them into a single undifferentiated control mode (Li et al., 13 May 2026).

In Stage S2, the low-level skills are frozen and a high-level skill selector is trained. This selector receives a short history of proprioceptive observations, excluding the skill indicator, and predicts which skill should be active. It therefore chooses $\zeta_t$ automatically and enables autonomous transitions among recovery, rolling, and climbing modes based only on internal state (Li et al., 13 May 2026).

This hierarchy distinguishes MUJICA from systems based on manual switching or multiple separate experts. The low-level controller is shared across tasks, but the tasks are explicitly disentangled by the indicator variable. The high-level selector then learns when to activate each mode. A plausible implication is that the framework treats multi-skill control not as simple reward aggregation, but as structured behavior composition under partial observability.

The paper reports that this mechanism is important in sequential tasks. In an ablation comparing policies without an indicator to the full system, the reported stage-wise success rates are 72%, 72%, 38% for w/o indicator, shared reward, versus 95%, 95%, 91% for with skill selector (ours) (Li et al., 13 May 2026). This suggests that explicit skill conditioning and learned switching are central to the system’s ability to execute heterogeneous task sequences.

3. Observation design, estimator, and blind control

Because MUJICA is deployed without cameras, depth sensing, or terrain reconstruction, it supplements the blind policy with a learned state estimator. The estimator uses the last $H=6$ frames of proprioceptive observations and processes them through an online encoder with a GRU:

$$\boldsymbol{f}_t=\operatorname{GRU}\left(\operatorname{NN}\left(\boldsymbol{o}_{t-H:t}\right),\boldsymbol{f}_{t-1}\right)$$

with output

$$\boldsymbol{f}_t = [\boldsymbol{\hat{v}_t,\hat{c}_t,\hat{u}_t,\hat{e}_t}]^T.$$

The estimator predicts base linear velocity, collision probabilities for robot components, wheel-ground distances, and a latent state vector (Li et al., 13 May 2026).

The reported motivation for these prediction targets is task-specific. Velocity helps distinguish stable from unstable locomotion. Wheel-ground distance helps infer roughness and avoid unnecessary leg lifting on flat ground. Collision probabilities help detect contact with obstacles and support adaptive behaviors such as using head collision to sense a ledge (Li et al., 13 May 2026). This is a structured privileged-to-blind transfer mechanism rather than a generic latent-state learner.

The estimator is trained with supervised prediction losses and a contrastive self-supervision term:

$\mathcal{L}^{\text{Estimate}=\mathcal{L}^{\text{Pred}+\mathcal{L}^{\text{SwAV}(\boldsymbol{e}_{t},\boldsymbol{\hat{e}_{t})$

$\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$0

where $\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$1 is mean squared error, $\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$2 is binary cross-entropy, and SwAV is the self-supervised contrastive loss from Caron et al. A reference encoder maps successor observations $\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$3 to a latent target $\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$4, which supervises the online encoder (Li et al., 13 May 2026).

Ablations reported in the paper indicate that removing any estimator component reduces performance; removing velocity estimation is described as particularly harmful for non-recovery locomotion. The comparison against DreamWaQ + P3O is reported to favor MUJICA, which the paper attributes to the chosen state targets and architecture being better suited to wheeled-legged tasks. Vanilla PPO can traverse some simple terrain, especially slopes, but cannot learn robust fall recovery without state estimation (Li et al., 13 May 2026).

4. Action space, low-level control, and constrained optimization

The action vector $\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$5 contains commands for all joints. MUJICA uses a heterogeneous low-level actuation interface: for leg joints, the policy outputs angular offsets from a default posture; for wheel joints, it outputs desired motor velocity. These outputs are converted to torques by a PD-style controller:

$\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$6

Thus wheels are velocity-controlled, whereas leg joints are controlled by position offsets (Li et al., 13 May 2026).

Training uses an asymmetric actor-critic. The actor sees the blind input stream plus the estimator output, while the critics receive privileged state information. Constraint handling is performed with P3O (Penalized PPO):

$\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$7

together with the reward and constraint surrogates stated in the paper (Li et al., 13 May 2026). This optimization setup is used to balance task performance against safety-related constraint violations.

The main reward and constraint terms listed in the system are:

$\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$8

$\boldsymbol{o}_t=\left[\boldsymbol{\omega}^b_t,\boldsymbol{g}^b_t,\boldsymbol{cmd}_t,\boldsymbol{q}_t,\boldsymbol{\dot{q}_t,\boldsymbol{a}_t,\zeta_t\right]^T$9

$$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$0

$$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$1

$$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$2

$$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$3

The task applicability is also explicit: $$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$4 is used for skills i and ii, $$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$5 for skill i, $$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$6 and $$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$7 for skill iii, $$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$8 for all tasks, and $$\boldsymbol{s}_t\triangleq \boldsymbol{o}^{priv}_t=\left[\boldsymbol{o}_t,\boldsymbol{v}_t,\boldsymbol{c}_t,\boldsymbol{u}_t,\boldsymbol{h}_t\right]^T.$$9 for i and ii (Li et al., 13 May 2026).

A particularly important modeling choice is the hard DC-motor constraint. The paper states that, at low speed, maximum torque is constant; at high speed, maximum torque decreases linearly with velocity; and calf joints also exhibit a position-dependent limit with cosine-like dependence. The authors describe this as being inspired by actuator operating-region data from the Unitree motor manual and prior actuator-constrained RL literature (Li et al., 13 May 2026). The reported ablation indicates that, without this constraint, simulated policies often leave feasible actuator regions, especially during high platform climbing, causing unsafe torques and motor faults on the real robot. With the constraint, the torque-velocity distribution remains inside the feasible region and violation rates drop to below 3.5% (Li et al., 13 May 2026).

5. Training pipeline, curriculum, and empirical results

Training is conducted in IsaacLab with 4,096 parallel environments, a single NVIDIA RTX 4090, 50 Hz control for both the low-level policy and skill selector, and 200 Hz physics simulation. Low-level training runs for 30,000 iterations, and high-level selector training for 10,000 iterations (Li et al., 13 May 2026).

The curriculum is defined on a 33 × 20 grid, where rows correspond to terrain or task assignments and columns correspond to difficulty levels. For Stage S1, the correct skill indicator is provided. For Stage S2, all environments are treated equally and the selector learns to choose skills autonomously (Li et al., 13 May 2026). The listed terrains are stairs, slope, discretized terrain, rough terrain, and a pit or depression used for platform climbing. The difficulty parameter $\zeta_t$0 modifies terrain parameters, including:

• Stairs: step width $\zeta_t$1 m, step height $\zeta_t$2 m
• Slope: $\zeta_t$3
• Discretized terrain: 30 bumps, heights $\zeta_t$4 m
• Rough terrain: height sampled from $\zeta_t$5 m
• Pit: $\zeta_t$6, depth $\zeta_t$7 m (Li et al., 13 May 2026)

Domain randomization is used for sim-to-real robustness. The randomized parameters include static friction in $\zeta_t$8, dynamic friction in $\zeta_t$9, base mass bias in $\zeta_t$0 kg, external force every 2–3 s in $\zeta_t$1 N, push every 8–12 s at $\zeta_t$2 m/s in $\zeta_t$3 or $\zeta_t$4, and motor gain multiplier in $\zeta_t$5 (Li et al., 13 May 2026).

The experimental platform is the Unitree Go2-W, with real-world computation on an NVIDIA Jetson Orin NX (Li et al., 13 May 2026). The paper evaluates individual skills, sequential multi-skill transitions, sim-to-real transfer, state-estimator ablations, DC-motor-constraint ablations, and comparisons against MUJICA, MUJICA w/o Velocity, MUJICA w/o Wheel Height, MUJICA w/o Collision, DreamWaQ + P3O, and Vanilla PPO.

The reported qualitative results include successful omnidirectional movement on ordinary and uneven terrain, stair recovery from upside-down states on 30° stairs, and high platform climbing on an 80-cm outdoor platform and a 1-m indoor platform, the latter described as near the robot’s mechanical and actuation limits (Li et al., 13 May 2026). The paper also reports that the robot can chain recovery, stair climbing, ramp traversal, and high-platform climbing in one continuous mission, with the high-level selector switching skills autonomously.

6. Interpretation, significance, and disambiguation

MUJICA occupies a specific niche within locomotion research: unified multi-skill, blind, constraint-aware control for wheeled-legged robots. Its defining combination is a shared low-level policy, an explicit skill indicator, a learned high-level selector, a state estimator driven by proprioceptive history, and physically informed DC-motor constraints (Li et al., 13 May 2026). This differs from single-task controllers, manually switched systems, and methods that simplify actuator feasibility.

The system’s practical significance, as presented in the paper, lies in showing that a wheeled-legged robot can recover from falls, roll, climb, and switch among these modes without external sensing and without leaving the safe motor envelope on real hardware (Li et al., 13 May 2026). A plausible implication is that the framework is aimed at operational regimes where exteroception is unreliable or unavailable, and where actuator-safe behavior is as critical as task completion.

The term “Mujica” also appears in a separate body of arXiv literature as the surname of Jorge Mujica, associated with research on holomorphic functions on Banach spaces, polynomial decompositions, and operator-theoretic structures. For example, recent work on monomial bases in spaces of holomorphic functions explicitly extends results of Dineen and Mujica (Grando et al., 7 Jul 2025), while several other papers invoke Mujica’s linearization theorem or related results in holomorphic Lipschitz spaces, holomorphic ideals, norm-attaining operators, and convolution dynamics (Aron et al., 2023, Cabrera-Padilla et al., 2022, Jung, 2021, Dantas et al., 2021, Caraballo et al., 2018). In the robotics context, however, MUJICA is an acronym for Multi-skill Unified Joint Integration of Control Architecture, not a reference to that functional-analytic literature (Li et al., 13 May 2026).