World models promise a paradigm shift in robotics, where an agent learns the physics of its environment once and then acquires behaviors efficiently. Yet the learned dynamics models at their core are typically morphology locked. In legged locomotion, a dynamics model trained on an ANYmal-D quadruped fails on a Unitree Go1 because it overfits to one robot's embodiment rather than capturing the locomotion dynamics shared across robots, so even a small change in actuator dynamics or limb length forces retraining from scratch. However, if we formalize a robot's unique physical traits into a morphology specification, a controller for a family of robots can utilize this blueprint in two ways. It can feed the specification to a model-free policy, or it can feed the specification to a learned dynamics model and extract the policy in imagination. We argue for the second route and introduce the Quadrupedal World Model (QWM), which conditions a single generative dynamics model on scale-invariant physical features and trains policies entirely inside it, through a physical morphology encoder, an adaptive reward normalizer, and morphology conditioning in the latent dynamics. Holding the morphology information identical, a model-free policy matches QWM on the training cohort but degrades on unseen morphologies, while QWM transfers zero-shot with no fine-tuning, adaptation, or warm-up in such cases. To our knowledge, this is the first world model to demonstrate zero-shot cross-embodiment transfer within the quadrupedal family.
Figures & tables
Figure 1: Overview of QWM. Left : The heterogeneous quadrupedal morphology cohort. Middle : Training pipeline, where robot data is processed by PME ( Subsec. 4.2 ) to extract the morphology vector μ , which together with environment observations trains QWM, while ARN ( Subsec. 4.3 ) keeps the reward signals consistent across diverse platforms. Right : Deployment pipeline, where for an unseen robot the same PME module from training extracts its morphology vector μ .
Figure 2: Training performance and dynamics fidelity on the heterogeneous cohort. Shaded regions denote std over 5 seeds. Left panel : Learning curves on the full heterogeneous cohort, with mean reward (top) and mean episode length (bottom). Right panel : Long-horizon dynamics prediction. Open-loop ground truth is compared against imagination over 85 steps, where QWM stays synchronized across scales with minor late deviations (circled). The plots report NMSE over the training prediction window (32 trajectories), normalized by each robot’s natural motion variance, where QWM accumulates error gradually but does not diverge.
Method
Metric
ANYmal-D
Unitree Go1
Unitree B2
PME-PPO
Rwd.
10.1±1.9
23.1±2.3
−0.2±3.2
Len.
530±8.2
602±14.9
337±23.1
QWM
Rwd.
18.2±2.3
35.5±3.4
12.1±4.2
Len.
948.6±12.1
974.4±6.2
925.4±17.2
Specialist
Rwd.
21.8±1.2
39.7±2.1
13.6±4.2
PPO
Len.
981.3±4.2
996.1±1.1
961.1±8.9
Table 1: Zero-shot generalization performance on unseen robots. Mean ± std over 5 seeds (128 envs/seed).
Table 2: Standard Reward Functions (ANYmal & Unitree Families)
Functional Term
Equation
Penalty Term
Equation
Base Lin. Vel (Ramp)
m⋅e−∥vxy−vxycmd∥/σv
Foot Slip
−∑∥vf,xy∥⋅Icontact
where m=
1+0.5(∥vxycmd∥−1)+
Base Motion
−(0.8vz2+0.2∥ωxy∥1)
Base Ang. Vel (Abs)
e−∣ωz−ωzcmd∣/σω
Base Orientation
−∥gproj,xy∥
Foot Clearance
e−∑(hf−0.1)2tanh(2vf)
Act. Smoothness
−∥at−at−1∥
Gait Enforcement
rsync⋅rasync
Air Time Var.
−(Var(tair)+Var(tcont))
where ri=
e−(Δtair2+Δtcont2)/σ
Joint Pos/Vel
−∥q−q0∥,−∥q˙∥
Appendix
Table 3: Spot-Specific Reward Functions
Reward Term
ANYmal-D
ANYmal-C
ANYmal-B
Unitree A1
Unitree Go1
Unitree Go2
Unitree B2
Lin. Vel. Tracking (XY)
1.0
1.0
1.0
1.5
1.5
1.5
1.0
Ang. Vel. Tracking (Z)
0.5
0.5
0.5
0.75
0.75
0.75
0.75
Feet Air Time
0.5
0.5
0.5
0.25
0.25
0.25
0.5
Lin. Vel. Z ( L2 )
-2.0
-2.0
-2.0
-2.0
-2.0
-2.0
-2.0
Ang. Vel. XY ( L2 )
-0.05
-0.05
-0.05
-0.05
-0.05
-0.05
-0.05
Flat Orientation ( L2 )
-5.0
-5.0
-5.0
-2.5
-2.5
-2.5
-5.0
Appendix
Table 4: Reward Weights for ANYmal and Unitree Robots
Reward Term
Weight
Reward Term
Weight
Gait Reward
10.0
Base Motion Penalty ( vz,ωxy )
-2.0
Base Linear Vel (Ramp)
5.0
Base Orientation Penalty
-3.0
Base Angular Vel
5.0
Action Smoothness
-1.0
Air Time Reward
5.0
Air Time Variance
-1.0
Foot Clearance
0.5
Foot Slip
-0.5
Joint Position Penalty
-0.7
Joint Acceleration
-1.0e-4
Appendix
Table 5: Reward Weights for Boston Dynamics Spot
Parameter
Value / Distribution
Static Friction
U(0.8,0.8) (Default)
Dynamic Friction
U(0.6,0.6) (Default)
External Disturbance
vpush∼U(−0.5,0.5) m/s (Every 10 to 15s)
Control Latency
20ms (Simulation dt=0.005 s, Decimation=4)
Observation Noise
Additive Uniform Distribution
Linear Velocity
U(−0.1,0.1) m/s
Appendix
Table 6: Physics and Observation Randomization Parameters
Parameter
Distribution / Range
Mass (Large Robots)
Base ±5.0 kg
Mass (Small Robots)
Base +[−1.0,3.0] kg
Center of Mass (Base)
±5 cm (X, Y), ±1 cm (Z)
Appendix
Table 7: Morphology Randomization Parameters
State
Initialization
Base Position
Offset ±0.5 m (X,Y)
Base Yaw
U(−π,π)
Base Velocity (Lin/Ang)
U(−0.5,0.5) (Zero for Small Unitrees)
Joint Pos (ANYmal, B2)
Nominal ×U(0.8,1.2)
Joint Pos (Small Unitree)
Nominal (Fixed)
Joint Pos (Spot)
Nominal +U(−0.05,0.05) rad
Appendix
Table 8: Reset Randomization
Robot Group
vx (m/s)
vy (m/s)
ωz (rad/s)
All Robots
[−1.0,1.0]
[−1.0,1.0]
[−1.0,1.0]
Appendix
Table 9: Velocity Command Ranges
Robot
Mean ∣z∣
Max ∣z∣
Worst feature
# ∣z∣>1.5
ANYmal-B
0.83
1.79
mtrunk/Mtot
3/9
ANYmal-C
0.90
1.61
μact
1/9
ANYmal-D
1.10
1.56
μact
1/9
Spot
0.77
2.30
lstance/wstance
1/9
Unitree A1
1.17
1.61
lthigh
4/9
Unitree Go1
1.09
1.49
mtrunk/Mtot
0/9
Appendix
Table 10: Per-robot LOO z -score summary over 9 continuous features.
Figure 5: Per-feature LOO z -scores for the robot cohort. Each cell reports the raw physical value (top) and its signed deviation from the LOO cohort distribution in standard-deviation units (bottom). Color intensity encodes ∣z∣ , saturating at 2.5σ . Features are grouped by category (Dynamics, Kinematics, Geometry). The discrete topology flag kcfg is excluded.
Figure 6: Ablation study on the heterogeneous cohort. We compare QWM against architectural ablations on morphology encoding (PME), reward normalization (ARN), and conditioning locations. Shaded regions denote standard deviation across 5 seeds. Key findings. (1) QWM achieves the fastest convergence and highest stable asymptotic performance. (2) Removing encoder conditioning allows initial learning but leads to a catastrophic late-stage collapse. (3) Removing explicit morphology (PME) maximizes episode length but plateaus at a suboptimal reward, showing explicit morphology is needed for refined gaits. (4) Removing ARN prevents learning, leaving a flatline near zero reward.
Figure 7: Latent disentanglement. PCA of ht (left) and zt (right), both colored by robot identity. The μ -conditioned recurrent state organizes into morphology-specific clusters. The stochastic state collapses into a single morphology-agnostic cloud. By morphology the silhouette score of ht greatly exceeds that of zt , quantifying the asymmetry the architecture is designed to produce.
Figure 8: PCA of QWM latent states [ht,zt] , morphology (a) vs. dynamic state gradients (b-e). Each point is one latent observation, and 32768 points are shown across all eight robots. (a) Coloring by robot identity reveals that the latent space organizes itself into morphology-specific clusters, even though robot identity is never provided as a supervised signal. (b-e) The same projection colored by four continuous dynamic variables shows smooth, consistent gradients within each morphology cluster, demonstrating that dynamic execution context is encoded orthogonally to morphological identity. PCA explained variance for the first two components is 7.9% , reflecting the high-dimensional nature of the full latent space.
Figure 9: t-SNE of QWM latent states [ht,zt] , morphology (a) vs. dynamic state gradients (b–e). t-SNE (perplexity =40 , 1000 iterations, KL divergence =1.73 ) is applied to the same 32768 -point dataset as Fig. 8 . (a) Each robot occupies a spatially distinct, often multi-lobed cluster, and the multi-lobe structure within a single robot reflects different locomotion regimes (e.g., slow walk vs. fast trot). (b–e) Dynamic variables exhibit strong intra-cluster gradients, confirming that within each morphology’s region the model encodes a continuous dynamic state manifold.
Figure 10: Dynamic-state probing of ht vs. zt . PCA projections of each latent component, colored by forward speed vx . The morphology-colored counterpart of this figure appears earlier in this section ( Fig. 7 ), and here we show the same projections under a dynamic-state coloring to complete the probing picture. Left ( ht ): Velocity gradients are visible within each morphology-specific region, indicating that ht tracks the evolving dynamic trajectory of each robot in its morphology-aligned subspace. Right ( zt ): The cloud that is unstructured by morphology (see Fig. 7 ), yet a smooth vx gradient persists across the full projection, confirming that zt retains the instantaneous dynamic state of the robot even as it sheds morphological identity.
Metric
Value
Interpretation
Silhouette score (morphology)
0.033
≈0⇒ no cluster separation
Between-class variance fraction
0.098
9.8% of variance is inter-morphology
Within-class variance fraction
0.901
90.1% of variance is intra-morphology
Appendix
Table 11: Quantitative disentanglement metrics computed on the stochastic state zt across all 32,768 collected points. Low silhouette score and high within-class variance fraction confirm that zt encodes negligible morphological information.
Method
δ
ANYmal-D
Unitree Go1
Unitree B2
PME-PPO
0.00
530±8.2
602±14.9
337±23.1
0.05
441±18.4
498±22.7
284±31.8
0.10
329±27.1
387±31.5
223±38.4
0.20
174±41.6
208±47.2
142±44.9
QWM
0.00
948.6±12.1
974.4±6.2
925.4±17.2
0.05
931.2±14.8
962.1±9.4
897.6±22.5
Appendix
Table 12: Robustness to μ perturbation (episode length under multiplicative noise on raw features). QWM degrades gracefully. PME-PPO fails sharply, lacking a latent correction channel. Mean ± std over 20 draws × 5 seeds. δ=0 reproduces Table 1 .
Perturb.
Level
Analytical- μ
QWM
Torque ( η )
0.0
0.03±0.01
0.11±0.03
0.1
0.34±0.06
0.16±0.04
0.2
0.72±0.11
0.24±0.05
Payload ( p )
0.0
0.03±0.01
0.11±0.03
0.1
0.28±0.05
0.15±0.03
0.2
0.61±0.09
0.21±0.05
Appendix
Table 13: Open-loop NMSE under unmodeled dynamics, averaged over the three held-out robots (45-step horizon, normalized by motion variance). Analytical- μ is near exact at nominal but diverges once the true world departs from spec. QWM stays bounded because zt reads the realized dynamics from observations.
Method
Level
ANYmal-D
Unitree Go1
Unitree B2
Torque Attenuation ( η )
PME-PPO
0.0
530±8.2
602±14.9
337±23.1
0.1
388±21.4
447±26.8
233±31.6
0.2
244±33.7
281±37.2
148±36.9
QWM
0.0
948.6±12.1
974.4±6.2
925.4±17.2
0.1
871.4±19.8
902.7±15.3
838.1±26.4
Appendix
Table 14: Robustness to unmodeled dynamics. Closed-loop episode length under torque attenuation and payload, with nominal μ . QWM degrades gracefully, since zt reads true dynamics from ot . PME-PPO, lacking a learned model, fails sharply. Mean ± std over 20 draws × 5 seeds. The 0.0 rows reproduce Table 1 .
Reinforcement learning has enabled robust locomotion for legged robots, but learned policies are typically tied to a specific robot morphology and require additional engineering when the number of legs, actuators, or mechanical properties change. We present PolyMorph, a morphology-conditioned framework that learns a single locomotion policy for heterogeneous legged robots. PolyMorph converts each robot's URDF into joint-link modules and combines a recurrent encoder, graph-based Transformer, and per-joint decoder to generate actions across variable leg counts and joint configurations. To improve training stability, PolyMorph uses morphology-normalized rewards, adaptive robot sampling, and URDF-derived PD-gain initialization. We evaluate PolyMorph in simulation across bipedal, tripedal, and quadrupedal robots, including unseen morphology variants and platforms, and deploy the policy on real quadruped platforms operated in multiple configurations. The results show that a single PolyMorph policy generalizes across diverse robot morphologies without per-robot policy retraining.
Generalizing learned locomotion policies across quadrupedal robots with different morphologies remains a challenge. Policies trained on a single robot often fail when deployed on embodiments with different mass distributions, kinematics, joint limits, or actuation constraints, forcing per-robot retraining. Prior works have approached this primarily either by scaling via training across real or generated embodiments or using large architectures producing transferable policies which are both storage and compute heavy. We argue that both of these approaches circumvent a key failure mode in actor-critic learning: a shared value function tends to average incompatible value targets across embodiments, yielding miscalibrated advantages and fixing that helps transfer and reduce the compute cost. We present MorFiC, a reinforcement learning approach for zero-shot cross-morphology locomotion which fixes this issue using multiplicative critic conditioning on a morphology latent. Trained with a single source robot with morphology randomization in simulation, MorFiC achieves zero-shot transfer to total seven robots and reaching forward velocity competitive or surpassing scaling-based and morphology-conditioned PPO baselines for examples 1.98 m/s on AlienGo, where as additive PPO baselines remain below 0.65 m/s. MorFiC achieves this using single training robot within 2 hours of training time compared to tens or hundreds of hours required for scaling baselines training on multiple embodiments. We diagnose critic calibration via three metric: Explained variance, advantage sign-flip rate and policy gradient cosine, confirming our critic conditioning as determinant for policy update direction. Finally, we demonstrate zero-shot deployment on Unitree Go1 and Go2 robots without fine-tuning.
Prakhar Mishra, Amir Hossain Raj, Xuesu Xiao +1
University of Maryland, College Park, MD, USA · George Mason University, Fairfax, VA, USA
Action-conditioned world models are promising learned simulators for robotic manipulation, yet evaluating them exclusively on training robots fails to reveal whether they capture physical dynamics or merely memorize visual patterns. To answer whether a model can faithfully render a robot it has never seen, we introduce XEWorld, a controlled cross-embodiment testbed for world models that isolates embodiments by evaluating held-out robots within physically identical scenes. Our systematic analysis uncovers a shared architectural bottleneck: current models act primarily as 2D visual pattern matchers whose generalization is governed by visual similarity rather than physical kinematic similarity. Driven by this limitation, they struggle to translate abstract numeric joint actions into coherent visual trajectories, and fail to predict dynamic visual changes from static initial observations. Consequently, successfully rendering an unseen embodiment zero-shot strictly requires heavily grounded cues, specifically pixel-space actions and explicit spatial-temporal alignment. Even when bypassing this zero-shot barrier via few-shot adaptation, the forced appearance recovery triggers catastrophic forgetting of seen embodiments. Together, these failures expose a critical inability to apply learned physical dynamics to novel visual appearances, highlighting that achieving true cross-embodiment generalization requires architectural innovations that decouple visual appearance from underlying physical dynamics.
Yixiang Chen, Jiabing Yang, Yuan Xu +10
New Laboratory of Pattern Recognition (NLPR), Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences · Shandong University +1