Morphology-Conditioned World Model for Cross-Embodiment Quadrupedal Locomotion

Mohamad H. Danesh; Chenhao Li; Amin Abyaneh; Anas Houssaini; Kirsty Ellis; Glen Berseth; Marco Hutter; Hsiu-Chin Lin

cs.ROApr 9, 2026

Morphology-Conditioned World Model for Cross-Embodiment Quadrupedal Locomotion

Authors: Mohamad H. Danesh, Chenhao Li, Amin Abyaneh, Anas Houssaini, Kirsty Ellis, Glen Berseth, Marco Hutter, Hsiu-Chin Lin

Organizations: McGill University, Canada · Mila - Quebec AI Institute, Canada · ETH Zurich, Switzerland · Universite de Montreal, Canada

Abstract

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 $\mu$ , 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 $\mu$ .

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.	$10.1\pm 1.9$	$23.1\pm 2.3$	$-0.2\pm 3.2$
PME-PPO	Len.	$530\pm 8.2$	$602\pm 14.9$	$337\pm 23.1$
QWM	Rwd.	$18.2\pm 2.3$	$35.5\pm 3.4$	$12.1\pm 4.2$
QWM	Len.	$948.6\pm 12.1$	$974.4\pm 6.2$	$925.4\pm 17.2$
Specialist	Rwd.	$21.8\pm 1.2$	$39.7\pm 2.1$	$13.6\pm 4.2$
PPO	Len.	$981.3\pm 4.2$	$996.1\pm 1.1$	$961.1\pm 8.9$

Table 1: Zero-shot generalization performance on unseen robots. Mean

\pm

std over 5 seeds (128 envs/seed).

Figure 3: Quantitative real-world results (10 trials $\times$ 60s per platform).

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Tracking Term	Equation	Penalty Term	Equation
Lin. Vel. Track (XY)	$e^{-\\|\mathbf{v}_{xy}-\mathbf{v}^{cmd}_{xy}\\|^{2}/\sigma_{v}^{2}}$	Lin. Vel. Z	$-v_{z}^{2}$
Ang. Vel. Track (Z)	$e^{-(\omega_{z}-\omega^{cmd}_{z})^{2}/\sigma_{\omega}^{2}}$	Ang. Vel. XY	$-\\|\bm{\omega}_{xy}\\|^{2}$
Feet Air Time	$\sum(t_{air}-0.5)\cdot\mathbb{I}_{landed}$	Flat Orientation	$-\\|\mathbf{g}_{proj,xy}\\|^{2}$
		Torque	$-\\|\bm{\tau}\\|^{2}$
		Joint Acc.	$-\\|\ddot{\mathbf{q}}\\|^{2}$
		Action Rate	$-\\|\mathbf{a}_{t}-\mathbf{a}_{t-1}\\|^{2}$

Appendix

Table 2: Standard Reward Functions (ANYmal & Unitree Families)

Functional Term	Equation	Penalty Term	Equation
Base Lin. Vel (Ramp)	$m\cdot e^{-\\|\mathbf{v}_{xy}-\mathbf{v}^{cmd}_{xy}\\|/\sigma_{v}}$	Foot Slip	$-\sum\\|\mathbf{v}_{f,xy}\\|\cdot\mathbb{I}_{contact}$
where $m=$	$1+0.5(\\|\mathbf{v}^{cmd}_{xy}\\|-1)_{+}$	Base Motion	$-(0.8v_{z}^{2}+0.2\\|\bm{\omega}_{xy}\\|_{1})$
Base Ang. Vel (Abs)	$e^{-\|\omega_{z}-\omega^{cmd}_{z}\|/\sigma_{\omega}}$	Base Orientation	$-\\|\mathbf{g}_{proj,xy}\\|$
Foot Clearance	$e^{-\sum(h_{f}-0.1)^{2}\tanh(2v_{f})}$	Act. Smoothness	$-\\|\mathbf{a}_{t}-\mathbf{a}_{t-1}\\|$
Gait Enforcement	$r_{sync}\cdot r_{async}$	Air Time Var.	$-(\text{Var}(t_{air})+\text{Var}(t_{cont}))$
where $r_{i}=$	$e^{-(\Delta t_{air}^{2}+\Delta t_{cont}^{2})/\sigma}$	Joint Pos/Vel	$-\\|\mathbf{q}-\mathbf{q}_{0}\\|,-\\|\dot{\mathbf{q}}\\|$

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 ( $L^{2}$ )	-2.0	-2.0	-2.0	-2.0	-2.0	-2.0	-2.0
Ang. Vel. XY ( $L^{2}$ )	-0.05	-0.05	-0.05	-0.05	-0.05	-0.05	-0.05
Flat Orientation ( $L^{2}$ )	-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 ( $v_{z},\bm{\omega}_{xy}$ )	-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	$\mathcal{U}(0.8,0.8)$ (Default)
Dynamic Friction	$\mathcal{U}(0.6,0.6)$ (Default)
External Disturbance	$v_{push}\sim\mathcal{U}(-0.5,0.5)$ m/s (Every 10 to 15s)
Control Latency	20ms (Simulation $dt=0.005$ s, Decimation=4)
Observation Noise	Additive Uniform Distribution
Linear Velocity	$\mathcal{U}(-0.1,0.1)$ m/s

Appendix

Table 6: Physics and Observation Randomization Parameters

Parameter	Distribution / Range
Mass (Large Robots)	Base $\pm 5.0$ kg
Mass (Small Robots)	Base $+[-1.0,3.0]$ kg
Center of Mass (Base)	$\pm 5$ cm (X, Y), $\pm 1$ cm (Z)

Appendix

Table 7: Morphology Randomization Parameters

State	Initialization
Base Position	Offset $\pm 0.5$ m (X,Y)
Base Yaw	$\mathcal{U}(-\pi,\pi)$
Base Velocity (Lin/Ang)	$\mathcal{U}(-0.5,0.5)$ (Zero for Small Unitrees)
Joint Pos (ANYmal, B2)	Nominal $\times\mathcal{U}(0.8,1.2)$
Joint Pos (Small Unitree)	Nominal (Fixed)
Joint Pos (Spot)	Nominal $+\mathcal{U}(-0.05,0.05)$ rad

Appendix

Table 8: Reset Randomization

Robot Group	$\mathbf{v}_{x}$ (m/s)	$\mathbf{v}_{y}$ (m/s)	$\bm{\omega}_{z}$ (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	$m_{\text{trunk}}/M_{\text{tot}}$	3/9
ANYmal-C	0.90	1.61	$\mu_{\text{act}}$	1/9
ANYmal-D	1.10	1.56	$\mu_{\text{act}}$	1/9
Spot	0.77	2.30	$l_{\text{stance}}/w_{\text{stance}}$	1/9
Unitree A1	1.17	1.61	$l_{\text{thigh}}$	4/9
Unitree Go1	1.09	1.49	$m_{\text{trunk}}/M_{\text{tot}}$	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 $2.5\sigma$ . Features are grouped by category (Dynamics, Kinematics, Geometry). The discrete topology flag $k_{\text{cfg}}$ 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 $h_{t}$ (left) and $z_{t}$ (right), both colored by robot identity. The $\mu$ -conditioned recurrent state organizes into morphology-specific clusters. The stochastic state collapses into a single morphology-agnostic cloud. By morphology the silhouette score of $h_{t}$ greatly exceeds that of $z_{t}$ , quantifying the asymmetry the architecture is designed to produce.

Figure 8: PCA of QWM latent states $[h_{t},z_{t}]$ , 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 $[h_{t},z_{t}]$ , 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 $h_{t}$ vs. $z_{t}$ . PCA projections of each latent component, colored by forward speed $v_{x}$ . 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 ( $h_{t}$ ): Velocity gradients are visible within each morphology-specific region, indicating that $h_{t}$ tracks the evolving dynamic trajectory of each robot in its morphology-aligned subspace. Right ( $z_{t}$ ): The cloud that is unstructured by morphology (see Fig. 7 ), yet a smooth $v_{x}$ gradient persists across the full projection, confirming that $z_{t}$ retains the instantaneous dynamic state of the robot even as it sheds morphological identity.

Metric	Value	Interpretation
Silhouette score (morphology)	$0.033$	$\approx 0$ $\Rightarrow$ 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

z_{t}

across all

32{,}768

collected points. Low silhouette score and high within-class variance fraction confirm that

z_{t}

encodes negligible morphological information.

Method	$\delta$	ANYmal-D	Unitree Go1	Unitree B2
PME-PPO	$0.00$	$530\pm 8.2$	$602\pm 14.9$	$337\pm 23.1$
	$0.05$	$441\pm 18.4$	$498\pm 22.7$	$284\pm 31.8$
	$0.10$	$329\pm 27.1$	$387\pm 31.5$	$223\pm 38.4$
	$0.20$	$174\pm 41.6$	$208\pm 47.2$	$142\pm 44.9$
QWM	$0.00$	$948.6\pm 12.1$	$974.4\pm 6.2$	$925.4\pm 17.2$
	$0.05$	$931.2\pm 14.8$	$962.1\pm 9.4$	$897.6\pm 22.5$

Appendix

Table 12: Robustness to

\mu

perturbation (episode length under multiplicative noise on raw features). QWM degrades gracefully. PME-PPO fails sharply, lacking a latent correction channel. Mean

\pm

std over 20 draws

\times

5 seeds.

\delta=0

reproduces Table 1 .

Perturb.	Level	Analytical- $\mu$	QWM
Torque ( $\eta$ )	$0.0$	$0.03\pm 0.01$	$0.11\pm 0.03$
	$0.1$	$0.34\pm 0.06$	$0.16\pm 0.04$
	$0.2$	$0.72\pm 0.11$	$0.24\pm 0.05$
Payload ( $p$ )	$0.0$	$0.03\pm 0.01$	$0.11\pm 0.03$
	$0.1$	$0.28\pm 0.05$	$0.15\pm 0.03$
	$0.2$	$0.61\pm 0.09$	$0.21\pm 0.05$

Appendix

Table 13: Open-loop NMSE under unmodeled dynamics, averaged over the three held-out robots (45-step horizon, normalized by motion variance). Analytical-

\mu

is near exact at nominal but diverges once the true world departs from spec. QWM stays bounded because

z_{t}

reads the realized dynamics from observations.

Method	Level	ANYmal-D	Unitree Go1	Unitree B2
Torque Attenuation ( $\eta$ )
PME-PPO	$0.0$	$530\pm 8.2$	$602\pm 14.9$	$337\pm 23.1$
	$0.1$	$388\pm 21.4$	$447\pm 26.8$	$233\pm 31.6$
	$0.2$	$244\pm 33.7$	$281\pm 37.2$	$148\pm 36.9$
QWM	$0.0$	$948.6\pm 12.1$	$974.4\pm 6.2$	$925.4\pm 17.2$
	$0.1$	$871.4\pm 19.8$	$902.7\pm 15.3$	$838.1\pm 26.4$

Appendix

Table 14: Robustness to unmodeled dynamics. Closed-loop episode length under torque attenuation and payload, with nominal

\mu

. QWM degrades gracefully, since

z_{t}

reads true dynamics from

o_{t}

. PME-PPO, lacking a learned model, fails sharply. Mean

\pm

std over 20 draws

\times

5 seeds. The

0.0

rows reproduce Table 1 .

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PolyMorph: Learning Locomotion Policy for Cross-Embodiment Adaptation

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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.

Abstract

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