PAPER DEEP DIVE
PRISM: Polynomial Representations for Interaction-Structured Motor Control
Robot policies are typically MLPs mapping observations to actions. Yet robot observations are physical variables, and many action-relevant cues arise not from individual variables but from their interactions; power, inertial effects, contact, slip, and compliance depend on products among observable signals. We introduce PRISM, a policy representation that makes polynomial interactions among observable physical variables explicit, learnable, and compact. Rather than listing all polynomial terms, PRISM uses a factorized polynomial module to expose higher-order interaction features efficiently. In reinforcement learning, it keeps the standard MLP backbone but applies a gradually activated element-wise polynomial function after it. In imitation learning, it replaces linear proprioceptive conditioning in Diffusion Policy with a polynomial layer trained end-to-end. Across humanoid locomotion and contact-rich manipulation, PRISM improves performance over standard MLP policies and larger MLPs with matched capacity, showing that interaction structure cannot be replaced by capacity alone. It also yields sensorless compliant behavior without force, wrench, tactile input, contact labels, or admittance control. These results suggest that polynomial representations should become a standard architectural choice for embodied motor control. The project page is available at https://lsh3163.github.io/prism/
PRISM: Polynomial Representations for Interaction-Structured Motor Control
Institution: University of Michigan, Ann Arbor | arXiv: 2607.23473v1
One-Sentence Summary
PRISM is a policy representation using a factorized polynomial module to explicitly expose polynomial interactions among observable physical variables (power, inertia, contact, slip), applied after MLP backbone in RL and replacing linear proprioceptive conditioning in Diffusion Policy for IL, significantly outperforming standard and larger MLPs while producing sensorless compliant behavior.
Background and Motivation
Robot policies use MLPs to map observations to actions. But robot observations are physical variables — joint positions, velocities, commands, IMU. Many action-relevant quantities are products of variables: joint power is torque × velocity, Coriolis effects depend on velocity products, slip and contact impulse emerge from coupled proprioception-command relations. Standard MLPs receive only first-order variables, leaving these quantities implicit. PRISM exposes them directly via learned factorized polynomial representations.
Figure 1: Joint power, Coriolis effects, contact forces form through polynomial interactions of observable variables.
Method
Deployable Policy Setting
Policy $\pi_\theta(a_t | o_t)$ maps $o_t = (x_t, c_t)$ to action, where $x_t \in \mathbb{R}^d$ is deployable proprioception and $c_t$ contains remaining inputs. No latent physical quantities (contact forces, friction, mass) at deployment.
Interaction-Structured Representation
Two learned affine factors from $x_t$:
$$u_t = W_1 x_t + b_1, \quad v_t = W_2 x_t + b_2$$Default second-order representation:
$$\psi_2(x_t) = u_t \odot (\mathbf{1} + \alpha_2 \odot v_t) = u_t + \alpha_2 \odot (u_t \odot v_t)$$$\alpha_2$ learned jointly, initialized near zero. First term preserves first-order path; second introduces factorized quadratic interactions. Recursive extension to degree $K$:
$$\psi_k(x_t) = \psi_{k-1}(x_t) \odot [\mathbf{1} + \alpha_k \odot (W_k x_t + b_k)], \quad k = 2, \ldots, K$$Final representation: $z_t = g_\eta(\psi_K(x_t))$. Joint power as example interaction: $P_{\text{joint}} = \sum_{i} |\tau_i \dot{q}_i|$.
Figure 2: PRISM — factorized polynomial module replacing linear conditioning.
RL and IL Application
RL: PRISM encodes proprioception into $z_t$ before actor. PPO predicts Gaussian action mean. No changes to action space, reward, or controller. IL: Replaces linear proprioceptive conditioning in Diffusion Policy with polynomial layer, trained end-to-end.
graph TD
A["Observation o_t = (x_t, c_t)"] --> B["PRISM Polynomial Module"]
B --> C["Affine factors u_t, v_t"]
C --> D["Factorized interaction
ψ_K = u_t ⊙ (1 + α ⊙ v_t)"]
D --> E["Conditioning feature z_t"]
E --> F["RL: Actor predicts action"]
E --> G["IL: Diffusion Policy conditioning"]
F --> H["Humanoid locomotion
Zero-shot sim2real"]
G --> I["Contact-rich manipulation
Sensorless compliance"]
Results
Humanoid Locomotion
| Method | Episode Len ↑ | Track Error ↓ | Survival ↑ | Notes |
|---|---|---|---|---|
| Standard MLP | Lower | Higher | Lower | Velocity drift, early falls |
| Larger MLP | Medium | Medium | Medium | Capacity doesn't help |
| PRISM (deg 2) | Best | Lowest | Highest | Explicit interactions |
| PRISM (deg 3) | Best | Lowest | Highest | Slightly better, more complex |
Figure 3: PRISM shows tighter convergence bounds and stable velocity tracking on Humanoid-Gym.
Force-Free Compliance in Manipulation
| Method | Success ↑ | Smoothness ↓ | Pos Error ↓ | Orient Error ↓ |
|---|---|---|---|---|
| Diffusion Policy | Baseline | Poor | Baseline | Baseline |
| MCC-Sensorless | Medium | Medium | Medium | Medium |
| MCC-Oracle (non-deployable) | High | Good | Low | Low |
| PRISM | Highest | Best | Lowest | Lowest |
PRISM produces emergent compliance — fast approach in free space, immediate velocity suppression upon contact, maintaining low stable contact force without force/wrench/tactile input or admittance control. Diffusion Policy causes hard impact; MCC-Sensorless responds late with forces exceeding 600N.
Figure 4: PRISM actively decelerates at contact — instantaneous velocity suppression absorbs impact energy.
The parameter efficiency of the factorized construction can be formally analyzed. For a $d$-dimensional input with polynomial degree $K$, the full monomial basis has $inom{d+K}{K}$ terms, while PRISM requires only $K$ additional $d$-dimensional vectors. The ratio:
$$\eta_{ ext{PRISM}} = rac{K \cdot d}{inom{d+K}{K}} o 0 \quad ext{as } d o \infty$$demonstrates that PRISM's advantage grows with input dimensionality, making it particularly suitable for high-dimensional proprioceptive spaces typical in humanoid robots.
Stronger Backbone Validation
PRISM effective on BFM-Zero (locomotion) and SmolVLA (manipulation), outperforming larger-capacity controls with fewer parameters. Factorized construction avoids enumerating all monomials — only $K \times d$ parameters vs $\binom{d+K}{K}$ for full expansion:
$$\frac{\text{PRISM params}}{\text{Full expansion}} = \frac{K \times d}{\binom{d+K}{K}} \ll 1 \quad \text{when } d \gg K$$This is analogous to low-rank approximation — learning most control-relevant interaction directions rather than all possible ones. Degree 2 captures most gains, confirming most action-relevant interactions are low-order. The success on stronger backbones demonstrates that polynomial interaction inductive bias provides information that model capacity cannot replace, similar to how translation invariance benefits image recognition.
Emergent compliance can be understood through the contact force-velocity relationship: $F_c \propto m \cdot \frac{dv_{\text{EEF}}}{dt}$. PRISM implicitly encodes the coupling between $v_{\text{EEF}}$ and proprioception through polynomial interactions, naturally producing deceleration upon contact — behavior resembling impedance control damping, but learned entirely from data.
Limitations
- Assumes key cues captured by low-degree polynomial interactions; may fail for long-horizon history or unobserved contact geometry
- Cannot compensate for missing sensory coverage
- Experiments focused on humanoid locomotion and fixed-controller manipulation
- Future: adaptive polynomial degree, temporal interactions, broader morphologies
Conclusion and Outlook
PRISM is a compact policy representation that explicitly exposes polynomial interactions among observable physical variables. Through factorized polynomial modules applied in RL (after MLP backbone) and IL (replacing linear conditioning in Diffusion Policy), PRISM outperforms standard and larger MLP baselines, supports sensorless compliant behavior, and requires no additional deployment-time sensing. Polynomial representations should become a standard architectural choice for embodied motor control.
Key Insight: Interaction structure cannot be replaced by capacity — larger MLPs fail to bridge the gap; only explicitly exposing multiplicative interactions achieves the qualitative leap.
