PRISM: Polynomial Representations for Interaction-Structured Motor Control

arXiv:2607.23473 · cs.RO · Submitted 2026-07-26 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "PRISM: Polynomial Representations for Interaction-Structured Motor Control".

Dev: The gist: PRISM introduces a policy representation that makes polynomial interactions among observable physical variables explicit, learnable, and compact to improve robot control performance.

Rosa: First, who's behind it and why it matters.

Paper summary: Rosa: So we're talking about this paper called "PRISM: Polynomial Representations for Interaction-Structured Motor Control". Basically, it looks at how robot policies usually work, which is typically using multilayer perceptrons or MLPs that just take observations and map them to actions.

Dev: Right, so the core idea here is that robot observations are physical variables—things like power or slip velocity—and a lot of what makes a good action depends on how those variables multiply together, not just looking at them individually.

Rosa: Exactly. The paper argues that standard MLPs only get first-order physical variables, but cues for actions like contact or stability really come from these multiplicative interactions <ref:2607.23473#pg1>. PRISM is introduced as a policy representation designed to make those polynomial interactions explicit, learnable, and compact.

Dev: What it claims is that instead of having to list every single possible polynomial term you could think of, PRISM uses something called a factorized polynomial module to expose higher-order interaction features efficiently <ref:2607.23473#pg1>. It keeps the standard MLP backbone but just adds this gradually activated element-wise polynomial function on top in reinforcement learning, or it replaces the linear conditioning in Diffusion Policy with a polynomial layer trained end-to-end in imitation learning.

Taro: That makes sense from an autonomy perspective. When the world misbehaves, you need a system that can handle those complex interactions when things get messy <ref:2607.23473#pg1>. The focus on these interaction structures sounds like it’s giving the robot better intuition about physics than just looking at raw sensor data <ref:2607.23473#pg1>.

Rosa: And the results they show are pretty strong across humanoid locomotion and contact-rich manipulation tasks <ref:2607.23473#pg1>. They found that PRISM improves performance from fifty-one point zero percent survival in locomotion up to ninety-two point five percent, which is better than even larger MLPs <ref:2607.23473#pg1>.

Dev: But what's really interesting, Rosa, is that this advantage doesn't just stay for the specific tasks they tested <ref:2607.23473#pg1>. They found that the performance advantage keeps holding up even when you change things outside of those original conditions—like changing the friction, changing the robot's mass, or changing the payload—without needing to retrain anything <ref:2607.23473#pg1>.

Taro: That persistence under out-of-distribution changes suggests that PRISM isn't just memorizing a specific solution for a specific setup; it’s learning a representation of the underlying physical laws and interactions themselves, which is what we need for real autonomy <ref:2607.23473#pg1>.

Rosa: It also showed some other cool things. In imitation learning settings, they found PRISM improves success rates in manipulation tasks by getting ninety-one point zero percent compared to sixty-three point eight percent for the Diffusion Policy <ref:2607.23473#pg1>. Plus, it learned a sensorless compliance-like behavior without needing any explicit admittance control or force inputs <ref:2607.23473#pg1>.

Dev: That compliance thing is significant because it means the robot can react physically in contact situations in a more stable way than older methods that might suffer from those destabilizing force spikes upon initial contact, like MCC-Sensorless does <ref:2607.23473#pg1>. It seems to be learning how to manage that physical interaction implicitly.

Paper summary: Rosa: So, putting it together, the main point of "PRISM: Polynomial Representations for Interaction-Structured Motor Control" is that you don't need to explicitly list every polynomial term when modeling robot control <ref:2607.23473#pg3>. PRISM makes those interactions explicit and learnable without needing extra deployment-time sensing overhead <ref:2607.23473#pg1>.

Dev: It suggests that for embodied motor control, using polynomial representations as a standard architectural choice is something we should start considering <ref:2607.23473#pg3>. The paper points out that the results suggest polynomial interaction modeling should become a standard architectural choice for embodied motor control <ref:2607.23473#pg1>.

Taro: I think what this means is that we might be able to build policies that are more robust to real-world physical variations just by encoding the structure of the physics into the network architecture rather than relying solely on sheer model size <ref:2607.23473#pg1>.

Rosa: Exactly. We're looking at how this paper, "PRISM: Polynomial Representations for Interaction-Structured Motor Control," fits into the bigger picture of embodied AI and robot control systems <ref:2607.23473#pg1>. It moves the focus from just mapping inputs to outputs to explicitly modeling how those physical variables interact during movement <ref:2607.23473#pg1>.

Dev: So, the paper shows that by making polynomial interactions explicit through these factorized modules, you can achieve better performance across locomotion and manipulation tasks without needing to overhaul your entire training pipeline or add new sensors <ref:2607.23473#pg1>. The numbers they show for survival rates and success metrics are substantial improvements over baseline MLPs <ref:2607.23473#pg1>.

Taro: It's interesting that they showed these results across different policy backbones, like BFM-Zero and SmolVLA, both of which are stronger than the original MLP setup <ref:2607.23473#pg1>. That means the benefit isn't just a trick for one specific network architecture <ref:2607.23473#pg1>.

Rosa: And that leads into where we are now with this paper, "PRISM: Polynomial Representations for Interaction-Structured Motor Control". It really seems to argue that polynomial representations should become a standard architectural choice for embodied motor control <ref:2607.23473#pg1>.

Dev: So, the implication here is that future work might focus on how these polynomial representations generalize even further to completely novel physical setups, beyond just changing friction or mass in the same ways they tested <ref:2607.23473#pg1>.

Taro: That sounds like a necessary next step if we want to move this from impressive lab results to reliable systems operating in unpredictable environments <ref:2607.23473#pg1>.

Rosa: Alright, that wraps up our look at "PRISM: Polynomial Representations for Interaction-Structured Motor Control". It’s a paper about using polynomial representations to explicitly model the multiplicative interactions among physical variables in robot control <ref:2607.23473#pg1>.

Dev: A representation that learns these interactions compactly, without enumerating every single term, and it works across both reinforcement learning and imitation learning settings <ref:2607.23473#pg1>.

Taro: It suggests we should look more closely at how other researchers are using factorized parameterizations to represent higher-order dynamics in neural networks <ref:2607.23473#pg1>.

Rosa: That’s it for this discussion on "PRISM: Polynomial Representations for Interaction-Structured Motor Control".

Conclusion: Rosa: So we’re wrapping up on PRISM, which stands for Polynomial Representations for Interaction-Structured Motor Control by these authors from LSH3163 and others <ref:2607.23473#pg1>.

Dev: They basically show how to make robot policies handle the messy multiplication of physical variables directly, instead of just feeding them simple single numbers.

Rosa: They use this technique where they recursively build up polynomial features by adding scaled bilinear terms, which is a way to capture those high-order interactions without having to list every single term manually <ref:2607.23473#pg1>.

Dev: The core idea is that it keeps the standard MLP structure but just modifies that first layer, making it plug-and-play for reinforcement learning or imitation learning without changing your whole training setup <ref:2607.23473#pg1>.

Rosa: And the results they show across humanoid walking and manipulation are pretty solid, improving survival rates significantly while outperforming even much bigger MLPs <ref:2607.23473#pg1>.

Dev: What’s really interesting is that this performance doesn't just stick to the specific conditions they tested; it holds up even when you change things like the robot’s mass or the friction coefficient without retraining <ref:2607.23473#pg1>.

Rosa: It also showed a way for AI agents to learn sensorless compliance, meaning they can handle contact situations in a more stable way without needing those extra force sensors or explicit admittance controllers <ref:2607.23473#pg1>.

Dev: So what this means is that we might be able to build policies that are much better at handling the complex, multiplicative physics of the real world just by encoding that structure into the network architecture <ref:2607.23473#pg1>.

Rosa: Yeah, it suggests polynomial representations could become a standard way to model motor control for embodied AI systems <ref:2607.23473#pg1>.

Dev: It opens up a whole new direction for how we design these policies, especially when dealing with contact and dynamic tasks where those interactions matter most <ref:2607.23473#pg1>.

University of Michigan

cs.RO

Submitted: 2026-07-26

Updated: 2026-10-14

Project page: https://lsh3163.github.io/prism

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 87/100

The gist: The gist: PRISM introduces a policy representation that makes polynomial interactions among observable physical variables explicit, learnable, and compact to improve robot control performance.

Key concepts

Multiplicative Interactions: These are relationships where the effect of one physical variable depends on the value of another variable multiplied by it. Standard MLPs struggle to learn these because they treat inputs somewhat independently, requiring them to implicitly discover these complex dependencies through many layers.
Factorized Polynomial Module: Instead of listing every possible polynomial term, PRISM uses a factorized approach. It computes two learned affine factors and combines them using an element-wise multiplication with a learned coefficient. This efficiently exposes higher-order interaction features without needing to enumerate all possible terms.
Recursive Polynomial Representation: PRISM builds its representation recursively. It starts with a second-order interaction term (bilinear) and then extends it to higher polynomial degrees (K) by repeatedly applying this structure. This allows the model to capture complex physical relationships up to degree K without having to explicitly define every single monomial.
Sensorless Compliance-like Behavior: PRISM can exhibit behavior similar to a compliant robot even when it doesn't have explicit force or admittance control inputs. It learns features that naturally lead the policy to maintain a stable and low contact-force profile during manipulation, improving failure recovery without needing extra hardware or explicit controllers.

Terminology

Admittance Control — 1× in this paper · explained in 1 episode
Reinforcement Learning — 1× in this paper · explained in 4 episodes

Summary

The gist: PRISM introduces a policy representation that makes polynomial interactions among observable physical variables explicit, learnable, and compact to improve robot control performance.

How it works

PRISM addresses the limitation of standard MLP policies that receive only first-order physical variables, while many action-relevant cues arise from multiplicative interactions <ref:2607.23473#pg2>. The paper introduces PRISM, a policy representation that makes polynomial interactions among observable physical variables explicit, learnable, and compact <ref:2607.23473#pg2>. Rather than listing all polynomial terms, PRISM uses a factorized polynomial module to expose higher-order interaction features efficiently <ref:2607.23473#pg2>. In reinforcement learning, PRISM keeps the standard MLP backbone but applies a gradually activated element-wise polynomial function after it <ref:2607.23473#pg2>.

Key Mechanisms

The core of PRISM involves modifying the proprioceptive conditioning pathway to explicitly represent interactions among deployment-available robot observations <ref:2607.23473#pg2>. The representation is constructed by first computing two learned affine factors, ut = W1xt + b1, vt = W2xt + b2 <ref:2607.23473#pg2>. The default second-order representation is defined as ψ2(xt) = ut ⊙ (1 + α2 ⊙ vt) = ut + α2 ⊙ (ut ⊙ vt), where the element-wise multiplication and learned coefficient α2 introduce factorized quadratic interactions <ref:2607.23473#pg2>.

The representation can be extended recursively to polynomial degree K, where ψk(xt) = ψk−1(xt) ⊙ [1 + αk ⊙ (Wkxt + bk)] for k = 2,..., K <ref:2607.23473#pg2>. This allows PRISM to contain learned interaction terms up to degree K without requiring an explicit expansion of the corresponding monomial basis <ref:2607.23473#pg2>. The resulting representation is then mapped to the conditioning dimension used by the downstream policy via zt = gη(ψK(xt)) <ref:2607.23473#pg2>.

Impact and Results

PRISM has been evaluated on humanoid locomotion and contact-rich manipulation tasks <ref:2607.23473#pg2>. In Humanoid-Gym [10], PRISM improves command tracking, episode length, and survival over both a standard MLP actor and a larger MLP with matched capacity, showing that interaction structure cannot be replaced by capacity alone <ref:2607.23473#pg2>. Specifically, Table 1 shows that PRISM achieves the best episode length, linear-velocity tracking error, and survival rate in humanoid locomotion compared to baselines <ref:2607.23473#pg2>.

In imitation learning settings on LIBERO [11], PRISM improves Diffusion Policy success and yields compliance-like contact behavior without force, wrench, tactile input, contact labels, or an explicit admittance controller <ref:2607.23473#pg2>. For instance, PRISM achieves a 91.0 success rate in manipulation tasks compared to the Diffusion Policy's 63.8 <ref:2607.23473#pg2>. Furthermore, PRISM demonstrates emergent compliance by maintaining a low and stable contact-force profile throughout the rollout, unlike MCC-Sensorless which suffers from destabilizing force spikes upon initial contact <ref:2607.23473#pg2>.

Validation and Conclusions

PRISM's effectiveness is validated across stronger policy backbones, such as BFM-Zero and SmolVLA <ref:2607.23473#pg2>. In both domains, PRISM outperforms the corresponding larger-capacity control while using fewer parameters <ref:2607.23473#pg2>. The analysis of latent physical interactions confirms that PRISM makes Newtonian quantities more linearly recoverable than baselines, with Table 5 showing emergent task-specific interaction proxies like joint-velocity coupling and state–velocity interactions are high-impact in locomotion <ref:2607.23473#pg2>. Overall, the results suggest that polynomial representations should become a standard architectural choice for embodied motor control <ref:2607.23473#pg2>. The paper concludes that PRISM improves over standard and larger MLP baselines, supports sensorless compliant behavior, and requires no additional deployment-time sensing or privileged physical inputs <ref:2607.23473#pg2>. 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Improvements for AI systems

Improve policy representation by introducing PRISM to explicitly model multiplicative interactions among observable physical variables, as it makes polynomial interactions among observable physical variables explicit, learnable, and compact. This allows the system to derive action-relevant cues like joint power is the product of torque and joint velocity directly from data rather than implicitly through an MLP.
Enhance sensorless compliant behavior by utilizing PRISM's learned polynomial features to achieve compliance without force, wrench, tactile input, contact labels, or admittance control. Specifically, it can transition to active yielding immediately upon contact by absorbing impact energy through instantaneous velocity suppression, which is demonstrated in manipulation tasks.
Improve performance in humanoid locomotion and manipulation by replacing standard linear proprioceptive conditioning with a polynomial layer that exposes higher-order structure. This leads to gains over larger MLPs, as PRISM improves command tracking, episode length, and survival while showing that interaction structure cannot be replaced by capacity alone.
Increase the robustness of learned features by training linear probes on frozen policy representations to predict mechanics-inspired responses. This allows for the discovery of task-specific interaction proxies such as joint-velocity coupling and state–velocity interactions, which are then used to drive control, as shown in Table 5 where these factors have the highest impact on action prediction.
Improve policy generalization across different controller architectures by integrating PRISM into stronger backbones like BFM-Zero and SmolVLA. This shows that the benefit of explicit interaction modeling is not tied to a specific architecture, as PRISM improves BFM-Zero tracking EMD under nominal dynamics, low friction, and payload-mass perturbations without requiring additional deployment sensing.

Abstract

Robot policies map observations to actions, typically using multilayer perceptrons (MLPs). Yet robot dynamics and control objectives involve multiplicative interactions among observable physical variables. MLPs can approximate these interactions but must learn them implicitly, at a representational cost. We introduce PRISM (Polynomial Representations for Interaction-Structured Motor Control), a physics-motivated yet data-driven representation that learns multiplicative interactions by recursively adding scaled bilinear features, without enumerating all polynomial terms. Replacing only the MLP's first affine layer, PRISM is plug-and-play for reinforcement and imitation learning, requiring no changes to training objectives, sensing, or controllers. Across humanoid locomotion and contact-rich manipulation, PRISM improves survival from 51.0% to 92.5%, outperforming even larger MLPs. More strikingly, its performance advantage persists under out-of-distribution changes in friction, robot mass, and payload, without retraining. PRISM also exhibits sensorless compliance-like behavior, increases failure-recovery success from 40.9% to 52.3% without explicit admittance control, improves real-robot manipulation, and learns features that better predict physical quantities. These results make PRISM a compelling alternative to standard MLPs for robot policy learning. The project page is available at https://lsh3163.github.io/prism/.