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A Study of Conditional Diffusion Models for Open-Loop Control under Dry Friction and Stiction
Eric Aislan Antonelo
TL;DR
The paper asks whether conditioning a diffusion prior can improve open-loop control when dry friction and stiction make effective action sequences sparse and temporally structured. It models conditional control sequences, evaluates them through known dynamics, and compares the proposal with random shooting and CEM. Action Diffusion achieves lower terminal error and fewer stuck steps, especially at small sample budgets, while remaining an open-loop method rather than an online feedback controller.
Problem
The paper addresses inefficient proposal sampling for non-smooth open-loop control, where stiction requires controls to cross a static-friction threshold with suitable direction, magnitude, and timing.
Method
Action Diffusion learns a conditional distribution over bounded control sequences given initial and target states, then rolls out and ranks sampled candidates using known dry-friction dynamics.
Results
Action Diffusion achieves lower terminal error and fewer stuck steps than uniform and dataset-prior random shooting, especially with small sample budgets; CEM narrows the gap with more computation.
Takeaways & Limitations
Conditional diffusion provides a structured open-loop proposal distribution by assigning higher probability to control sequences with suitable direction, magnitude, and timing.
Takeaways & Limitations
The method is not an online feedback controller but a piecewise composition of state-to-state open-loop plans, and the benchmark uses dry friction with stiction.
Abstract
from arXiv · showhide
Diffusion models have recently emerged as expressive generative priors for planning and control. This paper studies Action Diffusion, an action-sequence diffusion formulation used as an open-loop proposal distribution for a point-mass system with dry friction and stiction. In this benchmark, motion starts only when the applied input exceeds a static-friction threshold, so effective controls occupy a small and temporally structured subset of the action-sequence space. A compact conditional 1D U-Net generates bounded control sequences conditioned on initial and target states. We compare it with uniform random shooting, random shooting from the same structured dataset prior, and the Cross-Entropy Method (CEM). Results show that Action Diffusion reduces terminal error and stuck steps, especially in low-sample regimes. These results indicate that conditional diffusion provides an effective mechanism for generating temporally coherent control sequences that overcome stiction by conditioning and recombining structured control primitives from the training prior for state-to-state open-loop control.
1. Introduction
The paper investigates conditional diffusion for open-loop control in a simple dry-friction system, where stiction makes effective action sequences sign-, magnitude-, and timing-sensitive. It proposes Action Diffusion and compares structured conditional sampling with alternative proposal strategies.
- Diffusion-based control remains underexplored as an expressive prior for continuous physical control and action sequencing.
- A one-dimensional point-mass benchmark makes frictional structure explicit while avoiding the opacity of high-dimensional control settings.
- Static friction creates a dead zone, so effective controls must exceed a threshold with the correct sign, magnitude, and timing.
- Action Diffusion learns pθ(u0:H−1 | x0, x⋆), samples candidate control sequences, and ranks them after rollout through known dry-friction dynamics.The formulation diffuses control sequences directly rather than full state-action trajectories.
- The study tests whether conditioning a structured but unconditioned prior on initial and target states improves sample-efficient proposal generation.The prior already contains kick, medium-amplitude, and near-zero control segments.
- The evaluation compares Action Diffusion with uniform random shooting, dataset-prior random shooting, and CEM.The control analysis examines threshold-crossing, motion, and settling phases.
2. Related Work
Prior work applies diffusion to trajectory generation, conditional decision making, and robotic action sequences. This paper instead studies directly generated control sequences in a low-dimensional non-smooth system, alongside sampling-based control baselines.
- Diffuser denoises full state-action trajectories, whereas Decision Diffuser uses conditioning variables such as returns or constraints.
- Diffusion Policy generates action sequences for visuomotor robotic manipulation, extending diffusion-based control to high-dimensional observations and actions.
- Action Diffusion focuses on a low-dimensional non-smooth system where generated control timing can be inspected directly.It denoises only controls and uses known dynamics externally for rollout and ranking.
- Random shooting selects the best rolled-out candidate, while CEM iteratively refines its proposal distribution.MPPI-style methods similarly address nonconvex objectives without requiring differentiability.
- With stiction, proposal distributions are central because large action-space regions can be ineffective.
3. Method
The method learns conditional control-sequence distributions and uses them to generate and select open-loop plans through known dry-friction dynamics. A compact 1D U-Net denoises temporally structured controls conditioned on state information.
- 3.1. Action Diffusion Model for Control Sequences: Action Diffusion models a conditional distribution over control sequences using a compact 1D U-Net with state and diffusion-time conditioning.The network denoises one-channel sequences of length H = 64, with conditioning injected into residual blocks.
- 3.1. Action Diffusion Model for Control Sequences: Training data pair synthetic control sequences with initial, target, and rolled-out terminal states from the known dynamics.The denoiser is trained to model controls consistent with transitions between conditioned states.
- 3.2. Training Objective: The denoiser predicts injected Gaussian noise under the standard diffusion noise-prediction objective.The objective uses a clean control sequence, diffusion time index, and noisy sequence; the schedule is defined through β_t and α_t.
- 3.4. Sampling with DDIM and Classifier-Free Guidance: Classifier-free guidance combines conditional and unconditional noise predictions during deterministic DDIM-style sampling.Conditioning is randomly dropped during training, then the specified initial and target states are used for conditional sampling.
- 3.5. Open-Loop Control: For each target, candidate sequences are rolled out through known dynamics and ranked by terminal state error.The same selection rule is applied across Action Diffusion, random-shooting baselines, and samples within CEM iterations.
- 3.5. Open-Loop Control: Multi-step references use segmented open-loop planning, executing selected state-to-state segments sequentially rather than applying online feedback.Each segment is generated from the currently reached state to the next target state.
4. Experimental Setup
The experiments use a one-dimensional point-mass benchmark with dry friction, damping, and stiction, evaluating conditional control-sequence proposals against structured and optimization-based baselines. Evaluation varies sampling budgets and measures terminal accuracy, energy, stuck behavior, and computation time.
- Benchmark dynamics: The benchmark models a one-dimensional point mass with state x = [p, v]⊤ and scalar control u under Coulomb friction, viscous damping, and static-friction dead zones.
- Benchmark dynamics: When velocity is near zero and |u| ≤ fs, stiction keeps the mass at rest; motion requires exceeding the static-friction threshold.The system uses semi-implicit Euler integration, updating position with the post-friction velocity.
- Dataset and proposal distributions: The control dataset uses eight constant segments with kick, medium-amplitude, and near-zero modes to provide threshold-crossing sequences for motion initiation.
- Dataset and proposal distributions: Action Diffusion conditions the structured control prior on the transition from x0 to xH, whereas dataset-prior random shooting uses the same prior without target conditioning.
- Training and baselines: The Action Diffusion model trains on 50,000 transitions for 15,000 Adam gradient steps, using an exponential moving average of model weights.
- Training and baselines: CEM iteratively refines a Gaussian control-sequence distribution from elite candidates, and comparisons vary iterations against wall-clock time at candidate budgets K = 32 and K = 512.
- Evaluation metrics: Evaluation reports terminal error, control energy, stuck steps, and computation time across composed open-loop trajectories.Stuck steps count effectively stopped states with sub-threshold input that remain away from the desired position; multi-step results also report mean end error.
5. Results
Across single-target, error–compute, and segmented multi-step evaluations, Action Diffusion performs especially well when sample budgets are small, while its advantage over CEM narrows with additional refinement. Its conditioned proposal produces more temporally coordinated threshold-crossing controls than unconditioned or uniformly sampled alternatives.
- 5.1. Single-Target Sample Budget: The selected small-target trajectory first exceeds the static-friction threshold, then modulates control to settle near x⋆= (0.2, 0).This illustrates conditional selection and timing of high-amplitude and near-zero control primitives already present in the training prior.
- 5.1. Single-Target Sample Budget: Action Diffusion achieves low terminal error at small sample budgets, while uniform random shooting remains largely ineffective across K values.Uniform samples often fail to cross the static-friction threshold at the right time or produce poorly coordinated motion.
- 5.1. Single-Target Sample Budget: Terminal error and stuck steps are more informative than control energy because low energy can simply indicate ineffective motion.The benchmark therefore emphasizes successful movement over energy minimization alone.
- 5.2. Error–Compute Trade-off Against CEM: For K = 32, Action Diffusion attains lower terminal error at smaller compute than CEM; for K = 512, CEM narrows the gap through additional refinement.The comparison uses target (0.5, 0), initial condition (0, 0), and representative budgets K = 32 and K = 512.
- 5.3. Segmented Multi-Step Open-Loop Planning: In segmented multi-step control, Action Diffusion has the lowest final error, mean end error, and stuck-step count for every tested K.Dataset-prior sampling is faster but less accurate, while CEM improves with population size at substantially higher computation.
- 5.3. Segmented Multi-Step Open-Loop Planning: Action Diffusion also achieves the lowest mean end error for fs ∈ {0.6, 0.7, 0.8} at K = 128, although this is only an initial sensitivity check.The remaining system parameters are kept fixed, so the experiment is not a complete robustness analysis.
6. Discussion
Action Diffusion’s advantage comes from conditioning a shared structured control prior on the desired transition, particularly when candidate budgets are small. With larger budgets, CEM narrows the performance gap through iterative refinement.
- Action Diffusion and dataset-prior random shooting use the same underlying control prior, but only Action Diffusion conditions samples on the desired transition.The comparison isolates conditioning from the mere presence of high-amplitude kick segments.
- Low-budget regimes favor Action Diffusion because stiction creates flat or ineffective regions that make CEM refinement inefficient.
- At larger candidate budgets, CEM improves and Action Diffusion’s advantage becomes less pronounced.
7. Conclusion
The paper evaluates Action Diffusion as an open-loop proposal distribution for point-mass control with dry friction and stiction. It reports lower terminal error and fewer stuck steps than random-shooting baselines, with the strongest advantage at small budgets, while CEM reduces the gap at larger budgets.
- Action Diffusion achieves lower terminal error and fewer stuck steps than uniform and dataset-prior random shooting, especially with small sample budgets.
- The CEM comparison indicates that Action Diffusion’s advantage is strongest in low-budget regimes, while more computation can reduce the gap.
- Qualitative results show threshold-crossing, motion, and settling phases as conditional recombinations of structured segments already present in the training prior.
- Future work will examine richer mechanical systems, stochastic disturbances, additional learning baselines, and receding-horizon feedback execution.