# Constraint-Enhanced Reinforcement Learning Based on Dynamic Decoupled Spherical Radial Squashing

arXiv: https://arxiv.org/abs/2605.04185 · HTML: https://mlsanity.com/paper/2605.04185/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for robot locomotion and manipulation, one of 15 directions in Deep reinforcement learning methods.
- Builds on 29 of its 38 references that are in the map.
- First posted on arXiv 5 May 2026.

## Placed in

- Deep reinforcement learning methods / [RL for robot locomotion and manipulation](https://mlsanity.com/deeprl/rl-for-robot-locomotion-and-manipulation/)

## Builds on (most cited first, 29 of 29)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Learning Complex Dexterous Manipulation with Deep Reinforcement Learning and Demonstrations](https://mlsanity.com/paper/1709.10087/) (2017)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [Reward Constrained Policy Optimization](https://mlsanity.com/paper/1805.11074/) (2018)
- [Leveraging Demonstrations for Deep Reinforcement Learning on Robotics Problems with Sparse Rewards](https://mlsanity.com/paper/1707.08817/) (2017)
- [Neural Network Dynamics for Model-Based Deep Reinforcement Learning with Model-Free Fine-Tuning](https://mlsanity.com/paper/1708.02596/) (2017)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Residual Reinforcement Learning for Robot Control](https://mlsanity.com/paper/1812.03201/) (2018)
- [Safe Exploration in Continuous Action Spaces](https://mlsanity.com/paper/1801.08757/) (2018)
- [Closing the Sim-to-Real Loop: Adapting Simulation Randomization with Real World Experience](https://mlsanity.com/paper/1810.05687/) (2018)
- [OptLayer - Practical Constrained Optimization for Deep Reinforcement Learning in the Real World](https://mlsanity.com/paper/1709.07643/) (2017)
- [Data-driven Economic NMPC using Reinforcement Learning](https://mlsanity.com/paper/1904.04152/) (2019)
- [Penalized Proximal Policy Optimization for Safe Reinforcement Learning](https://mlsanity.com/paper/2205.11814/) (2022)
- [Constrained Update Projection Approach to Safe Policy Optimization](https://mlsanity.com/paper/2209.07089/) (2022)
- [Highly Dynamic Quadruped Locomotion via Whole-Body Impulse Control and Model Predictive Control](https://mlsanity.com/paper/1909.06586/) (2019)
- [A Survey of Constraint Formulations in Safe Reinforcement Learning](https://mlsanity.com/paper/2402.02025/) (2024)
- [Perceptive Locomotion through Nonlinear Model Predictive Control](https://mlsanity.com/paper/2208.08373/) (2022)
- [A Safe Hierarchical Planning Framework for Complex Driving Scenarios based on Reinforcement Learning](https://mlsanity.com/paper/2101.06778/) (2021)
- [FlowPG: Action-constrained Policy Gradient with Normalizing Flows](https://mlsanity.com/paper/2402.05149/) (2024)
- [Learning Sim-to-Real Humanoid Locomotion in 15 Minutes](https://mlsanity.com/paper/2512.01996/) (2025)
- [Benchmarking Actor-Critic Deep Reinforcement Learning Algorithms for Robotics Control with Action Constraints](https://mlsanity.com/paper/2304.08743/) (2023)
- [Efficient Action-Constrained Reinforcement Learning via Acceptance-Rejection Method and Augmented MDPs](https://mlsanity.com/paper/2503.12932/) (2025)
- [Safety-Critical Kinematic Control of Robotic Systems](https://mlsanity.com/paper/2009.09100/) (2020)
- [Leveraging Constraint Violation Signals For Action-Constrained Reinforcement Learning](https://mlsanity.com/paper/2502.10431/) (2025)
- [Improving Stochastic Action-Constrained Reinforcement Learning via Truncated Distributions](https://mlsanity.com/paper/2511.22406/) (2025)

## Abstract (arXiv)

When deploying reinforcement learning policies to physical robots, actuator rate constraints -- hard limits on how fast each joint can move per control step -- are unavoidable. These limits vary substantially across joints due to differences in motor inertia, power bandwidth, and transmission stiffness, creating pronounced heterogeneity that existing methods fail to handle geometrically: the per-joint feasible region forms a high-dimensional box in action-increment space, yet QP projection and spherical parameterization methods impose isotropic ball-shaped constraints, exponentially under-covering the true feasible set as heterogeneity grows. This paper proposes Dynamic Decoupled Spherical Radial Squashing (DD-SRad), which resolves this mismatch by computing a position-adaptive radius independently for each actuator, achieving tight alignment with the true per-joint feasible region. DD-SRad satisfies per-step hard constraints with probability~1, preserves well-conditioned gradients throughout training, and admits exact policy gradient backpropagation with zero runtime solver overhead. MuJoCo benchmark experiments demonstrate the highest task return at zero constraint violation -- matching the unconstrained upper bound -- with 30%--50% improvement in constraint-space coverage over spherical baselines. High-fidelity IsaacLab simulations with Unitree H1 and G1 humanoid robots confirm end-to-end optimality parameterized directly from official joint specifications, validating a systematic pathway from hardware datasheets to safe deployment.