An Introduction to Zero-Order Optimization Techniques for Robotics

Many robotic tasks, such as inverse kinematics, motion planning, and contact-rich manipulation, can be formulated as optimization problems. Solving these problems requires addressing inherent nonlinear kinematics, complex contact dynamics, long-horizon correlations, and multi-modal optimization landscapes, each posing distinct challenges for state-of-the-art optimizers. While existing methods tackle these issues through problem-specific strategies, such specialization inherently limits cross-task generalization, requires heavy engineering effort in problem reformulation, and hinders multi-task autonomy. Monte Carlo Tree Search (MCTS) offers a compelling framework that generalizes across diverse robotic tasks via strategic exploration of the solution space. However, it typically suffers from combinatorial complexity when applied naively, resulting in slow convergence and excessive storage space in high-dimensional domains. To address this limitation, we propose Tensor Train Tree Search (TTTS), which leverages tensor factorization to exploit implicit correlations among different branches within the decision tree. By utilizing the resulting compact, linear-complexity representation, TTTS significantly reduces both computation and storage overhead, thereby enabling highly efficient global decision making. Experimental results across inverse kinematics, motion planning around obstacles, legged robot manipulation, multi-stage motion planning, and bimanual whole-body manipulation demonstrate the efficiency of TTTS for generalized robot optimization over a diverse set of tasks.

Trajectory optimization (TO) under nonlinear dynamics, actuation limits and collision avoidance constraints is a fundamental problem in robotics, albeit especially challenging due to its highly non-convex nature. For this setting, Differential Dynamic Programming (DDP) is an efficient second-order shooting method, yet its local structure renders it vulnerable to suboptimal basins. Sampling-augmented variants mitigate this susceptibility through stochastic exploration, but often sample only around the few trajectories they retain for reoptimization, based solely on their cost which restricts exploration breadth. We introduce Pareto-Optimal Entropy-Regularized DDP (PER-DDP), an entropy-regularized population framework derived from the free-energy/relative-entropy inequality. Our method combines prior-guided sampling that shapes exploration around each retained trajectory, with expanded rollout evaluations that probe these sampling policies beyond the few retained candidates, and Pareto filtering for preserving task-constraint alternatives across iterations. This decouples sampling effort from the optimization population size and broadens exploration without sacrificing the second-order structure that makes DDP effective. Across multiple systems and hundreds of environments, PER-DDP achieves higher success rates than state-of-the-art sampling-augmented TO methods and finds reliable solutions in environments beyond the reach of all baselines.

Research on robotic manipulation has developed a diverse set of policy paradigms, including vision-language-action (VLA) models, vision-action (VA) policies, and code-based compositional approaches. Concrete policies typically attain high success rates on specific task distributions, but limited generalization beyond it. Rather than proposing another monolithic policy, we propose to leverage the complementary strengths of existing approaches through intelligent policy routing. We introduce RoboRouter, a training-free framework that maintains a pool of heterogeneous policies and learns to select the best-performing policy for each task through accumulated execution experience. Given a new task, RoboRouter constructs a semantic task representation, retrieves historical records of similar tasks, predicts the optimal policy choice without requiring trial-and-error, and incorporates structured feedback to refine subsequent routing decisions. Integrating a new policy into the system requires only a lightweight evaluation and does not incur training overhead. Across simulation benchmark and real-world evaluations, RoboRouter consistently outperforms individual policies, improving the average success rate by more than 3% in simulation and 13% in real-world settings, while preserving execution efficiency. Our results demonstrate that intelligent routing across heterogeneous, off-the-shelf policies provides a practical and scalable pathway toward building more capable robotic systems.