# Motion Planning by Learning the Solution Manifold in Trajectory Optimization

arXiv: https://arxiv.org/abs/2107.05842 · HTML: https://mlsanity.com/paper/2107.05842/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 69 references that are in the map.
- First posted on arXiv 13 Jul 2021.

## Builds on (most cited first, 30 of 30)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Diversity is All You Need: Learning Skills without a Reward Function](https://mlsanity.com/paper/1802.06070/) (2018)
- [An Algorithmic Perspective on Imitation Learning](https://mlsanity.com/paper/1811.06711/) (2018)
- [Data-Efficient Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1805.08296/) (2018)
- [The Option-Critic Architecture](https://mlsanity.com/paper/1609.05140/) (2016)
- [FeUdal Networks for Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1703.01161/) (2017)
- [Inference Suboptimality in Variational Autoencoders](https://mlsanity.com/paper/1801.03558/) (2018)
- [Stochastic Neural Networks for Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1704.03012/) (2017)
- [Near-Optimal Representation Learning for Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1810.01257/) (2018)
- [Neural probabilistic motor primitives for humanoid control](https://mlsanity.com/paper/1811.11711/) (2018)
- [Understanding and Improving Interpolation in Autoencoders via an Adversarial Regularizer](https://mlsanity.com/paper/1807.07543/) (2018)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Learning Disentangled Joint Continuous and Discrete Representations](https://mlsanity.com/paper/1804.00104/) (2018)
- [One Solution is Not All You Need: Few-Shot Extrapolation via Structured MaxEnt RL](https://mlsanity.com/paper/2010.14484/) (2020)
- [Continuous-Time Gaussian Process Motion Planning via Probabilistic Inference](https://mlsanity.com/paper/1707.07383/) (2017)
- [Directed-Info GAIL: Learning Hierarchical Policies from Unsegmented Demonstrations using Directed Information](https://mlsanity.com/paper/1810.01266/) (2018)
- [Universal Planning Networks](https://mlsanity.com/paper/1804.00645/) (2018)
- [Batch Informed Trees (BIT*): Informed Asymptotically Optimal Anytime Search](https://mlsanity.com/paper/1707.01888/) (2017)
- [Harnessing Reinforcement Learning for Neural Motion Planning](https://mlsanity.com/paper/1906.00214/) (2019)
- [Discovering Diverse Solutions in Deep Reinforcement Learning by Maximizing State-Action-Based Mutual Information](https://mlsanity.com/paper/2103.07084/) (2021)
- [Learning to Plan in High Dimensions via Neural Exploration-Exploitation Trees](https://mlsanity.com/paper/1903.00070/) (2019)
- [Multimodal Trajectory Optimization for Motion Planning](https://mlsanity.com/paper/2003.07054/) (2020)
- [Describing Physics For Physical Reasoning: Force-based Sequential Manipulation Planning](https://mlsanity.com/paper/2002.12780/) (2020)
- [Hierarchical Reinforcement Learning via Advantage-Weighted Information Maximization](https://mlsanity.com/paper/1901.01365/) (2019)
- [Riemannian Motion Policy Fusion through Learnable Lyapunov Function Reshaping](https://mlsanity.com/paper/1910.02646/) (2019)
- [Motion Planning Explorer: Visualizing Local Minima using a Local-Minima Tree](https://mlsanity.com/paper/1909.05035/) (2019)

## Built on by (most cited first, 13 of 13)

- [RDT-1B: a Diffusion Foundation Model for Bimanual Manipulation](https://mlsanity.com/paper/2410.07864/) (2024)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [Robometer: Scaling General-Purpose Robotic Reward Models via Trajectory Comparisons](https://mlsanity.com/paper/2603.02115/) (2026)
- [XR-1: Towards Versatile Vision-Language-Action Models via Learning Unified Vision-Motion Representations](https://mlsanity.com/paper/2511.02776/) (2025)
- [Exploring Embodied Multimodal Large Models: Development, Datasets, and Future Directions](https://mlsanity.com/paper/2502.15336/) (2025)
- [Discovering Multiple Solutions from a Single Task in Offline Reinforcement Learning](https://mlsanity.com/paper/2406.05993/) (2024)
- [Bridging the gap between Learning-to-plan, Motion Primitives and Safe Reinforcement Learning](https://mlsanity.com/paper/2408.14063/) (2024)
- [Tensor Train for Global Optimization Problems in Robotics](https://mlsanity.com/paper/2206.05077/) (2022)
- [Learning Solution Manifolds for Control Problems via Energy Minimization](https://mlsanity.com/paper/2203.03432/) (2022)
- [Open-Ended Diverse Solution Discovery with Regulated Behavior Patterns for Cross-Domain Adaptation](https://mlsanity.com/paper/2209.12029/) (2022)
- [Toward Generalist Neural Motion Planners for Robotic Manipulators: Challenges and Opportunities](https://mlsanity.com/paper/2603.24318/) (2026)
- [Optimal Motion Planning using Finite Fourier Series in a Learning-based Collision Field](https://mlsanity.com/paper/2312.09073/) (2023)
- [A Cross-Environment and Cross-Embodiment Path Planning Framework via a Conditional Diffusion Model](https://mlsanity.com/paper/2510.19128/) (2025)

## Abstract (arXiv)

The objective function used in trajectory optimization is often non-convex and can have an infinite set of local optima. In such cases, there are diverse solutions to perform a given task. Although there are a few methods to find multiple solutions for motion planning, they are limited to generating a finite set of solutions. To address this issue, we presents an optimization method that learns an infinite set of solutions in trajectory optimization. In our framework, diverse solutions are obtained by learning latent representations of solutions. Our approach can be interpreted as training a deep generative model of collision-free trajectories for motion planning. The experimental results indicate that the trained model represents an infinite set of homotopic solutions for motion planning problems.