Nullspace MPC

Mizuho Aoki

Nagoya University

Paper Code

Demonstration

Demonstrating agile and smooth navigation through a narrow environment with a swerve drive vehicle.

Overview

Nullspace Model Predictive Control (Nullspace MPC) is a novel, hierarchical motion planning framework designed for online, receding-horizon control. Its core motivation is to handle multiple objectives via a strict task hierarchy. This approach largely mitigates the difficult and often unreliable process of tuning weight parameters found in conventional single-cost-function methods. To resolve a diverse range of complex tasks with real-time efficiency, the framework intelligently decomposes the control problem according to the nature of the tasks. A sampling-based optimizer handles the global, nonlinear aspects of motion, such as roughly shaping the overall path to avoid arbitrarily shaped obstacles. Meanwhile, a gradient-based solver addresses local, linear tasks by strictly enforcing constraints like velocity/acceleration bounds and ensuring trajectory smoothness. As a result, Nullspace MPC provides navigation through narrow environments with an omni-directional vehicle that is demonstrably safer and more agile than the conventional method, MPPI.

Note: This project forms a core part of my Ph.D. dissertation, which was made publicly available on October 1, 2025.

Key Idea

This figure illustrates our novel method for complex motion planning, where objectives like avoiding obstacles (a nonlinear task) and driving smoothly (a linear task) must be managed according to strict priorities. Our approach integrates two powerful optimization paradigms. It utilizes a high-level search for optimal intermediate waypoints, which are called "via poses." This search is a sampling-based process founded on variational inference, a technique inspired by Model Predictive Path Integral (MPPI) control. For each potential via pose generated by this process, a complete and locally-optimal trajectory is calculated using a hierarchical optimizer (HQP). This optimizer systematically enforces the full list of prioritized tasks, a list that includes objectives like system dynamics and motion smoothness. The complete trajectories are then evaluated to identify which initial via pose produced the best overall outcome concerning the original nonlinear goals, such as successful obstacle avoidance. This hybrid strategy effectively delegates the complex nonlinear search to the variational inference component, while using the priority-aware HQP to construct the hierarchically optimal trajectory for each sampled scenario.

sahqp_algorithm

Comparison Between Previous Approach

One-shot Trajectory Optimization

This image compares our Nullspace MPC with the conventional MPPI, highlighting our key innovation in strategically reducing the search space for sampling-based optimization. While MPPI (left) must explore thousands of entire trajectories in a high-dimensional control space, our method (right) achieves a more structurally diverse exploration with only 100 samples. This is accomplished by sampling a few low-dimensional "via poses" rather than full control sequences. This dramatic reduction in search space dimensionality leads to qualitatively superior solutions that more reliably satisfy critical tasks, such as precisely reaching the goal state. A notable trade-off is that each of our samples requires solving a Quadratic Programming (QP) problem, making the computation per sample more intensive.

one_shot_traj_opt_mppi_nullspace_mpc

Navigation in a Narrow Maze

These videos showcases a side-by-side comparison of MPPI (left) and our Nullspace MPC (right) navigating a challenging maze. In comprehensive testing, Nullspace MPC demonstrated superior navigation performance, achieving a shorter trajectory length and shorter travel time while maintaining a higher success rate. ] This performance difference arises because MPPI must balance safety against speed using a single weighted cost function , whereas Nullspace MPC is structured to maximize velocity within the solution space of actions guaranteed to be safe . This hierarchical approach enables robustly safe yet fast navigation, particularly in narrow environments.

The source of this performance advantage is evident in the vehicle's behavior during abrupt directional changes. MPPI frequently employs a sequential motion strategy, where the vehicle first stops its forward movement to rotate in place and align with the new path before accelerating. In contrast, Nullspace MPC executes a concurrent motion pattern, capable of translating the vehicle along the desired trajectory while simultaneously adjusting its orientation in a fluid manner. This simultaneous execution of translation and rotation permits more continuous and efficient navigation by eliminating the interruptions inherent in sequential maneuvers.

Citation



  @phdthesis{mizuho2025phd,
    author    = {Mizuho Aoki},
    title     = {{Nonlinear Model Predictive Control for Autonomous Vehicles: Enhancement via Simplified Physics-Aware Prediction and Decomposed Optimization}},
    school    = {Nagoya University},
    year      = {2025},
    type      = {Ph.D. Dissertation},
    language  = {English}
  }

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