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Interpreting and Improving Optimal Control Problems with Directional Corrections

Trevor Barron, Xiaojing Zhang

TL;DR

The paper presents a directional-correction framework to interpret and improve optimal control problems (OCPs) in robotics by introducing a consistency score that measures how each cost component aligns with a desired plan change. It extends corrections to state-based directions, closed-loop MPC settings, and constrained OCPs, and analyzes cross-stage effects via a sensitivity map $F$. It further proposes an LP-based weight-optimization method to maximize consistency with a set of corrections, enabling automatic tuning of weights $w_k^{(r)}$ with a margin $m$ under nonnegativity and horizon-sum constraints. Through nonlinear unicycle experiments, the approach identifies inconsistent components (e.g., obstacle costs) and demonstrates open- and closed-loop parameter optimization that yields more consistent and desirable plan behavior, even in the presence of prediction noise. This framework provides a practical, data-driven means to diagnose and refine OCP formulations for robust robotic control.

Abstract

Many robotics tasks, such as path planning or trajectory optimization, are formulated as optimal control problems (OCPs). The key to obtaining high performance lies in the design of the OCP's objective function. In practice, the objective function consists of a set of individual components that must be carefully modeled and traded off such that the OCP has the desired solution. It is often challenging to balance multiple components to achieve the desired solution and to understand, when the solution is undesired, the impact of individual cost components. In this paper, we present a framework addressing these challenges based on the concept of directional corrections. Specifically, given the solution to an OCP that is deemed undesirable, and access to an expert providing the direction of change that would increase the desirability of the solution, our method analyzes the individual cost components for their "consistency" with the provided directional correction. This information can be used to improve the OCP formulation, e.g., by increasing the weight of consistent cost components, or reducing the weight of - or even redesigning - inconsistent cost components. We also show that our framework can automatically tune parameters of the OCP to achieve consistency with a set of corrections.

Interpreting and Improving Optimal Control Problems with Directional Corrections

TL;DR

The paper presents a directional-correction framework to interpret and improve optimal control problems (OCPs) in robotics by introducing a consistency score that measures how each cost component aligns with a desired plan change. It extends corrections to state-based directions, closed-loop MPC settings, and constrained OCPs, and analyzes cross-stage effects via a sensitivity map . It further proposes an LP-based weight-optimization method to maximize consistency with a set of corrections, enabling automatic tuning of weights with a margin under nonnegativity and horizon-sum constraints. Through nonlinear unicycle experiments, the approach identifies inconsistent components (e.g., obstacle costs) and demonstrates open- and closed-loop parameter optimization that yields more consistent and desirable plan behavior, even in the presence of prediction noise. This framework provides a practical, data-driven means to diagnose and refine OCP formulations for robust robotic control.

Abstract

Many robotics tasks, such as path planning or trajectory optimization, are formulated as optimal control problems (OCPs). The key to obtaining high performance lies in the design of the OCP's objective function. In practice, the objective function consists of a set of individual components that must be carefully modeled and traded off such that the OCP has the desired solution. It is often challenging to balance multiple components to achieve the desired solution and to understand, when the solution is undesired, the impact of individual cost components. In this paper, we present a framework addressing these challenges based on the concept of directional corrections. Specifically, given the solution to an OCP that is deemed undesirable, and access to an expert providing the direction of change that would increase the desirability of the solution, our method analyzes the individual cost components for their "consistency" with the provided directional correction. This information can be used to improve the OCP formulation, e.g., by increasing the weight of consistent cost components, or reducing the weight of - or even redesigning - inconsistent cost components. We also show that our framework can automatically tune parameters of the OCP to achieve consistency with a set of corrections.

Paper Structure

This paper contains 19 sections, 1 theorem, 8 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Consistency Analysis Framework
  4. Directional Corrections
  5. Consistency
  6. Properties of (In)consistent Costs
  7. Extension to Closed-Loop Analysis
  8. Extension to Constrained OCPs
  9. Applications of consistency scores
  10. OCP Consistency Analysis
  11. OCP Parameter Optimization Algorithm
  12. Experiments
  13. Experimental setup
  14. Analyzing Open-Loop Plans
  15. Analyzing Closed-Loop Trajectories
  16. ...and 4 more sections

Key Result

Proposition 1

Assume our system is linear, i.e., $x_{k+1} = A_k x_k + B_k u_k$. Then there exists a plan $\hat{\zeta} = (\hat{x}_{0:N}, \hat{u}_{0:N-1})$ that improves upon $\zeta^\star = (x_{0:N}^\star, u_{0:N-1}^\star)$ in terms of $\hat{J}(\cdot)$, that moves in the direction of the correction $a$ with respect

Figures (4)

  • Figure 1: Example of open-loop OCP corrections analysis. The top plot shows the open-loop evolution of the robot within a corridor (green boundary) with an improperly sensed obstacle near (40, 2.5). The center plot shows the speed profile of the robot when affected by the obstacle (orange) and the desired change to that speed profile. The blue profile is the result when the obstacle is removed. The bottom plot shows the computed costs and consistency scores of each component. The consistency analysis successfully detects the obstacle component as inconsistent (red dashed line) even though the magnitude of the obstacle cost is small.
  • Figure 2: Example of closed-loop OCP corrections analysis. The top plot shows the closed-loop evolution of the robot (blue circles) and lead agent (green triangles) within a reference corridor (green boundary). The orange trajectories indicate the computed -- but not executed -- plans of the robot. The center plot shows the speed profile of the robot and its response to an incorrect decelerating prediction of the lead agent. The bottom plot shows the consistency scores of three components of particular interest. The consistency analysis detects the headway and relative speed components as inconsistent at all times for which there is an incorrect prediction.
  • Figure 3: Illustration of the open-loop training process. The left plot shows optimal weights for each cost component aggregated over all stages. The right two plots show the improvement in reference path and reference speed error through the training process.
  • Figure 4: Illustration of the closed-loop training process. The left plot shows the optimal trajectory at each training iteration (orange), along with the predicted trajectory of the lead agent (light green), and the final closed-loop trajectories for the robot (blue) and lead agent (dark green). The center plot shows the error in desired headway of the closed-loop trajectory. This error trends to zero as the robot finds a parameter configuration that avoids deceleration. The right plot shows the optimized parameter values by stage for the three cost components of most relevance.

Theorems & Definitions (5)

  • Definition 1: Consistency Score
  • Definition 2: Consistency
  • Remark 1
  • Proposition 1
  • proof