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Rapid and Robust Trajectory Optimization for Humanoids

Bohao Zhang, Ram Vasudevan

TL;DR

The paper tackles the challenge of designing energy-efficient, dynamically feasible trajectories for high-DOF humanoids under closed-loop kinematic constraints. It presents RAPTOR, a gait optimization framework that optimizes only actuated joints using Bezier-parameterized trajectories and reconstructs the full state via inverse dynamics, coupled with a robust handling of contact and reset-map constraints. Key contributions include a comprehensive hybrid-dynamics formulation for single- and double-support phases, a multi-step offline gait generation formulation with Chebyshev-node constraint enforcement, and an open-source C++ implementation that demonstrates faster and more robust convergence than prior methods while achieving energy-efficient locomotion. The approach is validated on Digit, showing strong convergence properties, effective constraint satisfaction, and favorable energy metrics, with potential for rapid deployment in real humanoid systems.

Abstract

Performing trajectory design for humanoid robots with high degrees of freedom is computationally challenging. The trajectory design process also often involves carefully selecting various hyperparameters and requires a good initial guess which can further complicate the development process. This work introduces a generalized gait optimization framework that directly generates smooth and physically feasible trajectories. The proposed method demonstrates faster and more robust convergence than existing techniques and explicitly incorporates closed-loop kinematic constraints that appear in many modern humanoids. The method is implemented as an open-source C++ codebase which can be found at https://roahmlab.github.io/RAPTOR/.

Rapid and Robust Trajectory Optimization for Humanoids

TL;DR

The paper tackles the challenge of designing energy-efficient, dynamically feasible trajectories for high-DOF humanoids under closed-loop kinematic constraints. It presents RAPTOR, a gait optimization framework that optimizes only actuated joints using Bezier-parameterized trajectories and reconstructs the full state via inverse dynamics, coupled with a robust handling of contact and reset-map constraints. Key contributions include a comprehensive hybrid-dynamics formulation for single- and double-support phases, a multi-step offline gait generation formulation with Chebyshev-node constraint enforcement, and an open-source C++ implementation that demonstrates faster and more robust convergence than prior methods while achieving energy-efficient locomotion. The approach is validated on Digit, showing strong convergence properties, effective constraint satisfaction, and favorable energy metrics, with potential for rapid deployment in real humanoid systems.

Abstract

Performing trajectory design for humanoid robots with high degrees of freedom is computationally challenging. The trajectory design process also often involves carefully selecting various hyperparameters and requires a good initial guess which can further complicate the development process. This work introduces a generalized gait optimization framework that directly generates smooth and physically feasible trajectories. The proposed method demonstrates faster and more robust convergence than existing techniques and explicitly incorporates closed-loop kinematic constraints that appear in many modern humanoids. The method is implemented as an open-source C++ codebase which can be found at https://roahmlab.github.io/RAPTOR/.
Paper Structure (19 sections, 1 theorem, 19 equations, 5 figures)

This paper contains 19 sections, 1 theorem, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Single-Support Dynamics
  4. Double Support Dynamics
  5. Fully-Actuated Representation
  6. Multiple-step Periodic Gait Generation
  7. Cost \ref{['eq-offlineopt-cost']}
  8. Joints Limits \ref{['eq-offlineopt-con-jointlimits']} & Actuator Limits \ref{['eq-offlineopt-con-torquelimits']}
  9. Maintaining Contact \ref{['eq-offlineopt-con-contact']}
  10. Torso Constraints \ref{['eq-offlineopt-con-torso']}
  11. Swing Foot Constraints \ref{['eq-offlineopt-con-swingfoot']}
  12. Reset Map Constraints \ref{['eq-offlineopt-con-resetmap']}
  13. Simulation Experiments
  14. Digit Overview
  15. Implementation Details
  16. ...and 4 more sections

Key Result

Theorem 4

Given actuated joint position $q_a(t)$, velocity $\dot{q}_a(t)$, and acceleration $\ddot{q}_a(t)$, the velocity and acceleration of all joints are given as The reaction force $\lambda(t)$ and the control input $u(t)$ are then uniquely given as where $\Tilde{\tau}(t)$ is the full inverse dynamics vector: $\Tilde{\tau}_u(t)$ and $\Tilde{\tau}_a(t)$ are the collection of unactuated and actuated en

Figures (5)

  • Figure 1: This figure illustrates a physically feasible six-step periodic gait with different step lengths for humanoid robot Digit that is found by RAPTOR, the trajectory optimization algorithm developed in this paper. The duration of each step is fixed to be $0.4$ seconds, which yields a total duration of $2.4$ seconds of the whole gait. It only takes RAPTOR $16$ seconds to converge to a feasible solution.
  • Figure 2: This figure illustrates the closed-loop structure on the legs of Digit. The orange arrows show the rotation axes of all actuated joints (motors). The blue arrows show the rotation axes of all unactuated joints. The purple arrows show the rotation axes of all springs (shin and heel-spring), which are assumed to be fixed in this paper. The green lines show how these joints are connected to be closed-loops.
  • Figure 3: Pareto curve of the constraint violation with respect to the computation time. Note that all curves do not start from 0 in time, since we need to evaluate the problem for at least one iteration to get the constraint violation. In other words, the beginning of the curves indicates the computation time of the first iteration.
  • Figure 4: The absolute value of the difference between the desired step length and the actual step length. The crosses indicate the difference on the optimized trajectory. They may be away from 0 due to optimization not converging to a feasible solution. The circles indicate the difference on the trajectory in simulation when tracking the optimized trajectory using a controller. The circles may not be aligned with the crosses due to imperfect tracking of the controller.
  • Figure 5: Control energy consumption of 3 variants of aligator and RAPTOR in the simulation for 3 different desired step lengths

Theorems & Definitions (1)

  • Theorem 4