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High-speed control and navigation for quadrupedal robots on complex and discrete terrain

Hyeongjun Kim, Hyunsik Oh, Jeongsoo Park, Yunho Kim, Donghoon Youm, Moonkyu Jung, Minho Lee, Jemin Hwangbo

arXiv:2506.02835v1cs.RO

TL;DR

High-speed legged navigation over complex, discrete terrain is challenging because planning is nonconvex and long-horizon, while accurate foothold tracking is difficult. The paper combines a sampling-based planner with a learning-based tracker and competitive terrain generation; its evaluations report agile navigation and real-world tracking, with scope limited by 2.5D planning and motion-capture experiments.

  • Problem

    High-speed navigation on discontinuous terrain requires feasible long-horizon foothold plans and accurate tracking despite nonconvex planning and demanding robot dynamics.

    Source: Introduction · p. 2

  • Method

    A hierarchical pipeline combines a time-efficient sampling-based planner with a learning-based tracker, trained competitively with a generative map model.

    Source: Discussion · p. 8, Terrain generation · p. 10

  • Results

    The pipeline enabled fast, agile motions that were difficult with previous approaches, with approximately 5.3 cm average tracking error in real-world maps and real-time feasible planning on diverse terrain.

    Source: Discussion · p. 8

  • Takeaways & Limitations

    The planner and tracker together support automatic discovery and control of fast, agile motions on complex and discrete terrain.

    Source: Discussion · p. 8

    The planner searches 2.5D maps and cannot plan vertical-wall footholds, while motion-capture dependence prevented outdoor experiments.

    Source: Limitations and future work · p. 8

Abstract

from arXiv · show

High-speed legged navigation in discrete and geometrically complex environments is a challenging task because of the high-degree-of-freedom dynamics and long-horizon, nonconvex nature of the optimization problem. In this work, we propose a hierarchical navigation pipeline for legged robots that can traverse such environments at high speed. The proposed pipeline consists of a planner and tracker module. The planner module finds physically feasible foothold plans by sampling-based optimization with fast sequential filtering using heuristics and a neural network. Subsequently, rollouts are performed in a physics simulation to identify the best foothold plan regarding the engineered cost function and to confirm its physical consistency. This hierarchical planning module is computationally efficient and physically accurate at the same time. The tracker aims to accurately step on the target footholds from the planning module. During the training stage, the foothold target distribution is given by a generative model that is trained competitively with the tracker. This process ensures that the tracker is trained in an environment with the desired difficulty. The resulting tracker can overcome terrains that are more difficult than what the previous methods could manage. We demonstrated our approach using Raibo, our in-house dynamic quadruped robot. The results were dynamic and agile motions: Raibo is capable of running on vertical walls, jumping a 1.3-meter gap, running over stepping stones at 4 meters per second, and autonomously navigating on terrains full of 30{\deg} ramps, stairs, and boxes of various sizes.

Introduction

High-speed navigation over discontinuous terrain requires feasible foothold plans and accurate tracking, yet both planning and control are difficult under legged robots’ dynamics. The paper presents a map generator, tracker, and foothold planner, and demonstrates agile maneuvers on challenging terrain.

Introduction

  • Discontinuous-terrain navigation is difficult because foothold optimization is nonconvex and long-horizon dynamics make planning costly, while fast leg motion demands accurate, low-delay tracking.The introduction also notes that reinforcement-learning curricula are difficult to define for arbitrary challenging environments.Source: Introduction · p. 2

Related work

  • Optimization-based control supports precise foot placement through constraints, but real-time dynamically consistent planning is computationally intensive and may require simplifying assumptions that degrade high-speed performance.The cited approaches include fixed contact timing and approximated dynamics.Source: Related work · pp. 2–3
  • Reinforcement learning can adapt to terrain and robot variation, but curricula may stagnate when environments are too easy or difficult, and existing adaptive methods face dimensionality limits.Prior curricula use staged difficulty, evolving feasible-parameter particles, or grid-based sampling adjustments.Source: Related work · p. 3
  • Sampling-based navigation can be costly and neglect dynamics, while learning-based planners may generalize poorly to unseen environments and long horizons.Conventional navigation systems pair a high-level command or path generator with a low-level controller.Source: Related work · p. 3

Contributions

  • The work contributes a competitively evolving map generator, a controller for parkour maneuvers, and a foothold planner for dynamically consistent plans, demonstrated on Raibo.Reported demonstrations include a 1.3-m gap jump, wall running, stepping-stone speeds up to 4 m/s, tilted-pad jumps, and real-time route planning.Source: Contributions · p. 3

Results

The planner and tracker were validated in simulation and real-world experiments for high-speed navigation on complex, discrete terrain. The tracker handled several disturbances, while planner tests showed high success rates and planning times below the next-target deadline.

Experimental setup

  • Because high-speed state-estimation drift and occluded footholds precluded perception, experiments used pre-acquired height maps and Vicon poses retrieved when planning began.The planner and tracker ran on Raibo’s onboard computer.Source: Experimental setup · p. 4

Evaluation of tracker

  • On the double-wall run, Raibo reached about 4 m/s at T0 and used about 60 Nm of front-knee torque when placing its front foot on the first wall.The authors report that this torque provided propulsion toward the next wall.Source: Evaluation of tracker · p. 5
  • Across five trials per disturbance, the tracker landed on the next target after induced foot slip, a tilting foothold, and a 15-cm map-height error.Each test succeeded in five of five trials; simulated stable-landing success remained above 90% for map-height errors from -5 to 12 cm.Source: Evaluation of tracker · p. 5
  • Mean tracking error was 3.7 cm in simulation and 5.3 cm in real experiments across 12 scenarios.Real-world errors were larger after the 1.3-m gap and on inclined surfaces; the longer jump flight allowed state-estimation error to accumulate.Source: Evaluation of tracker · pp. 5–6

Evaluation of planner

  • The planner sequentially applies performance, roughness, and collision filters; a learned boundary estimator lets it reject collision-prone footholds without rollouts.The estimator’s root mean square error was 2.27 cm, and its error was within 3.34 cm with 95% probability.Source: Evaluation of planner · p. 6
  • With all filters enabled, planner success was at least 90% on every tested map across 500 trials.The collision filter was particularly important on step and slope-patch maps, where flight-phase terrain collisions were more likely.Source: Evaluation of planner · p. 6
  • The planner ran at least 6.35 times faster than the next-plan request interval across scenarios, with an average safety factor of 18.41.These measurements supported real-time planning; sparse maps and terrain with greater collision potential took longer to process.Source: Evaluation of planner · p. 7
  • Real-world planning navigated Stepscape right at speeds up to 2.54 m/s, and rollout selected the lowest-cost foothold plan from eight feasible candidates.The robot reached the goal by tracking the selected plan.Source: Evaluation of planner · p. 7

Discussion

The pipeline combines a time-efficient sampling-based planner with a learning-based tracker to produce fast, agile motion on complex and discrete terrain. Its planner generates feasible foothold plans in real time, while the tracker accurately follows targets; the planner’s 2.5D map and motion-capture setup constrain its scope.

Discussion

  • The pipeline combines a time-efficient sampling-based planner and robust learning-based tracker for fast, agile motions that were difficult with previous approaches.The study presents both a learning framework for high-speed navigation on discrete terrain and a method for real-time feasible foothold planning.Source: Discussion · p. 8, Limitations and future work · p. 8
  • Approximately 5.3 cm average tracking error on real-world maps suggests the tracker accurately steps on target footholds.Competitive generative map training expanded terrain difficulty within the robot’s physical limits, ultimately enabling wall-running behavior.Source: Discussion · p. 8
  • On diverse stepping-stone, stair, and ramp terrains, the planner provided safe, feasible foothold plans in real time on the onboard computer.Physics-simulation rollouts account for full robot dynamics to check plan stability, and the cost function can guide movement toward user-defined objectives.Source: Discussion · p. 8

Limitations and future work

  • The planner searches 2.5D maps, so it cannot plan vertical-wall footholds even though the tracker can perform multiple wall runs.Future work proposes 3D map representations, such as voxel maps, for planning wall runs.Source: Limitations and future work · p. 8
  • Motion-capture requirements prevented outdoor experiments; the proposed future system would add perception for occluded regions and high-speed acceleration.That perception system is intended to support autonomous outdoor navigation.Source: Limitations and future work · p. 8

Materials and methods

The system separates navigation into feasible foothold planning and accurate tracking, with a competitively trained map generator shaping tracker difficulty. Sequential filters narrow sampled plans before physics rollouts select a plan.

Overview

  • The tracker and planner divide navigation into foothold-plan generation and target tracking; searching foothold positions and reusing front-foot placements for rear feet reduce planning complexity.The tracker is trained concurrently and competitively with a map generator.Source: Overview · p. 9

Training procedure

  • The tracker uses PPO in 300 parallel Raisim environments, with 4.2-second episodes and 10 types of domain randomization.A target index updates after either foot contacts its target for 0.06 seconds.Source: Training procedure · p. 9

Terrain generation

  • The CVAE map generator learns from successfully overcome footholds and adjusts latent variance to tune difficulty; after 30 updates, maps included 90° wall running and 1.6-m, 60° steps, with over 95% success.At larger r, the generated φ range widens to −60° to 60°.Source: Terrain generation · p. 10

Input representation for Actor and State Estimator

  • The actor combines proprioception, observation history, foothold targets, and estimated velocity; simulations found two future targets a practical balance between tracking performance and planning time.The state estimator uses proprioception and the previous action as input.Source: Input representation for Actor and… · pp. 10–11

Reward function for Actor

  • Target rewards guide touchdown and progress, while constraint and style rewards shape safe, robust motion; the impact reward reduces touchdown speed after bouncing caused real-world instability and higher tracking error.Style rewards include penalties for torque, slip, and oscillatory actions.Source: Reward function for Actor · p. 11

Planner module

  • Sampling uses the tracker’s trained range, synchronized target updates, and rear-foot reuse; performance, spike, and collision filters reject footholds outside the trained range, rough regions, or predicted collision boundaries.The collision boundary estimator predicts the lowest point of the robot shape swept along the trajectory, rather than a single point’s trajectory.Source: Planner module · p. 12
  • Eight candidate plans are evaluated in parallel physics rollouts with the trained tracker and full Raibo model; invalid plans are rejected and the lowest-cost valid plan is selected.The cost can include survival, goal distance, direction, and elevation.Source: Planner module · p. 12

Statistical analysis

  • For Fig. 3D, the reported mean and standard deviation are calculated over the sample size N.The text defines μ as the mean and σ as the standard deviation.Source: Statistical analysis · p. 13

Network details

The networks divide control, state estimation, contact estimation, and terrain generation into distinct modules. The actor uses future foothold targets, while the conditional variational autoencoder represents stepping-stone configurations.

Actor

  • The actor maps a 167-dimensional observation of robot state, recent history, future foothold targets, and estimated velocity to 12 joint-position targets.Its MLP hidden layers have sizes [512, 128]; target information covers the next two targets for each foot.Source: Actor · p. 26

Critic

  • The critic uses the actor’s 167-dimensional observation but substitutes simulated linear velocity for the actor’s estimated velocity.It shares the actor’s MLP architecture, with hidden layers [512, 128].Source: Critic · p. 26

State estimator network

  • The supervised state estimator predicts the robot’s linear velocity from a 42-dimensional input comprising O_p and previous action; its single-layer, 128-dimensional GRU feeds an MLP.The MLP has hidden layers of sizes [64, 16].Source: State estimator network · p. 26

Contact estimator

  • The contact estimator outputs a contact state for each foot and updates a target index after contact has lasted at least 0.06 seconds.Its input matches the actor’s, and its MLP hidden layers are [256, 64].Source: Contact estimator · p. 26

Conditional Variational Auto Encoder

  • The CVAE trains with reconstruction and KL-divergence losses, while stepping-stone parameters specify positions and orientations for generated stones.The reconstruction loss compares input ψ with decoded ψ′; the loss weights KL-divergence by 0.04. Stone position uses (r_n, θ_n, ϕ_n), and orientation uses sequential yaw and tilt rotations.Source: Conditional Variational Auto Encoder · pp. 26–27, Components of 𝜓 · p. 27

Details of sampling and filtering footholds

The planner samples foothold parameters within ranges conditioned on previous parameters, then filters candidates for safety and planning efficiency.

Details of sampling and filtering footholds

  • The planner samples r, θ, and Δyaw sequentially within ranges modeled as polynomials of influential factors in the previous ψ.The polynomial ranges replace separate supervised networks to reduce network count and time complexity.Source: Details of sampling and filtering… · p. 27
  • The performance filter checks six peripheral samples around each foot target, using a 6 cm radius to balance stability and planning time.Samples lie on a ground-parallel circle and are recursively rotated by 60 degrees from the previous target’s movement direction.Source: Details of sampling and filtering… · p. 27
  • The spike filter reuses height measurements from the performance filter to reduce computation.Source: Details of sampling and filtering… · p. 27

Data collection process for boundary estimator

The boundary-estimator data collection records collision-body clearance over a grid aligned with the current-to-next-target direction, then derives left and right collision boundaries.

Data collection process for boundary estimator

Supplementary Results

The supplementary curriculum analysis compares ways to represent feasible stepping-stone sequences in a 19-dimensional parameter space. The proposed conditional map generator represents that space with substantially less memory than the cell- and particle-based alternatives.

Analysis of various curricula

  • The controller receives two future targets per front and rear foot, so generating a feasible next target must account for the preceding two targets and the last target.This recursive feasibility requirement defines the 19-dimensional environment representation; the supplementary algorithms describe adversarial training and deployment planning.Source: Analysis of various curricula · pp. 28, 35, 36
  • The fixed-order curriculum’s controller averaged 4.3 of 10 stepping stones, failing to progress beyond moderate difficulty as infeasible parameters became prevalent.Its parameter distribution evolved through training stages; Fig. S3 depicts that evolution.Source: Analysis of various curricula · p. 28, Figure S3 · p. 31
  • Baseline 2’s cell-based representation required approximately 1013 times more memory than the proposed method.Its memory demand grows exponentially with parameter-space dimension; Fig. S2 compares memory use and calculation cost across curriculum methods.Source: Analysis of various curricula · pp. 28–29, Figure S2 · p. 31
  • Baseline 3 represents complex parameter distributions, but its particle requirements and memory use increase substantially as dimensionality rises.Fig. S2 compares memory and computational load for particle-filter variants.Source: Analysis of various curricula · p. 29, Figure S2 · p. 31
  • The proposed map generator models feasible next-target parameters conditioned on the previous two targets and last target, representing high-dimensional spaces with significantly less memory.Feasible generation considers ψ_(n−2), ψ_(n−1), and T_last when generating ψ_n in the 19-dimensional space.Source: Analysis of various curricula · pp. 28–29
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