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Rolling in the Deep -- Hybrid Locomotion for Wheeled-Legged Robots using Online Trajectory Optimization

Marko Bjelonic, Prajish K. Sankar, C. Dario Bellicoso, Heike Vallery, Marco Hutter

arXiv:1909.07193v3cs.ROcs.SYeess.SY

TL;DR

Wheeled-legged robots could combine legged terrain negotiation with wheel-enabled mobility, but online robust planning for hybrid walking-driving with actuated wheels remains a gap. The paper separates wheel and base trajectory optimization and uses whole-body control to track the plans. ANYmal demonstrated hybrid locomotion on rough terrain and in tunnel runs, where it averaged 0.5 m/s—more than double the traditional legged system’s average speed.

  • Problem

    Online trajectory generation for robust hybrid walking-driving with actuated wheels across varied terrains is an identified research gap.

    Source: A. Related Work · p. 2

  • Method

    The framework decomposes planning into wheel and base trajectory optimizations, synchronizing them through wheel trajectories used in the base ZMP constraints.

    Source: §VI · p. 8

  • Results

    ANYmal negotiated rough-terrain steps up to 20% of its leg length and averaged 0.5 m/s in tunnel runs, more than double the traditional legged system’s average speed.

    Source: C. Rough Terrain Negotiation · p. 7, E. DARPA Subterranean Challenge:… · p. 7

  • Takeaways & Limitations

    The authors conclude that the decomposition enables more tractable real-time onboard MPC planning and increases robustness against unforeseen disturbances.

    Source: §VI · p. 8

Abstract

from arXiv · show

Wheeled-legged robots have the potential for highly agile and versatile locomotion. The combination of legs and wheels might be a solution for any real-world application requiring rapid, and long-distance mobility skills on challenging terrain. In this paper, we present an online trajectory optimization framework for wheeled quadrupedal robots capable of executing hybrid walking-driving locomotion strategies. By breaking down the optimization problem into a wheel and base trajectory planning, locomotion planning for high dimensional wheeled-legged robots becomes more tractable, can be solved in real-time on-board in a model predictive control fashion, and becomes robust against unpredicted disturbances. The reference motions are tracked by a hierarchical whole-body controller that sends torque commands to the robot. Our approach is verified on a quadrupedal robot with non-steerable wheels attached to its legs. The robot performs hybrid locomotion with a great variety of gait sequences on rough terrain. Besides, we validated the robotic platform at the Defense Advanced Research Projects Agency (DARPA) Subterranean Challenge, where the robot rapidly mapped, navigated and explored dynamic underground environments.

I. INTRODUCTION

Legged robots negotiate challenging terrain, while wheels can extend their speed for rapid, long-distance mobility. This work plans dynamic hybrid walking-driving motion on wheeled-legged robots across varied terrain.

I. INTRODUCTION

  • ANYmal with four non-steerable, torque-controlled wheels is shown traversing a plank and rough terrain, and operating in the DARPA Subterranean Challenge.The figure also shows chains on its wheels for muddy terrain.Source: Figure 1 · p. 1
  • The paper targets rapid, long-distance mobility on challenging terrain by combining wheeled speed with legged robots’ ability to negotiate varied terrain.The introduction notes that matching conventional legged robots’ natural counterparts remains elusive.Source: §I · p. 1
  • The central contribution is locomotion planning for dynamic hybrid walking-driving motions on a wheeled-legged robot.Here, hybrid locomotion means simultaneous walking and driving.Source: §I · p. 1

A. Related Work

Prior wheeled-legged work rarely demonstrated hybrid walking-driving, and existing planning approaches faced limits in online operation or real-robot validation. The paper identifies a gap in online, robust trajectory generation for hybrid motion with actuated wheels.

A. Related Work

  • Most wheeled-legged studies focused on statically stable driving with active suspension, without wheel lift-offs or sophisticated wheel-motion planning.The review describes hybrid walking-driving demonstrations as scarce.Source: A. Related Work · p. 2
  • Earlier work could perform walking and driving but not simultaneously, while a single base-and-wheel optimization ran at 50 Hz and showed no real-robot hybrid motions.The earlier robot had to stop and switch to pure walking to overcome obstacles.Source: A. Related Work · p. 2
  • Other approaches also left gaps: CENTAURO walked without using its wheels, while simulated driving-and-walking planning omitted robot dynamics,.Robosimian used passive wheels and was shown only in simulation; Skaterbots’ nonlinear program was too computationally demanding for receding-horizon online updates.Source: A. Related Work · p. 2
  • The identified gap is online trajectory generation for hybrid walking-driving with actuated wheels that is robust across terrains.The authors point to online dynamic MPC and trajectory-optimization results on conventional legged robots as approaches to extend.Source: A. Related Work · p. 2
  • Conventional legged-robot methods already demonstrated dynamic MPC and trajectory optimization, stair locomotion, desk-height jumps, and visual foothold adaptation –.The authors conjecture that extending these approaches to wheeled-legged systems can aid robust motion generation.Source: A. Related Work · p. 2

B. Contribution

The framework combines separate wheel and base trajectory optimizations with whole-body control to generate torque commands. It supports varied gaits and fast hybrid trajectories, and was demonstrated in the DARPA Subterranean Challenge.

B. Contribution

  • The framework separates wheel and base trajectory optimization; hierarchical whole-body control tracks the motions and computes joint torque commands.Wheel optimization accounts for rolling constraints, while base optimization uses the zero-moment point (ZMP) for balance.Source: B. Contribution · p. 2
  • The framework supports pure driving, statically and dynamically stable gaits, and gaits with full-flight phases.Source: B. Contribution · p. 2
  • Wheel and base trajectories for hybrid motion are generated in milliseconds, and the authors report robustness to unpredicted disturbances and DARPA Subterranean Challenge deployment.At the challenge, the robot autonomously mapped, navigated, and searched dynamic underground environments.Source: B. Contribution · p. 2

II. MOTION PLANNING

Motion planning decomposes the problem into wheel and base trajectory optimizations, followed by tracking control. The base plan uses ZMP stability constraints informed by optimized wheel trajectories, and the stages can run in parallel.

II. MOTION PLANNING

  • Wheel and base trajectory optimizations form separate tasks that are synchronized using previously generated wheel trajectories in the base ZMP constraints.The authors propose this decomposition to make high-dimensional wheeled-legged locomotion planning more tractable and suitable for real-time MPC.Source: §II · p. 2, Figure 2 · p. 3
  • Given a fixed gait and reference velocities, wheel optimization precedes base optimization, which satisfies a ZMP stability criterion; a controller tracks both plans with torque commands.The wheel optimization, base optimization, and tracking controller can run in parallel.Source: §II · p. 2
  • The ZMP-based motion planner considers optimized wheel trajectories and robot state, while hierarchical whole-body control tracks operational-space references and outputs torque references.The whole-body controller optimizes whole-body accelerations and contact forces.Source: Figure 2 · p. 3
  • Wheel trajectories are optimized individually in terrain-aligned frame W, while base trajectories use COM-centered frame B with W’s orientation.The illustrated wheel path spans one stride, with lift-off, maximum swing height, touch-down, and horizon times set by the gait pattern.Source: Figure 3 · p. 3

III. WHEEL TRAJECTORY OPTIMIZATION

Wheel trajectories are formulated as separate quadratic programs for each wheel. The optimization minimizes a quadratic objective over its variables subject to linear equality and inequality constraints.

III. WHEEL TRAJECTORY OPTIMIZATION

  • Each wheel’s x, y, and z trajectories in its wheel coordinate frame are found through a separate quadratic programming problem.The trajectories are defined with respect to the wheel coordinate frame W.Source: §III · p. 3
  • The optimization variable ξ is chosen to minimize the quadratic objective 1/2 ξ^T Qξ + c^T ξ.Source: §III · p. 3
  • The optimization is subject to the linear equality Aξ = b and inequality Dξ ≤ f constraints.Source: §III · p. 3

A. Parameterization of Optimization Variables

Wheel trajectories are parameterized as connected splines, with separate parameterizations for segments when the wheel is airborne and when it is in contact with the ground.

A. Parameterization of Optimization Variables

1) Wheel segments in air:

Each airborne wheel trajectory coordinate is represented by a quintic spline whose polynomial coefficients are optimized over that segment’s time interval.

1) Wheel segments in air:

2) Wheel segments in contact:

For contact segments, the wheel’s rolling-direction velocity is parameterized while lateral and vertical velocities are set to zero; the resulting position trajectory is obtained by integration.

2) Wheel segments in contact:

  • During contact, rolling-direction velocity is a quadratic polynomial, while velocity in the other directions is set to zero.This parameterization inherently captures the wheel’s no-lateral-slip velocity constraint.Source: 2) Wheel segments in contact · p. 3
  • The contact-segment position trajectory is obtained by integrating velocity and adding the trajectory’s initial x and y positions.Source: 2) Wheel segments in contact · p. 3
  • Assuming constant reference yaw rate over the optimization horizon makes the integration analytic and yields a linear expression in the segment coefficients.Source: 2) Wheel segments in contact · pp. 3–4
  • The segment’s velocity and acceleration trajectories are given by the first and second derivatives of its position expression.Source: 2) Wheel segments in contact · p. 4

B. Formulation of Trajectory Optimization

The wheel-trajectory optimization is repeatedly solved online in an MPC fashion over a finite horizon. It is formulated as a quadratic program, with a contact-specific case indicated.

B. Formulation of Trajectory Optimization

velocity

The framework optimizes wheel and base trajectories with objectives and constraints for smoothness, tracking, foothold placement, and stability. The base optimization extends ZMP constraints to account for moving wheel contacts.

C. Objectives

2) Spline continuity:

IV. BASE TRAJECTORY OPTIMIZATION

A. Parameterization of Optimization Variables and Formulation of Trajectory Optimization

  • The base optimizer represents COM position and base orientation with quintic splines, initializes from measured state, and adds smoothness, velocity tracking, junction, and ZMP constraints.Its trajectories are updated online for the free-floating base.Source: §IV · p. 5, A. Parameterization of Optimization… · pp. 5–6

V. EXPERIMENTAL RESULTS AND DISCUSSION

  • Experiments used ANYmal equipped with non-steerable, torque-controlled wheels.A video accompanies the reported results.Source: §V · p. 6

A. Implementation

Concurrent online optimizations and whole-body control support the robot’s hybrid gaits. Tests demonstrate autonomous mine traversal and rough-terrain negotiation, with hybrid-trotting cost comparisons and automatic gait switching left as a future direction.

A. Implementation

  • Concurrent threads run wheel and base optimizations alongside a whole-body controller that tracks trajectories with actuator torque commands and accounts for rigid-body constraints.The controller runs at 400 Hz. Cost-function weights are tuned manually, with one parameter set used for all shown motions.Source: A. Implementation · p. 6

B. Solver Time of Different Contact Scheduler and Gait Switching

  • Wheel and base optimizations run in milliseconds across gaits ranging from all-contact driving to gaits with full-flight phases.Table I reports time horizons and optimization times including model setup; wheel-optimization times are for one wheel.Source: B. Solver Time of Different Contact… · p. 6, Table I · p. 6

C. Rough Terrain Negotiation

  • ANYmal blindly negotiated varied rough terrain, including steps up to 20% of its leg length, without stopping to switch to pure walking.When a wheel collided with a step edge, the framework adapted by driving over the obstacle.Source: C. Rough Terrain Negotiation · pp. 6–7

D. High Speed and Cost of Transport

  • At 2 m/s, hybrid trotting yields mechanical COT 0.2 and 156 W, with COT twice pure driving but 42% below traditional trotting and 9% below skating.These comparisons are for flat terrain.Source: D. High Speed and Cost of Transport · p. 7

E. DARPA Subterranean Challenge: Tunnel Circuit

  • ANYmal traveled more than 100 m autonomously in the second mine run and averaged 0.5 m/s at the Tunnel Circuit, more than twice traditional ANYmal’s speed.The terrain included hills, mud, and puddles; after a wheel slipped during the first run, a chain was added to increase friction. Fig. 5 shows desired trajectories for about two strides.Source: E. DARPA Subterranean Challenge:… · p. 7, Figure 5 · p. 7

VI. CONCLUSIONS

  • The conclusion credits the wheel/base optimization split with tractable, on-board real-time MPC and disturbance robustness; automatic gait-timing optimization remains future work.The authors state that automating when to lift a leg can increase locomotion speed and robustness.Source: §VI · p. 8
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