Optimization-Based Quadrupedal Hybrid Wheeled-Legged Locomotion

Authors: Italo Belli, Matteo Parigi Polverini, Arturo Laurenzi, Enrico Mingo Hoffman, Paolo Rocco, Nikolaos Tsagarakis · Humanoids 2020

Licences

Open Access
Paper: CC BY 4.0

Everything below draws on the full paper under its CC BY 4.0 licence; its figure is credited to the authors.

TL;DR

Offline trajectory optimization using single rigid body dynamics plus a unicycle model per steerable wheel plans driving, walking and mixed gaits for the CENTAURO robot.

Why it matters

One framework, with cost weights held constant across all experiments, produced pure legged, pure wheeled and hybrid motions on a real robot with little authoring effort. The wheel-speed cost on feet in contact acts as a tuning element between car-like and walking behaviour, a useful handle for anyone planning motions for wheeled-legged platforms.

Problem

Wheeled-legged robots can combine the speed and stability of driving with the terrain ability of legged locomotion, but few planners existed for them. Earlier methods switched between driving and stepping, optimised footholds before the centre of mass, ignored angular momentum or fixed the vertical centre-of-mass trajectory in advance, or targeted robots with non-steerable wheels.

Proposed solution

The robot is modelled with Single Rigid Body Dynamics extended by a unicycle model for each leg, so each wheel has a Cartesian position and a steering angle and must satisfy a pure-rolling, no-lateral-slip constraint. An offline optimal control problem with contact, foot-workspace, steering-limit, friction-cone and boundary constraints, and a weighted cost on acceleration, goal distance, contact-force changes, foot placement, wheel speed and steering rate, is transcribed by orthogonal collocation and solved with CasADi and Ipopt. The planned Cartesian references are tracked on CENTAURO through CartesI/O and the OpenSoT hierarchical inverse kinematics library.

Key insights

Adding a unicycle model per leg to Single Rigid Body Dynamics represents steerable wheels and their nonholonomic rolling constraint while keeping the model small.
Because a rolling wheel has zero instantaneous slip velocity against the ground, contact forces can be bounded with the same linearised friction cones as in legged locomotion.
Joint limits, which the reduced model cannot represent, are approximated by a cuboid workspace around each foot's nominal position relative to the centre of mass.
Applied to feet in contact, the wheel-speed cost term spans behaviour between car-like driving and pure walking.
The final centre-of-mass position is left to the cost function instead of a hard constraint, because a hard goal constraint made the optimisation fail in unforeseen situations.

Main results

Legged: 10 s horizon, 19.0 s computation, 37 iterations; goal 70 cm ahead reached with 8 steps (shown in the accompanying video only).
Wheeled: 15 s horizon, 20.0 s computation, 21 iterations; goal 100 cm ahead and 60 cm to the left with a 90° counterclockwise base rotation.
Hybrid forward: 10 s horizon, 98.0 s computation, 60 iterations; goal 200 cm away.
Hybrid go&back: 16 s horizon, computation time listed as 63.0 with no unit in Table I, 170 iterations; 90° base rotation and a goal switched back to the origin halfway.
All plans were computed on an Intel Core i7 1.8 GHz laptop with one fixed set of cost weights (γacc = γspeed = 1, γgoal = 10, γforce = 5·10⁻⁴, γfeet = 20, γyaw = 1.5) and executed on the real CENTAURO robot.

Limitations

The optimisation runs offline, taking 19 to 98 s for 10 to 16 s horizons; the authors name reducing computation time to enable MPC as future work.
The gait sequence (a crawl gait) is set by hand; automatic gait sequence generation and optimising lift-phase durations are left to future work.
No sensors are integrated in the planner; the authors list sensor integration, e.g. for gap crossing, as future work.
The reduced model assumes rigid links, point contacts, constant full-body inertia and negligible momentum from joint velocities, cannot account for joint limits directly, and its kinematic constraints use only the base yaw angle, which suits reasonably flat ground.
Hardware validation is shown through snapshots and video; the paper gives no measured tracking error, speed or energy figures for the real robot.

Field impact

Finalist for the Humanoids 2020 Best Oral Paper Award. The paper releases no code or data; its pipeline is built on CasADi, the open-source solver Ipopt, XBotCore, CartesI/O and OpenSoT.

Abstract

This paper presents a trajectory optimization approach to the motion generation problem of hybrid locomotion strategies for a wheeled-legged quadrupedal robot with steerable wheels. To this end, traditional Single Rigid Body Dynamics has been employed and extended by adding a unicycle model for each leg, conveniently incorporating the nonholonomic rolling constraints. The proposed approach can generate hybrid locomotion strategies as well as pure driving and legged locomotion with minimum effort for the user. The effectiveness of the proposed approach has been experimentally validated on the humanoid quadruped CENTAURO, employing a hierarchical inverse kinematics engine to track the planned motions.

Open Source Robotics