Article
Vertical Jumping for Legged Robot Based on
Quadratic Programming
Dingkui Tian
1,2
, Junyao Gao
1,2,
*, Xuanyang Shi
1,2
, Yizhou Lu
1,2
and Chuzhao Liu
1,2
Citation: Tian, D.; Gao, J.; Shi, X.; Lu,
Y.; Liu, C. Vertical Jumping for
Legged Robot Based on Quadratic
Programming. Sensors 2021, 21, 3679.
https://doi.org/10.3390/s21113679
Academic Editor: Salvatore Pirozzi
Received: 18 April 2021
Accepted: 19 May 2021
Published: 25 May 2021
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4.0/).
1
School of Mechatronical Engineering, Intelligent Robotics Institute, Beijing Institute of Technology,
Beijing 100081, China; tiandingkui@bit.edu.cn (D.T.); shixuanyang@bit.edu.cn (X.S.);
3120180169@bit.edu.cn (Y.L.); 3120150091@bit.edu.cn (C.L.)
2
Beijing Advanced Innovation Center for Intelligent Robots and Systems, Beijing 100081, China
* Correspondence: gaojunyao@bit.edu.cn
Abstract:
The highly dynamic legged jumping motion is a challenging research topic because of the
lack of established control schemes that handle over-constrained control objectives well in the stance
phase, which are coupled and affect each other, and control robot’s posture in the flight phase, in
which the robot is underactuated owing to the foot leaving the ground. This paper introduces an
approach of realizing the cyclic vertical jumping motion of a planar simplified legged robot that
formulates the jump problem within a quadratic-programming (QP)-based framework. Unlike prior
works, which have added different weights in front of control tasks to express the relative hierarchy
of tasks, in our framework, the hierarchical quadratic programming (HQP) control strategy is used
to guarantee the strict prioritization of the center of mass (CoM) in the stance phase while split
dynamic equations are incorporated into the unified quadratic-programming framework to restrict
the robot’s posture to be near a desired constant value in the flight phase. The controller is tested
in two simulation environments with and without the flight phase controller, the results validate
the flight phase controller, with the HQP controller having a maximum error of the CoM in the x
direction and y direction of 0.47 and 0.82 cm and thus enabling the strict prioritization of the CoM.
Keywords:
hierarchical quadratic programming; quadratic programming; vertical jumping;
full dynamic
1. Introduction
Due to the constraints of underactuation, high dimensionality, and the ground reaction
force in legged robots, it is remarkably challenging to control a highly dynamic legged
robot’s jumping motion. In the stance phase, the most common solution for jumping
is to reduce the full dynamics to a canonical spring-loaded inverted pendulum (SLIP),
which renders the control for legged robots computationally tractable and predicts the
energy wave and ground reaction force during the jumping motion in the stance phase [
1
].
A nonlinear controller is employed to synchronize the biped dynamics and SLIP [
2
–
6
],
which is an effective solution, but makes it difficult to introduce constraints, such as the
stability of the robot and acceleration of the joints, into the controller or to guarantee
strict prioritization in the overconstrained objective. Additionally, the majority of previous
approaches focus on reducing the angular momentum in the center of mass (CoM) at the
launch phase [
7
], and few studies have paid close attention to how to adjust the position
and attitude of the robot during the flight phase.
1.1. Related Work—Hierarchical Task Controllers
In overconstrained tasks for robots, controlling the hierarchies of the tasks is very
important. There are two main solutions to solving overconstrained problems: adding
weights to the control tasks and hierarchical task control.
Sensors 2021, 21, 3679. https://doi.org/10.3390/s21113679 https://www.mdpi.com/journal/sensors
