Article
Simulation of Upward Jump Control for One-Legged Robot
Based on QP Optimization
Dingkui Tian, Junyao Gao *, Chuzhao Liu and Xuanyang Shi
Citation: Tian, D.; Gao, J.; Liu, C.;
Shi, X. Simulation of Upward Jump
Control for One-Legged Robot Based
on QP Optimization. Sensors 2021, 21,
1893. https://doi.org/10.3390/
s21051893
Academic Editors: Abolfazl Zaraki
and Hamed Rahimi Nohooji
Received: 12 January 2021
Accepted: 4 March 2021
Published: 8 March 2021
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4.0/).
School of Mechatronical Engineering, Intelligent Robotics Institute, Beijing Institute of Technology,
Beijing 100081, China; tiandingkui@bit.edu.cn (D.T.); 3120150091@bit.edu.cn (C.L.);
shixuanyang@bit.edu.cn (X.S.)
* Correspondence: gaojunyao@bit.edu.cn
Abstract:
An optimization framework for upward jumping motion based on quadratic programming
(QP) is proposed in this paper, which can simultaneously consider constraints such as the zero
moment point (ZMP), limitation of angular accelerations, and anti-slippage. Our approach comprises
two parts: the trajectory generation and real-time control. In the trajectory generation for the launch
phase, we discretize the continuous trajectories and assume that the accelerations between the two
sampling intervals are constant and transcribe the problem into a nonlinear optimization problem.
In the real-time control of the stance phase, the over-constrained control objectives such as the
tracking of the center of moment (CoM), angle, and angular momentum, and constraints such as the
anti-slippage, ZMP, and limitation of joint acceleration are unified within a framework based on QP
optimization. Input angles of the actuated joints are thus obtained through a simple iteration. The
simulation result reveals that a successful upward jump to a height of 16.4 cm was achieved, which
confirms that the controller fully satisfies all constraints and achieves the control objectives.
Keywords: upward jumping; QP; ZMP; CoM; angular momentum; anti-slippage
1. Introduction
Jumping enables more flexibility and stronger terrain adaptability for robots in un-
structured terrain. Therefore, jumping motion is an important athletic ability in humanoid
technology.
To improve a robot’s jumping ability, Raibert and et al. designed a very innovative
controller in the 1980s and realized the hopping motion of a hydraulic robot [
1
,
2
]. An
existing legged robot can adjust the footing point by adjusting the step length and achieve
jumping motion on flat ground [
3
]. Poulakakis and Grizzle developed a two-level hybrid
controller that can be used on an spring-loaded inverted pendulum and induce a provably
stable gait on an spring-loaded inverted pendulum [
4
]. Based on a point-foot robot with
elastic legs and compliant hip joints, Hyon proposed a controller that does not require robot
dynamics or any pre-planned trajectories, and used precise nonlinear dynamics to realize
the robot’s continuous jump [
5
]. Haldane analyzed the ability of several arboreal mammals
and robots, constructed a jumping robot using a leg mechanism that enhances the power
modulation, achieved 78% of Gallago’s vertical jumping agility, and demonstrated the
jumping ability of the constructed robot through experiments [
6
]. Yim achieved accurate
and reliable leaping and landing on a narrow foot with the small one-legged jumping
robot Salto-1P [
7
]. The above-mentioned robots have very light-weight legs, the torso of
the robot accounts for the major proportion of the total mass and the torso mass of the
robot is concentrated. Because these robots have point foot or negligible foot in size, these
approaches cannot include constraints, such as stability, non-slippage, and limitation of
angular acceleration in the launch or landing phase. Therefore, the robots in [
1
–
7
] cannot
satisfy the requirements of humanoid robots’ jumping motion.
Sensors 2021, 21, 1893. https://doi.org/10.3390/s21051893 https://www.mdpi.com/journal/sensors
