Abstract—Lateral impact disturbances upon a legged running
locomotion can result in the generation of an unexpected lateral
velocity, consequently cause the deviation of the locomotion
from its original direction of movement, even falling down to
destroy the system. Dealing with the influence of lateral impact
disturbances greatly increases the complexity of control in 3D
space. Inspired by biomechanical studies, this paper constructs a
control strategy based on the spring-loaded inverted pendulum
principle (SLIP) for legged locomotion under lateral impact
disturbances. This strategy, named 3D-HFC, is composed of
three core modules: touchdown angle control, body attitude
angle control and energy compensation. The first module
regulates the forward/lateral running velocities such that lateral
velocity recovers to zero after impact, the second maintains the
body posture without being influenced by external forces, and
the third compensates energy loss to ensure a desired hopping
height. These three parts operate commonly to achieve the
APEX state variables converging to the desired value in each
running cycle, so as for the system to keep stable periodic
motion. The simulations of running systems bearing different
force impacts are conducted to verify the effectiveness of the
3D-HFC strategy, indicating that the proposed approach can
reject impact disturbances effectively.
I. INTRODUCTION
In the past few decades, a simple spring-loaded inverted
pendulum model (SLIP) was proposed as a template to
approximate the dynamics of trotting, running, and hopping in
2-, 4-, 6-, and 8-legged animals [1]-[3]. The SLIP template is
attractive in its simplicity both to biomechanics and to
engineering. Many researchers have anchored this template
with endeavors in embodied legged robots [4]-[6] and various
control methods have been proposed. Buehler presented the
CPDR controller imposing desired trajectories via inverse
dynamics to reduce energy spent for locomotion [7]. Schmitt
and Clark proposed the AER control strategy based on active
energy addition and removal by leg compression and leg
extension [8]. Arslan and Saranli regulated system energy by
changing leg length and stiffness, and realized Deadbeat
control of hopping machines using three control methods [9].
The authors proposed the Hybrid Feedback Control (HFC)
This work was supported in part by the National Natural Science
Foundation of China under grant numbers 51121002, 51435006 and
61175097, and the China Postdoctoral Science Foundation under grant
number 2014M550055.
B. Han and T. Zhang are with the Department of Automation, School of
Information Science and Technology, Tsinghua University, Beijing, China
(e-mail: {binhan, taozhang}@tsinghua.edu.cn
).
X. Luo, Q. Liu and X. Chen are with the State Key Laboratory of Digital
Manufacturing Equipment and Technology, School of Mechanical Science
and Engineering, Huazhong University of Science and Technology, Wuhan,
China (e-mail: {mexinluo, lqy, chenxd}@hust.edu.cn
).
strategy for stability control of SLIP system in sagittal plane on
uneven terrain [10]. But most of them simplified motion
control only in the sagittal plane, consequently confining
motion only in forward direction.
However, moving in three-dimensional (3D) space is more
consistent with the actual situation for a legged running
locomotion, especially in the case that the locomotion is
bearing lateral impact disturbance. Lateral impact disturbances
upon a legged running locomotion can result in the generation
of an unexpected lateral velocity, consequently causes the
deviation of the locomotion from its original direction of
movement, even falling down to destroy the system. In this
case, a 2D planar motion should obviously be converted to the
3D space. Besides, the difficulty of algorithms and the
computational complexity both increase dramatically
compared to the 2D situation.
Most recently, there have been some researchers who
attempted to transfer the SLIP model to 3D space. Raibert built
3D monopod hoppers that requires feedback control for
stability in 1984 [11]. Carver developed deadbeat control
results for a 3D point-mass hopper [12]. Seipel and Holmes
derived an approximation to the stride-to-stride Poincaré
mapping describing the dynamics of a point mass running over
a horizontal plane in 3D space [13]. Later, with semi-analytic
results added, this work was extended and generalized to the
full 3D-SLIP model, including gravity [14]. Wu and Geyer
studied 3D-SLIP running and steering time-based deadbeat
control laws that provide terrain robustness to the template
[15]. Wensing and Orin applied the local deadbeat control to a
3D-SLIP template model, and this method could change speed
and recover from disturbances for the system [16]-[17]. While
these studies involved stability control of the 3D-SLIP system,
however, some of them ignore the damping of the leg spring
and the energy loss in contact, or define the energy loss at a
constant value. In the actual terrain, accurate energy loss is not
easy to get and the imprecise estimates may have a great effect
on the stability control, especially under lateral impact
disturbances.
The goal of this paper is to propose a strategy based on
3D-SLIP model to achieve balance control under force impact.
Considering the situation of the energy loss, the 3D-SLIP
system can quickly restore stability even under unknown
instantaneous external force impact, and the proposed strategy
has a good real-time performance in order to meet the needs of
the dynamic stability control.
II.
ESTABLISHMENT OF 3D-SLIP MODEL
A. The 3D-SLIP Model
A Control Strategy for SLIP-based Locomotion
under Lateral Impact in 3D Space
Bin Han, Member, IEEE, Xin Luo, Member, IEEE, Qingyu Liu, Student Member, IEEE
Tao Zhang, Senior Member, IEEE and Xuedong Chen, Member, IEEE