Energy Dissipation Rate Control Via a Semi Analytical Pattern
Generation Approach for Planar Spined Quadruped Bouncing
Robot Based on Property of Passive Dynamic Walking
Mohsen Azimi
M. R. Hairi Yazdi
M.Sc. Student, School of Mechanical Engineering
College of Engineering, University of Tehran
Tehran, Iran
[email protected]
Associate Professor, School of Mechanical Engineering
College of Engineering, University of Tehran
Tehran, Iran
[email protected]
Abstract—In this paper Energy Dissipation Rate Control
(EDRC) method is introduced which enables stable walking or
running gaits for legged robots just by controlling the robot’s
swing foot velocity before each Impact Phase (IP). To measure
the effectiveness of the proposed method, it is applied to a Four
Link Two Point Foot (4L2PF) quadruped robot model to realize
an active dynamic bouncing gait on level grounds. Az the pointfoot contact assumption for 4L2PF imposes one degree of under
actuation in the ankle joint, in this paper a new semi analytical
pattern generation approach is introduced which facilitates the
design of the robot’s active joint trajectories during each Single
Support Phase (SSP) by solving the robot’s inverse kinematic
and inverse dynamic equations in parallel. Furthermore, using
the Central Pattern Generator (CPG) controller units, a
connection among CPG’s output, the semi analytical pattern
generation and the robot’s foot placement is introduced which
enables us to realize stable gaits for robots with higher Degree Of
Freedom (DOF). Simulation results show that the proposed
methods in this paper are effective and the 4L2PF robot model is
able to exhibit stable dynamic bouncing gate on level ground.
has been lost in the previous IPs. But all these methods are
faced with two main drawbacks. Firstly, during each SSP the
robot’s mechanical energy level is a function of robot’s both
position and velocity which makes the computational efforts
huge and complicated. Secondly, putting constraint on robot’s
energy level during SSP makes the walking pattern almost
arbitrary. In this paper, based on the fact that during each IP the
level of robot’s mechanical energy is just a function of robot’s
velocity, we will introduce EDRC method, which proposes to
control the amount of energy which is lost in each IP just by
controlling the robot’s velocity before each IP. Also, by
eliminating the robot’s energy level constraint during each SSP
in this method, enables us to control the robot’s walking
pattern, which is desirable for designing robots to work on
irregular terrains.
Keywords—Passive dynamic walking, Inverted Pendulum
Model (IPM), Point foot contact, Semi analytical pattern
generation, Central Pattern Generator (CPG).
I. INTRODUCTION
Passive dynamic walking proposed by McGeer has been
thought as one of the most energy efficient approaches for
humanoid walking robots [1], [2], and has been considered by
several researcher [3]–[5]. In this approach, an unpowered
biped robot utilizes its natural dynamics in walking down a
gentle slope continuously and stably. But, deployment of such
kind of robots are faced with problems [6]. In one hand, the
steady gait cannot be obtained easily without suitable
parameter choice, and in the other hand this steady gait is
sensitive to the initial states and ground slope. To rectify these
problems, several passive based controllers have been
proposed to stabilize steady gait on inclined slopes while they
are unstable in the absence of minimal actuations-such as ankle
torque, hip torque and knee torque. Consequently, several
control methods like Energy Shaping Control [7]–[9], Virtual
Gravity Control [10]–[13], and Parametric Excitation Control
[14] have been proposed to employ the concept of passive
dynamic walking for walking on level grounds. For most of
these passive based controllers, the gait generation idea during
each SSP is based on restoring the mechanical energy which
As mentioned, the EDRC method demands an accurate
control of swing foot’s velocity during each SSP. But, for
legged robot with point-foot assumption, there is no actuation
at the ankle joint. Therefore, there is no direct control over the
stance-leg angle with the ground and it is not clear how to
specify the forward kinematics to define the swing limb
position and velocity as a function of actuated joint angles
[15]–[17]. In fact, in such mechanism the kinematic equations
of actuated element are coupled to the dynamic equations of
the under-actuated elements. So, in this paper a new
methodology is introduced which explains how the kinematic
and dynamic equations of a four-linked quadruped robot with
one degree of under-actuation in the ankle joint could be driven
as an Ordinary Differential Equation (ODE) to generate
necessary trajectories for actuated joint. This enables us to
control the robot’s swing foot state during each SSP and
predicts the robot’s under-actuated joint’s behavior. In
addition, uniqueness of this ODE’s answer avoids any
redundancy problem during pattern generation for a three DOF
robot.
The proposed semi-analytical pattern generation approach
in this paper is very powerful but it is limited to robots with
just three DOF during SSP. To overcome this problem, the idea
of using CPG controller units is proposed, such that for every
extra DOF more than three, there is one CPG unit to control
this extra DOF. In the state of art works of CPG inspired
methods, mostly the pre-designed trajectories can be acquired
by some offline optimization methods [18], [19]. However, if
ground condition changes, a pre-designed trajectory may not be
applicable anymore. Thus, this approach cannot solve the
problem of robots’ walking in an unknown environment [20].
Several researchers have proposed methods that mainly focus
on using the CPG controller units along with solving the
inverse kinematic problem [21], [22]. Using this concept, in
this paper a new analogy is introduced to extend the proposed
semi analytical pattern generation approach for robots with
more than three DOF during SSP.
B. Bouncing methodology
In this article, we have not considered any Double Support
Phase (DSP). Therefore, a complete cycle of bouncing gait is
composed of one Jump-Off (JO), one Flight Phase (FP), one
Touch-Down (TD), one SSP, and one IP. During JO the
angular velocity of fore limb and hind limb joints will increase
during a very short time. After that the robot will come into FP,
in which the whole robot will behave as a ballistic throw. FP
will end by a plastic collision between FF and ground as TD.
After that, during SSP the stance leg is considered as an
Inverted Pendulum Model (IPM) which rotates around its
contact point passively [23], [24], and the whole torso and hind
limb is considered as a Fully Actuated Triple Pendulum
(FATP) model mounted at the hip level of fore limb. In such
manner, during a SSP the hind limb and torso will come
forward fully actuated while the fore limb rotates around its
contact point passively. As soon as the HF reaches the ground,
SSP ends and IP takes place instantaneously. During IP, a
completely plastic collision occurs between HF and the ground,
and the velocity of elements change. This whole cycle will
repeat after each other to form a complete bouncing gait.
The rest of this paper is organized as follows: Section II
introduces the robot’s mechanical structure and the bouncing
methodology. Section III describes the dynamic equations of
the quadruped robot during SSP and IP. Section IV explains
the proposed semi analytical trajectory generation algorithm.
Section V presents the control law and stability criteria, and
finally, in section VI the effectiveness of the proposed method
is examined by numerical simulations. Results show that the
proposed under actuated quadruped robot can bounce on level
ground firmly and continuously.
II. MODEL DESCRIPTION
A. Mechanical structure
Fig. 1 shows the model of the planar quadruped robot
which we have dealt with in this paper. This mechanical
arrangement consists of one torso and two separate limbs, such
that, the torso is composed of two upper and lower separate
parts. Furthermore, there is a point foot for each limb−as Fore
Foot (FF) for fore limb and Hind Foot (HF) for hind limb.
Physical parameters for each point foot is the mass (mFF /
mHF ), and for each rigid link (torso’s parts / limbs) includes
the mass (mi ), the length (Li ), the distance between the center
of mass and distal point (a i ), and inertia moment (Ii ). Also this
robot is composed of four frictionless pin joints with the angles
(θi , 𝑖 = 1,2,3,4) which (θi , 𝑖 = 1,2,4) are measured with
respect to the horizontal line and θ3 is measured with respect to
the local coordinate system mounted on link 2. They are
introduced separately as: ankle joint (J1 ), connecting stance
foot and ground, fore limb joint (J2 ), connecting fore limb and
upper torso, spine joint (J3 ), connecting upper torso and lower
torso, and hind limb joint (J4 ), connecting lower torso and hind
limb.
III. DRIVING EQUATIONS
A. Dynamic equations of single support phase
During each SSP, the whole robot is considered as an open
kinematic chain with four links and four DOF. Therefore, the
dynamic equations of the model are obtained by the wellknown Euler-Lagrange approach as:
M(θ)θ̈ + C(θ, θ̇)θ̇ + g(θ) = ST
(1)
Where θ = [θ1 θ2 θ3 θ4 ]T is the generalized coordinate
vector of the robot, M(θ) = [4 × 4] is the inertia matrix,
C(θ, θ̇) = [4 × 4] is the centripetal and coriolis forces matrix,
and g(θ) = [4 × 1] is the gravitational effects vector.
T = [0 T2 T3 T4 ]T represents the generalized internal torque
vector, and S = [4 × 4] is a matrix which selects the actuator
torques for each segment such that T2 , T3 and T4 appear at the
shoulder joint (J2 ) spine joint (J3 ) and hip joint (J4 )
respectively.
0
0
𝑆=[
0
0
−1
1
0
0
0
−1
1
0
0
0
]
−1
1
(2)
B. Impact equation
In each IP, by assuming a perfectly plastic collision at foot
contact, the total angular momentum around contact point is
conserved, and the post-impact and the pre-impact robot’s
angular velocities (θ̇i , i = 1, 2, 3,4) are related, by a set of
linear transition equations.
Fig. 1. Four Link Two Point Mass (4L2PM) Model
2
θ̇1+
θ̇1−
+
θ̇2
θ̇−
2
=
Q
θ̇+
θ̇−
3
3
−
̇
[θ̇+
]
[
θ
4
4]
ÿ HF = q6 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈1 , θ̈2 , θ̈3 , θ̈4 )
(3)
Taking ẍ HL and ÿ HL as known parameters, Eq. (4) and Eq.
(7) make a three ODE and three unknown system, which can
be written in explicit form of θ̈1 ,θ̈2 and θ̈3 as:
Where −/+ indicate the angular velocities just before and
after collision respectively, while Q is completely a function of
robot’s position. Considering that during an IP the robot’s
position does not change considerably, Eq. (3) exhibit
completely a linear behavior for different velocities. Interested
readers for more details about deriving Q can refer to [25].
θ̈1 = f1 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈3 , ẍ HF , ÿ HF )
θ̈2 = f2 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈3 , ẍ HF , ÿ HF )
θ̈4 = f3 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈3 , ẍ HF , ÿ HF )
C. Inverted kinematic analysis
In other hand, by solving Eq. (5) according to θ2 and θ4 ,
and Eq. (6) according to θ̇2 and θ̇4 respectively:
IV. TRAJECTORY GENERATION
By using robot’s kinematic and dynamic equations during
SSP and writing them as an ODE system, in this section a very
new approach for generating robot’s joint trajectories is
presented.
θ2 = z1 (θ1 , θ3 , xHF , yHF )
θ4 = z2 (θ1 , θ3 , xHF , yHF )
θ̈1 = g1 (θ1 , θ3 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈3 , xHF , yHF , ẍ HF , ÿ HF )
θ̇2 = g 4 (θ1 , θ3 , θ̇1 , θ̇3 , xHF , yHF , ẋ HF , ẏ HF )
θ̈2 = g 2 (θ1 , θ3 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈3 , xHF , yHF , ẍ HF , ÿ HF )
θ̇4 = g 5 (θ1 , θ3 , θ̇1 , θ̇3 , xHF , yHF , ẋ HF , ẏ HF )
θ̈4 = g 3 (θ1 , θ3 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈3 , xHF , yHF , ẍ HF , ÿ HF )
(4)
θ̇1 = ∫ θ̈1
(7)
(12)
D. Swing foot trajectory
In this paper the swing foot trajectory is determined by a
spline function. This has been done based on a few parameters
of each SSP−like the initial swing foot position and velocity
and the final desired swing foot position and velocity.
Differentiating Eq. (5), the velocity and acceleration of the
swing foot are expressed as:
ẍ HF = q5 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈1 , θ̈2 , θ̈3 , θ̈4 )
dθ1
dt
Made us able to write Eq. (11) as a set of nonlinear state
space, which are solvable with known initial conditions
[θ1 θ2 θ3 θ̇1 θ̇2 θ̇3 ] and a specific smooth trajectory for swing
foot.
(5)
(6)
(11)
Also by considering that stance leg functions passively:
B. Kinematic analysis
As depicted in Fig. 1, during each SSP, the origin of the
local coordinate system is assumed to lie in the robot’s
supporting ankle. Considering [xHF , yHF ]T as the HF’s position
in this coordinate system:
ẋ HF = q3 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 )
ẏ HF = q4 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 )
(10)
Substituting Eq. (9) in Eq. (8) and Eq.(10) result in:
In this work, it is considered that during bouncing gait the
robot’s spine joint is controlled by a CPG controller unit in
which, it’s parameters are designed in an offline process
beforehand. So, it is possible to consider the variables θ3 and
it’s derivative θ̇3 and θ̈3 as known parameters in Eq. (4).
Therefore Eq. (4) could be known as an ODE with three
unknown and therefore, two more equations are necessary for
solving it.
xHF = q1 (θ1 , θ2 , θ3 , θ4 )
yHF = q2 (θ1 , θ2 , θ3 , θ4 )
(9)
θ̇2 = z3 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇3 , ẋ HF , ẏ HF )
θ̇4 = z4 (θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇3 , ẋ HF , ẏ HF )
A. Dynamic analysis
Since during each SSP the robot has four DOF, Eq. (1)
contains just four nonlinear coupled ODEs. Therefore,
substituting these four equations in each other, the three
unknown internal torques − T2 , T3 , and T4 − could be
eliminated. The result is an ODE with the generalized
coordinates ( θi , i = 1,2,3,4) and their derivatives.
f(θ1 , θ2 , θ3 , θ4 , θ̇1 , θ̇2 , θ̇3 , θ̇4 , θ̈1 , θ̈2 , θ̈3 , θ̈4 ) = 0
(8)
V. CONTROL AND STABILITY
A. Control
As depicted in Fig. 2 the robot’s spine joint’ angle is
directly controlled by the output signal of a CPG controller
unit. Meanwhile, the robot’s two other joints− shoulder joint
and hip joint− are controlled by a feed-forward controller
which is supplied by the trajectories designed via Eq. (11). In
this way the robot’s swing foot state during each single step is
3
controllable exactly and accurately just by putting the desirable
swing foot path in Eq. (11) and solving it for generating hip
and shoulder joint trajectories, this is while there is not any
actuation in robot’s ankle joint. Beside the robot’s spine joint
trajectory does not change at all and the offline designed
trajectory is applied to the joint without any modification.
VI. NUMERICAL SIMULATIONS
Using Simulink and SimMechanic toolboxes of
MATLAB program, in this section the simulation results of
the proposed methods is presented. The physical parameter
values used in numerical simulation are listed in table 1 which
are based on anatomical measurements of a real cheetah [27].
TABLE I: Parameters of the proposed robot
Quantity
m1
the forelimb mass(kg)
11
m2
The upper torso mass(kg)
24
m3
The downer torso mass(kg)
24
m4
the hindlimb mass(kg)
11
mFF , mHF
the point foot mass(kg)
0.5
L1
the forelimb length(m)
0.67
L2
the upper torso length (m)
0.46
L3
the downer torso length (m)
0.46
L4
the hindlimb length(m)
I1
Fig. 2. Architecture of the controller
I2
As depicted in Fig. 3 in this paper a feed-forward controller
is used for controlling the robot during each SSP which is a
model based controller with feed backs from both position and
velocity [26]. Usually using this kind of controllers for a fully
actuated system involves two steps. First, solving the inverse
kinematics problem to obtain the desired trajectories for the
joints and second, solving the inverse dynamics problem to
obtain the necessary torques which realize the desired
trajectories. As discussed in section IV, in this paper, these two
steps are accomplished just in one step via solving Eq. (11).
𝐕𝐚𝐥𝐮𝐞 𝐢𝐧
𝐬𝐢𝐦𝐮𝐥𝐚𝐭𝐢𝐨𝐧
Symbol
0.82
2
the forelimb Moment of inertia (kg. 𝑚 )
0.4115
2
the downer torso Moment of inertia (kg. 𝑚 )
0.4232
I3
2
the upper torso Moment of inertia (kg. 𝑚 )
0.4232
I4
the hindlimb Moment of inertia (kg. 𝑚2 )
0.6164
ai
The distance between COM and
distal point (m)
0.5 Li
Based on physical constrains, there are two restrictions
on the robot’s HF’s velocity in each IP. First, the horizontal
component of HF’s pre-impact velocity cannot be backward
and second, the vertical component of HF’s pre-impact
velocity cannot be upward which can be written as:
𝑥̇ 𝐻𝐹 − > 0
𝑦̇ 𝐻𝐹 − < 0
(13)
Based on the results of previous works which have
studied several quadruped bouncing robots, energetically it is
more efficient to equip such kind of robots to an active [28],
[29] or even deactive [30] spine joint. This is why in this
paper the robot’s extra DOF is assumed to be on robot’s torso
element which is directly controled by the output signal of a
pre-deigned CPG unit. In such a situation eventhough Eq. (11)
is derived for a quadruped robot with a flexible torso and four
DOF but by considering the robot’s spine joint locked during
the whole bouncing gait and substituting:
Fig. 3. Feed-forward control scheme
B. Stability
The main difficulty in controlling legged robots is the
problem of instability and the risk of falling. Mechanical
energy restoration or existence of the limit cycle is absolutely
critical for providing a stable and continuous walking. Using
the EDRC concept, this is ensured by controlling the swing
foot velocity just before each IP. Also it is assumed that during
each JO, TD, SSP, and IP the stance foot does not take off nor
slips on the ground.
𝜃3 = 0
(14)
Eq. (11) still can be used to design trajectory for the inflexible
torso quadruped bouncing robot with three DOF such that one
DOF in robot’s ankle joint function passively and two DOF
are controlled online via Eq. (11). So in the rest of this section
4
the simulation results for both an inflexible and a flexible
quadruped robot during two complete cycles of bouncing gait
are presented. Fig. 4.a and Fig 4.b. are the stick diagram of
these inflixible and flexible torso quadruped robot respectively
which show the robot’s behavior during a SSP.
goal of the numerical studying in this section can be set as
studing the robot’s impact dynamic equation to find the
smallest swing foot velocities which provides the desirable
after-impact velocities.
Fig. 4.a. Stick diagram during SSP for inflexible torso model
Fig. 5.a. The phase plain for inflexible torso model during two complete
cycles of bouncing
Fig. 4.b. Stick diagram during SSP for flexible torso model
Fig. 5.b. The phase plain for flexible torso model during two complete
cycles of bouncing
Fig. 5.a and Fig. 5.b. shows the phase plane of the inflixible
and flixable robot’s hind limb during these two cycles of
bouncing gait respectively. Realization of these limit cycles
are done by using EDRC concept which proposes the idea of
controlling the robot’s pre-impact velocity to provide
desirable robot’s after-impact velocity. So a numerical studing
on robot’s impact dynamic equations is very useful which help
us in choosing apropriate velocity for robot’s pre-impact
velocity. Totally existance of such an impact phenomenon in
control systems is not very desirable and provides disturbance
and even some time results in harming and damaging the robot
physically. As during an IP lower amplitude for robot’s swing
foot velocities csuses lower amplitude for impact forces, the
One interperation in using the EDRC concept for
providing stable gaits for quaruped bouncing robots can be: to
realize a stable limit cycle it is necessary for robot’s hind limb
to have the same initial angular velocity in the beginning of
each single cycle of bouncing gait. Based on this definition
and by using Eq. (3) all the HF pre-impact velocities [𝑥̇ 𝐻𝐹 − ,
𝑦̇ 𝐻𝐹 − ]𝑇 which provide the hind limb after-impact velocity of
𝜃̇4+ = −400 (𝑑𝑒𝑔/𝑠𝑒𝑐) for different fore limb angles (𝜃1 ) are
determined and depicted in Fig. 6.a. and Fig. 6.b. respectively
for an inflexible and a flexible torso model.
5
Fig. 6.a. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus stance leg angle (𝛉𝟏 )
for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Fig. 7.a. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus swing foot
position(𝐱𝐇𝐅 ) for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Fig. 7.b. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus swing foot
position(𝐱𝐇𝐅 ) for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Fig. 6.b Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus stance leg angle (𝛉𝟏 )
for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
As it is obvious in Fig. 7.a., for an inflexible torso
model, there is two area for swing foot position around 𝑥𝐻𝐹 =
−0.05 (𝑚) and 𝑥𝐻𝐹 = −0.27 (𝑚) which the swing foot
velocities is minimum. Furthermore, considering Fig. 7.b., it is
obvious that in the same situation and for providing the same
results, existance of spine joint localize all the curves in a
narow region very significantly and different position and
velocities for spine joint does not affect it so much, since the
result in Fig 7.b. is just depicted for one spine joint state.
Besides, comparing Fig. 7.a. and Fig. 7.b. shows that the
existance of the spine joint cut the apmilitude of vertical
element of necessary swing foot velocity to almost a half.
Another important factor which affect the necessary HF preimpact velocity for providing a specific hind limb after-impact
velocity is fore limb pre-impact velocity. Using Eq. (3), all the
HF velocities [𝑥̇ 𝐻𝐹 − , 𝑦̇ 𝐻𝐹 − ]𝑇 which provide the after-impact
hind limb velocity of 𝜃̇4+ = −400 (𝑑𝑒𝑔/𝑠𝑒𝑐) for different
fore limb velocity (𝜃̇1− ) are depicted in Fig. 8.a. and Fig. 8.b.
for inflexible an flexible torso model respectiely.
Considering Fig. 6.a., it is obvious that in the same
situation and for providing the same results, smaller stance leg
angle demands smaller swing foot velocities. Furthermore,
Fig. 6.b. shows: different spine joint positions and velocitis
shift all the graphs either towards right or left while localizing
them. Comparing three different group of graphs in Fig. 6.b
depicted for three different positions (𝜃3 ) and velocities (𝜃̇3− )
for the spine joint shows that: allocating smaller positions or
smaller velocities to spine joint shifts all the graphs toward
left. This means towards smaller fore limb position, which
demand smaller swing foot velocities. Same as the stance leg
angle, during IP the horizontal distance between FF and HF
has a significant effect on necessary HF velocity too, which
must be considered in each IP. Therefore, using Eq. (3) all the
HF pre-impact velocities [𝑥̇ 𝐻𝐹 − , 𝑦̇ 𝐻𝐹 − ]𝑇 which provide the
hind limb after-impact velocity of 𝜃̇4+ = −400 (𝑑𝑒𝑔/𝑠𝑒𝑐) for
different HF positions (𝑥𝐻𝐹 ) are presented in Fig. 7.a and Fig.
7.b for inflexible an flexible torso model respectiely.
6
Fig. 8.a. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus fore limb velocity (𝛉̇−𝟏 )
for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Fig. 9.a. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus spine joint position
(𝛉𝟑 ) for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Fig. 8.b. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus fore limb velocity (𝛉̇−𝟏 )
for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Fig. 9.b. Vertical swing foot velocity (𝐲̇ 𝐇𝐅 − ) versus spine joint velocity
(𝛉̇−𝟑 ) for different horizontal swing foot velocities (𝐱̇ 𝐇𝐅 − ).
Depicted graphs in Fig. 8.a. show that choosing smaller
amplitude for robot’s fore limb pre-impact angular velocity
provides the opportunity to have same after impact velocity
with very smaller velocity for robot’s swing foot. In addition
localizing all the graphs in a narrow region is a feature of
existance of spine joint in Fig. 8.b. too, which is very sensible.
Comparing Fig. 6.a. and Fig. 6.b., it is obvious that in the
same situation and for providing the same results, existance of
spine joint shift all the curves toward right-smaller fore limb
velocity-which is desirable, since it provides us more option
within the efficient area. Moreover position and velocity of
robot’s spine joint have a significant effect on robot’s dynamic
behaviour during IP too. All the HF velocities [𝑥̇ 𝐻𝐹 − , 𝑦̇ 𝐻𝐹 − ]𝑇
which cases the hind limb after impact velocity of 𝜃̇4+ =
−400 (𝑑𝑒𝑔/𝑠𝑒𝑐) for different spine joint angle (𝜃3 ) and
velocity (𝜃̇3− ) are presented in Fig. 9.a. and Fig. 9.b
respectively.
As we can see in Fig. 9.a. in the same situation and for
providing the same results, minimum swing foot velocities
realizes when the robot’s spine joint position is about 𝜃3 =
−30 (𝑑𝑒𝑔). Also Fig. 9.b. tells us that considering velocities
around 𝜃̇3− = 30 (𝑑𝑒𝑔/𝑠𝑒𝑐) for spine joint during IP, make us
able to provide desired after-impact velocities for robot with
smaller swing foot velocities. Comparing all the figures from
6 to 9, we can see; in all the figures which the horizontal axis
show one of the robot’s position-like 𝜃1 or 𝜃3 - the depicted
graphs exhibit an absolutely nonlinear behavior and in all the
figures which the horizontal axis show one of the robot’s
velocities-like 𝜃̇1− or 𝜃̇3− -the depicted graphs exhibit an
absolutely linear behavior. Finally, graphical results obtained
by numerical simulation in MATLAB program are presented
in Fig. 8.a and Fig. 8.b for inflexible and flexible quadruped
model during one complet cycle of bouncing respectively.
7
realization of a stable limit cycle has been easily achieved just
by controlling the swing foot velocity in each IP without
putting any restriction on robot’s initial conditions or the
ground slope [6]. Secondly, by removing the mechanical
energy constraints during SSP and developing an analytical
trajectory generation algorithm, the robot is able to move on
irregular trains or to pass barriers which is desirable for
providing a real time controller [31], [32]. Finally, as the
robot’s ankle joint is under-actuated, during a SSP, the stance
leg behavior is completely a function of swing-foot’s
trajectory. Therefore, finding a powerful method to obtain an
appropriate foot trajectory seems necessary, but this is the
subject for our future studies.
VIII. REFERENCES
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Fig. 10.a graphical results for inflexible torso model during a complete
cycle of bouncing.
Fig. 10.b graphical results for flexible torso model during a complete
cycle of bouncing.
VII. CONCLUSIONS
In this paper a 4L2PM model of planar quadruped robot
has been used to bounce on level ground in two separate
situations, i.e. flexible torso and inflexible torso. This has been
realized by proposing a new strategy named as EDRC which
ensures the existence of limit cycle by controlling the energy
dissipation rate during each IP instead of controlling the
energy restoration during SSP. Also a semi analytical
trajectory generation algorithm has been proposed which is
suitable for three-linked model of robots with one degree of
under-actuation in ankle joint. But, using CPG controller
units, makes it posible to be used for higher DOF robots.
Simulation results show that these improvements are effective
and the robot exhibits a stable dynamic bouncing gate with
minimum computational effort.
With respect to previous works, there are two main
advantages in this paper. First of all, using EDRC method,
8
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[31]
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IX. BIOGRAPHIES
Mohsen Azimi received his B.Sc. degree in Mechanical
Engineering from Shahid Bahonar University, Kerman, Iran in
2011. Currently he is pursuing his M.Sc. degree in the School
of Mechanical Engineering, University of Tehran, Tehran,
Iran. His main research interests include design, control and
simulation of linear/nonlinear dynamic systems, and active
control of mechanical vibration.
Mohammad Reza Hairi-Yazdi received his B.Sc. and
M.Sc. degrees in Mechanical Engineering from Amir Kabir
University of Technology, Tehran, Iran in 1985 and 1987
respectively. He received his Ph.D. degree from Imperial
College London in 1992 and since then he has been at the
University of Tehran, Tehran, Iran where he is an Associate
Professor at the School of Mechanical Engineering. His main
research interests include design, simulation, manufacturing
and control of dynamic systems.
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