Nearly Passive Walking of 7-DOF Biped Robot by Means of Controlled Limit Cycles
N. K. M'Sirdi ,
N.Khraief ,
O. Licer
and
Laboratoire de Robotique de Versailles (LRV)
University of Versailles Saint-Quentin en Yvelines
10, Avenue de l’Europe 78140 Vélizy CEDEX
France
[email protected]
Abstract – The main interest of this work is to steer and
stabilize walking motions (periodic behaviour) of a kneed
legged biped robot on a level ground, by means of energy
regulation. The nearly passive walking on a given slope is
exploited. Thus we will present some theoretical and
simulation results based on the use of an energy based
control (referred to as Controlled Limit Cycles).
I. INTRODUCTION
Robotic systems involve mechanical part generally
actuated by DC motors. This gives us processes composed by
passive subsystems which represent a very important family
of nonlinear plants involving mechatronics passive and
dissipative parts. The design of legged robots has to be
conducted in such way to make some kind of motions
appropriate and needing minimal consumption amount. A
biped robot is a two-legged robot, which may have knees and
it most closely resembles human locomotion, which is
interesting to study and is very complex. There has been
extensive research over the past decade to find mechanical
models and control systems which can generate stable and
efficient biped gaits. Figure 1 represent the “RABBIT”1
biped robot and structure of his a 7-dof model.
Fig.1: the model of a 7-dof biped robot downhill a slope
Dynamic gaits are difficult to realize with classical control
design methods (the error decaying time is not finite) when
dealing with legged robots. The control approach, proposed
here, is based on extension of Controlled Limit Cycles (CLC
designed for stabilization of hopping robot’s motion (closed
orbits) see [1-3][14]. For walking robots, the main objective
stands for stable periodic trajectories during regular
displacements. From a state space point of view, such
objective corresponds to achieving stable limit cycles (e.g.
[2]). In such cases, the major problem is to look for these
potential Limit Cycles [5] (i.e. to prove their existence), and,
if they exist, to stabilize the involved periodic or quasiperiodic trajectories (within a chosen state subspace of the
1
Rabbit website: http://robot-rabbit.lag.ensieg.inpg.fr.
N. Manamanni
Centre de Recherche en STIC (CReSTIC)
University of Reims
Moulin de la Housse BP 1039 51687 REIMS cedex 2
France
[email protected]
biped state space).
For instance, bipedal walking might be largely understood
as a passive mechanical process, as shown for part by
McGeer [5], Mochon and McMahon [4] respectively. Indeed,
McGeer [5] proved by both computer simulations and
experimental applications, that some legged systems can
walk on a range of shallow slopes with no actuation and no
control (energy lost in friction and impact is recovered from
gravity). Since then, many researchers have considered this
passivity based approach (e.g. Goswami et al. [6], Mariano et
al. [7] M.W.Spong [8], F.Asano [12] et al. M.Haruna [13],
and reference therein).
The notion of obtaining passive gaits from mechanical
models was pioneered by McGeer [5]. These gaits are only
powered by gravity. In passive human walking, a great part
of the swing phase is passive, i.e. no muscular actuation is
used. Activation is primarily during the double contact phase
and then it essentially turns off and allows the swing leg to
go through like a double pendulum [4].
However, to our knowledge, none have found a method to
define initial conditions under which passive dynamic
walking in a downhill slope is generated for an underactuated biped robot with knees and torso. This motivates the
main part of the present work which considers the walking
motion of a kneed legged biped robot on level ground,
imitating the passive walking on a given slope [11] [22].
In this paper we deal with the stabilization of periodic gaits
for planar biped robots. Indeed, we investigate the existence
and the control of passive limit cycles. The main goal is to
achieve a walking gait, imitating the passive walking (for
energy saving) on a given slope. We develop some theory
and simulation results based on the control method referred
to as Controlled Limit Cycles [23-25]. Moreover such control
also leads the system trajectories to converge towards stable
limit cycles. In this context, we present some results based on
both Poincaré map method and trajectory sensitivity analysis
[19] to efficiently characterize the stability of the almostpassive limit cycles. Moreover, we will show that under a
simple control applied to stabilize a posture, trajectories of
the biped robot can converge towards stable limit cycles. We
found that it is possible to develop energy mimicking control
laws which can make the biped gaits stable and robust under
various kinds of environment.
The gaits have to be chosen in order to minimize control
effort required. The biped robots may exhibits periodic
behaviour, and then the underlying continuous dynamics may
induce an asymptotically stable limit set (periodic cycle) or
Limit Cycles [23]. As these cycles do not always exist (for
all slopes), we propose to add a complementary control to
ensure, electrically driven, periodic motion gaits. In this
purpose, we use the CLC control method (Controlled Limit
Cycles) [2] [23], which considers the system energy for both
controller design and system stabilization and has for aim to
compensate power or energy loss (at each step) to maintain
the cycle. The main goal is to achieve a walking gait of an
under actuated biped robot on the level ground, imitating the
passive walking on a given slope (see Figure 2). A very
interesting idea is to use the torso and the slope to control
and stabilize the robot at a desired walking speed.
equations written as follows:
x =
⎡ q
⎤
d ⎡q ⎤
f ( x) + g ( x)u = ⎢ −1
⎥
⎢
⎥
dt ⎣ q ⎦
⎣ M (q)(− H (q, q ) + Bu ⎦
(1)
Where x = (q, q ) , u = ( u1 , u 2 , u 3 , u 4 ) the inputs, and
q = (q31, q32 , q41, q42 , q1 ) the angular positions:
Equation (1) is obtained from the dynamics between
successive impacts given by: M ( q ) q + H ( q , q ) = Bu . The
dimension of the inertia matrix M(q) is 5 and H(q) depicts the
Coriolis,
centrifugal
and
gravity
terms
(i.e.
)
while
B
is
a
constant
matrix.
The
H(q) = C(q, q) + G(q)
matrices M, C, G,and B are developed in [21][26].
C. Impact model
Fig. 2: From passive to semi passive motion on a slope
The paper is organised as follows. First, we will present the
model of the studied biped (7-dof biped kneed robot with
torso) by considering both the swing phase model and the
impact one in order to succeed to a hybrid form for the
overall model. The following section focuses on the study of
the passive dynamic walking of this robot, on inclined slopes.
Moreover, we will show that locking both knees allows
existence of periodic motions. This is made by means of a
simple PD controls applied to the actuated link between the
torso and the stance leg. Trajectories of the biped robot can
then converge towards stable Limit Cycles (figure 2). In this
context, we use some results based on Poincaré map method
and trajectory sensitivity analysis to efficiently characterize
the stability of the almost-passive limit cycles [11]. Then
we'll present the results obtained with the CLC method for
active dynamic walking on the level ground.
II. THE MODEL
A. Description
The robot (figure (1)) has seven degrees of freedom (the
five joint angles plus the Cartesian coordinates of the hips,
for example). It consists of a torso, hips, and two rigid legs,
with knees and no ankles, connected by a hinge at the hip.
This mechanism moves on a rigid ramp of slope γ. During
locomotion, when the swing leg is in contact with the ground
(ramp surface) at heel strike, it is assumed to have a plastic
(no slip, no bounce) collision and its velocity jumps to zero.
It is assumed also that walking cycle takes place in the
sagittal plane and the different phases of walking consist of
successive phases of single supports only. With respect to
this assumption the hybrid dynamic model of the biped robot
consists of two parts: the differential equations describing the
dynamic of the robot during the swing phase, and an
algebraic equation describing the impacts and model
commutations (the contact with the ground).
B. Swing phase model
The robot’ continuous dynamic is described by differential
The impact between the swing leg and the ground is
modelled as a contact between two rigid bodies. The model
used here is from [18], which is detailed in [16]. The
collision occurs when the following geometric conditions are
(2)
met.
x2 = z2tg (γ )
Where x2, z2 are Cartesian positions of the swing leg end
point and Li are the segments lengths:
x2 = L3 (sin q31 − sin q32 ) + L4 (sin q41 − sin q42 )
(3)
z2 = L3 (cos q32 − cos q31 ) + L4 (cos q42 − cos q41 )
From biped’s behaviour, this means that there is a sudden
exchange in the role of the swing and stance side members.
The overall effect of the impact and switching can be written
as:
h:S → χ
(4)
x + = h( x − )
where S = {( q , q ) ∈ χ / x2 − z2tg γ = 0} , with h as specified in
[21][26]. The superscripts (-) and (+) denote quantities
immediately before and after impact, respectively.
A.
Overall model
The overall 7-dof biped robot has as hybrid model:
⎧ x = f ( x ) + g ( x ) u
⎨ +
−
⎩ x = h( x )
x − (t) ∉ S
(5)
x − (t) ∈ S
This can be expressed as ⎧⎪x = f ( x, Λi )u + g( x, Λi ), ∀x(t) ∉C(ei )
⎨ +
−
⎪⎩x = h( x , ei )
,
∀x(t ) ∈C(ei )
Where the discrete state set is Λ= {ss1,ss2 } = {Λ1 , Λ2 } .
C(ei) is state change condition event after impact for the
event ei. The states are ss1 (leg n°1 is the support leg) and ss2
(leg n°2 is the support leg). The two events correspond to
single supports on one leg and then the other. The
commutation events or changes are in the set
E= {e1 , e2 } = {( ss1 , ss2 ) , ( ss2 , ss1 )} . We can consider now this
model and transform the system by a first partial feedback, in
order to allow existence of Limit Cycles corresponding to
walking downhill a slope.
III. PASSIVE WALKING ON THE DOWNHILL A
SLOPE
A. Outline of procedure
We examine the possibility that a kneed biped robot with
torso can exhibit a passive dynamic walking in a stable gait
cycle, downhill a slope. In our previous work [14] [22-25] on
a kneeless biped robot with torso, we have shown that such
systems can steer a passive dynamic walking on inclined
slopes, one time they have been transformed by a partial
backstepping feedback. Thus, the main idea of this work is to
transform the 7-DOF biped robot in a system for which a
Limit Cycle exists. The objective is then, to obtain nearly
passive Limit Cycles, which will be exploited further to get a
dynamic walking behaviour. This is possible when a torque
is applied to stabilize the torso and knees at fixed positions.
A partial feedback will be designed for, in the next section.
B. Input / output partial linearization
We use some results issued from under actuated systems
from [9-11] [27,28]. The objective is to get a control scheme
able to lock both knees and to stabilize the torso at a desired
position. To show this, for the trunk and knees stabilization,
we can write the dynamic equations system (1) as:
M 11θ1 + M 12θ2 + h1 + φ1 = 0
(6)
M θ + M θ + h + φ = u
21 1
Where θ1 = q1 and
22 2
2
2
θ 2 = (q31 , q32 , q41 , q42 )T and:
⎡M
M ( q ) = ⎢ 11
⎣ M 21
centrifugal terms, the vector functions φ1(q) ∈ R1 and φ2(q)∈R4
contain gravitational terms and u represents the input
generalized force produced by actuators. From (6) we obtain:
⎧⎪M11θ1 + h1 + φ1 = −M12θ2 = −M121θ21 − M122θ22
(8)
⎨ ⎩⎪θ2 = (θ21,θ22 ) = (v21, v22 ) ; M12 = [M121; M122 ]
Where θ21 = (q31, q32 )T and θ22 = (q41, q42 )T .
Definition [9] [10] [11]: The system (6)-(8) is said to be
Strongly Inertially Coupled2 if and only if :
n
For all q∈\
rank ( M12 (q)) = n − m = l
n is the dimension of the state vector, m is the dimension of
the control vector.
This definition is essentially a controllability condition and
ensures that the control of acceleration in (13) may be used
as an input to control the response of q1 .
We verify that if rank (M121 (q)) = 2 for all q ∈ R5 , the
system (6) is said to be strongly inertially coupled [9-11].
Under this assumption we may compute a pseudo-inverse
†
T
T −1
matrix M121
and define the new control
= M121
(M121M121
)
21
121
11 1
1
1
122 22
If q1d now represents a desired position for the torso, and
d
d
(q41
, q42
) the desired ones for both knees we may choose the
control inputs v1 and v22 as simple PD feedbacks:
v1 = q1d + kd1 ( q1d − q1 ) + k p1 ( q1d − q1 )
v22 = θ22d + kd2 (θ22d − θ22d ) + k p2 (θ 22d − θ 22 )
Note also that we can choose a desired trajectory for the
transfert leg to avoid hitting the ground, for example such as:
C. Finding gait cycles and step period
After locking both knees and stabilizing the torso, as
preciously developed, we proceed by simulating the motion
of the biped robot. The walker’s motion can now, after this
transformation, exhibit periodic behaviour. By virtue of work
done by McGeer, we can say that there exist Limit Cycles for
our system. Nearly passive Limit cycles are often created in
this way (downhill a slope). At the start of each step we need
to specify initial conditions ( q , q ) such that after T seconds
(T is the minimum period of the limit cycle) the system
returns to the same initial conditions at the start. A general
procedure to study the biped robot motion is based on
interpreting a step as a Poincaré map. Limit cycles are fixed
points of this function or map.
A Poincaré map samples the flow φx of a periodic system
behaviour once every period [20]. The concept is illustrated
in figure (3). The limit cycle Γ is stable if oscillations
approach the limit cycle over time. The samples xk, starting
from xo, provided by the corresponding Poincaré map
approach a fixed point x* (see figure 3).
(9)
Let
2
(12)
3
is the symmetric, positive definite inertia matrix, the vector
functions h1(q, q ) ∈ R1 and h2 (q, q) ∈ R4 contain coriolis and
variable v21 for the hip angular positions in (8):
v = − M † ( M θ + h + φ + M v )
for θ21 = (q31, q32 )T we can write the following system:
θ1 = v1 , θ22 = v22
(10)
M 21v1 + M 221v21 + M 222 v22 + h2 + φ2 = u
Thus we see that: first the passive degree of freedom q1
have been linearized and decoupled from the rest of the
system (transformed in a simple double integration) and
second that knees may be stabilized around some desired
values. The same holds for the knees positions. The actual
control u can then be given by combining equations (6) to
(10), after some algebraic combinations.
j 21v + M
j 222 v + h + φ
(11)
u=M
2
1
22
2
d
d
⎡ π
⎤
q 4d 2 = F a s i n ⎢
t ⎥ , q41 = q31 − ε and q1 = Cte
⎣ t m ax ⎦
(7)
M 12 ⎤
M 22 ⎥⎦
With this choice for the trunk and the knee angles of v21
The strong inertial coupling requires m ≥ l ; i.e. that the number of active degrees of
freedom be at least as great as the number of the passive degrees of freedom.
P =Σ→ Σ
Fig. 3: Poincaré map
P ( x k ) = x k +1 = φ x ( x k , T )
(13)
Where the Poincaré hyperplane is defined by:
d
d
The
Σ = (q, q ) ∈ χ / x2 − z2tgγ = 0, q1 = q1d , q41 = q41
, q42 = q42
{
}
point xk=P(xk-1) are intersections of the Poincaré hyperplane
Σ with the system’s trajectory. A non stable limit cycle
results in divergent oscillations, for such a case the samples
of the Poincaré map diverge. The fixed point x* (x* is the
initial conditions we are looking for) can be located by the
use of shooting methods [20].
D. Gait cycle stability
Stability in the Poincaré map [20], for the obtained cycle is
determined by linearizing P around the fixed point x*. This
leads to a discrete evolution equation: ∆xk+1=DP(x*) ∆xk
-abFig. 5: a) Nearly passive limit cycle of the center of mass
(Z) of the 7-dof biped robot, γ=3°.
b) γ=0.001rad.
The following results stand for a simulation with a slope
γ=0.001rad. These figures show the periodic cycles obtained
for each joint (knees left curves and hip right curve)
Fig. 4 Algorithm for the numerical analysis
The major issue is how to obtain DP(x) - The Jacobean
matrix- while the biped dynamics is rather complicated; a
closed form solution for the linearized map is difficult to
obtain. But one can be obtained by the use of a recent
generalization of trajectory sensitivity analysis [20] [14]. A
numerical procedure [19] is used to test the walking cycle via
the Poincaré map, it can be summarized by figure 4 and as
follows:
1. With an initial guess xo we use the multidimensional
Newton-Raphson method to determine the fixed point x* of
Fig. 6: Limit Cycles for the knees and hips of the robot.
In the following figure, we present the results obtained for
different values of the slope (with constant trunk angle=10.41°) and different values of the trunk angular position
(with constant slope γ=6.88°).
P + (immediately prior the switching event).
2. Based on this choice of x*, we evaluate the eigenvalues
of the Poincaré map after one period by the use of the
trajectory sensitivity.
A.
Simulation results of a nearly passive walk
Consider the model (1), we choose the hyper plane Σ as
the event plane. We let the biped robot on a downhill slope
with the control scheme (12). Starting with a suitable initial
guess we obtain the following results:
x*=[-2.63, -3.25, -0.213, -0.9, 0, 0.002, 0, 0.001, 0.0012]
We took Kp1=275,Kv1=30,Kp2=30,Kv2=25, q1d=π/7 and γ=3°.
Fig. 7: Energy levels for various slopes and trunk angles.
IV “CLC” FOR ACTIVE DYNAMIC WALKING ON
THE LEVEL GROUND
We saw the existence of nearly-passive limit cycles
downhill a given slope in section 3. However these may not
exist on all slopes, so some additional control is required. In
this section we present “CLC” (Controlled Limit Cycles) [13]. The main goal is to stabilize a periodic motion (walking
gait) by the mean of limit cycles. The control computation
needs the use of energy reference, which can be deduced
from the nearly passive behaviour obtained in the previous
section.
A.
Stability of walking Gaits (Limit Cycles)
In order to visualize the entire dynamics of the robot
over a gait cycle, it is useful to represent the dynamics by
means of phase space trajectories. In phase space, steady
robot gaits are seen as stable Limit Cycles. We consider the
notion of orbital stability to be the most appropriate in the
context of biped robot dynamics. In what follows we
introduce some basic definitions which cope with orbital
stability notion [1-3]. Let us consider a second order system
with state vector x∈Rn , initial conditions x0 and {tk } a
sequence of time instants with lim tk = ∞ : x = f ( x)
(15)
t →∞
j
j
With u = M 21 ( v1 + u ) + M 222 v 22 + h2 + φ 2
j 21v + M
j 222 v + h + φ
u ref = M
2
1
22
2
(23)
u in (23) represents the additional control. We choose the
nonlinear control law:
(24)
u = −Γsign(q T B ( E − Eref ))
{
Where Γ > 0 . It can be shown that the manifold defined by
}
θ1 = θ22 = 0 ∪ { E = Eref } is attractive for the closed loop
system and all trajectories converge towards the Almostpassive limit cycle which is a stable cycle exhibiting a
periodic walk motion of the biped robot.
C.
Simulation results with “CLC” Control
The control law (24) is applied to the system (6) on the
level ground ( γ = 0 ). The next figures show that the CLC
control leads to a stable walking motion of the kneed biped
robot with torso (7-dof).
Definition 1: For the system (32) a positive limit set Ω of a
bounded trajectory x’t) ( x ( t ) < µ , ∀ t > 0) is defined as
(16)
Ω = { p ∈ R n , ∀ ε > 0, ∃ {t k } / p − x ( t k ) < ε , ∀ k ∈ N }
Ω is a closed orbit (limit cycle or non-static equilibrium) for
the system trajectory. Note that for second order systems the
only possible types of limit sets are singular points and limit
cycles. Let us consider the behaviour of neighbouring
trajectories in order to analyze the orbital stability.
Definition 2: Orbital stability: the system trajectory in the
phase space R n is a stable orbit Ω if
∀ε > 0, ∃δ > 0 / x − Ω( x ) < δ ⇒
x(t ) − p < ε , ∀t > t . (17)
0
inf
0
p∈Ω
0
All trajectories starting near the orbit Ω stay in its vicinity. If
all trajectories in the vicinity of Ω approach it as t → ∞
then the limit cycle Ω is said to be attractive.
Definition 3: Asymptotic Orbital stability: the system
trajectory in the phase space R n is asymptotically stable if it
is stable and
(18)
x − Ω < δ ⇒ lim inf x ( t ) − p = 0
0
t → ∞ , p∈Ω
B.
“CLC” Control
In order to get a periodic walk of the biped robot on the
level ground we will use an additional control which will
drive the zero dynamic to a reference trajectory characterized
by the energy obtained from the Nearly-passive limit cycles.
We define a Lyapunov function as follows:
1
(19)
V = ( E − E )2
ref
2
Where E is the total energy of the system defined as:
t
1 T
(20)
E =
q M ( q ) q + q T G ( q ) d t
2
∫
0
and Eref is the energy which characterizes the Almostpassive limit cycle.
(21)
V = ( E − E ref )( E − E ref )
T
E = q B u
(22)
Where
E r e f = q T B u r e f
(a) Swing Leg
(b) Stance Leg
(c) Energies
Figure 8: γ = 0° , θ d = π , K p = 150, K v = 20, Γ = 0.05
t
7
We show in figure 8 (a) the active limit cycle of the swing
leg, in (b) the active limit cycle of the stance leg, and in (c)
the kinetic, potential and total energy.
V CONCLUSION
In this paper, the main objective was the stabilization and
control of legged systems for dynamic gaits. We present a
control law, for a 7-dof biped robot, which realizes a stable
continuous walking on slopes and the level ground to imitate
nearly passive walking on the downhill slopes and Controlled
Limit Cycles (level ground). The proposed control law
realizes a stable continuous walking on the level ground. The
last one is obtained by the use of a simple nonlinear feedback
control with a numerical optimization. The methodology,
developed here, uses Controlled Limit Cycles for
stabilization of a periodic walk. CLC have been previously
shown successful for trotting, hopping and jumping robots
[1-3], here it is extended for control of walking bipeds with
minimum energy. The gaits correspond to limit cycles or
region of attraction defined with the energy of the almost
passive walking robots.
The presented analysis emphasizes the fact that fast
dynamic gaits have to be formulated as closed orbits
expressed in function of the system energy to be correctly
optimized.
The proposed approach is appropriate for (implicit) on
line trajectory design and motion stabilization. The obtained
control law can be interpreted as non linear feedback in
position and velocity (non linearity is deduced from the
energy model). In this way the energy needed for control is
reduced.
Use of passive properties, energy of the system and non
linear feedback adjusted with energy regard, are shown to be
efficient. Simulation results emphasize performance,
efficiency and robustness of the proposed methodology.
Future research intends to implement the same control law
on the test bed biped robot RABBIT [26]. We intend also to
use the hybrid property of the overall model of the biped
which has a switching behaviour between the stance and the
flight phase. The goal is to find the class of signal which
ensure the stability of the cycle, identify some stable limit
cycle and switch between them to change the dynamic of the
robot.
Acknowledgements: many thanks are addressed to Mark
Spong for the helpful comments.
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