PERIODIC STABILIZATION OF A 1-DOF HOPPING
ROBOT ON NONLINEAR COMPLIANT SURFACE
C. Canudas , L. Roussel , A. Goswami Laboratoire d'Automatique de Grenoble
UMR-CNRS 5528, ENSIEG-INPG
B.P. 46, 38402, St Martin d'Heres, France
E-mail : canudas@lag.ensieg.inpg.fr
INRIA Rh^one-Alpes
ZIRST, 655 ave. de l'Europe
38330 Montbonnot St Martin, France
Abstract: This paper deals with the problem of characterizing and stabilizing
periodic orbits of a 1-DOF hopping robot moving over a nonlinear compliant surface.
It is shown, through implicit calculations of the Poincare map, that globally stable
periodic motions are possible via the injection of nonlinear damping.
Keywords: periodic stabilization, walking and hopping robots.
1. INTRODUCTION
The system considered is the classic simple point
mass bouncing on a surface or equivalently, an
idealized 1-dof hopping robot. The objective of
this work is to introduce compliance in the
robot/ground interface in order to control the
interaction. The model can be seen as a rigid robot
bouncing on a compliant surface or a compliant
robot bouncing on a rigid surface.
The current work stems from our eorts in the
modeling and control of legged robots. While
modeling a legged robot it is important to recall
its principal dierences with a conventional manipulator arm, the dierences which are of fundamental importance but are sometimes overlooked
and, in our opinion, not suciently emphasized.
While a conventional manipulator arm is permanently rigidly xed to the ground, a legged robot
is not. More importantly the \constraint" between
the robot foot and the ground is unilateral. Thus
given appropriate joint torques the robot is free
to leave the ground, which is an impossibility for
a conventional arm.
There are two common lines of approach in treating the interaction of such a robot with the
ground. The rst is a rigid body approach where
both the robot and the ground are idealized rigid
bodies. A second approach, the one adopted here,
is the modeling of the interaction as a compliant
phenomenon. In this approach, we typically place
a spring/damper module on one of the two interacting objects. The advantage of this model is that
it has continuous dynamics governed completely
by dierential equations and can be used to simulate a variety of interacting surfaces by changing
the spring/damper coecients.
In case of interactions involving non-zero relative velocities between the interacting objects a
linear damper produces a discontinuous jump in
the interaction force during the rst contact, a
phenomenon which cannot be physically justied.
Also, traditional spring-damper elements, which
are active both in extension and compression,
cannot correctly model the ground contact of either a bouncing ball or a legged robot since they
would generate articial tension forces pulling the
interacting objects towards each other.
In order to rectify the two rst problem, we adopt
the use of non-linear spring-damper elements following (?) and (Marhefka and Orin., April 1996).
The authors of both of these articles focus on the
performance of the non-linear models in terms
of their dynamic behavior. We use the model,
in addition, to formulate simple control laws to
stabilize a simple one-degree of freedom hopping
robot about a desired periodic orbit. The characteristics of the walking robot is preserved here
through the restriction of allowing only positive
feedback. Given the height of jump altitude, the
employed control law tries to bring the robot to
the bouncing trajectory. In this paper we present
a control law that globally stabilizes the robot
to any arbitrary jump altitude. We present two
alternative controllers: a dead-beat feedback and
an asymptotic one.
Raibert performed a detailed theoretical and practical investigation of the hopping robot (Raibert,
1986) and was capable of establishing a stable
vertical hopping motion of the robot. Several theoretical studies, chiey those from the groups of
Koditschek (Koditschek and Buehler, 1991) and
Burdick(A.F. et al., 1991)(McCloskey and Burdick., April 1991) followed from this. (Ahmadi
and Buehler., April 1995) studied passive dynamic
running with a simple hopping robot model similar to the ones mentioned above. This model takes
advantage of tuned passive elements, in addition
to the actuators to run economically. The proposed controller can stabilize any desired robot
speed. (Francois and Samson., April 1994) considered the problem of creating constant-energy
gaits of hopping robots.
2. MODELS
The dynamics of the hopping robot may be written as follow:
m_v = ?mg + u + F u 0
(1)
where m is the masse, v is the velocity of the
robot, g is the gravity, u 2 R+ is the control input,
and F is the contact force.
Nonlinear contact models for compliant surfaces
have been introduced in works of (Hunt and
Crossley, 1975), and studied further in Marhefka
and Orin' 96, to cope with the following drawbacks of the simple linear spring/damper model:
the model is physically incoherent since interacting bodies may exert tensile forces on
another before separation,
the equivalent restitution coecient depends
upon the masses of the colliding bodies and
not on the impact velocities,
contact force is discontinuous at the moment
of impact.
A quite general nonlinear contact model is (Hunt
and Crossley, 1975) is:
F = ?jxjnv ? kjxjn?1x
(2)
where F is the contact force, v = x_ is the velocity,
x the position, and , k are positive constants.
The surface x = 0 represents the contact surface.
The power n 1 is used to model dierent geometries of contact (i.e. a sphere impacting a Hertzian
plate ; n = 2=3). Two interesting properties have
been demonstrated with this model (Marhefka
and Orin' 96):
x
Dnc
v
Dc2
Dc1
Fig. 1. Natural dissipative motion.
x
v 0 = - vi
v0
vi
v
x
Fig. 2. Accessible set of output velocities by positive
damping injection.
(i) consider a body of mass m, in contact with
a surface modeled by (1) and (2), i. e.
m_v + jxjnv + kjxjn?1x = 0;
x0
then the position x can be solved explicitly
as a function of the actual velocity v, the
contact velocity vi , and the contact position
xi
x = (xi; vi ; v)
(3)
h
2
m
(
n
+
1)
4
2 ? 3vi 3(v ? vi ) ? 2 ln 2 ? 3v = ? 92 k
1
+(?xi )n+1 n+1
(4)
where = (2=3k).
(ii) For small vi , the equivalent restitution coefcient e,can be approximated as e 1 ? vi .
The complete system can be written as:
(
v_ =
?g
if x > 0
1 (?jxjnv ? kjxjn?1 x + u) ? g if x 0
m
(5)
where x = 0 describes the contact surface, and
u 2 R+ . Note that the control u, u 0 can
only apply forces in the upward direction and
that no control action is possible during the ight
phase. This is a common factor in the walking
mechanisms where the degree of freedom between
feet and ground is not actuated.
3. INVARIANT PERIODIC ORBITS
The control philosophy underlined in this paper
consists rst in the characterization, via an innerfeedback loop, of an invariant periodic orbit, and
then to design, through an outer-feedback loop, a
control strategy to stabilize such an orbit.
Since model (5) can only dissipate energy during
the contact phases ( the map v 7! ?F is dissipative), and energy is preserved during the free
motion phases, it is clear that \natural" periodic
orbits (with u = 0 ) cannot exist. This can be
seen by writing the following energy-like positive
continuous function V
8
< mgx + 0:5mv2
if x > 0
V = : mgx + 0:5mv2 + k jxjn+1 if x 0 (6)
n+1
which has its time-derivative evaluated along the
solutions (continuous) of (5) given as
0
V_ = ?jx0jnv2 ifif xx >
0
(7)
It is convenient to split the state-space into three
disjoint subsets in R2,
Dnc = f(x; v) : x > 0g
Dc1 = f(x; v) : x 0; v < 0g
Dc2 = f(x; v) : x 0; v 0g
(8)
(9)
(10)
This state-space partition is shown in Fig(1)(2),
together with the phase diagram of a natural
dissipative motion (u = 0). In this gure v0 and
vi stands for the exit velocity and the entrance
velocity, respectively. xd is the desired altitude
of the periodic motion. Note that to reach xd it
is necessary to inject energy via u 0 during
the contact phases. However, this is only possible
during the motion in Dc2 due to the restriction of
the positive sign of the control action.
For this we consider that u be given as:
u = g + 2m(vi ) jxjnv 0 if (x; v) 2 Dc2 (11)
where the rst term compensates for the gravity
and the second term is a nonlinear damper used to
inject the precise amount of energy to ensure that
v0 = ?vi . Note that 2 (vi ) 0 is constant during
the contact phase, but it depends on the entrance
velocity resulting from the free fall starting at xd .
The nonlinear term in u is homogeneous to the
nonlinear damper of the contact surface. Hence
the total damper coecient 1 (vi ) = 2 (vi ) ?
can be virtually modied within the following
bounds: ? 2 < 1.
With this rst inner-loop, the closed-loop equation becomes,
8
f (x; v)
>
> 1
>
>
>
> f2 (x; v)
<
= ?g if (x; v); 2 Dnc
= m1 (?jxjnv ? kjxjn?1x)
v_ = >
if (x; v); 2 Dc1
(12)
>
1
>
n
n
?
1
>
f3 (x; v) = m (2 (vi )jxj v ? kjxj x)
>
>
:
if (x; v); 2 Dc2
Problem 3.1. (Existence of invariant periodic orbits). Let vd be the desired entrance velocity corresponding to the desired jump altitude, and assume that x(0) = xd . An invariant orbit does
exist, if an entrance velocity-dependent constant
2 (vi ) > 0, can be found such that the exit velocity
vo be equal to the entrance desired velocity vd , i.e.
vo = ?vi = vd .
The natural system motion is dened as being a
particular case of the above closed-loop equation
with 2 = ?, ( or equivalent u = 0), and is
characterized by the explicit map () : v 7! x,
8(x; v) 2 Dc1 [ Dc2
x = (vi; v)
the exit velocity resulting from this natural dissipative motion, namely vo , can be computed from
the implicit equation
0 = (vi; vo )
which results in vo < vi .
By injecting a positive damping, as described by
the last equation (12), or equivalent by changing
2 within the allowed range determined above, we
can enlarge the set of accessible output velocities
vo to the open set [vo ; 1), which, by construction,
includes vi . This property is shown in Figure (2),
and it can be demonstrated as follows.
With this controller, the motion in contact is
explicitly characterized by the maps 1 : v 7! x
in Dc1 and 2 : v 7! x in Dc2 , i.e.,
x = 1 (vi ; xi; v) = 1 (vi ; 0; v)
(13)
x = 2 (2 ; v; x; v) = 2 (2 ; 0; x; v) (14)
where v = 0; x are the values of v and x, respectively, when crossing from Dc1 to Dc2 at v = 0.
The contact position is xi. In spite of the switching
from ? ! 2 when crossing from Dc1 to Dc2 , the
functions f2 (x; v); f3(x; v), are such that:
lim? ff2 (x; v)g = lim+ ff3(x; v)g = ?kjxjn?1x
x
v
! x?
!0
x
v
! x+
!0
Hence the system solutions as well the maps i()
are continuous. We can thus concatenate these
maps and write x = 2 1, 8(x; v) 2 Dc1 [
Dc2 . Following the above notation we have that
x = 1(vi ; 0; v), from which we get,
x = 2(2 ; 0; x; v)
(15)
= 2(2 ; 0; 1(vi ; 0; v); v)
(16)
4
= (2 ; vi; v)
(17)
The problem is now to nd 2 , such that it solves
the implicit equation (17) evaluated at x = 0,
v = ?vi , i.e.,
0 = (2 ; vi; ?vi )
(18)
which is equivalent to solving for 2 = (2=3k)2,
in the following transcendental equation:
a22 + b2 ? ln(2 + b2)2 + 2 ln2 = 0 (19)
h
i
with a = 12 3vi + ln 2?32vi 2 , and b = ?3vi ,
vi < 0 = (2=3k): By inspection of (19), we
can see that for the suitable range of 2 > 0,
this function is continuous and it has the unique
solution 2 = . This solution corresponds to a
symmetric positive damping injection, where the
energy lost during motion in Dc1 is recovered in
Dc2 by just inverting the sign 2 . An example of
an invariant motion is shown in Fig. (3) with a
bold line, where motion is conned to the shown
orbit. This nominal orbit will be denoted in the
sequel as
x = (v) = (2 ; vi ; v)jvi =?vd 2=2
?
where 2 is the value of 2 that solves for (19) with
vi = ?vd . The problem considered in this section
is to add a correction term to the nominal control
(11) such as to render attractive the invariant
orbit x = (v). For this we consider the following
structure for u 0
1 (vi ) jxjn v if (x;v); 2 D
c1
m
u=
>
(vi ) n
:
g + 2m
jxj v if (x;v); 2 Dc2
8
>
<g?
(20)
where 1 > 0, 2 > 0. The control u is always
positive, since the v is negative in Dc1 and positive
in Dc2 . However due to this sign restriction on
the control input, it is not possible to make the
desired orbit x = (v) attractive in the usual
sense. Instead, it is possible (as it will be shown in
the following subsection), to nd 1 and 2 , such
that the exit velocity series fvo (k)g converges to
desired value vd , i.e.
vo (k) ! ?vi (k) = vo (k ? 1) = vd
This can be performed either in one cycle or
asymptotically, as studied next.
3.1 Dead-beat Control
With the controller (20), the closed-loop equations
now become:
x
Dnc
x= Φ*(x)
v0= vd
vi= -vd
v
Dc1
Dc2
Fig. 3. Accessible sets in the (x ? v)-plane, and nominal
orbit.
v_ =
8
f1 (x;v) = ?g if (x; v); 2 Dnc
>
>
1
>
n
n?1 x)
>
< f2 (x;v) = m (1 jxj v ? kjxj
>
>
>
f (x;v) =
>
: 3
if (x;v); 2 Dc1
1 ( jxjn v ? kjxjn?1 x)
m 2
if (x;v); 2 Dc2
(21)
with 1 2 [?; ?1), 2 2 [?; 1) . By modifying
these constant with the allowed sets, we can have
the following properties:
(iii) The point x = (0) is accessible from any
vi 2 [?vd ; ?1), for motions in Dc1 .
(iv) The value of the exit velocity vo = vd is
accessible from any x 2 (0; ?x], on the axe
v = 0, for motions in Dc2 .
Property (iii) results from the ability of dissipating arbitrarily large amounts of energy in Dc1,
while Property (iv) comes from the fact that energy can be injected in Dc1 . Then, the shaded area
shown in Figure (3) represents the state space
domain where energy can be either dissipate or
injected throughout positive feedback.
These properties are equivalent to nding 1 ; 2
such that the following relationships hold:
x = 1 (1 (vi );vi ; 0); if vi 2 [?vd ; ?1);
0 = 2 (2 (xc);xc ; vd ); if xc 2 (0; x];
(22)
(23)
where xc is the position when crossing the
velocity-axis while in contact. In case that the
initial contact velocity vi , is smaller in magnitude
than vd , the value of 1 is set to be the natural
system damping since there is no need to add
more dissipation. Therefore, the maps (22)-(23)
are completely dened for any value of vi , i.e.
(
xc =
1 (1 (vi );vi ; 0) = x if vi 2 [?vd ; ?1)
1 (;vi ; 0);
if vi 2 (?vd ; 0)
0 = 2 (2 (xc); xc ;vd ) if xc 2 (0; x];
(24)
(25)
Since the 1 -map projects all the contact velocities to the bounded set (0; x], there is no theoretical need to consider the case where xc > x.
However, if this happens, due to possible model
uncertainties, then 2 is set to be equal to ?,
which will have the eect to bring the system
x = 2 (2 ; xc; v)
= 2 (2 ; 1(1 ; vi ; v); v)
4
=
(1 ; 2 ; vi; v)
(26)
Equations (24)-(25) imply that we are able to nd
1 , 2 such that it solves the implicitequation (26)
evaluated at x = 0, v = ?vd for all bounded vi,
i.e.
0 = (1 ; 2 ; vi; vd )
(27)
Phase Diagram
xi=1.03 m
1
0.8
0.6
Position (m)
motion towards the interior of the desired orbit
x = (v).
As before 1; 2 are continuous and can be concatenated to get x = 2 1, 8(x; v) 2 Dc1 [ Dc2,
or equivalent,
0.4
0.2
xi=0.11 m
0
−5
−4
−3
−2
−1
(1 (vo (k ? 1));2 (vo (k ? 1));vo (k ? 1);vo (k)) = 0
and by continuity of the function (), the
Poincare map : vo (k ? 1) 7! vo (k), can be
represented as:
vo (k) = (1 (vo (k ? 1));2 (vo (k ? 1));vo (k ? 1))
Introducing v~(k) = vo (k) ? vd , we get:
v~(k) =
(1 (vo (k ? 1)); 2(vo (k ? 1)); vo (k ? 1)) ? vd (28)
If 1 (vo (k ? 1)); 2 (vo (k ? 1)) satisfy (24) and
(25), we have (1 (vo (k ? 1)); 2 (vo (k ? 1)); vo (k ?
1)) ? vd = 0, which when substituted in the above
equation gives
v~(k) = 0
(29)
Hence, dead-beat control is achieved. The following theorem summarizes this result.
Theorem 3.1. Let vd be the desired velocity needed
to reach the desired altitude xd . Let the control law
be dened by
1 (vi ) jxjn v
m
u=
>
:
g + 2m(vi ) jxjn v
8
>
<g?
if
(x; v); 2 Dc1
if
(x; v); 2 Dc2
(30)
2
3
4
5
Fig. 4. Phase-plane for the dead-beat control. The parameters for this simulations are: m = 1kg; n = 1;
k = 1000N=m; xd = 0:46m; = 0:3
Phase Diagram
xi=1.03 m
1
0.8
0.6
Position (m)
The solution studied in the previous section is a
particular case of the above problem. Solution for
1 , in the Equation (24) as well as the solution for
2 , in the Equation (25) are involved (except if
xc = x, in which case the solution is 2 = ), and
need to be solved numerically by a root nding
algorithm. However, these solutions do exist and
are uniquely dened.
We can now consider the Poincare section S(x; v) =
f(x; v)jx = 0; v > 0g. The Poincare map between
the exit velocity at time instant t = k ?1, vo (k ?1),
and the exit velocity at the time t = k, vo (k) is
implicitly given by Equation (26), evaluated as
vi = ?vo (k ? 1), v = vo (k) and x = 0:
0
1
Velocity (m / s)
0.4
0.2
xi=0.11 m
0
−5
−4
−3
−2
−1
0
1
Velocity (m / s)
2
3
4
5
Fig. 5. Phase-plane motion with the asymptotic controller. The parameters for this simulations are: m =
1kg; n = 1; k = 1000N=m; xd = 0:46m; = 0:4;
= 0:3
with the contact velocity-dependent constants 1 (vi )
and 2 (vi ), being such as the relations (24) and
(25) hold. Then the system solutions are bounded
and the altitude xd is reached in one cycle, or
equivalently the exit velocity vo (k) ! vd , in one
step of nite duration.
Simulations are shown in Fig (4) starting from
two dierent initial position : x(0) = 1:03m and
x(0) = 0:11m.
3.2 Asymptotic stabilization
We consider now the problem of asymptotic stabilization of equation (28). The motivation for
this is twofold: rst dead-beat control is known
to result in large amplitude control signals, while
asymptotic stabilizers necessitate lower control
amplitudes. Second, the system robustness will be
enhanced.
Assume now that instead of solving for the equation
(1 (vo (k ? 1));2 (vo (k ? 1));vo (k ? 1);vo (k)) = 0
with vo (k) set to vd , we rather solve for,
vo (k) = (vo (k ? 1) ? vd ) + vd
where jj < 1, i.e.
Dead−beat control
200
which results in
150
Control (N)
(1 (vo (k ? 1));2 (vo (k ? 1));vo (k ? 1);v~(k ? 1) + vd ) = 0
50
0
0
v~(k) = (1 (vo (k ? 1));2 (vo (k ? 1));vo (k ? 1)) ? vd
= v~(k ? 1)
(31)
d
(32)
? vo (k ? 1) 2 (?vd ; 0)
0 = 2 (2 (k ? 1);xc (k ? 1);vd )
if
xc (k ? 1) 2 (0; x];
20
25
30
35
25
30
35
150
100
5
10
15
20
Time (s)
8
1 (1(k ? 1); ?vo (k ? 1); 0) = x
>
>
< if ? v (k ? 1) 2 [?v ; ?1)
if
15
200
0
0
the following relationships hold,
>
>
: 1 (;vo (k ? 1); 0);
10
50
The following theorem summarizes the result.
Theorem 3.2. Let vd be the desired velocity needed
to reach the desired altitude xd . Consider the
control law 30, with the contact velocity-dependent
constants 1 (vi ) and 2 (vi ), dened now such that
o
5
Asymptotic Stabilization
Control (N)
since jj < 1, we have that
lim v~(k) = 0
k!1
xc (k ? 1) =
100
(33)
with x being dened as x = 1(; ?vd ; 0). Then
the system solutions are bounded and satisfy:
lim v (k) = vd
k!1 o
Hence, the jump altitude xd is reached asymptotically with a convergence velocity dened by .
Remark Boundedness of the solution away from
the Poincare section follows from the fact that
during contact, the system motion is completely
characterized by the relationship x = (vi ; v).
From the continuity and boundedness of the map
(), and the fact that vi is bounded (no energy
is injected during the ight phase), we have that
system solutions cannot escape in nite time. Finally note that for any bounded vi , Equation (32)
projects the system trajectories to x, from which
the map (33) provides a bounded exit velocity,
while providing bounded (x; v) trajectories. The
above analysis shows that vd is the unique asymptotically stable equilibrium point for vo (k).
The phase-plane motion of the simulations are
shown in Fig (5) starting from two dierent initial
position: x(0) = 1:03m and x(0) = 0:11m, with
= 0:4. As mentioned before, the transient
magnitude of the control signal can be reduced,
as shown the Fig (6).
4. CONCLUSIONS
In this paper we have presented a solution for
the problem of characterizing the existence and
Fig. 6. Time-evolution of the control law. Comparison
between the dead-beat and asymptotic strategy.
stabilizing periodic orbits of an actuated 1-DOF
hopping robot moving on a nonlinear compliant
surface. It was shown that stable periodic motions
are possible via the injection of nonlinear damping
through nonlinear stabilizing feedback. This gives
rise to one-step or asymptotic stabilizers.
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