1

Control Approach for Hopping Robots: Controlled Limit Cycles N. K. M’Sirdi, N. Manamani and N. Nadjar-Gauthier LRP Laboratoire de Robotique de Paris 10-12, Avenue de l’Europe 78140 VELIZY FRANCE fax: 33 1 39 25 49 67 Email: [email protected] University of Versailles S.Q.-CNRS URA 1778 Abstract – In this paper we propose a viable approach for control of hopping robots. We use controlled limit cycles for gait stabilization (closed orbits). An analysis of robustness of the proposed control with respect to parameters variation and disturbances is presented and illustrated by numerical simulations. Keywords – Legged robots control, Hopping robots, Passive Feedback Systems, Controlled limit cycles.

is analyzed in section 2. This analysis is exploited to define the control target and an approach is proposed for the design of controlled limit cycles (CLC). Some discussions and main comments on simulation results are presented. Our future prospects, investigation and some conclusions in this work are finally given. II. Dynamics of hopping systems

I. Introduction Increasing attention is paid to legged robots and research has been focused on gait generation, and mechanism design among others. Dynamic model equations of robots interacting with environment are non linear and time-varying. When dealing with such variable structure systems or intermittent dynamic processes the control difficulties come from the fact that the robot is characterized by different descriptions in non-overlapping space regions. There is an important and growing literature on stability analysis and control of vertical hopping robots based on simplifying assumptions for the system equations and control. When dealing with legged robot, for fast motions, the complete dynamical model is rather complex and involves complex control laws and experiments of heavy procedure for on line trajectory determination and adaptation. When hopping or running, legged robots require that the desired trajectory must be processed on line for preservation of the system stability and energy optimization [1]. On the other hand, the robot performances depend on the used control law and legged machines. Involving complex non linear models complicate the control design and require increase of the computational power. In this paper, we propose a control structure with clarified interactions and interconnections involving passivity properties for energy shaping and optimization. Our main purpose is to show that an equivalent energy model can be used to stabilize hopping motion. With the proposed approach, the control system and energy shaping will then be simple and easy to design. Controlled Limit Cycles (denoted CLC in the following) allow implicit trajectory generation and optimization. The organization of the paper is as follows. After a brief bibliographic review on some works concerned with hopping robots, the problem formulation is presented. The case of a simple (1 DOF) mass spring system in free motion 0 This work is a part of our contribution to the PRC of the CNRS on control of legged robots. 0 IEEE-AVCS98, July 2-4, 1998, Amiens, France.

Raibert [2] has shown that a leg can be modeled by a spring mass system during running gaits[1]. This gives a powerful approach to legged robot design, analysis and control. Decoupled and partitioned control is applied for one legged planar hopper. To overcome equations complexity and limitations, simple mechanical systems have been considered for dynamic gaits and different approaches have emerged in literature [3] [4] [5]. In [6] bifurcation phenomena have been analyzed numerically and studied for a simplified hopper model. Strange attractions, bifurcations and dynamical behavior with existence of higher order limit cycles have been exhibited [7][8]. A mechanical system with unforced motion gives passive dynamics. This principle was exploited first by Raibert’s robot for the vertical motion [2]. The stiffness of the leg influences the vertical oscillatory behavior of the hopping system and governs the details of landing on the ground and taking off. Mc Geer [9] has built non-actuated (gravity powered) two-legged mechanisms for walking down inclines. In [10] a feedback control is designed to stabilize a desired limit cycle for a compliant leg composed with a spring mass system (1 DOF). Interaction with the ground is assumed without energy loss and the desired limit cycle is stabilized modulating the lift off velocity. Ahmadi [11] showed by simulation that by using the passive motion trajectory of a swinging leg at the current robot speed as the basis for motion planning, stability can be achieved. This study is made on a model based on Raibert’s experimental robot and later on another robot built to study the control and energy of electrically actuated variants. The authors investigated how a legged robot can be actively stabilized for fixed as well as varying velocity while exploiting the passive hip swinging to permit energy savings. In [12] a simplified conservative model (Raibert’s planar hopper) is considered for control of forward velocity. This problem is addressed also in [8] where simple control schemes are analyzed for forward motion under assumption that energy of the hopper is constant. In [13] effect of nonlinear friction in feedback is studied.

2

So as pointed out in [14], the classically used definitions have to be adjusted in the way that the gaits can be realized. The question to point out is how can we define a methodology for control dealing with legged machines? The approach needed, must help to define requirements, features and essential components for the system control: Definition of gaits, trajectory generation and adjustment, Definition of nominal positions and velocities versus time required for the gait and dynamical stability, Definition of behavior adjustment in function of environment reactions, Definition of the control objective that copes with the previous blocks requirements, The approach proposed here for stabilization of dynamic gaits is represented in figure (1). The control objectives have to be task oriented. We mean that precise trajectory tracking is not needed when dealing with walking robots. τ→

Energy Reference Stance

+

-

Gait Definition

Motion Evaluation

=•Εz-u)

Control

+

Robot

Fig. 1. Approach for Gait stabilization with CLC

III. Control approach for hopping systems In order to clarify the rationale behind of the proposed approach, we consider the case of a simple mass - spring system (Figure 2). This free system is used to analyze the energetic interaction between the robot, control and ground. This analysis will be exploited to avoid the use of modelling of ground interaction and simplify the limit cycle control and stabilization. Interaction with the ground is implicit; expression of constraints and restitution model for impacts are avoided. This simplified model has been shown efficient for our robot by experimental validation [15]. Exchange of energy (kinetic and potential) between the ground and different parts of the robot is of main importance when dealing with legged robots operating with high velocities (high level of kinetic energy). For prediction, supervision and control of the robot motion and behavior, we have to consider the power exchanges in the system (actuators, robot, ground and control system). A. Mass-Spring free system Let us consider the mass-spring system of figure (2), with mass M , stiffness constant k and zo the original length of the massless spring. The environment is assumed infinitely rigid: ke >> k. If this is not the case let kr be the stiffness of the spring and³k the equivalent stiffness of interaction ´ ke . The position of the mass with the ground k ' kkrr+k e M , in a frame attached to the ground, is noted z. The

.

gravity constant is g = 9.81ms−2 . Let vd =z d > 0 be the . lift off velocity and vc =z c< 0 the touch down velocity. M

M

k

k

Zo

Z X

Con trol u Zo

Z X

Fig. 2. Mass-spring system (free & controlled)

The dynamic interaction with the ground is composed by two phases: flying and stance phases [16] [17]. This system is composed by interconnection of three subsystems (mass, spring and ground) and energy evolutions are: Potential (g)→ Kinetic→ potential accumulation→ potential restitution→ Kinetic and so on. We assume the landing without rebounds and no energy loss. Note that the values used in simulations are estimations of equivalent coefficients for our robot SAP [15] (M ' 2.6kg; k = 1100). 1. Flight Phase: This phase of motion is conservative and uncontrollable. The system has a ballistic trajectory determined by knowledge of the mass lift off velocity . vd =z d (t = 0). M z¨ = −M g (1) Let us note Tv the flight phase duration. During the flight motion the system has a purely ballistic trajectory with as . initial velocity z d and position zd . The equation of this part of motion can be written: 1 . z(t) = zd + z d t − gt2 2 .

(2)

.

z (t) =z d −gt; z¨(t) = −g

(3) .

The maximal position zm is reached at time tm = zgd , where . . the velocity is zero z (t) =z d −gtm = 0. We can then deduce the flight period and maximum position value: zm

.2

z 1 = zd + z d tm − gt2m = zd + d 2 2g .

(4)

.

Tv = 2tm =

2vd 2 zd = g g

(5)

2. Contact Phase: Contact between the end point of spring and ground appears when z − zo ¹ 0; we can write: M z¨ + k(z − zo ) = −M g

(6)

Let us define the contact function ξ(z) which is 0 during flight and equal to 1 in contact phase: 1 ξ(z) = (1 − sign(z − zo )) 2

(7)

we can then write: M z¨ + ξ(z)k(z − zo ) + M g = 0

(8)

3

3. Contact period Tc : The system natural frequency, when in contact with the ground, q is imposed by mass and spring stiffness values: k M.

If friction are neglected, the system equation ω0 = can be rewritten, with initial conditions at contact zc and . vc =z c as follows: z¨ + ω 20 z = ω20 zo − g

(9)

z(t) =

g vc sin(ω 0 t) − 2 (1 − cos(ω 0 t)) + zc ω0 ω0

(10)

In this case, equality of positions at lift off and touch down can be assumed zc = zd = z(Tc ) and we can compute the contact period Tc : Tc =

2 vc ω0 )) (π − arctan(− ω0 g

The contact period Tc depends on mass M and the equivalent stiffness value k. The solution of the system can be written as follows: . ½ if ξ(z) = 0 zd + z d t − 12 gt2 z(t) = vc sin ω t − g (1 − cos ω t) + z if ξ(z) = 1 ω0

0

ω 20

0

c

and then ¡. ¢ g . z (t) = (1 − ξ(z)) z d −gt + ξ(z)(vc cos ω0 t − sin ω 0 t) ω0 (11) The oscillations correspond to a closed orbit as shown in figure 3. The obtained periodic orbit, for the free system, depends on initial conditions and is defined by the equation: 1 k . .2 Vf (z, z, z(0)) = M z +M g(z − zm ) + ξ(z) (z − zm )2 2 2 (12) which is plotted in figure 3. Equation (12) can be used to describe those arbitrary motions that cope with the system’s ability (admissible motions for the system).

2

1 .2

0

1

-2

0 .8

V e lo c ity o f M

4

1 .4

0

0 .5 1 T im e i n s e c

100

1 .5

-4

0

80

F o rc e

80

40

40

20

20

0 .5 1 T im e i n s e c

1 .5

A c c e le ra tio n

60

60

0

Solution of (9) is given by:

P o s it io n o f M

1 .6

0 0

0 .5 1 T im e i n s e c

1 .5

-2 0

0

0 .5 1 T im e i n s e c

1 .5

Fig. 4. Simulation of motion of mass-spring system

flight phase acceleration and velocity are linear. In contact phase, velocity has a sinusoidal form, acceleration is not constant and shows similarities with contact force. Then, in conclusion, as claimed in section 2, definition of gaits, can be done by use of an equation like (12). The closed orbit may be easily defined using potential and kinetic energy. This definition copes with the characteristics of the system (inertia or mass M and stiffness k) and the internal energy flow can be optimized. Tangential dynamics [18] with regard of the periodic orbit can be minimized (dynamical stability) [19]. These results show also that the jump height zm depends on initial conditions (cycle energy) and the motion period depends on the spring stiffness and mass. If k increases the period decreases and if M increases the period increases. B. Controlled Limit Cycles for mass-spring hopper The one degree of freedom (DOF) controlled system considered now is represented in figure (2b). The spring attach is assumed to have a controllable linear displacement u from the center of mass M (control input). The spring (with stiffness k) can be moved as shown in figure (2). Our main objective is now to define and stabilize a periodic motion by control. For vertical motion of the system we can write: M z¨ + ξ(z − u)k(z − zo − u) + Mg = 0

(13)

The system can be written in state space form ½

Fig. 3. Periodic Orbit (Free System)

Numerical simulations, with as initial position z = 1.5m and null initial velocity, gives a periodic motion (a closed orbit) as represented in figures 3 and 4. Figure(4) shows the existence of two regions (flight and stance). During the

.

x˙ 1 = x2 =z k x˙ 2 = z¨ = − M ξ(z − u)(z − zo − u) − g

(14)

It is clear from this representation that the system is uncontrollable when ξ(z − u) = 0, i.e. in the flight phase. So the control contribution, in the contact phase, has to preserve the periodic motion. The contact period may be very short if the stiffness k is very large and the mass small. Recall that the periodic orbit of the free system was defined by the equation (12) The oscillation is maintained by a periodic exchange, without dissipation, of energy stored in potential form (gravity) with inertia (kinetic energy) and then potential form in the spring. To obtain this orbit as the controlled

4

limit cycle, we have to solve two problems: Amplitude of oscillation depends on initial conditions and Robustness, a small perturbation or infinitesimal dissipation will cancel oscillations [20] (the system has a stable equilibrium zeq ). The input u has to be designed for this system such that zeq is either an unstable node or an unstable focus. In this case all trajectories starting near the origin would diverge away from it toward the desired limit cycle [19]. This corresponds to an ”active” behavior which means that the controller supplies energy [20]. The total energy of the system (our reference model equation), at any time can be written: Es =

1 k .2 M z +Mgz + ξ(z − u) (z − zo − u)2 2 2

(15)

The last term of the right hand side is zero in the flight phase, where the system is conservative and uncontrollable. We are interested by the contact phase, where the controller may be active or dissipative in order to control the limit cycle. Let us choose for our system as Lyapunov function candidate: k 1 .2 z +gz + (z − zo )2 (16) 2 2M . . At the touch down time instant we have z = zc and z=z d . and at the apex we have z = zm and z= 0 for the desired hopping motion. The maximum position during the flight corresponds to an energy quantity obtained by the lift off V =

.2

.

z

velocity z d . This can be defined by V0 = gzm = 2d . In order to obtain this motion, the value of the system energy Es (reference model) must be equal to M V0 at the lift off instant, despite energy lost, if any. This allows us to define the orbit that we want to stabilize for the system: ( ) .2 zd . 2 = gzm (17) Ωo = (z, z) ∈ R : V = V0 = 2 The main objective is to make Ωo invariant and attractive. To stabilize this orbit, we consider as control input u (for spring displacement) the following nonlinear feedback expression: .

u = −Λ (sign(V − V0 )) z

(18)

This control is equivalent to a force feedback because u is multiplied by the stiffness in the system equation (13). This control will then be able either to supply energy to the system or to reduce its energy level. It can be either active or passive (dissipative). The control computation needs the use of energy reference level as shown in figure (1). The energy reference is computed from gait definition and the system energy evaluation. So we can state the following theorem for the controlled limit cycles. Theorem 1: The mass-spring system of equation (13) has as limit cycle the closed orbit Ωo defined by the Lyapunov function value V0 (equations 16 and 17) if the gain Λ used in control of equation (18) is positive. Proof: Choosing equation (16) as Lyapunov function .2 to describe the energy evolution of the system, V = 12 z

k +gz + 2M (z − zo )2 , we obtain the following expression for . . k . z the Lyapunov function derivative: V˙ =z z¨ + g z + M . (z − zo ). From the system (13) we have V˙ = − z ξ(z − k k . z (z − zo ). Recall that the control (z − zo − u) + M u). M is considered only for the contact phase i.e. the case where . k uξ(z − u). By ξ(z − u) = 1, then we can write: V˙ =z M use of the control expression (18), we obtain µ ¶ k . . sign(V − V0 ) z ξ(z − u) (19) V˙ = − z Λ M

if z − zo − u > 0, ξ(z − u) = 0, the energy of the system is constant during the flight phase, the control is not needed and has no effect on the system (conservative phase) • if z − zo − u ¹ 0, ξ(z − u) = 1, we can easily . . k . z M check that: Λ (sign(V − V0 )) z (V − V0 ) = − V µ ¶ •

1 .2 z 2

+gz +

k (z 2M

− zo )2 −

.2

zd 2

V0 we have V˙ = − z M height is greater than zm , then energy is decreased, the feedback u has dissipative contribution. . ¡ k ¢ . Λ z> 0, the jump height is — if V < V0 we have V˙ =z M less than zm , then energy is increased by an active control u. .

We can then conclude, for the controlled limit cycle, that the periodic orbit Ωo (equations (16) and (17)) is a stable invariant set of the mass spring controlled system. The controlled limit cycle is obtained by a control expression which is activated only during the contact phase (the controllable region). This control is composed by a nonlinear velocity feedback which is either active or dissipative depending on the system energy with regard to the desired closed orbit Ωo . The CLC control is frozen in the flight phase (uncontrollable region). Rebounds may appear if the system has an energy level greater than the one of the orbit at initial configuration. C. Simulation of Controlled Limit Cycles The first simulation, represented in figure (5), shows that the CLC is stabilized after two oscillations with Λ = 0.05 used with a saturation function (with maximum and minimum values ±0.5) instead of the sign function (in order to . save simulation time). The initial conditions are z (0) = 0 z(0) = 2 and parameters are M/k = 500. The friction coefficient is β = 0 and the desired height of hopping is imposed to zm = 4. Periodic motion stabilization (with use of saturation function) when the system initial position is z(0) = 0.5; or 1; ....; 5.5, are represented in figure (6). The CLC is obtained after one or two jumps with Λ = 0.05. When initial position is near to zeq , the needed transient period to retrieve the required energy is as long as z(0) is near to equilibrium point zeq . Figure (7) illustrates the fact that rebounds may appear when the system energy, before contact, is greater than the one of the desired closed orbit Ωo (with β = 0, Λ = 0.05, M/k = 500, zm = 4, z(0) = 5, saturation control). These

5

P o s i t io n a n d V e l o c i ty o f M

10

6

6

C o n t a c t f u n c t io n

1

Z A x is

4

0 .6

0

4

3

0 .4

-5

3 Y A xis

0 .2 0

5

0

10

15 sec

P o s i ti o n a n d In p u t C L C

6

2

0

5

10

1

15 sec 0 10

In p u t F o r c e N

200

-5

0

V e l o c it y

5

0 -1 0

10

-5

0

V e lo c i ty5

10

150

4

100 2

Fig. 8. CLC for smal gain Λ = 0.0005 with β = 0.025, Sat and sign function control

50

0 -2

5

0 .8

5

10

X Y P lo t S ig n . C o n t r o l

Z A x is

X Y P lo t S a t . C o n tr o l

5

0 0

5

10

-5 0

15

8

0

5

15 sec

10

7

Z = f (Z p )

6 5

Z (m )

Fig. 5. Periodic Motion Stabilization, β = 0, Λ = 0.05, M/k=500, zm = 4, z(0) = 2, Sat.control.

4 3 2 1 0 -1 0

-5

0 Z p

6

L a m b da=0.05

Position of M ass

5

b) Z =f(Z p) ϒ =0.2

4

4

3 2

R ebound 2

1

Lift off 1

0

0

T ouch down -10 -8

-6

-4

-2

0

2

4

V elocity of M ass

6

8

10

L ift off

con ta ct -5

0

5

Fig. 6. CLC a) Without and b) With Frictions

rebounds can be avoided by reduction of the control gain Λ, but the transient time is increased (see figures (8) and (9)). P o s it io n , V e lo c it y

0 4

2

4

0

6

5

P o s itio n , I n p u t

2

0

sec

4

4

with u = −Λsign(V − V0 ) z, this leads to

3

2

.

2

0

1 0

2

4

0 10

6

5

0

5

10

.

M z¨ + ξ(z − u)k(z − zo + Λsign(V − V0 ) z) + β z +M g = 0 (23) 8

Fig. 7. Periodic Motion Stabilization, β = 0, Λ = 0.05, M/k=500, zm = 4, z(0) = 5, witrebound.

7

Z = f(Z p )

6 5 Z

(m )

4 3 2

D. Robustness of CLC

1 0 -1 0

Robustness to parameter changes is illustrated by figure (9). The orbit Ωo is retrieved despite the use of M/k ten times greater or smaller, provided that energy of Ωo is correct and no friction appears in the flight phase (the Lyapunov function (Vo ) value can be defined independent

-5

0 Z p

1 0

X

Y

P lo t

8 7

.2

6

.

u = −Λsign(V − V0 ) z

Y A xis

5 4 3 2

(20)

1 0 -1 0

-5

0 X

V =

5 ( m /s )

Fig. 10. CLC with frictions

z

of system’s parameters V0 = 2d = gzm ). Recall that the control input is given by:

with

(22)

.

0 2 0

4

2

.

M z¨ + ξ(z − u)k(z − zo − u) + β z +Mg = 0

0 6

5

6

R eb oun d

0 8

0

and that at the lift off, the position is z−zo = 0, in our case. The transient duration depends on the nonlinear feedback gain Λ. Note that introduction of frictions in simulation produces deformation of the orbit (symmetry is lost) and reduced the height of jumps as shown in figure (6) and figure (10). The height reduction is important for high frictions level, in figures (11) and (12) the viscous friction coefficient has been increased several times. If we still increase of friction level, the periodic orbit disappears, the origin becomes an asymptotically stable equilibrium point. . Let us consider a friction force β z in the system, we obtain the closed loop equations:

C o n ta c t f u n c tio n

1

5

10

10

Fig. 9. CLC for smaller control gain Λ = 0.001

3

10

5

( m /s )

5 Z A xis a) C LC for k/M =5 00, Zm =4 ,

k 1 .2 z +gz + (z − zo )2 2 2M

(21)

5

A x is

Fig. 11. CLC with High level frictions

1 0

6

a)

b)

p o s itio n

8 6

(m )

c)

v it e s s e

15

800

10

600

5

400

(m /s )

4

0

2

a c é lé ra rtio n

200

-5

0

(m /s²) 0

0

5 Tem ps en s fo rc e

10

d)

800

-1 0

0

e)

1

5 10 Tem ps en s R e bonds si =1

8

0 .8

600

(N )

-2 0 0

0

5 Tem ps en s

f)

10

P la n d e P h a s e (z p , z )

6

0 .6

400

4 0 .4

200 0

2

0 .2 0

5 Tem ps en s

10

0

0

5 Tem ps en s

10

0 -2 0

0

20

Fig. 12. System states for CLC

any structure of robot with fast dynamics. The presented analysis emphasizes the fact that fast dynamic gaits have to be formulated as closed orbits expressed in function of the system’s energy to be correctly optimized. This shows that compensation of inertial effects or introducing friction equivalent terms, by means of control, corresponds to energy wasting. These investigations show that the proposed approach is appropriate and efficient for (implicit) on line trajectory design, stabilization, energy shaping and optimization. Our current investigations on extension of this approach for application to a pneumatic legged robot and a biped are in progress and the first results confirm the efficiency of the CLC approach. Acknowledgement 2: Acknowledgments are addressed to J. P. Barbot and J. Hauser for their helpful discussions and comments.

or in the flight phase (ξ(z − u) = 0):

References

.

M z¨ + β z +Mg = 0

(24)

[2]

and contact phase (ξ(z − u) = 1): .

[1]

.

M.¨ z + kΛ(sign(V − V0 ) z) + β z +k(z − zo ) = −M g (25) The previous equation shows that when the amplitude of control input u is limited and if .

β > Λ max(k(sign(V − V0 ) z))

∀t > 0

(26)

then the system will be asymptotically stable at zeq near the origin and then no limit cycle exists. This is the limit that was crossed in the previous simulation trials. We can remark that this limit can be increased by increase of the non linear feedback gain Λ. Despite the friction existence, we can define and stabilize a limit cycle for our system by use of high feedback gain. In our simulation with β = 0.2, we show orbit deformations due to friction and increase of gain. This leads rebounds in figures (12) and (11), these are obviated by plotting the contact function ξ(z − u) (curve (e)) and their contribution in acceleration and force. In conclusion we can say that the system robustness can be preserved by use of our control method. IV. Conclusion This paper explores the notion that an equivalent energy model (mass-spring) can be a good deal for control of dynamic gaits for legged robots. Use of the passive properties, the energy of the system and non linear feedback adjusted with energy regard, have been considered and shown to be efficient for an automatic generation and stabilization of hopping and reaction to perturbations and unknown events (unknown ground, stumbling,...) which permit the robot to be able to keep stable the stance and motion. This may be done by using controlled limit cycles (CLC). Experimental application, of this control approach on a pneumatic legged robot is in progress and the first results are quite encouraging. The design of this approach has been done in order to simplify the study and to know in detail pertinent features in the behavior of legged robot when hopping. This method can be easily extended and applied for

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16]

[17] [18] [19] [20]

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