AN ENERGETIC CONTROL BASED ON LIMIT CYCLES FOR STABILIZATION OF FAST LEGGED ROBOTS

N. Nadjar-Gauthier, N.K. M'Sirdi, N. Manamani and D. El Ghanami. Laboratoire de Robotique de Paris 10-12, Avenue de l'Europe 78140 Velizy, France e-mail: [email protected] Control of legged robots with fast gaits is addressed in this paper. These kind of systems interact intermittently with environment. We propose a viable approach for control of hopping gaits for legged robots. This control approach is based on Controlled Limit Cycles (CLC) for stabilization of fast gaits (closed orbits) for legged robots. The designed control system generates the desired trajectories (on line) and control inputs. Robustness of the proposed control with respect to parameters variation and disturbances is illustrated by numerical simulations. Viable de nition of gaits and their admissibility are introduced. Key words: Fast legged robots, energetic control, stabilization, limit cycles1

Abstract|

INTRODUCTION For legged robots, the crucial problem is the design and control of gaits. Trajectory de nition and optimization involves heavy procedures (complex non linear models). An increasing attention is paid to these systems and their interaction with environment. Biological similarities such as the Groucho running pointed out the possibility to nd models retaining only pertinent informations for runner behavior [2][3][4]. Fast gaits are diÆcult to realize with classical control design methods, when dealing with complex models and ground interactions [5][6]. Fast dynamic gaits require that trajectories be processed on line for preservation of the system stability and energy optimization [7]. This problem has been generally considered in two steps: 1) trajectory generation [8] and 2) trajectory tracking and adjustment [9]. Interactions with the ground require de nition of force and position trajectories. The positions, velocities and accelerations must be compatible with the desired forces and ground interactions. This diÆcult task involves several problems, mainly continuity of trajectories and synchronizations when dealing with varying environment (unknown positions and stiness). The trajectories cannot be optimized if the environment is unknown and the control is not taken into account. The class of dynamic legged robots suers from lack of powerful techniques for simultaneous trajectory design and control. This is due to the nonlinear nature, presence of phase transitions (contact phase, ying phase, energy transformations

1.

1 This

work is a part of our contribution to the PRC of the CNRS on control of legged robots

[1].

1

and ow inside the system) and environment variations. The objective of the control system must be rede ned and oriented to energy optimization and control. The control must stabilize a behavior or a quasi periodic motion of the legged robot. The desired behavior depends on the required gaits and motion. The classically used methods do not cope with this objective. The question is how a methodology can be de ned for control of fast legged machines? The aim is to propose a control structure with clari ed interactions involving passivity properties for energy shaping and optimization. The proposed formulation emphasizes the fact that the system's behavior and trajectories are consequences of its dynamic, the control and ground interactions. In this paper, we show that the control of legged robots can be reduced to an equivalent energy model involving the robot, the ground and a simple feedback. The use of the passive properties and energy of the system is considered for an automatic generation of trotting. This may be done by use of Controlled Limit Cycles (CLC) [10][11]. The proposed approach is evaluated by application to the SAP2 robot presented in gure (1) [12]. Section 2 presents some preliminaries to introduce the problem and the model of our robot. In order to obtain a good problem formulation, appropriate de nitions are presented in section 3. Then we present the proposed methodology for control of dynamic gaits. This control approach is applied to generate trajectories and stabilization of the robot gait, in section 4. Some comments on results are presented. Our future prospects, investigations and some conclusions in this work will be given. 2.

BACKGROUND AND PRELIMINARIES

2.1. Previous work

The most of results, presented in literature for hopping or running systems, are generally limited to simple mechanisms,consider the environment well known and use model simpli cations[13]. Some eÆcient methods use the Poincare map for limit cycle stabilization [14][15]. This method is devoted to second order systems and need the knowledge of solution of the system dierential equations. Bifurcation phenomena have been analyzed numerically and studied for a simpli ed hopper model in [16][17]. Strange attractors, bifurcations and dynamical behavior with existence of higher order limit cycles have been shown [5] [6]. To overcome equations complexity and limitations, simple mechanical systems have been considered for gaits and dierent approaches, based on experimental knowledge, have emerged in literature [18] [19]. Rigid massless leg models have been used to study walking [20], but leg models that include mass and spring are important for understanding running and hopping. In [21] a simpli ed model of the Raibert's planar hopper is considered for control of forward velocity. Simple control schemes are analyzed for forward motion under assumption that energy of the hopper is constant [22]. 2.2. Dynamic Model for SAP Robot with one leg

Experimentations have been done, in our laboratory, on the robot called SAP (Pneumatic Actuated System), using one leg, to study hopping and running gaits. 2 SAP:

Pneumatic Actuated System

2

Two free wheels (with as diameter 2l1) are used to impose the motion to be in the sagittal plane (see gure 1). q1 correspond to the horizontal translation of the wheels. The system (unconstrained) is composed by three links (3 rigid bodies: platform and a 2 links with length l3) connected by two actuated joints as shown by gure 1. One side of the body is maintained on the ground by a free wheeled system. The dynamic equation of the rigid system can be written: M (q)q + C (q; q_)q_ + G(q) =  +  (1) e

Figure 1: Legged Robot SAP (Pneumatically Actuated System). Let q = [q3; q4] be the angular positions of actuated joints of the leg and q2 is the angular position of robot body. Let q = (q3; q4 ; q1; q2) be the joint position vector considered in the system equation (1), and recall that q_, q denote respectively joint velocities and accelerations. The torque input vector is  = [1; 2 ; 0; 0] , M (q) is the (4  4) generalized inertia matrix, G(q) and C (q; q_)q_ are the vectors of gravitational and centripetal and Coriolis forces. The ground reaction is considered as a nonlinear impedance with very high stiness (k = 105) (see [23] for the model). If the Cartesian variables are de ned as x = [x ; z; ] , the expression of the robot motion in Cartesian space can be obtained by use of geometric model (x = L(q)) and the corresponding Jacobian matrix J (which can be expressed for the end point of the leg). Then x_ = J q_ and the system can be written: M  x + C  (x; x_ )x_ + G (x) = F + F (2) The rst thing we have to deal with is stabilization of a stance when in contact with the ground. As it will be shown, this can be obtained by use of a simple PD feedback. T

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2.3. Control with Implicit ground interactions

A stabilization by means of pure position control can be applied at this step. Let x = [x ; z ;  ] be the desired position vector (equilibrium posture). This correspond to an equilibrium posture as represented by the scheme of gure (1). T

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A simple stabilizing control (PD feedback without gravity compensation and null derivative gain, K = 0) can be considered F = K (x x) (3) with this partial feedback control we obtain, en closed loop M  1 (F + K x G ) = x + M  1(C  x_ + K x) (4) We can remark that the ground reaction force F is weighted by M  1. If the mechanical impedance of ground is Z with stiness matrix K (F = Z
z ' K
z with
z the ground deformation) the equivalent stiness is K . This corresponds (as shown in gure 2) to the impedance equivalent to the ground and robot plus the control ones in serial connection [24]. If the ground is in nitely sti compared to control gain, the equivalent impedance reduces to the control one. Then we can note that the system characteristic pulsation is xed by inertia and the equivalent stiness K . v

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Figure 2: Equivalent Impedances When the robot, in contact with the ground, is excited by periodic signals corresponding to a vertical motion of the leg endpoint (z ), a resonance frequency appears at about 21 rad/s and gives a hopping motion [3]. This experimental result (periodic motion) have been retrieved in simulation and such a simulation is presented in gure 3. The phase portrait shows evolution of position versus velocity of the robot body (zplatform = f (z_platform)). Experimentally outside of the interval [10rad/s, 28rad/s], the robot stops hopping and has attenuated motion while the leg stay in contact with the ground [4]. These characteristics depend on the feedback gain and contact mechanical impedance (nature of the ground). The mechanical dynamics is up to about 30 rad/s [25]. This shows the existence d

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of natural gaits corresponding to the system frequency characteristics (this will be useful for energy optimization). In what follows, an approach which copes with

these features is developed. 0.5

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Figure 3: Behaviour of SAP with stance stabilization and square wave excitation. Despite equations complexity, this system has, as main features (or nominal behavior), the same physical characteristics as a simple mass-spring hopper like the one considered for CLC in [10]. Its motion can be characterized by three main points:  Existence of a contact phases (stance) and a ight phases (generally longer than contact ones),  During the ight phase the system is not completely controllable and has it motion due to gravity eect,  Interaction with ground appear in contact phase and its mechanical impedance results essentially from control gain K and ground stiness K (in serial connection to produce an equivalent stiness K ). Then the equivalent system can be written, if some damping terms are either neglected or supposed to be compensated by means of control, M  x + C  x_ + Kx = Kx G (x) (5) The applied partial feedback ensures existence of an equilibrium point (a nominal posture). If x is constant, this point will be stable. The equivalent system (robot+ground, see equation 5) is composed by interconnection of subsystems (masses or inertia, springs and ground see gure 2). The hopping motion is periodic, then we have periodic cycles. Let us describe the behavior of energy of the SAP robot during a cycle or a hopping period. Energy evolutions, during a motion p

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Potential Energy

Mgz Kinetic Energy

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Figure 4: Energy evolution cycle for a hopping robot. can be, for example: Potential (g)! Kinetic! potential accumulation! potential restitution ! Kinetic ! Potential (g) and so on (see gure 4). For prediction, supervision and control of the robot motion and behavior, we have to consider the power exchanges in the system (gravitational eect, energy storage & exchanges). Then for gait de nition and trajectory processing, inertia matrix and potential energy of the system have to be taken into account. In another hand, gaits must correspond dynamically to limit cycles, by means of energy exchanges. The gain K can then be optimized in function of the required behavior and the encountered ground stiness. This must be done in order to reduce sensitivity of the system (Robot+Control+ground) to variations of ground stiness and energy balance depending on motion. Controlled Limit Cycles allow implicit trajectory generation and optimization. The previous remarks are helpful for de nition of the control objective for legged robots. Thus the aim of our control approach is to stabilize periodic motion cycles by means of Controlled Limit Cycles (CLC). In what follows, we present some basic de nitions and the proposed approach for stabilization of fast periodic motion gaits. p

CLC APPROACH FOR LOCOMOTION CONTROL The approach proposed for control can be summarized by gure 5. It has two feedback loops, one for stabilization of nominal stance and one for regulation of the energy balance to stabilize a CLC. The latter correspond to a periodic motion cycle around a nominal stance. 3.

3.1. Motion and Stability de nitions

The required motion gaits can be de ned such that robot obeys some desired behavior. The main point is to ensure gaits and to maintain the behavior. In what 6

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Figure 5: Approach for Gait stabilization with CLC. follows, we introduce some basic de nitions which cope well with our problem [10][11][17]. Let us consider a free second order system (equation 6) with state vector x 2 R , initial condition x and ft g a sequence of time instants with lim t = 1: !1 x_ = f (x) (6) De nition 1 For the system (6) a positive limit set of a bounded trajectory x(t) (kx(t)k < ; 8t > 0) is de ned as = fp 2 R ; 8" > 0; 9ft g such that kp x(t )k < "; 8k 2 N g: This de nes 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 behavior of neighboring trajectories in order to analyze the orbital stability. n

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De nition 2 Orbital stability: The system trajectory in the phase space R is a stable orbit if 8" > 0 9Æ > 0 such that kx (x )k < Æ ) inf kx(t) pk < "; 2

8t > t n

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All trajectories starting near the orbit stay in its vicinity. If all trajectories in the vicinity of approach it as t ! 1 then the limit set has the property of a region of attraction: veri es the property of asymptotic orbital stability. De nition 3 Asymptotic orbital stability: The system trajectory in the phase space R is asymptotically stable if it is stable and kx k < Æ ) lim kx(t) pk = !1 inf 2

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This can be used for de nition of gaits, trajectory generation, obtention of nominal positions and required velocities for the gait and dynamical stability. 3.2. Gaits for legged robots

A regular motion of a legged machine correspond to a quasi periodic variation of the system state vector x. This allow us to de ne a closed orbit , in the phase plane, in order to determine the expected quasi periodic behavior. 7

De nition 4 De nition of gaits: A gait for a legged machine corresponds to a closed orbit , described by the system state vector x in the phase plane.

Strong relations exist between periodic trajectories and a limit cycles for the system. We can then adapt to our problem the stability properties. Proposition 5 Stable gait: A gait, corresponding to a closed phase trajectory

for the system state x; is stable if starting from any point in the phase space, near to , the system trajectory remains near to , as shown by de nition 2, then the gait is stable. Proposition 6 Asymptotically stable gait: If the system returns to the original gait, corresponding to , in spite of disturbances, as shown by de nition 3, then the gait is asymptotically stable.

For stability analysis, Lyapunov second method can be applied to prove the preceding propositions. 3.3. Admissible Gaits

Exchange of energy (kinetic and potential) between the ground and the robot parts is of main importance when operating with high velocities (high energy level). This point out that a crucial feature for gaits admissibility is to involve inertial and potential eects of the system. The energy ow in the system cannot be arbitrarily shaped (independently of system characteristics). This elucidates the encountered bad conditioning of some procedures for on line trajectory design and optimization when dealing with legged robots [3]. Mechanical impacts can either be modeled and taken into account or involved by mean of mechanical impedance [24][9][25]. Proposition 7 If we consider the general case like the robot of equation (4), the motion gaits which cope with the system dynamics have an energy of the following form Z 1 1  (7) V (x) = x_ M (x)x_ + G (s)ds + x Kx x

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Remark 8 It is worthwhile to note that the positions and velocities are strongly related through the robot, the control and the ground characteristics. Mechanical resonance of the system can be exploited by this way for energy optimization. The interaction robot-environment is implicitly involved by equivalent stiness (mechanical impedance) and cannot be constrained by imposing positions and forces independently. The gain K is devoted to stabilization and oscillation damping. Its use must be oriented to damp (only) dynamics which are transverse to the desired limit cycle (gait) [26]. This will avoid energy wasting. Equation (7) shows that (natural) gaits with constant energy can be obtained simply by aecting an energy to the system. The dissipations have to be compensated in order to stabilize the corresponding limit cycle. Choosing arbitrarily the gaits before control design may be inappropriate. These features point out the origin of main diÆculties encountered for trajectory generation and optimization. v

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3.4. Motion Evaluation

For motion evaluation in the case of hopping, the preceding analysis shows that the phase space (x; x_ ) have to be decomposed in two subspaces, one for the robot hopping and its transverse. For hopping, a limit cycle must be stabilized. Then the Lyapunov function of equation (7) is decomposed in two parts (one for the limit cycle and the second for the transverse motion). By this way the robot motion can be evaluated for control of the limit cycle. V (x) = V (z) + V (x) This de nition of the control objective copes well with the previous requirements, avoids use of ground interaction model and simplify the limit cycle control. For hopping, we can specify either the height of the jumps or the lift o velocity. This gives an energy value V (z) for the limit cycle and positions (and velocities) will result as consequence (Inertia matrix, gravitational eects and potential effect of contact are imposed by interaction and control K ). For hopping evaluation V (z ) can be computed and compared to V  (z ). This allow adjustment of desired trajectories with as Energy reference model V  (z). The jump height z depend, for a free system, on cycle energy and the motion period depend on the stiness and mass. By this way, the remaining trajectories will be imposed by energetic behavior of the system (robot+control+ground). They are solutions of equation (7). In what follows, we consider the Lyapunov function Z 1 (8) V (x) = x M  x + G (s)ds 2 lc

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3.5. Controlled Limit Cycles (CLC) with Energy reference

The objective is now to stabilize a periodic motion (vertical hopping) by mean of limit cycle control. The control computation need the use of energy reference level as shown in gure (5). Let us consider x = [x ; z ;  ] , the nominal position of the leg. If the maximum height of jumps is z , we can de ne the closed orbit and its Lyapunov function: R V =21 x_ M x_ + 0 G (s)ds (9)

= (z; z) 2 R  R : V = V x have to be designed for this system such that x the equilibrium point 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 [26]. This correspond to an "active" behavior which means that the controller supplies energy [27]. This can be realized by the following non linear control law:  sign(V V0 )x_ (10) x =x K This control is equivalent to a force feedback because x is multiplied by the equivalent stiness in the system equation (5). 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). So we can state the following theorem for the CLC. T

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Theorem 9 The system of equation (5) with the partial state feedback of equation (3), has as Limit Cycle the closed Orbit de ned by the Lyapunov function value V0 (equations 9) if the control of equation (10) is used with positive gain : o

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Derivation of the Lyapunov function (8), use of equation (5) and anti symmetry property (M_  2C ) gives: V_ = x_ (Kx Kx) (11) It can be shown that: (V V0 ) V < 0,  if V > V0 we have V_ < 0, the jump height is greater than z (the one of ), then energy is decreased, the feedback has dissipative contribution.  if V < V0 we have V_ > 0, the jump height is less than z , then energy is increased by an active control u.

Proof.

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Then the CLC de ned by = f(x; x_ ) ; V (x) = V0 g is orbitally asymptotically stable. We can then conclude, for the Controlled Limit Cycle, that the periodic orbit is a stable invariant of the system. o

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Remark 10 The Controlled Limit Cycle is obtained by a control expression which is either active or dissipative depending on the system energy with regard to the desired closed orbit . The CLC control is frozen in the ight phase (uncontrollable region). Rebounds may appear if the system has an energy level greater than the one of the orbit at initial con guration. [11][12] o

4.

APPLICATION TO THE SAP ROBOT

4.1. CLC: Robot Control and trajectory design

For legged robots with fast dynamic gaits, the aim of performance is that it can have fast and good transient behavior, maintain the system stability in its wide sense and dynamic stability (stable gait cycles). Let us de ne a stance for the robot as for example the one in gure (1). The angle between the segment de ned by contact point and the body is noted . If we de ne  as shown in gure (1), we obtain the angular relations q3 =   q2  and q4 = 2. If  = 0 the body (platform) is at a height from the ground equal to z = r = 2l3 cos() we can choose for example z = 1:8l3 < 2l3. This stance can be easily stabilized for the SAP robot by a state feedback. By de nition of stance angles  and , we can obtain the nominal desired vector  q =  q2  ; 2; 0; q2 for the robot position variables. For a gait corresponding to a vertical hopping, we can also de ne the following desired nominal positions. x = 0;  = 0; _ = 0;  = 0 (12) z =r =z +u An implicit stabilization by mean of pure position control can be applied at this step. Let q = q1 ; q2 ; ::::; q  corresponding to x = [x ; z ;  ] be the desired position vector. A simple stabilizing control (without gravity compensation) can be considered such as done in equation (3). b

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The control input  = [1 ; 2] (dim 2) can be computed as follows (with K = diag(k 1 ; k 2 ; 0; 0)).  = K (q q) K q_ (13) x = x  sign(V V0 ) x Following the approach of the preceding section, the input u (x = [0; z0 + u; 0]) is computed and applied to stabilize a hopping cycle with speci ed amplitude. The amplitude z correspond to V0 = M gz de ned for the maximum height of jumps. T

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4.2. Simulations and experimental results

The simulations show vertical hopping with as desired maximum height z = 0:5m. The control gains are K = 10, K = 1:5 and  = 0:3: m

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Figure 6: A snapshot of the simulation interface. The gures (7) shows that the CLC (for the body) is not as regular as for spring-mass hopper. This is due to the non linearity of the system, the control and coupling. The steady state error on the maximum height is due to the error (assumed for energy reference) on stiness and inertial parameters. An integral action can be used to compensate the error on the maximum height of the jumps. Running in forward direction (x < 0) is obtained by imposing a constant angle . The control input and positions of the body and leg are presented in gure (7). We can remark the regularity of hopping, control input and forward motion. The simulation results illustrate the eectiveness of the proposed. Experimental validation of this method has been done with the SAP robot ( gure 8 shows a sample of experimental results). 5.

CONCLUSION

In this paper, the main objective was control of dynamic gaits. We show that energy model is a good deal for control of legged robots with dynamic gaits and 11

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Figure 8: Experimental results on CLC. emphasizes pertinent features in the behavior when hopping and running. Use of the passive properties, the energy of the system and non linear feedback adjusted with energy regard, are shown to be eÆcient for an automatic generation and stabilization of hopping and running. The major contribution of this paper is the proposition of a simple and viable approach for analysis and control of legged systems. The methodology developed uses Controlled Limit Cycles for stabilization of gaits. The gaits correspond to limit cycles or a regions of attraction de ned with the energy. This approach simplify the study for robots with fast dynamics. The presented analysis emphasize the fact that fast dynamic gaits have to be formulated as closed orbits expressed in function of the system energy to be correctly optimized. This shows that compensation of inertial eects or introducing friction equivalent terms, by mean of control, correspond to energy wasting. The proposed approach is appropriate for (implicit) on line trajectory design, stabilization, en12

ergy shaping and optimization. The obtained control law can be interpreted as non linear feedback in position and velocity (non linearity is deduced from the energy model). The proposed methodology is applied to the SAP robot for hopping and running. Simulation results emphasize performance and eÆciency of the proposed approach. References

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