1

Control Approach for Legged Robots with fast Gaits N.K. M’sirdi+ , N. Manamanni∗ and D. El Ghanami+ +

LRP Laboratoire de Robotique de Paris University of Versailles, CNRS. 10-12, Avenue de l’Europe 78140 Vélizy. France. fax: 33 1 39 25 49 67 Email: {msirdi,ghanami}@robot.uvsq.fr ∗ LAM Laboratoire d’Automatique et de Microélectronique University of Reims. faculté des Sciences- Moulin de la house BP 1039 51687 Reims Cedex 2. France. fax: 33 3 26 91 31 06 Email: [email protected] Abstract A Constructive control approach is proposed for legged robots with fast dynamic gaits. These systems interact intermittently with the environment. Our approach is based on Controlled Limit Cycles (CLC) and stabilizes periodic system trajectories. The designed control law generates (on line) the desired trajectories and control input.

Contents I

Introduction

1

II Problem statement and preliminaries II-A Model definition and stabilizing feedback . II-A.1 Model definition . . . . . . . . . . II-A.2 Stabilizing feedback . . . . . . . . II-B System Properties and characteristics . .

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III CLC Approach for locomotion control III-AControl with Implicit ground interactions . . . . . . III-B Motion and Stability definitions . . . . . . . . . . . . III-C Gaits for legged robots . . . . . . . . . . . . . . . . . III-DAdmissible Gaits and motion control . . . . . . . . . III-E Motion Evaluation and gait definition . . . . . . . . III-F Controlled Limit Cycles (CLC) with energy reference

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5 5 7 8 8 9 10

IV Application to the SAP Robot

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V Conclusion

13 Keywords

Legged robots Control, Hopping robots, Passive Feedback Systems, Controlled Limit Cycles. 1

I. Introduction When dealing with complex mechanical models for legged robots and ground interactions [?][1][2], seemingly dynamic gaits are difficult to realize with classical control design methods. An increasing attention is paid to systems having interaction with environment such as legged robots with fast gaits. The use of the complete robot model is rather complex and the underlying problem is still challenging. The well known classical control methods require a priori defined reference trajectories. When dealing with legged machines, if the environment is not assumed perfectly known, trajectories must be computed or adjusted on line. This increases the motion tracking error (additional perturbations) complicating thereby the control task. Fast dynamic gaits require on line trajectory processing for preservation of the system stability and energy optimization [2][3] [4] [5]. 1 1

To appear (accepted) in JIRS : Journal of Intelligent & Robotic Systems (1999-2000) This work is a part of our contribution to the PRC of the CNRS on control of legged robots

2

Inputs =≥ F,

Perturbations

Robot

Outputs

Uncertainties

Reactions F,=≥ Interactions

Ground

Constraints

Variations Fig. 1. Robot / Environment Interactions

Otherwise the positions, velocities and accelerations trajectories must be compatible with the desired forces and ground interactions. Generally the error vanishing time is not finite (e → 0 as t → ∞) and to ensure performance (reducing the convergence period), the control become large enough and may violate limitation constraints of actuators. If the environment is unknown and the control is not taken into account, trajectories are difficult to optimize for such non linear systems. This problem has been generally takled in two steps, one for trajectory generation and the other for their tracking and adjustment [?] [6][7]. Very limited performances can be obtained (only for slow gaits) when classical control approches are used [8][9]. We have tried to control the system by use of non linear passive and adaptive methods [10]. The objective of control was to follow a priori designed trajectories for positions and forces [11]. The latter depending on the environment and synchronization with the robot dynamic was difficult to obtain. If the environment is assumed not completely known and varying, the control problem can be represented as in figure (1). The two complex systems in interaction have to be controlled by actions on the robot inputs only. Due to their nonlinear nature, the time variation of dynamics and the presence of phase transitions (several configurations in contact phase, flying phase, energy transformations and flow inside the system), dynamic legged robots suffers from lack of powerful techniques for jointly design and control. The classically used tools have to be revisited in the way that the desired task can be realized [12]. The objective of the control system has to be redefined and oriented to self trajectory generation and control. The control must stabilize a behavior or a quasi periodic motion of the legged robot. This has to be carried out such that the robot obeys some desired behaviors depending on the required gaits and motion. A genuine approach would be to propose a structure for the control problem decomposition in subsystems with clarified interactions. This formulation emphasizes the fact that the system’s behavior and trajectories are consequences of its dynamic, the control and ground interactions. The purpose is to perform energy shaping by use of an admissible control law. In this paper, we show that analysis and control of legged robots can be reduced to study an equivalent energy model involving the robot, the ground and a simple control law. The main interest of this paper is to intoduce an approach leading to this objective (stabilizing quasi periodic gaits). The energy of the system is considered for an automatic generation of trotting, and a reaction to perturbations and unknown events. In the proposed approach, trajectory generation is implicit and deduced from the dynamic behavior of the robot in order to avoid synchronization problems. This is done by means of a Controlled Limit Cycle (CLC) which allow regulation of the system energy and its control. The considered CLC correspond to the nonlinear second order system obtained by projection of the robot’s dynamic on the axis representing vertical motion (monitored by gravitation and potential effects). The projected motion correspond to a mass and spring system as those studied in literature [3][4][5][13][14] [15][16][17][18][19] [20][21]. This second order represent the desired nominal dynamic for the behavior of the robot in the vertical direction. Stabilization of a CLC for the project model leads to gaits obtained by use of natural periodic cycles of the system’s dynamic. The organization of the paper is as follows. Section 2 presents some preliminaries and appropriate definitions in order to obtain a good problem formulation. In section 3 we present the proposed methodology for control of dynamic gaits, this is done by taking one’s inspiration from the behavior of a mass and spring model for stabilizing a constant energy. The model of the robot is presented in section 4, where we’ll show that the

3

Energy Reference & Gait Definition

=•Ε z-u)

Nominal Stance

+

-

Stabilizing Control

Motion Evaluation

Robot

Fig. 2. Approach for Gait stabilization with CLC

system have a quasi periodic motion which tends to an orbit. This control approach is applied to generate trajectories and stabilization for hopping gaits. Some discussions and main comments on results are presented. Our future prospects, investigation and some conclusions in this work will be given. II. Problem statement and preliminaries Required motion gaits can be defined such that robot obeys some desired behavior. The main point is to ensure dynamic gaits and to maintain global behavior. The latter is the consequence of energy transformation and exchanges between different parts of the robot, the environment and the control. The approach proposed for control can be summarized by figure (2). It is composed by two feedback loops, one for stabilization of nominal stance (in order to obtain existence of quasi periodic behavior by introduction of elastic energy storage) and one for regulation of the energy balance to maintain a quasi periodic motion (CLC). In this paper, we consider the case of systems moving in the sagittal plane. In order to simplify presentation of our methodology, we shall consider the case of one leg only. The proposed approach can be extended to the case of 2 legs or more by alternating the control actions, this is the topics of our futur works. The role of each block will be detailed [21][10]. The following section presents the robot model and a preliminary feedback for stabilization of a nominal stance. A. Model definition and stabilizing feedback A.1 Model definition Let q = [q1 , ..., qn ]T be the vector of generalized coordinates. The robot dynamics is described by the Lagrange equations: d ∂T ∂T − = τ + τf + τe − g(q) dt ∂ q˙ ∂q where T (t) = 12 q˙T M(q)q˙ =

1 2

n P

(1)

mij (q)q˙i q˙j is the kinetic energy of the robot, τ is the control input vector,

i,j=1

τf are friction forces (function of q, q) ˙ and τe the environment reaction and internal perturbations of which we know only that they are bounded. g(q) is the vector of gravitational forces. The dynamic equation of the rigid system can be written: M (q)¨ q + C(q, q) ˙ q˙ + g(q) = τ + τf + τe

(2)

M(q) is the generalized inertia matrix, and C(q, q) ˙ q˙ is the vector of centripetal and Coriolis forces. The friction part τf can be compensated by control, they are neglected here for ease of presentation. The Cartesian variables can be defined, for a point in the robot’s body, as x = [xr , z, φ]T with respect to a reference frame fixed to ground. The horizontal and vertical coordinates of the body are respectively xr and z, the orientation is noted φ. The expression of motion in Cartesian space is obtained by use of geometric model x = L(q) and the corresponding Jacobian matrix J (x˙ = J q). ˙ The system can be written: M ∗x ¨ + C ∗ (x, x) ˙ x˙ + g ∗ (x) = F + Fe where M ∗ , C ∗ and g∗ are same as M, C and g but expressed in the Cartesian space.

(3)

4

The energy of the robot alone can be represented by the following Lyapunov function: 1 V (q) = q˙T M(q)q˙ + 2

Z

q

g(s)ds

(4)

0

The ground reaction (contact) can be assumed as a nonlinear impedance with high stiffness [22]. A.2 Stabilizing feedback The first thing we have to deal with is definition and stabilization of a stance (nominal robot position) when in contact with the ground. Energy exchanges with the ground can be performed by mean of elastic storage and restitution. This ability can be confined to the robot by use of a preliminary feedback loop (this makes the nominal position attractive)[23][24]. Let us introduce the following control functions (qd , xd and K), qd corresponds to the desired nominal position vector xd = [xrd , zd , φd ]T and defines a posture (we can take for example xrd = 0, φd = 0, zd = zo + u, u will be used for stabilization of hopping motion). An implicit stabilization by mean of simple feedback can be considered at this step (without gravity compensation): F = Kp (xd − x) − Kv x˙

(5)

x ¨ + M ∗−1 (C ∗ + Kv )x˙ + M ∗−1 Kp x = M ∗−1 (Fe + Kp xd − g∗ (x))

(6)

with this partial feedback we obtain:

Physical interpretation: The ground reaction force Fe has its effect weighted by the inertia matrix inverse expressed in Cartesian space M ∗−1 Fe . The control gain Kp contributes jointly with ground for the system excitation. This imposes the characteristic pulsation of the global system. If we assume that the mechanical impedance of ground is Ze with stiffness matrix Ke (or consider as reduced nominal impedance Fe = −Ze ∆z ' −Ke ∆z with ∆z the ground deformation) the equivalent stiffness is K = f1 (Kp , Ke , x). This corresponds to the equivalent impedance of the ground and control ones in serial connection [25]. The environment reaction force is implicitly involved with a nonlinear stiffness. Then the equivalent system can be written (Kv = 0): ¨ + C ∗ x˙ + Kx + g∗ (x) = Kxd M ∗x

(7)

B. System Properties and characteristics The applied partial feedback ensures existence of an imposed equilibrium for the system when in contact with the ground. If xd = (0, zo , 0) is constant this point will be stable. The equation (7) describes the legged system with a primary feedback loop which is able to have energy exchanges with the environment (introduction of elastic storage). It can be written also x˙ = f (x, xd , K) where xd and K can be used as control functions. The energy of this system can be represented by the following Lyapunov function: 1 V (t) = x˙ T M(x)x˙ + Pg (t) + Pk (t) ≥ 0 (8) 2 Rt Rx where potential energy due to gravity is Pg (t) = 0 x˙ T g ∗ (x)dt = 0 g(s)ds and potential energy due to R compound stiffness is Pk (t) = 0t x˙ T Kxdt. The motion speed depend on the power of the system and is modulated by the control gain Kp which imposes the characteristic pulsation of the global system (by means of stiffness change). The system (7) may have, as main features, the same physical characteristics as a simple mass-spring hopper [10]. Its motion can be characterized by three main points: • Existence of a contact phase (stance) and a flight phase, • During the flight phase the system is not completely controllable and has its motion due to gravity effect,

5 • Interaction with ground appear in contact phase and mechanical contact impedance results essentially from control gain Kp and ground stiffness Ke (in serial connection to produce an equivalent stiffness K). ˙ − C(x, x) ˙ is Property 1: The elements of the system’s model (7) are such that the matrix N(x, x) ˙ = 12 M(x) skew symmetric [26] [27]. R R Property 2: The mapping v = Kxd 7→ y = x, ˙ verifies the passivity property 0t y T vdt = 0t x˙ T Kxd dt ≥ −γ 2 [28]. Rt Rt ¨ + C ∗ x˙ + Kx +´g∗ (x)) dt ³ Proof:³ 0 x˙ T Kxd dt = 0 x˙ T (M ∗ x ´ Rt T R ˙ ˙ x˙ − N (x, x) ˙ x˙ + Kx + g∗ (x) dt = 0t x˙ T M ∗ x x˙ + Kx + g∗ (x) dt = 0 x˙ M ∗ x ¨ + 12 M(x) ¨ + 12 M(x) ´ ³ R Rt ˙ x˙ dt + 0t x˙ T (Kx + g∗ (x)) dt ¨ + 12 M(x) = 0 x˙ T M ∗ x Rt¡ d ¢ Rt¡d ¢ Rt¡ d 1 T ∗ ¢ ˙ M x˙ dt + 0 dt Pk (t) dt + 0 dt Pg (t) dt = 0 dt 2x = T (t) + Pg (t) + Pk (t) − V (0) = V (t) − V (0) ≥ 0 The total work done by the input force Kxd during the time interval [0, t] is equivalent to the system energy variation. . Property 3: Control functions xd = h(z, z) can be found such that the system (7) will be conservative: d V (t) = 0, ∀xo , ∀t ≥ 0. V (t) = Vo 6= 0 and dt The last property shows that for the system (7) an ’invariant limit set’ Ω with constant energy ( V (t) = Vo ) exists and trajectories x(t) ∈ Rn in Ω are bounded. If dissipation vanishes: energy exchanges are realized between parts of the system (transformation from kinetic to potential forms and reciprocally). For second order systems (1 DOF) this defines a closed orbit Ω (limit cycle or non-static equilibrium) for the system trajectory [29]. Note that for second order systems the only possible types of invariant limit sets are singular points and limit cycles. In our case, the system has 3 DOF (x = [xr , z, φ]T ), and then can be considered as an association of 3 coupled second order subsystems     (x, x˙ r )T f (x, xd , K) d  ¡ . ¢T   x fz (x, xd , K)  x˙ = f(x, xd , K) = (9) = z, z dt T ˙ f (x, x , K) d (φ, φ) φ˙

In our approach, we must stabilize two subsystems at (xr = 0, x˙ r = 0) and (φ = 0, φ˙ = 0) asymptotically and ¡ . ¢T control the one of vertical motion ( z,¨ z = fz (x, xd , K)) to obtain a CLC. Let us recall definition of limit sets. Definition 1: For the second order system ( x˙ = f(x, t)) a ’positive limit set’ Ω of a bounded trajectory x(t) ( kx(t)k < µ, ∀t > 0) is defined as Ω = {p ∈ Rn , ∀ε > 0, ∃{tk } such that kp − x(tk )k < ε, ∀k ∈ N}; where {tk } be a sequence of time instants with lim tk = ∞. k→∞

The first and the third subsystems can be maintained near the origin by the feedback of equation (5). Note that other control laws can be used, for instance the ones decoupling the 3 subsytems and robustly stabilizing the first and the third ones. III. CLC Approach for locomotion control Let us focus now on robust stabilization of a Controlled Limit Cycle for a second order (mass and spring) system and introduction of suitable definitions and features for gaits control. A. Control with Implicit ground interactions

In order to justify necessity of the previously introduced preliminary feedback for energy exchanges between robot and ground trough control, we consider the following simple system. Example of 1 DOF system: Let us consider the mass-spring system of figure 3, without frictions. The spring attach is assumed to have a controllable displacement u from the center of mass M (control input). The spring (with stiffness k) can be moved as shown in figure (3). The system equation can be written as follow: M z¨ + k(z − zo − u)ξ(z − u) = −Mg with ξ(z) = 12 (1 − sign(z − zo − u))

6

M

C on tro l u

k

Zo Z X

Fig. 3. Vertical Hopper (1 DOF)

equal to unity when the spring is in contact with ground and zero otherwise. This simple model allows to analyze energetic interactions between the robot, the control and ground. It is composed by interconnection of 3 subsystems (Mass, Spring and ground) and energy evolutions are: Potential ( g) →Kinetic → potential accumulation → potential restitution → Kinetic → potential and so on (for details see [21]). This shows existence of periodic cycles, corresponding to system oscillations, for free motion ( u = 0). This example emphasizes the necessity of potential energy storage (elastic behavior) for energy exchange between the ground and the mass M. For legged robots, this effect is introduced by mean of the control law. During the contact of the leg with the ground the compound systems robot+ controller + environment (ground) interact with each other by mutual application of forces. The commonly used models, either to investigate the impact of a robot with its environment [30] [31] or to analyze its dynamic stability [25] [32], are of type inertia-damping-stiffness. In the equivalent scheme of figure (4), Ke represents the effective stiffness of ground and Be the relevant damping coefficient. The whole {robot + control} can be ideally represented by the equivalent 2nd order impedance (Z(p) = Mp2 + Bp + K, with effective stiffness K and damping B = 0 for our example). During contact this impedance is connected to the environment one (ξ(z) = 1). The resulting equivalent impedance is:

Zequivalent (p) =

Z(p).Ze (p) (Mp2 + Bp + K)(Be p + Ke ) = Z(p) + Ze (p) Mp2 + (B + Be )p + K + Ke Z

M

B Robot Impedance

Z

Kp Fe

0

Z e

Ground

Be

Ke

Fig. 4. Contact Impedance model

K.Ke If the ground is very stiff (Ke >> K), the contact stiffness become K ∗ = K+K ' K. e The example of 1 DOF system verifies the preceding definition and properties, its equation

M z¨ + k(z − zo − u)ξ(z − u) = −Mg can be written in state space form: ½

(10)

.

x˙ 1 = x2 = z k ξ(x1 − u)(x1 − zo − u) − g x˙ 2 = z¨ = − M

(11)

7

Fig. 5. Closed orbit of example 1 DOF

If we take the control function u = 0, ∀t ∈ R+ , the system has obviously an equilibrium point and its solution describes a closed orbit Ωo (xo , t) corresponding to positive limit sets [33][34]. The non perturbed oscillations correspond to a closed orbit. The periodic orbit, obtained for the free system, depends on initial state xo (e.g. . (zm , z = 0)) and is defined by the following equation [21]: 1 .2 k . . Vf (z, z) = M z + Mgz + ξ(z) (z − zo )2 = Vo = V (zm , 0) = V (0, z d ) 2 2

(12)

which is plotted in figure ( 5). Equation (12) can be used to describe those arbitrary motions which cope well with the system’s ability (admissible motions for the system). We can verify that the system ( if u = 0, ∀t) . . is conservative (no energy dissipation), V (z, z) = 12 M z 2 + Mgz + ξ(z) k2 (z − zo )2 = Vo = Mgzm 6= 0 and . d dt V (z, z) = 0, ∀z, ∀t ≥ 0. B. Motion and Stability definitions In what follows, we adapt for our problem some basic stability definitions [21][23][13]. We consider the case of nonlinear (1 DOF) systems in the following form interacting with a rigid environment: .

M ∗ z¨ + C ∗ z + Kz z + gz∗ = Kzd

(13)

Stiffness and ground reaction force contribute to the nonlinear dynamic (K = Kp Ke (Ke + ξ(z)Kp )−1 ). Let us consider the behavior of neighboring trajectories in order to analyze the orbital stability. Definition 2: Orbital stability: The system trajectory in the phase space R2 is a stable orbit Ω if ∀ε > 0 ∃δ > 0 such that kzo − Ω(zo )k < δ ⇒ inf kz(t) − pk < ε, ∀t > to p∈Ω

This means that all trajectories z(t) starting near the orbit Ω (for the second order subsystem corresponding to the vertical motion) stay in its vicinity. If all trajectories in the vicinity of Ω approach it as t → ∞ then the limit set Ω has the property of a region of attraction: Ω verifies the property of asymptotic orbital stability. Definition 3: Asymptotic orbital stability: The system trajectory in the phase space R2 is asymptotically stable if it is stable and kzo − Ωk < δ ⇒ lim inf kz(t) − pk = 0 t→∞ p∈Ω

Let us illustrate these notions by use of our simple example. . Let us consider the previous example of 1 DOF system with as control function (with z d a constant for .2 .2 . landing velocity): u = − Λ2 (z − z d )z. It can be written in state space form: ½

.

x˙ 1 = x2 = z .2 k x˙ 2 = z¨ = − M ξ(x1 − u)(x1 − zo + Λ2 (x22 − z d )x2 ) − g

or equivalently: ½

M z¨ +

Λ .2 2 (z

z¨ = −g in the flying phase + kz = kzo − M g in contact phase

.2 . − z d )z

(14)

8

In the flying phase the system will have a ballistic trajectory depending on taking off velocity. This part of trajectory is symmetric and without loss of energy (conservative). During contact phase, the system behaves like the Van Der Pol oscillator [35]. The obtained stable invariant set is represented in figure ( 6). The system has an asymptotically stable orbit (energy level is regulated to desired level in contact phase). Z=f(Zp) Lim it Cy cle 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -10

-8

-6

-4

-2

0

2

4

6

8

10

Fig. 6. Stable Limit Cycle

The previous example is not controllable during the flying phase. The control function u has no effect on the system during this phase. These results can be used for definition of gaits, trajectory generation, obtention of nominal positions and required velocities for the gait and dynamical stability. C. Gaits for legged robots A regular motion of a legged machine x˙ = f(x, xd , K) (see equation (7)) correspond to a quasi periodic variation of the system state vector x in the phase plane. This allows to define, at least for the subsystem obtained by projection to the vertical direction, a closed orbit Ω, in the phase plane, in order to determine the expected quasi periodic behavior of the robot. Definition 4: Definition of gaits: A gait for a legged machine correspond to a closed orbit Ω, described . . by the state vector (z, z) of the subsystem (¨ z , z)T = fz (x, xd , K), corresponding to the vertical motion, in the phase plane. This establishes a strong relation between a periodic trajectory and a limit cycle for the controlled system. As consequence, we can 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, with a projection near to Ω, the system remain near to Ω, as shown by definition 2, then the gait is stable. Proposition 6: Asymptotically stable gait: If the system return to the original gait, corresponding to Ω, in spite of disturbances, as shown by definition 3, then the gait is asymptotically stable. For stability analysis, Lyapunov second method can be applied to prove the preceding propositions [29]. Remark 7: • The previous definition and properties establish strong relations between the invariant set of the vertical motion of the robot’s body and gaits. • The case of legged machines with several legs, the limit cycle have to be entertained by actions of the legs in a defined sequential order. • In the last case, several cycles (invariant set) can be considered with an appropriate management of the dynamic transitions. D. Admissible Gaits and motion control 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). The

9

system has, at least two kinds of energy accumulations. The equivalent system (robot+ground, see equation (7)) is composed by interconnection of subsystems (masses or inertia, springs, gravitational effect, energy storage elements and ground). Energy evolutions, during a motion can be, for example: Potential (g)→ Kinetic→ potential accumulation→ potential restitution→ Kinetic→Potential (g) and so on. Definition 8: Admissible Gait: A gait is admissible if the conditions for existence of the corresponding limit cycle or periodic motion is verified. This point out that a crucial feature for gaits admissibility is to involve inertial and potential effects of the system. The energy flow in the system cannot be arbitrarily shaped (independently of system characteristics). Thus for gait definition and trajectory processing, inertia matrix and potential energy of the system have to be taken into account. On the other hand, gaits must correspond dynamically to limit cycles, by mean of energy exchanges. Their definition have to be energy oriented to cope well with the objective of energy optimization. This elucidate the encountered bad conditioning of some procedures for on line trajectory design and optimization when dealing with legged robots [4]. Mechanical impacts can either be modeled [36] and taken into account or environment interaction is involved by mean of mechanical impedance [25]. Proposition 9: If we consider the system of equation (13), in the joint space, the motion gaits which cope with the system dynamics have an energy of the following form 1 V (q) = q˙T M(q)q˙ + Pg (q) + Pk (q) 2

(15)

or in Cartesian coordinates, the obtained orbit (or system trajectories) can be defined by 1 (16) V (x) = x˙ T M ∗ (x)x˙ + Pg (x) + Pk (x) 2 Note that this may be used to ensure the existence of periodic orbits and then periodic gaits. This will be done by choosing a constant energy level V (x) = Vo . We can define a closed orbit by use of Lyapunov function values: R Vo =© 12 x˙ Td M ∗ x˙ d + 0x g ∗ (s)ds ª . (17) Ωo = (z, z) ∈ R × R : V = Vo E. Motion Evaluation and gait definition

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). Interaction with the ground is implicit, expression of constraints and restitution model for impacts are avoided [36]. For motion evaluation in the case of hopping, the preceding analysis shows that the phase space (q, q) ˙ or . (x, x) ˙ have to be decomposed in two subspaces, one for the robot hopping (z, z) and its transverse (they will be denoted the hopping subspace and the transverse one). For hopping, a limit cycle must be stabilized. We have to deal with attenuation of motion perturbations and control of velocity of the robot (example: stabilization of the forward Cartesian velocity). Then the Lyapunov function of equation (16) 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 and anticipation on motion. V (x) = Vlc (z) + VT (x) 

       xr xrd m11 m21 m31 0 k11 0 x =  z , xd =  zd , M ∗ (x) =  m21 m22 m23 , K =  0 k22 0  φ 0 0 k33 φd m31 m23 m33 1 V (x) = x˙ T M ∗ (x)x˙ + 2

Z

0

x

1 1 1 . g∗ (s)ds + xT Kx = m22 (x)z 2 + m2 (x)gz + z T k22 (x)z + VT (x) 2 2 2

where m2 (x) is the equivalent massregarding the vertical motion.

(18)

10

This definition of the control objective copes well with the previous blocks requirements and can be exploited to avoid the use of modelling ground interactions and simplify the limit cycle control and stabilization. For . hopping, we can specify either the height of the jumps (zm ) or the lift off velocity (z d ). This gives an energy .2 value (computed at one of the two points: lift off or the maximum height) Vlc∗ (z) = 12 m22 (x)z d = m2 (x)gzm for the limit cycle and positions (and velocities) will result as consequence (potential effects (elastic) are imposed by interaction and control K). The jump height zm depend, for a free system, on cycle energy and the motion period depend on the spring stiffness and mass. Controlling the Limit Cycle allows implicit trajectory generation and energy optimization. For hopping evaluation Vlc (z) can be computed and compared to Vlc∗ (z). This allow adjustment of desired trajectories with as Energy reference model Vlc∗ (z). Note that VT (x) is implicitly chosen to be zero (minimal energy mean minimal variation of xr and φ around xrd and φd ). This means that the remaining trajectories xr and φ (or q) will be imposed by energetic behavior of the system (robot+control+ground). Remark 10: The hopping motion of the system is not completely controllable in the flying phase. . . . The energy reference is defined at one point of the orbit (z = 0, z = z d ) or (z = zm , z = 0). • • 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. • Accelerations are consequence of the possible energy flow in the system; they cannot be imposed arbitrarily. The interaction robot-environment is implicitly involved by equivalent stiffness (mechanical impedance) and cannot be constrained by imposing positions and forces independently. • Equations (15 and 16) show that (natural) gaits with constant energy can be obtained simply by affecting 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. . • xrd and φd can be chosen as functions of z and z to obtain an appropriate motion (e.g. for running . . φd = h1 (z, z), xrd = h2 (z, z) such that the velocity of contact point correspond to the running velocity [2]). These features point out the origin of main difficulties encountered for trajectory generation and optimization. F. 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 figure (2). Recall that xd = [xrd , zd , φd ]T = [0, zo + u, 0]T where u will be used for control of the limit cycle and the rest defines the nominal position of the leg corresponding to a robot posture either fixed (for hopping) or in function of body’s height. If the desired jumps height is zm , we can define a closed reference orbit by: ª © . (19) Ωo = (z, z) ∈ R × R : V (x) = Vo = Vlc∗ = m2 gzm Let us consider in what follows the Lyapunov function:

1 V (x) = x˙ Tl M ∗ x˙ l + Pg (x) + Pk (x) 2

(20)

The hopping gait can be realized by the following non linear control function zd = zo + u:

u = −zo −

Λ . ψ(V − Vo )z K

(21)

Where ψ(ξ) is any positive function of ξ (i.e. ψ(ξ).ξ > 0, ∀ξ 6= 0 and ψ(0) = 0). In this case all trajectories starting near the orbit converge toward the desired limit cycle. In other way if we consider the pure second order system (resulting from the projection of the motion on the z axis) we can define an orbit Ω0 the same as the one of the previous example. In what follows we’ll ensalve x and φ to the origin and strive to tend the

11

z motion towards the defined orbit to project it on x This control is equivalent to a force feedback because xd is multiplied by the equivalent stiffness in the system equation (7). 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. Theorem 11: The system of equation (7) with the partial state feedback of equation (5), has as Limit Cycle the closed Orbit Ωo defined by the Lyapunov function value Vo (equations 19) if the control of equation (21) is used with positive gain Λ. Proof: Differentiation, with respect to time, of the Lyapunov function (V (x) = 12 x˙ T M ∗ x+P ˙ g (x)+Pk (x)) 1 ˙ ∗ ∗ use of equation (7) and anti symmetry property ( 2 M − C ) gives: V˙ (x) = 12 x˙ T M˙ ∗ x˙ + x˙ T (Kxd − C ∗ x˙ − Kx − g∗ (x)) + (g∗ (x) + Kx)T x˙ , then V˙ = x˙ T Kxd

(22) .

To stabilize the periodic orbit (limit cycle), we need the following condition: (V − Vo )V ≤ 0, (V − Vo )x˙ Tl Kxd ≤ 0

(23)

if V > V0 ⇒ V˙ < 0, the jump height is greater than the one corresponding to Vo the one of Ωo , then energy is decreased, the feedback has dissipative contribution. • if V < Vo ⇒ V˙ > 0, the jump height is less than the one corresponding to Vo , then energy is increased by an active control u. ¡ . . ¢T Λ Λ ψ(V −Vo )z, we obtain V˙ (x) = x˙ T Kxd = x˙ T K 0, − K ψ(V − Vo )z, 0 replacing the control function by zd = − K and then ³ ´ ¡ . . . ¢T . Λ ψ(V − Vo )z, 0 = −(V − Vo )ψ(V − Vo )Λz 2 ≤ 0 (V − V0 ) x˙ r , z, φ K 0, − K Then the CLC defined by Ωo = {(x, x) ˙ , V (x) = Vo } is orbitally asymptotically stable. We can then conclude, for the Controlled Limit Cycle, that the periodic orbit Ωo is a stable invariant of the system. Remark 12: • 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 Ωo . • The CLC control may be frozen in the flight phase (uncontrollable region). • The positive function ψ(ξ) may used for introduction of system limitations an smoothness of control. • Rebounds may appear if the system has an energy level greater than the one of the orbit at initial configuration. [23] • This kind of control is robust versus parameters change and perturbations. The frictions impose a minimal value for the control gain Λ [21]. • Tangential dynamics [37] with regard of the periodic orbit can be minimized (dynamical stability) [38]. Remark 13: • The gain Kp (Section II.A.2) can then be optimized in function of the required behavior and the encountered ground stiffness. This can be done in order to reduce sensitivity of the system (Robot + Control + ground) to variations of ground stiffness and energy balance depending on motion. • The gain Kv materializes energy dissipation, its use is devoted to stabilization of motion. Its use must be oriented to damp (only) dynamics which are transverse to the desired limit cycle (gait) [37] [38]. This will avoid energy wasting. •

IV. Application to the SAP Robot The legged robot (SAP) considered, for evaluation of robustness of the proposed control approach, is represented in the following picture. It has six degrees of freedom with only four actuated joints (2 rotational joints by leg). In simulation, the model with 6 DOF is used for control and only one leg is considered, the other one is stabilized by state feedback to a fixed position such that it may be considered attached to the body (as shown in the following figure). We have designed a control for stabilization of hopping gaits, for the moving leg (with reduced complexity), with constant contact angle θ as defined in the following figure. The body’s height is zo = 2l3 cos(α) cos(θ). We choose as nominal posture θ = 0. This stance is first stabilized by a PD feedback for a constant value αd .

12

q

Zplateforme 3

l3

↓ 2∼ 2∼

q

Legged Robot SAP

∼

q1

l2

∼

° r o

q

2

Z

l3

4

r

SAP Robot Model with one leg

Following the proposed approach, the input u (zd = zo + u) is computed and applied to stabilize a hopping cycle with specified amplitude zm . This corresponds to Vo = m2 gzm defined for the maximum height of jumps. As shown by the following figure the computation of the control function is simplified in order to test the . .2 robustness of our approach in simulation: u = −zo − Λ.ψ(V − Vo )z with V (z) = 12 z + gz and Vo = gzm . As positive function ψ we use the saturation function. Note that by this way the system parameters are not needed. P2

0.5

Zp

V(z,zp)

9.81

Z

u Out

Lambda

V-Vo

Masse2

Zm

.8

.7

Saturation ÷

9.81

Fig. 7. CLC control function

The simulations show vertical hopping with as desired height zm = 0.55m. The control gains are Kp = 10, Kv = 1.5, and Λ = 0.3. The figures show the obtained limit cycle, the control and the motions of the body and leg end point. The figures (a to h) 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 system start hopping and moves in the forward direction (x < 0). The control is frozen during flying phase. The control is not frozen during the flying phase in figure b. (c and d) Λ =0.3, e) Λ = 0.4, f) Λ = 0.5, g) Λ = 0.5, the same control but applied with the other leg stabilized in the opposite position. h) same as g by CLC applied to z and to angular position of the other leg. X Y Plot

testm od XY Plot

0.54

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The steady state error on the maximum height is due to the error (assumed for energy reference) on stiffness and inertial parameters. An integral action can be used to compensate the error on the maximum height of the jumps. In simulations running in forward direction (x) is obtained by imposing a constant angle θ ( zm = 0.55, Λ = 0.5). The obtained CLC have the same aspect as the preceding one but has a more regular form. This is due to control damping effect (Kv = 4.5 and Kp = 40). We can remark the regularity of hopping, control input and forward motion. In conclusion the proposed approach is very simple and efficient for control of legged robots with fast gaits. The environment interaction is involved implicitly. The controlled system is robust versus inertia coefficients and stiffness variations (for ground and control). V. Conclusion In this paper, the main objective was the stabilization and control of the system for dynamic gaits. These investigations are promizing and appropriate for on line trajectory design, stabilization, energy shaping and optimization. We show that energy model is a good deal for control of legged robots with dynamic gaits and emphasizes pertinent features in the behavior when walking, hopping and running. Use of passive properties, energy of the system and non linear feedback adjusted with energy regard, are shown to be efficient 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 region of attraction defined with the energy. This approach simplify the study and can be easily applied 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 effects or introducing friction equivalent terms, by mean of control, correspond to energy wasting. 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). The proposed methodology is applied to the SAP robot for hopping and after for running. For our futur works we intend to apply this approach on the real system. Our investigations concern the application of this method on the pneumatic legged robot SAP for one leg (which is progress) end then extend the approach for the two legs after prooving the stability for the multi-legged robot by alterning the control actions for each leg. The authors are also intereseted by speed gradient control for energy shaping and multiple Lyapunov functions as tool for analyzing Lyapunov stability of such systems References [1]

M. Raibert, S. Tzafestas, and C. Tzafestas, “Comparatvie simulation study of three control techniques applied to a biped robot,” IEEE Conf. on Syst. Man and Cyb., Le Touquet, France, pp. 494—502, 1993.

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[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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