Minimum Energy Control based on Limit Cycles (CLC): Application for walking of a biped robot N. Khraief, N. K. M’Sirdi and O. Licer LRV: Laboratory of Robotics of Versailles Université de Versailles Saint Quentin en Yvelines 10, Avenue de l’Europe, 78140 Vélizy, France [email protected] , [email protected]

ABSTRACT: The main interest of this paper is to stabilize periodic motions of mechanical systems by means of energy regulation to save electrical consumption. Periodic motions correspond, in general, to periodic cycle or characteristic frequencies of the mechanical system. The main goal is to achieve a walking gait of an under actuated biped robot on the level ground, imitating the passive walking (for energy saving) on a given slope. We propose a viable approach for control of legged robots walking motion. This control approach is based on extension of Controlled Limit Cycles (CLC designed for stabilization of hopping robot’s motion (closed orbits) see [6-8]. We develop some theoretical and simulation results based on the control method referred to as Controlled Limit Cycles. Moreover such control also leads the system trajectories to converge towards stable limit cycles. In this context, we present some results based on both Poincaré map method and trajectory sensitivity analysis to efficiently characterize the stability of the almost-passive limit cycles.

I. INTRODUCTION Robotic systems always involve mechanical part generally actuated by DC motors or controlled electric power sources. This gives us processes composed by passive subsystems which represent a very important family of nonlinear plants involving passive and dissipative parts. The design of legged robots has to be conducted in such way to make some kind of motions appropriate and needing minimal consumption amount. Seemingly dynamic gaits are difficult to realize with classical control design methods (the error vanishing time is not finite), when dealing with complex mechanical models for legged robots and ground interactions. The control input may become large enough and may violate limitation constraints of actuators It has been claimed that impulsive control input are needed when fast gaits are considered, when dealing with classical control approaches. This is not obvious to ensure with actual electric drive motors and robustness has to be considered to preserve the robot. The approach that we propose here has, by definition, the intent to reduce the required power to minimum. This means that the biped designed as a mechanical system, has natural frequencies and particular features which can be exploited to stabilize periodic cycles by compensating only the lost energy during each cycle. When dealing with walking or hopping biped robots, one of the main objectives is to stand for stable periodic trajectories during regular displacements. From a state space point of view, such objective may correspond to achieve stable limit cycles (e.g. [1] [2] [3]). In such case, the major problem is to look for these potential limit cycles (i.e. to prove their existence), and, if existing, to stabilize the involved periodic or quasi-periodic trajectories (within a chosen state subspace of the biped state space). A brief survey of existing works points out that several approaches have been investigated to solve such problem. For instance, bipedal walking might be largely understood as a passive mechanical process, as shown for part by McGeer [5], Mochon and MacMahon [4] respectively. Indeed, McGeer [5] demonstrated by both computer simulations and experimental applications, that some legged systems can walk on a range of shallow slopes with no actuation and no control (energy lost in friction and impact is recovered from gravity). Since then, many researchers have considered this passivity based approach (e.g. Goswami et al. [6], Coleman, Garcia et al. [7] Mark.W.Spong [8], F.Asano [9] et al. M.Haruna [10], and reference therein). The gaits have to be chosen in order to minimize control effort required. The biped robots may exhibits periodic behaviour, and then the underlying continuous dynamics may induce an asymptotically stable limit set (periodic cycle) or Limit Cycles. As these cycles do not always exist (for all slopes), we propose to add a complementary

control to ensure, electrically driven, periodic motion gaits. In this purpose, we use the CLC control method (Controlled Limit Cycles), which considers the system energy for both controller design and system stabilization and has for aim to compensate power or energy loss (at each step) to maintain the cycle. The main goal is to achieve a walking gait of an under actuated biped robot on the level ground, imitating the passive walking on a given slope. However, to our knowledge, none have found a method to define initial conditions under which passive dynamic walking in a downhill slope is generated for an under actuated biped robot with knees and torso. This motivates the main part of the present work which considers the walking motion of a kneed legged biped robot on level ground, imitating the passive walking on a given slope (e.g. [11]). The paper will be organized as follows: First we will present the modelling of the 7-dof biped robot under consideration (that is a kneed robot with torso). Then, we will focus on the study of the passive dynamic walking of this robot, on inclined slopes. Moreover, we will show that before locked both knees and under a simple PD control applied to the lonely actuated1 link (link between the torso and the stance leg), trajectories of the biped robot can converge towards stable limit cycles. In this context, we will present some results based on Poincaré map method and trajectory sensitivity analysis to efficiently characterize the stability of the almost-passive limit cycles [11]. However, as such limit cycles may not exist for all ground configurations, a complementary control schemes is required. Thus, we will present some theoretical and simulation results based on the use of a recent control method (referred to as Controlled Limit Cycle [1] [2] [3]), which considers the system energy for both controller design and system stabilization. Finally, some potential extensions for future works will be discussed.

II. THE MODEL A. Description The dynamic model of a planar biped robot is considered in this section .It’s shown in figure (1). The robot has seven degrees of freedom (the five joint angles plus the Cartesian coordinates of the hips, for example). It consists of a torso, hips, and two rigid legs, with knees and no ankles, connected by a frictionless hinge at the hip. This linked mechanism moves on a rigid ramp of slope γ. During locomotion, when the swing leg contacts the ground (ramp surface) at heel strike, it has a plastic (no slip, no bounce) collision and its velocity jumps to zero. The motion of the model is governed by the laws of 1

As the other links are free of actuation.

classical rigid body mechanics. Following McGeer, we make the non physical assumption that the swing foot can briefly passes through the walk surface when the stance leg is near vertical. This concession is made to avoid the inevitable scuffing problems of stiff-legged walkers like the model analyzed in this paper. It’s assumed that walking cycle takes place in the sagittal plane and the different phases of walking consist of successive phases of single support. With respect to this assumption the dynamic model of the biped robot consists of two parts: the differential equations describing the dynamic of the robot during the swing phase, and the algebraic equations for the impact (the contact with the ground).

C. Impact model The impact between the swing leg and the ground (ramp surface) is modelled as a contact between two rigid bodies. The model used here is from [18], which is detailed by Grizzle & al. in [16]. The collision occurs when the following geometric condition is met. x2 = z2tgγ (3) Where

x2 = L3 (sin q31 − sin q32 ) + L4 (sin q41 − sin q42 ) z2 = L3 (cos q32 − cos q31 ) + L4 (cos q42 − cos q41 ) (4)(4) Yet, from biped’s behaviour, there is a sudden exchange in the role of the swing and stance side members. The overall effect of the impact and switching can be written as:

h:S → χ

x + = h( x − ) (5)

Where

S = {(q, q& ) ∈ χ / x2 − z2tgγ = 0} , with h is

specified in [21].The superscripts (-) and (+) denote quantities immediately before and after impact, respectively. Figure 1 the model of a 7-dof biped robot downhill a slope

D. Overall model The overall 7-dof biped robot model is written as follows:

B. Swing phase model During the swing phase the robot is described by differential equations written in the state space as follows:

x& = f ( x ) + g ( x )u

With:

x − (t ) ∈ S

(6)

A. Outline of procedure ⎤ d ⎡ q ⎤ ⎡ q& = In a last work [14] and about the study of a kneeless ⎢ ⎥ dt ⎢⎣ q& ⎥⎦ ⎣ M −1 ( q)( − H ( q, q& ) + Bu ⎦ biped robot with torso, we demonstrate that such systems

Where x = ( q, q& ) . (1) Is derived from the dynamic equation between successive impacts given by:

M ( q ) q&& + H ( q, q& ) = Bu

(2) Where

q = (q31 , q32 , q41 , q42 , q1 ) ,

u = (u1 , u2 , u3 , u4 ) , M (q) = [5 × 5] is matrix and H (q, q& ) = term

x − (t ) ∉ S

III. ALMOST PASSIVE DYNAMIC WALKING ON THE DOWNHILL A SLOPE

(1)

f ( x ) + g ( x )u =

⎧ x& = f ( x ) + g ( x )u ⎨ + − ⎩ x = h( x )

the

inertia

[5 ×1] is the coriolis and gravity

(i.e.: H ( q ) = C ( q, q& ) + G ( q ) )

constant matrix . The matrices developed in [21].

while B is

a

M , C , G , B are

can steer an almost passive dynamic walking on inclined slopes. Thus, the basic idea in this work is to do some transformation for the 7-dof biped robot. The main objective is to obtain nearly passive limit cycles, which will be exploited further to get active dynamic walking behaviour. At heel strike, the impact is plastic, some energy is dissipated and support is transferred instantaneously. Because the model has a torso then an impulsive torque must be applied against the post-transfer stance leg to hold the torso in a desired region [1] [2]. Then we decide to examine the possibility that a kneed biped robot with torso can exhibit a passive dynamic walking in a stable gait cycle, downhill a slope, when torque is applied to stabilize the torso by a nonlinear feedback given in the next section (after locked the knees of both legs).

B. Non collocated (partial)

input/output

linearization

In this section we use some results from [9] [10] [11], the objective is to get a control scheme able to lock both knees and to stabilize the torso at a desired position. To show this we may write the dynamic equations system (2) as:

D11θ&&1 + D12θ&&2 + h1 + φ1 = 0 (7) D θ&& + D θ&& + h + φ = u 21 1

Where

θ1 = q1

22 2

2

T

D12 ⎤ D22 ⎥⎦

(8)

functions

h1 ( q, q& ) ∈ R1 and

h2 ( q, q& ) ∈ R 4 contain coriolis and centrifugal terms, the vector functions

we may choose the additional controls v1 and v22 as:

v1 = q&&1d + kd1 (q&1d − q&1 ) + k p1 (q1d − q1 ) v22 = θ&&22d + kd2 (θ&22d − θ&22d ) + k p2 (θ 22d − θ 22 )

(13)

C. Finding period one gait cycles and step period

is the symmetric, positive definite inertia matrix, the vector

d d ( q41 , q42 ) represents the desired positions of both knees

2

and θ 2 = ( q31 , q32 , q41 , q42 ) and:

⎡D M (q) = ⎢ 11 ⎣ D21

d

If q1 now represents a desired position of the torso, and

φ1 ( q ) ∈ R1 and φ2 (q ) ∈ R 4 contain

gravitational terms and u represents the input generalized force produced by the actuators. From (9)-(10) we obtain:

⎧⎪ D11θ&&1 + h1 + φ1 = − D12θ&&2 = − D121θ&&21 − D122θ&&22 (9) ⎨ && ⎪⎩θ 2 = (θ&&21 ,θ&&22 ) = (v21 , v22 ) Where

θ 21 = (q31 , q32 )T

After locking both knees and stabilizing the torso we proceed to simulate the motion of the biped robot. The walker’s motion can exhibit periodic behaviour. Nearly passive Limit cycles are often created in this way (downhill a slope). At the start of each step we need to specify initial conditions ( q, q& ) such that after T seconds (T is the minimum period of the limit cycle) the system returns to the same initial conditions at the start. A general procedure to study the biped robot model is based on interpreting a step as a Poincaré map. Limit cycles are fixed points of this function. A Poincaré map samples the flow φx of a periodic system once every period [20].The concept is illustrated in figure (2). The limit cycle Γ is stable if oscillations approach the limit cycle over time. The samples provided by the * corresponding Poincaré map approach a fixed point x . A non stable limit cycle results in divergent oscillations, for such a case the samples of the Poincaré map diverge.

θ 22 = (q41 , q42 )T We verify that rank ( D121 ( q )) = 1 for all q ∈ R , the 5

system (9) is said to be strongly inertially coupled [9] [10] [11]. Under this assumption we may compute a pseudo-inverse matrix

† T T D121 = D121 ( D121 D121 ) −1 and

define v21 in(9): † v21 = − D121 ( D11θ&&1 + h1 + φ1 + D122 v22 ) (10)

With this choice of v21 we can write the following system:

θ&&1 = v1 ,θ&&22 = v22 D21v1 + D221v21 + D222 v22 + h2 + φ2 = u

(11)

Thus we see that the passive degree of freedom q1 have been linearized and decoupled from the rest of the system. The actual control u is given by combining (9) and (10), after some algebra as:

u = D 21v1 + D 222v22 + h%2 + φ% 2 (12) M 21 , h%2 and φ% 2 are given in Appendix B.

Figure 2 Poincaré map Let

P=Σ→Σ P ( xk ) = xk +1 = φx ( xk , T )

(14) Where the Poincaré hyperplane is defined by: d d Σ = ( q , q& ) ∈ χ / x2 − z 2tg γ = 0, q1 = q1d , q41 = q41 , q42 = q42

{

*

The fixed point x (initial conditions) can be located by the use of shooting methods [20]. D. Gait cycle stability Stability of the Poincaré map (20) is determined by * linearizing P around the fixed point x , leading a discrete evolution equation:

}

∆xk +1 = DP ( x * ) ∆xk (15) The major issue is how to obtain DP ( x ) - The Jacobean matrix- while the biped dynamics is rather complicated; a closed form solution for the linearized map is difficult to obtain. But one can be obtained by the use of a recent generalization of trajectory sensitivity analysis [19] [20] [14]. E. Numerical procedure A numerical procedure [19] [20] [14] is used to test the walking cycle via the Poincaré map, it’s resumed as follows: 1. With an initial guess we use the multidimensional Newton-Raphson method to determine the fixed point *

+

x of P (immediately prior the switching event). * 2. Based on this choice of x , we evaluate the eigenvalues of the Poincaré map after one period by the use of the trajectory sensitivity.

Figure 4 Nearly passive limit cycle of the center of mass (Z) of the 7-dof biped robot We took K p1 = 275, K v1 = 30, K p1 = 145, K v1 = 25

γ = 3° and q 1d = π we obtain:

7 x* = [ -2.63 -3.25 -0.213 -0.213 -0.9 0 0.002 0 0.01 0.0012]

IV “CLC” FOR ACTIVE DYNAMIC WALKING ON THE LEVEL GROUND We saw the existence of Nearly-passive limit cycles in section III. However these may not exist on all slopes, so some additional control is required. In this section we present “CLC” (Controlled Limit Cycles) [1] [2] [3], which considers the system energy for both controller design and system stabilization.

A. “CLC” Control

Figure 3 Algorithm for the numerical analysis F. Simulation results Consider the model (1), we choose the hyper plane Σ as the event plane. We let the biped robot on a downhill slope with the control scheme (16). Starting with a suitable initial guess we obtain the following results:

In order to get a periodic walk of the biped robot on the level ground we will use an additional control which will drive the zero dynamic to a reference trajectory characterized by the energy obtained from the Nearlypassive limit cycles. We define a Lyapunov function as follows:

1 V = ( E − Eref )2 (16) 2 Where E is the total energy of the system defined as follows: t

1 E = q& T M ( q) q& + ∫ q& T G ( q)dt (17) 2 0 and E ref is the energy which characterizes the Almostpassive limit cycle.

V& = ( E& − E& ref )( E − Eref ) (18) Where

E& = q& T Bu (19) E& = q& T Bu ref

ref

With

u = D 21v1 + D 222 v22 + h%2 + φ% 2 + u (20) u = D 21v + D 222 v + h% + φ% 1

ref

22

2

2

u in (27) represents the additional control. We choose the nonlinear control law: u = −Γsign(q& T B( E − Eref )) (21) Where Γ > 0 .It can be shown that the manifold defined by {θ&1 = θ&22 = 0} ∪ { E = Eref } is attractive for the closed loop system and all trajectories converge towards the Almost-passive limit cycle which is a stable cycle exhibiting a periodic walk motion of the biped robot. B. Simulation results The control law (27) is applied to the system (7) on the level ground ( γ = 0 ). The next figures show that the CLC control leads to a stable walking motion of the kneed biped robot with torso (7-dof). Limit cycle of the center of mass of the biped robot 0.3 z

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4 0.44

0.445

0.45

0.455

0.46

0.465

0.47

0.475

0.48

0.485

0.49

dz/dt

Figure 4 The active limit cycle of the center of mass (Z) of the 7-dof biped robot

V CONCLUSION In this paper, the main objective was the stabilization and control of the system for dynamic gaits. We present a control law, for a 7-dof biped robot, which realizes a stable continuous walking on the level ground to imitate a controlled nearly passive walking on the downhill slope. The last one is obtained by the use of a simple nonlinear feedback control with a numerical optimization. Next, the methodology developed uses Controlled Limit Cycles for stabilization of a periodic walk. CLC have been previously shown successful for trotting, hopping and jumping robots [1-3], here it is extended for control of walking bipeds with minimum energy. The gaits correspond to limit cycles or region of

attraction defined with the energy of the almost passive walking robots. In this way the energy needed for control is reduced. 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 approach is applied to the kneed biped robot with torso. Use of passive properties, energy of the system and non linear feedback adjusted with energy regard, are shown to be efficient. Simulation results emphasize performance and efficiency of the proposed methodology. Future research intends to implement the same control law on a test bed biped robot.

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