Methodology for control of a legged robot with fast dynamics∗ N. Manamani, N. K. M’Sirdi and N.Nadjar-Gauthier Laboratoire de Robotique de Paris 10-12 Avenue de l’Europe 78140 Vélizy France Email:{manamani,msirdi,nadjar}@robot.uvsq.fr

Abstract: In this paper, we study the control of hopping robots via an energetic approach. We show that an equivalent energy model can be used to stabilize hopping motion. Controlled limit cycles (CLC) are used implicitly define gaits and stabilize them. Thus the control system and energy shaping will be simple and easy to design.

1. Introduction The main purpose of this paper is to define a robust and easy to design control method for the hopping and the running of a legged robot. The objective is to be able to apply this method to other robotic systems with the same complexity and constraints. Several studies in literature pointed out the behavior analysis of one or two legged robot for the walking and running phases [1] [2] [3] [4] [5]. Some experiments showed that the dynamic behavior of a leg during the running motion is similar to a mass-spring system characterized by its mass and stiffness [2] [6]. Our bibliography study shows that, the control of energy for a hopper is a difficult problem to solve, and all the authors have the same conclusion: the study of the motion stabilization or limit cycles stabilization through the system energetic behavior is the best way to study locomotion. It allows to avoid the use of the complete nonlinear model of the robot and thus to obtain implicit trajectory generation. We studied in [7] an adaptive impedance control for the walk of a biped robot. The implementation of this control showed some limitations essentially due to the complexity during the computation and the necessity of force sensors. Despite of its robustness and adaptation to the variations of the ground characteristics, it seems insufficient for fast dynamic gaits like running or hopping. In fact, during these gaits where the contact phase is too short, the force control will compromise the dynamic postural stability. These reasons lead us to study the case of fast dynamic gaits (we intend existence of flight and contact phases during the motion). In our knowledge, a control methodology for fast legged robots without regarding their structure and their specific actuators does not exist. The studied systems are often simplified or optimized for particular trajectories. ∗

To appear on Ro-Man-Sy 98

Figure 1.1: One leg hopping robot (SAP) In this paper, we study first these problems through experiments and observations on our robot called SAP and presented in figure (1.1). The previous experiments lead us to compare the behavior of our robot to the one of simple mass-spring system, in an energetic point of view. The latter system admits natural periodic cycles. Thanks to a theoretical study, a methodology to achieve controlled limit cycles (CLC) is derived [?] [9]. The proposed approach is evaluated by application to the SAP robot presented in figure (1.1).The organization of the paper is as follows. Some experimental observations are presented in section 1. These experiments led to the analysis of a simple (1 DOF) mass spring system in free motion. This analysis is compared for the design of controlled limit cycles (CLC). Some discussions and main comments on simulation results and preliminary study for application for the robot SAP are presented. We end by some conclusions and our investigation on this work.

2. Equivalent model for the hopping motion 2.1. Introduction Dynamic model equations of robots interacting with environment are nonlinear and timevarying. So, for the sake of simplicity, we develop a new approach which consists of the approximation of the leg model by a simple mass-spring model in order to study the analogy of both systems, in their energetic behavior. We will show that this approximation is valid in case of periodic movements of the system. 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). First, the robot is stabilized (one leg in contact with the ground) by a feedback control law. In this configuration, we give as desired trajectory a periodic excitation (vertical motion according to z) with a small amplitude and adjustable pulsation ω. The desired trajectory is computed [10] to correspond to a vertical displacement of the end of the leg. In fact, at ω = 18rad/s a hopping movement occurs. Our experiments show that the behavior of the robot during this motion is like the one of a mass-spring system. The system thus defined presents, while in movement, a contact phase and a flight phase. 2.2. Study of a mass-spring system The simulated system for the behavior analysis is represented in figure (2.1), 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

³

´

ke spring and k the equivalent stiffness of interaction with the ground k ' kkrr+k . Let e . . vd =z d > 0 be the lift off velocity and vc =z c < 0 the touch-down velocity.The dynamic M k

Zo

Z X

Figure 2.1: Mass-spring system interaction with the ground is composed by two phases: flying and stance phases [11] [2]. 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 [10]. Flight phase During this phase, the system is not in contact with the ground and follows a ballistic trajectory determined by knowledge of the mass lift-off velocity z˙d = z(t ˙ = 0). This motion phase is without loss of energy and uncontrollable. The equation of the system can then be given by : M z¨ = −M g (2.1) Contact phase The contact between the end point of the spring and the ground occurs when z−zo ≤ 0. In this case we can write: M z¨ + k(z − zo ) = −M g, and

x¨ = 0, x˙ = 0

(2.2)

Let us also define the contact function ξ(z) which is 0 during the flight and 1 during the contact phase: 1 (2.3) ξ(z) = (1 − sign(z − zo )) 2 By the use of this nonlinear function and equations (2.1) and (2.2), we can write: M z¨ + ξ(z)k(z − zo ) + Mg = 0

(2.4)

The contact is supposed to be punctual and without slip or loss of energy. The vertical velocity of the mass when z = 0 at the initial time t = 0 is vc . The oscillations correspond to a closed orbit. The obtained periodic orbit, for the free system, depends on initial conditions and is defined by the following equation where zm is the maximal position: 1 k . .2 Vf (z, z, z(0)) = M z +M g(z − zm ) + ξ(z) (z − zm )2 2 2

(2.5)

2.3. Simulation results for the mass spring model Numerical simulations, with as initial position z = 1.5 m and null initial velocity, give a periodic motion (a closed orbit) as represented in figure (2.2). This figure shows the existence of two regions (flight and stance). During the flight phase acceleration and

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Figure 2.2: Simulation of motion of mass-spring system velocity are linear. In contact phase, velocity has a sinusoidal form, acceleration is not constant and shows similarities with contact force.These results show 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. 2.4. Experimental results of the robot SAP After stabilizing the robot with a feedback control law [?], as explained in section 2.1, one gives a desired trajectory corresponding to a vertical displacement of the end point of the leg (vertical motion according to z). Figure (2.3) shows the behavior of the platform velocity, which is practically the same as the one of the mass-spring system (figure (2.2)). We can see in figure (2.3) the platform position which also presents an analogy with the mass-spring system. The behavior of the end point of the leg presents also a short hopping phase and a contact phase. The last curve in figure (2.3) shows in the phase plane the periodic cycle for the robot’s platform, which presents a flight and contact phase like the mass-spring system. So the comparison of figure (2.2) and figure (2.3) shows the strong analogy in the nominal energetic behavior of both systems. That’s why our main objective is to control the limit cycle obtained for the mass-spring system during hopping motion and then transpose the study for the case of the one leg hopping robot. 0 .0 3 5

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Figure 2.3: Experimental results for the robot SAP

3. Controlled limit cycle for the mass-spring system The experimental observation done previously and the comparative results with the massspring system incline us to introduce a new approach which consists of establishing a limit cycle in the system and stabilize it. In this part, this approach will be considered for the mass-spring system. Then we will adapt the method for hopping motion of the SAP robot.The one degree of freedom (DOF) controlled system considered now is represented M k

C o n tro l u Zo

Z X

Figure 3.1: controlled mass-spring system in figure (3.1). 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 (3.1). Our main objective is now to define and stabilize a periodic motion by mean of the control function u. For vertical motion of the system we can write: M z¨ + ξ(z − u)k(z − zo − u) + M g = 0

(3.1)

The proposed approach allows us to maintain a prescribed limit cycle whic is defined by the system’s energy function corresponding to the maximal height of the hop or function of the lift-off velocity vd . So let consider the Lyapunov function given by: 1 V = M z˙ 2 + Mgz 2

(3.2)

We can write V˙ using ξ(z, u) : V˙ = −z˙ (k(z − zo − u)ξ(z, u) + M g) + Mg z˙

or V˙ = −zk(z ˙ − zo − u)ξ(z, u)

(3.3)

If z − zo − u > 0, the energy is preserved by the system, and during contact phase. If z − zo − u ≤ 0, we have to ensure by the choice of u that the system maintains the corresponding limit cycle. This can be done by an appropriate choice of the term ˙ − zo − u). V˙ c = −zk(z . . 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 velocity z d . This can be defined by: . z 2d V0 = gzm = (3.4) 2 So to stabilize the cycle, we have to impose the following conditions: if V > V0, we impose V˙ c < 0 : if the hop is higher that zm we have to reduce the energy, and if V < V0 , we impose V˙c > 0 : if the energy is low, we increase V .

• Condition We can gather these two conditions such as: . 1 . .2 .2 V c (V − V0 ) = − z k(z − zo − u)( M z +Mgz − M z d ) < 0 2

(3.5)

To ensure this condition, let us consider as control law u: u = (z − zo ) − Λ (sign(V − V0 )) z˙

(3.6)

This control is equivalent to a force feedback: u is multiplied by the stiffness in the system equation (3.1) ( Λ is a positive gain). This control will then be able either to supply energy to the system or to reduce its energy level in function of the running cycle energy. It can be either active or passive (dissipative). The control computation needs the use of energy reference level ( 3.4). 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 3.1. The mass-spring system of equation (3.1) admits as limit cycle the closed orbit Ωo defined by the Lyapunov function value V0 (equation (3.4)) if the gain Λ used in control of equation (3.6) is positive. We can then conclude (see [?] [?] [?] [9] for the proof of the theorem), for the controlled limit cycle, that the periodic orbit Ωo (equations (3.2)) 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. 3.1. Simulation results for the CLC .

Let us take into account a friction force β z in the system. We obtain the closed loop equations: . M z¨ + ξ(z − u)k(z − zo − u) + β z +Mg = 0 (3.7) with the same control defined in (3.6), this leads to : .

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

(3.8)

For simulations, we have considered both cases, with frictions (β 6= 0) and without (β = 0). Periodic motion stabilization (with use of saturation function) when the system initial position is z(0) = 0.5; or 1 to 5.5, are represented in figure (3.2). The desired height of hopping is imposed to zm = 4. 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)

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a) C LC for k/M =5 00, Zm =4 , L a m b da=0.05

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Figure 3.2: CLC a) Without and b) With Frictions is near to equilibrium point zeq .Note that introduction of frictions in simulation produces deformation of the orbit (symmetry is lost instead of high function) and reduced the height of jumps. The height reduction is important for high frictions level (the viscous friction coefficient has been increased several times). If we still increase friction level, the periodic orbit disappears, the origin becomes an asymptotically stable equilibrium point. Despite the friction existence, we can define and stabilize a limit cycle for our system by use of high feedback gain.

4. Application to the robot SAP By considering the dynamic model of the robot [?] M(q)¨ q + h(q, q) ˙ = M (q)¨ q + C(q, q) ˙ q˙ + G(q) = τ + τ e

(4.1)

M (q) is the (2 × 2) generalized inertia matrix, G(q) is the (2 × 1) vector of gravitational forces and C(q, q) ˙ q˙ is the (2 × 1) vector of centripetal and Coriolis forces. τ is the applied torque and τ e the external torque. The system energy can be given by the following Lyapunov function: 1 V (q) = q˙T M (q)q˙ + Ep (q) 2

where Ep (q) =

Z

q

G(s)ds

(4.2)

0

When applying such a control law, τ = Fp (qd − q) − Fv q. ˙

(4.3)

in closed loop the system becomes : M (q)¨ q + (C(q, q) ˙ + Fv )q˙ + Fp q = τ e + Fp qd − G(q)

(4.4)

The considered system is more complex than the mass-spring one but presents the same characteristics [9]: • A phase without contact (flight) and a contact phase, • In the flight phase, the system is not completely controllable, it supports the gravity effects.

• The spring effect owe to the control will have an effect only during the contact phase. The equations (4.4) and (3.7) present some analogies, so the same approach as the mass-spring system can be considered for the robot to have a controlled limit cycle.

5. Conclusions We have shown in this paper the advantage of an energetic approach through controlled limit cycles. An experimental study allows us to show the behavior analogy between our robot and a mass-spring system during hopping motion. By a simple control law depending on the system energy, we can define and maintain a stable limit cycle called Controlled Limit Cycle (CLC). This result allows us to focus on the choice of the cycles to define the robot gaits without carrying trajectory generation, [?] [?] [9]. These cycles can be stabilized for legged robots and defined to give to the robots quasi-periodic gaits. In this approach we avoid explicit trajectory generation. This one will be obtained implicitly through the control of limit cycles (CLC) exploiting the energetic properties of the system. In our future works, we will show that for hopping motion, the dynamic model of a pneumatic legged robot can be reduced to the study of an equivalent mass and spring model. The application to a biped is the next step of the future investigation.

References [1] B. Espiau and A. Goswami. Compass gait revisited. 1994. symposium of robot control. Capri, Italy. [2] G.Valiant T.A.McMahon and E.C.Frederick. Groucho running. Journal of Applied Physiology, 62:2326—2337, June 1987. [3] P.R.Greene T.A.McMahon. Fast running tracks. Scientific American, 239:148—163, 1978. [4] M.H.Raibert. Legged robots that balance. MIT Press Cambridge. MA., 1986. [5] M.H.Raibert and I.E.Sutherland. Machine that walk. Scientific American, 248:44—53, 1983. [6] A.Vernon R.McN.Alexander. The mechanics of hopping roos(macropodidas). Journal of Zool. London, 177:265—303, 1975.

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[7] C. Tzafestas, N.K. M’Sirdi, and N. Manamani. Adaptive impedance control applied to a pneumatic legged robot. 1997. Journal of Intelligent and Robotic Systems. Invited Paper. [8] N. Manamani. Commandes et observateurs pour systèmes pneumatiques méthodologie pour la locomotion à pattes. Thèse université de Paris 6, LRP, Paris France, 1998.

[9] N.K. M’Sirdi, N. Manamani, and N. Nadjar-Gauthier. Methodology for control of legged robots with fast gaits. IROS98, Canada, 1998. [10] N.Manamani, N.K.M’Sirdi, N.Nadjar-Gauthier, and L. Alvergnat. Simplified modelization and control of a two link hopping robot. ECPD Conf. on advanced robotics and intelligent automation, September 1996. [11] M.D.Berkmeier and K.V.Desai. Design of a robot leg with elastic energy storage, comparison to biology, and preliminary experimental results. IEEE International Conference on Robotics and Automation, 1996. [12] N.K. M’Sirdi, N. Manamani, and N. Nadjar-Gauthier. Control approach for hopping robots: Controlled limit cycles. Proc. IEEE AVCS98, Amiens, France, 1998. [13] N.K. M’Sirdi, N. Manamani, and N. Nadjar-Gauthier. Controlled limit cycles approach for control of legged robots. Proc. IFAC Motion Control98, Grenoble, France, 1998.