**1. Introduction**

The zero moment point (*ZMP*) and linearized inverted pendulum have continued to play a fundamental role in the control of postural equilibrium of biped robots and lower-limb exoskeletons since their introduction by Vukobratovic [1]. Vukobratovic showed that the center of pressure (*CoP*) of reaction forces under the feet soles on a flat horizontal surface coincides with a point he called the zero moment point (*ZMP*) and that postural equilibrium can be guaranteed if the *CoP* (alias *ZMP*) is maintained inside the convex hull of the surface encompassing the supporting foot (or feet in double stance). Moreover, a very simple relationship, based on the linearized inverted pendulum, links the *ZMP* and the center of gravity (*COG*) projection on the ground of the mechanical chain. The goal is to control the *COG* acting on the joint angles as both are algebraically linked by the kinematics of the chain, the target objective being the *ZMP*.

If motion maintaining postural equilibrium is desired (e.g., during a step in gait), to control the *ZMP* requires a certain degree of anticipation. Hence, in the so-called preview control framework [2], transition of *COG*, *COG* velocity, and *ZMP* were precomputed in advance and applied in open loop as reference to the biped control. However, this approach has not been able to model a closed-loop system and does not face the problems of disturbance rejection and stability.

A solution to track the preview trajectories in closed loop was successively proposed by Choi [3]. The preview *COG* − *ZMP* trajectories during rectilinear gait were reviewed, and a closed-loop strategy was devised and proven, using Lyapunov techniques, to guarantee closed-loop stability and a bounded error tracking of the *COG* and *ZMP* preview references. The loop was closed from the actual values of *COG* and *ZMP*, generating a feedback signal to control the *COG* velocity. Then, a speed control for the joints from the *COG* velocity was designed using a *COG* Jacobian with specified embedded motion. The output measures were not detailed, but presumably, *COG* was assessed by measuring the joint angles from the weight distribution on the kinematic chain, and *ZMP* was measured by pressure sensors under the feet of the biped. The strategy, according to theory, guarantees closed-loop stability. However, when the authors tested it in simulations and practical examples, it showed a lack of robustness to disturbances or poor damping of the closed-loop dynamics. Lyapunov theory guarantees stability but does not say how much the resulting closed-loop poles will be damped.

If real-time measures (let us call them *COGm* and *ZMPm*) of *COG* and *ZMP* are independently available, it is reasonable to assume that before closing the loop, a filtering is performed, fusing both data. However, if these are generated, as stated before, the latter by direct measures and the former indirectly from joint measures and a priori information, they are not always consistent with the relationship stated by the linearized inverted pendulum. The main reasons are: uncertainties in the model parameters (especially in the weight distribution when dealing with an exoskeleton interacting with a patient), external forces acting on the biped (crutches or a chair in a sit-to-stand exercise), centrifugal forces in the frontal plane when motion is not rectilinear.

This study was motivated by the intention to improve postural equilibrium in lower-limb exoskeletons for rehabilitation. The same approach is at the basis of all applications needing balance control of biped robots, such as biped walking in rectilinear [4,5] and curved trajectories [6], in haptic lower-limb exoskeletons [7], and in performing sit-to-stand exercises [8], described by the authors in other papers.

#### *1.1. Paper Contributions*

The main contribution of the paper is the development of a feedback control more robust than the one offered by Choi. In order to achieve this result, a detailed understanding of the closed-loop dynamics generated by controlling an inverted pendulum is presented, with particular attention devoted to the control design techniques and the engineering problems in closing the loop. Then, in order to make the filtering effective and close the loop from *COGm* and *ZMPm*, ensuring compatibility, the proposed approach operates at two levels: a nonlinear algebraic function and a a linear dynamic model.

The nonlinear function is a simplified mapping from joint angles to *COG*, called statistically equivalent serial chain (SESC) [9], to be identified in a priori experiments. As this identification is based on the same force sensors under the feet used for measuring the *CoP*, it also resolves any calibration mismatch.

The linear model is an extended system based on the inverted pendulum from input *u*, the reference *COG* velocity, and output *COGm* and *ZMPm*, used to estimate, along with the *COG*, *ZMP*, and external disturbances affecting the *CoP*, the model states. Then, using the estimated states, the loop from *COGm* and *ZMPm* can be closed applying robust control theory. The reasons for estimating the unknown external force disturbances are twofold: (1) to take into account real external disturbances, especially in exoskeletons but also in biped robots (e.g., centrifugal forces in turning

while walking); (2) robustness in the *COG* − *ZMP* joint estimation, accommodating modeling errors, parameter uncertainties, and the simplifications introduced by the linearized inverted pendulum.

Still recently, the linearized inverted pendulum has continued to be at the basis of the models for balance control ([10,11] and references therein). However, to the authors' knowledge, there are no works introducing, for robustness, an extended system to estimate disturbances, or the need for a SESC identification.

The proposed control is a non-conventional tracking problem, as two separate model states are tracked. Two different control design techniques are proposed and tested to control the extended system: (1) computing a robust estimator and solving the output feedback problem from the estimated states using a numerical approach based on the Levenberg–Marquardt algorithm [12,13]; (2) solving the standard robust regulator, adapted to deal with the preview signal tracking.

In order to test the approaches, three different experiments were performed. First, both observer and state feedbacks were implemented and compared with Choi's original feedback through simulation of the 2D linearized inverted pendulum. In a standing position, a preview shift of the *COG* from the heels to the tips of the feet was imposed, while in the meantime, an external force disturbance was applied. Then, a real lower-size mechanical mock-up was considered, composed of foot, leg, thigh, and trunk, with three degrees of freedom (DOF) in the sagittal plane to represent the real exoskeleton for implementing the sit-to-stand exercise. The SESC model of this simplified kinematics was identified with a priori experiments and used in the proposed feedback control through the *COG* Jacobian of the chain. Finally, a non-linear simulation of the full-scale exoskeleton with patient was run on the same exercise executed by the linearized inverted pendulum, emulating the first phase of a stand-to-sit exercise. A complete sit-to-stand exercise with the presence of a chair and switching dynamics exploiting the same control technique can be found in [8].

The paper is organized as follows. Section 2 reviews Choi's results. Section 3 introduces the main contribution of the paper: a *COG* − *ZMP* model of the linearized inverted pendulum and an extended system, embedding in the model external disturbances, for applying robust estimation and robust control. This model is also used in the Appendix to show the limitations of Choi's feedback. Sections 3.1 and 3.2 present the robust estimator–estimate state feedback and the standard robust regulator. Sections 3.3 and 4 contain simulated and real control experiments. In particular, Section 4.1 approaches the identification of the SESC model, and Section 4.2 presents the simulation of a stand-to-sit exercise. Section 5 concludes the paper. The appendix discusses the limitations of Choi's original method.

#### **2. Choi's Approach**

As in [3], the 3D linearized inverted pendulum is split into two separate, independent 2D models for the sagittal and the frontal planes. However, in this paper only the sagittal plane will be considered, with axes *x* (horizontal) and *z* (vertical). The equation linking *COG* and *ZMP*, adopting Choi's notation, is

$$p = c - (1/w\_n^2)\ddot{\mathfrak{e}},\tag{1}$$

$$w\_n \triangleq \sqrt{\lg/c\_z} \tag{2}$$

where *p* is the coordinate of the *ZMP*, *c* is the projection of the *COG* on the ground, *cz* is the height of the *COG*, and *g* is the acceleration of gravity. *wn* is the only parameter of the model of the simplified biped walking system.

Let *pd*, *cd*, and *c*˙*<sup>d</sup>* indicate the desired preview trajectories of the *ZMP*, of the *COG*, and of its derivative during a postural exercise, and assume that the pendulum joint servo is controlled in speed by an input signal *u* according to

$$
\dot{\mathfrak{c}} = \mathfrak{u} + \mathfrak{e},\tag{3}
$$

where accounts for the speed-tracking error and process disturbances and *u* is given by the following feedback law:

$$
\varepsilon\_c = \varepsilon\_d - \varepsilon, \varepsilon\_p = p\_d - p,\\
u = \varepsilon\_d + k\_c \varepsilon\_c - k\_p \varepsilon\_p. \tag{4}$$

Then, Choi's results prove, with Lyapunov theory ([3], Theorem 1), that if *kc* > *wn* and 0 < *kp* < *wn*, the closed-loop system is bounded disturbance () - bounded errors (*ec*,*ep*) stable.

Anyway, in spite of stability, a feedback implemented using Choi's equations has a poor damping of the closed-loop dynamics in practical operating conditions. A proof and discussion about this topic can be found in Appendix A.
