**1. Introduction**

Robotic rehabilitation started with passive exoskeletons [1], or devices that impose (on the interested limbs) a forced motion. An early example for lower limbs is Lokomat [2]. It evolved into systems that create force tunnels to address the motion generated by the patient [3]. These were further extended with the introduction of a feedback from the patient, to offer cooperative controls (also called hybrid control) not only to guide but also to contribute to the efforts of the patient [4]. An extensive state-of-the-art of cooperative exoskeletons for rehabilitation is contained in [5]. Following this line of approach, we proposed a haptic exoskeleton where the joints are actuated using admittance control based on the patient's Electromiographical (EMG) signals [6].

One classical exercise for postural rehabilitation performed in a fixed position is the "sit-to-stand". Then, a haptic exoskeleton able to guide the patient to perform this exercise is highly desirable. The study of the motion of the body during this apparently simple, but in reality not so simple, exercise has attracted interest for a long time [7], not only to understand the human physiological behavior, but also to mimic the control for autonomous biped robots or for actively cooperating exoskeletons.

The majority of available studies are related to the analysis of the human physiological behavior [8,9], but also examples of synthesis of the control based on optimization are available [10,11]. A recent comprehensive review can be found in [12]. However, none discussed the key determinants at the root of the exercise. Here, we follow a different approach. Recognizing that the human motion in performing the exercise is the direct consequence of the respect of physical laws of dynamics, these laws are analyzed and a feedback control based on them is synthesized. This also offers an explanation of well know physiological results such as the "Alexander STS technique" [11].

The exercise is composed of two dynamical phases: phase 1, when still sitting on the chair, the trunk, through the hips, is moved forward to gain balance on the feet, and phase 2, when the balance is maintained moving from the chair to an erect posture. In both phases, postural balance plays a key role, however, in phase 1, the coordination between the motion of the trunk and the torques on ankles and knees to release the load from the chair are also important. From the understanding of these dynamics, an automatic control can be synthesized. However, in the case of a lower limb exoskeleton for rehabilitation, such as in [6], where an automatic postural feedback operates in the *Cartesian* space and the patient controls the joints in the *joint* space, the interaction between the two players has to be considered. This paper proposes to program the exoskeleton to perform the exercise autonomously, then, with an innovative approach, to blend the two actions, moving seamlessy during the evolution of the rehabilitation, under the direction of the physiotherapist, from purely automatic to completely under the control of the patient. Moreover, according to the needs of the rehabilitation, some of the components of the coordinates of the *Cartesian* space, indicated here as elemental postural tasks, can be actuated by the automatic control and the remaining components, through selected joints by the patient, keeping the two groups separat from postural tasks without interfering between each other. Section 2 contains a background on the dynamics of the exercise. Section 3 presents the problem, describes the general adopted model and the autonomous control. Details of control during both phases are in Section 3.3. Section 4 applies the approach to a haptic exoskeleton. The results of a simulation are discussed in Section 5. The conclusions, mentioning future ongoing researches, in Section 6 complete the paper. Details of the control algorithms are contained in Appendices A–C.

#### **2. Background on the Dynamical Behaviour of the Exercise**

Two main aspects characterize the exercise: balance and coordination, briefly introduced in the following subsections.

#### *2.1. Balance*

The dynamic of a linearized inverted pendulum is a fundamental element to understand the balance of biped systems. It was introduced by Vukobratovic [13] for controlling exoskeletons, but actually has been widely exploited in autonomous biped robotics. He argued that balance is guaranteed if the center of pressure (*CoP*) of the reaction forces exerted by the ground on the feet and is maintained in the convex hull containing the surface of the feet. This point is coincident, on a flat horizontal surface, to a point he called *ZMP* (zero moment point-where the reaction from the constraint is a pure force with zero moment), a result from classical equivalence and replacement [14] in mechanics. Moreover, a linearized inverted pendulum is a really good approximation of the more complex kinematic chain of a biped, but also, adopting this approximation, and *ZMP* motion is linked by a linear relationship to position and acceleration. In a simplified 2D environment, (the paper considers motion only on the sagittal plane) the relationship is

$$\text{ZMP}\_{\text{x}} = \text{COG}\_{\text{x}} - \text{COG}\_{\text{z}} / \text{g} \cdot \text{COG}\_{\text{x}} \tag{1}$$

where *ZMPx* and *COGx* are the motion coordinates of *ZMP* and *COG* on the ground, *COGz* is the height of the barycenter and *g* the gravity acceleration.

Hence, to perform a transition, the *COG* must be moved controlling the joint angles to which it is algebraically linked, having as objective the *ZMP* position. As this requires a certain degree of anticipation, usually it is achieved by tracking some special pre-computed references of the *COG* (preview control [15]). Choi in [16] showed that a closed loop feedback from the measures of *COG* and *ZMP* position is able to track a preview reference signal. This feedback has been improved in [17] by introducing in the loop a state estimator of *COG* − *ZMP*, extended to external disturbance forces.

## *2.2. Coordination*

The biped in the sit-to-stand exercise is a dynamical system with changing non-holonomic constraints: the contacts of ground–feet are always present and the chair–pelvis is only present in the first phase only. This has consequences on the number of degrees of freedom of the multibody chain, and in the role of torque on the joints, that are defined in [18,19] as *Position* and *Auxiliary*. The former contributes to the motion, and the latter only to the reaction forces on the constraints. The balance with tracking of a preview reference of the *COG* − *ZMP*, based on the measures of the center of pressure under the feet and the pelvis–chair contact, and the coordination of *Position* and *Auxiliary* torques for motion and constraint force control are the elements of a "sit-to-stand" transition.
