Simulation of a Walking Robot-Exoskeleton Movement on a Movable Base

The paper studies the problem of movement of a two-legged walking machine on a movable base. This task is relevant for design rehabilitation and mechanotherapy complexes for people with impaired functions of the musculoskeletal system. Presents a mathematical model that allows obtaining the kinematic and dynamic parameters of the movement of the executive units of the device under study. The paper presents a method for planning the trajectory of exoskeleton links, its algorithmic and software implementation. The paper proposes the structure of the automatic link position control system, which ensures the movement of the executive links along a given trajectory. A mathematical apparatus is proposed for studying the dynamics of the controlled movement of the links of the human-machine system of the exoskeleton. The article presents the results of numerical experiments on the movement of the lowlimb exoskeleton leg in the one step mode and analyzes them.


I. INTRODUCTION
Exoskeletal devices are widely used for rehabilitation of patients with disorders of the musculoskeletal system [1][2][3][4][5][6][7][8]. The maximum effect is achieved when the patient and the exoskeleton form an integrated Human-Machine System (HMS), the effectiveness of which is determined by the degree of consistency (synchronicity) of the elements of this system, including the human and active exoskeleton [9][10][11][12]. To meet this requirement, the system contains a human-machine interface (HMI), which is a technical means that provides interaction between the patient and the exoskeleton. This system, consisting of the patient, the exoskeleton, and HMI was named bio-electromechanical system (BEMS). The BEMS structure consists of an exoskeleton, a humanmachine interface, a patient, and a support surface. At an early stage of rehabilitation, the patient occupies a passive position, and the movements of the lower limbs are performed using an exoskeleton along program-defined trajectories in accordance with the rehabilitation exercises performed, for example, the patient's walking on a path. At later stages, a combined mode of BEMS operation is possible, in which at certain times the patient takes an active position and the exoskeleton implements the movements set by the patient, and at another time the operator switches to a passive mode, and the exoskeleton performs the movements set by the program. Various control strategies are implemented in these modes [13][14][15]. In the first case, we can talk about a tracking control system, and in the second one -about a combination of tracking and copying control strategies. As the review of information sources shows, most scientific works consider walking on a fixed base [16][17][18], which in most cases reflects the real conditions of the use of walking machines, but in the case of creating a rehabilitation complex, the task of studying walking on a movable base, synchronizing the movement of the walking device and the moving surface is relevant [19][20]. The design of the exoskeleton considered in this work is equipped with special sensors system that allows assessing the interaction forces between the operator and the exoskeleton [19]. Also, an important feature of the work is the use of piecewise polynomial functions when specifying the exoskeleton foot movement trajectory. The device studied in the article is intended for mechanotherapy for patients with lower extremities musculoskeletal system dysfunctions [6,17]. The main contributions of this paper are: -the mathematical models of BEMS executive units movement, -the new trajectory planning method -algorithms of the system motion control that provides collaborative functioning of the human exoskeleton and the movable supporting surface -the numerical modeling results, proving the proposed methods applicability in the exoskeleton practical implementation.

II. MATHEMATICAL MODEL AND FORMULATION OF THE SIMULATION PROBLEM
The paper deals with the lower limbs' exoskeleton during the walk implementation along the moving belt of a treadmill (Fig. 1). The upper part of the exoskeleton 1 is attached to a fixed base -the frame. The femoral links 2 (left leg) and 4 (right leg) by means of active joints that contain electric drives creating moments, that can have an assisting effect on the patient, are attached to the body. To control the relative position of the links, the active hinges are equipped with angle sensors and torque sensing systems. In a similar way, the femoral links are connected to the lower leg links 3 (left leg) and 5 (right leg).

. Design diagram of the device in the process
The task of the system that controls drives of the hip and knee joints of the device is to realize this relative motion of links, at which points 3 L O and 3 R O move along the desired trajectory, implementing a walking motion of the exoskeleton. An important feature of the desired trajectory is to ensure that there is no slippage or impact at the moment of contact with the surface, which is provided by the choice of appropriate boundary conditions. In this work there is a number of assumptions: the trajectory of the foot attachment point is symmetrical with respect to the vertical axis, the movable surface is parallel to the horizon, the size and shape of the feet are neglected, considering that the contact of the robot's links with the surface occurs through the points 3

. Planning the trajectory of moving links of a walking machine
Since the trajectory is symmetrical, it is convenient to use a piecewise polynomial function, splitting it into 3 fragments: AB , BD, DA.
In order to provide the continuity of the trajectory setting functions, as well as the lack of collision of the links on the surface, the conditions must be met at each point. Next, an example related to the function for left leg is presented performed (for the right leg the function is going to be similar):    the trajectory becomes elongated, due to the synchronization requirements imposed to the links movement, which are composed of a movable base (collision is not stipulated). Using the trajectory shape coefficient  k results possible to obtain the required law of movement of links. This law is necessary to establish the parameters of the mechanotherapy of a specific patient, which simplifies the configuration and preparation of equipment.

IV. NUMERICAL SIMULATION OF THE MOVEMENT KINEMATIC OF THE ACTING LINKS OF A TWO-LEGGED WALKING MACHINE-EXOSKELETON ON A MOVABLE BASE
Mathematical modeling of the device movement at walking on a movable base stage has been performed. The length of the links of the device (0.5 m), as well as the step parameters has been specified.  During the modeling process, absolute and relative angles related to the movement of the links were also obtained when the symmetrical trajectory of movement of the links of the acting mechanism of the two-legged walking machine on a horizontal movable base is performed.

V. CONTROL SYSTEM AND HUMAN-MACHINE SYSTEM DESIGN
In order to setup the coefficients of the regulators, adjust the Human-Machine System (HMS) and other system parameters, it is necessary to perform a mathematical modeling of the control system of the links of the exoskeletal system. Since the control loops for the left and right legs are the same and differ only in the phases of the trajectories, we will describe one leg in the simulation process. Also, we will not consider the interaction of the patient's foot and the corresponding nodes of the exoskeletal system. In many rehabilitation devices designed to simulate walking and rehabilitation of the hip and knee joints, the ankle joint does not contain actuators, in this cases the foot is simply fixed on elastic elements that provide comfort to the patient [12,20].

FIGURE 7. Design diagram of the device and control system structure
The control task is reduced to working out the required trajectory of movement. The principle of operation of the Human-Machine System is as follows. The patient or the exoskeleton operator selects the necessary mode and the parameters of the exercise, and starts its execution. Next, the trajectory construction unit synthesizes the laws of changing the coordinates of the foot joint -coefficients of the PD controller. The coefficients of the regulator were selected as result of computational experiments. Thus, the legs of the exoskeleton and patient can be represented by two-coordinate pendulums with a common base and connected each other by elastic elements that simulate the operation of the cuffs, while the links of the exoskeleton will be driven by the moments of electric drives equipped with a position control system. In this case the system of differential equations describing the leg of the exoskeleton will take the form:  For the patient's leg, described as a two-link mechanism, we can write similar equations:

VI. NUMERICAL SIMULATION OF THE MOVEMENT DYNAMICS OF THE ACTING LINKS OF A TWO-LEGGED WALKING MACHINE-EXOSKELETON ON A MOVABLE BASE
In this part of the work, we will perform a numerical simulation of the system by ignoring the torques in the patient's joints ) 0 (  hsi M . The system parameters threshold ranges, as well as the mathematical model used parameters are presented in the form of a table.  From the simulation results we can observe that the actuators reach forces up to 100Nm, with a maximum force of 200N fixed in the ankle cuff, when performing a step. The selection of movement parameters and suspension parameters allows reducing the effort and moments, which can be useful when setting up a rehabilitation unit.

VII. CONCLUSION
A mathematical model of the movement of a two-legged walking mechanism on a movable base of a treadmill has been carried out. A method for studying this mechanism has been proposed. Also, trajectory planning algorithms have been developed, as well as the equations for solving the inverse kinematics problem for the mechanism links have been written. In order to set the shape of the trajectory, a coefficient  k is proposed. This coefficient represents the relationship between the speed values of movement of the legs on the surface and above the surface. Also, this coefficient in conjunction with other step parameters, such as length, height and duration, allows getting the required movement of the links of the walking mechanism, which are necessary for conducting a mechanotherapy treatment. The presented method is approved by mathematical modeling, which demonstrates the applicability and appropriate operation of the developed algorithms for trajectory planning for robot-exoskeleton devices. In this paper, a block diagram of the control system is proposed. This control system diagram describes the strategy of generating the required supply voltages for the drives of the exoskeleton when working out the trajectory. The presented mathematical model allows taking into account the This article has been accepted for publication in a future issue of this journal, but it is not yet the definitive version. Content may undergo additional copyediting, typesetting and review before the final publication. Citation information: Sergey Jatsun, Andrei Malchiko1, Andrey Yatsun, Al Manji Khalil, Andres Santiago Martinez Leon, Simulation of a Walking Robot-Exoskeleton Movement on a Movable Base, Journal of Artificial Intelligence and Technology (2021), DOI: https://doi.org/10.37965/jait.2021.0009 7 dynamics of the Human-Machine System, as well as to obtain the load diagrams for the drives of the device in walking mode, and estimate the value of the forces arising between the exoskeleton and the patient due to the existence of inertia and gravity. In the future researches, it is planned to develop a prototype of a rehabilitation complex for practical investigation of the proposed models and algorithms, as well as their modernization to meet the requirements of rehabilitation doctors.