**1. Introduction**

The trend of using robots in all spheres of life, especially in industry, is constantly growing. We are witnessing an increasingly massive scale of production processes robotization in order to relieve people from performing periodic operations or directly creating processes that are physically impossible for humans. In addition to industrial robotics, we can currently observe an increase in interest in experimental robotics. Humanoid robots such as Atlas [1], Digit [2] or many others [3,4] represent the current pinnacle of human effort and intelligence in this technical field.

More and more progress in the field of human walking emulation also brings the seeds of practical use in terrain inaccessible to other mobile robots. Apart from uneven natural surfaces, these are mainly urban and industrial environments designed for the movement of people, not wheeled, or flying robots.

The area of interest in our contribution is to provide an analytical view of human walking from a mechanical point of view and, based on it, to design robot models at different levels of abstraction. The different levels cover models from the most primitive model—a mathematical graph of the robot, which provides information about the degrees of freedom of individual joints and the links between them, and this model is a source of data for the creation of another—a kinematic model. The next one—three-dimensional model containing the physical and material properties of the future real model, such as the weight and density of the construction material, can be considered as another level of abstraction. This model serves as a source of information about specific dimensions for the kinematic model. Based on these dimensions, a movement model of the robot is created with utilization of the trajectories of individual joints in space during walking. A

**Citation:** Polakoviˇc, D.; Juhás, M.; Juhásová, B.; Cerve ˇ ˇ nanská, Z. Bio-Inspired Model-Based Design and Control of Bipedal Robot. *Appl. Sci.* **2022**, *12*, 10058. https:// doi.org/10.3390/app121910058

Academic Editors: Luis Gracia and Carlos Perez-Vidal

Received: 8 August 2022 Accepted: 30 September 2022 Published: 6 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

simulation model is a combination of motion and three-dimensional models. The diagram of the simulation model represents the physical relationships acting on individual parts of the robot, as well as the control interface. The final product after the simulation is finished is the physical model. The key design model is a motion model of the robot, and the model is designed based on the linear inverse pendulum method. An important part of the design is the control of the robot and the construction of its physical model, based on which the main idea—identification and transfer of the key features of human walking into the engineering design of technical equipment—can be verified.

#### **2. Analysis of the Problem Area**

#### *2.1. Dynamic Side of the Problem—Human Walking*

Human biomechanics deals with the analysis of gait as part of the movements of human beings. In order to analyze human gait, the human walking was divided into the unique atomic periodic movements called step phases (Figure 1). These phases are further divided into two primary step phases, which are made up of eight secondary step phases, referred to as BAC 1–8 (Basic Action Concept). A set of two primary or eight secondary phases constitutes a gait cycle. The cycle can be measured from the start of any phase back to the start of the same phase of the same leg. A cycle is defined by two basic parameters: the cycle time and the spatial measures of the step. Based on the parameters of the cycle, the symmetry, variability and quality of the step are further determined. It is obvious that the step quality parameters are variable for each individual [5,6].

**Figure 1.** Visual representation of the gait cycle [7].

Two primary phases of the step are identified as the stance and the swing. During the stance phase, the tracked leg is in contact with the ground at all times. In the swing phase, the tracked leg is in the air all the time. In secondary phases, it is possible to observe three basic events [5].

**The weight transfer** makes up 12% of the total gait cycle. This event can be observed in phases BAC 1 (3%) to BAC 2 (9%). During this event, the initial contact of the heel with the ground takes place and damping also occurs. Damping factors are in ascending order of total contribution: bending at the knee, rolling over the heel and elasticity of the skin, muscles and tendons in the leg. During weight transfer, there is an initial increase in the moment of force M acting on the person's center of gravity [5,6].

**The support of one leg** makes up 38% of the total gait cycle. This event can be observed in phases BAC 3 (19%) to BAC 4 (19%). During this action, there is a movement of the upper part of the body and an overall increase in stability. At the end of the BAC 3 phase, the system (human body) acquires the greatest stability during the walking cycle. At the end of BAC 4, the person's center of gravity is so deviated that it is not possible to finish the step stably. In general, single-limb stance as an event is best defined by ankle rotation [5].

**The swing** makes up 50% of the total gait cycle. This event can be observed at BAC 5 (12%), BAC 6 (13%), BAC 7 (12%) and BAC 8 (13%) phases. Three sub-events occur during this event.


Based on detailed standardized video footage of real human walking cycle considering previous movement analysis, data were experimentally collected from the figure's walking cycle, which describes the movement of the hip, knee and ankle in space and time. A virtual measurement unit for distance *j* was involved in this video footage. During one phase, three measurements were taken, in which the position of the joint was monitored on the x and y axes (α-plane—side view of the figure) and in the z and y axes (β-plane—front view of the figure). The beginning of the measurement is in phase BAC 3, because the measurement also dealt with starting from the rest position, when the joints are located on one vertical axis.

From the graphical representation of the data in Figures 2 and 3, the critical region and the damping region were identified. Relatively complicated events take place in these areas and cause significant deviations from the otherwise simply describable trajectory, especially in the ankle. The activity in this joint has a direct effect on the stability of the step. Furthermore, a deviation between the cycle lengths in different parts of the gait was observed from the data. Based on these deviations, the cycle stability function for the given figurant was visualized. The observed trajectory forms a set of desired values of joint positions in time. According to the deviations between the observed trajectory and the generated trajectory of the robot, it is possible to clearly indicate the complexity of the created model.

In order to control the gait cycle of the robot, the zero-movement point (ZMP) concept is implemented in the linear inverse pendulum movement model (LIPM).

The place of operation of the pressure force is directly in the place of contact with the surface on which the robot walks. Therefore, it is a point on the robot foot that occurs during the physical interaction of the foot with the ground. The principle of the ZMP concept is that a new walking cycle is planned based on a reference trajectory. In the case of uncontrolled passive-walking robots, the ZMP forms the point of contact of the foot arch with the ground. The position of the ZMP during the stance phase is not stationary, but changes from heel to toe. This phenomenon causes complications in the simulation of flat-footed robots because the stability changes by leaps and bounds after the application of the pressure force moves beyond the tip of the flat edge. From the point of view of control, movement based on the ZMP concept forms one of several optimal trajectories. Walking appears visually fluid and energetically efficient [8].

**Figure 2.** Ankle trajectory (the first step).

**Figure 3.** Ankle trajectory (the second step).

By default, the robot step is modeled according to the inverted pendulum model (LIPM). Unlike an ordinary pendulum, an inverted pendulum does not make an oscillating motion, but tends to fall to a certain side. The moving base of the pendulum tries to compensate for the fall and keep it in a vertical position. The vertical position is the unstable equilibrium position of the pendulum [9–11]. The body of the robot represents the weight of the pendulum, which is supported by the leg, which represents the arm of the pendulum, and rotates around the ankle, which represents the base of the pendulum. The step cycle is thus formed by the controlled fall of the robot body in a certain range, after exceeding which the second arm of the pendulum—the other leg—hits. A system modeled in this way is considered passive, since the dynamics of the system depends on the force of gravity and the momentum of the system. To complete one walking cycle, the momentum of the system is required to produce forward motion. A system modeled by LIPM is inherently unstable, but for each input there are initial conditions from which the system will converge to a stable output. Likewise, there is a stabilization input for all initial conditions [9,10].

The LIPM model provides some variability in the robot design process, as it is independent of the dimensions and kinetic model of the structure, or the number of robot legs. However, it assumes that the weight of the robot's body is much greater than the weight of the legs. In an ideal scenario, the weight of the legs is negligible, which is highly unlikely with a physical model. The LIPM model does not further limit the robot construction in the mutual position of the center of gravity and ZMP—the length of the pendulum arm. The implementation of the model is also possible with a higher weight of the legs, which, however, significantly reduces the stability of the gait cycle. LIPM limits the physical model only in the number of degrees of freedom in the leg, which must be at least six [12].
