Excerpt

## Contents

1 Introduction

2 Modelofhuman balancing

2.1 Mechanical model

2.2 Controllers

2.2.1 PD controllers

2.2.2 PD controllers with dead-zone

2.2.3 Model-predictive energy based controller

3 Measurement data and comparison with the simulations

3.1 Simulated and measurement data

3.2 Comparison by means of the stabilometry measures

4 Results and conclusions

## Acknowledgement

I would like to thank Dr. Zelei Ambrus for providing his valuable guidance, remarks and suggestions during the whole process

Budapest University of Technology and Economics

## Abstract

Human balancing is an important issue in many fields of everyday life, such as walking, running, cycling, carrying objects and even standing. The understanding of the balancing process is very important, especially from the point of view of elderly people. However, there are a lot of open questions about the working principle of the neural system.

We focus on the mathematical modelling of the neural process, which flows in the human neurotic system during standing still. There are several approaches in the literature. One approach is to apply a linear compensator (such as PD, PID, PIDA) in the model. Besides, model based predictive controllers are also feasible, when the human acts using a pre-learned control input pattern in the certain situations.

In our study, we compare the operation of the two control approaches, by comparing stabilometry measures, such as typical vibration frequencies, average velocity of the center of mass of the body and maximum/average tilt angle.

## Osszefoglalas

Az emberi egyensu´lyozas az elet majdnem minden teru¨leten nagyon fontos kerdes, ilyenek peldaul a jaras, a futas, a biciklizes, targyak mozgatasa kezzel, de meg ide tartozik az egy helyben allas is. Az egyensu´lyozasi folyamatok megertese nagyon fontos, elsosorban az iosodo emberek szempontjabol. Ennek ellenere, meg mindig nagyon sok nyitott kerdes vetodik fel az idegrendszer mu˝kodesevel kapcsolatban.

Az idegrendszeri folyamatok matematikai modellezesere fokuszalunk, amelyek az egyhelyben all´as kozben zajlanak. A szakirodalomban tobbfele megkozel´ıtes is talalhato. Az egyik megkozel´ıtes linearis szabalyoz´o (mint peldaul a PD, PID es PIDA) alkalmazasa a modellben. Emellett a modell alapu´ predikt´ıv szabalyozok is szoba johetnek, melyek alapjan az ember elore megtanult mozgasmintakkal avatkozik be az egyes sz-ituaciokban.

Munkankban a ketfele szabalyozasi alapelvet hasonl´ıtjuk ossze stabilometriai meroszamok seg´ıtsegevel, mint peldaul az oszcillacio tipikus frekvenciaja, a su´lypont mozgasanak atlagos sebessege es a test maximalis vagy atlagos dolesszoge.

## 1 Introduction

The importance of feedback control is significantly high in the area of science and engineering. Since human-balancing is a feedback control too, the same mathematical apparatus can be used for its analysis. Balancing task is a great example to help us investigate and understand the underlying control mechanism of human brain. In spite of the huge amount of already existing scientific results, the nature and the attributes of the feedback process utilized by the central nervous system (CNS) is still a subject of discussions.

Understanding the mechanism of human balancing can result in mitigating the risk of accidental death or morbidity which derives from balance-related accidents of the elderly. While the average lifetime of Earth’s population increases drastically, putting a spotlight on research in this field can be very promising in terms of saving lives. The median age of the world population is shown in Fig. 1 based on Reference 1.

Balancing might seems an easy task to do for first look, however it definitely is a complex process, which is carried out by the human brain. The brain receives many different kind of signals from the sensory organs. These signals involve pose and velocity data, acceleration, contact forces. The balance system works with the visual and the musculoskeletal systems too in order to maintain orientation. Visual signals are also conveyed to the brain about the body’s position in relation to its environment are processed by the brain and compared to information from the vestibular, visual and skeletal systems. Finally, brain takes an action by sending signals to the musculoskeletal system. This results a necessary movement.

Figure 1: Median age of world population through the years

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In this paper, the inverted pendulum is studied as a simple model to describe human postural balancing, similarly, as in 2. This model is widely used in the literature, e.g. 3 together with different kind of controllers. A typical approach is to use linear compensators, such as the proportional-derivative (PD), the proportional-integrator-derivative (PID) or the proportional-integrator-derivative-acceleration (PIDA). Another approach is to use some kind of model-predictive controllers. In our study, different kind of controllers (PD and model-predictive energy-based) will be applied for the mechanical model through the process. The different controllers will be compared with measurement results of human standing still.

## 2 Model of human balancing

Our overall mathematical model of the postural balancing consists of the equations of motion for an inverted pendulum and the equations that formulate the control law. The model allows us to compare the operation of different type of controllers applied on the same mechanical model.

### 2.1 Mechanical model

The mechanical model of the human body is the inverted pendulum, which is shown in Fig. 2. This model can be taken as the model of steady postural human balancing during standing. We assume that the relative motion of the body segments are negligible. The ankle is the only joint of which the motion is considered.

The inverted pendulum is a prismatic homogeneous bar with mass m and length l. The mass m corresponds to the whole human body weight and the length l corresponds to the body height. Since the bar is homogeneous and prismatic, the gravitational force G = *mg* is acting in the geometric centre of the bar. The general coordinate of this 1 DoF system is chosen to be γ(t). The bar is connected to the ground via a frictionless ideal pin joint. This joint corresponds to the ankle.

We model balancing as a control system, which has control inputs in general. In our model, the input torque is applied to balance the pendulum in the upright position, where γd is zero.

The equation of motion of the mechanical model is derived in the followings. The kinetic energy of the system is the following

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where Θa is mass moment of inertia of the pendulum which equals to:

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The control torque is taken into account in the general force *Q = T.* The control torque does not have potential function, therefore the potential energy of the pendulum is given by the gravitational terms as follows:

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The Lagrange equation of the second kind is as follows:

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where *E* d *is* is the dissipative energy, which is zero in our case, since there is no mechanical damping in the system at all.

After applying Lagrange equation of second kind, our equation of motion is:

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Based on Eq. (5), the angular acceleration is obtained:

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Figure 2: Inverted pendulum model of balancing during standing still

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Eq. (6) is the actual mathematical model of the inverted pendulum. This model has to be augmented with the model of the controller in order to obtain the entire postural balancing model.

### 2.2 Controllers

Three different kind of controllers, which are possible candidates of human balancing models, are compared in this study. The first two are both proportional-derivative (PD) controllers, which are often referred as linear compensator in the literature 4. The first controller considers the sensory dead-zone regarding the angular position measurement only, while the second one considers the dead-zone for angular velocity too. The third controller uses a concept which absolutely not uses the approach of linear compensators. Instead, the third controller considers the dynamics of the controlled plant and generates proper control impulses in order to reach the desired state. The details are given in the subsections below.

#### 2.2.1 PD controllers

PD controllers have been widely used for the last few decades. Abbreviation PD refers to the fact that these controllers have proportional and derivative parts.

An output value that is proportional to the current error value is produced by the proportional term. The proportional response can be adjusted by multiplying the error by a constant P, which is the proportional gain constant. A high proportional gain produces a large leap in the output for a given difference in the error. If the proportional gain is too high, the system can turn out to be unstable which should be avoided. This kind of instability appears in the presence of time delay or time digitization. In contrast, a small gain results in a small output response to a large input error, and less sensitive controller. If the proportional gain is too low, the control action will not be high enough while reacting to system disturbances. Proportional gain should be most contributing one to the output, it means it should be selected properly in order to get satisfactory results.

Derivative part anticipates the system behavior and thus improves settling time and stability of the system. The time derivative of the error is computed by finding the slope of the error over time and multiplying the rate of change by D (derivative gain). All in all, PD controller is given by:

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where *T* is the input torque which is applied to keep the pendulum from falling and γ0 is desired angular displacement value about which angle we want to keep the pendulum tilted.

PD controller and other linear compensators, such as PID, PDA or PIDA controllers are widely used for the modelling of the neural process of human balancing [5, 6]. In ideal case, when there is no sensory dead-zone, no delay, no digital effects, these controllers make the controlled variables settle down without oscillation. However, a small amplitude oscillation is always present in case of humans. In the following Section we introduce the sensory dead-zone into the model in order to obtain human-like behaviour with oscillations that never settle down completely.

#### 2.2.2 PD controllers with dead-zone

In general, dead-zone is a group of input values in the domain of a transfer function in a control system where the output is zero. Dead-zone regions are used to impede oscillation or repeated activation-deactivation cycles Ref. 6. Good example for this can be air-conditioners which keep the desired temperature while having dead-zone of couple degrees.

In the case of our human balancing model, we apply a dead-zone for the angle and the angular velocity. We assume that there is a small angle or angular velocity, which is not sensible by the human sensory organs. The output signal with dead-zone is shown by blue curve in Fig. 3. This curve can be obtained as a combination of Heaviside functions.

Using Heaviside functions would drive us to deal with piecewise-smooth dynamic systems, which would highly increase the complexity of our model. In order to avoid this problem, the Heaviside function is replaced by a similar but smooth function.

Smoothing is used in our Heaviside step function in order to avoid rapid/sharp turns and noise phenomena 7. The created approximating function tries to grasp the important patterns while gets rid of sharp edges. In smoothing, the data points of a signal are modified so individual points are diminished, and points that are lower than the adjacent points are upserged making a much smoother signal. We can set the sharpness of smoothing by changing the parameter inside. Our smooth Heaviside function is the following:

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where, *d* is a parameter for tuning the sharpness and *x* is the independent parameter. Finally, the smoothed

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Figure 3: Output torque in case of dead-zone with and without smoothing

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dead-zone function is composed as:

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which is shown by red curve in Fig. 3. Here, *ε* is the width of the dead-zone and *T(x)* is the control signal without considering the dead-zone.

**PD controller with a dead-zone of the angle**

Here, the dead-zone for the angular position is considered only. We consider γ in dead-zone function, where γ equals to the angular displacement error relative to the upward vertical position and εP is the dead-zone width. The smoothed dead-zone function (9) is applied only for γ (angular displacement error), therefore the output torque is

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The numerically obtained solution by using a the PD controller with dead-zone for the angle can be seen in Fig. 4. The corresponding phase-space diagram is shown in Fig. 5. The time history of the control torque is shown in Fig. 6.

As the graphs show in Figures 4., 5. and 6., the oscillations settle down after the pendulum starts to move from a tilted position. This behaviour is not typical for humans; therefore we do not consider the application of this PD controller for modeling the neural processes. Instead, in the next section, we detail the result of a controller that considers dead-zone for the angular velocity too.

Depending on the tuning of the positive gain parameters *P* and *D,* the settling of oscillations could be faster or slower, and the overshooting at the beginning could be eliminated, but the overall behaviour would

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Figure 4: Solution in time for PD controller with a dead-zone of an angle

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Figure 5: Phase Space diagram for PD controller with a dead-zone of an angle

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Figure 6: Torque in time for PD with dead-zone of an angle

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**[...]**

- Quote paper
- Mirroyal Ismayilov (Author), 2018, Predictive Control Algorithms in Human Balancing, Munich, GRIN Verlag, https://www.grin.com/document/541268

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