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Файл:The design of the exoskeleton. Monograph
.pdf
41
2.1.2. Мodeling Methods and motion study of the dynamics of
the exoskeleton suffrage
In the works on the dynamics of anthropomorphic walking robots,
significant assumptions are often made.
In the work of A. M. Formalismo [4] the problem of managing the walk
paticipating anthropomorphic mechanisms is studied through the
application of pulsed impacts and modeling of the process of moving in a
symmetric motion and ballistic movements of the body (i.e., inertial
motion), and solves the problem of impulse control for the linearized
equations of motion.
In the works of V. V. Beletsky [5, 6] investigated the dynamic
characteristics with predetermined functions of the parts, when the
suspension point of the legs is at a constant height relative to the surface
walking, and the motion is uniform and rectilinear (the so-called
comfortable movement).
In V. E. Berbyuk's works [6, 7] the comfortable and uncomfortable
walking of a five-step walking robot on the given functions of time of the
generalized coordinates is considered. The problem of stabilization of the
robot body is solved under the assumption that the motion of the limbs is
known and the nature of the motion does not change in time.
2.1.3. Equation of motion and dynamic characteristics
The equations of motion of the studied exoskeleton are constructed
on the basis of the Lagrange equations of the second kind, which are
written in the formula (2.1) [4, 5, 6, 7]:
, (2.1)
In equations (2.1): q – vector of generalized coordinates, Q – vector of
generalized nonconservative force, L = T – P – Lagrange function, T – kinetic
energy, P – potential energy system. For the considered system n = 5.
42
In order to find the kinetic energy of the system T, find the kinetic
energy of each link separately. Thus, in the generalized coordinates it has
the form presented in the formula (2.2):
(2.2)
for the i-th link kinetic energy is calculated by the formula (2.3)
(2.3)
Speed of center of mass
(2.4)
, (2.6)
where: x, y – Cartesian coordinates of the suspension point of the
legs and are presented in formulas (2.7) and (2.8), and M – the mass of
the five-link mechanism and is represented in (2.9):
(2.7)
(2.8)
(2.9)
Обобщенные силы найдём из элементарной работы всех
сил, приложенных к системе:
(2.10)
.
43
The system of Lagrange differential equations of the second kind for
a five-stage exoskeleton is presented in (2.11):
(2.11)
The coordinates of the center of mass of the mechanism , have
the following expressions [4]:
(2.12)
( 2.13)
The equation of motion of the center of mass of the mechanism can
be written as follows:
(2.14)
(2.15)
The equations governing the moments at the joints of the robot are
calculated according to formulas (2.16):
(2.16)
44
In the process of walking there are two phases, namely phase on two
supports (double support phase) and phase single-bearing alternator
movement (phase transfer). During on two supports of the movement both
feet are on the surface, during odnoimennogo – only one foot is on the
surface of the support, and the other is in the process of migration. While
walking, these two phases alternate. In the single-foot phase of walking, for
example, if the first leg (j = 1) is a support leg, then 2 = 2 = 0 and we
can calculate the control moments in the robot joints.
However, the definition of these moments is ambiguous in the twosupport phase. To determine the eight unknowns
, , ,
(j= 1,2) we
have only five equations. It is necessary to determine the part of the
unknown (semi-inverse method), for example, you can set the reaction
force of one of the two pillars as follows:
(2.17)
(2.18)
where: 1, 2, 3, 1, 2, 3 - coefficients determined from the time
parameters and desired States of the mechanism at the end of the bisupport phase.
2.2. Mathematical model of the actuator of the exoskeleton
All joints of the exoskeleton are rotational kinematic pairs of the 5th
class.
Electric drives are built on contactless motors with permanent
magnets. A feature of these engines is the ability to provide a very high
torque at a relatively low speed, which allows you to remove the gearbox
from the system design. Gearless drive has undeniable advantages for use
in exoskeletons.
The design of the brushless motor with permanent magnets [8] is
shown in figure 2.2:
45
Figure 2.2 – a DC Motor with permanent magnets
The valve motor is a synchronous motor based on the principle of
frequency regulation with self-synchronization, the essence of which is to
control the magnetic field vector of the stator depending on the position
of the rotor. Valve motors (in the English literature BLDC or PMSM) are also
called brushless DC motors, because the controller of such an engine is
usually powered by DC voltage.
This type of motor is designed to improve the properties of DC motors.
High requirements for actuators (in particular, high-speed micro-drives
precise positioning) led to the use of specific DC motors: brushless threephase DC motors (BDP or BLDC).
Structurally, they resemble synchronous AC motors: the magnetic
rotor rotates in a charge stator with three-phase windings. But rpm is a
function of the load and voltage on the stator. This function is implemented
by switching the stator windings depending on the rotor coordinates.
There are plug-in versions with separate sensor on the rotor without
separate sensors. Hall sensors are used as separate sensors. If the
execution without separate sensors, the stator windings act as a fixing
element. When the magnet rotates, the rotor induces EMF in the stator
windings, resulting in a current. When one winding is turned off, the signal
that was induced in it is measured and processed. This algorithm requires
46
a signal processor. For braking and reverse BDPS do not need a bridge
circuit reverse power supply - enough to apply the control pulses to the
stator windings in reverse sequence.
In the valve motor, the inductor is on the rotor (in the form of
permanent magnets), the armature winding is on the stator (synchronous
motor). The supply voltage of the motor windings is formed depending on
the position of the rotor. If the DC motors for this purpose was used the
manifold, the valve engine it is the function of the semiconductor switch
(DPR inverter).
The main difference between the VD and the synchronous motor is its
self-synchronization with the help of DPR, as a result of which the VD, the
rotation frequency of the field is proportional to the rotor speed.
The principle of operation of the VD is based on the fact that the VD
controller commutes the stator windings so that the magnetic field vector
of the stator is always orthogonal to the magnetic field vector of the rotor.
With the help of pulse width modulation (PWM), the controller controls the
current flowing through the windings of the VD, i.e. the vector of the
magnetic field of the stator, and thus regulates the torque acting on the VD
rotor. The sign at the angle between the vectors determines the direction
of the moment acting on the rotor.
C in the calculation of the electric. They are smaller than the geometric
degree to the number of pairs of poles of the rotor. For example, in a VD
with a rotor having 3 pairs of poles, the optimal angle between the vectors
will be 90°/3 = 30° Switching is performed so that the excitation flow of the
rotor — F0 is maintained constant relative to the anchor flow. As a result
of the interaction of the flow of the armature and excitation, a torque M is
created, which tends to expand the rotor so that the flow of the armature
and excitation coincide, but when the rotor rotates under the action of the
DPR, the windings switch and the armature flow turns to the next step.
In this case, the resulting current vector will be shifted and stationary
relative to the rotor flow, which creates a moment on the motor shaft.
In the motor mode, the MDS stator is ahead of the MDS rotor by an
angle of 90°, which is supported by the DPR. In the brake mode of the MDS
47
MDS stator behind the rotor, an angle of 90° are also supported with the
help of the DWP.
Consider the mathematical model of a brushless motor with
permanent magnets. The motor rotor with permanent magnets usually
has from two to sixteen pairs of poles. We investigate a three-phase
brushless motor with four pairs of poles.
The mathematical model of the engine is based on the following
conditions:
• The voltage on the windings of the armature:
(2.19)
• EMF is taken in the form of a rotor position function and has an
ideal trapezoidal shape, where
(2.20)
Consider the dependence of the inductance of the angle of rotation
of the rotor in figure 2.3:
Figure 2.3 – the Laws of change of phase inductance
When the rotor is rotated due to its apparent polarity, the magnetic
resistance between the rotor and the stator changes, which in the model
is taken into account by an equivalent change in the inductance of the
stator windings, i.e. the inductance in the model depends on the angle of
rotation.
(2.22)
48
(2.23)
(2.34)
Assume that the inductance of the stator windings will be determined
by the expressions (2.22), (2.23) and (2.24), where is the coefficient of
dependence.
2.3. Formation of the General model of the exoskeleton
2.3.1. Features of mechanics and electric drives of the
exoskeleton robot
Despite the fact that the gearless drive has advantages for use in
exoskeletons, but, on the other hand, leads to the fact that the
imperfections of the engine begin to affect the movement of the links of
the exoskeleton.
First of all, such imperfections in yavnopolyusnyh engines include
changing the parameters of mutual inductance rotor and stator
depending on their mutual angular position q, as shown in figure 2.4).
49
Figure 2.4 – Diagram of the stator and rotor of a permanent magnet
motor
Thus, the parameters of electromagnetic processes in the object
depend on the configuration of the mechanical structure. In our case, we
present this dependence as written in the formula (2.25):
, (2.25)
where
э
– electric angle of rotation of the engine (considering the
number of couples poles'.)
In the absence of a reducer, this dependence can lead to unacceptably
large fluctuations in the drive torque when the robot is moving. In General,
the construction of a high-accuracy model requires a joint consideration of
mechanical and electromagnetic processes from the very beginning of the
model formation.
50
Figure 2.5 Interconnection of the mechanical design of the robot and
electric drives of its degrees of mobility.
Figure 2.5 schematically shows the kinematic and dynamic
relationship of the mechanical structure and drives, where BLDC –
brushless DC motor, Power inverter – power Converter, Controller –
controller.
2.3.2. Building a common model
Let us introduce the vector of generalized coordinates of
mechatronic systems:
= [
м
;
Э
] consisting of a mechanical
м
and electric
Э
variables.
The Lagrange–Maxwell equations have the form presented in the
formula (2.27) [7, 9]:
(2.27);
is a function of Lagrange
– Maxwell; = 1, : – the rotation angle of the i-th link;
= – the
moment of forces of the i-th link; = ,2:
= – the total current of the i-
th motor;
= – electric voltage.
At
Y
X
O
BLDC
MOTOR
BLDC
MOTOR
Power
inverter
Power
inverter
Controller
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