SIMULATION OF HEXAPOD WITH A VECTOR-CONTROLLED HYBRID STEPPER MOTOR

Yuri Zhukov, Nikita Slobodzyan

VOENMEH Baltic State Technical University named after D.F. Ustinov1-th Krasnoarmeyskaya str., 1, 190005 St. Petersburg, Russia

E-mail: zh_kv@mail.ru

Abstract This paper proposes an algorithm that solves the inverse kinematics for hexapod with six legs having universal joints at the top and bottom, and screw joints in the stepper drives transmission. In this paper a numerical model of hexapod based on linear drives with vector-controlled stepper motor is presented. The description covers the main components of the model, namely a block hexapod mechanics, an electromagnetic subsystem stepper motor, a model of elasto-plastic friction, and vector-controlled stepper drives. The complex models are implemented in MATLAB® and SIMULINK®. The result of control modes is verified on a numerical example. Conclusions are drawn based on the simulation results for hexapod with linear stepper drives.

INTRODUCTION VOENMEH BSTU (Baltic State Technical University) and JSC ISS named after M.F. Reshetnikov have been performing a team-work study in creating a number of multi-stage mechanisms with parallel kinematics to ensure precise positioning and stabilisation of on-board instrumentation and devices for aerospace application. Successful creation of the said systems is impossible without preliminary computer modelling that allows making qualitative assessment of any structural, functional, or algorithmic decisions occurring in the course of the design process.

1 HEXAPOD KINEMATICS Hexapod as a positioning system with parallel kinematics has many advantages over serial kinematics stages, such as lower inertia, improved dynamics, smaller package size and higher stiffness. The hexapod positioners are often referred to as Stewart Platforms [1, 2]. Hexapod (as demonstrated in Figure 1) is based on a system having six linear actuators with stepper motor arranged in parallel between the top 1 and bottom 2 platforms. The top platform has six degrees of freedom with regard to the bottom platform. Each leg has the function of a linear drive and consists of two lower and upper legs 3 и 4 coupled by universal joints with the base and platform. In the

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lower leg, there is a stepper (step drive) 5 responsible for the legs’ displacement. Translatory and rotational degrees of freedom between the upper and lower legs are ensured by a telescopic joint.

Figure 1 – Hexapod design sketch

The ‘screw-nut’ pair transforms rotation angle of the drive drα , i.e. the screw position

relative to the lower leg, as it is shown in Figure 2, into the linear displacement l, i.e. the leg’ length.

Figure 2 – Kinematic diagram of the screw-nut pair

Kinematic diagram of each hexapod leg is given in Figure 3. Each leg consists of the lower and upper legs coupled by the universal joint based of the screw-nut pair with the generalised coordinates l and γ. The lower leg is connected with the hexapod base by the joint A with the generalised coordinates – angles , and a aα β , while the upper leg is

connected to the mobile platform by the biaxial joint B with the generalised coordinates – angles , and b bα β .

Figure 3 – Kinematic diagram of the hexapod leg

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The required position and orientation of the hexapod’ mobile platform is pre-set by the

vector having six coordinates [ ]T , , , , , X Y Z ϕ θ ψ=q , where: X,Y,Z are Cartesian

coordinates; , , ϕ θ ψ are orientation angles of the mobile platform (for instance, Euler angles). The leg’ length is calculated in the course of the inverse kinematics solving [3] based on the expression

a bL P P= − , (1)

where [ ]T, ,a a a aP x y z= and [ ]T

, ,b b b bP x y z= are positions of the universal joints A and

B centres calculated according to the pre-set position of the mobile platform orientation. Moreover, knowing the initial orientation of the biaxial joints A and B, it is possible to compute rotational angles of the universal joints α ,β ,α ,βa a b b and the lower and upper

legs’ axial rotation angleγ . So, to calculate the required complete rotational angle of the screw pair’ nut we shall get

02 ,idr i i

L L

hα π γ−= − i=1...6, (2)

where iL is the required length of i-leg corresponding to the pre-set position and

orientation of the hexapod’ mobile platform; 0L is length of the leg in ‘zero’ position;

h is the screw pair step; andiγ is the axial rotation angle.

In the MATLAB® mathematical modelling environment the software has been developed to implement hexapod’ inverse and direct kinematics problem solution. In the course of the studies, ANSI / ISO compatible C and C++ code of hexapod kinematics solutions has been developed to be used in creating a software for the specific hardware-software platform. Solution of the hexapod kinematics problems has been produced using microcontroller of the STM32F103x family on the basis of ARM Cortex M3 architecture. Microcontrollers of the 1986ВЕ9x family manufactured by Milandr Company are the domestic analogues of the above-mentioned microcontroller.

2 HEXAPOD DYNAMICS MODEL

A hexapod simulation model including stepping motor control model and hexapod mechanics model considering frictions model has been developed. The mathematical model of an electromagnetic system of a hybrid stepper with two motor winding is recorded in the state-space as a system of the following non-linear differential equations [4]:

( )( )

( ) ( ) ( )

sin

cos ,

sin cos sin 2

a a a m

b b b m

e m a m b dm

LI U I R K p

LI U I R K p

T K I p K I p T p

ω θ

ω θθ θ θ

= − + = − − = − + −

ɺ

ɺ (3)

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where aI , bI are currents in the stator windings; aU , bU are input voltages on the

stator windings; R is the stator winding resistance; L is the stator winding induction; mK is the torque constant; p is the number of the pole pairs; θ is the

motor shaft angular position; ω is the motor shaft rotation speed; dmT is the residual

motor breaking (drag) torque; eT is the output electromagnetic torque created by the

electric motor on the shaft. Through applying Park transformation to the measured currents in the motor windings

aI , bI , we shall get currents in the d-q coordinates space

sin( ) cos( ).

cos( ) sin( )d a b

q a b

I I p I p

I I p I p

θ θθ θ

= − + = +

(4)

In the d-q space, the desired current value in d coordinate makes Irefd=0, while the

required current value in q coordinate is formed by a controller based on the control error. The desired current values are used for calculating the

desired phase voltages in the current controllers, Urefd and Uref

q, while applying the inverse Park transformation we shall get values of phase voltages Uref

b and Urefb. The

driving voltages in the motor windings are formed by pulse width converters. To achieve fine kinematic precision in linear drive design, backlash take-up mechanisms are used, which results in generation of substantial drag forces (friction) in motion. Thus, for instance, practical experience gained at creating breadboard models of linear drives proved the drag force ability at the motion start (‘start-up’) to reach 20-30 percent of the rated force developed by the linear drive. It is proposed to simulate the said effect using the model of elasto-plastic friction [5] defined by the equations:

( )( )

0 1 2 ,

,1

tr

ss

F z z x

z x zz x

f x

σ σ σ

α

= + +

= −

ɺɺ

ɺɺɺ

ɺ

, (5)

where xɺ is the object’ linear velocity; z is elastic deformation of the ‘fibres’; 0σ is

stiffness of deformation; 1σ is the dissipation factor; 2σ is viscous friction

coefficient; ( )ssf xɺ is Stribeck function approximated by the equation

( ) ( ) s

x

vss c ba cf x f f f e

−= + −

ɺ

ɺ , (6)

where cf the ‘dry’ friction is force; baf is breakaway force; vs is threshold breakaway

speed; ( ),z xα ɺ is the function of sideslip threshold defined based on the system

( )ref refq p dI K Kθ θ θ= − − ɺ

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( )

0 ,

( )

20.5sin 0.5,

,

1,

0, ( ) ( )

ba

ss ba

ba ssss ba

ss

at z z

z zz

at z z zz x z z

at z z

at sign x sign z

π

α

≤

+ − + < ≤= −

>

≠

ɺ

ɺ

, where (7)

zba- breakaway threshold; zss – sideslip threshold. The model allows calculating the total friction force trF in the linear drive based on the

signal – speed of the hexapod leg length change – sent to the input port. Based on the equations (3-7), the model of dynamics of hexapod leg’ linear drive control system shown in Figure 4 was realised in MATLAB® and SIMULINK® environment using the objects of SimMechanics’ toolbox of mechanic systems modelling.

Figure 4 – Model of hexapod leg’ linear drive

In the model, linear drive force Fe is formed as a result of the ideal transformation of the stepper’ electromagnetic torque Те by the ‘gear – screw – nut’ transmission:

e w eF K T= ⋅ , where

2w pK i

h

π=is the linear drive’ transmission coefficient, where h is

the screw pair step; pi is the reduction ratio.

Angle θ and angular velocity θɺ of the stepper shaft rotation are related to the leg’ linear displacement by the equations:

xɺ

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,

w w

w

dL dxK K

dt dtK L

dL dxL

dt dt

θ ω

θ

= = ⋅ = ⋅

= ⋅ = =

ɺ

ɺ

(8)

where L is the leg length; Lɺ is the drive’ linear velocity.

Figure 5 – Hexapod control system model structure

The required extension of the Legs_ref hexapod’ linear drives is delivered to the system’ input, while at the output, the platform position is indicated – vector of X, Y, Z coordinates and the platform orientation defined by the R turn matrix used as a basis for calculating the platform orientation angles.

3 HEXAPOD SIMULATION

The modelling object is assessment of the required precision of the hexapod positioning control. The hexapod control system must perform positioning control of the mobile system accompanied by calculation of the setting actions for the linear drives based on the inverse kinematics problem solution [3], and the control error is minimised according to the feedback transducers’ signals. In compliance with the requirements of the technical specifications for the precision hexapod design, the linear positioning accuracy should be within ±0.01 mm, while that of the angular positioning is to be within ±30 angular sec. Input data for the hexapod model: Hexapod having the following mechanical parameters is analysed: initial height h=0.4 m; base diameter – 0.4 m; platform diameter – 0.3 m; interval between the attachment points of the adjacent joins on the base and the platform – 0.05 m; the platform mass including inertia load – 100 kg; the principal moments of inertia of the platform – Jxx=4900 kg.m2, Jyy=4900 kg.m2, Jzz=6300 kg.m2. Mass moments of inertia parameters for lower and upper legs are calculated through presenting them as cylinders, the data on the material density and the upper and lower legs’ dimensions being available.

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( – , – , – )dx X X dy Y Y dz Z Z= = =r r r

The stepping motor with the rated parameters of FL57STH56 Model: p=50; R=1.8 Ohm; L=0.0025 H, Km=0.5 Nm/A, U=30 V, Tdm=0.05 Nm. Transmission factor of the gear with screw pair –Кw=1.57.105 rad/m. Parameters of the model of the linear drive’ total mechanical friction:

7 30 1 2

0 0

5 10 / , 3 10 sec/ , 10 sec/ ,

20 N, 30 N, 0.05 / sec, 0.7 / , 1.2 / .c ba s ba c ss c

N m N m N m

f f v m z f z f

σ σσ σ

σ= ⋅ = ⋅ ⋅ = ⋅= = = = =

Figures 6 and 7 demonstrate the results of hexapod modelling at vector control of the steppers. Figure 6 shows errors typical for the hexapod legs’ length control, while Figure 7 shows positioning errors and

angular errors in the course of the platform position control at the setting action

[ ]TT , , , , , 0.0001 , 0.0001 , 0.4001 , 0.035º , 0.035º , 0.0[ ] 35ºX Y Z m m mϕ θ ψ= = − − −r r r r r r rq

and the hexapod’ initial ‘zero’ state T[0,0,0.4 ,0,0,0]m=q .

Figure 6 – Errors at hexapod legs control

Figure 7 – Kinematic errors of hexapod position control

( , , )d d dϕ ϕ ϕ θ θ θ ψ ψ ψ= − = − = −r r r

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The modelling results prove successful adjustment of the setting action for hexapod position control at vector control of linear drives’ steppers.

CONCLUSIONS

Thus, in the MATLAB® and SIMULINK® mathematical modelling environment, the simulating model has been developed that allows making the following: to assess modes of control of the hexapod with linear drives based on the stepping motors (precision, threshold velocities), as well as effect of the hexapod’ design parameters, inertia load, and friction non-linear forces on the transition processes quality; to choose the preferred solutions in the course of the hexapod designing. The study is used directly in the design of the actual hexapod system at the research laboratory of robotic and mechatronic systems at the VOENMEH Baltic State Technical University named after D.F. Ustinov.

REFERENCES

1. Gough, V.E. Universal tire test machine / V.E. Gough, S.G. Whitehall // In

Proceedings 9th Int. Technical Congress F.I.S.I.T.A. – 1962. – V.117. – pp. 117-135. 2. Stewart, D.A platform with six degrees of freedom / D. Stewart //

Proceedings of the Institution of mechanical engineers. – 1965. – V.180, рt.1, №15. – pp. 371-385.

3. V.Gavrilenko, Y.Zhukov, A.Moroz Kinematics Solutions on the ARM microprocessor for mechatronic control systems of hexapods // Voprosy radioelektroniki 2016, V.6, p.92 – ISSN 2218-5453.

4. Takashi Kenjo Stepping Motors and Their Microprocessor Controls (Monographs in Electrical and Electronic Engineering) - Oxford University Press, 1984,244p.

5. P.Dupont, B.Armstrong, V.Hayward Elasto-Plastic Friction Model: Contact Compliance and Stiction // Proceedings of the American Control Conference, Chicago, Illinois, June 2000.

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