Modeling and simulation of a hybrid actuator

Modeling and simulation of a hybrid actuator

Mechanism and Machine Theory 38 (2003) 395–407 www.elsevier.com/locate/mechmt Modeling and simulation of a hybrid actuator L.C. D€ ulger a, Ali Kirec...

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Mechanism and Machine Theory 38 (2003) 395–407 www.elsevier.com/locate/mechmt

Modeling and simulation of a hybrid actuator L.C. D€ ulger a, Ali Kirecßci a

b,*

, M. Topalbek_ıroglu

b

Department of Mechanical Engineering, University of Gaziantep, Gaziantep 27310, Turkey b Department of Textile Engineering, University of Gaziantep, Gaziantep 27310, Turkey

Received 31 January 2001; received in revised form 4 September 2002; accepted 9 October 2002

Abstract Modeling and kinematic analysis of a hybrid actuator is presented in this study. The dynamic behavior of hybrid actuator is studied by applying numerical simulation on the whole system. Lagrangian mechanics is applied to derive equations of motion. Simulation results are presented to demonstrate the ability of model developed with PID controller action. Similar following characteristics are obtained from numerical simulation, and the performance of the control system with PID controller is quite effective. In hybrid actuator configuration, a seven link mechanism having an adjustable crank is used. One input of the mechanism is given by a gearbox coupled to brushless DC servo motor, and the second input having an adjustable crank is provided by a permanent magnet DC servo motor coupled to a lead screw mechanism. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction A hybrid actuator is a configuration that combines the motions of two characteristically different electric motors by means of a mechanism to produce inexpensive programmable output. Where one of the motion coming from a constant speed motor provides the main power, a small servo motor introduces programmability to the resultant actuator. Hybrid actuator may be considered as an improving alternative to the direct drive servo motor, because servo motor driven systems may have some serious drawbacks. Initial investment of servo systems is relatively much more higher despite developments in electronic industry. The other drawback is higher power requirements, since such systems are not energy regenerative ones such that some of the energy will be converted to the heat energy, and transferred environment through the fins. The idea of hybrid machines is a field of study with full potential. Demand for greater machine productivity with improved quality, diversity of product, competition on market, and industrial *

Corresponding author. Tel.: +342-360-1200/2512; fax: +342-360-1100.

0094-114X/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0094-114X(02)00129-5

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automation have accelerated needs for new alternative ideas like hybrid machines to generate programmable output motion. Such machines will introduce to users greater flexibility with programmability option, and energy utilization will be realized at maximum. Some points are partially explored, but there is still a need for more studies to guide potential users for possible industrial applications with hybrid machines. Cut to length machine, printing machine, and stamping press are the industrial examples where hybrid machine idea has already been used with different hybrid machine configurations. Earlier research in kinematics and dynamics of hybrid machine configurations can be found in some studies and publications. Tokuz [1] has used a hybrid machine configuration to produce a reciprocating motion. A slider crank mechanism was driven by a differential gearbox having two separate inputs; constant speed motor and a pancake servo motor to simulate a stamping press. A mathematical model was developed for the hybrid machine-motor system. The model results were compared with the experimental ones, and model validation was achieved. Later Kirecßci and D€ ulger [2] have proposed a different hybrid arrangement driven by two permanent magnet DC (PMDC) servo motors and a constant speed motor. In the configuration, a slider crank mechanism having an adjustable crank was used together with two power screw arrangements to produce a motion in x–y plane. A mathematical model was prepared and the simulation results were presented for different motion requirements including electrical drive properties. Conner [4] has studied on the synthesis of a hybrid mechanism, also performing kinematic analysis. D€ ulger [5] was studied dynamic behavior of brushless DC (BLDC) and PMDC servo motors with changing load for the purpose of understanding these electrical drive systems in detail. A mathematical model representing the motor-load structure was prepared, performance characteristics of two motors were explored with the experimental results obtained. Greenough et al. [3] have later presented a study on design of hybrid machines, and a Svoboda linkage is considered as a two degree of mechanism. Kinematic analysis of Svoboda linkage was presented with inverse kinematics issues. Kirecßci [6] has then offered a hybrid drive system involving a servo motor and a constant speed motor for a printing machine. Lastly, Kirecßci and D€ ulger [7] had given description of a different hybrid actuator configuration consisting of a servo motor driven seven link mechanism with an adjustable crank is starting point of this study presented here. The experimental setup and experimental results were included in [7]. The main purpose of the paper is to present kinematic analysis and investigate the dynamic behavior of hybrid actuator by deriving its mathematical model. Lagrangian formulation is performed. The differential equations of the system are obtained. The resulting nonlinear dynamic equations are solved by using fourth order Runge Kutta integration. Simulation results are obtained for two different motion profiles. The optimum setting for the controller gains are achieved by running the simulation with real mechanism parameters and motor datas. So the integration of mechanical and electrical elements is completely achieved in this system herein.

2. Hybrid actuator description Fig. 1 represents seven link mechanism configuration having all revolute joints except one slider on link 4. Fig. 1a and b show the positional and the geometrical relationships respectively. Notations shown in Fig. 1a and b are applied throughout the study. The hybrid actuator can be

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397

Fig. 1. Configuration of seven link mechanism.

considered here as two four bar mechanisms coupled together in which the second four bar mechanism has an adjustable crank designed to include a power screw mechanism for converting rotary motion to linear motion by means of a small slider. The slider is assumed to move on a frictionless plane. The crank (input link––notated by a2 , h2 ) is driven by BLDC servo motor through a 1/30 reduction gearbox. The slider (r) is driven by a lead screw coupled PMDC servo motor. Here BLDC servo motor is applied as a constant speed motor for the purpose of easiness during measurements, and the constant speed motor motion profile is applied. Point-to-point positioning is certainly achieved for both DC servo motors, and the system output is taken from the last link (notated by a7 , h7 ). a1 , a2 , a3 , a4 , a5 , a6 , a7 link lengths of the mechanism h1 , h2 , h3 , h4 , h5 , h6 , h7 angular displacement of the links including the ground link (rad) h_2 , h_3 , h_4 , h_6 , h_7 angular velocity of the links (rad/s) h€2 , h€3 , h€4 , h€6 , h€7 angular acceleration of the links (rad/s2 ) r2 , r3 , r4 , r6 , r7 positions to the centre of gravity (mm) m2 , m3 , m4 , m6 , m7 masses of the links (kg) j2 , j3 , j4 , j6 , j7 link moment of inertias (kg mm2 )

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Table 1 Mechanical properties of seven link mechanism a1 ¼ 375 mm a5 ¼ 375 mm r2 ¼ 19 mm r7 ¼ 45 mm m2 ¼ 3:9 kg m7 ¼ 25:186 kg j2 ¼ 12300 kg mm2 j7 ¼ 131500 kg mm2 P ¼ 1 mm

a2 ¼ 102 mm a6 ¼ 375 mm r3 ¼ 218 mm

a3 ¼ 375 mm a7 ¼ 120 mm r4 ¼ 145 mm

a4 ¼ 167 mm r6 ¼ 178 mm

m3 ¼ 1:385 kg mr ¼ 0:425 kg j3 ¼ 24900 kg mm2 jr ¼ 263 kg mm2

m4 ¼ 13:295 kg

m6 ¼ 1:1 kg

j4 ¼ 34300 kg mm2 jg ¼ 15000 kg mm2

j6 ¼ 18700 kg mm2

Table 2 Brushless and PM servo motor data Motor types Rotor moment of inertia (kg m2 ) Motor torque constant (N m/A) Motor voltage constant (V/krpm) Winding inductance (mH) Winding resistance (X)

BLDC servo motor Jm1 ¼ 6:8  104 Kt1 ¼ 0:76 Ke1 ¼ 90 L1 ¼ 5:8 R1 ¼ 0:8

PMDC servo motor Jm2 ¼ 1:3  103 Kt2 ¼ 0:63 Ke2 ¼ 38 L2 ¼ 1:1 R2 ¼ 0:45

jm1 , jm2 , jg motor and gearbox moment of inertias (kg mm2 ) mr , jr mass and moment of inertia of the slider (kg, kg mm2 ) P the pitch of the lead screw (mm) r, r_ , €r displacement, velocity, and acceleration of the slider on link 4 (mm, mm/s, mm/s2 ). Mechanical properties of seven link mechanism; link lengths, positions to the centre of gravity of each link, link masses, and link inertias are shown in Table 1. BLDC and PMDC servo motor data taken from motor manufacturers catalogues [8] are given in Table 2. 3. Kinematic analysis Kinematic analysis of seven bar linkage is needed before carrying out derivations for the mathematical model. The mechanism is shown with its position vectors in Fig. 2. Position analysis is performed by applying complex number method by using loop 1 and loop 2 one by one.

Fig. 2. Schematic diagram of seven link mechanism.

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By referring to ! ! AB þ BC ¼ ! ! ED þ DF ¼

Fig. 2, the loop closure equations is written as: ! ! CE þ EA ! ! FG þ GE

399

ð1Þ ð2Þ

The above equations may be written in complex polar notation: a2 eih2 þ a3 eih3 ¼ a4 eih4 þ a1 eih1

ð3Þ

reih4 þ a6 eih6 ¼ a7 eih7 þ a5 eih5

ð4Þ

Intermediate steps for the kinematic analysis can be seen in Appendix. Implicit relations necessary for h3 and h4 , h6 and h7 are found using Freudenstein equation [9]. By solving first vector loop equation (1), angular positions of the link 3 and 4 are obtained. Similarly, solution of second loop equation has given the angular positions of link 6 and link 7. Having found the angular displacements of each linkage as h3 , h4 , h6 , and h7 in the seven bar linkage, time derivatives can be taken to find angular velocity and accelerations. They are definitely needed during the analysis of dynamic model. Motor input angles and velocities are needed to be used as reference motion during simulation. The output of system is dependent on two seperate motor inputs and the geometry of seven bar mechanism. Referring to Fig. 2, the output is given by h7 , the configuration represents in-line mechanism, and h1 ¼ h5 ¼ 0°, they have no effect in derivations since the ground is fixed. Initially the output motion profiles h7 , h_7 , h€7 are designed for the system, called as Motions 1 and 2 by using a motion design package called MOTDES [10]. Motion 1 is designed to perform simple motion character involving only rise and return without any dwell. However, Motion 2 is designed to achieve more difficult output with dwell portions introducing heavier changes in motion. Inputs necessary from the crank (h2 ) and the slider (r) are then calculated by using inverse kinematics. The crank input is later converted by using coupling ratio introduced, BLDC servo motor input (h2m ) is found. The motion of slider (r) is then converted into PMDC servo motor input (hs ) using lead screw relation.

4. Mathematical model of hybrid actuator The complete mathematical model of the hybrid actuator is characterized by highly coupled and nonlinear second order differential equations. Simplifying assumptions are required while developing the mathematical model. Friction in all joints and transmission losses (gearbox etc.) are neglected. The mechanism operates in vertical plane and gravity effects are included. LagrangeÕs equations [11–13] are used while developing of the equations of motion of the system. 4.1. Dynamic system equations In general, the model of a motor and a mechanical system can simply be considered as inertias rigidly coupled to the motor shafts. Through the gearing between BLDC servo motor and the input link-the crank, the inertias reflected on the motor shaft are found [12–14]. The angular

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displacement of BLDC servo motor is notated by h2m and the angular displacement of the crank is given by h2 . The motion of BLDC servo motor to the motion of crank is defined as coupling ratio: N¼

h2m h2

ð5Þ

Similarly, the angular displacement of PMDC servo motor is notated by hs . The relation between the angular displacement of PMDC servo motor and the linear displacement of slider, r (link 5) is given for a lead screw: hs ¼

2p r P

ð6Þ

The electrical equation of a BLDC and PMDC motor [14,15] is: Lk

dIk þ Rk Ik þ Kek h_n ¼ Vk dt

ð7Þ

where Lk is the motor inductance, Rk is the motor resistance, Kek is the motor voltage constant, Ik is the current passing through the armature, and Vk is the voltage applied to the armature. Here k is taken as an index 1 and 2 for BLDC and PMDC servo motors respectively. hn is replaced by two generalized coordinates; h2m and hs during simulation. The mechanismÕs total kinetic energy T is defined by T ¼ 12 Jm1 h_22m þ 12 Jg h_22m þ 12ðJ2 þ m2 r22 Þh_22 þ 12ðJ3 þ m3 r32 Þh_23 þ 12ðJ4 þ m4 r42 Þh_24 þ 12mr r_ 2 þ 12 Jm2 h_2s þ 12ðJ6 þ m6 r62 Þh_26 þ 12ðJ7 þ m7 r72 Þh_27

ð8Þ

The total potential energies V due to gravity are defined by V ¼ m2 gr2 sin h2 þ m3 gða2 sin h2 þ r3 sin h3 Þ  m4 gr4 sin h4 þ mr gr sin h4 þ m6 gðr sin h4 þ r6 sin h6 Þ þ m7 gr7 sin h7

ð9Þ

The Lagrangian function, L ¼ T  V where T and V are expressed in terms of the generalized coordinates. Two angle coordinates are necessary for the complete definition of the system. LagrangeÕs equation can be written as   d oL oL ¼ Qk ð10Þ  dt oq_ k oqk where k ¼ 1; 2 representing each degree of freedom and Qk representing the generalized forces or torques on the system. Since the hybrid actuator is a two degree of freedom system, the generalized coordinates are the positions of BLDC (link 2) and PMDC servo driven links (link 4), q1 ¼ h2m , q2 ¼ hs . The joints are rotational and actuated by electrical servo motors, the Lagrangian dynamic model becomes a torque balancing equation for each axis. The resulting equations of motion are decoupled and they are given as the following:

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The equation of motion for the first input axis: " !# " !#    _3 _4 m2 r22 þ J2 o h o h h€2m þ ðm3 r32 þ J3 Þ Jm1 þ Jg þ h€3 þ ðm4 r42 þ J4 Þ h€4 N2 oh_2m oh_2m " !# " !# " !# _6 _7 _3 o h o h d o h h€6 þ ðm7 r72 þ J7 Þ h€7 þ ðm3 r32 þ J3 Þh_3 þ ðm6 r62 þ J6 Þ dt oh_2m oh_2m oh_2m " !# " !# " !# _4 _6 _7 d o h d o h d o h 2 2 2 þ ðm6 r6 þ J6 Þh_6 þ ðm7 r7 þ J7 Þh_7 þ ðm4 r4 þ J4 Þh_4 dt oh_2m dt oh_2m dt oh_2m          h2m 1 h2m 1 oh3 oh4 þ m2 gr2 cos  m4 gr4 cos h4 þ m3 g a2 cos þ r3 cos h3 N N N N oh2m oh2m          oh4 or oh4 oh6 þ mr g sin h4 þ m6 g r cos h4 þ r6 cos h6 þ mr gr cos h4 oh2m oh2m oh2m oh2m   oh7 ¼ Qm1 ðt; h2m ; h_2m Þ ð11Þ þ m7 gr7 cos oh2m The equation of motion for the second input axis: "   # !# !# " " 2

2 oh_6

2 oh_7 P þ Jr þ Jm2 h€s þ m6 r6 þ J6 h€6 þ m7 r7 þ J7 h€7 mr 2p oh_s oh_s " !# " !#   _6 _7 d o h d o h or 2 2 _ _ þ ðm7 r7 þ J7 Þh7 þ mr gr sin h4 þ ðm6 r6 þ J6 Þh6 dt oh_s dt oh_s ohs        oh4 oh4 oh6 þ mr gr cos h4 þ m6 g r cos h4 þ r6 cos h6 ohs ohs ohs   oh7 ð12Þ ¼ Qm2 ðt; hs ; h_s Þ þ m7 gr7 cos h7 ohs The generalized torques developed by the servo motors are given by: Qm1 ðt; h2m ; h_2m Þ ¼ Kt1 I1

ð13Þ

Qm2 ðt; hs ; h_s Þ ¼ Kt2 I2

ð14Þ

Where Kt1 and Kt2 are the motor torque constants for BLDC and PMDC servo motors respectively. 4.2. Controller action used in model Several controller actions may be applied to servo control systems to obtain accurate position tracking while providing system stability. Many studies can be found on applications using proportional plus derivative and proportional plus derivative plus integral (PID) controllers on servo systems [16,17]. PID controller utilizes a control signal involving position, velocity errors and the integral of position error function. Here the objective is to establish PID control action to see how it can improve the transient and steady-state responses of the system. Actually PID

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controller action is chosen to make similarities between the experimental set-up given in [7]. A motor control card simulating PID action is used with some modifications in control software. So the motor armature voltage given in Eq. (7) can be calculated using PID controller as Z t eðtÞ dt ð15Þ Vk ¼ Kgk eðtÞ þ Kvk e_ ðtÞ þ Kik 0

where Kgk , Kvk and Kik represent the proportional, derivative and integral gains for the controller, error functions are eðtÞ ¼ Kgk ðhnc  hn Þ and e_ ðtÞ ¼ Kvk ðh_nc  h_n Þ

ð16Þ

representing the controller outputs. hnc and h_nc represent the reference motion profiles found from inverse kinematics studies referring to Motions 1 and 2. Angular positions, velocities, and accelerations to be followed bv each motor are notated by using subscripts as 2m and s. PID controller action is used while calculating armature voltages of DC servo motors, and Ziegler– Nichols tuning rules are applied to obtain the optimal PID values referrring to the experimental set-up used [7]. Since there is integral action, a steady-state error is expected to be reduced. But there can be still steady-state error due to variations in the load and high nonlinearity introduced in the model equations. The derivative action simply adds damping into the system and reduces the maximum percent overshoot during transient time response. 4.3. Solution of system equations of motion Explicit forms of Eqs. (11) and (12) are lengthly formulas in terms of time and partial derivatives, they are not included throughout the paper because of space limitation. The servo motordrive-mechanism system is completely described by Eqs. (7), (11)–(14). Eqs. (7), (11) and (13) represent the first axis, whereas Eqs. (7), (12) and (14) give the second axis. Eq. (13) is substituted into Eq. (11), and Eq. (14) is substituted into Eq. (12) representing the generalized torques. Thus Eqs. (7) and (11) represent electrical and dynamic equations for the first axis, and Eqs. (7) and (12) give the electrical and dynamic equations for the second axis. Finally there are one first order and one second order nonlinear differential equations for each axis. State space form of the system equations are required for solution. By putting Eqs. (7), (11) and (12) for each motor into the state variable form, six first order differential equations are resulted. Numerical solution techniques are required for their integration. Fourth order Runge Kutta method is then applied to integrate the systems of differential equations. A computer program performing kinematic analysis of hybrid actuator and solving model equations is prepared in Pascal. The dynamic coupling of seven link mechanism with BLDC servo motor with a reduction unit and a PMDC servo motor represent a highly nonlinear problem. All system equations are to be solved together because of the coupling of axes. In the program, the system parameters and initial conditions are to be specified. First motor electrical datas, mechanism parameters; link lengths, link masses, link positions to the center of gravity, link moment of inertias, and the controller gains representing PID control are given as input parameters. The output is the current, the angular displacement, the angular velocity and the angular acceleration of both motors.

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5. Simulation results The servo motors operate under continuously varying inertias. Through the gearing (between BLDC servo motor and the crank), the mechanism drive requirements are reflected on the motor shaft. Several motion profiles are studied, however simulation results are carried out by using these two motions. Both motion profiles are performed in 3 seconds. During simulation optimum setttings for the controller gains are established by trial and error. Initially, the proportional gain is set, the derivative gain is then added to smooth out the motor responses obtained. Finally the integral gain is tried to be set. So angular displacement and velocity curves for BLDC and PMDC servo motor are satisfactorily simulated with a real feedback effect (PID). Fig. 3a–d represent the simulation results for Motion 1. Fig. 3a and b show the angular displacement and the angular velocity for BLDC servo motor. This motor rotates at 600 rpm, and introducing 1/30 reduction unit, the crank has 20 rpm at the end. The constant speed motor data representing its angular displacement and velocity are read from a data file created by the user. While designing the motion r, r_ , €r are taken into consideration, and these points are converted into hs , h_s , h€s by using Eq. (6), they are made ready before simulation. The proportional, derivative, and integral gains used to get this simulation are Kp1 ¼ 8, Kv1 ¼ 0:75 Volt/rad/s,

Fig. 3. Simulation results for BLDC and PMDC servo motors for Motion 1.

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Fig. 4. Simulation results for BLDC and PMDC servo motor for Motion 2.

Ki1 ¼ 1:0 V/s respectively for the first axis (BLDC servo motor). Later, these results are multiplied by gear ratio to get the angular displacement and velocity of the crank. Fig. 3c and d show the angular displacement and the angular velocity for PMDC servo motor driving the slider on the lead screw. The proportional, derivative, and integral gains used for this motor are Kp2 ¼ 70:0, Kv2 ¼ 0:5, and Ki2 ¼ 1:0. Acceptable speed response curves are obtained. Similarly, the designed motion, Motion 2 is loaded from another data file for PMDC servo motor. Fig. 4a, b, c, and d show the simulation results for Motion 2 which is more difficult to implement compared to Motion 1. Fig. 4a and b demonstrate the angular displacement and velocity for BLDC servo motor. The angular displacement and velocity of PMDC servo motor are given in Fig. 4c and d. The optimum gains found for BLDC and PMDC servo motors are Kp1 ¼ 8:0, Kv1 ¼ 0:75, Ki1 ¼ 1:0, and Kp2 ¼ 70:0, Kv2 ¼ 0:5, Ki2 ¼ 1:0 respectively. Simulation results are presented only for one motion cycle of representing 3 s here. Observations are made for coming cycles by running simulation program many times at the beginning, output speed response curve for BLDC servo motor is regarded as oscillatory. This behavior is simply caused by initial conditions given as zero. Later transients caused by initial conditions are disappeared completely after the first motion cycle. In other words, the system reaches a desired steady-state after the first cycle. Simulated results are well correlated with the reference points. Tracking performances for angular velocity curves of PMDC servo motor is very good in the first cycle.

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6. Conclusions This paper has described theoretical investigation of a hybrid actuator with its kinematics and dynamics.A basis for the analysis of the real system response has been formed with the mathematical model developed. Although some simplifications are made during derivations, this study illustrates how good the real system is modelled, and controller values are tuned in simulation. It was seen that the dynamic behavior of the seven link mechanism-motor-driver is highly dominated by the coupling effects of other axis. A PID controller is adequate for position control for varying load on the mechanism. Parameters of the controller are tuned during simulation, the optimum settings for the controller gain are found and acceptable system responses are obtained. With these controller settings, the steady state speed is nearly reached in the first cycle with an acceptable amount of overshoot during application of in Motions 1 and 2. So the actual system of the hybrid actuator, with this model and settings behaved much as predicted by the simulation. Power calculations for both motors can be considered when a detailed study on energy is performed, some calculations can be added to model equations. Other improvements can also include the complete mathematical model with inclusion of friction on all joints, servo motors, and transmission efficiencies on this subject.

Appendix A Referring to Fig. 2, h3 can be expressed in terms of h2 , input angle by using loop 1 and eliminating h4 . Following constants are found during the calculation of angle h3 ; a1 a1 a2  a21  a22  a23 ; K23 ¼ ; K33 ¼ 4 a2 a3 2a2 a3 A3 ¼ K33 þ K23 cosðh2  h1 Þ; B3 ¼ sin h2  K13 sin h1 ; C3 ¼ cos h2  K13 cos h1 " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B3  C32 þ B23  A23 u3 ¼ ðA3 þ C3 Þ K13 ¼

h3 ¼ 2 arctan½u3 Similarly, h4 can be found by using loop 1, following constants are needed for h4 . a1 a1 a2  a21  a22  a24 ; K24 ¼ ; K34 ¼ 3 a2 a4 2a2 a4 A4 ¼ K34 þ K24 cosðh2  h1 Þ; B4 ¼  sin h2 þ K14 sin h1 ; C4 ¼  cos h2 þ K14 cos h1 " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B4 þ C42 þ B24  A24 u4 ¼ ðA4 þ C4 Þ K14 ¼

h4 ¼ 2 arctan ½u4

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During the calculation of angles h6 and h7 , loop 2 is applied, and following constants are found. a5 a5 a2  a26  a25  r2 ; K26 ¼ ; K36 ¼ 7 r a6 2ra6 A6 ¼ K36 þ K26 cosðh4  h5 Þ; B6 ¼ sin h4  K16 sin h5 ; " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B6  C62 þ B26  A26 u6 ¼ ðA6 þ C6 Þ K16 ¼

C6 ¼ cos h4  K16 cos h5

h6 ¼ 2 arctan ½u6 and a5 a5 a2  a27  a25  r2 ; K27 ¼ ; K37 ¼ 6 r a7 2ra7 A7 ¼ K37 þ K27 cosðh4  h5 Þ; B7 ¼  sin h4 þ K17 sin h5 ; " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B7  C72 þ B27  A27 u7 ¼ ðA7 þ C7 Þ

K17 ¼

C7 ¼  cos h4 þ K17 cos h5

h7 ¼ 2 arctan ½u7 where h1 ¼ h5 ¼ 0° representing in line mechanisms without any offset in the hybrid actuator configuration used.

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