development of a zero gravity emulating test-bed for spacecraft...
TRANSCRIPT
Progress Report
Development of a Zero Gravity Emulating
Test-bed for Spacecraft Control
Mehrzad Namvar and Chun-Yi SuFaculty of Engineering & Computer Science, Concordia University
[email protected], [email protected]
September 10, 2005
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Chapter 1
Overview of zero gravityemulation concept
This report describes development of a 3D zero-gravity emulation scheme forspacecraft motion control and its implementation on a test-bed consisting ofa prototype spacecraft located on the tip of a 6-DOF hydraulic manipula-tor, Schilling Titan II. It provides description of overall emulation strategyand a design procedure for velocity control of the manipulator. Extensiveexperimental results are reported to illustrate system performance.
1.1 Introduction
A major problem in testing a spacecraft attittude/translation control is toemulate zero gravity condition in a ground-based 1-g environment. Simula-tion is widely used for characterizing the functional behavior of spacecraftcontrol systems. However, it is of vital importance to be able to test andvalidate the system performance under realistic conditions and through max-imum usage of hardware. Therefore, in the aerospace industry, it is highlydesirable to incorporate actual hardware, as much as possible, in the simu-lation loop.
Validation and testing of the functional capability of spacecraft atti-tude/translation control systems with real physical units on the ground posesmany challenges due to the effects of gravity. Ground testbed facilities havebeen used for spacecraft control hardware/software verification since spaceprograms began almost half a century ago. Due to the high cost of launch and
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operations associated with on-orbit repair, a spacecraft must work adequatelyonce it is placed in orbit. Therefore, a realistic testing of spacecraft prior tothe launch, ideally with all of its hardware/software in place, has to be un-dertaken to ensure that the spacecraft functions as it was intended. Thesefacts have motivated building different testbed facilities in various govern-ment and university laboratories for the analysis and testing that spacecraftundergo. Gas-jet thrusters, ion thrusters, and balanced reaction wheels arecommonly employed as actuators for spacecraft attitude and/or translationcontrol. Simulation is widely used for characterizing the functional behaviorof spacecraft control systems. However, it is of vital importance to be able totest and validate the system performance based on actual sensors and actu-ators which often have complex characteristics. Therefore, in the aerospaceindustry, it is highly desirable to incorporate actual hardware in the test-ing, as much as possible. Motion table systems, that replicate the motion ofa satellite, such as multi-axis motion controlled tables, have been used forHardware-In-The-Loop (HIL) testing of an attitude control system (ACS).The motion table testing system allows the incorporation of real sensors of asatellite such as gyro and star tracker in the simulation loop. However, actu-ators such as reaction wheels, torques, or gas-jet thrusters must be simulatedin this method. The concept of the HIL methodology has also been utilizedfor design and implementation of various laboratory testbeds to study thedynamics coupling between a space-manipulator and its spacecraft operat-ing in free space. In these approaches the spacecraft actuators have to besimulated as well.
There exists many technologies to solve the problem of reproducing themicro-gravity space environment: air bearings, underwater test tanks, free-fall tests, and magnetic suspension systems. However, only the air bearingshave proven useful for testing spacecraft. Underwater test tanks have beenused extensively for astronaut training, but they are not suitable for testingsatellites. Although a free-fall test can achieve 0-g in a 3-D environment,this can be sustained for only a short period of time. Magnetic suspen-sion systems provide only a low force-torque dynamics environment with asmall range of motion. Thus far only air-bearing systems have proven theuseful technology for spacecraft simulator. Air-bearing tables (also known asplanar air-bearing) and spherical air-bearings are commonly used for ground-based testbeds for testing the translation and attitude control systems of aspacecraft. An emulation of 0-g translational motion can be achieved by anair-bearing table where the spacecraft can navigate along a surface perpen-
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dicular to the gravity vector while being floated on a cushion of compressedair with almost no resistance. This technique is widely used for testing vari-ous space systems such as formation flying, free- flying space robots, orbitalrendezvous and docking, capturing mechanisms of spacecraft, and free- flyinginspection vehicles. Although the air-bearing table system can be utilized totest physical units of spacecraft control systems including the sensors andactuators, this system is limited to a two-dimensional planar environment.Spherical air-bearings have been used for spacecraft attitude determinationand control hardware/software verification for many years. The earliest de-velopment and design of a satellite simulator based on spherical air-bearingwith three axis of rotation was reported in . That system has been evolvedinto a modern testbed facility. A spherical air-bearing yields minimum fric-tion and hence offers a nearly torque-free environment if the center of mass iscoincident with the bearings center of rotation. The main problem with theair bearing system is the limited range of motion caused by the equipmentaffixed to the bearing. Also, spherical airbearing are not useful for simu-lating spacecraft having flexible appendages (such as solar panels), becausethe location of center-of-mass of such spacecraft is not fixed. To this end,although one can envisage combining the two air-bearing technologies for re-producing both the rotation of and translation of motions, a useful testbedthat has complete freedom in all six axis degrees is an unlikely achievementby air-bearing systems.
An alternative to air-bearing systems that uses robotic technology is pre-sented here. This report describes a ground-based robotic-testbed for 0-gemulation of spacecraft dynamics in a 1-g laboratory environment. Unlikethe air-bearing systems, the robotic testbed can emulate motion of a space-craft with complete freedom in all six axis degrees. It can also deal with theemulation of spacecraft having flexible appendages. Fig. 1.1 schematicallyillustrates the concept of the emulation system. A ground spacecraft is at-tached rigidly to a manipulator. A six-axes force-moment sensor is installedat the interface of the spacecraft and the manipulator, for sensing an echo ofthe external forces for instance, firing thrusters superimposed by gravita-tional and inertial forces. Upon measurements of the wrist force-moment andthe joint angles and velocities, the signals are fed back to a control systemthat moves the manipulator and the ground spacecraft accordingly.
The actuation process in a typical spacecraft motion control system is usu-ally carried out by jet-gas or ion thrusters which exhibit complex dynamicbehavior. Therefore, in order to test the performance of thruster actuators
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Azimuth
Shoulder
Elbow
Pitch
Yaw
Wrist
Figure 1.1: Schilling, Titan II manipulator
and spacecraft control algorithms, a spacecraft prototype equipped with sim-ilar thrusters is built and tested in a ground-based 1-g environment. Thereexists several technologies for creation of a 0-g environment in a ground-basedlaboratory such as motion table system or air bearing system. However, thesetechnologies suffer mostly from limitations in the range of achievable motionfor the spacecraft or limitations in degrees-of freedom.
The goal of the present work is to implement a novel method for 3Demulation of dynamic behavior of a real spacecraft in 0-g, by using a space-craft prototype located in a ground-based laboratory. The testbed consistsof a prototype spacecraft located on the tip of a 6-DOF hydraulic manip-ulator which is controlled such that the dynamic behavior of the prototypespacecraft, as seen by its thruster actuators, becomes equivalent to a realspacecraft in 0-g environment.
1.2 Emulation scheme
In this section we describe general scheme of the emulating system togetherwith its different components.
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1.2.1 Spacecraft dynamics
The dynamics of a rigid spacecraft can be described by a second order equa-tion of the form
Msv + Cs(v)v = fe (1.1)
where v ∈ R6 represents linear and angular velocity of the spacecraft center-of-mass with respect to the world frame. Also, Ms > 0 represents spacecraftinertia matrix. The equivalent force fe is expressed by fe = fa + BT λ whereBT λ represents force/moments exerted to the spacecraft due to nonholonomicconstraints on spacecraft velocities expressed by Bv = 0, and fa representsforce/moments exerted to the spacecraft by actuation thrusters. Also, C(v)vrepresents Coriolis and Centrifugal forces. We assume that Ms and Cs areknown.
1.2.2 Dynamics of spacecraft prototype
We consider a spacecraft prototype installed on the tip of a rigid manipulatorwhose center-of-mass with respect to the world frame has the same velocityas the real spacecraft and acts under the same force/moment fe
Mpv + Cp(v)v + gp + AT fs = fe (1.2)
where gp is the gravity vector acting on the prototype and is a functionof prototype center-of-mass cartesian position. fs is the force exerted fromthe manipulator to the prototype and A is the prototype Jacobian matrixmapping manipulator tip velocity to prototype center-of-mass velocity. Weassume that Mp, Cp, gp and A are known and that fs is measurable.
1.2.3 Velocity trajectory generation
Having an identical velocity, the real and prototype spacecraft dynamics aresaid to be equivalent for any fe if and only if fs(t) and v(t) satisfy
(Ms −Mp)v + (Cs(v)− Cv(v))v − gp − AT fs = 0 (1.3)
The above equation specifies all admissible trajectories for velocity and con-tact force that ensure equivalence of prototype and real spacecraft dynamics.For given fs, measured by a force/moment sensor, the robot manipulatorgrasping the prototype spacecraft should be controlled such that the proto-type center-of-mass tracks v(t).
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1.2.4 Hydraulic Manipulator dynamics
The manipulator used for emulation purpose is a 6-DOF hydraulic manipu-lator with rigid links. The robot link dynamics is described by
Mr(q)q + Cr(q, q)q + gr(q) = J(q)T fs + τ (1.4)
where q represents vector of joint angles and Mr(q) > 0 represents robot in-ertia matrix. The Coriolis and Centrifugal forces are represented by C(q, q)qand gr(q) is the vector of gravitational forces. Robot Jacobian matrix isdenoted by J mapping joint angle velocities to prototype center-of-mass ve-locity: v = J(q)q. It is assumed that J(q) is known. The vector τ representsactuator torque/forces and is expressed by
τ = τI − τf
where τI is the internal toque of the actuators which is a function of ac-tuator chamber pressures and it is measurable. The joint friction torque isrepresented by τf and is not measurable and is expressed by
τf = cq + FCsgn(q) (1.5)
where c and FC represent viscous and Coulomb friction coefficients, respec-tively. The internal torque of hydraulic actuators is a state variable governedby a second order nonlinear dynamics of actuator valves
τI = Q(pH , pL, sgn(u), q)u− P (q)q − L(q)τI
where Q(pH , pL, q) > 0 acts as actuator valve gain and is a function of cham-ber pressures and joint angles. P (q) > 0 represents the effect of joint velocityon the internal torque (similar to back EMF effect in electric machines) andL(q) > 0 represents internal leakage. In general, Q, P and L are functions ofchamber geometry and hydraulic parameters which are not available in caseof Schilling manipulator. For this reason we assume that:
Assumption 1 The variation of chamber pressures and joint angles in eachactuator is sufficiently small such that Q, P and L are constant.
This assumption enables us to adopt standard adaptive control approach foronline estimation of unknown parameters P , Q and L.
On the other hand, due to the lack of a model structure for link dynamicsof the Schilling manipulator, we need to make the following assumptions:
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Assumption 2 The variations of joint angles and joint velocities are suffi-ciently small such that Mr, Cr and gr are constant. Moreover, Mr is diagonal.
By this assumption the design of a velocity controller for the manipulator isreduced to the design of velocity controllers for each actuator, separately.
1.2.5 Manipulator control design
In view of Assumption 1 and 2, the dynamic model of each joint of themanipulator can be written by
mw + g + cw + FCsgn(w) = JTi fs + τI (1.6)
τI = Qu− Pw − LτI (1.7)
where w is the joint velocity (component of q corresponding to each joint)and Ji is the i-th column of the manipulator Jacobian which is consideredknown.
Problem 1 Let v be the solution of (1.3) and define qd = J−1v. Let wd
be the component of the vector qd corresponding to i-th actuator. Then, theemulation problem is equivalent to the design of a control law u for the system(1.6), (1.7) such that w tracks wd.
Recall that in the system dynamics (1.6), (1.7), m, g, c, FC , Q, P, L areunknown constants and τI , w, fs are measurable. The presence of a nonlin-ear term resulting from the friction force prevents us from using standardadaptive control for linear systems. Alternatively, we adopt a backstepingapproach for control design.
Step 1
In this stage we consider (1.6) and calculate a desired torque τId such thatw tracks wd:
τId = mwd + cwd + g + FCsgn(w) + JTi fs −K1(w − wd) (1.8)
where K1 > 0 is controller gain and the estimated parameters are computed
by ˙m = −lmwd(w − wd), ˙c = −lcwd(w − wd),˙
FC = −lF sgn(w)(w − wd) and˙g = −lg(w − wd) and lm, lc, lF , lg are some positive gains.
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Step 2
Having determined τId in Step 1, we consider actuator dynamics (1.7) andcalculate a control command such that τI tracks τId.
u = Q−1(Pw + LτI −K2(τI − τId) + τId) (1.9)
where K2 > 0 is the controller gain and the estimated parameters are given
by˙Q = lQu(τI−τId),
˙P = −lP w(τI−τId),
˙L = −lLτI(τI−τId) where lQ, lP , lL
are some positive gains.Note that the command signal depends on the derivative of the desired
torque τId which is a function of joint acceleration and force derivative fs.It is possible to design an adaptive velocity observer for estimation of jointacceleration, however, we will not follow this approach for simplicity andinstead, we use numerical differentiation for calculation of τId
τId =as
s + a∗ τId
where 2π < a < 4π. Moreover, we replace sgn(w) with tanh(bw) where100 < b < 400.
1.3 Schilling manipulator
The Schilling Titan II hydraulic manipulator is equipped with 5 Rotary-vaneand one linear hydraulic actuators named: Azimuth, Shoulder, Elbow, Pitch,Yaw and Wrist. Depending on their type, the actuators can generate maxi-mum torque of 68 Nm. to 2100 Nm. The maximum velocity of actuators isbetween 64 deg/sec and 360 deg/sec. The manipulator weights 75 kg. andits links are made of Titanium. It can stretch about two meters and at fullextension, the arms lift capacity is 110 kg. Joint angle of each actuator is mea-sured by a 16-bit resolver with sampling rate of 1kHz. Each joint is equippedwith 2 analog pressure transducers (read in volts and non calibrated) and arate (velocity) sensor. Noise level in joint rate measurement is about 0.005rad/sec. which is a limiting factor in friction compensation. In addition, therate reading of Elbow joint has significant jumps with duration of 1-2 msec.which might have been resulted from mechanical deficiencies in the resolver.The manipulator is also equipped with a six axis JR3 force/moment sensorin its wrist.
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The MATLAB-Simulink model named ”schilling actuator model.mdl” givesaccess to any signal coming from and going to the robot. The model con-sists of a SC-console block available in online operations and used for signalmonitoring, and SM-master unit containing controller blocks.
1.4 Gravity model for the prototype
The first step in construction of the trajectory generator (eq. (3)) is theidentification of the gravity term gp for the prototype model (2). Assumingthat the trajectory generator is written with respect to a frame attachedto the prototype, then the objective is to extract all gravity terms in theforce/moment readings fs. For this purpose assume that c ∈ R3 is the coor-dinates of prototype center of mass with respect to force sensor coordinates.Assuming that both of these coordinates are parallel then the transformationmatrix A becomes
A =
[R 0
−R× c R
]
where R is the rotation matrix of prototype with respect to fixed inertialframe. The vector gp has the form gT
p =[
mg 0]
where m is prototypemass and gT = 9.81[0, 0, 1] m/s2. On the other hand due to practical con-siderations sensor offsets are not exactly known and besides they vary basedon temperature and hysteresis effects, so we consider f0 as the vector ofunknown sensor offsets. It is now possible to locate the robot in differentpositions such that prototype velocity and acceleration becomes zero anduse gp = AT (fs − f0) to identify the unknown parameters c, m and f0 bystandard linear optimization techniques.
Figures 1.2 and 1.3 show the comparison between the measured force-moments and the estimated ones defined as
fs := A(c)−T gp(m) + f0
The achieved relative identification error is 1.11 percent while the additiveestimation error is around 0.9N and 0.02N.m for force and moments, respec-tively. Precision of identification depends highly on force sensor noise andamount of force disturbances caused by slight movements of the prototype. Itis recommended that sensor is frequently calibrated by the method discussedabove to preserve the fidelity of the emulation system.
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0 100 200 300 400 500 600−100
−50
0
50
x
Forces in N
0 100 200 300 400 500 600−40
−20
0
20
40
y
0 100 200 300 400 500 600−50
0
50
100
z
Figure 1.2: Blue: Measured forces. Red: estimated forces
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0 100 200 300 400 500 600−1
−0.5
0
0.5
1
x
Moments in N.m
0 100 200 300 400 500 600−3
−2
−1
0
1
y
0 100 200 300 400 500 600−0.05
0
0.05
0.1
0.15
z
Figure 1.3: Blue: Measured moments. Red: estimated moments
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Chapter 2
Experiments
In his section we illustrate experimental verification of the emulation scheme.The experiment is conducted by applying an external force to the prototypeby hand and recording prototype and manipulator motion. As discussed inthe previous report, the applied force-moments generate desired velocitiesfor manipulator that if is tracked satisfactorily will ensure validity of theemulation system.
Experiments are conducted with two different forces. In the first exper-iment we have used impulsive force moments similar to those generated byjet thrusters and shown in Fig. 2.1.
2.1 Impulsive force-moments
The duration of each impulse is approximately 0.2 seconds. The mass of theprototype is identified as m = 9 kg. and it has an cylinder shape with octag-onal cross-section. The satellite is considered to have the same dimension asthe prototype but with the mass of 150 Kg.
Fig. 2.2 demonstrates joint velocity tracking for the hydraulic manipu-lator. The accuracy of velocity tracking is slightly different in joints whereAzimuth and Yaw actuators exhibit more friction than other joints. Thereis also some flexibility in actuators which limits controller gain for achievinghigh accuracy. The average relative error in velocity tracking (and in lowvelocities where friction is dominant) defined by
eq :=‖qd − q‖‖qd‖
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60 80 100 120 140 160 180 200−50
0
50
N
fx
60 80 100 120 140 160 180 200−100
0
100
N
fy
60 80 100 120 140 160 180 200−50
0
50
N
fz
60 80 100 120 140 160 180 200−20
0
20
N.m
mx
60 80 100 120 140 160 180 200−20
0
20
N.m
my
60 80 100 120 140 160 180 200−10
0
10
N.m
Time(s)
mz
Figure 2.1: Impulsive force-moments
is about 15 percent which is satisfactory for our relatively aged manipulator.
Pressure torque tracking which an essential step for good velocity trackingin hydraulic systems is shown in Fig. 2.3. The average relative error in torquetracking is about 1.8 percent which is very satisfactory. The quality of torquetracking is an indication of how well the controller handles nonlinearity inhydraulic systems. Note that the torque shown in the figure is not calibratedand is in volt.
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60 80 100 120 140 160 180 200−10
0
10
deg/
sec Yaw
60 80 100 120 140 160 180 200−10
0
10
deg/
sec Pitch
60 80 100 120 140 160 180 200−10
0
10
deg/
sec Elbow
60 80 100 120 140 160 180 200−5
0
5
deg/
sec Shoulder
60 80 100 120 140 160 180 200−5
0
5
deg/
sec
Time(s)
Azimuth
Figure 2.2: Joint velocity tracking in case of impulsive force-moments. Blue:desired velocity and Green: actual velocity
2.2 Smooth force-moments
Similar experiment but by applying smooth force-moments to the prototypeby hand has been conducted and the results are shown in Figs. 2.4 to 2.6.In this case the average relative joint velocity tracking is about 14 percentwhile the average relative torque tracking is around 2 percent.
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60 80 100 120 140 160 180 200−5
0
5
60 80 100 120 140 160 180 2000
5
10
60 80 100 120 140 160 180 2000
5
60 80 100 120 140 160 180 2000
2
4
60 80 100 120 140 160 180 200−2
0
2
Time(s)
Azimuth
Shoulder
Elbow
Pitch
Yaw
Figure 2.3: Joint pressure torque tracking in case of impulsive force-moments.Blue: desired torque and Green: actual torque
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−20
0
20
N
fx
−50
0
50
N
fy
−20
0
20
N
fz
−10
0
10
N.m
mx
−10
0
10
N.m
my
110 120 130 140 150 160 170−5
0
5
N.m
Time(s)
mz
Figure 2.4: Smooth force-moments
20
−10
0
10
deg/
sec Yaw
−20
0
20
deg/
sec Pitch
−20
0
20
deg/
sec Elbow
−5
0
5
deg/
sec Shoulder
110 120 130 140 150 160 170−10
0
10
deg/
sec
Time(s)
Azimuth
Figure 2.5: Joint velocity tracking in case of smooth force-moments. Blue:desired velocity and Green: actual velocity
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−5
0
5Yaw
−5
0
5Pitch
0
5Elbow
0
2
4Shoulder
110 120 130 140 150 160 170−2
0
2
Time(s)
Azimuth
Figure 2.6: Joint pressure torque tracking in case of smooth force-moments.Blue: desired torque and Green: actual torque
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Chapter 3
Conclusion and progressassessment
The Schilling manipulator used for construction of the 0-g emulation test-bed is a relatively aged manipulator with significant amount of friction injoints such that the friction is 70 percent of the gravity force in most joints.Friction phenomena combined with large amount of noise in velocity signalswhich is resulted from numerical differentiation of low resolution encodersignals, makes the velocity control system particularly challenging. To dealwith friction we have used an adaptive scheme with high adaptation gainstogether with a nonlinear velocity controller gain (K1 in (8)) which ensuressatisfactory friction compensation while avoiding undesirable oscillations invelocity due to actuator or payload flexibility effects. The value of the timevarying positive gain K1 is essentially a function of desired velocity. Thefiltering of the velocity signal is also a significant impact on the performanceof the velocity controller and is adjusted for each actuator based on extensiverounds of experiments.
On the other hand pressure torque control problem is mainly affectedby nonlinearity of the hydraulic system and the designed adaptive torquecontroller exhibits good level of performance. This is largely due to relativelyclean measurement of pressure signals and adaptive nature of the controller.
In the previous report and presentations we had mentioned three technicalproblems as: 1. Poor force-moment measurement, 2. Existence of someimpulsive jumps in Elbow velocity and 3. Mechanical stiction in Wrist joint.
The first problem has been resolved by installation of a high precisionforce sensor inside the prototype. The second problem has been partially
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dealt with by using a combination of rate limiter and filtering. However thethird problem has not been dealt with yet.
An important improvement in velocity controller has been brought byconstruction a gravity model for manipulator. This gravity model helped todecrease the adaptation gain of the gravity term in the velocity controllerand hence decrease the oscillations.
All steps mentioned in the previous report has been accomplished. Theonly remaining part is the direct compensation of external force-moments inthe velocity tracking which is normally insignificant but can be done easilyin the next phase of the project.
As a perspective we can consider integration of jet thrusters to the proto-type and design of a satellite motion controller capable of bringing the pro-totype to specified position and orientation by applying appropriate forcesthrough jet thrusters.
As a more technical perspective we can mention the problem of estimatingjoints velocities by using measurement of force-moments and joint encodersignals. Since there is not a clean and specific solution for this problem inthe literature, this problem can potentially bring some contributions in robotcontrol theory level.
Chapter 4
Appenxix: Simulink Models
The Simulink model use for real-time simulation and implementation of thecontrol system consists of two main parts: Master and Console (see Fig. 4.1).Console is accessible during real-time operation and used for changing systemparameters and signal monitoring. The operation unit in Console (shown inFig. 4.2) consists of 6 blocs corresponding to 6 joint velocity controllersand a number of switches for different operation modes. Parameters of eachvelocity controller can be modified in blocs as shown in Fig. 4.3.
Master consists of the blocks which are not available during real-timeoperation and any parameter found in Master can be changed through aparameter dialogue box in opal-RT. Master consists of a velocity controllerunit and a kinematics modeling unit. Trajectory generation equations to-gether with gravity compensation are carried out in a bloc shown in Fig. 4.4.Structure of the joint velocity controller consisting of an adaptive torque andvelocity controller are shown in Fig. 4.5.
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Figure 4.1: Master and Console
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Figure 4.2: Operation unit in Console
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Figure 4.3: Operation unit for each joint in Console
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Figure 4.4: Kinematics conversions and trajectory generation in Master
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Figure 4.5: Joint controller in Master