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In: Mechanical Engineering

Derive and explain the effect of dynamic motor on force control and Hybrid position for coordinated...

Derive and explain the effect of dynamic motor on force control and Hybrid position for coordinated robots Answer expected: i) Definition, ii) Derivation, and iii) remarks

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Expert Solution

Answer 1:-

A new conceptually simple approach to controlling compliant motions of a robot
manipulator is presented. The "hybrid" technique described combines force and
torque information with positional data to satisfy simultaneous position and force
trajectory constraints specified in a convenient task related coordinate system.
Analysis, simulation, and experiments are used to evaluate the controller's ability to
execute trajectories using feedback from a force sensing wrist and from position
sensors found in the manipulator joints. The results show that the method achieves
stable, accurate control of force and position trajectories for a variety of test
condition.

Answer 2.

The Hybrid Controller :-
The basic hybrid control idea is an architectural concept
that links the constraints of a task requiring force feedback to
the controller design. The transformation form (C) to the
joints of the manipulator is such that, for the general case,
control of one manipulator joint involves every dimension in
(C):
Qi =Qi(Xi,X2 XN) (1)
where:
q, = position of/th joint of manipulator
fi, = inverse kinematic function
Xj = position of /'th degree of freedom in {C)
Therefore in hybrid control the actuator drive signal for
each joint represents that particular joint's instantaneous
contribution to satisfying each positional and each force
constraint. The actuator control signal for the /'th joint has N
components - one for each force controlled degree of freedom
in [C], and one for each position controlled degree of
freedom:
(2)
where:
Tj = torque applied by the rth actuator
A/y = force error in/th DOF of {C]
&Xj = position error in/th DOF of {C)
Tjj and i/-,-, = force and ..position compensation functions,
respectively, for the /'th input and this rth
output
Sj = component of compliance selection vector.
The compliance selection vector, S, is a binary iV-tuple that
specifies which degrees of freedom in [C\ are under force
control (indicated by Sj = 1), and which are under position
control (Sj = 0). (In this paper it is assumed that the number
of manipulator joints equals N < 6.)
For example:
if S = [0,0, 1,0, l,l] r

then
T, = +n (Ax,) + fe(Ax2) + r/3(A/3) + iMA*4)
+r,,(A/ 5 )+r/ 6 (A/ 6 )
Though the total number of active control loops is always N,
the type mix will vary as the task geometry and natural
constraints change.
We also define the compliance selection matrix [S]:
~S, 0 "
S2
[S] = diag(S) = S3
0 "" SN_
Figure 2 illustrates a hybrid control system that in-
corporates these ideas. The two complementary sets of
feedback loops (upper-position, lower-force), each with its
own sensory system and control law, are shown here con-
trolling a common plant, the manipulator. Notice that sen-
sory signals must be transformed from the coordinate system
of the transducer, [q] for position and [H] for force, into
(C) before errors are found and the compliance selection
vector is applied:
where:
cX = A(q)
c
F=[gr]«F
X = position of manipulator hand
A = kinematic transform from [q] to (C)
F = force on manipulator hand

Modeling :-
The mechanical system of manipulator, force-sensing wrist,
manipulator hand, and reaction surface was modelled as
shown schematically in Fig. 5. The manipulator linkage has
one revolute and one sliding joint, driven by pure torque and
force sources, respectively. The hand of mass M3 is supported
by webbing of the force-sensing wrist of elasticity Kw. Also
acting on the hand is reaction force fx produced through
contact with an environmental surface. The state equations
for this system are as follows:
1
Ate)
[Tl+Bi(q{) + q2Kww2+lKww1]
<72 =
1
M, +M
[T2+B2(q2) + Kww1]
(5)

w
2
= TT \-~K
^
w
7. +Asin(<?i)l M3
Reaction surface model:
fx=K,.(Cx CXf)
Cx = ^cosfa,) + /sinfa,)
(7)
(8)
(9)
(10)
Where
Cv = location of reaction surface
, = stiffness of wrist sensor
Kr
= stiffness of reaction surface
M, ,M2 Mi = masses of manipulator links
/ = link offset
Bx (q\),B2(q2) = manipulator joint friction (see below)
7,fa2) = moment of inertia at joint 1
The surface with which the manipulator interacts is
modeled as a spring with stiffness Kr
between the hand and
ground. A typical value for Kr
is 106
Nt/m. The force sensor
is also represented as a spring, K„ = 8 x 105
Nt/m, between
the hand and the arm.
It is difficult to model frictional damping properly. The
primary frictional component in the JPL Scheinman arm is
due to Coulomb sliding force. However, since force servoing
often involves very small positional changes made at low
velocity, static friction forces are also important. The model
includes a simplified static friction term plus the Coulomb
force:

Bi(q1) =
r-sgn(<7,.)[min(Ti];,lT,-l)]
L-sgn(<7i)[Tc,i]
where:
TS = static friction constant
TC = Coulomb friction constant
r = actuator torque


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