The controllers in Figure 1.15 perform hybrid position-force control or position control alone, depending on the nature of the tasks. However, it is desirable to have as small a change as possible in the controllers for different tasks. This is one of the important characteristics of an intelligent robotic system.
It was shown in the last section that the dynamics of the joint motors play an important role in hybrid position-force control. It is important to include the dynamics in the system's dynamic model.
Let the ith robot in the system have n~ joints. Its dynamic model, including the joint motor dynamics, can be written as
f Di(qi)o'i + Ci(qi, [ti) nt- Gi(qi) + rfi dzi _ - 1
dt - - Ti Ti -'[- K eiqi -[- lgi i = 1 , 2 , . . . k
T i
(1.45)
where
r : = joint torque that produces output force f/
T~ = n x n diagonal matrix whose entries are the time constant of the joint motors Kez = voltage constant of the m o t o r (back electromotive force)
u i = m o t o r armature voltage and qi, ri ~ Rni"
The general task space for the ith robot is defined as
4 EVENT-BASED PLANNING AND CONTROL FOR MULTIROBOT COORDINATION 33
where
0ffi (~ ~ 6 is the general position space.
c~/6 ~ 6 is the general force space.
~z e IR "i is the general r e d u n d a n t joint space.
Yi
~i = f/ ~ 1 2 +ni i
and @ denotes the orthogonal direct sum. For any given task, it wil| be a subspace of the general task space.
Because only ni joints are available, the robot can be controlled on|y in a subspace of ~ ,
~ e Rat Hence, the actual output is in ~-~ic.
3-~ c is defined as
where
&c = ~c @~c @4%
@//c ~ ~lil
~iic ~ ~ li2
~ic ~ ~ni -- (lil + li2)
The task projection operator Bic" ~ ~ ~ c is the basis of ~ c and is described as
Bic= [gic Fic Zic]
The joint space torque that produces the o u t p u t force f can be written as [39]
f = (jr(q)) #'of where " # " denotes a pseudoinverse.
Because of redundancy, the joint torque is not unique for a given f~ and can be formulated as [39]
"cf, = J~(qi)fi + [I,,~ - Jf(qi)(jr(qi))#]r/~
= 9 " # " denotes where I,, is an n~ x n i identity matrix; Ji(qi) is a Jacobian matrix, ?g Ji(qi)[li,
a pseudoinverse; Fz, is any ni x 1 vector; and [I,, - J~(qi)(J ~(qi)) #]Ff~ is a vector in the null space of (J[(qi)) #, which describes the r e d u n d a n c y of the robot.
By a similar argument,
{l'i = Jff(qi)(f'i- Ji(q)Oi) + [ I , , - Jff(qi)Ji(qi)]F,~
where F,, is any n i x 1 vector and [I,, - J[(qi)J~(q~)]F,, is a vector in the null space of Ji(q~),
which also describes the r e d u n d a n c y of the robot.
By choosing Fr, = 0 and FI, =//'i, (1.45) can be written as
I Di(q,)J?(qi)(('i-Ji(qi)dli)+ Ci(qi. (I,)+ Gi(qi)+ Jf (qi) fi + [l.i--jT(qi)( J f (qi)) #]qi = zi d'ci _ - 1
d t - - Ti z i -+- N e l l -+- u i Let
(1.46)
gil = D i ( q i ) J { # ( q i ) ( f " i - J i ( q i ) q i ) -Jr- C i ( q i , q,) Jr- G i ( q i ) Ti2 -- J f (qi) f i
Ti3 : [ini __ j T ( q i ) ( j T(qi)) # ] q i Then r i = ril + gi2 @ I7i3.
In addition, let
dzi 1
+ T/- 1-Cil - Keiqi U~ l = dt
d'c i2
+ T//- 1"ci2 ui2 = dt
d~i3
-~- T/- 1Ti3
ui3 =" dt
Then u i = Uil _qt_ Ui 2 .71_ Ui3. As a result of these definitions, the dynamic model (1.46) has been separated in the general task space ~/.
In the general position space ~i"
f Di(qi)J?(i"i Ji(qi)qi) + C,(qi, Oi) -Jr- Gi(qi) = zi, d'Ci l __ __ gi - 1
d t - ~c i 1 _11_ K ei O i -Jr- bl i l In the general force space ~ "
(1.47)
f Jri (qi) f i = "ci2
dgi2 - 1
7 i = - r ,
In the general redundant joint space ~i"
~i2 nt- Ui2
(1.48)
f [I,i -- J~(qi)(J ~(qi)) #]qi = zi3
dzi3 _ - (1.49)
dt - - Ti 1~i3 + ui3
When the motor dynamics is considered, acceleration measurements are necessary for linearization and decoupling in the general position and redundant joint spaces [25]. In practice, it is very difficult to get accurate acceleration information. Therefore, the motor dynamics is considered only in the general force space and can be ignored in (1.47) and (1.49).
4 EVENT-BASED PLANNING AND CONTROL FOR MULTIROBOT COORDINATION 35
After ignoring the motor dynamics in the general position and redundant joint spaces, (1.47) and (1.49) can be rewritten as
Di(qi)Jie(qi)(f"i- Ji(qi)gli) + Ci(qi, Oi) -[- Gi(qi) = TiKei[ti + TiUil (1.50) and
[in, __ jT(q,)(j T(q,)) #]0", = riui3 (1.5~)
Based on the well-known nonlinear feedback technique, the nonlinear controls Uil = Ti- l ( D i ( q i ) J i e ( q i ) V i l - Di(qi)Jf~(qi)Ji(qi)Oi + Ci(qi, dli) + Gi(qi) - TiKeidli ) Ui2 ~- ( r i - l j r ( q i ) n t- ~jr(qi)) f i n t- j r ( q i ) g i 2
ug3 = T~- ~[In, -- j r ( q z)(j r(q ~)) #] V~ 3
can linearize and decouple the systems (1.47), (1.48), and (1.49) in ~i, ~ , and ~i. The linearized models are
/: = V/1 f = V/2 q', = V,~
where Vzl, V/2 ~ R 6 and V~3 ~ R ni are auxiliary inputs.
Since only ni controls are available, this linearization and decoupling are not actually feasible in ~ . But if we consider only the controllable subspace ~ c , the linearization and decoupling can be achieved.
It can be proved that, if the position and force are not controlled in the same task space axis for a robot, then the nonlinear feedback control,
u~ = T~- ~(D~(q~)J~(q~)~a - D~(q~)J~(q~)J~(q~)4~ + C~(q~, (1~) + Gi(q i) -- TiKeiOi ) "t'- T i- 1jT(qi) + jr(qi))fi
+ jT(qg)~2 + T - ~ [ I , , , _ dT(qg)(dr(q~))#]~3
(1.52)
linearizes and decouples the dynamic system (1.46) in Yc. The linearized and decoupled system is
~i ~ V//1 f~= v~2
where
ri ri
g/~ q
and
V//1 V/1
Vi = B i T ~i2
Then the linear controllers
~ = r'~(s) + ki~(r~(s) - r i ) + kip(rd(s) --l'i)
~2 = ~d(s) + ki~(~d(s) -- ~)
= + k , o ( 4 - + ki (Of( ) - O)
(1.53)
can be used to stabilize and control position, force, and redundant joints. All desired values in (1.53) are given by event-based task planners.
Several Remarks
1. Because the preceding control law can directly take task space commands, it is a task-level controller.
2. For a different task or a different system configuration, the structure of controler (1.52) is same. Changes must be made only in the task projection operator Bic. In this respect, the controller structure is task independent and is suitable for multirobot systems working on complex tasks.
3. The management of redundancy consists of designing Bic to achieve certain goals. In addition, once the redundant joints are determined, their motion can be planned and controlled based on some secondary optimization criteria, such as obstacle avoidance and load sharing [24, 59].