An approach is presented for the resolution of the redundancy of a seven-axis manipulator arm from the AI and expert systems point of view. This approach is heuristic, analytical, and globally resolves the redundancy at the position level. When compared with other approaches, this approach has several improved performance capabilities, including singularity avoidance, repeatability, stability, and simplicity

Chen, I.

English

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N9o: 738 
RESOLUTION OF SEVEN-AXIS MANIPULATOR REDUNDANCY-
A HEURISTIC ISSUE
I. Chen
Computer and Information Sciences Department,
Alabama A & M University,
Normal, AL 35762, U.S.A.
ABSTRACT
This paper presents an approach for the resolution of the
redundancy of a seven-axis manipulator arm from the AI and expert
systems point of view. This approach is heuristic, analytical, and
globally resolves the redundancy at the position level. When
compared with other approaches, this approach has several improved
performance capabilities, including singularity avoidance,
repeatability, stability, and simplicity.
1 . INTRODUCTION
With an addition of one more degree of freedom (d.o.f.), a yaw
motion of the forearm, to the typical six d.o.f, articulated arms,
the shoulder of the three-link manipulator will have pitch, yaw,
and roll motions (Figure I) . These seven-axis articulated
manipulator arms when compared with the typical six d.o.f, robot
arms may have many advantages, including flexibility,
manipulability, obstacle avoidance, singularity avoidance,
stability, and optimal control. In addition, because of the
similarity to human arm configuration, the seven-axis, three-link
manipulator arm is the best candidate for the master-slave
teleoperated robot useful in hazardous environments. However, the
resolution of the kinematics, control, and dynamics with
redundancy is not an easy task. With the redundant d.o.f., there
will be an infinite number of arm configurations kinematically for
each desired hand position. The motion that maximizes a specific
performance index is the optimal motion for that special
condition. However, the optimal solutions are not easily
accessible due to a highly nonlinear relationship between the
joint angle space and the hand position. Because of the complex
nature of this problem, each approach seems to have its own
drawbacks and limitations. In order to resolve this redundancy
with the already complicated problem, the author employed ([5],
[6] ) human arm motion heuristics to analytically derive the
Support for this research is provided by a grant NAG 023 from NASA
Marshall Space Flight Center.
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https://ntrs.nasa.gov/search.jsp?R=19900018015 2020-03-19T21:28:22+00:00Z
inverse kinematics (see the Appendix) .The purpose of this paper is
to present this heuristic concept and to compare the approach
with other existing methods.
2. STATEMENT OF THE APPROACH
By observing the human arm motions, two heuristic rules are
concluded:
I) The travel distance of the wrist joint from a hand
position to a new hand position should be a minimum for
each arm motion.
2) The travel distance of the elbow joint from an arm position
to a new arm position should be a minimum for each arm
motion.
In addition, there are two meta-level rules:
I) The first rule can be applied only when the second rule is
satisfied.
2) If the movement of the robot arm violates the joint limit
constraints or obstacle constraints, then a configuration
which satisfies the above two domain specific rules and the
constraints is the solution.
The first two rules state the fact that both the joint travel
distances have to be minimized in order to obtain the kinematic
solution. The first meta rule defines the relationship between
these two domain-specific rules. The first rule is subject to the
constraint of the second rule, and the first rule will be applied
only when the second rule is satisfied. The purpose is to minimize
the use of the lower arm as much as possible since each movement
of the lower arm causes the movement of the forearm and hand
together. This consumes more energy because a bigger moment of
inertia is involved in the movement. The same principle is also
applied to the movement of forearm and hand. In other words, the
travel distance of the wrist joint should also be maintained to a
minimum to ensure the minimization of the energy consumption. The
main idea is to minimize the lower link of the manipulator since
the motion of the lower link causes movement of all the upper
links together. A bigger movement of the lower link will in turn
consume more energy because of the bigger moment of inertia
involved in the motion. The motion with the least amount of energy
consumption is the most comfortable to the body. Besides, for a
given kinetic energy, the optimal arm motion which obeys these
rules is the minimum-time motion and is the fastest way to reach
the new hand position. The second meta rule governs the condition
whenever the arm motions are prohibited because of the violation
of the joint limit constraints or obstacle constraints. Although
this strategy has not yet been developed, a suboptimal
configuration, which relaxes the constraint free minimum-distance
solution according to the imposed constraints, is accessible. Note
that from the point of view of AI and expert systems the second
meta rule is of second-order since it governs all the rest of the
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rules, including the first meta rule. This relaxation strategy may
be an iterative process by relaxing the related joint angles to
the subsequent neighbors according to a prescribed small joint
interval.
The motion control of this seven-axis manipulator arm may be
restated as follows:
Minimize the following performance index,
j = KI* DI2 + k 2 * D22 (i)
subject to the kinematic equations of the manipulator arm and
the joint limit and obstacle avoidance constraints.
Here, D 1 is the elbow joint travel distance and D 2 is the wrist
joint travel distance. K 1 and K 2 are the corresponding weighting
factors with K 1 >> k 2 to make sure that the minimization of the
travel distance of the elbow joint D 1 before minimizing the travel
joint distance of the wrist, D 2. The squares of D 1 and D 2 assure
that D 1 and D 2 will be positive throughout the converging process.
3 . COMPARISONS
This approach is compared to other approaches in the following
manner:
3.1 INVERSE KINEMATIC METHOD VS. RESOLVED MOTION METHOD
Research has been carried out on the control of redundant arms
mostly through a pseudoinverse matrix, also known as the Moore -
Penrose generalized inverse. The instantaneous joint displacements
are computed from the joint velocities by using the pseudoinverse
J+ of the Jacobian matrix J. For a given hand position, the
resolved motion methods cannot provide the corresponding joint
angles. This implies that the resolved motion technique cannot
directly map the workspace to the joint angle space of the arms.
In other words, the approach is numerical and not analytical.
Because of the instantaneous resolution of the redundancy, this
type of technique also inherits other drawbacks (to be discussed
later). The proposed approach has advantage over the resolved
motion methods by resolving the redundancy analytically and
globally ( not incrementally ) . Benati's approach [2] is
recursive, partly analytical, and partly numerical. The
application of the heuristics rules to the derivation of the
inverse kinematics of a seven-axis manipulator arm is shown in the
Appendix.
3.2 RESOLUTION LEVEL
This approach resolves the redundancy at the position level
rather than resolving the redundancy at the velocity level or at
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the acceleration level.
The resolved motion method has many interesting
characteristics [4]. A desired performance criterion function can
be incorporated in the general solution for the avoidance of joint
limits ([13], [17]), the improvement of the repeatability for
repeated end effector motions ([i], [ii]) , and the obstacle
avoidance in work space (e.g. [7], [12], [14], [17]) . Another
advantage is the least square property [3] of the pseudoinverse
method which minimizes the sum of squares of joint velocities
which in turn approximately minimizes kinetic energy. However,
this approach has an intrinsic inaccuracy because of the error
accumulation of the linear approximation of the Jacobian matrix.
Therefore, lack of repeatability is the major drawback of this
method (e.g. [4], [8]) . The approach involves the instantaneous
resolution of the redundancy at the velocity level where the sum
of the squares of the joint velocities is minimized. This means
that the kinetic energy is approximately minimized. This method
kinematically resolves the redundancy at the velocity level. Chang
[4] proposed a method called the criterion function method which
has resulted in improved repeatability for end effector motions.
Many researchers (e.g. [8], [i0] , [16]) extended the kinematic
resolution method from the velocity level to the acceleration
level by incorporating the generalized inverse into dynamics.
While this method resolves the redundancy at the kinetic level, it
may lead to stability problems. Local tampering with the
energetics of movement has led to global disaster [8].
3.3 SINGULARITIES AND AVOIDANCE
Some arm configurations are singular where the joint angles or
instantaneous joint velocities are impossible to realize. For the
resolved motion approach or other approaches, the joint velocities
for some manipulator configurations sometimes approach
mathematical infinity in the derivation. The joint velocities
that close to the singular points are also too large and are very
difficult to realize. This establishes forbidden regions around
the singular points. A substantial fraction of the workspace is
lost and the degree of freedom of the manipulator is functionally
reduced ([9], [15]).
Unlike other approaches, no singularity and singularity
avoidance consideration are necessary for this approach since the
resolution of redundancy is at the position level, and joint
angles are determined globally not incrementally (Refer to the
Appendix.) This approach is a geometrical approach where only
three inverse matrices appear in the derivation, two rotational
inverse matrices, Rz-I and Rx-l, and one translation inverse
matrix, T -I. These three are nonsingular matrices since their
determinants are non-zero.
That no singularity exists in this approach implies
improvement of the performance capabilities of the manipulator arm
because singular points functionally reduce manipulator degree of
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freedom. Besides, there is no need to consider the singularity
avoidance in the control or arm movement planning.
3.4 COMPLICATION OF APPROACH
Another major attraction of this approach is the simplicity of
the derivation (Refer to the Appendix.) The resolution of the
redundancy at the position level does not require incorporation of
the dynamics, pseudoinverse, etc. in the derivation. Compared to
other approaches, the derivation is drastically reduced. In
addition, this approach is analytical and geometrical. This means
that the resolution of the redundancy is not incremental or local,
but global. Thus, control and trajectory planning of the
manipulator are much more simplified than those of the other
approaches.
3. 5 OPTIMALITY
This approach geometrically applies the heuristic rules for
the resolution of the redundancy. For the resolved motion method,
the kinetic energy is approximately minimized; however, it has the
drawback of poor repeatability. Other extended methods of this
type which attempt to realize the energy minimization have other
problems also, such as stability. It would be a significant
research task to verify the assertion that heuristic application
of these rules does result in minimization of the energy
consumption for manipulator arm movements.
4. SUMMARY AND CONCLUSION
Resolution of seven-axis manipulator redundancy is a very
interesting and important research topic. Unlike most of the
approaches, the redundancy is resolved through the implementation
of heuristic rules in the derivation. It seems to the author that
this approach resolves the very complicated seven-axis redundancy
very easily and has no known drawbacks. In addition to this,
although yet to be verified theoretically, this approach may have
the most desirable feature of true minimization of the kinetic
energy. Since the resolution of the redundancy is at the position
level, the corresponding joint angles are derived analytically and
geometrically. Thus, the resulting joint angles are determined
globally not incrementally. This implies that, unlike other
approaches, the joint angles for a given hand position are given
directly or the joint angle space is a direct mapping from the
workspace. Another very significant advantage of the approach is
that there is no singular point in the derivation. Therefore,
compared with other approaches, this approach has a greater degree
of freedom functionally and has no singularity avoidance problem
in the workspace.
5. APPENDIX
Since the forward kinematics for redundant arms is always
feasible, these heuristics are applied only to the derivation of
inverse kinematics for the seven-axis anthropomorphic arms.
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According to the arm motion characteristics as described above,
the inverse kinematics of a two-link manipulator will be derived.
The two-link strategy for lower arm and forearm is then extended
to a three-link strategy with hand included ([5], [6]).
5.1 THE TWO-LINK STRATEGY
According to Figure 2,the arm positions are represented by the
following three points : the origin of the coordinates
Po(O,O,O), Pl(Xl,Yl,Zl) and P2(x2,Y2,Z2) .The point P2, (x2,,Y2,,z2,)
represents the desired position. When P2 reaches P2', the
collection of all possible locations of P1 forms a circle ( denoted
by C ) with center at Pc(xc, Yc, Zc) and radius r. The objective is
to find the point Pl*(Xl*,Yl*,Zl*) in C such that the distance
between P1 and PI* is minimized. Homogeneous coordinates should be
employed to simplify the required calculation. First of all, PcP2 ,
coordinates of PI* with respect to the original coordinates.
[ Xl*,Yl*,Zl*,l ]t = T-IRz-IRx-I [ Xl.^,yl.A, Zl.^,l ]t
5.2 THE THREE-LINK EXTENSION
By referring to Figure 3, let Lp = distance (P0, P3*) • If Lp >
11 + 12 + 13 or, Lp < 11 - ( 12 + 13 ) and 11 > 12 + 13 , then
there is no solution. Evaluate dl= ABS ( 12 - 13 ) , d 2 = 12 + 13,
and, D = distance (PI, P3*), If d I <= D <= d2, then do not move
PI, i.e., P1 = PI* Apply the "two-link" method (Section 5.1) to
find P2* If D < dl, then apply the "two-link" method with length
of link I = 11 and length of link2= ABS( 12 - 13 ) or d I to find the
new location PI* If D > d2, then apply the "two-link" method with
length of link I = 11 and length of link2= 12 + 13, or d2 to find
new PI*.
Because the joint limits are not imposed, PI*P2*P3* is always
in a straight line for the last two cases..
6. REFERENCES
[I] J. Beillieul, J. M. Hollerbath, and R. Brockett, "Programming
and Control of Kinematically Redundant Manipulators," 23rd
IEEE Conf. on Decision and Control, Dec, 12-14, 1984.
[2] M. Benati, P. Morasso, and V. Tagliasco, "The Inverse
Kinematic Problem for Anthropomorphic Manipulator Arms," ASME
J. of Dynamic Systems,Meas., and Control, Vol.104, pp.
110-113, Mar. 1982.
[3] A. Ben-Israel, and T. N. E. Greville, "Generalized Inverse
Theory and Applications," New York, Robert E. Krieger
Publishing Co.
540
[4] P. H. Chang, "A Closed-Form Solution for the Control of
Manipulators with Kinematic Redundancy," IEEE Int'l Conf. on
Robotics and Automation, pp. 9-14, 1986.
[5] I. Chen, T. C. Chen, and H. Saha, "Motion Simulation of an
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[6] I. Chen, T. C. Chen, and H. Saha, "A Strategy for Deriving
Inverse Kinematics for a Seven-Axis Manipulator Arms," 17th
Southeastern Symposium on System Theory, pp 124-127, March
24-26, 1985.
[7] H. Hanafusa, T. Yoshikawa, and Y. Nakamura, "Analysis and
Control of Articulated Arms," XIV-78- 83, Aug. 1981.
[8] J. M. Hollerbach, and K. C. Sub, "Redundant Resolution of
Manipulators through Torque Optimization," IEEE Int'l Conf. on
Robotics and Automation, pp. 1016-1021, Mar. 25-28,1985.
[9] J. M. Hollerbath, "Optimal Manipulator," 2nd Int'l Symp. on
Robotics Research, Kyoto, Japan, Aug. 20- 23,1984
[i0] O. Khatib, "Dynamic Control of Manipulators in Operational
Space," 6th Congress on Theory of Machines and Mechanism, Dec.
1983, New Delhi.
[ii] C. Klein, and C.Huang,"Review of Pseudoinverse Control for Use
with Kinematically Redundant Manipulators," IEEE Trans.on
System, Man and Cybernetics, Sol SMC-13, pp.245-250, 1983.
[12] C. A. Klein, "Use of Redundancy in the Design of Robotic
Systems," Preprints of the 2nd Int'l Symp. of Robotics
Research, Kyoto, Japan, pp. 58-65, Aug. 20-23, 1984.
[13] A. Liegeois, "Automatic Supervisory Control of the
Configuration and Behavior of Multibody Mechanisms," IEEE
Trans on System, Man and Cybernetics, Vol. SMC-7, No. 12,
1977.
[14] Y. Nakamura,and H. Hanafusa, "Task Priority Based Redundant
Control of Robot Manipulators," Preprints of the 2nd Int'l
Symp.of Robotics research, Kyoto, Japan, pp.357-364, August
20-23, 1984.
[15] R. P. Paul, and C. N. Stevenson, "Kinematics of Robot
Wrists," The Int'l Journal of Robotics Research, Vol. i, No.
2, 31-38, 1983.
• "A Dynamic approach to[16] M. Vukobratovic, and M Kircanski,
Nominal Trajectory Synthesis for Redundant Manipulators," IEEE
Trans. Systems, Man, Cybern., SMC-14, pp. 580-586, 1984.
[17] T. Yoshikawa, "Analysis and Control of Robot Manipulators with
Redundancy," The ist Int'l Symp., ed. Brady, M. and Paul, R.,
Cambridge, Mass., MIT Press, pp.735-748,1984.
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roll pitch roll
pitch
roll
Figure 1. An anthropomorphic arm
P1 [2 p2'(x2',y2',z2')
/ / l j) _* ")\
/ .- .- _ ) P2(x2,y2,z2)
eo (o,o,o) j_.__._f ---tz
[.! Pl(xl,yl ,zl)
Figure 2.
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P3*
/ \
/ \
/ _ P3
Lp / _D
/
Po
11 [2
P1
Figure 3.
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Resolution of seven-axis manipulator redundancy: A heuristic issue

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