human placement for maximum...
TRANSCRIPT
1
Human Placement for Maximum Dexterity
Karim Abdel-Malek and Wei YuDepartment of Mechanical Engineering
The University of IowaIowa City, IA 52242Tel. (319) 335-5676
[email protected]@engineering.uiowa.edu
Jerry DuncanHuman Factors/Ergonomics
Deere & Company Technical Center3300 River DriveMoline, IL 61265
Tel. [email protected]
Placement in ergonomic design is the problem concerning the specification of the
position of a human with respect to a pre-existing work environment. In an assembly line,
for example, it is advantageous to position a worker in such a manner to maximize his/her
dexterity, minimize the person’s stress on each joint, and maximize their reach. This
paper presents a rigorous mathematical approach that utilizes kinematic formulations
from robotics and optimization theory to define the placement problem. The concept
underling this approach is that the ergonomic design process is indeed an optimization
problem with many parameters. While only dexterity is presented in this paper, the
formulation is broadly applicable and can be generalized to the ergonomic design process
for any objective function or combination thereof. A measure of dexterity is developed
and examples are illustrated. The work is a part of a long term vision to establish a
fundamental formulation for ergonomic design.
Keywords: Human placement, dexterity measure, reachability, and ergonomics.
2
Introduction
In this paper, we characterize the ergonomic design process as an optimization problem
with many variables. We further believe that in order to achieve a rigorous approach to
this iterative process, the problem must be defined in terms of a multidisciplinary
approach and as a result mathematically obtain definitions for the various cost functions
that represent the measures of human performance.
Ergonomic design has traditionally depended on empirical data and rules of thumb as
evidenced by the many works that address the ergonomic design process. For example, a
method for the determination of ergonomic parameters that relate people to objects in
space was proposed by Costa, et al. (1997). The authors state that mathematical models
of human movements are complex to define and hard to solve and suggest the use of
Artificial intelligence in Neural Systems (ANS) as an approach to the problem. Indeed,
collection of data for simulation of human movement has been done by many researchers
(e.g., Ciungradi, et al. 1998), but there has never been a fundamental rigorous approach
to the problem. The study of human motion has led designers to produce ergonomic-
based workstations, by addressing work postures, work height, adjustable chairs,
foot/hand use, gravity, momentum within normal work area all in an experimental
manner (Kong 1990), which is costly and is difficult to use in a design scenario. The
proposed work will provide a viable venue to address such issues.
More rigorous work has recently appeared that has employed a mathematical model of
limbs as four-link systems consisting of trunk, upper arm, lower arm, and hand, being
3
regarded as a redundant manipulator with a total of eight degrees of freedom (DOF)
(Jung, et al. 1992; 1997; Jun and Kee 1996). The authors stated that inverse kinematics
were used for solving the system and that the joint range availability was used as a
performance function in order to guarantee local optimality (Jung and Park 1994).
Inverse kinematics is an expression given to the mathematics used in calculating joint
variables given the position and orientation of the end-link (e.g., hand). While inverse
kinematics of an eight DOF system is not only difficult to obtain, but is also unreliable
because of the many redundant solutions that may arise. Indeed, the concept of inverse
kinematics in ergonomic analysis and design should only be considered for non-
redundant systems. We first define the placement problem, introduce the needed
mathematics, and then demonstrate its applicability to ergonomic design.
Problem Definition
We will first describe the problem in rather general terms, followed by a more rigorous
approach quantifying our models. Consider the design of an assembly line in a
manufacturing environment where it is required to position a number of operators, each
to achieve a specific task. In order to maximize the output efficiency, it is necessary to
place the operators with respect to the assembly line while maximizing their dexterity. It
is also possible that the design would depend on a different maximization/minimization
function such as effort, stress, reachability, repetitive strain, or force. The formulation
developed in this paper will limit itself to the development of a measure for dexterity as
the driving cost function (so-called in the field of optimization) to optimize the
ergonomic placement of a person in a work environment.
4
Consider the reach envelope (shown in Fig. 1a) of a human that is known in closed form
(i.e., the equations of the envelope are known). Also consider a number of targets that
must be touched by the human in the work environment, which will be called target
points, as shown in Fig. 1b.
Reach envelope
Target points
Initialconfiguration
Configurationafter placement
Fig. 1 (a) A human and the reach envelope (b) The reach envelope positioned to includethree target points and the corresponding position of the human
It is required to position and orient the human in such a manner to touch all three points
while maximizing the dexterity at each point. To achieve this task, we will manipulate
the reach envelope and specify its position and orientation, which will be characterized
by the six coordinates w = x y zTα β γ . The first three coordinates identify the
envelope’s position and the last three its orientation.
In order to formulate the problem as an optimization problem, we define a function (also
called a cost or objective function) as f Dexterity= ( )w as a function of the w variables
5
(also called design variables), whereby this function must be minimized or maximized
subject to some constraints. These constraints characterize the target points falling
within the reach envelope, the joints being within their ranges of motion, and the person
being finally in the upright position. Once defined in this manner, an iterative numerical
optimization algorithm can be called upon to make the necessary calculations.
Plan of Work
In order to address the above problem in a mathematically rigorous manner, we have to
formulate the problem as an optimization algorithm (Arora 1989), whereby the design
variables characterize the position and orientation of the person and the design function is
a quantifying measure for dexterity. Therefore a measure for dexterity must be
developed. Furthermore, constraints imposed on the human’s final position include the
following: (a) Target points must be within reach (i.e., inside the reach envelope of the
person) and (b) Target points are not on the boundary of the workspace envelope, (c)
Joint ranges of motion must be considered, and (d) The final position of the human within
a finite space.
Modeling Scheme
In order to obtain a systematic representation of the workspace produced by the motion of
a point of interest (typically called a point on the end-effector), we will use the Denavit-
Hartenberg method adopted from the field of robotics (Denavit and Hartenberg 1955;
Abdel-Malek, et al. 1997; 1999). Consider Fig. 2 where three consecutive links are
shown.
6
Link m+1
Link m
Link m-1
Figure 2 Define the joint reference frames for the D-H representation
Let Zi-1 and Zi represent fixed axes at either end of link i-1, about which or along
which links i-1 and i move, respectively (as shown in Fig. 2). Let axes Xi-1 be defined
from Zi-1 to Zi and perpendicular to both. Let Yi-1 represent the unique axis that together
with Xi-1 and Zi-1 completes a right-hand Cartesian coordinate system. Let Z i’ represent
a vector from Oi-1 parallel to Zi . Let X i’-1 represent a vector from Oi parallel to Xi-1 as
illustrated in Fig. 3.
Zi-1
Zi
Joint i
Joint i-1
Yi
Xi-1
Oi
Oi-1
ai
Xi
di
θi
αi
Fig. 3 The relation between two consecutive coordinates
7
The following four ordered operations completely specify the configuration of the frame
i coordinate system relative to the frame (i-1) coordinate system:
(a) A constant twist of α i degrees about axis Xi-1 of Zi-1 into Z i’ , here Z i
’ is parallel to
Zi , α i is the angle from Zi-1 to Z i’ , with transform matrix Tai :
Taii i
i i
=
-
�
!
"
$
####
1 0 0 0
0 0
0 0
0 0 0 1
cos sin
sin cos
α αα α
(1)
(b) A constant displacement of bi units along Xi-1 from Zi-1 to Zi , with transformation
matrix Tbi given:
Tbi
ib
=
�
!
"
$
####
1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
(2)
(c) A rotation of θ i degrees about Zi of X i’-1 into Xi , here θ i is the angle from X i
’-1 to
Xi , X i’-1 is parallel to Xi-1 , with transformation matrix Tci given as:
Tci
i i
i i=
-�
!
"
$
####
cos sin
sin cos
θ θθ θ
0 0
0 0
0 0 1 0
0 0 0 1
(3)
(d) An offset of di units along Zi from the Xi-1 - Zi intersection to Oi , with
transformation matrix Tdi as:
Tdiid
=
�
!
"
$
####
1 0 0 0
0 1 0 0
0 0 1
0 0 0 1
(4)
8
Note that the four parameters iiii ad αθ ,,, completely define the relation between any two
consecutive frames. These values are entered in a table, which is typically known as the
DH Table. The overall Denavit-Hartenberg coordinate transformation matrix from frame
i coordinate system relative to the frame i 1- coordinate system is then given by:
T T T T Ti 1i
ai bi ci di
i i i
i i i i i i i
i i i i i i i
b
d
d-
= =
-
- -
�
!
"
$
####
cos sin
cos sin cos cos sin sin
sin sin sin cos cos cos
θ θα θ α θ α αα θ α θ α α
0
0 0 0 1
(5)
Similarly, for an n-DOF model of a limb, the global joint and end-effector frames using
Eq. (5) are restated using n-homogeneous transformation matrices
T , T , T , T , , T01
12
23
34
n 1n
L
-
. The transformation matrix from the end-effector frame to
global frame is then obtained by pre-multiplying each matrix in series as:
T q T T T T0n
01
12
23
n 1n
nq q q q( ) ( ) ( ) ( ) ( )=-1 2 3 L (6)
where q= [ ... ]q qnT
1 are the generalized coordinates (joint variables) of the limb and
where the resulting transformation matrix TR X
00n 0
n0n
1 3
=
�!
"$#�
1 contains the ( 3 3� ) R0
n
rotation matrix and ( )3 1� X0n position vector, which represents every point that can be
touched by the index finger (or any other specified part). Therefore, the reach envelope is
described by the vector
x X q= =[ ] ( )x y z 0n (7)
where x R R: n�
3 is a smooth vector function defined as a subset of the Euclidean space.
However, the boundary of the reach envelope is not yet known. In order to determine
the boundary, we apply a rank deficiency condition to n0X . The so-called Jacobian is
obtained from differentiating the position vector n0X as follows:
9
& &x X q q= � � (8)
where &x represents the absolute velocity of the hand and &q represents the vector of joint
velocities. Therefore, the Jacobian X X qq = � � relates both velocities. It was shown by
Abdel-Malek, et al. (1997; 1999) that consecutive application of a rank deficiency
condition on the Jacobian qX yields singular sets denoted by is , which define the
boundary to the reach envelope. The benefits of this method is two fold:
(a) The reach envelope boundary is determined in closed form.
(b) The reach envelope boundary is exact.
Surface patches on the boundary of the reach envelope of human limbs are delineated in
closed form by substituting the singular sets into the equation for the reach envelope.
Y( ) ( ) ( , )i
iu X s q= (9)
where u are the remaining variables (recall that s is a set of constants). These Y( )i are
indeed surface patches in closed form and characterize the boundary of the reach
envelope (Abdel-Malek, et al. 2000). For example, if the base coordinates are embedded
in the shoulder, and we are seeking the reach envelope of the tip of the finger with respect
to the shoulder, then the surface patches are obtained and shown in Fig. 4.
10
Point sp
WCS
q1x(q)
Fig. 4 The reach envelope of the upper extremity
Consider a number of target points p( ) ; ,...,j j =1 l , located in space, whereby it is
required to place the human such that these target points are within reach yet maximizing
the dexterity function (to be developed in the following section). To ascertain that target
points p( )j are inside the reach envelope, the absolute value of the distance between a
target point and the boundary should be greater than a specified value ε (a specified
tolerance). This will guarantee that the target points are located inside of the reach
envelope but not on boundary.
In order to track the position and orientation of the envelope, we shall use a set of 6
generalized coordinates w = x y zw w w
Tα β γ , where a position vector
v( , , )x y zw w w will be used to track the position and a rotation matrix R( , , )α β γ will be
used to track the orientation.
The distance between all target points p( )j and all surface patches Y( ) ( )i u should be
greater than a specified minimum value such as
11
p u w( ) ( ) ( )( , )j i ij− ≥Y ε where j =1,...,l and i m=1,..., (10)
where Y x( ) ( )( , ) ( , , ) ( ) ( , , )i iw w wx y zw u R u v= ⋅ +α β γ (11)
and where R is the rotation matrix that will be used to orient the reach envelope, v is the
position vector that will be used to locate the envelope, and where ε j > 0 are specified
constants. If a target point satisfies both conditions of Eq. (4) and (5), then this point is
internal to the reach envelope (i.e, have placed the envelope in a configuration such that
all points can be reached.
In order to move the human to a new position, we will move the reach envelope towards
the target points, subject to the following constraints:
(1) Reach envelope at least covering the target points (shortest distance between the
target points):
gii≡ − ≤min ( , )( )p q wG β for i =1,...,l (12)
where G F( , ) ( , , ) ( ) ( , , )w q R q v= +α β γ x y zw w w and β is a very small positive number
and subject to the ranges of motion or joint limits as
q q qkL
k kU
� � for k n=1,..., (13)
(2) Embedding the target points inside the reach envelope (a minimum distance between
target points and surface patches).
gkj i
j≡ − ≥p u w( ) ( ) ( , )Y ε for i m=1,..., and j =1,...,l, k m= �1,..., ( )l (14)
where ε j is the depth of the target point inside the reach envelope. There are
l l+ � +( )m n total number of constraints. We have now defined all constraints but have
12
not provided for a cost function to optimize. We will address this task in the following
section.
A Measure of Human Dexterity
In this section, we define a cost function that is based on maximizing the dexterity at
target points. Indeed, to mathematically formulate this problem, it is necessary to use a
dexterity measure at specific target points and that is a function of the design variables w.
Such a measure must account for the ranges of motion for each joint. Because of the
need for an analytical expression that can be used in the proposed optimization approach,
we define a new dexterity measure.
Because human joints are constrained, we must characterize each joint limit by an
inequality constraint in the form of q q qiL
i iU
� � . In order to include ranges of motion in
the formulation, we have used a parameterization (see Appendix A) to convert
inequalities on qi to equalities qi i= L( )λ , where the new variables are defined by
l = ∈λ λ λ1 2, ,..., n
T nR . For the hand at a given location xh (i.e., for the hand at a
specific position that can be reached), X x 00n
h- = must be satisfied. Moreover, the
parameterized constraints of the ranges of motion must also be satisfied as L- =q 0 .
Therefore, the general constraint can be obtained by augmenting both equations to obtain
the ( )n + 3 constraint vector as
G qX q x
q0( )
( )*
( )
=
-�!
"$#
=
+ �
0
3 1
nh
nL l( ) −
(15)
13
where the augmented vector of generalized coordinates is q x q* [ ]= T T T Tl . By
defining a new vector z q= [ ]T T Tl (input parameters), the augmented coordinates can
be partitioned as
q x z*=
T T T(16)
The set defined by G q( )* is the totality of points in the reach envelope that can be
touched by the hand. The so-called extended Jacobian of G q( )* is obtained by
differentiating G with respect to z as
GX 0
Izq
=
�!
"$#L
l
(17)
which is an ( ) ( )n n+ ×3 2 matrix, where Xq = ∂ ∂X qi j is a ( )3× n matrix, I is the
( )n n× identify matrix, and Ll= � �Λi jλ is an ( )n n× diagonal matrix with diagonal
elements as Λλ λ) cosii i ib= . We define Gz as the augmented Jacobian matrix.
Since the extended Jacobian Gz inherently combines information about the position,
orientation, and ranges of motion of the hand, it is a viable measure of dexterity.
Furthermore, because of the simplicity in determining an analytical expression of Gz , it is
well-suited as a cost function for an optimization problem. We define the dexterity
measure as
D T= G Gz z (18)
Note that the measure characterized by Eq. (18) takes into consideration all ranges of
motion and singular orientations for a given arm, limb, or any serial chain.
14
The Placement Problem as an Optimization Algorithm
Given l target points p( ) ( , , )ii i ix y z for i =1 2, , ,L l defined in space, we introduce a
dexterity measure at each point denoted by D i( )( )p , where it is necessary to place a
human to achieve maximum dexterity at each target point.
Suppose the vector describing the point on the index finger of a human arm is given by
X q( ) and the boundary envelope of workspace are determined as closed-form surface
patches denoted by Y( ) ( )( ), , , ,j j j lu =1 2L . A mathematical model of the placement of
the robot base subject to maximizing the dexterity at specified target points is
characterized by the following optimization problem.
Cost function
Maximize f Dii
i
( ) ( , )( )w w p=
=
Êω 1
l
(19)
Subject to the following constraints:
(1) Target points are inside the workspace volume
P w u( ) ( ) ( )( , )i j iij- �G ε (20)
where i j m= =1 2 1 2, , , , , ,L l L and .
(2) Target points are not on the boundary of the workspace envelope
min ( , ) , , , ,( )P X w qii i l- � =δ where 1 2L (21)
where δ i i, , , ,=1 2L l, ε ij i, , , , ,=1 2L l and j m=1 2, , ,L are positive constants.
(3) Ranges of motion are imposed
q q qL U� � (22)
15
(4) The motion is within a finite space
w w wL U� � (23)
Because this is a multi-objective optimization problem, we will assign the dexterity at
each point a weight so as to transfer the problem into a classical optimization problem
where w i i l, , , ,= 1 2L are weights at each point.
We now introduce the general iterative numerical algorithm for solving the optimization
problem in Fig. 5. Input to the algorithm is a complete definition of the human (i.e., the
DH Table). For all practical purposes, the dimensions and joint ranges of motion of an
upper extremity is enough to manipulate the placement. Also as an input, we define in
terms of equations, the surface patches that characterize the boundary of the reach
envelope. These surfaces are defined with respect to the coordinate system in the
shoulder. We also define the cost function as a measure of dexterity and define the
constraints for this optimization problem. The numerical algorithm iteratively moves the
reach envelope towards including the target points within the envelope yet attempting to
maximize the cost function (i.e., dexterity). The algorithm will stop when all of the
tolerances are satisfied.
16
Define target points Reach envelope has been identified
Cost function
Constraints (no need for inverse kinematics)
Move boundary of reach envelope
Satisfies Tolerance
Stop
Iterate
w = w + ∆wIterative algorithm to move the workspace
Define human (dimension and ranges of motion)
Defined by the six generalized coordinates w that characterize its position and orientation
Maximize Dexterity
Fig. 5 Algorithm for Placement
Simple Example
Consider a model of the arm that is restricted to move on a planar surface (e.g., the
surface of a table). This example will be used to illustrate the theory and will be followed
by a more realistic model of the upper extremity. There are three target points, namely,
T]1014[)1( =P , T]50100125[)2( =P , and T]75125125[)1( =P , which must be
touched by the human hand with the ability to reach these points from many directions
(i.e., maximizing the dexterity of the upper limb at these points).
17
The arm shown in Fig. 6 is modeled as three revolute joints and restricted to planar
motion (e.g., on the surface of a table).
Fig. 6 Model of the arm with three revolute joints
The DH table is readily determined and presented in Table 1.
Table 1: DH Tableθ i di α i ai
1 q1 0 0 42 q2 0 0 23 q3 0 0 1
Substituting each row into Eq. () and performing the multiplication yields the coordinates
of the tip of the index finger are given by
++++++++++
=)sin()sin(2sin4
)cos()cos(2cos4)(
321211
3212110 qqqqqq
qqqqqqn qX (24)
For simplicity, ranges of motion on each joint are defined as 33 ππ ≤≤− iq ; 3,2,1=i .
Results of the reach envelope determination yield the following boundary curves (note
that curves are generated because we have restricted the arm to planar movement. The
boundary curves are defined by the following sets:
]0 ,3[ );0,,3( 22 ππ −∈qqx , ]3,0[ );0,,3( 22 ππ ∈qqx ,
] 3,3[ );0,0,( 21 ππ−∈qqx ]0,3[ );,3,3( 33 πππ −∈−− qqx
]3,0[ );,3,3( 33 πππ ∈qqx ]3,3[ );3,3,( 11 ππππ −∈−− qqx
18
and ]3,0[ );3,3,( 11 πππ ∈qqx .
Subsitituting the singular sets into Eq. () yields equations of curves shown in Fig., which
is the reach envelope as shown in Fig. 7.
x
1.0
2.0
3.0
4.0
1.02.03.04.0
y
Fig. 7 The exact reach envelope of the arm (restricted to planar motion)
As a result of the iterative algorithm, the six design variables representing the position
and orientation of the reach envelope are calculated as
[ ] [ ]TTyx 674.0663.4714.10== αw and shown in Fig. 8. The measure of
dexterity at each point is calculated as
D 80.79)( )1( =P
D 040.28)( )2( =P
D 914.16)( )3( =P
19
Note that any configuration that would have included the three points is a solution,
however, the solution calculated using this method yields the position of the arm that
would maximize the dexterity at all three points.
-10 -5 5 10 15 20
-10
-5
5
10
15
20
Fig. 8 The initial and final position of the arm.
Example: A Realistic Model of the Arm
Consider the upper extremity shown in Fig. 9 and modeled as a total of 9DOF (5DOF in
the shoulder, 1DOF in the elbow, and 3DOF in the wrist). Note that we have accounted
for two translations of the shoulder joint and three rotational motions. Also note that in
order to allow for realistic motion of the shoulder, we have coupled the translational and
rotational joints by a linear equation such that the behavior of the motion of one joint is
dependent upon the other.
20
q1
q2
q3
q4
q5
q6
q7
q8
q9
xo
yo
X(θ)
Fig. 9 Model of the upper extremity
The D-H parameters for this arm are presented in Table 2.
Table 2: DH Table for the armθ i di ai α i
1 π 2 q1 0 -π 22 -π 2 q2 0 π 2
3 0+q3 0 0 π 2
4 π 2+q4 0 0 π 2
5 0+q5 0 20 cm -π 26 0+q6 0 25 cm π 2
7 0+q7 0 0 -π 28 q8-π 2 0 0 -π 29 0+q9 0 0 0
21
Ranges of motion for this arm are as follows (note that the first two joints are
translational): - � �38 381. .q cm ; - � �38 382. .q cm ; - � �π π2 23q ;
- � �11 8 2 34π πq ; - � �π π2 25q ; 0 5 66� �q π ; - � �π π3 37q ;
- � �π π9 98q ; and - � �π q9 0.
It is required to place the human such that the following three target points are touchable
and dexterity is maximized.
T]50100100[)1( =P , T]50100125[)2( =P , and T]75125125[)1( =P .
The position of the index finger
x c c s s s c c c s s s c c s c c c c c s
s s c c s c s c s s s c c c c c s s s s s
( )[ ] ( ) ( ( ( (
) ) ( ) ) ( ( ) )
q 1 20 20 10 10 51 3 2 1 3 4 1 3 2 1 3 1 2 4 6 5 4 1 3 2
1 3 1 2 4 3 1 1 2 3 5 1 2 4 1 3 2 1 3 4 6
= - + + - + - + -
+ - + + + - - +
x( )[2] ( ) ( ( ( (
) ) ( ) ) ( ( ) )
q = - - + - - - + -
- - + - + + - - - -
20 20 10 10 53 1 2 1 3 4 3 1 2 1 3 2 1 4 6 5 4 3 1 2
1 3 2 1 4 1 3 1 2 3 5 2 4 1 3 1 2 1 3 4 6
c s s c s c c s s c s c s s c c c c s s
c s c s s c c s s s s c c s c s s c s s s x q [ ]3( ) ( ( ( ) )
( ) )
= + - + - -
+ - -
20 10 10 52 3 2 3 4 2 4 6 5 2 3 4 2 4 2 3 5
4 2 2 3 4 6
c c c c c s s c c c c c s s c s s
c s c c s s
where c q1 1= cos , s q1 1= sin , and qT
= q q1 7... . Note that the first two joints are
coupled and therefore the model is reduced to 7DOF. The boundary surfaces are
delineated and shown below in Fig. 10 (a total of 22 boundary surfaces):
22
-4-2024X
-100
-75
-50
-25
0
Y
-4-2024Z
-4-2024X
-100
-75
-50
-25
0
Y
-4-2024X
-50
-25
0
25
50
Y
-60
-40
-20
0
Z
-4-2024X
-50
-25
0
25
50
Y
-4-2024X
-50
-25
0
25
50
Y
0
20
40
60
Z
-4-2024X
-50
-25
0
25
50
Y
0
20
40
60X
-50
-25
0
25
50
Y
-4-2024Z
0
20
40
60X
-60
-40
-20
0X
-50
-25
0
25
50
Y
-4-2024Z
-60
-40
-20
0X
010
2030X
0
25
50
75
100
Y
-20
0
20
Z
010
2030X
0
25
50
75
100
Y
-30-20
-100X
0
25
50
75
100
Y
-20
0
20
Z
-30-20
-100X
0
25
50
75
100
Y
-4-2024X
0
25
50
75
100
Y
-20
0
20
Z
-4-2024X
0
25
50
75
100
Y
010
2030X
0
25
50
75
100
Y
-20
0
20
Z
010
2030X
0
25
50
75
100
Y
-30-20
-100X
0
25
50
75
100
Y
-20
0
20
Z
-30-20
-100X
0
25
50
75
100
Y
-4-2024X
0
25
50
75
100
Y
-20
0
20
Z
-4-2024X
0
25
50
75
100
Y
-20
0
20X 0
25
50
75
100
Y
0
10
20
30
Z
-20
0
20X
-20
0
20X 0
25
50
75
100
Y
-4-2024Z
-20
0
20X
-20
0
20X 0
25
50
75
100
Y
-30
-20
-10
0
Z
-20
0
20X
-20
0
20X 0
25
50
75
100
Y
-4-2024Z
-20
0
20X
-20-10
010
20X
0
25
50
75
100
Y
-20
-10
0
10
20
Z
-20-10
010
20X
0
25
50
75
100
Y
-50-25
0
25
50X
-50
-25
0
2550
Y
-50
-25
0
25
50
Z
-50-25
0
25
50X
-50
-25
0
2550
Y
-20
0
20X 0
25
50
75
100
Y
-30
-20
-10
0
Z
-20
0
20X
-20
0
20X 0
25
50
75
100
Y
0
10
20
30
Z
-20
0
20X
Fig. 10 Surface patches
These surface patches are combined, the 9DOF reach envelope is calculated, and shown
in Fig. 11.
23
-40-20
020
40
X
-200
20
Y
-40
-20
0
20
40
Z
-40
-20
0
20
40
Z
-40-2002040
X
-200
20Y
-40
-20
0
20
40
Z
-40-2002040
X
Fig. 11 Reach envelope of the upper extremity
As a result of the iterative algorithm, the six design variables representing the position
and orientation of the reach envelope are calculated as
[ ]T736.0463.1850.0171.94444.83235.107 −=w
The measure of dexterity at each point is maximized and its value is
D 83.11687907)( )1( =P
D 47.18419793)( )2( =P
D 54.13962690)( )3( =P
The initial and final configurations of the reach envelope of the arm are shown in Fig. 12.
24
Fig. 12 Initial and final configurations of the reach envelope
Conclusions
A rigorous mathematical formulation for placement of humans in a work environment
while maximizing dexterity has been introduced. Ergonomic design has traditionally
been dependent upon empirical data and rules of thumb, mainly because of the many
variables associated with the design process. We believe this approach is an initial step
towards making the ergonomic design process more rigorous and introducing well-
established methods from optimization. It was shown that optimization algorithms can
be used in an iterative manner to calculate the optimum position and orientation of a
human while considering a multitude of constraints. Furthermore, it was shown that
25
realistic ranges of motion are considered in the formulation and a new performance
measure (cost function) was developed and used to evaluate dexterity at given target
points.
References
1. Abdel-Malek, K. and Yeh, H.J., 1997, “Analytical Boundary of the Workspace forGeneral Three Degree-of-Freedom Mechanisms,” International Journal of RoboticsResearch. Vol. 16, No. 2, pp. 198-213.
2. Abdel-Malek, K., 1996, “Criteria for the Locality of a Manipulator Arm with Respectto an Operating Point,” IMEChE Journal of Engineering Manufacture, Vol. 210 (1),pp. 385-394.
3. Abdel-Malek, K., Adkins, F., Yeh, H.J., and Haug, E.J., 1997, “On the Determinationof Boundaries to Manipulator Workspaces,” Robotics and Computer-IntegratedManufacturing, Vol. 13, No. 1, pp.63-72.
4. Abdel-Malek, K., Yeh, H-J, and Khairallah, N., 1999, “Workspace, Void, andVolume Determination of the General 5DOF Manipulator, Mechanics of Structuresand Machines, 27(1), 91-117.
5. Abdel-Malek, K. and Yeh, H. J., (2000) "Local Dexterity Analysis for OpenKinematic Chains," Mechanism and Machine Theory, Vol. 35, pp. 131-154.
6. Abdel-Malek, K., Yang, J., Brand, R., and Vannier, M., (submitted) “Understandingthe Workspace of Human Limbs”, International Journal of Ergonomics.
7. Arora, J.S., 1989, Introduction to Optimum Design, McGraw-Hill Book Co., NewYork.
8. Chair, P.; Chaffin, D.B., “Human simulation modeling - will it improve ergonomicsduring design? “, Proceedings of the 1997 41st Annual Meeting of the Human Factorsand Ergonomics Society. Part 1 (of 2) v 1 1997 Albuquerque, NM, pp. 685-687.
9. Chou, H.C.; Sadler, J.P., 1993, “Optimal location of robot trajectories forminimization of actuator torque”, Mechanism and Machine Theory, Vol. 28, No. 1,pp. 145-158.
10. Ciungradi, B.; Costa, M.; Pasero, E.; Macchiarulo, L., 1998, “Recurrent network fordata driven human movement generation”, Proceedings of the 1998 IEEEInternational Joint Conference on Neural Networks. Part 1 (of 3) May 4-9 1998 v 11998 Anchorage, AK, pp. 216-220.
11. Costa, M.; Crispino, P.; Hanomolo, A.; Pasero, E., “Artificial neural networks and thesimulation of human movements in CAD environments”, Proceedings of the 1997IEEE International Conference on Neural Networks. Part 3 (of 4) Jun 9-12 1997 v 31997 Houston, TX, pp. 1781-1784.
12. Denavit, J., and Hartenberg, R.S. 1955 A kinematic notation for lower-pairmechanisms based on matrices. Journal of Applied Mechanics, ASME, Vol. 22, pp.215-221.
26
13. Design Optimization Toolkit (DOT) (1999) Users Manual, VR&D,http://www.vrand.com.
14. Feddema, J.T., 1995, “Kinematically optimal robot placement for minimum timecoordinated motion”, Proceedings of SPIE – Vol. 2596, Philadelphia, PA, , USA,Sponsored by : SPIE - Int Soc for Opt Engineering, Bellingham, WA, pp. 22-31.
15. Feddema, J.T., 1996, “Kinematically optimal robot placement for minimum timecoordinated motion”, Proceedings of the 1996 IEEE 13th International Conferenceon Robotics and Automation, Part 4, pp. 22-28.
16. Hemmerle, J.S.; Prinz, F.B., 1991, “Optimal path placement for kinematicallyredundant manipulators”, Proceedings of the 1991 IEEE International Conference onRobotics and Automation, Vol. 2, pp. 1234-1244.
17. Ji, Z., 1995, “Placement analysis for a class of platform manipulators”, Proceedingsof the 1995 ASME Design Engineering Technical Conferences, Sep 17-20 1995, Vol.82, No. 1, pp. 773-779.
18. Ji, Z., Li, Z., 1999, “Identification of placement parameters for modular platformmanipulators”, Journal of Robotics Systems, Vol. 16, No. 4, pp. 227-236.
19. Jung, E.S.; Choe, J.; Kim, S.H., “Psychophysical cost function of joint movement forarm reach posture prediction”, Proceedings of the 38th Annual Meeting of the HumanFactors and Ergonomics Society. Part 1 Oct 24-28 1994 v1 1994 Nashville, TN, pp.636-640.
20. Jung, E.S.; Kee, D., “Man-machine interface model with improved visibility andreach functions”, Computers & Industrial Engineering v30 n3 July 1996, pp. 475-486.
21. Jung, E.S.; Kee, D.; Chung, M.K., “Reach posture prediction of upper limb forergonomic workspace evaluation”, Proceedings of the 36th Annual Meeting of theHuman Factors Society. Part 1 (of 2) Oct 12-16 1992 v 1, 1992 Atlanta, GA, pp. 702-706.
22. Jung, E.S.; Park, S., “Prediction of human reach posture using a neural network forergonomic man models”, Proceedings of the 16th Annual Conference on Computersand Industrial Engineering, Mar 7-9 1994 v27 n1-4 Sep 1994 Ashikaga, Japan, pp.369-372.
23. Konz, S., “Workstation organization and design”, International Journal of IndustrialErgonomics, v 6 n 2 Sep 1990 p 175-193.
24. Lu, Y.C. “Singularity Theory and an Introduction to Catastrophe Theory”, Springer-Verlag, New York, 1976
25. Molenbroek, J. F.M., “Reach envelopes of older adults”, Proceedings of the 199842nd Annual Meeting ’Human Factors and Ergonomics Society', Oct 5-Oct 9 1998 v1 1998 Chicago, IL, pp. 166-170.
26. Nelson, B., Pedersen, K., and Donath, M., 1987, “Locating assembly tasks in amanipulator’s workspace”, IEEE Proceedings of the International Conference onRobotics and Automation, pp. 1367-1372.
27. Pamanes, G.J.A.; Zeghloul, S., 1991, “Optimal placement of robotic manipulatorsusing multiple kinematic criteria”, Proceedings of the 1991 IEEE InternationalConference on Robotics and Automation, Vol. 1, pp. 933-938.
28. Pamanes, J.; Zeghloul, S.; Lallemand, J., 1991, “On the optimal placement and taskcompatibility of manipulators”, Proceedings of the ICAR Fifth InternationalConference on Advanced Robotics - '91 ICAR, pp. 1694.
27
29. Papadopoulos, E. and Gonthier, Y., 1995, “On manipulator posture for planning forlarge force tasks”, Proc. Of the IEEE Int. Conf. On Robotics and Automation,Nagoya, Japan.
30. Richards, J. 1998, “The Measurement of Human Motion: A Comparison ofCommercially Available Systems”, Proceedings of the 5th International Symposiumon the 3-D analysis of Human Movement, Chabanacoque, Tennessee
31. Roth, B., 1991, “On the number of links and placement of telescopic manipulators inan environment with obstacles”, Proceeding of the Fifth International Conference onAdvanced Robotics - '91 ICAR, pp. 988.
32. Seraji, H., 1995, “Reachability analysis for base placement in mobile manipulators”,Journal of Robotic Systems, Vol. 12, No. 1, pp. 29-43.
33. Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.34. Tu, Q.; Rastegar, J., 1993, “Determination of allowable manipulator link shapes; and
task, installation, and obstacle spaces using the Monte Carlo Method”, Journal ofMechanical Design, Transactions of the ASME, Vol. 115, No. 3, pp. 457-461.
35. Vincent, T.L. Goh, B.S. and Teo, K.L., 1992, “Trajectory-following algorithms formin-max optimization problems”, Journal of optimization theory and applications,Vol. 75, No. 3, pp. 501-519.
36. Zeghloul, S. and Blanchard, M. A, 1997, “SMAR: A robot modeling and simulatingsystem Robotica, 1997, Vol. 15, pp. 63-73.
37. Zeghloul, S.; Pamanes-Garcia, J.A., 1993, “Multi-criteria optimal placement of robotsin constrained environments”, Robotica, Vol. 11, pt 2, pp. 105-110.
38. Zhang, et al. (1997) used dynamic human models that are empirically validated for atypical range of driving tasks and for people of varied anthropometry, gender and age.
39. Zhang, X.; Chaffin, D.B.; Thompson, D., “Development of dynamic simulationmodels of seated reaching motions while driving”, SAE Special PublicationsProgress with Human Factors in Automotive Design: Seating Comfort, Visibility, andSafety 1997 v 1242, Detroit, MI, pp. 101-105.
Appendix ARanges of motion are imposed in terms of inequality constraints in the form of
q q qiL
i iU≤ ≤ (a.1)
where i n= 1, ... . We transform the inequalities above into equalities by introducing a
new set of generalized coordinates l = [ ... ]λ λ1 nT such that
q q q q qi iL
iU
iU
iL
i= + + −( ) ( ) sin2 22 7 2 7 λ i n= 1,..., (a.2)