Table Of Content16th International Conference SAER-2002, sept.20-22,2002, Varna, Bulgaria 95
APPLICATION-ORIENTED LANGUAGE
FOR DESCRIBING OPEN KINEMATIC CHAINS
Kaloyan Yankov, Ph.D.
Medical Faculty, Thracian University,
Armejska str., 11, Stara Zagora 6000, Bulgaria
e-mail: [email protected]
Abstract: In this paper an application-oriented language for describing the spatial open
kinematics chains is presented. The program interpreter generates the kinematics model of the
chain and its motion equations in analytical form. The language is used as a user interface in
system CINDY for modeling and studying the human limbs and robot manipulators.
Key words: Biomechanics, Computer Aided Design, Programming Languages, Robotics,
Symbolic Computation.
1. INTRODUCTION
The modeling of lokomotory system of human beigs and animals is a significant for various
applications:
- Studying and evaluation of kinematics characteristics of limbs;
- Definition of biomechanical parameters of regional and local human motions in norma and
pathology;
- Evaluation of motion power for patients with peripheral neuropathy and prescription for
rehabilitation;
- Control and regulation of computer-aided motion of invalids;
- Design of supervisor control of mechanisms;
The lokomotory system of living organisms could be modeled as open kinematics chain (OKC).
OKC is a system of rigid bodies having two or more pairing elements, which connect it to other
bodies for the purpose of transmitting force or motion. A pair is a joint between the surfaces of two
rigid bodies that keeps them in contact and relatively movable. OKC consists of kinematic pairs,
which are revolute (R) and prismatic (Р). They define independent kinematics parameters -
generalized joint coordinates:
q = (q ,q ,....,q )T (1)
1 2 N
q is the OKC - configuration. Тo express the motions the kinematics equations are necessary. A
local Cartesian coordinate system is established for each rigid body. To the firm solid a base
coordinate system O X Y Z is attached, and for i-th kinematic pair the frame is OXYZ (i=1...N,
o o o o i i i i
N is the number of joints). Let
B = (x ,y ,z ,1)T (2)
N N N N
is the characteristic end-effector point in N-th frame. Its homogeneous coordinates in O X Y Z
o o o o
frame are:
П k=1,2,…,N (3)
B = А (q )*B
0,N k-1,k k к
where А is the transformation matrix from (k-1)-th to k-th frame. That is so called “right
k-1,k
kinematic problem”. The kinematic equations are very important in design and control of OKC. We
can obtain them in numerical or analytical (symbolic) form. The analytical one gives us quantitative
estimation of OKC. Some of advantages of symbolic form are:
- Explicit view of kinematic parameters and ability to appreciate its importance in the model [1];
- Increasing the precision during integration of dynamics differentiation equations [2];
- Decreasing the computation time for solving the right and inverse kinematics problem [3];
- Explicit view of inertia and dissipation parameters [4];
16th International Conference SAER-2002, sept.20-22,2002, Varna, Bulgaria 96
The deriving of analytical equations for hyper-redundant chains is time-consuming and human
error is very probable. The systems for symbolic computations, general or specialized, are useful
aid in this case [5,6]. An advantage in such system is the availability of application– oriented
language.
In this paper a language for describing the spatial open chains and generation of its analytical
equations is presented. The language is a part of user interface in the CINDY simulation system
[7,8].
2. CHARACTERISTIC EQUATIONS OF THE OPEN KINEMATICS CHAIN
Denavit-Hartenberg notation is widely used in the transformation of coordinate systems of
linkages [9]. It can be used to represent the transformation matrix between links as shown in the
Fig1:
a) axis Z coincide with the axes of the i-th kinematics pair;
i
b) axis X is a common perpendicular to the Z and Z : X = Z x Z;
i i-1 i i i-1 I
c) axis Y is chosen to define right-oriented frame OXYZ.
i i i i i
Relative position of two coordinate systems attached to adjoining joints is described with fourth
parameters (Denavit-Hartenberg parameters):
- θ is the included angle of axes X and X . When the
i i i–1
joint is revolute, θ is variable and its value changes
i
between restrictions [θmin,θmax];
i i
- s – the distance connecting the axis X и X ; This
i i i–1
distance will be variable for a prizmatic joint. Its value
changes between restrictions [smin,smax];
i i
- a is the shortest distance connecting the axis Z and Z;
i i-1 i
- α - is enclosed by the axes z and z;
i i-1 i
The last two parameters are constant and depend on
Fig.1 chain structure. Transformation from (i-1)-th to i-th frame
get up with four elementary transformations:
1. Rotation on angle θ around Z until X and X get parallel. This operator is denoted by
i i –1 i i –1
RQ if θ is measured in radians, and by RQD if the angle dimension is in degrees.
2. Translation of О along Z at distance s until X и X get in line;
i i –1 i i i –1
cosθ -sinθ 0 0 1 0 0 0
i i
RQ(θ) = sinθ cosθ 0 0 (4) TS(s) = 0 1 0 0 (5)
I i i i
0 0 1 0 0 0 1 S
i
0 0 0 1 0 0 0 1
3. Translation of О along X at distance a until О and О. coincide;
i i –1 i i-1 i
4. Rotation on angle α around X until Z and Z coincide;
i i –1 i i –1
1 0 0 a 1 0 0 0
I
TA(a) = 0 1 0 0 (6) RA|RAD(α) = 0 cosα -sinα 0 (7)
i i i i
0 0 1 0 0 sinα cosα 0
i i
0 0 0 1 0 0 0 1
The multiplication of these four operators gives the transformation matrix А in (3). It is
k-1,k
clear that it is difficult to receive the analytical form of the end-effector coordinates in function of
parameters (θ,α,s,a). The problem complicates if these parameters on the other hand are functions
of other, for instance constructive or manufacturing parameters. That defines the necessity of
creation of means for OKC-configuration description and computer-assisted equation generation.
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3. LANGUAGE ELEMENTS FOR DESCRIBING OKC
This is description-like language. The language symbols are:
• letter::=”A”|...|”Z”|”a”|…|”z”
• digit::=”0”|”1”|”2”|”3”|”4”|”5”|”6”|”7”|”8”|”9”
• arithmetic operator::= "+" | "-" | "*" | "/"
• delimiters::=”(“|”)”|”,”|”:”|”.”|”//”|” “
To indicate the OKC-parameters and other user-defined parameters are used identifiers:
• identifier ::= letter { letter | digit }
Identifier substitution with numerical values is possible only for joint identifiers. It is possible
to use arithmetic expressions according to PASCAL-language requirements.
The OKC is described with a program. The program consists of one or more sentences,
separated using “;” and the terminal symbol is “.”. The sentence represents a single kinematic pair.
It consists of some words delimited with commas “,”.
The first word in each sentence defines the type of the kinematical pair - “R|RD|Р” and the
identifier of the generalised coordinate. The joint restrictions must put in parenthesis. The following
words (if they exist) describe the spatial transformation toward to the previous frame. The
transformation operators (RQ,RQD,TS,TA,RA,RAD) are used followed by transformation
parameter: number, identifier or arithmetic statement in parenthesis. The possible comment follows
the sentence and it begins with “//”.
The fourth Denavit-Hartenberg’s parameters may present as a number, user-defined identifier or
arithmetic statement. They are shown on Fig.2. The missing syntax elements correspond to the
requirements of PASCAL language.
program
sentence .
;
sentenc
RD
e
joint frame
R ,
identifier transformation
P
joint
joint identifier identifier constraints
arithmetic
statement
joint
constraints
( min value , max value )
min value max value
number number
frame
transformation
RQD
number
RQ
RA
( identifier )
RAD
TS arithmetic
statement
TA Fig.2. Syntax diagrams
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4. SYNTAX ANALYSIS AND SYMBOLIC GENERATION
The goal of program interpretation is to create the kinematics model of the OKC [10,11] and to
derive the end-effector analytical equations. For each recognized joint identifier the corresponding
data structure [11] and the transformation matrix in analytical form is generated. For each
transformation word in a sentence the corresponding transformation matrix (4-7) is created. Its
arguments are numbers, user identifiers or arithmetic statements. The algorithm order follows
below.
1. Analytical multiplication of matrices using procedure MULT_MATR. This algorithm puts
parentheses “(“|”)”, signs “+|- “ and detect multiplication with “0” ,“-1”, “1” [5].
2. Algebraic reducing using procedure ALGEBRA_REDUCE. This procedure removes equal
terms with contrary signs and unifies terms with equal trigonometric and different algebraic parts. It
is part of SYANA system for symbolic computation [6].
3. Output of the result on the appropriate data medium: screen, printer, file.
4. Postfix notation generation of the analytical matrices using procedure
SYNTAX_ANALYSE. Postfix notation is a part of kinematics model of OKC [11]. They are
necessary for numerical-analytical kinematics analysis [3].
5. EXAMPLE
Human arm is an open kinematics chain beginning with the
shoulder joint and ending with the wrist joint (Fig.3). The shoulder
represents first link in a kinematical chain of levers. The possible
motions are [12]:
- flexion/extension in the sagital plane: [-20..90] degrees;
- elevation in the scapular/frontal plane: [0.. 100] degrees;
- rotation about the long axis of the humerus: [-30..60] degrees;
The elbow joint allows two degrees of freedom in motion:
- flection/extension: [10..140] degrees;
- pronation/supination in the horisontal plane: [-90..90] degrees;
- The length of the humerus is denoted by “H4”.
The articulus of the wrist joint allows motion in two planes:
-flection/extensionin in the saggital plane [-70.. 90] degrees;
- radial/ulnar deviation in the frontal plane: [-20..20] degrees;
- The length of the forearm is denoted by “U5”.
The last frame is O X Y Z . It defines the wrist approach. The
8 8 8 8
length of the wrist is denoted by “K8”.
On fig.4 the dialog window is shown. In includes the program that
Fig.3 describes the arm and the generated equations of the wrist. In the
equations “S” and “C” denote functions “SIN” and “COS”
respectively.
6. CONCLUSIONS
A language for description of open kinematics chain is presented. The language allows a
precise determination of 3D joint structures. Moreover, it provides a way to collect objective data
on the functional anatomy of joints and could be used to study its motions. It may be directly
applied to other species and may be used by researchers interested in discreet articular movements.
REFERENCES
1. Genova P., R. Kazandjieva, K.Yankov, Computer-mathematical modeling in symbolic form and
its application in robotics. First Int.Cong. on Technology and Robotics “TECHRO’88”, sept.19-
26, 1988, Diuni, Bulgaria (in Bulgarian).
