three-dimensional rigid body guidance using gear … · 2008. 4. 1. · 6-6 gear synthesis results...
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THREE-DIMENSIONAL RIGID BODY GUIDANCE USING GEAR CONNECTIONS IN A ROBOTIC MANIPULATOR WITH PARALLEL CONSECUTIVE AXES
By
JAVIER AGUSTIN ROLDAN MCKINLEY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
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© 2007 Javier Agustín Roldán Mckinley
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To the memories of Joseph Duffy and Ali Seireg
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ACKNOWLEDGMENTS
My greatest gratitude in the attainment of this goal is toward my supervisory committee
chair, Dr. Carl Crane III. This work was possible thanks to his continuous advisership, patience
and support in both the academic and personal aspects.
I also thank my committee member Dr. David Dooner. His trust in my work made possible
that I became part of CIMAR Lab. His guidance, support and encouragement throughout this
research will always be greatly appreciated.
The effort and support of my supervisory committee members: Dr. Antonio Arroyo, Dr.
William Hager, Dr. John Schueller, and Dr. Gloria Wiens are also appreciated.
I want also to remark on the support of close friends and colleagues, namely Jaime
Bestard, Jahan Bayat, and Jean-Francois Kamath.
I gratefully acknowledge the financial support provided by the Department of Energy via
the University Research Program in Robotics (URPR), grant number DE-FG04-86NE37967.
Finally, I express special deference to my family and parents for their undying love,
patience and support.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS ...............................................................................................................4
LIST OF TABLES...........................................................................................................................8
LIST OF FIGURES .........................................................................................................................9
ABSTRACT...................................................................................................................................11
CHAPTER
1 INTRODUCTION ..................................................................................................................13
1.1 Preliminary Spatial Mechanisms ..................................................................................13 1.1.1 Kinematic Link .................................................................................................14 1.1.2 Revolute Joint ...................................................................................................15 1.1.3 Mobility Criterion in Spatial Manipulators and Closed-Loop Mechanisms.....16
1.2 Preliminary Non-Circular Gears ...................................................................................17 1.3 State of the Art ..............................................................................................................18 1.4 Motivation.....................................................................................................................19 1.5 Previous Work...............................................................................................................21 1.6 Objective and Scope......................................................................................................23 1.7 Overview of the Manuscript .........................................................................................24
2 SPATIAL KINEMATIC BASIC CONCEPTS ......................................................................26
2.1 Standard Link Coordinate System ................................................................................26 2.2 Transformation Matrix between Standard Systems ......................................................27 2.3 Forward Kinematic Analysis of a 6-Link 6R Open Loop Spatial Mechanism.............30 2.4 Reverse Kinematics Problem Statement for 6L-6R Manipulator .................................31 2.5 Close-the-Loop Solution Technique .............................................................................32
2.5.1 Determination of the Close-the-Loop Parameters.............................................34 2.5.1.1 Twist Angle α71...................................................................................34 2.5.1.2 Joint Angle θ7......................................................................................35 2.5.1.3 Angle γ1 ..............................................................................................36 2.5.1.4 Distances a71, S7 and S1 ......................................................................37
2.5.2 Special Cases for Closed-the-Loop Solution Technique ..................................39 2.5.2.1 S1 and S7 are Parallel or Antiparallel ..................................................39 2.5.2.2 S1 and S7 are Collinear........................................................................40
2.6 Spherical Closed-Loop Mechanisms.............................................................................40 2.7 Mobility of Spherical Mechanisms and Classification of Spatial Mechanisms............42 2.8 Joint Vectors Expressions in the Spherical Mechanism ...............................................42 2.9 Link Vectors Expressions in the Spherical Mechanism................................................45 2.10 Orientation Requirements Specification: X-Y-Z Fixed Angles....................................48
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3 NON-CIRCULAR GEARS FUNDAMENTALS ..................................................................49
3.1 Pitch Curve and Pitch Point ..........................................................................................49 3.2 Input/Output Non-Circular Gear Relationship..............................................................50
3.2.1 Input/Output Relationship with the Connecting Link Grounded......................51 3.2.2 Input/Output Relationship with All Links Movable .........................................51
3.3 Complete Profile Synthesis: Non-Circular Relationships for the Entire Rotation........52 3.4 Pitch Curve Coordinates ...............................................................................................57
4 REVERSE KINEMATIC ANALYSIS OF THE 1-DOF SPATIAL MOTION GENERATOR ........................................................................................................................61
4.1 Mobility Analysis of the Spatial Motion Generator......................................................62 4.2 Grouping of the 7L-7R Spatial Mechanism with Consecutive Axes Parallel:
Equivalent Spherical Mechanism..................................................................................63 4.3 Reverse Kinematic of the 6L-6R Open-Loop Mechanism with Consecutive Axes
Parallel ..........................................................................................................................65 4.3.1 Problem Statement ............................................................................................65 4.3.2 Solution of the Equivalent Spherical Quadrilateral ..........................................66
4.3.2.1 Solving for θ1Q ....................................................................................67 4.3.2.2 Solving for θ2Q ....................................................................................68 4.3.2.3 Solving for θ3Q ....................................................................................69
4.3.3 Vector Loop Equation of the 7L-7R Parallel Axes Mechanism .......................69 4.3.4 Tan-Half-Angle Solution for θ5 ........................................................................74 4.3.5 Solution for θ3 ...................................................................................................81 4.3.6 Solution for θ2 ...................................................................................................83 4.3.7 Solution for θ1, θ4 and θ6 ...................................................................................83 4.3.8 Solution Tree.....................................................................................................83
4.4 Reverse Kinematics Analysis for the Complete Spatial Motion Generator..................84
5 SYNTHESIS OF THE NON-CIRCULAR PITCH PROFILES.............................................89
5.1 I/O Non-Circular Gear Relationships for the Spatial Motion Generator ......................89 5.2 Sequence Parameter Approach .....................................................................................91
5.2.1 Polynomial Interpolation of Joint Angles: Discarded First and Last Points.....92 5.2.2 I/O Relationship Expressions from Sequence Parameter Approach.................94
6 RESULTS: ILLUSTRATIVE EXAMPLES ..........................................................................96
6.1 Reverse Kinematics for a Single Position of the End Effector: Single Position, Half Mechanism Case ...................................................................................................96
6.2 Motion Generation along Discrete Path and Orientation Requirements: Complete Mechanism Case .........................................................................................................100
7 CONCLUSION AND FUTURE WORK .............................................................................111
7.1 Conclusion ..................................................................................................................111 7.2 Future Work ................................................................................................................112
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APPENDIX
A JOINT AND LINK VECTORS EXPRESSIONS IN THE SPHERICAL MECHANISM ..114
B POSITION AND ORIENTATION SPECIFICATION IN EXAMPLE 6.2.........................118
LIST OF REFERENCES.............................................................................................................120
BIOGRAPHICAL SKETCH .......................................................................................................123
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LIST OF TABLES
Table page 4-1 Direction cosines of a closed-loop 7L-7R mechanism expressed in a coordinate
system were the x and z axes are aligned with the vectors a23 and S3, respectively..........71
6-1 Constant mechanism parameters for numerical example. .................................................96
6-2 Desired position and orientation requirements. .................................................................96
6-3 Calculated close-the-loop parameters. ...............................................................................97
6-4 Sums of consecutive joint angles obtained from the equivalent spherical quadrilateral solution...............................................................................................................................97
6-5 Joint angles corresponding to the sixteen solutions of the numerical example. ................98
6-6 Constant mechanism parameters for the closed-loop example........................................102
B-1 Continuous position and orientation needs specification.................................................118
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LIST OF FIGURES
Figure page 1-1 A kinematic link, the simplest representation of a spatial link. .........................................14
1-2 Revolute joint and joint angle. ...........................................................................................16
1-3 Non-circular gears..............................................................................................................18
1-4 Robotic manipulators used in industry. .............................................................................18
1-5 1-DOF planar motion generator using non-circular gears. ................................................19
1-6 1-DOF spatial motion generator implementing non-circular gears. ..................................24
2-1 Standard i-th coordinate system attached to the link ij. .....................................................27
2-2 Coordinate systems of two successive links ij and jk. .......................................................29
2-3 Given information in the reverse analysis. ........................................................................32
2-4 Hypothetical closure link. ..................................................................................................33
2-5 Vector loop to find the distances a71, S7 and S1. ................................................................38
2-6 Spherical quadrilateral. ......................................................................................................41
3-1 Pitch point and pitch curves of two bodies in mesh...........................................................50
3-2 Non-circular input and output bodies in mesh...................................................................51
3-3 Arbitrary working and non-working sections and angles in a complete non-circular gear.....................................................................................................................................53
3-4 Boundary conditions for the whole synthesis of the Input/Output relationship. ...............54
3-5 Rectangular coordinate systems x-yo and x-yi for a gear pair in mesh. ..............................58
3-6 Local coordinate systems x-yi and x-yo in the synthesis of the pitch point with input, output and connecting bodies movable..............................................................................59
4-1 1-DOF spatial motion generator. .......................................................................................61
4-2 Labeling of the spatial motion generator for mobility analysis. Parts with the same number label are one single link. .......................................................................................63
4-3 Kinematic labeling of the 6L-6R open-loop mechanism...................................................64
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4-4 Equivalent spherical mechanism........................................................................................68
4-5 Solution tree of the spherical quadrilateral. .......................................................................69
4-6 Sixteen solutions tree for 7L-7R closed-loop mechanism with consecutive pairs of parallel joint axes. ..............................................................................................................84
4-7 Kinematic labeling of the 6L-6R open-loop mechanism 1-12→7.....................................86
4-8 Transformation of the fixed coordinate systems xF1-yF1-zF1 and xF2-yF2-zF2. ....................87
4-9 Disposition of the two sixth coordinate systems................................................................87
5-1 Non-circular gear connection’s labeling............................................................................90
5-2 Labeling of the h non-circular gear connection. ................................................................90
5-3 Introduction of the sequence parameter u to discretely express the motion generation requirements along [xo,xf],[yo,yf],[zo,zf].........................................................................93
6-1 Real solutions A-H in Table 6-5, corresponding to the A solution of the spherical quadrilateral in Table 6-4...................................................................................................99
6-2 Real solutions I-P in Table 6-5, corresponding to the B solution of the spherical quadrilateral in Table 6-4.................................................................................................100
6-3 Motion needs....................................................................................................................102
6-4 Motion needs versus the sequence parameter in each fixed coordinate system. .............103
6-5 Joint angles for the closed-loop mechanism. ...................................................................105
6-6 Gear synthesis results for connection 234 in half mechanism 0-1→6.............................106
6-7 Gear synthesis results for connection 456 in half mechanism 0-1→6.............................107
6-8 Centrode, output angle and Input/Output gear relationship versus the input angle in half mechanism 0-11→6..................................................................................................108
6-9 Closed-mechanism at two different motion needs in Table B.1. .....................................110
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
THREE-DIMENSIONAL RIGID BODY GUIDANCE USING GEAR CONNECTIONS IN A ROBOTIC MANIPULATOR WITH PARALLEL CONSECUTIVE AXES
By
Javier Agustín Roldán Mckinley
August 2007
Chair: Carl Crane III Major: Mechanical Engineering
Robot manipulators are often employed in industry to perform repetitive motions.
Examples are pick-and-place operations that occur during packaging processes and positioning
operations that occur during assembly. Often, six degree-of-freedom open-loop robotic
manipulators are used in these operations. A disadvantage of using these manipulators for
repetitive tasks is twofold. First, the hardware associated with the six actuators that are required
for a six degree-of-freedom chain is expensive. Second, the coordinated control of these six
actuators that is needed for precision motion is complicated.
In our research, a new approach is presented to address repetitive motion operations such
that they can be accomplished by a one degree-of-freedom device. The end effector that is to be
positioned and oriented along a specified path (or through a set of specified poses) is attached to
the middle link of a spatial closed-loop mechanism that is comprised of 12 links (including
ground) that are interconnected serially by 12 revolute joints. Five pairs of non-circular gears are
then designed and inserted into the mechanism to reduce the degree-of-freedom of the device to
one.
To make the approach more practical, the geometry of the closed-loop mechanism was
chosen so that pairs of adjacent joint axes are parallel. For example, the first joint axis is parallel
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to the second, the third is parallel to the fourth, and so on. The reason for this is that planar non-
circular gears can be designed to be inserted into the mechanism. A kinematic analysis of this
analysis is presented and it is shown that an open-loop chain comprised of six joints where
adjacent joint axes are parallel (in effect half of the closed-loop mechanism) have a total of
sixteen possible solution configurations that will position and orient the distal link as specified.
The kinematic analysis of this special geometry and the design of the working and non-working
sections of the gear pairs are the main contributions of our research. Numerical examples are
presented.
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CHAPTER 1 INTRODUCTION
Robotic systems and mechanisms exist to perform two kinematical tasks: function and
motion generation. In both cases a body, namely the coupler, or a point of this body, must
occupy certain positions. In motion generation, the coupler must traverse a particular path and
orientation with respect to a certain reference frame, while function generation requires that a
point of the coupler describes a certain path and the orientation of this body is of no interest. If
the task is performed in the two-dimensional space then it is referred to as a planar case, while a
task performed in three-dimensional space is referred to as a spatial case.
Some authors consider closed loop linkages as mechanisms, while open loop multi- degree
of freedom (DOF) manipulators –serial manipulators with programmable motion- are considered
as robots or robotic systems. In this work, both open and closed loop linkages are considered
indistinctively as mechanisms.
The thrust of this research is the integration of non-circular gear pairs into a serial closed
chain spatial mechanism to fulfill motion generation requirements through a single degree of
freedom mechanism in three-dimensional space. In order to present the idea and further its
analysis, some preliminary concepts about spatial mechanisms and non-circular gearing are
considered. Later in this chapter is the justification of this project, the objective statement, and a
complete review of related previous work.
1.1 Preliminary Spatial Mechanisms
Basic concepts related to spatial mechanisms are presented here: mobility or degrees of
freedom, kinematic link, revolute joint, joint axis, link axis, and joint and link distances. For
further information about, see Crane and Duffy [1].
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1.1.1 Kinematic Link
In this work, the spatial mechanism of interest is to be formed by a series of links (assumed
to be rigid bodies) and joints forming a closed chain, where one link is connected to ground.
These links are to be represented in their simplest form: kinematic links, which can have in
reality many equivalent configurations. Figure 1-1 shows a kinematic link with his joint vectors,
Si and Sj, which are unit vectors along the two consecutive joint axes i and j.
Figure 1-1. A kinematic link, the simplest representation of a spatial link.
The relative position of the skew joint axes is defined by the link length aij and the twist
angle αij, depicted in Figure 1-1. The link distance aij is the mutual perpendicular distance
between the skew joint axes, and the twist angle αij is the angle between the vectors Si and Sj.
The vector aij is defined as the cross product of the vectors Si and Sj, defined as SixSj =aij sinαij,
then the direction and sense of aij is defined by the right hand rule applied to the vectors Si and
Sj. In the same way, the sense of the angle αij is obtained by applying the right hand rule, with the
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thumb pointing along aij and the fingers sweeping from the direction of Si to the direction of Sj to
form the angle αij.
It is evident that the direction of the vectors Si and Sj presented in Figure 1-1 is arbitrary
because either Si or Sj can be drawn in the opposite direction; however aij will always be
determined here by the cross product of Si and Sj according to their direction.
1.1.2 Revolute Joint
The term joint was used without any previous definition in the last section. In general, a
joint is the connection existing between a pair of successive links that determines the nature of
their relative motion.
The revolute joint, denoted by the letter R, is the simplest and most common joint. The
revolute joint presented in Figure 1-2 connects the link ij with the link jk. Link jk can rotate
about the joint vector Sj, relative to the link ij. As a must, the vector Sj of link ij and the vector Sj
of the link jk are to be parallel and not antiparallel.
The link jk presented in Figure 1-2 can only rotate about Sj relative to the link ij, thus the
link jk has one degree of freedom with respect to link ij. A new concept is presented here, the
joint angle θj, which allows measurement of the relative motion of link jk relative to link ij.
From Figure 1-2, the joint angle θj is the angle between the unit vectors aij and ajk,
measured in a right-hand sense with the thumb pointing along the joint vector Sj, defined as
aijxajk=Sjsinθj. A joint distance associated with the joint vector is also defined here; this is the
joint or offset distance Sj. As depicted in Figure 1-2, Sj is the mutual perpendicular distance
between the vectors aij and ajk. For the revolute joint, the offset distance Sj is a constant because
the link jk cannot translate along the link ij, it can only rotate about it, changing only the joint
angle θj.
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Figure 1-2. Revolute joint and joint angle.
1.1.3 Mobility Criterion in Spatial Manipulators and Closed-Loop Mechanisms
The mobility or degrees of freedom (DOF) of a mechanism is the number of independent
inputs required to determine the position of all links of the mechanism with respect to ground. A
shorter concept for the DOF in a mechanism is the number of inputs that need to be provided in
order to define the output. Complete information about mobility in planar and spatial
mechanisms can be found in Diez-Martínez et al. [2].
The Kutzbach mobility equation for spatial linkages where one link is connected to ground
is1
54321 ff2f3f4f51)6(nM −−−−−−= (1-1)
1 Some particular geometry may exist in special mechanisms where the real mobility does not match the value obtained by Equation 1-1.
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where:
M: mobility or DOF
n: number of spatial links or connected bodies
fi: number of joints with i relative degrees of freedom.
A more compact format of Equation 1-1 for closed spatial mechanisms is presented as
∑=
−=n
1ii 6fM
(1-2)
where:
M: mobility or DOF
n: number of spatial links or connected bodies
fi: number relatives degrees of freedom permitted by the i-th joint.
The purpose of this subsection is to introduce the mobility concept. A mobility analysis is
later presented in Section 2.1.
1.2 Preliminary Non-Circular Gears
Non-circular gears are toothed bodies whose pitch curves (or primitive curves) are not
common circles. These primitive curves can have any functional shape, derived from the
necessity of varying output position, velocity, and/or acceleration. Figure 1-3A shows a non-
circular gear profile, with the teeth and pitch profiles. Figure 1-3B presents a spur non-circular
gear pair in mesh. For detailed information, see Dooner and Seireg [3].
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Figure 1-3. Non-circular gears. A) Non-circular teeth and pitch profiles. B) Spur non-circular gear pair in mesh. Courtesy of Dr. David Dooner.
1.3 State of the Art
The spatial path and motion generation tasks are currently fulfilled by open-loop robot
manipulators, being the PUMA 560, GE P60 (see Figure 1-4) and the Cincinnati Milacron T3 the
most used configurations. Further information about the reverse kinematic of these industrial
manipulators can be found in Crane and Duffy [1].
Figure 1-4. Robotic manipulators used in industry. A) Puma robot. B) GE P60. Courtesy of Dr. Carl Crane.
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Current applications of non-circular gears include printing presses, packaging machines,
conveyors, and low speed precision instruments; their combined use with linkages to accomplish
motion generation tasks is currently limited to the planar case. The 1-DOF six-link mechanism
motion generator depicted in Figure 1-5 incorporates two non-circular gear pairs and was
proposed to achieve any path and orientation requirements in the planar scenario, subject to
feasibility of the non-circular gear profiles. For further information, see Roldán Mckinley et al.
[4]. There are no known references related to 3D2 spatial motion generation using non-circular
gears.
Figure 1-5. 1-DOF planar motion generator using non-circular gears.
1.4 Motivation
There are applications in industry where it is necessary for a body in a mechanism to
repeatedly traverse a particular path and orientation with respect to a certain reference frame.
Planar and spatial applications include automotive suspension systems, garbage truck delivery,
automotive welding and painting processes, and shoe testing machines. The path and/or
orientation requirements can be met through function and motion generation.
2 Three dimensional or spatial scenario.
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Currently, these spatial kinematic tasks are fulfilled by open and closed-loop mechanisms.
Figure 1-4A and 1-4B are examples of open loop mechanisms and Figure 1-5 shows a closed-
loop planar mechanism. The inverse kinematics involved in three dimensional kinematic tasks is
solved via analytical methods (in some cases combined with numerical iterative solutions). One
degree-of-freedom three-dimensional closed-loop linkages can be designed to satisfy a finite
number of values for function, path, and motion generation.
Robot manipulators are versatile 3D motion generators. These open loop manipulators are
widely accepted in industry; however, because of their mobility or degrees-of-freedom,
specification of the actuators is necessary. Typical industrial manipulators such as the PUMA
and GE robots possess six degrees-of-freedom, where six actuators must be synchronized for a
single motion of the working tool. Typically, actuation is achieved via motors (stepper or servo
with controls). The cost and complexity associated with achieving this coordinated computer
control can be eliminated if a 1-DOF mechanism can generate the desired repetitive motion. A
single actuator can be used and the mechanism response is exactly quantified.
To elaborate on the last statement, the cost of the mechanism can be reduced due to the
elimination of five actuators and their corresponding electronic equipment. This becomes the
main saving when comparing the 6-DOF and the 1-DOF mechanisms. Although there is a cost
associated with the 1-DOF spatial mechanism building, such as the manufacture of the non-
circular gears, the cost of the electronic equipment in a six degree-of-freedom mechanism is
anticipated to be higher than the non-circular gear manufacture and assembling costs. The
elimination of actuators in the system can also have additional benefits: less maintenance and
lower operation costs.
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To summarize, the implementation of a 1-DOF spatial motion generator is suited for
repetitive tasks and can reduce the fabrication, installation, programming, operation, and
maintenance costs of multi-degree-of-freedom robotic manipulators.
1.5 Previous Work
The four-bar spatial mechanism was the first linkage used to accomplish path and motion
generation tasks in the three dimensional scenario. Better results were obtained in just path
generation due to some limitations of this four-link mechanism. Suh [5] proposed the synthesis
of an RSSR3 function generator through a generalized matrix for the description of finite
functional displacement, based on a kinematic inversion technique. Suh [6] also worked on the
rigid body guidance through finitely separated multipositions by the synthesis of the RSSR four-
bar and the RSSR-SS six-bar spatial mechanisms, including again a generalized matrix derived
from the constraint equations.
Spatial path and motion generation have been traditionally solved by parallel and serial
robot manipulators. Significant contributions in this field were done by Rooney and Duffy [7] by
proposing the closure of spatial linkages as a direct method to solve the displacement analysis.
The same authors, Duffy and Rooney [8], presented their landmark work in robotics by
developing a unified procedure for the analysis of spatial four-link, five-link, six-link, and seven-
link mechanisms, by stating that there are only three fundamental loop equations for any
spherical polygon: sine, sine-cosine, and cosine laws.
Later, Duffy and Rooney [9-11] expressed the input-output displacement equation as a
degree eight polynomial in the half-tangent of the output angular displacement for spatial six-
link 4R-P-C mechanisms; they realized it for the RCRPRR, RCRRPR, and RRRPCR inversions.
3 R: revolute joint, P: prismatic joint, C: cylindrical joint, and S: spherical joint.
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Duffy and Rooney [12] also deduced the input-output displacement equation of degree 16 for
spatial six-link 5R-C mechanisms, in particular for the RRRRCR and RRCRRR inversions. The
displacement spatial seven link 5R-2P mechanism was also solved by Duffy [13]; an eight
degree input-output displacement equation was derived for the RPPRRRR, RRRPPRR,
RPRRRPR, RPRRPRR, and RPRPRRR inversions.
Separately, Sandor, Kohli, and Zhuang [14] considered a RSSR-SRR spatial mechanism
for motion generation with prescribed crank rotations, limited to four precision points. A single-
actuator RS-SRR-SS adjustable spatial motion generator was synthesized by Sandor et al. [15],
where the authors also presented the advantages of this mechanism in comparison with the multi-
degree-of-freedom parallel robotic manipulators when performing highly repetitive tasks. As a
disadvantage, the mechanism was limited to only two exact prescribed positions. Lee and Liang
[16] also performed the displacement analysis of spatial mechanisms: the inversions RRPRRRR,
RRRPRRR, and RRRRRPR of the spatial 7-link 6R-P linkage. The input-output equations of
degree 16 in the half-tan-angle of output angular displacements were presented.
Additional contributions are now presented. Sandor, Weng, and Xu [17] synthesized an
RPCPRR spatial mechanism for motion generation without branching defect. Premkumar, Dhall,
and Kramer [18-21] developed path and function generation with diverse spatial four-bar
mechanisms using the selective precision synthesis method.
Duffy and Crane [22] obtained a 32nd degree input-output equation in the tan-half-angle of
the output angular displacement for 7-link 7-R mechanism. The “Mount Everest of kinematic
problems” 4 was later presented by Lee and Liang [23], based on [22], as a 16th degree
polynomial input-output equation in the tan-half-angle of the output angular displacement. The
4 As the General Spatial 7-Link 7-R Mechanism was described by Professor Ferdinand Freudenstein.
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inverse kinematic analytical solution of the General Spatial 7-Link 7-R Mechanism is of
particular interest for this project.
Non-circular gears have not been included into spatial motion generation tasks, only into
planar path and motion generation. Dooner [24] proposed utilizing a 1-DOF eight-link
mechanism and optimized non-circular gear elements with application to automotive steering to
fulfill path generation requirements. Later, Roldán Mckinley [25] developed the planar 1-DOF
six-link mechanism with two pairs of non-circular gears to achieve path and orientation meets,
and finally Roldán Mckinley et al. [4] obtained non-circular gear pitch profiles for a motion with
a path that was approximated as a quadratic function.
More recently, Mundo, Liu and Yan [26] utilized non-circular profile synthesis to integrate
cams and linkages to obtain a precise planar path generator. Gatti and Mundo [27] also proposed
a six-link mechanism incorporating two prismatic joints and two cams to achieve planar rigid
body guidance.
1.6 Objective and Scope
The objective of this work is to develop a 1-DOF spatial mechanism, depicted in Figure 1-
6, to achieve generalized three-dimensional motion generation. This mechanism will allow
satisfying any generalized path f(x,y,z) of point P in link 7 and orientation ζ (ζ x, ζ y, ζ z) of end
effector link 7. To this end, it is first necessary to solve the inverse kinematic of the 1-DOF
spatial mechanism involving parallel axes in the geared transmissions. Next, the non-circular
pitches curves are synthesized.
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Figure 1-6. 1-DOF spatial motion generator implementing non-circular gears. A) Position requirement f(x,y,z. B) Orientation requirements ζ(ζx, ζy, ζz).
In addition to the attainment of link lengths, joint offset distances, link angles, and joint
angles required to synthesize the spatial mechanism, attainment of the pitch profiles for the non-
circular gears are of interest in the present project.
1.7 Overview of the Manuscript
The aim of this research was presented in this chapter along with basic introductory
concepts. This dissertation has seven chapters, where Chapters 2 and 3 include basic concepts in
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spatial kinematics; and non-circular gearing and screw theory, respectively. Chapter 4 details the
reverse kinematic analysis of the mechanism utilized in this research. The synthesis of the non-
circular pitch profiles is presented in Chapter 5, based on gearing concepts previously presented
in Chapter 3. The results and comments section is presented in Chapter 6. This dissertation ends
with a conclusion and description of future work in Chapter 7.
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CHAPTER 2 SPATIAL KINEMATIC BASIC CONCEPTS
The reverse kinematic solution technique used in this work is based on the unified theory
proposed by Crane and Duffy [1] to solve spatial robotic manipulators. In this chapter, the
fundamentals to apply the unified theory are presented. The totality of the topics here presented
Standard coordinate systems and their transformation matrices are presented followed by the
forward analysis.
The loop closure technique is also explained, where an open loop manipulator is
transformed into a closed loop mechanism by introducing a virtual closure link. Next, based on
the closed loop mechanism, equivalent spherical mechanisms are introduced. The solution of the
equivalent spherical mechanism reverse kinematics satisfies the solution of the spatial closed
loop mechanism. The chapter ends with the fixed angles specification for the orientation of a
rigid body with respect to a fixed reference frame, which will be used when specifying the
orientation needs for the robotic manipulator.
2.1 Standard Link Coordinate System
This section describes the methodology proposed by Crane and Duffy [1]. For the analysis
of robot manipulators, it is common to attach a coordinate system to each link. A standardized
approach will be used to select these coordinate systems. First, the coordinate system will have
its origin located at the intersection of the link and joint unit vectors, see Figure 2-1. If the link ij
is to be considered, then the origin of its coordinate system is located at the intersection of aij and
Si. Second, the direction of the x axis is parallel to the link unit vector aij. Finally, the direction of
the z axis is parallel to the joint unit vector Si.
27
The direction of the y axis is to be determined through a cross product of the vectors i and
k (or aij and Si): kxi = j¸ where i, j and k are unit vectors along the x, y, and z axes. It is
important to note that the vectors aij and Sj are unit vectors.
Figure 2-1. Standard i-th coordinate system attached to the link ij.
2.2 Transformation Matrix between Standard Systems
This section describes the methodology proposed by Crane and Duffy [1]. In a general
sense, spatial kinematics is based on the transformations of coordinates between standard
coordinate systems attached to the mechanism links. The objective here is to find the
mathematical expression that relates two successive standard coordinate systems. To start,
consider two consecutive links, the links ij and jk, as shown in Figure 2-2.
The transformation matrix that relates the standard coordinate systems i (xi-yi-zi) and j (xj-
yj-zj) can be obtained as:
1) Start with the two coordinate system aligned.
2) Translate the i-th coordinate system the distance aij along the xi axis. The translation
matrix is
28
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
100001000010a001 ij
1M . (2-1)
3) Rotate the i-th coordinate system an angle αij about the xi axis. The rotation matrix is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
10000cs00sc00001
ijij
ijij2M (2-2)
where cij=cos(αij) and sij=sin(αij).
4) Translate the i-th coordinate system a distance Sj along the zj axis. The translation matrix
is given by
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1000S10000100001
j3M . (2-3)
5) Rotate the i-th coordinate an angle θj about the zj axis. The rotation matrix is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
1000010000cs00sc
jj
jj
4M (2-4)
where cj=cos(θj) and sj=sin(θj).
6) Systems xi-yi-zi and xj-yj-zj are aligned.
29
Figure 2-2. Coordinate systems of two successive links ij and jk.
It is possible to relate Equations 2-1 to 2-4 to obtain the transformation matrix that relates
the standard coordinate systems i and j. This matrix is
4321ij MMMMT = (2-5)
The explicit expression for this transformation matrix is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−−
−
=
1000SccscssSsscccs
a0sc
jijijijjijj
jijijijjijj
ijjj
ijT . (2-6)
The inverse of this transformation can be readily obtained as,
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
−
=
1000Scs0asscccsacsscsc
jijij
ijjijjijjj
ijjijjijjj
jiT . (2-7)
One more transformation matrix is needed for completeness; this is the matrix that relates
the fixed coordinate system (ground) and the first link, link 12. The ground coordinate system is
selected with its origin coincident with that of the coordinate system attached to the first link.
30
The z axis of the ground coordinate system is parallel to the first joint axis. Thus the
transformation matrix that relates the first and ground coordinate system can be obtained as a
rotation of an angle φ1 about the z axis,
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
1000010000cossin00sincos
T 11
11
F1
ϕϕϕϕ
(2-8)
where φ1 is the angle measured from the x axis of the ground coordinate system to the x axis of
the coordinate system attached to the first moving link, measured in a right-hand sense about S1.
2.3 Forward Kinematic Analysis of a 6-Link 6R Open Loop Spatial Mechanism
This section describes the methodology proposed by Crane and Duffy [1]. The forward
analysis is the inverse process of the reverse kinematic analysis. Because this analysis is much
simpler than the reverse kinematic analysis, it will be presented first as follows. Given the
manipulator joint angles, i.e. φ1 through θ6, and the position of the tool point (attached to the end
effector) measured in the coordinate system attached to the sixth (distal) link, 6Ptool, determine
the position of the tool point and the orientation of the sixth link measured with respect to
ground. The orientation of the last link is defined by the vectors Fa67 and FS6 which are the
directions of the x and z axes of the sixth coordinate system measured with respect to the fixed
coordinate system.
The position of the tool point measured with respect to the fixed coordinate system is
obtained as
tool65
645
34
23
12
F1tool
F PTTTTTTP = . (2-9)
31
The orientation of the sixth link as measured in the fixed coordinate system can be
obtained as the first and third column of the rotation matrix (upper left 3×3 submatrix) of the
product TTTTTT 56
45
34
23
12
F1 .
2.4 Reverse Kinematics Problem Statement for 6L-6R Manipulator
This section describes the methodology proposed by Crane and Duffy [1]. The reverse
analysis for the 6L-6R determines all possible sets of the six joint angles that satisfy any position
and orientation of the end effector, given the desired coordinates of the tool in the fixed and 6th
coordinate system, the geometry of the open-loop mechanism (link lengths, twist angles, and
joint offsets), and the desired orientation of the end effector as defined by the vectors FS6 and
Fa67. Figure 2-3 summarizes the reverse analysis given information and based on this the reverse
kinematic problem statement can be written as
Given:
1) The constant mechanism parameters: link lengths a12 to a56, twist angles α12 to α56, and
offset distances S2 to S5
2) Offset distance S6 and the direction of the a67 relative to the vector S6
3) Position and orientation of the end effector: FPtool, FS6 and Fa67
4) Location of the tool point in the 6th coordinate system: 6Ptool
Find:
The angle between the fixed coordinate system and the first standard coordinate system, φ1, and
the joints angles θ2 to θ6.
The transformation matrix that relates the sixth coordinate system and ground is given by
32
⎥⎦
⎤⎢⎣
⎡=
1000orig6
FF6F
6PR
T (2-10)
where:
[ ]6F
67F
6F
67FF
6 SaSaR ×= (2-11)
( ) ( ) ( ) 6F
tool6
67F
6F
tool6
67F
tool6
toolF
orig6F SkPaSjPaiPPP ⋅−×⋅−⋅−= . (2-12)
The analytical solution to this problem involves two main steps. The first step is to find the
hypothetical closure link that connects an imaginary joint axis with the first joint axis. This step
is presented in the immediate section. The second step is obtaining the angles θ1 to θ6 via
analysis of the equivalent spherical closed-loop mechanism.
Figure 2-3. Given information in the reverse analysis.
2.5 Close-the-Loop Solution Technique
This section describes the methodology proposed by Crane and Duffy [1]. The close-the-
loop solution technique will introduce a hypothetical closure link that connects an imaginary
joint axis with the first joint axis. For this work, there are six links in the open-loop mechanism,
then the hypothetical closure link will connect an imaginary joint axis S7 with the first joint axis
33
S1. The direction and location of the joint axis S7 must be selected first, and it is required that the
vector S7 is perpendicular to the link vector a67. An arbitrary value for the link angle α67 will be
set as 90 degrees, which will define the direction of S7. After this conscious selection of α67, S7 is
calculated as
6F
67F
7F SaS ×= . (2-13)
The vector FS7 is selected to pass through the origin of the standard sixth coordinate
system, where from the link distance a67=0. An interesting result derived from the closed-form
solution technique is that the offset distance S1 is introduced as the perpendicular distance
between the vectors a71 and a12 measured along S1. Figure 2-4 shows the hypothetical joint axis
S7 and the hypothetical link 71. Also the offset distance S1 is depicted along with all the close-
the-loop parameters to be determined: a71, S7, S1, α71, θ7, γ1. As shown in the figure, the angle γ1
is the angle swept from a71 to the x axis of the fixed coordinate system as measured in a right-
hand sense about S1. It is apparent that the relationship between γ1, φ1, and θ1 is θ1 = γ1+φ1.
Figure 2-4. Hypothetical closure link.
34
2.5.1 Determination of the Close-the-Loop Parameters
The problem statement for the close-the-loop parameters is as follows.
Given:
Joint vectors FS1, FS7 and FP6orig, the coordinates of the origin of the sixth coordinate system
measured in the fixed system,
Find:
Joint distances S1, S7; link distance a71; joint angle θ7; twist angle α71, angle γ1
First the direction of the vector a71 measured with respect to the fixed coordinate system is
to be found. By definition, a71 is perpendicular to S7 and S1, then
1F
7F
1F
7F
71F
SSSS
a×
×= . (2-14)
The vector FS7 is known from Equation 2-13, and the vector FS1 is parallel to the z axis of
the fixed coordinate system, then
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100
1F S . (2-15)
Substituting Equations 2-13 and 2-15 into Equation 2-14, Fa71 leaves
( )
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
=
100100
6F
67F
6F
67F
71F
Sa
Sa
a . (2-16)
2.5.1.1 Twist Angle α71
The general definitions for the sine and cosine of the twist angle αij are
35
ijjiijs aSS ⋅×= (2-17)
jiijc SS ⋅= . (2-18)
The twist angle α71 can be obtained by knowing its sine and cosine, by substituting
Equations 2-13, 2-15 and 2-16 into Equations 2-17 and 2-18, to obtain
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅×=
100
c 6F
67F
71 Sa (2-19)
( )( )
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××=
100100
100
s
6F
67F
6F
67F
6F
67F
71
Sa
Sa
Sa . (2-20)
2.5.1.2 Joint Angle θ7
The sine and cosine expressions for the joint angle θj are
jjkijjs Saa ⋅×= (2-21)
jkijjc aa ⋅= . (2-22)
The joint angle θ7 can be found by substituting Equations 2-13 and 2-16 into Equations 2-
21 and 2-22, to obtain
( )
( )6
F67
F
6F
67F
6F
67F
67F
7
100100
s Sa
Sa
Sa
a ×⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
×= (2-23)
36
( )
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⋅=
100100
c
6F
67F
6F
67F
67F
7
Sa
Sa
a . (2-24)
2.5.1.3 Angle γ1
The sine and cosine of γ1 can be determined based on the definition of θj. The angle γ1 is
the angle between the vector a71 and the x axis of the fixed coordinate system (as depicted in
Figure 2-4, Equations 2-21 and 2-22 give
1FF
71F
1sinγ Sxa ⋅×= (2-25)
xa F71
F1cosγ ⋅= . (2-26)
The vector x, denoted as Fx, is by definition
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
001
F x . (2-27)
The substitution of Equations 2-15, 2-16 and 2-27 into Equations 2-25 and 2-26 gives the
final expressions for the sine and cosine of γ1
( )
( ) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
=100
001
100100
sinγ
6F
67F
6F
67F
1 o
Sa
Sa
(2-28)
37
( )
( ) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡××
=001
100100
cosγ
6F
67F
6F
67F
1 o
Sa
Sa
. (2-29)
2.5.1.4 Distances a71, S7 and S1
Figure 2-5 shows a vector loop starting at the origin of the fixed coordinate system and
involving the unit vectors FS7, Fa71 and FS1. The vector loop in Figure 2-5 is expressed as
0SaSP =+++ 1F
171F
717F
7orig6F SaS . (2-30)
The only unknowns in Equation 2-30 are the distances S7, a71 and S1. The distance S7 is
obtained by forming a cross product of Equation 2-30 with FS1
0SaSSSP =×+×+× 1F
71F
711F
7F
71F
orig6F aS . (2-31)
From Equation 2-18
71F
711F
7F s aSS =× . (2-32)
Substituting Equation 2-32 into Equation 2-31, and forming now a scalar (dot) product with Fa71,
it can be solved for S7 to obtain
71
71F
orig6F
1F
7 sS
aPS o×= . (2-33)
38
Figure 2-5. Vector loop to find the distances a71, S7 and S1.
Similarly, the distance a71 is obtained by first forming a dot product of Equation 2-31 with
FS7 to obtain
0a 7F
1F
71F
717F
1F
orig6F =×+× SSaSSP oo . (2-34)
The scalar triple product in the a71 term, can be expressed as
1F
7F
71F
7F
1F
71F
7F
1F
71F SSaSSaSSa ×=−×=× ooo . (2-35)
Equation 2-32 can be again substituted into Equation 2-35 to obtain
717F
1F
71F s−=× SSa o . (2-36)
The substitution of Equation 2-36 into Equation 2-34 allows solving for a71, to obtain
71
7F
1F
orig6F
71 sa
SSP o×= . (2-37)
Finally, the expression for S1 is obtained by forming a cross product of Equation 2-30 with
FS7, to obtain
39
0SSSaSP =×+×+× 7F
1F
17F
71F
717F
orig6F Sa . (2-38)
Next, a scalar product on Equation 2-38 is taken with Fa71 to obtain
0S 71F
7F
1F
171F
7F
orig6F =×+× aSSaSP oo . (2-39)
Equations 2-32 and 2-35 are substituted into Equation 2-39 and the expression for S1 is
71
71F
1F
orig6F
1 sS
aSP o×= . (2-40)
2.5.2 Special Cases for Closed-the-Loop Solution Technique
Equations 2-33, 2-37 and 2-40 become useless when the twist angle α71 is 90 degrees, this
occurs when S1 and S7 are collinear, parallel or antiparallel.
2.5.2.1 S1 and S7 are Parallel or Antiparallel
When the joint vectors S1 and S7 are parallel or antiparallel, the choice S7=0 gives a
solution to the problem. Substituting S7=0 into Equation 2-30 and following a similar procedure
to obtain S1 and a71, it is found
1F
orig6F
1S SP o=− (2-41)
1F
1orig6F
71 Sa SP −−= . (2-42)
The vector Fa71 is also unknown in this case, and its unique value is give by
71
1F
1orig6F
71F
aS SP
a−−
= . (2-43)
The angles θ7 and γ1 are found using Equations 2-23 and 2-24, and Equations 2-28 and 2-
29, respectively.
40
2.5.2.2 S1 and S7 are Collinear
When a71=0 is obtained from Equation 2-42, the joint vector S1 and S7 are collinear. In this
case, the angle θ7 is chosen as zero, making parallel the vectors a71 and a67. The angle γ1 is
calculated from Equations 2-28 and 2-29.
The closed-the-loop solution technique ends here. Pointing to the reverse kinematic
problem, the angle γ1 was obtained. The other parameters obtained here will be used when
finding θ1 to θ6 by solving the resulting closed-loop mechanism. This is explained in the next
sections of this chapter.
2.6 Spherical Closed-Loop Mechanisms
This section describes the methodology proposed by Crane and Duffy [1]. In the last
section, it was shown how any open-loop serial mechanism can be transformed into a closed-
loop spatial mechanism by adding a hypothetical closure link. In this section, a new equivalent
spherical mechanism is formed from the closed-loop spatial mechanism in order to generate
equations relating the joint angles and twist angles that will assist in the reverse kinematic
analysis problem.
The mechanism in this research involves only revolute joints, and thus the creation of the
equivalent spherical mechanism will be focused on this type of joint. To facilitate the
explanation, a spatial closed-loop quadrilateral will be considered. The first step to create the
equivalent spherical mechanism is to translate all the joint vectors (unit vectors by definition), Si
(i=1,…,4), of the revolute joints so that their tail is coincident with a common point O, as shown
in Figure 2-6.
41
Figure 2-6. Spherical quadrilateral. Courtesy of Dr. Carl Crane.
The second step consists in drawing a unit sphere centered at the point O. The unit vectors
Si (i=1,…,4) will meet the sphere at a sequence of points labeled 1,…,4, respectively. This was
expected because the joint vectors have unit norm. Because the joint vectors have maintained the
same direction, the angles between them, the twist angles, αij (ij=12, 23, 34, 41), remain
unaltered. Finally the equivalent spherical mechanism is formed by connecting adjacent links
(α12-α23, α23-α34, α34-α41, α41-α12) with the joints. If an original joint in the spatial mechanism was
a revolute joint, the equivalent joint in the spherical mechanism is also a revolute joint. Figure 2-
6 presents a complete spherical quadrilateral, equivalent to a 4L-4R spatial closed-loop
mechanism.
According to the way as the adjacent links have been connected, the link angles αij and the
joint angles θj are defined to be the same for the closed-loop spatial mechanism and the
equivalent spherical quadrilateral. As a consequence, any equation that relates twist and joint
angles of the equivalent spherical mechanism is also valid for the corresponding closed-loop
spatial mechanism.
42
2.7 Mobility of Spherical Mechanisms and Classification of Spatial Mechanisms
This section describes the methodology proposed by Crane and Duffy [1]. A link of a
spherical mechanism has three degrees of freedom. Only three angles are needed to completely
specify the position and orientation of the spherical link in the unit sphere. By analogy to
Equation 1-2, the mobility of a spherical mechanism with one link fixed to ground is
3fMj
1iiS −= ∑
=
. (2-44)
where:
Ms: mobility of the equivalent spherical mechanism
fi: relative degrees of freedom of the i-th joint
j: number of revolute joints.
Knowing the mobility of the equivalent spherical mechanism will allow for classification
of the closed-loop spatial mechanism. All one degree of freedom spatial closed-loop mechanisms
are classified based on the mobility of their equivalent spherical mechanism. It has been shown
that the complexity of the solution of a one degree of freedom spatial mechanism (where one
joint angle is given) is directly related to the degree of mobility of its equivalent spherical
mechanism. One degree of freedom mechanisms with the same equivalent spherical mechanism
mobility have similar solution techniques for solution of the reverse kinematics.
2.8 Joint Vectors Expressions in the Spherical Mechanism
This section describes the methodology proposed by Crane and Duffy [1]. Expressing the
direction cosines of each unit joint vector, S1 to S7, in each of the standard coordinate systems is
important in the analysis of both spherical and spatial mechanism. In this section, the joint
vectors S1 to S3 will be expressed in the first standard coordinate system. Later, useful recursive
notations derived from the joint vectors’ expressions are presented.
43
All the joint vectors are unit vectors aligned with the z axis of its associated standard
coordinate system, where for example
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100
11S (2-45)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100
22 S . (2-46)
From the result obtained in Equation 2-6, the rotational part of the transformation matrix
that relates the first and second coordinate systems will allow expressing the second joint vector
presented in Equation 2-46 in the first standard coordinate system as
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
12
12
12122122
12122122
22
21
cs0
100
cccssscccs0sc
S . (2-47)
Exchanging the subscripts in Equation 2-47, the expression for the third joint axis in the
second standard coordinate system can be obtained by making 1→2 and 2→3, it is found
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
23
2332
cs0
S . (2-48)
Now, 1S3 can be obtained as
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−==
223122312
223122312
223
23
23
12122122
12122122
22
321
231
csscccsccs
ss
cs0
cccssscccs0sc
R SS . (2-49)
A recursive shorthand notation is introduced in Equation 2-49
44
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
223122312
223122312
223
2
2
2
31
csscccsccs
ss
ZYX
S . (2-50)
And the recursive expressions for the terms in Equation 2-50 are
jjkj ssX = (2-51)
( )jjkijjkijj csccsY +−= (2-52)
jjkijjkijj cssccZ −= . (2-53)
The next expression to be found is the joint vector S4 in terms of the first standard
coordinate system, 1S4. First, the term 2S4 is obtained by exchanging subscripts in the first two
terms of Equation 2-50
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
3
3
3
42
ZYX
S . (2-54)
1S4 is obtained by
( )( )⎥
⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++−+
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−==
312232312
312232312
2323
3
3
3
12122122
12122122
22
321
241
Zc)cYsXsZs)cYsXc
sYcX
ZYX
cccssscccs0sc
R SS . (2-55)
From Equation 2-55, another recursive notation can be obtained. The objective of this
section is not to deduce all the expressions, which is an extremely lengthy process. The objective
is just to introduce the recursive notations resulting from expressing the joint vectors in a
standard coordinate system different to the system attached to that joint. The summary for the
joint vectors expressed in the first standard coordinate system is
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100
11S ;
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
12
1221
cs0
S ;⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
2
2
2
31
ZYX
S ;⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
32
32
32
41
ZYX
S ;…⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
−−
−−
−−
2,...,2,1
2,...,2,1
2,...,2,1
ZYX
nn
nn
nn
n1 S . (2-56)
45
The result obtained in Equation 2-56 can be extended for expressing the joint vectors in
any standard coordinate system (n=2,…,7). Along the process of finding the results in Equation
2-56, Crane and Duffy [1] found recursive notations when expressing the joint vectors in
different standard coordinate systems, which are useful for solving consciously the reverse
kinematic problem. The expressions are presented in Appendix A, Equations A-1 to A-38.
2.9 Link Vectors Expressions in the Spherical Mechanism
This section describes the methodology proposed by Crane and Duffy [1]. Expressing the
direction cosines of each link vector, a12 to a67, in each of the standard coordinate systems is
important in the analysis of both spherical and spatial mechanisms. In this section, the joint
vectors a12 to a34 are expressed in the first standard coordinate system. Later, recursive notation
derived from the link vectors’ expressions is presented.
All the link vectors are unit vectors aligned with the x axis of its associated standard
coordinate system, and thus
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
001
121a (2-57)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
001
232 a . (2-58)
From the result obtained in Equation 2-6, the rotational part of the transformation matrix
that relates the first and second coordinate systems will allow expressing the link vector a23,
presented in Equation 2-58, in the first standard coordinate system as
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
212
122
2
12122122
12122122
22
321
sscs
c
001
cccssscccs0sc
a . (2-59)
46
Exchanging the subscripts in Equation 2-59, the expression for the link vector a34 in the
second standard coordinate system can be obtained, by making 1→2, 2→3 and 3→4, it is found
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
323
233
3
432
sscsc
a . (2-60)
Now, 1a34 can be obtained as
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−==
233
233
3
12122122
12122122
22
4321
2431
sscsc
cccssscccs0sc
R aa
( )( ) ⎥
⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++++−
−=
2332321223312
2332321223312
233232
341
csccssssccsccscsss
csscca . (2-61)
A recursive shorthand notation is introduced in Equation 2-61
( )( ) ⎥
⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++++−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= ∗
2332321223312
2332321223312
233232
321
321
32
341
csccssssccsccscsss
csscc
UU
Wa . (2-62)
The next expression to be found is the joint vector a45 in terms of the first standard
coordinate system, 1a45. First, the term 2a45 is obtained by exchanging subscripts in the first two
terms of Equation 2-62
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= ∗
432
432
43
452
UU
Wa . (2-63)
The link vector 1a45 is obtained by
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−== ∗
432
432
43
12122122
12122122
22
4521
2451
UU
W
cccssscccs0sc
R aa
47
and a new recursive notation is identified as
( )( ) ⎥
⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−−
+= ∗
∗
∗
∗
1432
1432
243
43243222143221
43243222143221
4324322
451
UU
W
WsUcsUcWsUccUs
WcUsa . (2-64)
By following the same procedure, expressions for all the link vectors can be expressed in
the first standard coordinate system, to obtain
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
001
121a ;
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
212
212
2
231
sscs
ca ;
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= ∗
132
132
32
341
UU
Wa ;
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= ∗
1324
1324
432
451
UU
Wa ;
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= ∗
13254
13254
5432
561
UU
Wa ;
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= ∗
132654
132654
65432
671
UU
Wa . (2-65)
The results presented in Equation 2-65 can be extended for finding the link vectors in any
standard coordinate system. This is done by properly exchanging the subscripts in Equation 2-65.
Crane and Duffy [1] found recursive notations when expressing the link vectors in different
standard coordinate systems, which are useful for solving consciously the reverse kinematic
problem. The recursive notations are presented in Appendix A, Equations A-39 to A-60.
The joint and link vectors can be expressed in any coordinate system, then there is a set
consisting of seven joint vectors (S1 to S7) and six link vectors (a12 to a67) expressed in each of
the standard coordinate systems based on the recursive equations presented, seven sets in total. In
addition, there are sets that are combinations of the standard coordinate systems: the x and z axes
can be aligned with the a and S axes or two different standard coordinate systems. All the sets
for the closed loop mechanisms can be obtained at the appendix section of Crane and Duffy [1].
A useful characteristic of the notation here presented is that when the X, Y, or Z notation
terms are fully expanded, they only contain the sine or cosine of the joint angles whose
48
subscripts appear in the shorthand notation. This will be used when solving the reverse kinematic
of the mechanism in Chapter 4.
2.10 Orientation Requirements Specification: X-Y-Z Fixed Angles
The desired orientation of the body to which the end effector is attached is specified by the
angles ζx, ζy and ζz. In this method, two frames, [A: xA-yA-zA] and [B: xB-yB-zB], are initially
aligned. Frame [B] is then rotated an angle ζx about xA, then it is rotated an angle ζy about yA
and finally it is rotated an angle ζz about the axis zA. The transformation matrix resulting from
the successive rotations is obtained by
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
333231
232221
131211
AzAyAx
rrrrrrrrr
)z,(ζ)y,(ζ)x,(ζ RRR (2-66)
where the right-hand side matrix in Equation 2-66 is given by Equation 2-11. Given the
resulting transformation matrix, the angles can be obtained through the atan2 function as
( )221
21131y rr,r2tanaζ +−= (2-67)
( )y11y12x cosζ/r,cosζ/r2tanaζ = (2-68)
( )y33y32z cosζ/r,cosζ/r2tanaζ = . (2-69)
Further information about representation of the orientation of rigid bodies, see Craig [28].
49
CHAPTER 3 NON-CIRCULAR GEARS FUNDAMENTALS
This chapter introduces non-circular gear fundamentals that will support the non-circular
gear synthesis proposed in Chapter 5. Gears are bodies in direct contact that have been toothed
to ensure motion transmission without slipping. Gear pairs are used to transmit motion either
from a rotating shaft to a body which rotates or from a rotating shaft to a body which translates -
which is considered to be rotating about an axis at infinity. This chapter presents basic gearing
concepts such as the pitch point, pitch curve, Input/Output5 non-circular gear relationship, and
input and output angles. Geometric synthesis of the pitch profile in Cartesian coordinates closes
this chapter.
3.1 Pitch Curve and Pitch Point
Figure 3-1 shows a direct-contact three-link mechanism. The line N-N is the common
normal to the contacting surfaces and it intersects the centers line O2O3 at point P. The points O2
and O3 are instant centers and the line N-N is called line of action. The Arnold-Kennedy theorem
states: “Any three bodies having plane motion relative to one another have exactly three instant
centers, and they lie on the same straight line”. As a consequence of this theorem, the point P is
the third instant center (See Dooner and Seireg [3] and Erdman, Sandor and Kota [29]). An
instant center is (1) a point in one body about which some other body is rotating either
permanently or at the instant, or (2) a common point to two planar bodies in motion which has
the same velocity in both magnitude and direction (e. g., see Martin [30]).
Uniform motion transmission between two parallel axes is only possible if the line of
action passes through a fixed point, known as the pitch point -Dooner and Seireg [3]-, this is the
point P in Figure 3-1. Now, for each different position of the links 2 and 3, the locus of pitch
5 Also written as I/O.
50
point determines the pitch circle. The path of the moving instant center -pitch point- generates
the pitch curve or centrode. In the case of circular gears the distances O2P2 and O3P3 are
constant. In the case of non-circular gears, the distances O2P2 and O3P3 vary.
Figure 3-1. Pitch point and pitch curves of two bodies in mesh.
3.2 Input/Output Non-Circular Gear Relationship
The transmission function for two bodies in mesh is defined as the relationship between
the angular position of the input element and the corresponding angular position of the output
element. From this concept, the instantaneous gear ratio, g, is written as the ratio between the
infinitesimal displacement of the output body and the corresponding infinitesimal displacement
of the input body. Accordingly, it can be written as
i
o
δdδd
g ≡ (3-1)
where:
g: instantaneous gear ratio or I/O relationship
51
dδo: instantaneous angular displacement of the output body
dδi: instantaneous angular displacement of the input body.
3.2.1 Input/Output Relationship with the Connecting Link Grounded
When the connecting link is fixed to ground, as presented in Figure 3-2A, the I/O
relationship is given directly by Equation 3-1. The I/O relationship, g, in Equation 3-1 is defined
as positive according to the senses of the input and output angles, δi and δo, depicted in Figure 3-
2A.
Figure 3-2. Non-circular input and output bodies in mesh. A) Grounded connecting link. B) All links moving.
3.2.2 Input/Output Relationship with All Links Movable
Figure 3-2B depicts the geared links with the links j-1, j, and j+1 all movable which is the
general case for two bodies in mesh. For non-circular gears in mesh attached to movable links,
the recurrent equation for the I/O non-circular relationship is given by Equation 3-2. The angles
σj-1, σj, and σj+1 are the net angular displacements. These angles are measured with respect to the
fixed x axis.
)σd(σ)σd(σ
g1jj
j1jj
−
+
−
−= (3-2)
52
where:
gj: I/O relationship of the transmission
dσj: instantaneous angular displacement of the j-th link.
If the connecting link in Figure 3-2B is fixed to ground, Equations 3-1 and 3-2 are the
same since σj becomes zero. For the case where the connecting link is grounded, dσj in Equation
3-2 is zero, then the expression for gj in Equation 3-2 looks to be negative. This apparent
difference is explained if it is noticed that in Figures 3-2A and 3-2B, the input and output angles,
σi and σo, are measured respect to the x fixed axis in clockwise and counterclockwise sense,
respectively. On the other hand, all the link angles in Figure 3-2B are measured in clockwise
sense; this explains the apparent difference between Equations 3-1 and 3-2. In the current work,
the recurrent expression presented in Equation 3-2 will be used and referred to as the local
coordinates approach model, in which the reference system x-y is perpendicular to the rotation
axes of the links (bodies) in mesh pointing out of the page in Figure 3-2A and 3-2B.
3.3 Complete Profile Synthesis: Non-Circular Relationships for the Entire Rotation
Some real applications of toothed bodies are fulfilled by partial circular or non-circular
gears, where the covered angle is less than 2π. However, a complete profile is sometimes
required for manufacturing purposes –in particular for the simulation of the cutter, Dooner and
Seireg [3]-. To this end, working and non-working sections of the centrode are now introduced.
The working section of the gear is used to perform a task. The non-working section converts the
working slice into a complete disk. Figure 3-3 shows a complete non-circular gear with arbitrary
working and non-working sections.
53
Figure 3-3. Arbitrary working and non-working sections and angles in a complete non-circular gear.
If the working section is circular, then the complete profile becomes a conventional
circular gear. If the working section is non-circular, additional geometric considerations must be
taken into account to achieve a complete non-circular pitch profile.
The I/O non-circular relationship corresponding to the working section is written as gw and
its attaintment was presented in Equations 3-1 and 3-2. Similarly, the I/O relationship for the
non-working section is labeled by gnw. The geometric considerations to obtain the non-working
section are discussed below.
When the I/O relationship is not specified for the complete rotation of the input gear, the
non-working I/O relationship can be achieved by introducing a polynomial in the input angle,
whose order and coefficients are derived from the geometry constraints of the gearing. Figure 3-
4 illustrates four constraints required to guarantee a continuous profile where δint makes
reference to the final value of the input angle for the working section.
54
Figure 3-4. Boundary conditions for the whole synthesis of the Input/Output relationship. A) g boundary conditions. B) Slope boundary conditions.
The I/O non-circular gear relationship must satisfy another particular constraint to
successfully synthesize a complete profile. The integral shown in Equation 3-3 must be always
rational; otherwise, the output gear cannot maintain an indefinite number of cycles with the
desired functional relationship (Dooner and Seireg [3]). Therefore
∫=π2
0 iδdgπ2 (3-3)
where:
g: instantaneous gear ratio or I/O relationship
dδi: instantaneous angular displacement of the input body.
55
From Figure 3-4 and the constraint imposed by Equation 3-3, the five boundary conditions
are
)(δg)(δg intnwintw = (3-4)
)(δg')(δg' intnwintw = (3-5)
π)2(g)0(g nww = (3-6)
)0(g')0(g' nww = (3-7)
∫ ∫−=π2
δ
δ
0 iwinwint
int dδgπ2dδg (3-8)
where in general, idδ
dgg'= . This is the derivative of the I/O relationship respect to the input
angle, for both working and non-working sections.
Equation 3-8 is not evident from Figure 3-4 and its attainment is explained as follows.
Figure 3-3 presents the working and non-working angles for an arbitrary non-circular gear. From
Figure 3-4 is apparent that working and non-working angles satisfy
π2δδ nww =+ . (3-9)
Equation 3-9 is valid for both input and output angles. If Equation 3-9 is used for the output
angle –output body-, it becomes
π2δδ nwowo =+ . (3-10)
From Equation 3-1, it can be solved for dδo to obtain
io dδgδd = , (3-11)
where integration gives the solution for the output angle δo. This integral is divided into two
terms: the working and the non-working sections whose integration limits are evident from
Figure 3-4. The working and non-working output angles are, respectively
56
∫=intδ
0 iwwo δdgδ (3-12)
∫=π2
δ inwnwoint
δdgδ . (3-13)
Substituting Equations 3-12 and 3-13 into Equation 3-10 gives
π2δdgδdgπ2
δ inw
δ
0δ iwint
int
0
=+ ∫∫ =, (3-14)
which is the same as Equation 3-8. In summary, Equation 3-8 is derived from Equation 3-3 and
the linearity principle for the integral.
A conscious selection of the degree of the polynomial in the input angle of the non-
working profile is degree 4. There are five coefficients that can be obtained from the five
Equations 3-4 to 3-8. In addition, this selection of gnw makes the integral in Equation 3-8 always
rational and guarantees an indefinite number of cycles. The 4th degree polynomial in the velocity
or g domain is given by
0i12
i23
i34
i4inw cδcδcδcδc)(δg ++++= (3-15)
where:
δi: input angle
gnw: non-working I/O gear relationship
ci: fourth-degree polynomial’s coefficients.
The derivative of the polynomial is
1i22
i33
i4inw cδc2δc3δc4)(δg' +++= . (3-16)
The substitution of Equations 3-15 and 3-16 into Equations 3-4 to 3-8 forms a set of five
linear equations representing the five unknown coefficients c0,…,c4 of the polynomial presented
in Equation 3-15. The solution for the coefficients can be written in the matrix format as
57
ΗΩC 1−= (3-17)
where:
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
0
1
2
3
4
ccccc
C ,
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
=
∫intδ
0 iiw
'w
int'w
intw
dδ)(δgπ2(0)gπ2
)(δg)(δg
H ,
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−−
=
int
2int
23int
34int
45int
5
23
234int
2int
3int
int2int
3int
4int
δπ22
δπ)(23
δπ)(24
δπ)(25
δπ)(201π)2(2π)3(2π)4(21π)(2π)(2π)(2π)(201δ2δ3δ41δδδδ
Ω . (3-18)
At this point, a (not unique) g has been completely derived for the non-working profile.
For additional information about the non-working gear relationship synthesis, see Dooner [24].
In the next section, a method to obtain the Cartesian coordinates of the pitch point is presented.
3.4 Pitch Curve Coordinates
Equations that relate the input angles, output angles, and the I/O relationships in order to
obtain the pitch curve coordinates are presented here. Figure 3-5 introduces the center distance,
E, as an additional parameter. This plot also defines two rectangular coordinate systems used to
describe the pitch curves of the input and the output bodies. The input and output radii are related
by
Err io =+ . (3-19)
The I/O relationship is introduced to add one more equation. For the two bodies in mesh
shown in Figure 3-5, g becomes
58
o
i
rr
g = . (3-20)
Equations 3-19 and 3-20 form a system of two equations in the two unknowns, ro and ri.
The radii expressions are
g1Ero +
= (3-21)
g1gEri +
= . (3-22)
Finally, the Cartesian coordinates for the output and input pitch profiles (deduced from
Figure 3-5) are, respectively,
)sinδr,δcos(rR ooooo = (3-23)
)sinδr,cosδ(rR iiiii = . (3-24)
Figure 3-5. Rectangular coordinate systems x-yo and x-yi for a gear pair in mesh.
In this work, non-circular gear transmissions with movable links are considered, as
presented earlier in Figure 3-2(b). If this is the case, the pitch profile Cartesian coordinates are
easily obtained by attaching the abscissa of the rectangular coordinate system in Figure 3-5 to the
connecting link j in Figure 3-2(b), as presented in Figure 3-6.
59
Expressions for the input and output angle are now presented. For the working section, the
output and input angles can be directly obtained from the numerator and denominator of
Equation 3-2, to obtain
j1jwo δδδ −= + (3-25)
1jjwi δδδ −−= . (3-26)
Figure 3-6. Local coordinate systems x-yi and x-yo in the synthesis of the pitch point with input, output and connecting bodies movable.
In reference to the non-working section, from the abscissas in Figure 3.4, the input angle is
set as the portion from the end of the working profile, δint, to the end of the whole disk: 2π. The
input angle of the non-working section can be represented as the interval
[ ]π,2δδ intnwi = . (3-27)
The output angle of the non-working profile is found by using Equation 3-11 for the non-
working section, to obtain
nwinwnwo dδgdδ = . (3-28)
60
A substitution of the polynomial expression for g in the non-working section, Equation 3-
15, into Equation 3-28 will give
( )∫ ++++=nwiδ
i0i12
i23
i34
i4nwo dδcδcδcδcδcdδ (3-29)
where δnwi is varied from Equation (3-27).
After integrating Equation 3-21, the compact format equation for the output angle at the non-
working profile is
[ ]∑=
++ −+
=4
0i
1i1iinwo δπ)(2
1ic
δ (3-30)
where:
ci: coefficients in polynomial of Equation 3-15, found in Equation 3-17
δ: non-working input angle given by the interval presented in Equation 3-27.
Here end the gearing considerations for the synthesis of the non-circular profiles.
Input/Output non-circular gear relationship, and output and input angles synthesis have been
presented for both working and non-working profile sections.
61
CHAPTER 4 REVERSE KINEMATIC ANALYSIS OF THE 1-DOF SPATIAL MOTION GENERATOR
The closed-loop spatial mechanism previously presented in Figure 1-7 can be separated
into two open loop spatial mechanisms. Consider the two fixed coordinates systems xF1-yF1-zF1
and xF2-yF2-zF2 as depicted in Figure 4-1. Two paths from the origins of the new coordinate
systems, OF1 and OF2, to the point P can be easily identified, they are 1-2-3-4-5-6-7 and 1-12-11-
10-9-8-7, respectively.
Figure 4-1. 1-DOF spatial motion generator.
Each path defines an open-loop mechanism formed by six links (not counting ground) and
six revolute joints. The reverse kinematic analysis of the closed-loop spatial motion generator
will be solved by performing the reverse kinematic analysis for each of the two open-loop
mechanisms separately. In this way the joint angles for the left and right-side open loop chains
can be determined for a specific position of point P in link 7 and for the orientation of link 7.
In this chapter, the reverse kinematic solution of an open-loop mechanism comprised of six
links and six revolute joints with consecutive pairs of joint axes parallel is presented. A closed-
62
form solution technique is used where a hypothetical link that connects the first and sixth joint
axes is introduced which converts the open-loop chain to a closed-loop mechanism.
Before starting the reverse kinematic analysis, a brief analysis of the overall mechanism
mobility is presented; followed by the introduction of the standard coordinate systems, forward
kinematic analysis, reverse kinematic problem statement, close-the-loop technique, and spherical
mechanisms. For detailed explanation about the topics presented in this chapter, see Crane and
Duffy [1].
The solution of the complete closed-loop mechanism is obtained by combining the result
obtained in the solution of the open-loop mechanism with a coordinate transformation.
4.1 Mobility Analysis of the Spatial Motion Generator
Figure 4-2 presents a detailed labeling of the spatial motion generator. The non-circular
gears do not represent extra links because they are attached to the kinematical links. Links 1
(fixed link), 3, 5, 7, 9 and 11 have non-circular gears attached to them. The mobility equation for
this closed-loop mechanism can be written as
∑∑==
−−=k
1jj
n
1ii g6fM (4-1)
where:
M: mobility or DOF
n: number of spatial links or connected bodies
fi: number of relative degrees of freedom permitted by the i-th joint
gj: relative degrees of freedom that removes the j-th modification
k: number of modifications introduced to the mechanism.
63
Figure 4-2. Labeling of the spatial motion generator for mobility analysis. Parts with the same number label are one single link.
When Equation 4-1 is applied to the spatial motion generator in Figure 4-2, there are
twelve links (n=12); every joint is a revolute joint with one relative DOF (fi=1; i=1,…,12); there
are five geared connections or modifications (k=5); and each modification allows one relative
DOF (gj=1; j=1,…5). Thus the mobility of the mechanism is one (1-DOF).
4.2 Grouping of the 7L-7R Spatial Mechanism with Consecutive Pairs of Parallel Joint Axes: Equivalent Spherical Mechanism
The 6L-6R6open-loop spatial mechanism forms a 7L-7R closed-loop spatial mechanism
after the close-the-loop technique is used. Figure 4-3 presents the kinematic labeling for the
open-loop spatial mechanism. A special geometry is used whereby the twist angles α12, α34 and
α56 are all zero. This results in the pair of axes S1-S2, S3-S4 and S5-S6 being parallel.
6 Mechanism comprised by six links and six revolute joints.
64
Figure 4-3. Kinematic labeling of the 6L-6R open-loop mechanism. A) Joint angles. B) Link angles. C) Link and joint distances.
65
In Section 2.7 it was explained how the equivalent spherical mechanism is formed by first
translating the joint vectors to the center of a unit sphere. If two joint vectors are parallel, then
they will be coincident on the sphere. As a result, the joint vectors S1 and S2 will be coincident,
similarly for the pairs of vectors S3-S4 and S5-S6. This implies that the equivalent spherical
mechanism associated with the 7L-7R parallel axes is a spherical quadrilateral with only four
revolute joints, as presented in Figure 2-6, where S1, S2, S3 and S4 in the quadrilateral represent
the joint vectors S1-S2, S3-S4, S5-S6 and S7, respectively.
The mobility equation of the spherical mechanisms was presented in Equation 2-44. There
are four revolute joints in the spherical quadrilateral, from Equation 2-44 the mobility of this
mechanism is one. This result is interesting because it classifies this 7L-7R mechanism with
parallel axes as Group 1, different to the conventional 7L-7R mechanism which is Group 4,
which means that two spatial mechanisms with the same number of links and revolute joints are
to be solved by different solution techniques.
The next section presents the solution of the equivalent spherical quadrilateral, which is the
step following the closure-the-loop technique for solving the reverse kinematic problem.
4.3 Reverse Kinematic of the 6L-6R Open-Loop Mechanism with Consecutive Pairs of Parallel Joint Axes
4.3.1 Problem Statement
It is assumed that the close-the-loop parameters are already found (close-the-loop
technique was extensively explained in Section 2.5) and are known parameters at this stage. The
problem statement is reduced to
Known
1) The constant mechanism parameters: link lengths a12 to a56, twist angles α12 to α56, and
offset distances S2 to S5.
66
2) Offset distance S6 and the direction of the vector a67 relative to the vector S6
3) Position and orientation of the end effector: FPtool, FS6 and Fa67
4) Location of the tool point in the 6th coordinate system: 6Ptool
5) Close the loop parameters: Offset distances S1, S7; link distance a71; joint angle θ7; angle
γ1.
Find
The joint angles θ1,…,θ6 of the closed-loop 7L-7R spatial mechanism.
Because the equivalent spherical mechanism is a quadrilateral, solving this spherical
quadrilateral will be the first step. The second part of the reverse kinematics is to express the
joint and link axes of the vector loop equation (for the 7L-7R) in a set of joint and link axes
expressed in a convenient coordinate system.
4.3.2 Solution of the Equivalent Spherical Quadrilateral
Figures 4-4A and 4-4B present the spherical quadrilateral with the “quadrilateral labeling”
and “heptagon labeling”, respectively, where the subscript Q stands for quadrilateral and the
subscript H stands for heptagon. Comparing Figures 4-4A and 4-4Bit is apparent that
H7Q4 θθ = (4-2)
H67H56Q34 ααα += (4-3)
H71Q41 αα = . (4-4)
It was stated that this is the second stage of the solution. The first step was the close-the-
loop procedure, where the joint angles θ7H, α71H and α71H were found. From Equations 4-2 to 4-4,
the joint angle θ4Q and the link angles α34Q and α41Q are known parameters. The angle θ4Q is
chosen as the input angle for being a known parameter. When comparing Figures 4-4A and 4-4B
is also evident that
67
H23H12Q12 ααα += (4-5)
H45H34Q23 ααα += . (4-6)
The twist angles α23H and α45H are known parameters which depend on the geometry of the
spatial mechanism, see Problem statement in Section 2.4, then the angles α12Q and α23Q are also
known parameters from Equations 4-5 and 4-6.
At this point, the only unknowns in the spherical mechanism are the joint angles θ1Q, θ2Q
and θ3Q. Crane and Duffy [1] easily solved the spherical quadrilateral by writing an appropriate
spherical cosine law which contains the angle θ1Q as the only unknown. This solution is
presented next.
4.3.2.1 Solving for θ1Q
The spherical cosine law Z41=c23 is used (defined in Equation A-6)
23412Q14Q1412 cZc)cYs(Xs =++ . (4-7)
Rearranging terms in Equation 4-7 yields
0DBsAc Q1Q1 =++ (4-8)
where:
412YsA = , 412XsB = , 23412 cZcD −= , X4, Y4 and Z4 are expanded using Equations 2-57 to 2-
59, respectively.
The trigonometric solution of Equation 4-8 gives
( )22Q1
BADγθcos+
−=− (4-9)
where a unique value for γ is determined from
22 BABsinγ+
= and 22 BAAcosγ+
= .
68
Because two values satisfy cos(θ1-γ) in Equation 4-9, then two values of θ1Q will also be
obtained, θ1QA and θ1QB.
Figure 4-4. Equivalent spherical mechanism. A) Spherical notation. B) Heptagon notation.
4.3.2.2 Solving for θ2Q
In order to find a unique value for θ2Q, the sine and the cosine expressions are obtained
from
Q22341 ssX = (4-10)
Q22341 csY = . (4-11)
Substituting Equations 2-60 and 2-62 into Equations 4-10 and 4-11, it is solved for the sine and
cosine as
23
Q14Q14Q2 s
sYcXs
−= (4-12)
23
412Q14Q1412Q2 s
Zs)cYs(Xcc
−+= (4-13)
where X4, Y4 and Z4 are expanded using Equations A-4 to A-6, respectively.
69
A unique value for θ2Q is obtained for each value of θ1Q (θ1QA and θ1QB), they are θ2QA and
θ2QB.
4.3.2.3 Solving for θ3Q
A suitable choice for finding the sine and cosine of the θ3Q is
Q32314 ssX = (4-14)
Q32314 csY = . (4-15)
Substituting Equations 2-64 and 2-66 into Equations 4-14 and 4-15 and solving for sine and
cosine of θ3Q yields
23
Q41Q41Q3 s
sYcXs
−= (4-16)
23
134Q41Q4134Q3 s
Zs)cYsX(cc
−−= (4-17)
where 1X , 1Y and 1Z are obtained from Equations 2-51 to 2-53, respectively.
A unique value is for θ3Q each value of θ4Q (θ4QA and θ4QB), they are θ3QA and θ3QB. A
solution tree of the spherical quadrilateral solution obtained angles is Figure 4-5.
Figure 4-5. Solution tree of the spherical quadrilateral.
4.3.3 Vector Loop Equation of the 7L-7R Parallel Axes Mechanism
Some known parameters will be first rewritten. According to the geometry of the
mechanism, the values of the twist angles are
0α12 = (4-18)
70
0α34 = (4-19)
0α56 = . (4-20)
The zero offset distances are
0S3 = (4-21)
0S5 = . (4-22)
The twist angle α67 and the link distance a67 are free choices that were chosen before the
close-the-loop procedure respectively as
o67 90α = (4-23)
0a 67 = . (4-24)
On the other hand, when comparing Figures 4-4A and 4-4B, as a direct consequence of the
parallel axes geometry, it is apparent that
Q1H2H1 θθθ =+ (4-25)
Q2H4H3 θθθ =+ (4-26)
Q3H6H5 θθθ =+ . (4-27)
From now on, the notation will be simplified, the joint angles, θ1H,…, θ6H will be denoted
by θ1,…,θ6. Equation 4-28 is the vector loop equation of the closed-loop mechanism
0aSaSaSaSaSaSaS
=+++++++++++++
7171776767665656554545
44343433232322121211
aSaSaSaSaSaSaS
. (4-28)
The first reduction of Equation 4-28 is made by substituting Equations 4-21, 4-22 and 4-24, to
yield
0aSSaaSaaSaS
=++++++++++
717177665656454544
3434232322121211
aSSaaSaaSaS
. (4-29)
71
Table 4-1. Direction cosines of a closed-loop 7L-7R mechanism expressed in a coordinate system were the x and z axes are aligned with the vectors a23 and S3, respectively.
Joint vectors Link vectors S3 (0, 0, 1) a23 (1, 0, 0) S2 (0, s23, c23) a12 (c2, -s2c23, U23) S1 (X2, -Y2, Z2) a71 (W12, U*
123, U123) S7 (X12, -Y12, Z12) a67 (W712, U*
7123, U7123) S6 (X712, -Y712, Z712) a56 (W6712, U*
67123, U67123) S5 (X6712, -Y6712, Z6712) a45 (W56712, U*
567123, U567123) S4 (X56712, -Y56712, Z56712) a34 (c3, s3, 0) Crane and Duffy [1].
Equation 4-29 is conveniently expressed in a particular coordinate system where the x-axis
is aligned with the link vector a23 and the z-axis is aligned with the joint vector S3 (See Table 4-
1), to obtain
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
000
UUW
aZYX
SZYX
SUUW
aUUW
a
ZYX
S0sc
a001
acs0
SU
csc
aZYX
S
123
*123
12
71
12
12
12
7
712
712
712
6
67123
*67123
6712
56
567123
*567123
56712
45
56712
56712
56712
43
3
3423
23
232
23
232
2
12
2
2
2
1
. (4-30)
Three scalar equations are extracted from Equation 4-30 for the x, y and z projection of the
vector loop equation onto Set 13. These equations are
0WaXSXSWaWaXScaacaXS
127112771266712565671245
5671243342321221
=+++++++++
(4-31)
0UaYSUaUa
YSsasScsaYS*123717126
*6712356
*56712345
5671243342322321221
=+−++
−++−− (4-32)
0UaZSZSUaUaZScSUaZS
1237112771266712356
56712345567124232231221
=++++++++
. (4-33)
The theory of analysis developed in robot manipulators also allows expressing some
recurrent terms in different forms; with different subscripts by using fundamental and subsidiary
72
equations for a spherical and polar heptagon, further information see Crane and Duffy [1]. It was
explained before that the joint angles associated with the subscripts in the shorthand notation also
appear in the expansion of that term, then some substitutions are going to be made in order to
obtain expressions that involve the known values, so far the sums of the angles (θ1+ θ2), (θ3+ θ4),
(θ5+ θ6), and the closure angle θ7. Based on this, the x-projection written in Equation 4-31 can be
rewritten if some fundamental and subsidiary equations are used, they are
356712 XX = (4-34)
4356712 WW = (4-35)
5436712 WW = . (4-36)
The substitution of Equation 4-34 to 4-36 into Equation 4-31 allows expressing the
projection onto a23 in Set 13 as
0WaXSXSWaWaXScaacaXS
1271127712654345
4345343342321221
=+++++++++
. (4-37)
Similarly, the substitutions
33456712 csY = (4-38)
43*567123 VU −= (4-39)
543*67123 VU −= (4-40)
into Equation 4-32 will convert the y-axis of Set 13 into
0UaYSYSVaVa
csSsasScsaYS*123711277126543564345
33443342322321221
=+−−+−−
−++−−. (4-41)
For the projection onto S3 of Set 13, the substitutions
3456712 cZ = (4-42)
43567123 UU = (4-43)
73
54367123 UU = (4-44)
into Equation 4-33 yield
0UaZSZSUaUacScSUaZS
12371127712654356
4345344232231221
=++++++++
. (4-45)
Equations 4-37, 4-41 and 4-45 are going to be transformed by using the recursive notation
presented in Appendix A. Consider first Equation 4-37; the expansions for X2, 3X , W43, W543,
X712, X12 and W12 are going to be taken from Equations A-4, A-1, A-44, A-48, A-15, A-7 and A-
41, respectively. Next, the substitution of Equations 4-18 to 4-20 -the link angles α12, α34 and α56-
and 4-23 –α67- into Equation 2-37 yields
0cassS)ccss(cScaascsaccacaca
2171712177712172164345
23545435654356334212
=++++++−++
+++++
++ . (4-46)
The shorthand notation ci+j=cosθicosθj-sinθisinθj and si+j=sinθicosθj-cosθisinθj has been
used in Equation 4-46 so that the terms (θ1+ θ2), (θ3+ θ4), and (θ5+ θ6) (previously found in the
spherical quadrilateral solution) have been introduced.
Similarly, Equations 4-18 to 4-20 and 4-23 are substituted into Equation 4-41and 4-45.
Next, the expansion of the shorthand terms in Equations 4-41 and 4-45 using the recursive
notation presented in Appendix A, yields for the y-projection and S3 projections on Set 13,
respectively
0S)csscssccc(ccsasaS)sccc(sscccs
sSsScsasccasasca
623721771232371217232171
4345771212371237121237123
23223154356545435633422312
=−−+−++++
+++++−
+++
+++
++
(4-47)
0)ccscs(SssaScScS)cscssscscc(Sssassa
712371212372321714232231
7712323721712372165455622312
=+−+++++−+−++
++
++ . (4-48)
In the next section, Equations 4-46 to 4-48 are solved for the three joint angles θ2, θ3 and
θ5.
74
4.3.4 Tan-Half-Angle Solution for θ5
Equations 4-46 to 4-48 are conveniently rewritten as
0AsAcAcAcA 554533221 =++++ (4-49)
where:
121 aA =
342 aA =
43563 caA +=
4543564 csaA +−=
2171712177712172164345235 cassS)ccss(cScaaA +++++ +++++= ,
0BsBcBsBsB 554533221 =++++ (4-50)
where:
23121 caB −=
342 aB =
43563 saB +=
4543564 ccaB +=
6771232372171217232171
4345771212371232322315
S)cssc]sscc([ccsasaS)sccc(ssSsSB
−−+−++++=
+++
++ ,
0DsDsD 35221 =++ (4-51)
where:
23121 saD =
45562 saD =
75
)ccscs(SssaScScS)cscs]ssccc([SD
712371212372321714232
23177123237217172163
+−+++++−+−=
++
++ .
The purpose of this section is to strategically manipulate Equations 4-49 to 4-51 so that a
single equation where only s5 and c5 appears, and later substituting the tan-half-angle identities.
From Equation 4-49 it is solved for c3
2
55453213 A
AsAcAcAc
+++−= . (4-52)
The same can be done from Equation 4-50 for s3 to obtain
2
55453213 B
BsBcBsBs
+++−= . (4-53)
The next step is squaring both sides of Equations 4-52 and 4-53, adding term to term,
applying the trigonometric identity 1cs 23
23 =+ , and multiply by 2
222BA in order to eliminate the
denominator, to obtain
0EsEsEcEcE 52422322
221 =++++ (4-54)
where:
22
211 BAE = ,
325225212 EsEcEE ++= ,
where:
223112 BAA2E =
224122 BAA2E =
225132 BAA2E =
21
223 BAE = ,
435425414 EsEcEE ++= ,
76
where:
312214 BBA2E =
412224 BBA2E =
512234 BBA2E = ,
6555555455532552
25515 EsEcEscEsEcEE +++++= ,
where:
22
23
23
2215 BABAE +=
24
22
22
2425 BABAE +=
)BAABB2(AE 224343
2235 +=
)BBABA2(AE 5322
225345 +=
)BBABA2(AE 5422
225455 +=
25
22
22
22
22
2565 BABABAE +−= .
The strategy now is to eliminate the terms s2 and c2 in Equation 4-54 in order to have the
single unknown θ5, in the terms s5 and c5. To this end, first Equation 4-54 is rewritten as
22524223
221 cEEsEsEcE −=+++ . (4-55)
Next, Equation 4-55 is squared both sides to obtain
0csEE2scEE2sEE2sEcE
sEE2s)EEE(2c)EEE(2E22241
22
2231
3243
42
23
42
21
25422
2453
22
2251
25
=+++++
+++−+. (4-56)
On the other hand, from Equation 4-51 it is solved for s2
1
3522 D
DsDs
+−= . (4-57)
Using the basic trigonometric identity 2 22 2c 1 s= − , from Equation 4-57 it is obtained for c2
2
77
2
1
35222 D
DsD1c ⎥
⎦
⎤⎢⎣
⎡ +−= . (4-58)
The substitution of Equations 4-57 and 4-58 into Equation 4-56 will arise a trigonometric
equation involving only the sine and cosine of the joint angle θ5. The substitution of the tan-half-
angle identities
25
55 x1
x2s
+= (4-59)
25
25
5 x1x1
c+−
= (4-60)
into the trigonometric equation in sine and cosine of θ5 yield an 8th degree polynomial in x5,
which after being divided by 425 )x(1+ is expressed as
0CxCxCxCxCxCxCxCxC 055152525
3535
4545
5555
6565
7575
8585 =++++++++ (4-61)
where:
C05 = [-2E1E56-2E51E54-2E1E54-2E1E51+2E23E21-E542+E23
2-E12-2E51E56-2E54E56-E56
2-
E512+E21
2]D14+(2E41E56+2E43E56+2E51E43+2E1E43+2E51E41+2E54E43+2E1E41+2E54E41)D3
D13+[-E43
2-2E3E56-2E54E3-2E41E43+2E1E54+2E12-2E1E3+2E1E51-E41
2-E232-E21
2+2E1E56-
2E51E3-2E23E21]D32D1
2+(2E3E43-2E1E43+2E3E41-2E1E41)D33D1+[-E3
2+2E1E3-E12]D3
4
C15 = [-4E1E55-4E53E56-4E54E55-4E51E53+4E22E21-4E51E55-4E55E56-4E1E53+4E22E23-
4E53E54]D14+[4E43E56+4E1E41+4E1E43+4E54E43+4E41E56+4E54E41+4E51E41+4E51E43]D2+
(4E53E43+4E43E55+4E54E42+4E41E55+4E42E56+4E1E42+4E53E41+4E51E42)D3D13+[4E43
2-
8E51E3-4E212-4E23
2-4E412+8E1E56-8E1E3-8E3E56-8E41E43+8E1E54-
8E54E38E23E21+8E1E51+8E12]D3D2+(-4E22E21+4E1E53-4E22E23-4E41E42+4E1E55-4E3E55-
4E53E3-4E42E43) D32D1
2+[-12E1E41+12E3E43+12E3E41-12E1E43]D32D2+(4E3E42-
4E1E42)D33D1+(-8E1
2+16E1E3-8E32)D3
3D2
78
C25 = [-8E51E52-8E52E54-8E1E56-4E54E56-8E53E55+4E51E54+4E232-4E1
2+4E23E21-4E1E54-
4E562+4E51
2-4E532-4E55
2+4E222-8E1E52-
8E52E56]D14+[8E41E55+8E43E55+8E53E43+8E54E42+8E51E42+8E53E41+8E42E56+8E1E42]D2
+(4E1E41+8E43E56+4E54E43+4E41E56+ 8E1E43+8E42E55-
4E51E41+8E52E41+8E53E42+8E52E43)D3D13+[-4E43
2-8E51E3-4E212-4E23
2-4E412+8E1E56-
8E1E3-8E3E56-8E41E43+8E1E54-8E54E3-8E23E21+8E1E51+8E12]D2
2+(16E1E53-16E22E21-
16E53E3-16E41E42+16E1E55-16E42E43-16E22E23-16E3E55)D3D2+[8E1E52-4E422-4E41E43-
4E54E3-4E222-8E1E3+8E1
2-4E432-8E52E3-4E23
2+4E1E54+8E1E56-8E3E56-
4E23E21]D32D1
2+[24E3E43-24E1E43-24E1E41+24E3E41]D3D22+(24E3E42-
24E1E42)D32D2+.[-4E1E41+4E3E41+8E3E43-8E1E43]D3
3D1+[-24E12+48E1E3-
24E32]D3
2D22+(-4E3
2+8E1E3-4E12)D3
4
C35 = [-16E52E55-12E1E55-4E53E56+4E51E55-12E55E56-16E52E53+12E51E53+4E22E21-
4E1E53+12E22E23+4E53E54-
4E54E55]D14+[12E43E56+16E52E41+4E41E56+16E42E55+4E54E43+16E53E42-
12E51E41+16E52E43+12E1E43+4E1E41-4E51E43-
4E54E41]D2+(12E43E55+12E42E56+4E53E43+12E1E42+4E41E55-4E51E42-
4E53E41+16E52E42+4E54E42)D3D13+[16E1E53-16E22E21-16E53E3-16E41E42+16E1E55-
16E42E43-16E22E23-16E3E55]D22+(24E1
2+8E1E54-8E54E3+4E412-32E52E3+24E1E56-8E23E21-
24E3E56-16E222-12E43
2+32E1E52-16E422-12E23
2-8E1E51+4E212+8E51E3-24E1E3-
8E41E43)D3D2+[12E1E55-12E22E23-12E3E55-12E42E43+4E1E53-4E41E42-4E53E3-
4E22E21]D32D1
2+[16E3E41+16E3E43-16E1E43-16E1E41]D23+
(-48E1E42+48E3E42)D3D22+[12E3E41+36E3E43-36E1E43-12E1E41]D3
2D2+
(-12E1E42+12E3E42)D33D1+[-32E3
2+64E1E3-32E12]D3D2
3+(-24E12+48E1E3-24E3
2)D33D2
79
C45 = [-16E1E52+16E51E52-8E552+4E51E56-16E52
2+2E542+8E22
2+6E232-6E1
2+4E1E51-12E1E56-
6E562-6E51
2-2E212+8E53
2-16E52E56]D14+[-16E51E42+32E52E42-
16E53E41+16E1E42+16E43E55+16E42E56]D2+(16E52E43-4E51E43-
4E54E41+12E43E56+16E42E55+12E1E43)D3D13+[-8E23
2+8E412-32E52E3+32E1E52-
8E432+8E21
2+16E1E56-16E1E3-16E3E56-16E1E51+16E51E3-16E222-16E42
2+16E12]D2
2+
(-32E42E43-32E22E23-32E3E55+32E1E55)D3D2+[-6E432-8E42
2+2E212+12E1E56-4E1E51-
12E3E56-12E1E3+2E412-8E22
2-16E52E3+16E1E52+12E12-6E23
2+4E51E3]D32D1
2+[32E3E42-
32E1E42]D23+(48E3E43-48E1E43)D3D2
2+[-48E1E42+48E3E42]D32D2+(12E3E43-
12E1E43)D33D1+[32E1E3-16E3
2-16E12]D2
4+(-48E12-48E3
2+96E1E3)D32D2
2+[12E1E3-
6E32-6E1
2]D34
C55 = [-16E52E55+4E51E55+4E54E55+4E1E53+12E22E23+4E53E54+4E53E56-12E51E53+16E52E53-
12E55E56-4E22E21-12E1E55]D14+[12E51E41-16E52E41+12E43E56-4E54E41-4E51E43-4E41E56-
16E53E42+12E1E43+16E42E55-4E54E43+16E52E43-
4E1E41]D2+(12E42E56+12E43E55+16E52E42-4E54E42-4E51E42-4E53E43+12E1E42-4E41E55-
4E53E41)D3D13+[-16E42E43-16E1E53+16E22E21-16E3E55-
16E22E23+16E1E55+16E53E3+16E41E42]D22+(4E41
2+32E1E52-32E52E3-16E222+24E1E56-
24E1E3-24E3E56-8E1E51-12E232+8E41E43-16E42
2-
12E432+24E1
2+4E212+8E51E3+8E54E3+8E23E21-8E1E54)D3D2+[-12E42E43-12E22E23-
4E1E53+4E41E42+4E22E21+4E53E3+12E1E55-12E3E55]D32D1
2+[16E1E41-16E3E41-
16E1E43+16E3E43]D23+(-48E1E42+48E3E42)D3D2
2+[-12E3E41+36E3E43+12E1E41-
36E1E43]D32D2+(-12E1E42+12E3E42)D3
3D1+[-32E32+64E1E3-32E1
2]D3D23+
(-24E12+48E1E3-24E3
2)D33D2
80
C65 = [4E1E54-4E23E21+8E53E55-8E52E56-8E1E52-8E1E56+8E52E54+4E232-4E1
2-8E51E52+4E54E56-
4E562+4E51
2-4E532+4E22
2-4E552-4E51E54]D1
4+[8E1E42-8E53E43-
8E54E42+8E53E41+8E51E42+8E42E56+8E43E55-8E41E55]D2+(-8E53E42+8E43E56-
4E41E56+8E1E43-4E1E41+8E42E55+4E51E41-8E52E41+8E52E43-
4E54E43)D3D13+[8E1E51+8E1
2+8E23E21-4E432+8E41E43-4E41
2-8E1E3-8E3E56-4E232-
8E1E54+8E54E3-8E51E3-4E212+8E1E56]D2
2+(-16E42E43-16E1E53+16E22E21-16E3E55-
16E22E23+16E1E55+16E53E3+16E41E42)D3D2+[4E23E21-8E52E3-8E1E3-
4E232+8E1E56+4E54E3-4E42
2-4E1E54+8E12-4E43
2+8E1E52+4E41E43-8E3E56-
4E222]D3
2D12+[24E1E41+24E3E43-24E1E43-24E3E41]D3D2
2+(24E3E42-24E1E42)D32D2+
[-4E3E41+4E1E41-8E1E43+8E3E43]D33D1+[-24E1
2+48E1E3-24E32]D3
2D22+(8E1E3-4E3
2-
4E12)D3
4
C75 = [4E1E53+4E22E23+4E54E55+4E53E56+4E51E53-4E53E54-4E55E56-4E22E21-4E1E55-
4E51E55]D14+[4E51E43-4E41E56+4E54E41-4E51E41-4E54E43+4E1E43-4E1E41+4E43E56]D2+
(-4E41E55+4E1E42+4E42E56-4E53E43-4E54E42+4E51E42+4E53E41+4E43E55)D3D13
+[8E1E51+8E12+8E23E21-4E43
2+8E41E43-4E412-8E1E3-8E3E56-4E23
2-8E1E54+8E54E3-
8E51E3-4E212+8E1E56]D3D2+(4E53E3-4E42E43+4E22E21+4E1E55-4E1E53-4E22E23+4E41E42-
4E3E55)D32D1
2+[12E3E43+12E1E41-12E3E41-12E1E43]D32D2+(4E3E42-4E1E42)D3
3D1+[-
8E12-8E3
2+16E1E3]D33D2
C85 = [2E51E54-2E51E56-2E1E56-2E23E21+E232-E1
2-2E1E51-E512-E56
2+E212-
E542+2E54E56+2E1E54]D1
4+(2E51E43+2E43E56-2E51E41+2E54E41-2E1E41-2E54E43+2E1E43-
2E41E56)D3D13+[-E21
2-2E1E3-E232+2E1E56-E41
2+2E54E3-2E51E3-E432+2E1
2+2E41E43-
2E3E56-2E1E54+2E23E21+2E1E51]D32D1
2+(2E1E41-2E1E43-2E3E41+2E3E43)D33D1+[-E1
2-
E32+2E1E3]D3
4.
81
To obtain the joint angle θ5, the roots of the polynomial in x5 presented in Equation 4-61
are first found. Next, Equations 4-59 and 4-60 are used to obtain the sine and cosine of the angle,
then a unique value for θ5 is obtained for each root of x5.
The coefficients C05 to C85 are function of the known parameters and the new sum of
angles found in the spherical quadrilateral solution: (θ1+ θ2), (θ3+ θ4), and (θ5+ θ6). According to
the solution tree for the spherical quadrilateral presented in Figure 4-5, and according to
Equations 4-25 to 4-27, there are two solutions for the spherical quadrilateral, they are (θ1+ θ2)A,
(θ3+ θ4)A, and (θ5+ θ6)A; and (θ1+ θ2)B, (θ3+ θ4)B, and (θ5+ θ6)B, which implies that there is a total
of sixteen solutions for θ5, eight for A solution and eight for the B solution.
4.3.5 Solution for θ3
At this point the joint angle θ5 is known, which makes possible to write Equations 4-49 to
4-51, respectively, as
63221 AcAcA =+ (4-62)
where:
)AsAc(AA 554536 ++−= ,
63221 BsBsB =+ (4-63)
where:
)BsBc(BB 554536 ++−= ,
421 DsD = (4-64)
where:
)Ds(DD 3524 +−= .
Equation 4-62 is initially rearranged as
32621 cAAcA −= , (4-65)
82
and both sides are squared to yield
36223
22
26
22
21 cAA2cAAcA −+= , (4-66)
which after substituting the trigonometric identity 2i
2i s1c −= for i=2,3 yields
36223
22
26
22
21 cAA2)s(1AA)s(1A −−+=− . (4-67)
On the other hand, from Equation 4-63 is solved for the sine of θ3
2
2163 B
sBBs
−= . (4-68)
Substituting Equation 4-68 into Equation 4-67 and solving for c3 yields
62
22
21
2
2
21622
26
3 AA2
1)(sAB
sBB1AA
c
−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎥⎦
⎤⎢⎣
⎡ −−+
= . (4-69)
From Equation 4-64 it is solved for the sine of θ2
1
42 D
Ds = . (4-70)
Substituting Equation 4-70 into Equation 4-69 will allow expressing c3 in terms of known
parameters,
221
22
21
24
21
221
21
22
26
3 ]B[DB]D[DAAFDBA
c−++
= (4-71)
where:
24
214161
26
21
21
221 DBDDBB2BDDBF −+−= .
The value for the sine of θ3 is to be obtained in order to define a unique value for θ3, s3 is
found by substituting Equation 4-70 into Equation 4-68 to yield
12
41163 DB
DBDBs
−= . (4-72)
83
Knowing the cosine as sine values for θ3 from Equations 4-71 and 4-72, respectively,
defines a unique angle θ3 for each value of θ5.The solutions A and B were obtained for θ5, then
also A and B solutions are obtained for θ3.
4.3.6 Solution for θ2
Once the values of θ3 and θ5 are known, the sine and cosine values that determine a unique
value for θ2 are easily obtained. The value for s2 was already defined in Equation 4-70. The value
for c2 is obtained directly form Equation 4-62
1
3262 A
cAAc
−= . (4-73)
The values for s2 and c2 defined by Equations 4-62 and 4-73, respectively, define a unique
angle θ2 for each set of angles θ3 and θ5. A and B solutions stand for θ2.
4.3.7 Solution for θ1, θ4 and θ6
Directly from Equations 4-25 to 4-27, it is solved for θ1, θ4 and θ6 to obtain
2Q11 θθθ −= (4-74)
3Q24 θθθ −= (4-75)
5Q36 θθθ −= . (4-76)
4.3.8 Solution Tree
The closure of the solution of the spatial 7L-7R closed-loop with three pairs of parallel
axes is a solution tree as presented in Figure 4-6 (the Q subscript stands for quadrilateral), to
illustrate the way as the sixteen solutions were obtained.
84
Figure 4-6. Sixteen solutions tree for 7L-7R closed-loop mechanism with consecutive pairs of parallel joint axes.
The reverse kinematic analysis of the half of the closed-loop motion generator has been
performed. In the next section, the closed mechanism reverse kinematic analysis is completed.
4.4 Reverse Kinematics Analysis for the Complete Spatial Motion Generator
So far, the twelve-link spatial closed-loop mechanism has been divided into two open loop
six-link mechanisms (see Figure 4-1) whose reverse position kinematic analysis has been solved.
The two resulting open-loop six-link mechanisms are not identical but are similar in geometry.
Based on this similarity, a simple transformation of coordinates is presented for completing the
85
reverse kinematic analysis of the twelve-link closed-loop mechanism. To this end, consider the
two fixed coordinate systems xF1-yF1-zF1 and xF2-yF2-zF2, attached to the two fixed joint axes as
presented in Figure 4-1.
The solution of the position reverse kinematic analysis of the half mechanism
corresponding to links 1-12-11-10-9-8-7 in Figure 4-1 starts by presenting the respective
kinematic labeling in Figure 4-7. The subscript (2) in the nomenclature is used for distinguishing
these parameters from the ones previously used in solving the half mechanism1-2-3-4-5-6-7 (i.e.,
S1(2) makes reference to the link distance S1 in the fixed coordinate system x2-y2-z2 and θ7(2)
makes reference to the joint angle that closes the loop with the part of the link 1 attached to the
x2-y2-z2 coordinate system). The problem statement is
Given
1) The constant mechanism parameters: link lengths a1,12 to a87, twist angles α1,12 to α87, and
offset distances S1(2) to S9.
2) Offset distance S8 and the direction of the vector a87 relative to the vector S8
3) Position and orientation of the end effector: F2Ptool, F2S8 and F2a87.
4) Location of the tool point in the 8th coordinate system: 8Ptool.
5) Close the loop parameters calculated in the x2-y2-z2 coordinate system: Offset distances
S1(2), S7(2); link distance a71(2); joint angle θ7(2); angle γ1(2).
Find
The joint angles θ1(2), θ12, θ11, θ10, , θ9 and θ8 of the closed-loop 7L-7R spatial mechanism 1-12-
11-10-9-8-7-1.
86
Figure 4-7. Kinematic labeling of the 6L-6R open-loop mechanism 1-12→7.
Solution
In the above problem statement it was assumed that the close the loop parameters are
already known. This problem statement is identical to the problem stated in Section 4.3.1 for the
half mechanism 1-2-3-4-5-6-7, so the solution is taken in the x2-y2-z2 fixed coordinate system.
It is possible to obtain the transformation matrix that relates the coordinate systems xF1-yF1-
zF1 and xF2-yF2-zF2 (see Figure 4-8), as a designer’s convenient choice based on the physical
space available. This transformation matrix is defined as T1F2F , which contains the translation
and the rotation information that relates the second fixed coordinate system (xF2-yF2-zF2) as seen
in the first fixed coordinate system (xF1-yF1-zF1). This transformation matrix is defined as
⎥⎦
⎤⎢⎣
⎡=
10002OF
1F1F2F1F
2FPR
T (4-77)
where:
R1F2F : rotation information that relates the 1 and 2 fixed coordinate systems
87
2OF1F P : position vector of the origin of the second fixed coordinate system OF2 measured with
respect to the first coordinate system.
Figure 4-8. Transformation of the fixed coordinate systems xF1-yF1-zF1 and xF2-yF2-zF2.
The tool coordinates as seen in the xF2-yF2-zF2 fixed coordinate system need to be obtained as
⎥⎦
⎤⎢⎣
⎡=⎥⎦
⎤⎢⎣
⎡
11tool
1F2F1F
tool2F P
TP
. (4-78)
The tool coordinates as seen in the 6th coordinate system are transformed based on Figure
4-9, where the orientation of the z axes of the each coordinate system are chosen to be
antiparallel to ensure the configuration o the sixth link as shown in the same figure, where it is
remarked that F2S8 = F2S6(2) and F2a87=F2a6(2).
Figure 4-9. Disposition of the two sixth coordinate systems.
88
From Figure 4-9, the transformation matrix that relates the two 6th coordinate systems is obtained
from a rotation of 180 degrees about the x axes which are collinear, to obtain
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
=
1000010000100001
)1(6)2(6 T . (4-79)
The transformation matrix that relates the orientation of the x and z axes of the 6th coordinate
system as seen in the xF1-yF1-zF1 coordinate system is obtained by
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡×=
1000000
61F
671F
61F
671F
1F)16(
SaSaT . (4-80)
Combining Equations 4-77, 4-79 and 4-80, yields the vectors F2S8 and F2a87,
( ) ( )( )TTTSaSa )1(6)2(6
1F)16(
11F2F
82F
872F
82F
872F
1000000
−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡× . (4-81)
One more transformation matrix is needed to express the tool point coordinates as seen in
the new introduced coordinate system xF2-yF2-zF2,
⎥⎦
⎤⎢⎣
⎡=⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
111tool
)1(6(2)6(1)6
tool)2(6
tool8 P
TPP
. (4-82)
Known parameters F2Ptool, F2S8-F2a87 and 8Ptool have been specified in the fixed xF2-yF2-zF2
coordinate system, in Equations 4-78, 4-81 and 4-82, respectively. The solution presented in
Section 4.3 is now appropriate for solving the half mechanism comprised for the links 7-8→1.
Here ends the position reverse kinematic solution of the spatial manipulator.
89
CHAPTER 5 SYNTHESIS OF THE NON-CIRCULAR PITCH PROFILES
A numerical approach for finding the working section centrodes of the non-circular gear
connections are presented in this chapter. This approach introduces a sequence parameter to
define the instantaneous joint angles required to calculate the I/O non-circular gear relationship.
First, the position and orientation specification for each position are associated with a unique
sequence parameter. Next, each joint angle is associated with its respective parameter and
numerical derivatives of the joint angles are obtained. Finally, the ratios of the numerical
derivatives yield the expressions for the I/O gear relationships.
5.1 I/O Non-Circular Gear Relationships for the Spatial Motion Generator
Figure 5-1 depicts the non-circular gear connections’ labeling of the spatial motion
generator. The five non-circular gear relationships are distinguished from each other by the
subscript associated with the connecting link: g3, g5, g7, g9 and g11.
Figure 5-2 presents four consecutive links f, g, h, and i; where links g and i have the non-
circular gears attached to them, and the joint axes Sh and Si are parallel (αhj=0°) or antiparallel
(αhj=180°). When using Equation 3-2 to find the I/O relationship, it must be recognized that
every joint angle θ is, by definition, taken relative to the link vector of the previous link (as
defined in Figure 1-2). Based on this, the expression for the I/O relationship for the geared
connection in Figure 5-2, with connecting link h, is given by
h
ih dθ
dθg = (5-1)
where:
dθi: infinitesimal change of the i joint angle
dθh: infinitesimal change of the h joint angle.
90
Figure 5-1. Non-circular gear connection’s labeling.
Figure 5-2. Labeling of the h non-circular gear connection.
Equation 5-1 is used to find the instantaneous speed ratio of the non-circular gears depicted
in Figure 5-1. For the half mechanism 0-1-2-3-4-5-6, the I/O relationships for the connecting
links 3 and 5 are
3
43 dθ
dθg = (5-2)
91
5
65 dθ
dθg = , (5-3)
respectively, where the output body is the link after the connecting link, following the sequence
0-1→6. Analogously, following the sequence 0-11→6 for the half mechanism 0-11-10-9-8-7-6,
the output bodies are links 10, 8 and 6 and the I/O relationships are, respectively,
)2(1
1011 d
dθg
ϕ= (5-4)
9
89 dθ
dθg = (5-5)
7
67 dθ
dθg = (5-6)
where φ1(2) is the angle between the x axes of the second fixed coordinate system and the first
standard coordinate system. In the next sections, two approaches to calculate the value of the
instantaneous gear ratio g in Equation 5-1 are presented.
5.2 Sequence Parameter Approach
The path requirements f(x,y,z) can be discretely represented as a set of points
(xj,yj,zj)/j=1,…,n along the intersection of the intervals [x1,xn];[y1,yn]; and [z1,zn], depicted in
Figure 5-3(a). The discrete representation of the path requirements allows one to introduce a
non-negative integer sequence parameter, u, associated with each point (xj,yj,zj), see Figure 5-
3(b). Thus there is a single sequence parameter associated with each position giving a total of n
parameters: uj=j (j=1,…,n).
Figures 5-3(c) and 5-3(d) depict the discrete representation of the position and orientation
needs, respectively, related to the introduced sequence parameter. On the other hand, due to the
solution technique used to solve the reverse kinematic position analysis (Chapter 4), the motion
92
variables x, y, z, ζx, ζy and ζz are not explicitly presented in the expressions obtained for the joint
angles; i.e., the joint angle θ5 is obtained from the roots of an 8th degree polynomial in the tan
half angle. However, the numerical value of each joint angle depends on the numerical value of
the six motion generation variables written above. Based on this, it is possible to relate each joint
angle with the sequence parameter u, as presented in the next section.
5.2.1 Polynomial Interpolation of Joint Angles: Discarded First and Last Points
Section 3.1.5 presented the polynomial to approximate the centrode corresponding to the
non-working section. In reference to the working section’s centrode, its smoothness is going to
depend on the interpolation method used to relate the joint angles with the sequence parameter.
In order to evaluate the first derivatives of the I/O relationship in Equations 3-5 and 3-7
that guarantee infinite number of cycles for the gears in mesh, the order of the polynomial
approximation used to relate the joint angles and the sequence parameter is to be at least
quadratic. This is apparent from Equation 5-1. A cubic polynomial interpolation would also
guarantee the smoothness of the entire profile. Cubic spline functions are proven to be smooth
functions with which to fit data, and “when used for interpolation they do not have the
oscillatory behavior that is characteristic of high-degree polynomial interpolation”, Atkinson
[31]. Based on this, a natural cubic spline is suitable for the piecewise interpolation. The last
reference can be consulted for information about cubic spline interpolation.
93
Figure 5-3. Introduction of the sequence parameter u to discretely express the motion generation requirements along [xo,xf],[yo,yf],[zo,zf]. A) Original path requirements x,y,z. B) Sequence parameter, u, along the path. C) Equivalent representation of the x, y and z coordinates of the path versus the sequence parameter u. D) Extension of this representation to the orientation requirements ζx, ζy and ζz.
94
The lack of information about the slope of the end points (first and last points) would lead
to the assumption of “reasonable” slope values in order to calculate the derivatives of the joint
angles with respect to the sequence parameter at those points. If there is no information, any
assumption made is a speculation that might affect the smoothness of the centrode. To preserve
the smoothness of the pitch curve, the first and last points are discarded after the natural cubic
spline interpolation is performed. The first derivative of the joint angles with respect to the
sequence parameters is numerically approximated as
( ) ( )u
uθuuθlimdudθ
0u ΔΔ
Δ
−+≅
→ (5-7)
where:
θ : joint angle approximated using cubic natural spline interpolation
u: sequence parameter
Δu: infinitesimal increment of u
( )uθ : value of approximated joint angle evaluated at u
( )uuθ Δ+ : value of the approximated joint angle evaluated at u+Δu
dθ/du: numerical derivative of the joint angle respect to the sequence parameter u.
The approximation described above in Equation 5-7 is also applicable for the angle φ1.
5.2.2 I/O Relationship Expressions from Sequence Parameter Approach
Once the joint angles are parameterized using spline polynomials in terms of the sequence
parameter u, the value of the I/O relationship can be found by introducing the unit ratio du/du
(chain rule) into Equation 5-1 to yield
dudθdudθ
gh
i
h = (5-8)
95
where the subscript h makes reference to the connecting link in Figure 5-2 and the derivatives of
the joint angles respect to u are obtained through Equation 5-7. Equations 5-2 to 5-6 are modified
by the unit ratio, to yield
dudθdudθ
g3
4
3 = (5-9)
dudθdudθ
g5
6
5 = (5-10)
dud
dudθ
g)2(1
10
11 ϕ= (5-11)
dudθdudθ
g9
8
9 = (5-12)
dudθdudθ
g7
6
7 = . (5-13)
96
CHAPTER 6 RESULTS: ILLUSTRATIVE EXAMPLES
6.1 Reverse Kinematics for a Single Position of the End Effector: Single Position, Half Mechanism Case
The objective of this example is to illustrate the reverse kinematic analysis presented in
Chapter 4. The constant mechanism parameters for the example case are presented in Table 6-1.
The twist angle α67 and the link length a67 were selected as 90° and 0 in respectively. The free
choice values for the offset S6, the hypothetical link length a67, and the hypothetical twist angle
α67 are presented in Table 6-1.
Table 6-1. Constant mechanism parameters for numerical example. Offset Distance [in] Link Lengths [in] Twist Angles [deg]
S2 = 3.4947 a12 = 14.2368 α12 = 0 S3 = 0 a23 = 0.7411 α23 = 59.2992 S4 = 1.3465 a34 = 12.9009 α34 = 0 S5 = 0 a45 = 6.1349 α45 = 76.8924 S6* = 6.0 a56 = 10.3782 α56 = 0 a67* = 0 α67* = 90 * Free choice.
The desired position and orientation of the end effector are summarized in Table 6-2. In
this example, the orientation requirements are specified by the x and z axes of the sixth
coordinate system (attached to the end effector) as seen in the fixed coordinate system, i.e. the
vectors Fa67 and FS6. The close-the-loop parameters obtained for this case are summarized in
Table 6-3. The solution for the equivalent spherical quadrilateral for the sums θ1+θ2, θ3+θ4, and
θ5+θ6 is presented in Table 6-4.
Table 6-2. Desired position and orientation requirements. Fa67 [-0.4771 -0.5994 -0.6428] FS6 [ 0.7393 -0.6692 0.0752] [ζx, ζy, ζz][deg]* [135.1142, -47.6716, -63.5884] FPtool [in]** [10.1041 -8.0151 0.5516] 6Ptool [in] [5 7 8] * X-Y-Z Fixed Angles ** [x, y, z] position needs
97
Table 6-3. Calculated close-the-loop parameters. Distance [in] Angle [degrees]
a71 = 4.2174 α71 = 40.3277 S7 = 0.8123 θ7 = -96.6756 S1 = -9.119 γ1 = -132.7510
Table 6-4. Sums of consecutive joint angles obtained from the equivalent spherical quadrilateral solution.
Sum of Angles
Solution A [degrees]
Solution B [degrees]
θ1+2 -7.5924 -162.2098 θ3+4 -87.2244 87.2244 θ5+6 -80.4042 160.6752
Fourteen real solutions and two complex solutions were found for this numerical example
for the joint angles φ1 through θ6. Figures 6-1 and 6-2 illustrate the real configurations for the
open-loop manipulator in this numerical example. Figures 6-1 and 6-2 were generated using
MatLab. A forward analysis was performed as a check for all 16 solutions, including the
complex solutions D and E. Each solution positioned and oriented the end effector as desired.
98
Table 6-5. Joint angles corresponding to the sixteen solutions of the numerical example. Sol φ1[deg] θ2[deg] θ3[deg] θ4[deg] θ5[deg] θ6[deg] A 119.6877 5.4709 177.7643 95.0113 -177.7795 97.3753 B 281.5373 -156.3786 -6.1702 -81.0542 145.8062 133.7896 C 155.4960 -30.3374 -175.0900 87.8656 136.4939 143.1019
-5.6494 130.8063 31.5585 -118.7799 -127.9873 47.5841 D
+31.5413 i -31.5413 i +42.3301 i -42.3301 +39.4481 -39.4481 -5.6494 130.8063 31.5585 -118.7799 -127.9873 47.5841
E -31.5413 i +31.5413 i -42.3301 i +42.3301 -39.4481 +39.4481
F 174.3520 54.1905 114.2585 158.5171 -115.1178 34.7136 G 173.63906 -48.4804 149.8232 122.9524 79.5779 -159.9822 H 262.0020 -136.8434 47.4029 -134.6273 64.8370 -145.2412 I 129.0418 -158.5006 19.6492 67.5752 150.2720 10.4032 J 185.5112 145.0300 52.55648 34.6679 -140.0509 -59.2739 K 2.4264 -31.88519 172.7676 -85.5432 136.2312 24.4440 L 21.5798 -51.0386 -136.5833 -136.1922 83.5839 77.0913 M 100.3648 -129.8236 -47.9943 135.2187 79.1644 81.5108 N 270.2382 60.3030 159.7512 -72.5268 -89.8907 -109.4341 O 273.2697 57.2715 172.7296 -85.5052 -75.1938 -124.1310 P 171.1326 159.4086 -43.9788 131.2032 -21.9573 -177.3675
99
Figure 6-1. Real solutions A-H in Table 6-5, corresponding to the A solution of the spherical quadrilateral in Table 6-4.
100
Figure 6-2. Real solutions I-P in Table 6-5, corresponding to the B solution of the spherical quadrilateral in Table 6-4.
6.2 Motion Generation along Discrete Path and Orientation Requirements: Complete Mechanism Case
Consider a hypothetical manufacturing process where a tool traverses 3-D space as
illustrated by the position and orientation specifications shown in Figure 6-3A and 6-3B.
101
Discrete data corresponding to these figures are given in Table B-1 (Appendix B), where the
fixed coordinate system xF2-yF2-zF2 is selected to be the reference system on which the data in
Table B-1 is given. The origin of the fixed coordinate system xF1-yF1-zF1 was selected to be
located a distance of 300.0mm along the z axis, and the coordinate axes are parallel to the xF2-
yF2-zF2 coordinate system. The transformation matrix T2F1F that relates the two fixed coordinate
systems xF1-yF1-zF1 and xF2-yF2-zF2 is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
10000.300100
00100001
2F1F T . (6-1)
The transformation matrix given in Equation 6-1 and the motion information in the xF2-yF2-
zF2 coordinate system is used to express the motion needs as seen in the xF1-yF1-zF1 coordinate
system by using methodologies in Sections 2.10 and 4.4. The orientation requirements are given
in Figure 6-3B, where since the fixed reference systems xF1-yF1-zF1 and xF2-yF2-zF2 are parallel,
the X-Y-Z Fixed-Angles representations for the orientation needs are the same in both fixed
coordinate systems. Representations of the position and orientation requirements as a function of
the sequence parameters are presented in Figures 6-4A and 6-4B, respectively. From now on,
the nomenclature for links and joint angles follows the labeling in Figure 5-1. The mechanism
parameters and free choices for the closed-loop mechanism are summarized in Table 6-6.
The position angles obtained from the reverse kinematic stage are summarized in Figure 6-
5. For the half mechanism 0-1→6, solution I was chosen as this configuration was able to
position and orient the end effector at every pose along the path with real solutions for the joint
angles. These real joint angles are presented in Figure 6-5A. On the other hand, real joint angles
102
corresponding to solution P satisfied the motion needs for the half mechanism 0-11→6; they are
shown in Figure 6-5B.
Figure 6-3. Motion needs. A) Position requirements. B) Orientation requirements.
Table 6-6. Constant mechanism parameters for the closed-loop example. Offset Distance [mm] Link Lengths [mm] Twist Angles [deg]
S2 = 0 a12 = 277.44 α12 = 0 S3 = 0 a23 = 203.20 α23 = 67.75 S4 = 168.95 a34 = 331.5 α34 = 0 S5 = 0 a45 = 171.37 α45 = 108.78 S7 = 0 a56 = 349.22 α56 = 0 S8 = 50 a11,10 = 100.00 α11,10 = 0 S9 = 0 a10,9 = 200.00 α10,9 = 30.00 S10 = 0 a98 = 105.00 α98 = 0 S6(1)* = 70.00 a76 = 95.00 α87 = 0 S6(2)* = 60.00 a67(1)* = 0 α67(1)* = 90.00 a67(2)* = 0 α67(2)* = 90.00
* Free choice
103
Figure 6-4. Motion needs versus the sequence parameter in each fixed coordinate system. A) Position needs. B) Orientation needs.
The value of the Input/Output relationships along the working profile section can be
obtained since the joint angles are known at this stage. The gear synthesis results regarding the
half mechanism 0-1→6 are depicted in Figures 6-6 and 6-7 for gear connections 234 and 456,
respectively. These figures show the variation of the Input/Output relationship and the output
angle versus the input angle along the working section. The values of the Input/Output
relationship for the working section in Figures 6-6B and 6-7B were calculated using the
sequence parameter approximation in Section 5.3.
104
Feasibility of the gear connections in this work is limited to external-external gears in
mesh configuration. Feasible external-external pitch profiles were not obtained for the working
sections, since the Input/Output relationship presents asymptotic behavior as depicted in Figures
6-6B and 6-7B. At different positions of the motion needs, both Input/Output gear relationships
behaved asymptotically. The asymptotic behavior lead to a sign change in both cases, therefore
the gear connection changed from external-external to external-internal configuration at this
point. This is due to the double-value shape that the output angle presents, as seen in Figures 6-
6A and 6-7A.
105
Figure 6-5. Joint angles for the closed-loop mechanism. A) Half mechanism 0-1→6. B) Half mechanism 0-11→6.
106
Figure 6-6. Gear synthesis results for connection 234 in half mechanism 0-1→6. A) Output angle versus the input angle. B) Input/Output relationship versus the input angle.
Better results were obtained for the remaining gear connections. Figures 6-8A, 6-8B and
6-8C show the gear synthesis results corresponding to the half mechanism 0-11→6, for gear
connections 10110, 8910 and 678, respectively. The values of the Input/Output relationship
along the working section were obtained using the sequence parameter approach. Feasible
external-external pitch profiles were obtained since the Input/Output relationships neither
reached negative values nor presented changes of sign. The non-working profile approximation
model also would allow the simulation of the cutter, wherefrom the synthesis of gear connections
10110, 8910 and 678 were successful. Three-dimensional animations that show the gear
connections in Figure 6-8 in mesh is available upon request at the CIMAR Laboratory at the
University of Florida.
107
Figure 6-7. Gear synthesis results for connection 456 in half mechanism 0-1→6. A) Output angle versus the input angle. B) Input/Output relationship versus the input angle.
108
A
Centre distance a11,10
Centre distance a98
B
Centre distance a76
C
Figure 6-8. Centrode, output angle and Input/Output gear relationship versus the input angle in half mechanism 0-11→6. A) Connection 10110. B) Connection 8910. C) Connection 678.
109
Finally, two views of the closed-loop mechanism are presented in Figure 6-9, where the
Figures 6-9A and 6-9B correspond to positions 15 and 35 in Table 6-6, respectively. A rigid
body has been attached to the end effector link. A MatLab animation of the example here
presented is available upon request to the CIMAR Laboratory at the University of Florida. Here
ends the successive points illustrative example.
110
Figure 6-9. Closed-mechanism at two different motion needs in Table B.1. A) At position 5. B) At position 30.
111
CHAPTER 7 CONCLUSION AND FUTURE WORK
7.1 Conclusion
In this research, the reverse kinematics and the gear connections synthesis of a spatial
closed-loop manipulator mechanism comprised of 12 links, 12 revolute joints, and 5 non-circular
gear connections were analyzed. Two similar open-loop mechanisms comprised of 6 links and 6
revolute joints with consecutive pairs of joint axes parallel were derived from the original
mechanism. The first joint axis is parallel to the second, the third is parallel to the fourth, and so
on. A closed-loop mechanism comprised of 7 links and 7 revolute joints was obtained after the
hypothetical closure loop was inserted in each open-loop mechanism. The equivalent closed-
loop spherical mechanism possesses a single degree-of-freedom. The reverse kinematic analysis
was reduced to the solution of an open-loop robotic manipulator comprised of 6 links and 6
revolute joints where consecutive pairs of axes are parallel. A transformation of coordinates that
relates the fixed coordinate systems, and the position and orientation needs of each open-loop
mechanism completed the kinematic analysis.
It was deduced that for the 6 link 6 revolute joint open-loop mechanism that for a single
position and orientation of the end effector there are 16 possible configurations that satisfy the
motion needs. This finding was one of the contributions of this research as the particular
manipulator geometry had not been presented in the prior literature. One example was included
to illustrate the degree of the solution, supported by the graphical display of 14 real solutions. A
forward analysis of the remaining two imaginary joint angles solutions positioned and oriented
the end effector as desired. In theory, for the closed-loop 12 link-12 revolute joints mechanism
there are 256 possible configurations that result when combining the degrees of the solutions of
each open-loop mechanism.
112
The reverse kinematic analysis was also successfully proven on a discrete set of specified
successive position and orientation needs to be traversed by the end effector. One single solution
(out 16 possible) was found for each open-loop mechanism in which real joint angles satisfy the
successive position and orientation needs. This case was presented as a second illustrative
example and was supported by a computer animation.
A methodology of synthesis of the pitch curves was also tested in the second example.
Each motion requirement was associated to a monotonic sequence parameter, and the joint
angles were numerically related to this parameter in order to numerically approximate the gear
relationship values for each gear connection. The pitch curves of three gear connections were
successfully synthesized with feasible pitch profiles (external-external gears in mesh).
Numerical values of the remaining gear connections were obtained; however, the geometric
synthesis was unsuccessful since negative values and asymptotic behavior were present. As a
result, the resulting mechanism was not a one degree-of-freedom as desired, but a three degree-
of-freedom that would still require three actuators to work. As a disadvantage of this numerical
approach, initial and final motions needs are discarded after approximations are performed.
Extra points in the vicinity of the end needs should be included.
This work is intended to be the starting platform for future analysis that would lead to the
implementation of this technology in highly repetitive tasks manufacture industry. Non-feasible
gear connections and intersection of the links were some of the unresolved problems in this
research that point to future considerations. They are presented in the next section.
7.2 Future Work
Subjects that deserve future analysis in order to implement this project into industry are
presented here. Some of the proposed problems were difficulties faced along this research and
others are complimentary studies that would make the project marketable.
113
The implementation of a routine that combines the selection of the mechanism parameters
(link lengths, link angles, and joint distances) and evaluate the gear connections’ feasibility,
other than just randomly checking and discarding would be the next step in this research.
The evaluation of the links’ intersection is also another algorithm to implement. The
shortest distance between two non-consecutive links must be evaluated according to the real
shape of the links. Also the intersection of the links with the gear sections must be considered.
The effect of the relative position and orientation of the two fixed coordinates systems in
the number of real solutions obtained in each half open-loop mechanism is to be evaluated. The
physical space available would be the main constraint when evaluating this condition.
The design of generic gear trains that replace the single external-external gear connections
when negative values and asymptotic behaviors arise should be explored as a solution to the
unfeasibility of the gear connections.
To minimize the centre distance on the gear connections in order to reduce the area of the
gears and the shaking moment is another aspect to be considered. Also, the minimization of the
non-circularity of the pitch profiles by bounding the value of the gear relationship is necessary.
A more general and elaborated approach in order to obtain the values of the non-circular
gear ratios along the working profile should be considered. This additional method is currently
under investigation based on screw theory analysis.
A general methodology to evaluate the economical feasibility of this technology that
indicates in what specific cases the implementation of this technology is advantageous according
to the market conditions, is required in order to implement this technology in the industry. The
number of tasks to be performed, the fabrication costs, the maintenance costs, and the operation
costs would be the main aspects to consider.
114
APPENDIX A JOINT AND LINK VECTORS EXPRESSIONS IN THE SPHERICAL MECHANISM
(CRANE AND DUFFY [1])
Joint Vectors Expressions:
Recursive Notations for One Subscript
jjkj ssX = (A-1)
( )jjkijjkijj csccsY +−= (A-2)
jjkijjkijj cssccZ −= (A-3)
jijj ssX = (A-4)
( )jijjkijjkj csccsY +−= (A-5)
jijjkijjkj cssccZ −= (A-6)
X, Y and Z Recursive Notations for Two Ascending Subscripts
jijiij sYcXX −= (A-7)
jijiij cYsXX +=∗ (A-8)
ijkijjkij ZsXcY −= ∗ (A-9)
ijkijjkij ZcXsZ += ∗ (A-10)
X, Y and Z Recursive Notations for Two Descending Subscripts
jkjkkj sYcXX −= (A-11)
jkjkkj cYsXX +=∗ (A-12)
kijkjijkj ZsXcY −= ∗ (A-13)
kijkjijkj ZcXsZ += ∗ (A-14)
X, Y and Z Recursive Notations for Three Ascending Subscripts
115
kijkijijk sYcXX −= (A-15)
kijkijijk cYsXX +=∗ (A-16)
ijklijkklijk ZsXcY −= ∗ (A-17)
ijklijkklijk ZcXsZ += ∗ (A-18)
X, Y and Z Recursive Notations for Three Descending Subscripts
ikjikjkji sYcXX −= (A-19)
ikjikjkji cYsXX +=∗ (A-20)
kjhikjihikji ZsXcY −= ∗ (A-21)
kjhikjihikji ZcXsZ += ∗ (A-22)
X, Y and Z Recursive Notations for Four Ascending Subscripts
khijkhijhijk sYcXX −= (A-23)
khijkhijhijk cYsXX +=∗ (A-24)
hijklhijkklhijk ZsXcY −= ∗ (A-25)
hijklhijkklhijk ZcXsZ += ∗ (A-26)
X, Y and Z Recursive Notations for Four Descending Subscripts
hkjihkjikjih sYcXX −= (A-27)
hkjihkjikjih cYsXX +=∗ (A-28)
kjighkjihghkjih ZsXcY −= ∗ (A-29)
kjighkjihghkjih ZcXsZ += ∗ (A-30)
X, Y and Z Recursive Notations for Five Ascending Subscripts
116
lhijllhijkhijkl sYcXX −= (A-31)
lhijklhijkhijkl cYsXX +=∗ (A-32)
hijklmhijkllmhijkl ZsXcY −= ∗ (A-33)
hijklmhijkllmhijkl ZcXsZ += ∗ (A-34)
X, Y and Z Recursive Notations for Five Descending Subscripts
hlkjihlkjilkjih sYcXX −= (A-35)
hlkjihlkjilkjih cYsXX +=∗ (A-36)
lkjighlkjihghlkjih ZsXcY −= ∗ (A-37)
lkjighlkjihghlkjih ZcXsZ += ∗ . (A-38)
Link Vectors Expressions:
U, V and W Recursive Notations for Two Ascending Subscripts
ijiij ssU = (A-39)
( )ijijijij csccsV +−= (A-40)
ijijijij cssccW −= (A-41)
U, V and W Recursive Notations for Two Descending Subscripts
ijjji ssU = (A-42)
( )ijjijiji csccsV +−= (A-43)
ijjijiji cssccW −= (A-44)
U, V and W Recursive Notations for Three Ascending and Descending Subscripts
jkijjkijijk sVcUU −= (A-45)
117
jkijjkijijk cVsUU +=∗ (A-46)
ijkijkkijk WsUcV −= ∗ (A-47)
ijkijkkijk WcUsW += ∗ (A-48)
U, V and W Recursive Notations for Four Ascending and Descending Subscripts
jkhijjkhijhijk sVcUU −= (A-49)
jkhijjkhijhijk cVsUU +=∗ (A-50)
hijkhijkkhijk WsUcV −= ∗ (A-51)
hijkhijkkhijk WcUsW += ∗ (A-52)
U, V and W Recursive Notations for Five Ascending and Descending Subscripts
jkghijjkghijghijk sVcUU −= (A-53)
jkghijjkghijghijk cVsUU +=∗ (A-54)
ghijkghijkkghijk WsUcV −= ∗ (A-55)
ghijkghijkkghijk WcUsW += ∗ (A-56)
U, V and W Recursive Notations for Six Ascending and Descending Subscripts
jkfghijjkfghijfghijk sVcUU −= (A-57)
jkfghijjkfghijfghijk cVsUU +=∗ (A-58)
fghijkfghijkkfghijk WsUcV −= ∗ (A-59)
fghijkfghijkkfghijk WcUsW += ∗ (A-60)
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APPENDIX B POSITION AND ORIENTATION SPECIFICATION IN EXAMPLE 6.2
Table B-1. Continuous position and orientation needs specification. Sequence x [mm] y [mm] z [mm] ζx [rad] ζy [rad] ζz [rad]
1 705.9128 -16.8861 119.6788 1.299305 -0.14600 0.215781 2 704.4675 2.304591 124.3691 1.296416 -0.17747 0.258972 3 702.3611 21.38499 129.0285 1.293071 -0.20890 0.302415 4 699.6003 40.32938 133.6546 1.289252 -0.24030 0.346144 5 696.1929 59.11239 138.2450 1.284940 -0.27166 0.390194 6 692.1478 77.70905 142.7973 1.280113 -0.30296 0.434604 7 687.4749 96.09491 147.3090 1.274746 -0.33419 0.479414 8 682.1852 114.2460 151.7779 1.268810 -0.36534 0.524667 9 676.2905 132.1390 156.2017 1.262273 -0.39640 0.570409
10 669.8039 149.7511 160.5781 1.255098 -0.42735 0.616690 11 662.7392 167.0603 164.9050 1.247245 -0.45819 0.663563 12 655.1112 184.0451 169.1803 1.238668 -0.48889 0.711085 13 646.9355 200.6849 173.4019 1.229317 -0.51943 0.759319 14 638.2286 216.9599 177.5677 1.219136 -0.54981 0.808332 15 629.0077 232.8510 181.6757 1.208061 -0.57999 0.858199 16 619.2907 248.3399 185.7241 1.196023 -0.60995 0.909000 17 609.0965 263.4095 189.7110 1.182944 -0.63967 0.960822 18 598.4444 278.0433 193.6346 1.168738 -0.66913 1.013763 19 587.3542 292.2257 197.4931 1.153309 -0.69828 1.067927 20 575.8465 305.9423 201.2849 1.136552 -0.72709 1.123430 21 563.9424 319.1795 205.0084 1.118350 -0.75553 1.180399 22 551.6633 331.9248 208.6620 1.098575 -0.78355 1.238971 23 539.0311 344.1666 212.2443 1.077085 -0.81111 1.299296 24 526.0681 355.8945 215.7538 1.053727 -0.83814 1.361537 25 512.7970 367.0988 219.1893 1.028334 -0.86458 1.425870 26 499.2404 377.7713 222.5494 1.000729 -0.89037 1.492482 27 485.4216 387.9044 225.8329 0.970720 -0.91543 1.561571 28 471.3637 397.4920 229.0387 0.938112 -0.93967 1.633345 29 457.0901 406.5286 232.1658 0.902701 -0.96299 1.708013 30 442.6241 415.0100 235.2132 0.864290 -0.98528 1.785783 31 427.9891 422.9331 238.1799 0.822692 -1.00643 1.866850 32 413.2087 430.2957 241.0652 0.777744 -1.02631 1.951386 33 398.3059 437.0968 243.8682 0.729324 -1.04477 2.039522 34 383.3040 443.3360 246.5883 0.677369 -1.06166 2.131330 35 368.2260 449.0145 249.2249 0.621897 -1.07684 2.226801 36 353.0945 454.1340 251.7773 0.563030 -1.09014 2.325822 37 337.9321 458.6974 254.2452 0.501013 -1.10142 2.428157 38 322.7609 462.7084 256.6281 0.436228 -1.11053 2.533433 39 307.6028 466.1718 258.9257 0.369200 -1.11736 2.641136 40 292.4791 469.0933 261.1377 0.300577 -1.12182 2.750625 41 277.4108 471.4792 263.2640 0.231112 -1.12384 2.861158 42 262.4185 473.3369 265.3044 0.161615 -1.12342 2.971936 43 247.5221 474.6745 267.2588 0.092901 -1.12057 3.082151 44 232.7412 475.5009 269.1274 0.025742 -1.11536 3.092142 45 218.0946 475.8259 270.9101 -0.039180 -1.10789 2.985244
119
46 203.6006 475.6597 272.6071 -0.10131 -1.09828 2.880888 47 189.2769 475.0135 274.2187 -0.16022 -1.08669 2.779483 48 175.1406 473.8989 275.7450 -0.21564 -1.07326 2.681300 49 161.2081 472.3282 277.1865 -0.26741 -1.05818 2.586481 50 147.4949 470.3144 278.5435 -0.31549 -1.04160 2.495059 51 134.0161 467.8708 279.8165 -0.35992 -1.02369 2.406979 52 120.7859 465.0113 281.0060 -0.40080 -1.00460 2.322121 53 107.8177 461.7504 282.1125 -0.43830 -0.98446 2.240322 54 95.12436 458.1027 283.1367 -0.47258 -0.96341 2.161389 55 82.71782 454.0834 284.0792 -0.50385 -0.94157 2.085113 56 70.60926 449.7079 284.9408 -0.53231 -0.91904 2.011282 57 58.80912 444.9922 285.7222 -0.55815 -0.89592 1.939686 58 47.32703 439.9522 286.4242 -0.58157 -0.87229 1.870120 59 36.17186 434.6041 287.0477 -0.60274 -0.84824 1.802390 60 25.35166 428.9644 287.5936 -0.62183 -0.82384 1.736316 61 14.87375 423.0497 288.0628 -0.63900 -0.79914 1.671727 62 4.744633 416.8766 288.4563 -0.65439 -0.77421 1.608467 63 -5.029950 410.4620 288.7751 -0.66813 -0.74910 1.546391 64 -14.44500 403.8225 289.0202 -0.68034 -0.72385 1.485366 65 -23.49630 396.9751 289.1927 -0.69113 -0.69851 1.425271 66 -32.18040 389.9363 289.2937 -0.70059 -0.67312 1.365994 67 -40.49460 382.7231 289.3244 -0.70881 -0.64772 1.307432 68 -48.43670 375.3519 289.2858 -0.71587 -0.62234 1.249492 69 -56.00550 367.8391 289.1792 -0.72184 -0.59701 1.192087 70 -63.20040 360.2013 289.0057 -0.72678 -0.57177 1.135139 71 -70.02140 352.4543 288.7666 -0.73075 -0.54664 1.078574 72 -76.46910 344.6143 288.4630 -0.73380 -0.52164 1.022326 73 -82.54500 336.6969 288.0962 -0.73598 -0.49682 0.966333 74 -88.25090 328.7175 287.6675 -0.73733 -0.47219 0.910536 75 -93.58940 320.6912 287.1782 -0.73789 -0.44778 0.854884 76 -98.56360 312.6331 286.6294 -0.73769 -0.42360 0.799327 77 -103.1770 304.5575 286.0224 -0.73677 -0.39969 0.743819 78 -107.4340 296.4788 285.3586 -0.73515 -0.37607 0.688318
Points to be discarded after numerical approximations.
120
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BIOGRAPHICAL SKETCH
Javier Roldán Mckinley studied to become a mechanical engineer at his home town school,
University of Atlántico at Barranquilla-Colombia, with machine design as his area expertise. He
earned his MSc degree from the University of Puerto Rico at Mayagüez in the field of planar
kinematics. Javier continued his research at the University of Florida where he earned his PhD
degree in 2007 in the area of robotics and spatial kinematics. His research interests also include
the design of non-circular gear connections, computer graphics, and computer animations of
robotic manipulators. During his experience at the corrugated board industry in his home
country, Javier participated and led projects in diverse areas of mechanical engineering. The
attainment of the Professional Engineer (PE) license with a concentration in mechanical
engineering is his immediate professional goal.