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Robots that Duplicate Themselves: Theoretical Principles and Physical Demonstrations by
Kiju Lee
A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy.
Baltimore, Maryland July, 2008
© Kiju Lee 2008 All rights reserved
UMI Number: 3340034 Copyright 2008 by Lee, Kiju
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Abstract This dissertation primarily focuses on robotic self-replication including a theoretical framework and quantitative measures that can be applied to self-replicating systems. A new descriptive model for physical replicating systems is introduced based on three sets of components: an initial functional system, a set of resources, and a set of external elements. Robotic self-replication is viewed as a process by which an initial functional robot duplicates itself given a set of resource modules in a bounded environment. In order to assess physical self-replicating systems with respect to their structural properties and performance, two quantitative measures are defined. The first is the degree of self-replication which is a combined measure of structural complexity distribution across the modules and the ratio of the robot's complexity to the complexity of the modules. This quantifies an intuitive notion of many simple parts versus a few complex parts. The second is configurational entropy changes resulting from the self-replication process. Configurational entropy is used to measure the amount of
uncertainty in the locations of modules. Entropy is also applied for articulated chains by using Fixman's method to compute the mass-metric tensor determinant (MMTD). This dissertation also presents a further extension of Fixman's method to compute ii
the inverse of the generalized mass matrix for serial manipulators and polymer chains. Building on previous experimental work, two new self-replicating robots are constructed and presented in this dissertation. The first prototype contains an initial functional robot, a set of modules to form a replica, and a structured environment including tracks, barcodes, contact codes, etc. This system duplicates itself in a similar fashion to the previous prototypes developed in the Robot and Protein Kinematics Laboratory at Johns Hopkins University. However, it shows a progression toward a robot consisting of an increased number of modules while performing more complex tasks. The second prototype duplicates itself through mitosis. This system has several unique properties including its mechanical design and self-replication strategy that distinguish it from any other existing systems.
Dissertation Advisor: Professor Gregory S. Chirikjian Dissertation Committee: Professor Gregory S. Chirikjian, Mechanical Engineering Professor Edward R. Scheinerman, Applied Mathematics and Statistics Professor Noah Cowan, Mechanical Engineering
m
Acknowledgement s I would like to acknowledge many people for helping me during my doctoral studies. First of all, I wish to thank my family: my parents, Chanam, Seung-Ho, Dong-Sun, Young-Ju, Dong-Hoon, and my lovely nephew Hyun-Joon, for their unconditional support, encouragement and enthusiasm. I am very grateful to my advisor, Prof. Gregory S. Chirikjian, for his guidance and support. He encouraged and taught me to develop independent thinking and research skills, and shared his experience and ideas with me. I would like to thank him for his time and continual support during the last five years. I would like to thank to my defense committee, Prof. Edward R. Scheinerman and Prof. Noah Cowan, for their insightful comments that have contributed to the improvement of this dissertation. Prof. Allison Okamura has been a great help during my doctoral work. I am also thankful to Dr. Mehran Armand for his support and for writing reference letters. I am thankful to Georgios Kaloutsakis for his unconditional support and encouragement. Also, I would like to acknowledge my friends and colleagues at Johns Hopkins University, Jusuk Lee, Wooram Park, Jin Seob Kim, Michael Kutzer, Yingyu Wang, Brian Weibeler, Whitney Hastings, and Eun-Jung Rhee. My special thanks
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go to Matt Moses for sharing his experience and expertise on machining and manufacturing and to Kevin Wolfe for reading my dissertation and correcting the English, The work presented in Chapter 9 was developed in collaboration with Dr. Yunfeng Wang, and I thank her for this. I extend my thanks to Prof. Kwee-Bo Sim and Prof. Hong-Ta,e Jeon at ChungAng University in Korea for my undergraduate education and training. I am also very grateful to my friends in Korea.
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Contents Abstract
ii
Acknowledgements
iv
List of Tables
ix
List of Figures
x
1 Introduction 1.1 Overview 1.2 Mathematical Background 1.2.1 Group and Semigroup 1.2.2 Counting Problems and Graph Theory 1.2.3 Rotations 1.2.4 Rigid-Body Motions 1.3 Organization
1 2 6 6 6 8 11 13
2 Principles of Robotic Self-replication 2.1 Overview of Related Works 2.1.1 Automata Theory 2.1.2 Kinematic Self-replicating Systems 2.2 Descriptive Frameworks 2.2.1 Descriptive Model for Physical Replicating Systems 2.2.2 Categorization of Replicating Systems
16 18 18 21 27 27 29
3 Structural Complexity and Reliability for SRRs 3.1 Measures of Structural Complexity 3.1.1 Active Elements 3.1.2 Interconnections 3.1.3 The Degree of Self-replication 3.2 Reliability 3.2.1 Graph Representation of Robotic Self-replication 3.2.2 Reliability Ratio
35 36 36 38 39 43 43 45
VI
4
Configurational E n t r o p y 4.1 Definitions and Background 4.1.1 Discrete Entropy 4.1.2 Continuous Entropy 4.1.3 Partition Function and Thermodynamic Entropy 4.1.4 Entropy Difference 4.1.5 Entropy of a Weighted Sum of PDFs 4.2 Configurational Entropy of Rigid Objects 4.2.1 Sanderson's Parts Entropy Method 4.2.2 Continuous Entropy on Rigid Objects 4.3 Configurational Entropy of Modular SRRs 4.3.1 Entropy in an Unstructured Environment 4.3.2 Entropy in a Structured Environment 4.3.3 Entropy in an Assembled System 4.3.4 Complexity of a Structured Environment 4.3.5 Complexity of Robot Tasks 4.4 Configurational Entropy for Serial Chains 4.4.1 Information-theoretic Entropy 4.4.2 Entropy of Molecular Chains 4.4.3 Entropy of Mechanical Chains 4.4.4 Fixman's Method to Compute the MMTD 4.5 Modeling Excluded Volume Effects 4.5.1 Volume Effects on Entropy Computation 4.5.2 Examples 4.5.3 Upper Bound Estimation
5
P r e v i o u s P r o t o t y p e s a n d their C o m p l e x i t i e s 5.1 Phase I and Phase II: Preliminary Experiments 5.2 Phase III: SRRs with Computer Control 5.2.1 Prototype III-l 5.2.2 Prototype III-2 5.3 Phase IV: SRRs without Computer Control 5.3.1 Prototype IV-1 5.3.2 Prototype IV-2 5.4 Discussions on Previous SRRs
93 94 96 96 100 104 105 109 112
6
P r o t o t y p e IV-3: S R R in a Partially Structured E n v i r o n m e n t 6.1 Experimental Overview 6.2 Robot 6.3 Environment 6.3.1 A Partially Structured Environment 6.3.2 Landmarks 6.4 Complexities and Reliability 6.4.1 Degree of Self-Replication 6.4.2 Configurational Entropy Changes
114 115 120 125 126 127 129 129 130
Vll
48 49 49 51 52 53 55 58 58 60 61 62 63 65 67 68 69 69 70 71 72 76 76 78 90
6.4.3
Reliability Ratio
133
7 Prototype V: Mitosis SRR 7.1 Conceptual Overview 7.2 Experimental Overview 7.3 System Description 7.3.1 Robot 7.3.2 Environment 7.4 Complexities and Reliability 7.4.1 Degree of Self-replication 7.4.2 Configurational Entropy Changes 7.4.3 Reliability 7.5 Extension of Mitosis SRR
136 137 139 144 144 152 153 153 153 156 157
8 0(n) 8.1 8.2 8.3
Mass-Matrix Inversion for Serial Chains of Point Masses 161 Related Works and Background 163 Efficient Inversion of the Mass Matrix 165 Examples with Point Masses 168 8.3.1 The One-Link Pendulum with Mass Concentrated at Tip . . . 168 8.3.2 The n-link Planar Manipulator with Point Masses 172 8.3.3 The n-link Polymer Chain with Point Masses 175
9 0(n) Mass-Matrix Inversion for Serial Chains of Rigid Bodies 9.1 Lie Derivatives 9.1.1 Differential Operators for SO{3) 9.1.2 Differential Operators for SE(3) 9.2 Simple Examples with Rigid Bodies 9.2.1 The Single-Link Planar Rigid Body 9.2.2 The Two-Link Planar Revolute Manipulator 9.3 Extension to Chains of Rigid Bodies 9.3.1 D-H Parameterizations and Jacobians 9.3.2 Lie Derivatives of Generalized Coordinates 9.4 More Examples with Rigid Bodies 9.4.1 PUMA 560 Robot Arm 9.4.2 A Polypeptide Chain 9.4.3
177 178 178 180 182 182 185 188 189 194 195 195 199
Computational Time
203
10 Conclusions and Future Work
206
Bibliography
209
Vita
220
viii
List of Tables 3.1
Number of active elements of some common electronic components . .
5.1 5.2 5.3 5.4 5.5
Phases of physical development Components and the number of Components and the number of Components and the number of Components and the number of
6.1 6.2
Components in six modules Number of Active Elements in Each module
119 129
7.1 7.2 7.3
Components in M A -type modules Components in M B -type modules The time required for self-replication (trep) and the time the parent robot stays in the checkerboard before crossing the boundary without replicating itself (tout)
145 147
D-H parameters of PUMA 560 Arm [59] D-H parameters of a Polypeptide chain [56]
196 201
9.1 9.2
active active active active
IX
elements elements elements elements
for for for for
Prototype Prototype Prototype Prototype
III-l III-2 IV-1 IV-2
[72] [57] [19] [45]
36 94 98 101 106 110
155
List of Figures 1.1
A directed graph with four vertices
2.1 2.2
An example of a finite state system with three states, SI, S2 and S3. Examples in each categories of replicating systems
3.1
Interconnections among the modules: (a) a robot consisting of six modules and (b) a robot composed of seven modules 37 Ds vs. n, the upper line indicates the upper bound and the lower line indicates the lower bound of Ds in (3.9) for Ctotai — 330 and Itotai — 30. 42 Example 1: self-replication process with n modules represented by a directed graph 44 Example 2: if the robot fails to collect any of the first k modules, then it repeats the procedure until it completes the process successfully. . . 44 Example 3: when the robot fails in any of n steps, it returns to the initial step ('State 0') and repeats the process 45
3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6
Placing a block in a 1-D, bounded area Placing two blocks in a 1-D, bounded area Placing three blocks in a 1-D, bounded area QH for n = 2,5,10 when the block size is A QH for k = 1, 2, 3 and n = 3 1-D example with 2 parts (d = 2) in a 1-D bounded environment with various lengthes 4.7 Shaded region indicates the area of all possible positions of the center of Disk 2 when d11258[sec]
124
124
125 127
128
134
135
7.1
Cell cycle of eukaryotic cells: the cell division process is divided into two processes, interphase and mitosis 7.2 Mitosis SRR: a parent robot and unassembled modules are placed in a checkerboard environment 7.3 The moving directions of the parent robot and the behaviors of three wheels: two motors on the right side are always on while batteries are connected and the motor on the left side is controlled by the light sensor output based on the color patterns on the floor 7.4 MA- and M B -type modules. Each contact surface is equipped with an EM installed in male/female couplers, a light sensor and interconnection ports 7.5 Top and bottom views of MA containing a light sensor on the bottom to detect black and white colors on the checker board. Four ball casters are installed in each module to reduce friction 7.6 Robotic mitosis with time lapse sequence 7.7 Silicon mold and aluminum couplers for casting plastic couplers . . .
141 142 144
7.8
Circuit diagram and internal connections in M^
146
A
146 148 148
7.9 Circuit diagram and internal connections in M 7.10 Circuit diagram and internal connections in Mf 7.11 Circuit diagram and internal connections in M®
xn
137 138
139
140
7.12 Schematic of the parent robot composed of four modules. Two interconnection ports in each of the front and back sides (indicated with arrows) are closed when four resource modules are fully assembled signaling the completion of the expansion process. The closed interconnection results in the separation of the parent robot into two daughter robots 7.13 Interconnections across four resource modules form an open loop which is connected to the parent robot as shown in Fig. 7.12 7.14 Top view of Mitosis SRR in a checkerboard 7.15 Two different configurations of the parent robot consisting of four modules. There exists more than one possible configuration for a given number of modules 7.16 Examples of different configurations with two types of cubic modules for a given number of modules, n = 2,4, •••, 12 7.17 The degree of self-replication, Ds vs. the number of module sets, n. . 8.1 8.2 8.3 8.4
9.1
9.2
9.3 9.4 9.5
150 151 152
157 158 160
A massless link with distal point mass; soft variable 0 and hard variable rl69 A massless link with distal point mass; soft variable 9 and hard variable 5170 An n-link planar manipulator composed of point masses with rotational joints, $i, ••• , 0 n , and fixed link lengths, L1: ••• , Ln 172 An n-link polymer chain with point masses at each joint; soft variables 4>i and hard variables 6*j and Lj. Lt is ith link length defined in the same way as Fig. 8.3 175 A 2DOF manipulator consisting of one rigid link with distributed mass; soft variables x, 6 and hard variable y. {S} is the frame of reference and {B} is the frame attached to the rigid body. 182 A 2DOF revolute manipulator consisting of two rigid links with distributed mass; soft variables 6i, 92 and hard variables xi, yi,x2, y2. {S} is the frame of reference and {Bi} and {B2} are the body fixed frames attached to the first and second links respectively. 185 th Tk and Tf; are the transformations from the frame of reference to the k joint and to the center of mass (c.o.m.) of the kth rigid body respectively. 190 A polypeptide chain with alanine side chains 200 Computational time [sec] vs. the number of links, n. The time required for inverting M using a Matlab built-in function, 'inv(-)' (dashed line), and using the extended Fixman's method (solid line), for n = 1, • • • , 400.204
xm
Chapter 1 Introduction This dissertation presents two different topics in robotics: (1) robotic self-replication including theoretical principles and physical experiments, and (2) an 0{n) forward dynamics algorithm developed for serial chains. Self-replicating robots are based on modular robotics. An initial functional system consisting of several modules duplicates itself by locating and assembling modules which are provided as resources. In order to quantify the amount of information associated with the robot's replication process, the concept of configurational entropy is introduced. Configurational entropy can be used to describe the uncertainty not only in multiple separated parts (resource modules in self-replicating systems) but also in other types of systems, such as articulated chains. Fixman's method to calculate the mass-metric tensor determinant (MMTD) for serial chains is adapted in entropy computation for such systems. This dissertation also presents a further extension of this method used to compute the inverse of a mass matrix for serial manipulators and polymer chains composed of both point masses and rigid bodies. 1
1.1
Overview
Biological Inspiration Many aspects of robotics research are inspired by biological systems with regard to mechanical design, locomotion, and control algorithms. Among the many unique capabilities of living creatures, replication is one of the most fundamental and distinctive features that can be observed at various length and time scales. Examples include DNA transcription with polymerase enzymes and associated proteins, RNA self-replication, cell-division, virus reproduction within a host cell, etc. Environmental conditions and external elements are essential in the replication process. These environmental conditions may include passive tools used by the system and returned to the environment, such as catalysts, and/or external factors, such as temperature, humidity, pH, etc. This dissertation presents robotic systems that duplicate themselves using available resources in structured environments of varying complexity. A key to mimicking biological replication within a robotic system is modularization. The current state of the art of this concept is based on modular robotics where an initial functional robot duplicates itself by collecting and assembling another set of modules provided as resources. Over the past decade, several prototypes have been developed which have demonstrated different scenarios of robotic self-replication. However, no proper measures have been introduced to examine how much order is created by the robot's replication process relative to the order inherent in the initial resources.
2
Mathematical Tools for Self-replicating Robots To assess and examine physical self-replicating systems in terms of their structural properties and performance, two quantitative measures are defined in this dissertation: (a) the degree of self-replication and (b) the configurational entropy reductions resulting from the environmental structures and the robot's self-replication process. The degree of self-replication is based on the structural complexity of the system. It is a measure defined by a combination of the complexity distribution across the modules and the complexity ratio of the robot and the individual modules provided as resources. The structural complexity is quantified by counting the number of fundamental electrical components and moving mechanical parts. In addition, mechanical and electrical connections between adjacent modules are counted as a part of the structural complexity of the robot. The configurational entropy is used to measure the amount of information or uncertainty associated with the replication process. The entropy reduced by structuring the environment is related with the amount of information embedded in the environment. The change in entropy resulting from the self-replication process is associated with the complexity of the task and the amount of information handled by the robot. This dissertation also presents a method for computing the configurational entropy for articulated chains such as serial manipulators 1 or polymer chains. Fixman's method for calculating the MMTD is used in entropy computation for articulated chains. Also, this method is revisited and extended for an efficient forward 1
In self-replicating systems, an initial functional robot can take the form of a serial chain or a snake-like robot consisting of several serially assembled modules.
3
dynamics algorithm in the last two chapters of this dissertation.
Two New Self-replicating Robots In addition to the theoretical developments, two new self-replicating robots (SRR) were designed and built to demonstrate different levels of self-replication. The first SRR is composed of six heterogeneous modules with relatively low complexity. Selfreplication is performed by collecting and assembling these modules one by one in a structured environment. Information about the location of individual modules is embedded in environmental structures and the robot's trajectory is determined by the tracks. The robot duplicates itself by following the tracks and reading information from the environmental landmarks. This prototype is an extension of previous prototypes which are fully autonomous and based on a line-tracking algorithm. The second SRR is called a Mitosis SRR. The parent robot (an initial functional robot) composed of four heterogeneous modules grows by attaching the resource modules to itself. After collecting a second set of modules, it separates into two daughter robots by splitting in the middle. The environment where the robot duplicates is defined by a black-and-white checkerboard. Unlike the structured environments used in previous prototypes, it does not contain any information about the modules.
An Efficient Forward Dynamics Algorithm for Serial Chains The next topic involves the mass-matrix inversion for constrained serial chains. Fixman's method, which is adapted for the configurational entropy computation in
4
articulated chains, is further extended to compute the inverse of the mass matrix. The forward dynamics problem in serial chains is to compute accelerations from given forces/torques, r, such that q = M-1(q)(r-C(q,q)-G(q)) where q is a vector of generalized coordinates, M is the generalized mass matrix, C(q, q) are the Coriolis and centrifugal terms, and G(q) is the gravity term. A significant amount of computational complexity results from the mass matrix inversion. This dissertation focuses on computing q = M _ 1 b for constrained serial manipulators and polymer chains based on the Fixman's method. The vector b is any vector with an appropriate dimension. In Fixman's method, a vector of generalized coordinates is partitioned into a vector of variables and a vector of constraints. Fixman's algorithm to efficiently calculate the MMTD is extended to compute the inverse of the mass matrix for serial chains consisting of point masses. In general, robotic manipulators are composed of rigid bodies whose motions must be defined differently from those of point masses. Polymer structures containing side chains can be also treated as rigid bodies rather than point masses. To implement the developed algorithm for systems consisting of rigid bodies, the inverse of the Jacobian matrix (which is essential to computing the inverse of the generalized mass matrix) is obtained by using Lie derivatives instead of regular partial derivatives. The Denavit-Hartenburg (D-H) parameterization is used to describe the rigid-body motions of each link. In the following chapters, we will first review some preliminary mathematics used throughout this dissertation. 5
1.2
Mathematical Background
1.2.1
G r o u p a n d Semigroup
A group is defined as a nonempty set together with a binary operation 'o' satisfying the following properties [13]: • Closure: For any gi,g2 € G, then g\ o g2 € G and g
