UFDC Home  myUFDC Home  Help 



Full Text  
MANIPULABILITY BASED PATH AND TRAJECTORY PLANNING FOR CLIMBING MODE OF A HYBRID MOBILITY ROBOT By JAIME JOSE BESTARD A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Jaime Jose Bestard This work is dedicated to my parents. ACKNOWLEDGMENTS The author profoundly thanks his advisor, Doctor Gloria J. Wiens. The author additionally extends his appreciation to his committee, Doctors Carl Crane and John Schueller. The author thanks the Department of Mechanical and Aerospace Engineering for the opportunity to complete the Master of Science degree. The author deeply thanks his parents, whose overall support and commitment to a fruitful education were essential for the completion of this work. The author additionally thanks his peers, Frederick Leve, Andrew Waldrum, Shawn Allgeier, Javier Roldan McKinley, Gustavo Roman, Kaveh Albekord, Jessica Bronson, Nick Martinson, Takashi Hiramatsu, Sharanabasaweshwara Asundi, Sharath Prodduturi, JeanFrancois Kamath, and Daniel Jones, for their support. Finally, the author would like to express his appreciation and thanks to Natasha M. Elejalde, Joaquin A. Bestard, Juan P. Bestard, Carolina C. Bestard, Juan M. Fernandez, Rolanda M. G6mez, Mireya Llense, Enriqueta Perez, Maria S. Valdez, Martin R. Rosales, Roberto and Nancy Cachinero, Anshley Sardifias, and Yadnaloy Acosta. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES .............. ................. ........... ................ .......... vii LIST OF FIGURES ......... ........................................... ............ ix A B S T R A C T .......................................... ..................................................x iii 1 IN TR O D U C T IO N ....................... .......................... .. ........ .............. 1.1 Types of Climbing Mechanisms and Their Algorithms ..................................3 1.1.1 Clim bing Inside Tubular Structures.....................................................5 1.1.2 Climbing Outside Tubular Structures..................................................7 1.1.3 Clim bing A bout Structural N odes ........................................ ..............8 1.1.4 Clim bing Sm art Structures.................................. ....................... 9 1.2 M anipulability ........................................................................... 11 1.3 Definitions and Terminology................................................. ............... 12 1.3.1 Path Planning ......................................... .......... .. .... ........ .... 12 1.3.2 Trajectory G generation ...................................................................... 12 1.4 Motivation and Scope of the Research .........................................................12 2 HYBRID MOBILITY ROBOT CLIMBING MECHANISM.................................15 2.1 Fundamental Kinematics: The Jacobian Matrix .............................................16 2.2 Lagrangian D ynam ics ......................................................... .............. 19 2.3 M anipulability M measure ............................................................. ................... 24 2.3.1 Definition and Derivation of Manipulability Ellipsoid and Kinematic M anipulability M easure..................................... ..... ............... 25 2.3.2 Definition of the Dynamic Manipulability Measure............................26 2 .4 P ath P planning .............................. .......................... .. ........ .... ..... ...... 2 8 2.5 Trajectory Generation ......... ...... ......... ..... ............... 29 3 PRELIMINARY CLIMBING OF 3DTP....................... ...............31 3.1 Design and Configuration of the 3DTP .......... .... ..................31 3.2 K inem atics of the 3D TP ........................................................ ............... 33 3.3 D ynam ics of the 3D TP.......................................................... ............... 38 3.4 Climbing M ethods ................................ ... .. ........ ............ 38 v 3.4.1 Docking/Undocking Maneuvers for the 3DTP.............................. 38 3.4.2 Flip Climbing M maneuver for 3D TP........................... .....................40 3.4.3 Side Climbing Maneuver for 3DTP......... ..............................41 3.4.4 Rectilinear Climbing Maneuver for 3DTP .......................................43 3.5 Optim ization via Exhaustive Search ............... .................. ...............44 3.6 Flip Climbing Optimization via Minimal Torque Approach........................46 3.7 C lim bing T rajectories ..................... ... ................................... ... 47 3.7.1 Minimal Torque Optimized Flip Trajectory .............. .. ................48 3.7.2 Preliminary Flip Climbing ............ .............................................. 53 3.7.3 Prelim inary Side Clim bing ...................................... ............... 56 3.7.4 Preliminary Rectilinear Climbing ............................... .................59 3.7.5 Further Comparison Results on Preliminary Climbing .....................62 3.8 P relim inary O b servations...................................................... .....................63 4 OPTIM IZED CLIM BING OF 3DTP.................................... ......................... 65 4.1 Docking/Undocking Optimization........................................ ............... 65 4.2 Flip and Rectilinear Climbing Optimization........................................66 4.3 Side C lim bing O ptim ization ............................... .......... ............... .... 69 4.4 Trajectory Generation Using Manipulability Optimized Paths ....................70 4.4.1 Flip C lim bing M ethod...................................... ........................ 70 4.4.2 Side Clim bing M ethod ................................................. .............. 71 4.4.3 Rectilinear Clim bing M ethod ................................... ............... ..73 4.5 Inverse Dynamics Simulation of 3DTP .................... ......................... 74 4 .6 O b serve atio n s ................................................................... 8 1 5 C O N C L U SIO N S ......................................................... .................................. .. 86 A MATLAB FUNCTIONS AND SCRIPTS....................... ...... ..............89 A.1 Recursive Lagrangian Dynamics for Serial Manipulators ..............................89 A .2 Inv erse T an g ent........... .............................................................. .... .... .. ....9 1 B JACOBIAN AND TRANSFORMATION MATRIX ELEMENTS FOR 3DTP ........92 B .1 Jacobian M atrix Elem ents of 3D TP ............................................ ................92 B.2 Transformation Matrices Elements of 3DTP............................................94 C TRAJECTORY GENERATION WITHOUT SEGMENT LINEARIZATION ..........96 D A D D IT IO N A L T A B L E S................................................................ .....................101 L IST O F R E FE R E N C E S ........................................................................ ................... 116 B IO G R A PH IC A L SK E T C H ................................................................ .....................119 LIST OF TABLES Table page 2.1 M echanism param eters for 3D TP .................................................. .....................20 3.1 Actual mechanism configuration parameters for 3DTP ........................................33 3.2 Actual mechanism inertia parameters for 3DTP as computed by ADAMS ............33 3.3 Docking and undocking maneuver parameters for 3DTP simulations ...................39 3.4 Flip maneuver boundary configurations for numerical example.............................41 3.5 Side maneuver boundary configurations ...................................... ............... 43 3.6 Rectilinear maneuver boundary configurations for 3DTP................... .......... 44 3.7 Actual mechanism inertia parameters for 3DTP including motors ........................46 3.8 Flip climbing minimal torque approach optimal configurations for 3DTP..............47 3.9 Effects of segment linearization on joint maximum torques and maximum net power of different 3DTP manipulability path optimized trajectories....................48 3.10 Summary of preliminary simulation results for 3DTP climbing ............. ..............64 4.1 Optimal manipulability configurations of 3DTP at postundocking and predocking ste p s ....... .. .......... .... ........... ...... ................................................. . 6 6 4.2 Optimal manipulability configurations of 3DTP at select intermediate steps for the flip clim bing m aneuver......... .................................... ..................... ............... 67 4.3 Optimal manipulability configurations of 3DTP at select intermediate steps for the rectilinear clim bing m ethod ......................................................... .............. 68 4.4 Optimal manipulability configuration of 3DTP at intermediate step for the side clim bing m aneuver.......... ....... ................................ .. ............ 69 4.5 Summary of preliminary, torque, and path optimized results for 3DTP ..................81 4.6 Performance of different climbing optimizations for 3DTP as compared to previous re se a rc h ........................................................................... 8 3 D.1 Preliminary flip climbing joint torques for 3DTP............. .........................101 D.2 Preliminary side climbing joint torques for 3DTP .........................................103 D.3 Preliminary rectilinear climbing joint torques for 3DTP.................... ........ 106 D.4 Optimized flip climbing joint torques for 3DTP................................ ...............108 D.5 Optimized side climbing joint torques for 3DTP............................... .............. 110 D.6 Optimized rectilinear climbing joint torques for 3DTP ................ ....... ........113 LIST OF FIGURES Figure page 1.1 Alicia3 robot over outdoor concrete wall. ......................................... ...............2 1.2 M echanical adaptations of an SG platform ........................................ ....................2 1.3 X33 concept vehicle boarded by 3DMP robots................. ................3 1.4 Climbing parallel robot (CPR) sequence of postures evading a structural node ........5 1.5 Hyperredundant robot clim bing via bracing................................. .....................6 1.6 Experimental results of the application of the kinematics control algorithm to the clim bing parallel robot (CPR ) prototype ........................................ .....................7 1.7 B ase m odular clim bing robot........................................................... ............... 8 1.8 CPR robot ...................... .................. ........................ 1.9 Climbing structure .................................. .............. ....... ...... 10 1.10 Isometric and front view of the docking mechanism.................... .................11 2.1 The 3DTP robot ................................. ............... .. ............16 2.2 The 3DTP robot's link and joint axes kinematic parameters...............................20 3.1 Different hybrid m obility kinem atical designs ............................... ............... .32 3.2 Closeloop m echanism param eters ........................................ ........ ............... 34 3.3 Configuration of endeffector 2 for 3DTP robot....................................................39 3.4 Flip clim bing m maneuver .................................... ....................... ............... .41 3 .5 Side clim bing m aneuv er ........................................ .............................................42 3.6 Rectilinear climbing maneuver.......................... ................ .......... 45 3.7 Minimal torque optimized flip trajectory of 3DTP robot: joint angles .................49 3.8 Minimal torque optimized flip trajectory of 3DTP robot: joint velocities ...............49 3.9 Minimal torque optimized flip trajectory of 3DTP robot: joint accelerations..........50 3.10 Minimal torque optimized flip trajectory of 3DTP robot: joint gravity effects........50 3.11 Minimal torque optimized flip trajectory of 3DTP robot: joint torques................ 51 3.12 Minimal torque optimized flip trajectory of 3DTP robot: joint power ............... 51 3.13 Minimal torque optimized flip trajectory of 3DTP robot: net power ..................52 3.14 Minimal torque optimized flip trajectory of 3DTP robot: manipulability................52 3.15 Preliminary flip climbing of 3DTP robot: joint angles...............................53 3.16 Preliminary flip climbing of 3DTP robot: joint velocities............................54 3.17 Preliminary flip climbing of 3DTP robot: joint accelerations .............................54 3.18 Preliminary flip climbing of 3DTP robot: joint gravity effects.............................55 3.19 Preliminary flip climbing of 3DTP robot: joint torques ................ ...............55 3.20 Preliminary flip climbing of 3DTP robot: joint power ...................................56 3.21 Preliminary side climbing of 3DTP robot: joint angles.......... .......................56 3.22 Preliminary side climbing of 3DTP robot: joint velocities................. ............57 3.23 Preliminary side climbing of 3DTP robot: joint accelerations ..............................57 3.24 Preliminary side climbing of 3DTP robot: joint gravity effects ............................58 3.25 Preliminary side climbing of 3DTP robot: joint torques ................................58 3.26 Preliminary side climbing of 3DTP robot: joint power .............. ...............59 3.27 Preliminary rectilinear climbing of 3DTP robot: joint angles .............. ...............59 3.28 Preliminary rectilinear climbing of 3DTP robot: joint velocities.............................60 3.29 Preliminary rectilinear climbing of 3DTP robot: joint accelerations......................60 3.30 Preliminary rectilinear climbing of 3DTP robot: joint gravity effects ....................61 3.31 Preliminary rectilinear climbing of 3DTP robot: joint torques ..............................61 3.32 Preliminary rectilinear climbing of 3DTP robot: joint power ...............................62 3.33 Preliminary climbing of 3DTP robot: net power ...................................................62 3.34 Preliminary climbing of 3DTP robot: manipulablity..............................................63 4.1 Path optimized flip method for 3DTP robot: joint angles ............. ..............70 4.2 Path optimized flip method for 3DTP robot: joint velocities .............................71 4.3 Path optimized flip method for 3DTP robot: joint accelerations...........................71 4.4 Path optimized side method for 3DTP robot: joint angles.....................................72 4.5 Path optimized side method for 3DTP robot: joint velocities..............................72 4.6 Path optimized side method for 3DTP robot: joint accelerations...........................73 4.7 Path optimized rectilinear method for 3DTP robot: joint angles...........................73 4.8 Path optimized rectilinear method for 3DTP robot: joint velocities......................74 4.9 Path optimized rectilinear method for 3DTP robot: joint accelerations ................74 4.10 Path optimized flip method for 3DTP robot: gravitational load effects ...................75 4.11 Path optimized flip method for 3DTP robot: joint torques.............. ............ 75 4.12 Path optimized side method for 3DTP robot: gravitational load effects ................76 4.13 Path optimized side method for 3DTP robot: joint torques...................................76 4.14 Path optimized rectilinear method for 3DTP robot: gravitational load effects.........77 4.15 Path optimized rectilinear method for 3DTP robot: joint torques .........................77 4.16 Path optimized 3DTP robot climbing methods: manipulability ............................78 4.17 Path optimized flip method for 3DTP robot: joint power .....................................79 4.18 Path optimized side method for 3DTP robot: joint power.....................................79 4.19 Path optimized rectilinear method for 3DTP robot: joint power...........................80 4.20 Path optimized climbing methods for 3DTP robot: net power ..............................80 C.1 Path optimized flip climbing of 3DTP robot without segment linearization: joint an g le s ...............................................................................9 6 C.2 Path optimized side climbing of 3DTP robot without segment linearization: joint an g le s ...............................................................................9 7 C.3 Path optimized rectilinear climbing of 3DTP robot without segment linearization: joint angles ............. .... .............................................................. 97 C.4 Path optimized flip climbing of 3DTP robot without segment linearization: joint torques ............... ............ .... ................... ........................ 97 C.5 Path optimized side climbing of 3DTP robot without segment linearization: joint torqu es ................ ..................................... ........................... 9 8 C.6 Path optimized rectilinear climbing of 3DTP robot without segment linearization: joint torques ........................................... ........................... 98 C.7 Path optimized flip climbing of 3DTP robot without segment linearization: joint p ow er ......... .. .... ..... ......... ....................................... ..............................99 C.8 Path optimized side climbing of 3DTP robot without segment linearization: joint pow er ......... .... .............. ..................................... ...........................99 C.9 Path optimized rectilinear climbing of 3DTP robot without segment linearization: joint pow er ............. ..... ..... ............................... ....................... .. 100 C.10 Path optimized climbing of 3DTP robot, no segment linearization: net power......100 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MANIPULABILITY BASED PATH AND TRAJECTORY PLANNING FOR CLIMBING MODE OF A HYBRID MOBILITY ROBOT By Jaime Jose Bestard December 2006 Chair: Gloria J. Wiens Major: Mechanical Engineering Inspection robots used in a wide range of applications require the ability to efficiently climb while carrying the necessary loads which may include sensors, special tooling, and batteries. Therefore, it is essential to develop power efficient climbing algorithms for the overall design and analysis of autonomous multiagent robotic systems. The vast majority of such robots are currently able to climb using only one method with no alternate or hybrid methods available, in part due to the geometric constraints of the mechanism. Hybrid mobility robots consist of wheeled locomotion when traversing horizontal terrains with modest grades and roughness. Upon encounter of vertical terrains, two hybrid robots join to reconfigure into a climbing mode on a smart structure. To this avail a six degreeoffreedom mechanism was developed for the simulation of different methods of climbing while maintaining constant geometric and inertia properties, thus providing grounds for comparison. This mechanism is capable of transforming into the different climbing configurations for the hybrid mobility robot developed in prior research. In this thesis, the research focuses on development of efficient path and trajectory planning algorithms for the climbing mode of the hybrid mobility robot. Three different climbing methods were compared on the basis of their resulting maximum torques and instantaneous power requirements. These quantities were selected since they directly affect the selection of joint motors. To achieve optimal trajectories, manipulability performance indices were implemented as a means for identifying configurations of maximum manipulability (minimal torque) at viapoints along a nominal trajectory. Piecewise trajectories were then generated using quintic splines between the viapoints of optimal configurations. The effect of the number of viapoints selected on the resulting torqueminimal trajectories was investigated. Limitations on the optimality were found to be heavily weighted by the trajectory planning portion of the algorithms. In spite of this, the manipulabilitybased approach was shown to be effective in generating paths and trajectories that exhibit overall low torque and power requirements. Of the three climbing methods, the rectilinear climbing method proved to be the most versatile in not only providing ample room for battery loads, but for additional loads such as sensors. However, in hybrid mobility robotic systems, multiple climbing methods may still be required simply due to the geometric and/or physical constraints. The impact of the presented research is demonstrated optimality in presence of physical and trajectory planning constraints as well as a foundation for future design optimization of climbing hybrid mobility robotic systems. CHAPTER 1 INTRODUCTION The thrust of this thesis is to develop path planning and trajectory generation techniques that optimize the climbing capabilities of hybrid mobility multiagent robotic systems. A hybrid mobility multiagent robotic system consists of robots within a team of robots that each have two or more locomotion types (e.g., wheeled and climbing) or acquire multiple locomotion types by reconfiguring and/or joining with other robots. These systems are applicable for tasks defined by urban surveying, inspection and reconnaissance, which may be found in both groundbased and space environments. Typical operating environments for climbing mechanisms vary extensively throughout the planet. In the civilian environment, applications for climbing mechanisms range from the cleaning of high rise buildings, as shown in Figure 1.1, to the inspection of sewers and plumbing systems for leaks and maintenance (Figure 1.2). Space applications for climbing robots are also extensive, and analogies on the developed climbing algorithms can be applied to several maneuvers and applications in the cosmos. An example of this is the maneuvering about a Resident Space Object (RSO), e.g. satellite, on the space station and/or on space trusses, to be serviced in space where several robots must cooperate, and optimal trajectories about the RSO are necessary. Climbing robots have also been envisioned for in space inspection and servicing applications [Men05]. In each of these applications, it is important that the joint actuators are sized for compactness, leaving room for clamping, grapple, or adhesive mechanisms, but at the same time performing as necessary with minimal joint torque requirements. Figure 1.1 Alicia3 robot over outdoor concrete wall [Lon06]. Figure 1.1 Alicia3 robot over outdoor concrete wall [LonO6]. Figure 1.2 Mechanical adaptations of an SG platform for (a) climbing the exterior of tubes; (b) an open universal joint; and (c) climbing the interior of tubes [Ara06]. An ideal application for climbing robots for space applications is for groundbased inspection and/or repair of the space shuttle or a shuttle concept vehicle by three dimensional modular platform (3DMP) robots [Cle03, Cle04]. In the research of Clerc [Cle03], the conceptualized implementation was for these robots to gain access to different areas of the shuttle via joining together to form a single modular robot capable of climbing a smart structure, as illustrated in Figure 1.3. Within each area the 3DMP lb) (c) robots would separate and inspect the spacecraft in accordance to previously developed inspection/ coverage algorithms [But03]. In this work, a smart structure consisted of a series of automated ports located along the robot's climbing trajectory in which each end of the climbing robot would dock securely and undock as it climbed. X33 ro ea Concept Vehicle Smart Structure Zoom of Carto Area T \\ o i oin ied ,DN lP robots Smart Stnmture Figure 1.3 X33 concept vehicle boarded by 3DMP robots [Cle03] 1.1 Types of Climbing Mechanisms and Their Algorithms Climbing algorithms have been developed for different situations and resource capabilities. For climbing inside tubular structures, there exist parallel mechanisms based in the StewartGough platform [Ara06] as well as serial mechanisms that used properties of contact to climb [Gre05], among others. Similarly, the same parallel mechanisms developed to climb inside tubular structures can be used to climb outside tubular structures with slight modifications to their design or configuration. Parallel mechanisms may also be used for climbing outside any other structure, including a smart structure [Cle03], as demonstrated in related research [Ara06]. However, the use of serial configurations is more common for climbing on the outside of structures, as shown earlier in Figure 1.1. Other adhesion based surface climbing robots were developed by researchers of references [Men05, RipOO]. All these mechanisms have the same purpose; however, they differ in approach to achieve it and in their overall performance. A subclass therefore can be made of all these mechanisms where their climbing algorithm is the dividing criteria. Three groups exist then, where for one group climbing is done in a fashion similar to worms or snails, by contracting or shortening the distance between docking points and then extending towards the next docking point, this being named rectilinear locomotion or climbing. The other group is that which climbs in a fashion similar to a slinky, which is labeled as flip climbing in this thesis. And the final group climbs in a similar fashion to quadrupeds like geckos, where the climbing portion revolves about the stationary portion to climb. This group is called side climbing in this thesis. Climbing about nodes, as shown in Figure 1.4 is another facet important in the investigation of climbing. A node has been defined as a corer in a structure, or a location at which the climbing path changes direction abruptly [Sal05]. The performance of the climbing robot changes completely at this point, due to changes in the climbing algorithm and the complexity of the task. Careful path planning is necessary to adequately address this situation and should be considered. Therefore, a category of climbing can also be defined as the climbing about nodes. For example, Figure 1.4 illustrates a climbing parallel robot (CPR) following a sequence of moves around a corner node. (a) Pos1 (b) Pos1d (ce Pos2a (d) Pos2 Figure 1.4 Climbing parallel robot (CPR) sequence of postures evading a structural node [Sal05] In both the serial and parallel configurations different climbing tasks can classify further the design of a robot, mainly the climbing inside and outside tubular structures, climbing on smart structures, and climbing about structural nodes. Additionally, the climbing method used, further defines and classifies the robot. 1.1.1 Climbing Inside Tubular Structures Climbing inside tubular structures is one important category of climbing mechanisms, where an extensive range of applications can be performed. The medical field can benefit vastly from machines developed for autonomous or supervised inspection and perhaps the repair of parts of the human body. Another application for this kind of mechanism is the exploration of oil and gas pipelines. Investigations on the usage of parallel mechanisms to this avail are ongoing [Ara06]. However, the question remains: How efficient can these parallel mechanisms be in different aspects, especially energywise, but also considering the storage and transportation of these machines, and the time they take to perform their respective tasks? All of these questions, are out of the scope of this thesis, and depend on the application. Albeit an ongoing research topic, parallel machines of the StewartGough kind are additionally out of range for the climbing application of reconfigurable inspection multi agents as in previous research [Cle03]. These machines are difficult to implement in a reconfigurable fashion. Serial mechanisms for the same application provide a different scope to the aforementioned questions. Regarding the question of efficiency, this thesis focuses on the development of new path optimization algorithms for the climbing mode of a hybrid mobility robot. To this avail, several methods will be explored where only serial mechanism configurations will be simulated. Serial mechanisms for this purpose may also provide favorable answers to the other relevant questions regarding climbing [Kot97]. Climbing via bracing [Gre05] is an interesting approach that may provide a very feasible solution for specific tasks. Though a rectilinear climbing method, the fact that contact is used at both ends of one or more links as shown in Figure 1.5, deviates this topic from the main methods to be investigated. That is, to conform to the focus of this thesis, the hyperredundant robot mechanism would have to be grounded at one end and free to move in the other which is not the case shown in Figure 1.5. 1n2 n  ti_ Figure 1.5 Hyperredundant robot climbing via bracing [Gre05] 1.1.2 Climbing Outside Tubular Structures Climbing outside of tubular structures is a category that may benefit from the results of this thesis. Parallel mechanisms can and have been developed to this avail for climbing the trunks of palm trees [Alm03], as shown in Figure 1.6. Other applications may include the repairing of electric wiring in potentially harmful situations. Figure 1.6 Experimental results of the application of the kinematics control algorithm to the climbing parallel robot (CPR) prototype [Alm03] Serial mechanisms have also been developed for climbing outside tubular structures. There have been many achievements and designs in this group, and consideration to modular and reconfigurable designs have been given (Figure 1.7) [RipOO]. Climbing outside a tubular structure can be done in all three climbing mode subclasses. The problem arises on the selection of the most efficient of these climbing algorithm subclasses. This dilemma exists because of the need of adding gripping and climbing motors [Pac97, Yan97]. Therefore, the topic of climbing outside tubular structures with serial mechanisms, reconfigurable in cases, will benefit from the methods presented in this thesis where efficient algorithms for climbing are developed. (a) \ \ (b) Figure 1.7 Base modular climbing robot: (a) Single robot (b) Coupling of two modules of climbing robot to form new capabilities of negotiating nonstraight vertical motion [RipOO] 1.1.3 Climbing About Structural Nodes Climbing about structural nodes is a topic that is of utter importance to any climbing robot; however, this topic is not in the scope of this investigation and is only used to consider and explain most situations encountered by a climbing robot. Most structures climbed will contain structural nodes of some sort. Nodes may be evaded using path planning algorithms; however, their presence affects the simple climbing algorithm on a vertical surface. The most explicit descriptions of this issue occur in designs of parallel mechanisms as the CPR [Sal05]. As shown in research with StewartGough platform based mechanisms, these machines though excellent in certain aspects of climbing, will lack the capability structurally of maneuvering about a corer node, as climbing a wall that ends into a flat roof or any kind of roof for that matter without the aid of additional appendages. Parallel mechanisms developed to this avail become more complex due to their joints requiring a greater range of mobility. For example, in the CPR [Sal05], new spherical and universal joints had to be developed to accomplish 900 configurations, as shown in Figure 1.2 and Figure 1.8. Thus, simply considering joint limitations, serial mechanisms are more adept structurally for the node kind of obstacles. Henceforth, serial mechanisms are definitely superior in the matter of climbing about structural nodes. Another example is that even inside a room where the walls and ceiling need to be inspected, the serial mechanism can out perform a parallel mechanism. In general, climbing about nodes is a matter of special interest and may be easier to analyze using serial mechanisms, though such behavior will not be considered in this thesis. B  (a) (b) Figure 1.8 CPR robot: (a) Posture where it is necessary to achieve 90 between both plates of StewartGough platform (b) Modifiedjoints [Sal05] 1.1.4 Climbing Smart Structures Climbing smart structures is the main scenario of research for this thesis. These structures offer a favorable level of independence regarding the development of climbing algorithms and the analysis of torque and power used just for climbing and not for gripping. A structure, for the purposes of this research, is labeled smart when it has the capabilities to hold, or grip, a climbing mechanism, as previously demonstrated [Cle03]. An example of such a climbing structure is shown in Figure 1.9 and Figure 1.10. In such example, the smart structure is composed of docking points, where a mechanism is in charge of detecting an end of the climbing robot and clamping it or docking it to the smart structure. When the docking is completed, the other end of the robot is freed autonomously by the climbing structure, and the robot proceeds to climb to the next desired docking point. Previous research on climbing the smart structure considered climbing in the flip fashion. However, in this thesis three different methods will be considered and evaluated for determining which is the most efficient for climbing such vertical structures. Figure 1.9 Climbing structure [Cle03] Isometric View Servomotor /1 Locking Mechanism Front View Female Docking Locking Pin Limit Switch Docking Motion Figure 1.10 Isometric and front view of the docking mechanism [Cle03] 1.2 Manipulability Manipulability is a measure that determines the ease of changing the position and orientation of the endeffector in the sense of the required joint torques being lower, thus easier to maneuver the endeffector. The manipulability ellipsoid, or vice versa, the manipulatingforce ellipsoid [Yos90], is a visual description of manipulability and is in turn inversely proportional to the manipulability measure. The manipulability ellipsoid delineates via its principal axes the directions in which a maximum manipulability and corresponding minimal manipulating force are required to generate motion. Non obstante, these concepts are devoid of dynamic considerations and are further developed in [Yos90] into the dynamicmanipulability ellipsoid and measure. This thesis investigates the development of path planning and trajectory generation algorithms using manipulability measures. The goal is to provide efficient paths and trajectories generated while capturing the dynamics of the system per the tasks to be achieved. Path planning and trajectory generation algorithms can be modified by following the direction in which a minimal manipulating force can be generated as given by the manipulability measure towards a final configuration. 1.3 Definitions and Terminology 1.3.1 Path Planning Path planning refers to the ability to determine a path in task or configuration space in order to move a robot to a final position while simultaneously avoiding collisions with obstacles in its path or workspace. Paths generated in this fashion are independent of time, and as such, neither the velocities nor accelerations along the planned path are considered [Spo05]. 1.3.2 Trajectory Generation Trajectory generation is the development of reference trajectories considering the time history of a robot throughout a certain path. These reference trajectories are usually provided in joint space as polynomial functions of time [Spo05]. 1.4 Motivation and Scope of the Research Climbing for any purpose is a complex procedure that requires the utmost efficiency on the climber's side. Climbing can be performed in different fashions according to the geometry of the robot. The adequate selection of a climbing method can decrease the required size of motors or simply provide excess actuation that may be used to carry an additional load. In the design and development of these autonomous climbing capable robots, the constraints and limitations are dominated by the torque, mobility and maximum power required, thus the sizing of the actuators is critical. To address these issues, a comparative study on different climbing strategies was conducted. These strategies were evaluated in terms of their maximum torque and maximum power requirements for given climbing transversals done in a given amount of time. To improve the resulting performance, path generation algorithms were then developed based sub optimally on robot manipulability metrics and minimal torque. The effectiveness of these algorithms is evaluated in terms of the maximum torque, maximum power required, and robustness to inertia/payload variations. In addition, the effects of implementation variations, such as number of intermediate viapoints and boundary conditions used for the trajectory planning and the curve fitting selection (quintic, cubic, linear), were investigated. Hybrid mobility inspection mechanisms have previously been designed [Chi94, Mer05], and path planning and obstacle evasion algorithms were developed [Cle03]. However, climbing ability for these mechanisms is in a primitive stage. Regardless of the previous design, though mainly using the geometry, several climbing algorithms will be analyzed and ranked accordingly. Additionally, a generalized climbing methodology where path planning and trajectory generation are constrained by manipulability concepts will be developed. This novel methodology is a building block of climbing, for it optimizes the climbing path. Dynamic simulations using this algorithm will provide empirical results for the comparison of different methods of climbing. In Chapter 2, an overview of the generalized concepts used throughout the investigation is exposed in detail. The chapter covers the definition of the Jacobian matrix, to be used finally in the chapter for the definition of the manipulability measure. The necessary concepts of Lagrangian dynamics are also explained and an approach avoiding the necessity of symbolic computations is also presented. The concept of manipulability is also introduced and its quantification is partly derived. Finally, path and trajectory planning concepts are introduced. Chapter 3 contains the development of the hybrid mobility climbing robot 3DTP as a design and as it regards its geometric configuration. The basic solution for the closed loop form of the climbing robot is also introduced and used extensively as a method of obtaining configurations. The dynamics of the mechanism are reiterated to be obtained via Lagrangian methods. Chapter 3 introduces the three different climbing methods to be explored as well as the docking and undocking maneuvers. Additionally, in chapter 3 the preliminary paths and trajectories are developed and tested, leading to initial conclusions regarding the behavior of each method. Chapter 4 tests the optimization using the concept of manipulability and observations are drawn regarding improvements in performance using this method. Additionally, chapter 4 explores using static or quasidynamic methods [Cle03] for finding preliminary path viapoints and optimizing them using manipulability concepts for flip climbing. Chapter 4 also delves into the different methods considering motor loads and the effects that will have in carrying additional loads. Finally, chapter 5 draws conclusions regarding the behavior of each method and possible applications of each method for climbing while providing a path for future investigations. CHAPTER 2 HYBRID MOBILITY ROBOT CLIMBING MECHANISM In this thesis, the hybrid mobility robot has two basic modes of operation, as a pair of wheeled locomotion robots and as a single climbing robot. In today's research community, wheeled robots have become the platforms of choice for not only developing path planning for inspection, reconnaissance, and surveying, but for testing control algorithms involving multiagent and cooperative teams of robots. However, these robots are limited to relatively flat terrains. Via hybrid mobility, achievable by introducing reconfigurability, this next generation of robots also has the ability to climb. In prior research [Cle03], an initial path planning algorithm for a hybrid mobility robot was developed based on a quasidynamic approach. In this chapter, a generalized mechanism is defined which captures the climbing kinematics of the work of Clerc [Cle03] and others. In addition, the generalized mechanism easily transforms to accommodate all three climbing methods studied in this thesis while maintaining the same geometric and inertial properties. The following sections detail the supporting theoretical derivations that demonstrate the kinematics, dynamics, and manipulability of a generalized serial mechanism. Following the naming convention of the previous research [Cle03], the 3DTP (Three Dimensional Test Platform) denotes a robot capable of climbing out of the plane of wheeled mobility. The 3DTP's reconfigured workspace is defined by kinematics of a sixjointed serial mechanism and that of a closedloop mechanism depending on the various climbing configurations, refer to Figure 2.1. As opposed to the 3DMP (ThreeDimensional Mobile Platform), the 3DTP was developed with the purpose of comparison of different climbing methods and performs only climbing tasks via a smart structure. In this work, a smart structure consists of a series of automated ports located along the robot's climbing trajectory in which each end of the climbing robot would dock securely and undock as it climbed. .1Aitti I lu//il 2 .1h11t/I 4 .Jo,,,i 5 . Figure 2.1 The 3DTP robot: (a) extended, free end (serial mechanism); and (b) docked to a smart structure after performing a climbing step (closedloop mechanism) 2.1 Fundamental Kinematics: The Jacobian Matrix The use of the Jacobian matrix provides a relationship between the velocities of the endeffector and the joints of the 3DTP. As will be defined later in section 2.3, the Jacobian is an essential part of the manipulability measure. For the definition of the Jacobian, as shown in [Asa86, Cra05], the vector v = [OPT T] is used when expressing the velocity of the endeffector, where p refers to the absolute linear velocity component and 0a, refers to the absolute angular velocity component. The relation between velocities is given as S= J(4q)q (2.1) The subscript e denotes end effector and the superscript 0 denotes the vector is expressed in terms of the inertial frame 0, and is the vector of joint velocities. Using the Denavit Hartenberg notation, the angular and linear velocities of the endeffector with respect to the ground reference frame can be expressed as Pe= [00 1]TJx oRi +1i (2.2) 0ae = o0R[0O 0 1]Tq, (2.3) J=1 Where OR is the rotation matrix describing the orientation of frame relative to frame 0. The (*)z denotes the transpose of (*), x denotes the crossproduct of two vectors, n is the number of links in the system, and the q, denotes the velocity of theth joint. The 'pf, is the position of i+l frame's origin in the ith reference frame where for n+1 frame's origin is also the point on the end effector for which the end effector velocities are defined. Considering the following definitions, S=OR [O 0 1] (2.4) n 1=j IJ = ofe op, (2.5) The generalized expression for the Jacobian matrix becomes 0 FlX0 0 02 0 02x 0 S Z ,1 2 X ,2 ... e,n (2.6) 0ZI 0ZI 0. "n I The vector oz represents the unit vector of joint axisj, whilst 0P, refers to the vector from a point on joint axisj to the endeffector, nominally taken as a vector from the origin of thejth frame to the end effector. Additionally, considering the homogeneous transformation matrix from the ground reference frame to joint FO 0^ 0^' 0^' O OT x= J yJ z J (2.7) S 0 0 1 Where op is thejth frame's origin relative to the fixed frame 0. The O%, oj and 02, are the unit vectors of the /h frame affixed to link j, determined according to Denavit Hartenberg notation [Cra98]. Furthermore, oT = T'1T2 .".Y (2.8) And cos O sin 8 0 a1 , T sin0 cosa 1 cosO cosa sin a sin S (2.9)S S sin sin a,, cos O, sin a,, cos a 1 cos a jS, 0 0 0 1 The OJ is the joint angle for revolute joint. The aJ is the orientation ofjointj axis relative toj1 axis about the XJ, axis. The S, is the distance between XY and XJ, axes along Zi axis. The aj1 is the shortest distance between Zi and ZJ, axis, forming the X, axis. 2.2 Lagrangian Dynamics To demonstrate the effectiveness of each climbing strategy and path/trajectory planning algorithm of the later chapters, the equations of motion of the 3DTP robot are introduced. These were obtained in a generalized fashion using a recursive Lagrange formulation. In robotics literature [Yos90], this method has been integrated with the DenavitHartenberg definitions such that differentiations with respect to joint variables can be achieved via matrix multiplications. The Lagrangian function is L=KP (2.10) Where K denotes the kinetic energy and P the potential energy of the system. By definition, Lagrange's equations of motion are expressed as Q (2.11) dt Qqj, Oq, Oq, Next, the equations of motion are found using the homogeneous transforms between link frames attached to each link. The docked end of the manipulator is grounded, and as such is referenced as the 0 frame. The relationship between each link reference frame and the 0 frame is then given by equation (2.8), which is labeled the homogeneous transform from the 0 to thejth reference frame. Additionally, each recursive transform between links is defined by equation (2.9). In equation (2.9), OJ and Si are the fh link joint angle and offset respectively, while aj and aj_ are the twist angle and link length, respectively. In the 3DTP mechanism all the joint angles OJ define the joint variables, since all joints are revolute. Refer to Figure 2.2 for a visual description of the joint variables. The overall model parameters are provided in Table 2.1, the remaining parameters necessary for accurate computation of the Jacobian will be further provided in Chapter 3. Table 2.1 Mechanism parameters for three dimensional test platform (3DTP) robot i Link lengths (m), aI1 Twist angles, 1 Joint offsets (m), S' Joint angles, , 1 0 900 L0 +L, Variable 2 0 900 0 Variable 3 L2 0 0 Variable 4 L3 0 0 Variable 5 L4 0 0 Variable 6 0 900 L5 + L6 Variable 7 0 0 0 0 J, J2 J3 J4 x4 X J6 J5 x, X6 X5 Figure 2.2 The 3DTP robot's link and joint axes kinematic parameters Next, consider a point on link i, this point is labeled as 'T with respect to the reference frame i. The location of this point with respect to the ground, 0 reference frame, is given by 0r 0 TI Relative to frame i, r is constant and d'r/dt = 0. Therefore, dor dt ( a"T J1 dqj 0 0^u 1 Considering equation (2.13) and 0rT r tr( rr"T), where tr denotes the trace of a matrix, then rr =ZZtr ]=1 k=l (2.14) Referencing to equations (2.10) through (2.14), it is possible to compute the kinetic energy K, and the potential energy IP of link i. The former is found as follows K dK OT 10r pdv K =f dK = nk 2 pdv h' Jlnk hA Jlk 2 (2.15) 12 a ("T, "TT1 >1 tr TH' 4q,4k 2 j1 k= 9q k Where H' refers to the pseudo inertia matrix [Yos90] given by H' = f1k 'rTpdv t ink t I' + + I ixx lyy 2 H' H' m nrCMx I' I' +1' 2 H' m rCMly H' Ixx yy i'zz 2 m rCMA To express the relation between the pseudo inertia matrix and the inertia tensor consider FI" l I"X I" 7 I, I, , (2.17) (2.12) (2.13) aqT TTq Oq, O8q, k (2.16) imn rcMx m rCMy m, nalr a Where I 1,, and I, are the principal inertias of link i about its bodyfixed frame and the remaining offdiagonal terms are the products of inertia for link i defined about the bodyframe i axes. Then the following relations hold JI =x +m (rcMy2+ rcMI 2) (2.18) H' = Imxy + CMx rCMly (2.19) The same notational form is followed for I', I1,, H' H' Additionally, the remaining parameters are 'cM = [rc,,x rcTM rCM" (2.20) Where 're refers to the vector from the origin of link i to its center of mass in the coordinates of the link. The potential energy, P,, of link i is computed as follows P = mkgTO, cr (2.21) Considering g = [g gy gz 0] as the gravitational acceleration in the ground, 0, reference frame. In this fashion the Lagrangian, as expanded from equation (2.11) becomes L= (K,P,) (2.22) Substituting the previous equation into equation (2.11) and rearranging, one obtains the dynamic equations for each joint torque as n k (T T TT r, = tr " H' kO q k=i j=1 I aq7) S q (2.23) n k k 2 OTk T _T2kT a Tq j + C tr/ k H T TH' O , k=i ,= l = 1q;m m1 ]= q The generalized equation of motion is of the vector form S= M(j)q + v(c,q) +G (c) (2.24) Where M(q) refers to an n x n inertia matrix that can be further defined as (,T TT" Mr= tr kH H k (2.25) k=max( ) I q, Additionally, V(q, q) is an ndimensional vector that contains the centrifugal and Coriolis forces and its ith element is expressed as n n n a 2T O TkT =y tr k H' ,, (2.26) j=1 m=1k=max((,,mi) q qm q ) The G(q) vector contains the load due to gravity and is given by G, = m/ ] rc (2.27) o 2q, Furthermore, the matrices iT,/cq, and 2 'TI/(q,0q )=9 ''2 /(Sqkqj ) can be found using the relationship = ''TA, (2.28) 9q, Which comes from the fact that the transformation matrix is a function of the generalized coordinate. The parameter A is defined as follows 0 1 0 0 1 0 00 0 0 for a revolute joint 0 00 0 000 (2.29) 0 0000 0 000 S0 0 for a prismatic joint 0 0 0 0 The parameter A, provides derivative definitions as shown 0T= TAj JT (2.30) 8qJ 02 0T OTA JTkAk kT ,i>k > j (2.31) 8q, cq 0 ,max (j,k)>i Thus, these equations avoid the need for symbolic manipulation when finding the equations of motion. In addition, it should be noted that the above derivation of the equations of motion are for the robot in its serial mechanism configuration. For the closedloop mechanism configuration, the additional constraints of the end effector with the structure must be accommodated. For this thesis, the mode of mobility for which the optimization requires use of the equations of motion is only the serial mechanism mode. 2.3 Manipulability Measure An important factor in the selection and design of any robot manipulator is the facility of changing position and orientation of the endeffector [Yos90]. A quantitative measure for this facility has been developed from the kinematics and dynamics viewpoint. In this thesis, the kinematic manipulability measure will be used for the purpose of pathplanning, whereas the use of the dynamic manipulability measure will be left for future research. 2.3.1 Definition and Derivation of Manipulability Ellipsoid and Kinematic Manipulability Measure The relationship between the endeffector velocity i and the joint velocity q was previously given in equation (2.1). From this relationship and per reference [Yos90], the definition and procedure to obtain the manipulability measure and its properties is provided below. First, consider the set of all endeffector velocities possible by the joint velocities in such a fashion that the norm of q satisfies S4 12 + q2 +...2+ 42 <1 (2.32) The set provides an ellipsoid. From the relationship between the endeffector velocity v and joint velocities 4 given in equation (2.1), the ellipsoid's major axis is the direction where the endeffector can move at high speeds (high manipulability), whereas the minor axis satisfies the contrary. Analogous to this definition, one can also use this measure to denote the direction in which large manipulating force can be generated in the direction of poor manipulability. The size of the ellipsoid is also an indicator of the overall speed at which the endeffector can move. This definition is essential to the future pathplanning process. The ellipsoid, since it represents the ability to maneuver is named the manipulability ellipsoid [Yos90], which can be further labeled the kinematic manipulability ellipsoid (KME). For the purposes of this thesis, the most useful measure derived from this ability for manipulation and the respective KME is the volume of the former. The measure is derived from the mathematical definition of the volume of an ellipsoid and is directly proportional to the overall named manipulability measure [Yos90] for a certain configuration and labeled as w. It is formulated as wk = det J(q)(J(q)) (2.33) However, when the degrees of freedom of the manipulator is equivalent to the number of joints, n = m (refer to Section 2.2.1), the measure further reduces to Wk= det(J(4))l (2.34) The manipulability measure is a property that also relates to the distance of the manipulator configuration from being singular. Generally the manipulability measure is greater than zero, wk > 0, and is only zero, wk = 0 iff rank(J (cq)) 2.3.2 Definition of the Dynamic Manipulability Measure The dynamic manipulability measure (DMM) is a similar concept to that of the KMM, and is further refined by considering the manipulator dynamics. Additionally, it provides suitable properties for highspeed and highprecision motion control [Yos90]. While not implemented in the optimization algorithm, its definition is presented here for completeness and as an option for future research. To derive the dynamic manipulability ellipsoid (DME) and measure (DMM) consider initially the manipulator dynamics as given by (1.24). Differentiating the relationship between joint and endeffector velocities given by (1.1) yields v= J( )q+J(j)q (2.36) The second term of the equation can be interpreted as the virtual acceleration and redefined as r (q,q ) J()q (2.37) From (2.24) and (2.36) ( J JJ) =JM'(f V(,q) (q)+MJ i (2.38) Where J+ is the pseudoinverse matrix of J. Further introducing the vectors Z= r(,i)G(q)+MJ +d (2.39) v=v(IJ J) (2.40) Equation (2.38) can be rewritten as v=JM 'f (2.41) Similar to the definition of the KME, the DME is constrained by the inequality r <1 1. The ellipsoid is described in Euclidean space as v (J +) MTMJ+ <1 (2.42) The dynamic manipulability measure (DMM) is defined similarly to the KMM w, = det ((M'M) 1JT) (2.43) Once again, similarly to the KMM, the DMM is further reduced when the manipulator is not redundant, n = m, as given by det(J) wd det(M) (2.44) Wk (2.44) det (M) The denominator of (2.44) contains the effects of the dynamics of the manipulator while the numerator refers to the effects of the kinematics on the manipulability. The physical interpretations regarding singular configurations remain. 2.4 Path Planning Path planning provides a geometric description of the motion of a given manipulator [Spo05]. However, it does not provide any dynamic aspects of the given motion. The joint velocities and accelerations while traversing a given path are obtained using a trajectory generator. In the case of the 3DTP while climbing a smart structure, there is not a set of obstacles to be avoided. However, the manipulator must achieve a certain configuration for docking and undocking. For these maneuvers, the manipulator behaves entirely as a closedloop four bar mechanism and as such the inverse kinematics become trivial. For this reason, throughout the path, the user is compelled to obtain the necessary boundary joint configuration by using the analysis of a fourbar closed loop mechanism. The path in between the boundaries of the manipulator for the purposes of this thesis, is optimized using a method similar in concept to those developed in [Zhe96], both of which used manipulability to plan the desired path. In reference [Zhe96], the authors use a map of a target zone and form a nominal path, which in the future is called a preliminary path, a new path is obtained by using a cost function, J, weighting both manipulability and the preliminary path as shown in [Zhe96] J =W ds W 1 (2.45) Where md (s) is the manipulability (dynamic or cinematic), at the given point and W, and Wr are, respectively, weights deciding if the cost function to be minimized must adhere to the preliminary path or to the manipulability based path more. The cost function provided beforehand is also computationally intensive for the 3DTP, since it was developed and demonstrated for a simple two revolute joint manipulator. For the purposes of this research, the complexity of using the given cost function is avoided by first defining endpoints to the desired path and at those endpoints obtaining the optimal manipulability configuration, which in flip and side climbing are endpoints, but in the case of rectilinear climbing are mainly viapoints. From there it is only a problem of generating a convenient trajectory and deciding which climbing method is more appropriate. Objectively, such minimization was done in previous research [Cle03] in the form of qj,optmal = q(min(max(r (t)))),= 1,2,...,6 (2.46) In the case of this research, optimization is done on the basis of maximum manipulability as opposed to minimal maximum torque as shown next q,optimal = q(max w) (2.47) Such optimizations are all further explained in the next chapter for the 3DTP. 2.5 Trajectory Generation A trajectory is a function of time from a previously obtained path [Spo05]. A path only provides a sequence of points, which are named viapoints, on the path. For the case of the 3DTP, the viapoints are the initial and final points for each climbing step, including undocking and docking. Using cubic polynomial trajectories will provide discontinuities along the acceleration and as such are useless. Using quintic polynomials is then a logical choice for planning the trajectory, suggested by further research [Spo03, Atk78]. Therefore, if possible, it will be useful to use quintic splines along the computed paths. Trajectory generation is the intermediate step connecting the viapoints and a simulation over time. As such, it is critical to use the proper fit for the viapoints and to constrain it accordingly, since otherwise, the trajectory generator is free to choose trajectories that are well outside of the geometrical and physical capabilities of the manipulator. With the knowledge of the different concepts of path planning/optimization and trajectory generation it is then proper to continue to the next chapter in order to perform some preliminary simulations and generate a comparison database for optimized paths. CHAPTER 3 PRELIMINARY CLIMBING OF THREE DIMENSIONAL TEST PLATFORM (3DTP) ROBOT The previous chapter provided the theoretical background necessary for the development of a climbing algorithm exhibiting torqueminimal paths. In this chapter, three different climbing methodologies are explored as candidate methods of climbing for which optimal paths will be generated leading to corresponding trajectories. The general characteristics of each climbing method are quantified for downselecting each method used for further algorithm development. 3.1 Design and Configuration As previously stated, the 3DTP was designed to take the form of multiple climbing robot designs, in particular the 3DRP and 3DMP designs of reference [Cle03]. In doing so, one is able to readily switch between designs without experiencing a change in the link inertia tensors while investigating different climbing methods. The 3DTP has a six joint configuration where the first and last joint axes are perpendicular to the other four joint axes, which are in turn all parallel. When two 3DMPs are joined they form a four joint serial manipulator with all joint axes parallel (e.g. the 3DTP with the first and last joints locked). On the other hand, the 3DRP also has four joints, however, the first and last are perpendicular to the other two joints (e.g. the 3DTP with two intermediate joints locked). Figure 3.1 shows all three robots, 3DTP, 3DRP, and 3DMP demonstrating common features and differences. Y Sl / a12 834 S2 S3S3 U 23 S, (a) (b) Figure 3.1 Different hybrid mobility kinematical designs: (a) 3DTP on smart structure, (b) 3DRP joint axes and offsets [Cle03], (c) 3DMP joint axes and offsets [Cle03] For a numerical analysis, the 3DTP is modeled out of acrylic, which features the properties, E = 3.8 GPa (Young's modulus), v = 0.36 (Poisson's ratio), and p = 1190 ky3 (density). For 3DTP kinematics, the length parameters are specified as /im L0 = 0.05 m, L, = 0.05 m, L, = 0.15 m, L, = 0.075 m, L4 = 0.15 m, L, = 0.05 m, L6 = 0.05 m. These parameters render Table 2.1 as Table 3.1. The inertia parameters of the 3DTP mechanism shown in Figure 2.1, were computed by the ADAMS dynamic modeling software and are listed in Table 3.2. Table 3.1 Actual mechanism configuration parameters for robot iI Link lengths (m), a, Twist angles, a, Joint offsets (m), S, Joint angles, 0, 1 0 900 0.1 Variable 2 0 900 0 Variable 3 0.15 0 0 Variable 4 0.075 0 0 Variable 5 0.15 0 0 variable 6 0 900 0.05 variable 7 0 0 0.05 0 Table 3.2 Actual mechanism inertia parameters for 3DTP robot as computed by ADAMS Inertia tensor (kgm2 i Center of mass (m), rC,, Mass (kg), m Inertia tensor 1 [0 0 2.34*10 2] 2.64e2 1.02e5 1.01e5 1.62e6 2 [0.075 0 0] 7.17e2 5.56e6 1.13e4 1.12e4 3 [0.0375 0 0] 4.31e2 2.81e6 3.10e5 3.10e5 4 [0.075 0 0] 7.17e2 5.56e6 1.13e4 1.12e4 5 0 2.34*10 2 0T 2.64e2 1.02e5 1.62e6 1.01e5 6 [0 0 2.34*102]T 1.71e2 3.45e6 3.45e6 8.32e7 3.2 Kinematics In this thesis the path planning and trajectory generation assumes the scenario of the 3DTP robot climbing on a smart structure. The path can be segmented into docking and undocking from the smart structure sequences with an arbitrary path between two docking ports. The arbitrary path will be optimized using a manipulability based exhaustive search algorithm. For docking and undocking sequences, the first and last joints are set as 0, = 0 = +90 throughout the motion sequence. The remaining joints can be found by considering the mechanism as a closedloop kinematic chain and having four revolute joints parallel to each other. to this avail the kinematics reduces to the analysis of a fourbar mechanism, illustrated in Figure 3.2. Furthermore, the constraint 02 + 0 +04 +05 = 180 must be satisfied given = 0 + 90, 3 = 4 +3, and = 3 + 4. Refer to Figures 2.2 and 3.2. 66 ze / ZL 6 \ 2 = 60+90 0 FI lo Figure 3.2 Closeloop mechanism parameters Next, consider the closed loop equations 01, +12 cos +/3 cos +/4 cos s/56 = 0 (3.1) 12 sin 2 + 13 sin 3 + 14 sin 4 Ah = 0 (3.2) Setting one variable (such as 43) as a known parameter, the following derivations show that an instantaneous closedform solution can be obtained for 4 and 4, given 1,0 and 156 instantaneous values. These values are defined at incremental docking and undocking steps. Next, move the known parameters to one side of the equation as shown 12 COs 2 +/4 cos4 = / 56 1 1 cos 3 (3.3) 12 sin 2 +/4 sin = Ah 13 sin 43 (3.4) Define x (3) = 15 1, cos 3 and y () = Ah13 sin 43. Then solving for cos 2 and sin 2 x 4 COS4 cos2 = o (3.5) 2 y^ sin^ sin=2 y 4 sin4 (3.6) 12 Squaring and adding the previous two equations x2 +2 '+2214 (xcos +ysin4) cos2 2 + sin2 2 2 = 124 = 1 (3.7) 2 Therefore, x2 + Y2 + 2 2 xcos+ +ysin44 = 2 4 2 (3.8) 214 X2 y2 +2 2 Defining z(43)= 4 12 And considering that 2/4 z ( :) = xcos4 +y sin4 = x2 y cos 4 arctan j (3.9) One can easily solve for 4, as 4 (43) = arccos + arctan (3.10) y2)J Then referring to equations (3.5) and (3.6) 2 (43) = arctan ) (3.11) After obtaining 2 and 04 as a function of 43 only, configurations immediately prior to docking and undocking each end of the robot are obtained for a given 3 As a result, this derivation yields a set of feasible configurations and corresponding endpoints to the climbing robot's paths. This provides a simplification basis for the future algorithmic search for optimal manipulability. The Jacobian matrix for the 3DTP is found as specified in Chapter 2 and contains only elements due to revolute joints, therefore simplifying to 0 0o 0 0 Z ... Z6 The Jacobian matrix provides a basis for further optimization of climbing methods using the manipulability measure. Every time the Jacobian matrix is square, the definition of manipulability given in equation (2.34) applies. However, for certain climbing methods the 3DTP behaves as a four revolute joint manipulator, and as such there is additional redundancy. During such climbing sequences, the Jacobian must be reduced for meaningful manipulability measures and in that case it is necessary to use equation (2.33). The Jacobian matrix in equation (3.12) can be further expanded as shown in equation (B.1). For flip and rectilinear climbing, the Jacobian is reduced to the following J22 J23 J4 J25 Reduced =J32 J33 J34 J35 (3.13) 42 43 J44 J45 Which reduced equation (2.1) to q2 SJreducedq (3.14) q5 This Jacobian contains only the components affecting motion in the y, z, and 0, directions due to the 2nd, 3rd, 4th, and 5th joints. On the other hand, the Jacobian for side climbing is reduced to J11 J12 J13 J14 J15 Reduced = 31 J32 J33 J34 J35 (3.15) J51 J52 J53 J54 J55 Which once again reduced equation (2.1) to q, = Reduced 43 (3.16) _q5 Due to the motion in the x, y, and c6, directions imparted by the 1st, 2nd, 3rd, 4th, and 5th joints. Though depending on the climbing step, this motion may be due instead to the 2nd, 3rd, 4th, 5th, and 6th joints. 3.3 Dynamics The dynamics of the 3DTP, similarly to the kinematics, are based on the procedures described in Chapter 2. The inverse dynamics problem is easily formulated as in equation (2.30). This definition provides the joint torques necessary for the specified climbing maneuver and for the comparison of optimized climbing using manipulability. 3.4 Climbing Methods This section will explore the three climbing methods proposed. The first method to be explored is the flip climbing, followed by the side climbing, and finally the rectilinear climbing. The flip method was previously optimized using a quasidynamic method, thus not considering the concept of manipulability [Cle03]. The side method was also proposed in the previous research; nevertheless, it was not fully evaluated. To form a uniform basis for comparison, the 3DTP was required to climb the same distance in the same time interval for all climbing methods investigated. In the ensuing sections, each climbing method is evaluated without any optimization aside from the flip climbing using the method from previous research in order to set a level for comparison. 3.4.1 Docking/Undocking Maneuvers While docking and undocking, the 3DTP behaves as a closedloop mechanism as shown in Section 3.2. For every climbing maneuver, both 3DTP endeffectors are initially docked, followed by one of them going through an undocking maneuver, and then performing a climbing maneuver (e.g. moving the undocked end to a new docking location). At the end of this climbing maneuver, the free end is docked. These docking/undocking maneuvers are illustrated in Figure 3.3 and further summarized in Table 3.3 for the 3DTP designed using the parameters from Section 3.1. Table 3.3 Docking and undocking maneuver parameters for 3DTP robot simulations Docking Undocking Endeffector Parameters Docking Undockin Initial Final Initial Final S L 0.05 m 0.05 m 0.05 m 0.05 m 1, free end L' 0.05 m 0.025 m 0.025 m 0.05 m L, 0.05 m 0.05 m 0.05 m 0.05 m 2, grounded end  L6 0.025 m 0.025 m 0.025 m 0.025 m L, 0.05 m 0.05 m 0.05 m 0.05 m 1, free end L0 0.025 m 0.025 m 0.025 m 0.025 m L6 0.05 m 0.05 m 0.05 m 0.05 m 2, grounded end L6 0.05 m 0.025 m 0.025 m 0.05 m Docking ~~ ..b (a) (b) Figure 3.3 Configuration of endeffector 2 for 3DTP robot (a) docked and (b) undocked The L' and L, parameters refer to the distance from the first or last joint to the smart structure being climbed, respectively. In this fashion, the joint parameters can be found as in Section 3.2 and used to obtain a path and a trajectory from a docked to an undocked configuration and vice versa. For the purpose of consistency in comparing the different climbing methods, all docking and undocking maneuvers are assumed to take the same amount of time (i.e. 1 second for the simulated 3DTP). Such selection was made to minimize the number of variables that could cause an inconsistent evaluation and subsequent conclusion of the comparative results, especially when considering the rectilinear climbing maneuver. This maneuver performs several docking maneuvers as opposed to the other two methods, which only undock and dock once. 3.4.2 Flip Climbing Maneuver As stated in Chapter 1, and as previously demonstrated for the 3DMP robot [Cle03], flip climbing is a method of climbing for which the manipulator undocks one of its ends and follows a path by flipping to arrive at the next desired docking point. The method can be further visualized in Figure 3.4. The flip maneuver is bounded by the closedloop mechanism configurations illustrated by Figure 3.4 (a) and (c). In these two configurations, the corresponding joint angles are obtained from the four bar mechanism analysis detailed in Section 3.2. Table 3.4 provides the set of joint angles that bound the flip maneuver in the preliminary simulations. End effector 2 T m SEnd effector 2 (a) (b) (c) Figure 3.4 Flip climbing maneuver (a) Endeffector 2 postundocked (b) Sample sequence of flip steps (c) Endeffector 2 predocked Table 3.4 Flip maneuver boundary configurations for numerical example Endeffector 2 pre Endeffector Joint Endeffector 2 Endeffector 2 post Edefeco 2 e Endfcto docked (Fig. 3.4 2 docked angles docked (initial) undocked(Fig. 3.4 (a)) dcd (. finc (c)) (final) ,1 900 900 900 900 02 138.590 132.650 47.350 41.410 03 41.410 47.350 47.350 41.410 04 41.410 34.670 34.670 41.410 05 41.4100 34.670 214.670 221.410 06 0 0 0 0 3.4.3 Side Climbing Maneuver The side climbing maneuver is a method of climbing which mainly utilizes the motion of the first and/or last joints. In this thesis, the allowable docking points are assumed to be vertically placed above one another. Hence, this climbing method is constrained to maneuver the net vertical displacement as for the flip method. Alternate smart structures can potentially enhance the performance of this climbing method if intermediate docking ports were provided in vertically located but horizontally offset parallel to the current line of ports. This consideration is left for future research. /\ /1 /i &I *I / A f 9 1 '/I SI I II / / (a) (c) Figure 3.5 Side climbing maneuver (a) Endeffector 2 postundocked (b) Sample sequence of side steps (side view) (c) Endeffector 2 predocked A visualization of the side climbing method is shown in Figure 3.5, where Figure 3.5 (b) demonstrates the 900 to 900 rotation of the first joint showing intermediate configurations from the side view. Table 3.5 provides the set of joint angles that bound the side climbing maneuver used in the numerical analysis. Note, for the preliminary investigations of the side climbing method, the configuration of all inner joints is arbitrarily held constant. Upon introducing an optimization algorithm such as the manipulability based approach, these joint conditions are bound to change. Table 3.5 Side maneuver boundary configurations Endeffector Endeffector Joint 2 d ed Endeffector 2 post Endeffector 2 pre 2 docked angles docini d undocked (Fig. 3.5 (a)) docked (Fig. 3.5 (c)) final (initial) (final) 81 900 900 900 900 02 138.590 132.650 132.650 138.590 03 41.410 47.350 47.350 41.410 04 41.410 34.670 34.670 41.410 05 48.590 55.330 55.330 48.590 06 0 0 0 0 3.4.4 Rectilinear Climbing Maneuver The rectilinear climbing maneuver is the most complex method of climbing, but in turn it may prove the most efficient method. For this maneuver the manipulator utilizes only the four inner joints, therefore, once again, redundancy is an issue. For comparison, the maneuver, from initial docked configuration to final docked configuration, is generated so that it takes the same time span as that taken when using the other climbing methods. The rectilinear maneuver consists of two cycles of climbing steps to achieve the same goals as that defined for the other climbing methods. As illustrated in Figure 3.6 (steps (a) through (e), and steps (g) through (k)). In the simulated cases, the endeffector is raised half the initial distance between the two docked endeffectors (Figure 3.6 (b)). Endeffector 2 then docks (Figure 3.6 (c)), followed by a release of endeffector 1 (Figure 3.6 (d)). The robot continues to climb by raising endeffector 1 the user defined height once again (Figure 3.6 (e)). Endeffector 1 docks at the new location. This sequence of climbing maneuvers is repeated until the final position is reached, as in Figure 3.6 (k). For analysis, the numerical example used only two sequences of the climbing maneuvers. In addition to the initial postundock and final predock configurations shown in Figures 3.6 (a) and (k), respectively; the rectilinear climbing yields additional boundary configurations, listed in Table 3.6. These configurations provide boundaries for all climbing steps and optimizations. Table 3.6 Rectilinear maneuver boundary configurations for 3DTP robot Endeffector Endeffector 2 pre Endeffector 2 Endeffector 1 post 1 ped ed Joint angles docked (Fig. 3.6 (b) docked (Fig. 3.6 undocked (Fig. 3.6 3.6 (e and (h)) (c) and (i)) (d) and (j))d ()) and (k)) 18 900 900 900 900 02 86.840 104.480 123.710 145.330 03 93.160 75.520 56.290 34.670 04 56.290 75.520 93.160 47.350 05 56.290 75.520 93.160 47.350 6 0 0 0 0 The initial undocking and final docking steps are not shown in Table 3.6 as in the previous methods due to space constraints. However, they are the same (or mirror the values) of the corresponding configurations listed in prior Tables 3.4 and 3.5. 3.5 Optimization via Exhaustive Search Throughout this investigation, exhaustive search methods were used to find the optimal manipulability configurations. This method of finding the maximum manipulability configurations is based on searching throughout the robot's workspace, that is, searching the entire range of motion for each joint combination in the form of q,,mm < q ma ., where q: = q 1 + (q,, qO = q, mm q, = q, max In this fashion, solutions can be refined by decreasing the step size 8q, for finding different joint values at the cost Ai 'I 'I 'I 'I~ 4 'II 'Ic 'II 'i 'I Figure 3.6 Rectilinear climbing maneuver (a) Endeffector 2 postundocked (b) through (j) steps including intermediate dockings (k) Endeffector 1 predocked (k) Sample sequence of rectilinear 'i A, /1 /i  (g) of increasing computation time. This section details the different optimization scenarios used throughout this investigation. 3.6 Flip Climbing Optimization via Minimal Torque Approach The flip via the minimal torque approach was performed as a basis for comparing all other methods. The 3DTP used in this method is simulated with motors from Hitec RCD USA, Inc, model number HS645MG, which can output a maximum torque of 0.9413 Nm and has a mass of 0.055 kg each, therefore modifying the mass and inertia properties in Table 3.2 to those of Table 3.7. For this method, an exhaustive search was performed initially in the post undocking position to find an optimal configuration using the closedloop form of the mechanism, varying 3. Secondly, the same method was used to find the optimal configuration at predocking. Afterwards, these two configurations set boundaries for an intermediate step which involved performing an exhaustive search in the serial mode of the mechanism varying 03, 04 and 05 to determine the climbing configuration yielding minimal (max joint torques) at the midpoint of joint motion 0,. Table 3.7 Actual mechanism inertia parameters for 3DTP robot including motors Inertia tensor (kgm2) i Center of mass (m), rcm, Mass (kg), m, Inertia ten 1 [0 0 2.34*102]T 8.14e2 3.37e5 3.37e5 1.25e5 2 [0.075 0 0] 0.182 2.73e5 7.53e4 7.53e4 3 [0.0375 0 0] 4.31e2 2.81e6 3.10e5 3.10e5 4 [0.075 0 0] 0.182 2.73e5 7.53e4 7.53e4 5 0 2.34 *10 2 T 8.14e2 3.37e5 3.37e5 1.25e5 6 [0 0 2.34*102T 1.71e2 3.45e6 3.45e6 8.32e7 Table 3.8 Flip climbing minimal torque approach optimal configurations for 3DTP robot Joint Endeffector 1 post Flip step 1 (Fig. Endeffector 1 predocked variable undocked (Fig. 3.4 (a)) 3.4 (b)) (Fig. 3.4 (c)) q, 900 900 900 q2 132.160 900 47.840 q3 92.840 77.370 92.840 q4 32.230 26.860 32.230 q5 12.770 154.360 167.230 q6 0 0 0 3.7 Climbing Trajectories The preliminary joint trajectories for climbing along the smart structure according to each method are presented in this section, laying the groundwork for the optimization performed in Chapter 4. Using the boundary configurations defined in previous sections and intermediate steps as in the case of the minimal torque optimized flip method, the joint trajectories are generated using quintic splines. It is important to note that using quintic spline fits between these given configurations will provide continuity in the position, velocity, and finite accelerations of the resulting trajectories. However, if the difference in viapoints is large, the results can be physically limiting since the joints will experience a wavy maneuver through undesired configurations, such as going through the smart structure or through manipulator links as can be seen in Appendix C, Figures C. 1 through C.3. To this avail in this thesis, more viapoints are obtained in between the ones optimized by simply using linear interpolation between the boundary configurations. This method is further named segment linearization since segments of data are interpolated linearly. The method of segment linearization reduces the joint power required to perform the maneuver, however, in some cases this is at the cost of increasing joint torques as seen in Table 3.9 with a combination of results exposed in Chapter 4 and results obtained without segment linearization. Table 3.9 Effects of segment linearization on joint maximum torques and maximum net power of different 3DTP manipulability path optimized trajectories Climbing method No segment linearization Segment linearization Flip 0.95179 1.0641 "max (Nm) Side 1.091 1.0137 Rectilinear 0.46016 0.40237 Flip 0.83289 0.26146 Pnetmax (W) Side 1.9589 0.1274 Rectilinear 0.18787 0.12837 As it can be seen, without segment linearization, in addition to the trajectories obtained being impossible to perform due to physical constraints, the maximum net power is quite large. When segment linearization is applied, though not constantly, joint torques decrease slightly, while net power decreases considerably in all methods and the resulting trajectories are realistic as will be seen in ensuing chapters. Afterwards, this denser amount of viapoints is interpolated using quintic splines in order to obtain a joint trajectory and the corresponding velocities and accelerations. The inverse dynamics are then used to obtain the corresponding joint torques by applying the interpolated trajectories. The instantaneous values for manipulability are also obtained. From this pool of data generated, several comparisons are drawn regarding the maximum joint torque, the maximum joint power, the maximum overall power required, the effect of gravity (for comparison with joint torques and with the quasidynamic optimization method [Cle03]), and the variation of manipulability with respect to time. 3.7.1 Minimal Torque Optimized Flip Trajectory The following figures demonstrate trajectory parameters interpolated. 49 Flip  ' 150 1 1 0 0     .. . . .       5 .. ........ 4... ...... .. 4 O0       0 i 50 0    100 ......... .  ........................ .............. 0 5 10 15 20 25 30 Time (sec) Figure 3.7 Minimal torque optimized flip trajectory of 3DTP robot: joint angles Flip 1 5       .......... 3 1  1 t1 S      05 ................ .. 0 l.  5  I , . . _1  . j   . :. 0 5 10 15 20 25 30 Time (sec) Figure 3.8 Minimal torque optimized flip trajectory of 3DTP robot: joint velocities 2 .  ., J  . . I        I I I SI .     y    5_ .. .... .. .. .... .. .. .... .. .. ..J ... .. .... .. .. .. .. .. .... .. .. ... ... . ... .. ........  ... .3 I 4 " 1 02 .......... CL3 i a5 06 0 5 10 15 20 25 30 Time (sec) Figure 3.9 Minimal torque optimized flip trajectory of 3DTP robot: joint accelerations The joint torques were computed using the inverse dynamics as mentioned beforehand. The manipulability measure was also computed along these joint trajectories. Additional measures for comparison were also computed from these trajectories and are shown in the following set of figures. Flip, gmax=1.1061 Nm 0 8   . . . ... 06 .   ,   .. . .. . _ S .. ........... 0 5 10 15 20 25 30. ..... .. ......... 02   0 5 10 15 20 25 30 Time (sec) Figure 3.10 Minimal torque optimized flip trajectory of 3DTP robot: joint gravity effects 9g 92 .......... 93 94 95 g6 51 Flip, Tmax=1.1059 Nm I iI 1 .... . ..  ... ............ .. ......... ................ ................ 1 g 0 .2 . .. . . .. . .. . .. . . r^ ^ \   . .  . . . . 22 0.2          02      0   02 ...... ..... 02 / ;  r .: ; ^ ,.^ _ 0 5 10 15 20 25 30 Time (sec} Figure 3.11 Minimal torque optimized flip trajectory of 3DTP robot: joint torques Flip, Pmax=0.35917 W 035 P       : P2 .......... P3 i P4 0.25   . .     P 0 02 P6 P n 0.15 .  .  .  0.1   .. ., "'..     .  005          01 .. ..... 0 5 10 15 20 25 30 Time (sec) Figure 3.12 Minimal torque optimized flip trajectory of 3DTP robot: joint power Net power for 3DTP climbing methods 15 Time (sec) Fip, Pnet, m 074262 W Figure 3.13 Minimal torque optimized flip trajectory of 3DTP robot: net power Manipulability measures for 3DTP climbing methods 0.045 i F ipl 0.045 0.04  0.035  E 0.03 S0.025 0.02  0.015 001  0 5 10 15 20 25 30 Time (sec) Figure 3.14 Minimal torque optimized flip trajectory of 3DTP robot: manipulability As it can be seen, using the minimal torque optimization, the maximum joint torque exceeds the capacity of the selected motors. A failed attempt to clarify this issue and try to minimize the torque was done by extending the time span for the simulation. However, only the power requirements decreased. In addition, the joint torques at the specific instant where the maximum occurs did not decrease due to the fact that there will always 53 be a need to flip the arm. Furthermore, the manipulator configuration can only be realistically contracted up to its geometric limits; therefore, maximum reduction of the moment arm is not possible in real life. Even though the results indicate that the current motors will not be able to perform the climbing task by flipping, this optimization is a valid base for comparison with other methods. 3.7.2 Preliminary Flip Climbing Preliminary flip climbing refers to climbing using only the boundary conditions with linear interpolation in between those boundary conditions to constrain oscillatory (wavy) motion from resulting in the quintic spline trajectory generation. This means that no optimization is done whatsoever. The trajectories for these preliminary simulations are shown in the following figures, followed by computed joint torques and other useful measures of the flip climbing method. Flip I I Ia 100 2 S6.......... 6 60    . . .. . 0 ....'"" 5 0       L    100      0 5 10 15 20 25 30 Time (sec) Figure 3.15 Preliminary flip climbing of 3DTP robot: joint angles 0I1 0 1 .    . .  0 05   I I  Figure 3.16 Preliminary % 3 Figure 3.17 5 10 15 20 25 Time (sec) flip climbing of 3DTP robot: joint velocities 0 5 10 15 20 25 Time (sec) Preliminary flip climbing of 3DTP robot: joint accelerations a1 02 .......... C 3 "4 C5 CO CaC "2 .......... cc3 O5 %6 E z 7, Figure 3.18 P z E F Figure 3.19 Flip. 9max=1.1551 Nm 0.8 1         0     0 6  0o     U    .'   " I  _ 0 2  0.2  t 1 I    0 5 10 15 20 25 3( Time (sec) Preliminary flip climbing of 3DTP robot: joint gravity effects Flip, tmax=11551 Nm 1 6 04* * 02         0.4 _     I  .  04 .;  ......................................   0. ^ 0.2 . 1r  r   0 5 10 15 20 25 3( Time (sec) Preliminary flip climbing of 3DTP robot: joint torques 91 92 .......... 93 94 96 g6  12 "6 56 Flip, Pmax=0061418 W 0.06     P 0.05      .......... P P 0.04     P SP6 .00 3 " ...... , 0.03   . ..  "*. ...    0.02 0.02         . 0 o  ." ^ r^: ..... .', : %'*,: ... 0 5 10 15 20 25 30 Time (sec) Figure 3.20 Preliminary flip climbing of 3DTP robot: joint power 3.7.3 Preliminary Side Climbing In this section, simulation results are presented for the preliminary side climbing method. Side 60     60   ;. . . B 40 40 <       0   20   S0    20   .8 0        .  .  .  .  .  .    100  ,      1 00  T    I H   LL 0 5 10 15 20 25 30 Time (sec) Figure 3.21 Preliminary side climbing of 3DTP robot: joint angles 57 Side 01  a 01 0    " , " 2" 0 0 .. *. r .. .. ............... ............... ............ .. 0.05     0   0.1  T  ,  1 .  I I I 0 5 10 15 20 25 30 Time (sec) Figure 3.22 Preliminary side climbing of 3DTP robot: joint velocities Side I I I 0.3       a 02    01      a 01  ja o.2 .. **. ....... r.. ......... 4, I 01 . S 02    S03  ~~ 4~     0 5 10 15 20 25 30 Time (sec) Figure 3.23 Preliminary side climbing of 3DTP robot: joint accelerations 58 Side, gmax=0.86389 Nm 02 I I 0.2    .  I..  g 501 1   0.2 . . ; ...............,...............,.............. ,......... .... "1 94 5 0 4  . 06 S0.6 7  04  I  .  . ^ ^    .. ... .. .. 08  0 5 10 15 20 25 30 Time (sec) Figure 3.24 Preliminary side climbing of 3DTP robot: joint gravity effects Side, T =086389 Nm 0 .2 F. .  ,    0.1       2 0 . . .. 3 0.1           0    > 0 5   ^     .... ..   07   Ee0 6  02k 0.8  0 5 10 15 20 25 30 Time (sec) Figure 3.25 Preliminary side climbing of 3DTP robot: joint torques 59 Side, Pmax=0.096928 W P 0.09 0.07 4       P4 P 0.05 o 5.... .. .P. . a 0.04  0.03  0.02       0 01      0.02 j   I  I   IV :"' 004     .  .  0o *******.  .* .* . *** ***.  0 5 10 15 20 25 30 Time (sec) Figure 3.26 Preliminary side climbing of 3DTP robot: joint power 3.7.4 Preliminary Rectilinear Climbing This section presents the trajectories and simulation results of the 3DTP climbing via the rectilinear method. Rectilinear 50  5 E 504 iI I S100 '*i,  . ^"   ':   .I    :'. _ 3 100 ......l.... . 100~ L L,  0 10 15 20 25 30 Time (sec) Figure 3.27 Preliminary rectilinear climbing of 3DTP robot: joint angles 60 Rectilinear 03  ... 0.2   ....... ....... ;........... O 3 .0 1         I I4 013     I      C0 . .................. . . 02 Sa...... .. 0 53               ; ;  1  0 5 10 15 20 25 30 Time (sec) Figure 3.28 Preliminary rectilinear climbing of 3DTP robot: joint velocities Rectilinear I 05   l .   .l [..JC I'      a 1        I    0 5 10 15 20 25 30 Time (sec) Figure 3.29 Preliminary rectilinear climbing of 3DTP robot: joint accelerations z 0 Figure 3.30 Figure 3.31 Rectilinear, gmax=0 34146 Nm I     .     ''   . . . ... .. . . . .. . .  ^                         0 5 10 15 20 25 30 Time (sec) Preliminary rectilinear climbing of 3DTP robot: joint gravity effects Rectilinear, T m=0.34148 Nm U    .   U        1~ I i i 0.1      0.2    02 ~   .,   0 3   .. . :    . .. ..... . 0 5 10 15 20 25 3 Time (sec) Preliminary rectilinear climbing of 3DTP robot: joint torques 91 92 ....... 93 94 95 96 1 62 Rectilinear, P x=0.12857 W 0.12 .. . .  I.. .. I.. . ....   ....  P1 0.12   : : 4 008    P   5 P 006   ............ 0 5 10 16 20 25 30 Time (sec) Figure 3.32 Preliminary rectilinear climbing of 3DTP robot: joint power 3.7.5 Further Comparison Results on Preliminary Climbing For the trajectories presented in the above sections, the following plots are used to further delineate the comparison of the different climbing algorithms. Additional data was obtained regarding the maximum instantaneous power for different climbing methods as shown in Figure 3.33. Net power for 3DTP climbing methods 0.03 : .............. i .......... Rectilinear, P 0.36309 W 0.25 02 002 .1" 001 01 0 05i t 0 5 10 15 20 22 30 Time (sec) Figure 3.33 Preliminary climbing of 3DTP robot: net power Figure 3.33 Preliminary climbing of 3DTP robot: net power In this plot it can be clearly seen that the rectilinear climbing method is the one that will require a higher power input. Additionally, for further exploration, a plot of the manipulability measures for the three climbing methods is shown in Figure 3.34. Manipulability measures for 3DTP climbing methods 0.05 Flip ........ Rectilinear 0.04  S 0.035 \ ............... S0.03 E 0.025  0.02 S 0.015  0.01  0.005  0 I I 0 5 10 15 20 25 30 Time (sec) Figure 3.34 Preliminary climbing of 3DTP robot: manipulablity 3.8 Preliminary Observations A summary of the critical information from the previous sections is provided in Table 3.10. As it can be seen in this summary, the flip method is the least efficient method, in fact, its climbing ability using the motors available (refer to Section 3.5) is questionable. Net power requirements, however, are significantly less for the flip method than for the rectilinear method, while the side method has the lowest of all power requirements and an intermediate torque requirement. Compared to the torque optimization it can be seen that all methods but preliminary flip climbing perform better overall. Table 3.10 Summary of preliminary simulation results for 3DTP robot climbing Method I max (Nm) Pmax (W) I etmax (W Flip (minimal torque optimization) 1.1059 0.35917 0.74262 Flip 1.1551 0.061418 0.11075 Side 0.86389 0.096928 0.096928 Rectilinear 0.34148 0.12857 0.36309 These observations reflects in the design process in the ability of the robot to be autonomous and completely able to roam and inspect by itself, for if maximum joint torque is well below the maximum provided by the motors, climbing can be easily achieved with additional loads (e.g. batteries, sensors, et cetera). On the other hand, higher net power translates into bigger batteries to run the robot the same amount of time. Overall, the rectilinear method is the method of choice for climbing a smart structure, though it may require larger batteries. The side method is feasible but cannot carry a large load if it can carry a load at all. The flip method is the least realistic of all. In the next chapter, the optimization of all these methods using the manipulability measure will be explored to investigate whether or not the above conclusions of this chapter will remain valid upon application of the optimization algorithms. CHAPTER 4 OPTIMIZED CLIMBING OF THREE DIMENSIONAL TEST PLATFORM ROBOT One of the goals of this investigation is to analyze the effects of maximizing manipulability throughout the path being climbed and observe the effects on joint torques. A comparison was done to previous research [Cle03], where quasidynamic exhaustive searches were done to find optimal maximum torque configurations throughout the workspace of a 3DMP. Additionally, viapoints selected from the resulting minmax torque configurations were used for trajectory planning. In doing so, an important issue observed throughout the trajectory planning process is exposed regarding the number of viapoints chosen and oscillations on the curve fit. 4.1 Docking/Undocking Optimization Using solutions found via closedloop analysis, as presented in Chapter 3, the maximum manipulability configuration (q when w = wma ) defines optimal docking/undocking configurations. Docking/undocking optimization was done using an exhaustive search as introduced in Chapter 3 considering 180 < <180', where 3 = 03 (0, 03) per the previous chapter, and 53 = 1. In addition, the feasible solution space is reduced by the introduction of the physical constraint that the mechanism cannot pass through the smart structure being climbed. Considering the steps throughout this method, the resulting constrained optimal configurations for predocking and post undocking are shown in Table 4.1. Table 4.1 Optimal manipulability configurations of 3DTP robot at postundocking and predocking steps Postundocking Predocking Method and description Joint variables Postundocking Pred in (Ah = 0.3 m) (Ah = 0.3 m) q1 900 900 q2 134.210 45.790 Flip climbing and side q3 40.79 40.790 climbing cases, end effector 2 free '4 41.42 41.42 q5 36.420 143.580 q6 0 0 Postundocking Predocking (Ah = 0.3 m) (Ah = 0.15 m) q, 900 900 q2 134.210 94.970 Rectilinear climbing case, q3 40.790 76.030 endeffector 2 free q 41.420 74.200 q5 36.420 65.200 q6 0 0 Postundocking Predocking _(Ah = 0.15 m) (Ah = 0.3 m) q1 900 900 q2 114.800 143.580 Rectilinear climbing case, q3 74.200 41.420 endeffector 1 free q4 76.030 40.790 q5 85.030 45.790 q6 0 0 4.2 Flip and Rectilinear Climbing Optimization Flip and rectilinear climbing share a common workspace where the Jacobian uses only its y, z, and oa components (refer to Figure 2.2, equation (1.1), and equation (3.13)), while only being actuated by the second through fifth joints. Such workspace reduces the Jacobian to a threebyfour matrix. In order to optimize for flip climbing, the joint angles where searched in the following ranges, 134.21 < 02 <45.79, (40.79 360) < 03 < 40.79, 41.42' 360 < 04 < 41.42', and 143.58 360 <05 < 143.58 with the steps 802 = 22.103", 803 = 5", 804 = 20', and 805 = 40. The first step ensures that 02 stays within the range provided by postundock and predock configurations provided in Table 4.1, while obtaining optimal configurations at three intermediate 02. The remaining steps provide for full rotations of the corresponding joint variables in order to maximize manipulability performing the exhaustive search. In this fashion, the intermediate configurations are found as shown in Table 4.2 for flip climbing. Table 4.2 Optimal manipulability configurations of 3DTP robot at select intermediate steps for the flip climbing maneuver Joint variable q = q2 q q2 q2 = q2 q, 900 900 900 q2 134.210 900 67.900 q3 40.790 44.210 44.210 q4 41.420 58.580 58.580 q5 36.420 103.580 63.580 q6 0 0 0 Rectilinear climbing is a more complex method for optimization. Referring to Figure 3.6, the first intermediate steps to be optimized are between Figures 3.6 (a) and (b). In these steps the joint angles where searched in the ranges given by, 134.21 < 02 < 94.97', 76.03 < 03 < 40.79, 74.20 < 04 < 41.42', and 36.42 < 0, < 65.20 with the steps 802 =19.62, 303 = 0.49", 04 = 1.82, and 8,0 = 3.20 These steps provide a desired range of operation for the manipulator throughout the rectilinear climbing maneuver, obtaining one very accurate configuration for maximum manipulability in between the undocking/docking steps. The second intermediate steps to be optimized are between Figures 3.6 (d) and (e). In these steps the joint angles where searched in the ranges given by, 143.58 < 02 < 114.80, 74.20 < 03 < 41.42', 76.03 < 04 < 40.79", and 45.79 < 0, < 85.03 with the steps 802 = 14.39", 803 = 0.46', 804 =1.96', and 805 = 4.36'. The optimal configurations repeat again as shown in Figure 3.6. Table 4.3 shows the two optimal intermediate configurations found. Table 4.3 Optimal manipulability configurations of 3DTP robot at select intermediate steps for the rectilinear climbing method Des n Intermediate climbing step between Intermediate climbing step between Description ____ Fig. 3.6 (a) and (b) Fig. 3.6 (c) and (d) Joint 1 I q2 =2 2 = 42 variable 2 2 2 2 q, 900 900 q2 114.590 129.190 q3 51.070 46.430 q4 50.530 52.540 q5 62.000 80.670 q6 0 0 As it can be seen, only one intermediate configuration was found for the rectilinear climbing maneuver as opposed to three in the flip climbing maneuver. The reason for this is due to the range of motion necessary to go from one postundocked position to the next predocking position in both maneuvers. Additionally, it can be easily observed that in the flip climbing maneuver, while performing the exhaustive search, the joints are searched through a whole revolution in order for the optimal manipulability configuration to be found. On the other hand, for the rectilinear climbing maneuver, the joints are only searched in the range of motion provided by boundary postundocking and predocking optimized configurations. This reduced configuration search space was necessary to avoid collisions against the smart structure. While performing the flip operation it can also be observed that collision with the smart structure does not occur for all possible configurations of joints 3, 4, and 5. Thus, due to the geometry of the 3DTP in flip motion full joint revolutions in the named joints is possible. 4.3 Side Climbing Optimization Side climbing, unlike flip and rectilinear climbing, has a reduced Jacobian shaped by a workspace based on x, z and oy,, and joints 1 through 5 or 2 through 6, depending on the free and grounded endeffector (in this case, 1 through 5, because endeffector 1 is simulated as grounded). In side climbing the joint angles are optimized in the ranges of 90 < ,0 < 90, 134.21 < 02 <134.21 +360, 40.79 <03 < 40.79 +360, 41.42 < 04 41.42 + 360', and 36.423 360 < 05 < 36.423 The steps for these ranges are 80, = 45', 802 = 15', 803 = 15', 804 = 30', and 803 = 30 The optimal manipulability configuration found with these steps is given in Table 4.4. Table 4.4 Optimal manipulability configuration of 3DTP robot at intermediate step for the side climbing maneuver Joint variable q = q1 q1 00 q2 224.210 q3 34.210 q4 48.580 q, 96.420 q6 0 70 4.4 Trajectory Generation Using Manipulability Optimized Paths The trajectories for the optimized paths are generated similarly to the preliminary simulations. The segments where data is absent in between optimized viapoints are populated by linear interpolation. This is followed by utilizing quintic spline interpolation to guarantee continuity in positions, velocities and accelerations between the optimized and the additional segment viapoints. The results obtained in this fashion are shown graphically in the subsequent sections. 4.4.1 Flip Climbing Method The trajectory obtained for the flip method and the ensuing simulations are shown in this section. As it will be seen, the results lack the symmetry of preliminary simulations. Flip 100     Z 2 100  0 ... .. ... .. .. . .. ."" 0 ......... ... ................... .................................... .. . .. . . . . .50 " I   I ::*   100  T  .. . rII I I 0 5 10 15 20 25 30 Time (sec) Figure 4.1 Path optimized flip method for 3DTP robot: joint angles 71 Flip 0.25  025    I    .O 2 0.15  4 2 ) 00, 0.05 :0.05    .      .... ..... i.. .... 0 05 0 1 0.15     0 5 10 15 20 25 30 Time (sec) Figure 4.2 Path optimized flip method for 3DTP robot: joint velocities Flip 0.8  106  T    I  I    " S::::::::::::::::::: 0.4 .. . S0.4 .............. I      .    o M  0 .  1.  T.  I  0 5 10 15 20 25 30 Time (sec) Figure 4.3 Path optimized flip method for 3DTP robot: joint accelerations 4.4.2 Side Climbing Method The side climbing method trajectories generated from the optimal path previously obtained are shown in this section. 72 Side 50S        ._8 i 4 0   L   0..... ...... 1 0 .. . :    .,     50  200  I I I I I 0 5 10 15 20 25 30 Time (sec) Figure 4.4 Path optimized side method for 3DTP robot: joint angles Side 01   I  I  0. .................. ..... ... ........................... .... u i .......... 03 006    0) CO S002      II I   002     S0.02   00     0.02 ...0.04 ..... .................. 0 5 10 15 20 25 30 Time (sec) Figure 4.5 Path optimized side method for 3DTP robot: joint velocities 73 Side 3  I I  I a a2 I~as 2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..       ..................... ....................... c1 3 2  .. ... .. . ......... .".  1  .   3  .   I I     I 0 5 10 15 20 25 30 Time (sec) Figure 4.6 Path optimized side method for 3DTP robot: joint accelerations 4.4.3 Rectilinear Climbing Method This section depicts the trajectories generated via the optimal configurations obtained for the rectilinear climbing method. Rectilinear 50        S............................ 3 28 100 AE 6     100         0 5 10 15 20 25 30 Time (sec) Figure 4.7 Path optimized rectilinear method for 3DTP robot: joint angles 74 Rectilinear 02               O   0.0 . .... .. 0 5 10 15 20 25 30 S .5    :::   .  ;:  0 2 .. . .. . . . .. . .. . ..... . . .. . . . 0.15 II  I I   02      . .0 0      0 5 10 15 20 25 30 Time (sec) Figure 4.8 Path optimized rectilinear method for 3DTP robot: joint velocities Rectilinear SI I i 0.6 , , 0        0    I.I 0 5 10 15 20 25 30 06    r    I     o 5 10 15 20 25 30 Time (sec) Figure 4.9 Path optimized rectilinear method for 3DTP robot: joint accelerations 4.5 Inverse Dynamics Simulation As in the previous chapter, once the trajectories exist, then it is possible to perform inverse dynamics simulations in order to obtain joint torques and other measures dependent on these. As mentioned, a Lagrangian approach is used to obtain the equations of motion from which the joint torques are calculated along the resulting trajectories at each instant. The results of such simulations are shown in the plots following. Flip, gmax=1.0632 Nm 1 .      . .  g91 92 0 8     ...   .......... g93 zI o \ .94    g 0    L'''  02   ,    0.2 ............... .     ............ 02 7 7  i I r 1  " 0 5 10 15 20 25 30 Time (sec) Figure 4.10 Path optimized flip method for 3DTP robot: gravitational load effects Flip, rmax=1.0641 Nm 1     1 T2 S  ............ T4 0 5 10 15 20 25 30 0    : : .  ..............  Time (sec} Figure 4.11 Path optimized flip method for 3DTP robot: joint torques E z 76 Side, gmax=10135 Nm 0 2    i 1   1.2 ...  ............................. ,,  ........ 0.4 ............. i ............. 4' ............. i .............. .............. . 0.4..............i ............. ..........................i............ ..... 1 . . 0.27 02 0 5 10 15 20 25 Time (sec) Path optimized side method for 3DTP robot: gravitational Side, max=1.0137 Nm ... 9g 92 .......... g 95g4 g6 30 load effects z Figure 4.13 02     08 . U  ..  ........ ........ 0.4       0.6      /              0 5 10 15 20 25 3 Time (sec) Path optimized side method for 3DTP robot: joint torques  T 1: 77 Rectilinear, gmax=0.40067 Nm 094 0.1      2 012        g 0.3     , .......... . I I 5 i 0 0. 0 1  .', ','    0   E, 0 .2 *, " .. .. .. . .. .. ... .. .. .  .,:'"   . _.   . . .  .. . 0 5 10 15 20 25 30 Time (sec) Figure 4.14 Path optimized rectilinear method for 3DTP robot: gravitational load effects Rectilinear, max=0.40237 Nm 04  1 0.3        12 0 1.......... U ~~'i ~ ~ ~tL: U? I 6 .0 1  . u   j  0.2   't    0.3      ';    0 6 10 15 20 25 30 Time (sec) Figure 4.15 Path optimized rectilinear method for 3DTP robot: joint torques Manipulability measures for 3DTP climbing methods 0.05 5 Flip 0.045 _, .... .. ........... Side .......... Rectilinear 0.04 , S 0.035 ...... S0.03 E 0.025  I 0.02 S 0.015  0.01  0.005  0 I I 0 5 10 15 20 25 30 Time (sec) Figure 4.16 Path optimized 3DTP robot climbing methods: manipulability As it can be seen from the previous set of results, throughout all climbing methods gravity has the most effect on joint torques. This is an indicator that for optimization to be more efficient the effect of gravity must be taken into account. In this thesis only the kinematic manipulability was used for optimization, its value over time shown in Figure 4.16. Some may argue that the dynamic manipulability index may have improved performance, but the previous results prove that the largest opposing element in climbing is gravity. Additionally, it is observed that after being path optimized, the rectilinear method continued to outperform flip and side climbing in keeping minimalmaximum torques. Such performance indicates that the rectilinear climbing method is the most capable method for autonomous climbing due to the ability to carry additional loads such as batteries and sensors. Power requirements, another measure of climbing method performance was also obtained for the optimized paths and is shown in the following plots. 79 Flip, Pma=0.11335 W 0.1 ~~    j  01 .08 ... 0.08      . 0 .06   I I  .  0 .. 0 5 10 15 20 25 30 Time (sec) Figure 4.17 Path optimized flip method for 3DTP robot: joint power Side, Pmax=011374 W a 0.1  .         _ 008      S  .. 0.06  f   ;  004 _.y . .  I   0.02 0 0 5 10 15 20 25 30 Time (sec) Figure 4.18 Path optimized side method for 3DTP robot: joint power P1 Pi P2 P4 P1 P2 .......... P3 P4  P4 P6 80 Rectilinear, Pmax=00734 W 0.07  ......................................... ............. P SP2 r P4 004 .P 0.04    .     ................. 0.02         0.01     .  0 0 5 10 15 20 25 30 Time (sec) Figure 4.19 Path optimized rectilinear method for 3DTP robot: joint power Net power for 3DTP climbing methods 0.25 Flip, Pnet, m =026146 W Side, Pnet, max=01274 W S........ Rectilinear, Pn, =0.12837 W 02 0.15 S.y i ..*" 0.1 1 0.05 0 0 5 10 15 20 25 30 Time (sec) Figure 4.20 Path optimized climbing methods for 3DTP robot: net power Comparing all results with those of Chapter 3, it can be observed that after the path was optimized, the overall power requirement decreased for the rectilinear motion and localized decreases were observed for flip motion. The side climbing method exhibited an increase. The flip preliminary results indicated lower power requirements over the trajectory overall with the exception in the region (around 25 seconds) where the combination of the joint configurations resulted in the 3DTP acquiring a flipped configuration. Overall; however, the maximum values for each method may have increased for all methods except for the rectilinear which saw a significant reduction in maximum power required. 4.6 Observations From the path optimized results, it can be seen that manipulabilitybased optimization did not consistently yield improved torque characteristics and maximum power results throughout the generated trajectories. Further observing, it can be seen that in fact, the path optimization provided larger maximum torques in some cases. To further delineate these observations, Table 4.5 provides a summary of the results obtained. Table 4.5 Summary of preliminary, torque, and path optimized results for 3DTP robot Climbing method max (Nm) Pmax (W) Pjetmax (W) Optimization Flip 1.1059 0.35917 0.74262 Side Minimal torque Rectilinear Flip 1.1551 0.061418 0.11075 Side 0.86389 0.096928 0.096928 No optimization Rectilinear 0.34148 0.12857 0.36309 Flip 1.0641 0.11335 0.26146 Side 1.0137 0.11374 0.1274 Maximum manipulability Rectilinear 0.40237 0.0734 0.12837 Of all the climbing methods, the flipping method had less constraints on joint motion over which optimization could occur. As such Figures 4.16 and 3.34 indicate trajectories with greater ranges of high manipulability values. Side climbing on the other hand, the manipulability exhibited decreased values. As it can be seen from these results, of all the climbing methods, by using manipulability only flipping had a large enough range to be optimized properly. Additionally, considering the initial manipulability measures from the preliminary simulations, flip climbing had a larger range for manipulability. This facilitated the optimization. Side climbing on the other hand had a fairly steady manipulability, and as such when optimization was performed, the deviations from those values was towards singularity or w = 0, thus increasing joint torques. Rectilinear climbing was largely optimized and in fact, the small optimization achieved increased the joint torques. Regarding resulting power conditions, the maximum power required to achieve the maneuver decreased considerably for the rectilinear climbing method while for the other methods increases were observed. With these results in mind, conclusions of Chapter 5 are made on the effectiveness of path optimization via the manipulability measure and its effects overall. The various parameters used for comparison are "normalized" by using the flip method optimized as presented in previous research as a baseline for comparison as shown %r, (4.1) P %P P P (4.2) Where r, and PI refer to the baseline method, in this case the flip method optimized statically. Performing these operations, Table 4.6 is then a modification of Table 4.5 to demonstrate these values as comparison indices. The results are mostly negative, indicating an increase in performance in most cases. First observe that flip climbing was preliminarily worse torquewise than compared to the quasidynamic optimization. Regarding power requirements, flip climbing was actually much better preliminary than the baseline. After being path optimized, flip climbing performed better than its static optimization baseline. Side climbing was actually considerably a better method than flip climbing torque and power wise. Side climbing, once path optimized, and once its trajectory was generated performed slightly worse overall. This was due to the constraints placed by segment linearization. Rectilinear climbing was overall the better method of the three. Preliminarily, rectilinear climbing performed much better than the other two methods. After being path optimized, rectilinear climbing still outperformed its counterparts, in all indices. Table 4.6 Performance of different climbing optimizations for 3DTP robot as compared to previous research Climbing method % rmax %Pmax /netmax Optimization Flip 4.45 82.9 85.1 Side 21.9 73.0 86.9 No optimization Rectilinear 69.1 64.2 51.1 Flip 3.78 68.4 64.8 Side 8.34 68.3 82.8 Maximum manipulability Rectilinear 63.6 79.6 82.7 The actual values from simulations as shown in Table 4.5 indicate that flip climbing is not feasible when considering torque, since it requires more than that provided by the actual motors used for simulation. Regarding power, since flip climbing is not feasible it is observed that even after optimized, the ability to carry additional loads may not exist. Side climbing is an intermediate method in the sense that it outperforms flip climbing preliminarily and while statically optimized when regarding torque. Its power requirements are also significantly improved as opposed to flip climbing. After optimized, side climbing is actually not a feasible method in part due to the segment linearization as can be observed in Table 4.5. Rectilinear climbing is once again by far the best method of climbing, it requires the least torque by a very large margin while requiring comparable net power. Once optimized, torque requirements increase slightly, but power requirements decrease significantly. These results are indicators that rectilinear climbing outperforms the other two methods by a far margin, adding the capability of carrying larger loads and being autonomous. In order for the 3DTP, 3DMP, and all other robots designed for the purpose of inspection and hybrid mobility, the capability of being autonomous is the most important concern. The loading capacity of these robots increases if they use rectilinear climbing as the preferred method when on a vertical surface. This is an indicator of requiring smaller batteries for a comparative time span of operation or having the ability to operate longer with the same size of batteries as those used if climbing in other methods. Lower power requirements as prescribed by Tables 4.5 and 4.6 are such indicators. This relationship between power required and operational time comes from the relationships between power required, voltage, and discharge current defined by Peukert's law and the definition of the volt, where the former is C = Ikt (4.3) And the latter is defined as the potential difference across a conductor when a current of one ampere dissipates one watt of power. C is the capacity, I is the discharge current, k is a dimensionless constant (usually close to 1) and t is the discharge time. Therefore, the relationship between power and discharge time is given by CV Poc (4.4) t The capacity to weight ratio depends on the kind of battery and behaves linearly for all types, (e.g. alkaline dry cells, nickelcadmiun, lithiumion). This means that as the 85 capacity of a battery increases so does its weight as long as the battery type remains the same, and therefore, so does the time of operation. Therefore, as previously observed and discussed the rectilinear method will be the most useful for autonomous operations, for carrying additional loads as sensors, and for having the longest operational time if its battery is chosen to be of higher capacity. CHAPTER 5 CONCLUSIONS This research presents an alternative approach to the optimization of climbing methods using the concept of manipulability and proposing different methods of climbing. The system developed for this research was designed as a hybrid between previous designs satisfying all the kinematic configurations of those designs while maintaining a constant mass and moment of inertia throughout all the elements of the manipulator. The purpose of the design was to compare the different methods proposed for climbing and select the most efficient for smart structure climbing. Additionally, the concept of manipulability was used to optimize the path being climbed at viapoints. Trajectories were generated using quintic splines throughout the path in order to apply initial conditions for velocity and acceleration while maintaining continuity to the second derivative. The three methods proposed, flip, side, and rectilinear climbing, provided what the author considers all possibilities to climbing. Preliminary simulations showed that the flip method was the easiest method to implement alongside with the side method, while the rectilinear method for climbing is more challenging to implement. Additionally, preliminary simulations demonstrated that the rectilinear method is more efficient than the flip and side methods as observed when comparing the maximum torques and maximum power peaks. However, in terms of net power consumption the flip and rectilinear methods are comparable. In addition, it should be noted that all three methods are not set in stone, not only due to the possibility of combining these methods into a 