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## Material Information- Title:
- Manipulability Based Path and Trajectory Planning for Climbing Mode of a Hybrid Mobility Robot
- Creator:
- BESTARD, JAIME JOSE (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Aircraft maneuvers ( jstor )
Docking ( jstor ) Jacobians ( jstor ) Kinematics ( jstor ) Linearization ( jstor ) Robots ( jstor ) Smart structures ( jstor ) Torque ( jstor ) Trajectories ( jstor ) Velocity ( jstor )
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- University of Florida
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- University of Florida
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- Copyright Jaime Jose Bestard. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 3/1/2007
- Resource Identifier:
- 649810213 ( OCLC )
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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, Jean-Francois 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 post-undocking and pre-docking 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 S-G platform ........................................ ....................2 1.3 X-33 concept vehicle boarded by 3DMP robots................. ................3 1.4 Climbing parallel robot (CPR) sequence of postures evading a structural node ........5 1.5 Hyper-redundant 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 Close-loop m echanism param eters ........................................ ........ ............... 34 3.3 Configuration of end-effector 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 multi-agent 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 degree-of-freedom 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 via-points along a nominal trajectory. Piece-wise trajectories were then generated using quintic splines between the via-points of optimal configurations. The effect of the number of via-points selected on the resulting torque-minimal 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 manipulability-based 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 multi-agent robotic systems. A hybrid mobility multi-agent 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 ground-based 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 S-G 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 ground-based 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. X-33 ro ea Concept Vehicle Smart Structure Zoom of Carto Area T \\ o i oin ied ,DN lP r-obots Smart Stnmture Figure 1.3 X-33 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 Stewart-Gough 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) Pos-1 (b) Pos-1d (ce Pos-2a (d) Pos-2 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 energy-wise, 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 Stewart-Gough 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 hyper-redundant 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 Hyper-redundant 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 non-straight 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 Stewart-Gough 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 Stewart-Gough 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 end-effector in the sense of the required joint torques being lower, thus easier to maneuver the end-effector. The manipulability ellipsoid, or vice versa, the manipulating-force 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 dynamic-manipulability 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 via-points 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 quasi-dynamic methods [Cle03] for finding preliminary path via-points 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 multi-agent 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 quasi-dynamic 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 six-jointed serial mechanism and that of a closed-loop mechanism depending on the various climbing configurations, refer to Figure 2.1. As opposed to the 3DMP (Three-Dimensional 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 (closed-loop mechanism) 2.1 Fundamental Kinematics: The Jacobian Matrix The use of the Jacobian matrix provides a relationship between the velocities of the end-effector 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 end-effector, 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 end-effector 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 cross-product 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 I-J- = 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 end-effector, 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 toj-1 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 Denavit-Hartenberg definitions such that differentiations with respect to joint variables can be achieved via matrix multiplications. The Lagrangian function is L=K-P (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 J-1 dqj 0 0^u 1 Considering equation (2.13) and 0rT r tr( rr"T), where tr denotes the trace of a matrix, then r-r =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 j-1 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 body-fixed frame and the remaining off-diagonal terms are the products of inertia for link i defined about the body-frame 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' k-O 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 m-1 ]= 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 n-dimensional 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= T-Aj 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 closed-loop 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 end-effector [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 path-planning, 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 end-effector 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 end-effector 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 end-effector velocity v and joint velocities 4 given in equation (2.1), the ellipsoid's major axis is the direction where the end-effector 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 end-effector can move. This definition is essential to the future path-planning 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 high-speed and high-precision 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 end-effector 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 pseudo-inverse matrix of J. Further introducing the vectors Z= -r(,i)-G(q)+MJ +d (2.39) v=v-(I-J 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 closed-loop 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 four-bar 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 via-points. 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 via-points, on the path. For the case of the 3DTP, the via-points 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 via-points and a simulation over time. As such, it is critical to use the proper fit for the via-points 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 torque-minimal 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 down-selecting 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 (kg-m2 i Center of mass (m), rC,, Mass (kg), m Inertia tensor 1 [0 0 -2.34*10 2] 2.64e-2 1.02e-5 1.01e-5 1.62e-6 2 [0.075 0 0] 7.17e-2 5.56e-6 1.13e-4 1.12e-4 3 [0.0375 0 0] 4.31e-2 2.81e-6 3.10e-5 3.10e-5 4 [0.075 0 0] 7.17e-2 5.56e-6 1.13e-4 1.12e-4 5 0 -2.34*10 2 0T 2.64e-2 1.02e-5 1.62e-6 1.01e-5 6 [0 0 2.34*102]T 1.71e-2 3.45e-6 3.45e-6 8.32e-7 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 closed-loop kinematic chain and having four revolute joints parallel to each other. to this avail the kinematics reduces to the analysis of a four-bar 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 Close-loop 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 closed-form 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 () = Ah-13 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 '+2-214 (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 quasi-dynamic 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 closed-loop mechanism as shown in Section 3.2. For every climbing maneuver, both 3DTP end-effectors 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 End-effector 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 end-effector 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 closed-loop 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) End-effector 2 post-undocked (b) Sample sequence of flip steps (c) End-effector 2 pre-docked Table 3.4 Flip maneuver boundary configurations for numerical example End-effector 2 pre- End-effector Joint End-effector 2 End-effector 2 post- Ed-efeco 2 e- End-fcto 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) End-effector 2 post-undocked (b) Sample sequence of side steps (side view) (c) End-effector 2 pre-docked 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 End-effector End-effector Joint 2 d ed End-effector 2 post- End-effector 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 end-effector is raised half the initial distance between the two docked end-effectors (Figure 3.6 (b)). End-effector 2 then docks (Figure 3.6 (c)), followed by a release of end-effector 1 (Figure 3.6 (d)). The robot continues to climb by raising end-effector 1 the user defined height once again (Figure 3.6 (e)). End-effector 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 post-undock and final pre-dock 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 End-effector End-effector 2 pre- End-effector 2 End-effector 1 post- 1 pe-d 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) End-effector 2 post-undocked (b) through (j) steps including intermediate dockings (k) End-effector 1 pre-docked (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 HS-645MG, 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 closed-loop form of the mechanism, varying 3. Secondly, the same method was used to find the optimal configuration at pre-docking. 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 (kg-m2) i Center of mass (m), rcm, Mass (kg), m, Inertia ten 1 [0 0 -2.34*102]T 8.14e-2 3.37e-5 3.37e-5 1.25e-5 2 [0.075 0 0] 0.182 2.73e-5 7.53e-4 7.53e-4 3 [0.0375 0 0] 4.31e-2 2.81e-6 3.10e-5 3.10e-5 4 [0.075 0 0] 0.182 2.73e-5 7.53e-4 7.53e-4 5 0 -2.34 *10 2 T 8.14e-2 3.37e-5 3.37e-5 1.25e-5 6 [0 0 2.34*10-2T 1.71e-2 3.45e-6 3.45e-6 8.32e-7 Table 3.8 Flip climbing minimal torque approach optimal configurations for 3DTP robot Joint End-effector 1 post- Flip step 1 (Fig. End-effector 1 pre-docked 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 via-points 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 via-points 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 via-points 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 quasi-dynamic 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--- -----------t--------------1--------------------------- S-------------- -------------- --------------------- ------------- -- --- 0-5 ................ .. 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 -------------- --------------- .--------------- .--------------- -------------. --.---.---.--- 0-6 -----------. --- ----------- ---, ------------- ---------------- -.---.--- .------ --.---.--- .---- _ 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 ------------ -- ------------ -- ------- ---- ------------- ---- ------- 0-2 -------- --------- ------------------- ------- -------- 0 - - 02 -...... ..... -0-2 --/- ----------------; --------------- 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 0-1 -------------- ----------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 0-1 ------ --------j-------a--------------------- o.2 .. **. ....... r.. ......... 4, I 01 . S -02 -- ---- -- S-0-3 -- ---------~---------------~-- 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 5-01 -1- --- | 0.2 .- -. ; ...............,...............,.............. ,......... .... "1 94 5 -0- 4 ---- --. -06- S-0.6 7 ------ -0-4 ------------- 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 =0-86389 Nm 0 .2 F.------------ ------.------------------------ -------------- ------,-- --- --- - 0.1 -------------- ----------- -------- ---------- ----------- ----------- 2 0 -. .- .. 3 0.1 -- -- ----- --- --- ----- -- --- ----- - 0 ---- --------- -------- -> 0- 5 -------------- -- ^-- -- - ------ --------------- ....- ..--- --- --------------- -07 ---------- -- Ee-0 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--------------- ------------ -- I----------V ----:"'- 0-04 ------------- -------------- -------------- --------------- ---------------. -------- .--- - 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 S-100 -'-*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--- S-a...... .. -0 5-3 -- - ---- -- -- ---------- -- -- --- --- ---- - - - ----------;-- -----------;--- ------------ -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 --------- ---------- --------------- -0-2 -~ ---------- -------------- -----------------.,------------ -------------- --------------- -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 0-06 ---------- ---- ............ 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 quasi-dynamic exhaustive searches were done to find optimal maximum torque configurations throughout the workspace of a 3DMP. Additionally, via-points selected from the resulting min-max 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 via-points chosen and oscillations on the curve fit. 4.1 Docking/Undocking Optimization Using solutions found via closed-loop 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 pre-docking and post- undocking are shown in Table 4.1. Table 4.1 Optimal manipulability configurations of 3DTP robot at post-undocking and pre-docking steps Post-undocking Pre-docking Method and description Joint variables Post-undocking Pre-d 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 Post-undocking Pre-docking (Ah = -0.3 m) (Ah = -0.15 m) q, 900 900 q2 -134.210 -94.970 Rectilinear climbing case, q3 -40.790 -76.030 end-effector 2 free q -41.420 -74.200 q5 36.420 65.200 q6 0 0 Post-undocking Pre-docking _(Ah = -0.15 m) (Ah = -0.3 m) q1 900 900 q2 -114.800 -143.580 Rectilinear climbing case, q3 -74.200 -41.420 end-effector 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 three-by-four 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 post-undock and pre-dock 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 post-undocked position to the next pre-docking 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 post-undocking and pre-docking 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 end-effector (in this case, 1 through 5, because end-effector 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 via-points 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 via-points. 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 .. . S-0.4 -..............----- 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 50-S ----------- ----- ------ -- ------------------------------ ----------- ---- -------.--_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 0-2 ------- --------- ---- --- -------- ------ ---- --- ----- ---- ----- ---- -------- ----- O ---- ---- --0.0- -. .... .. 0 5 10 15 20 25 30 S .5 ------------ ------ -- -:-:----: --- ----- -. --------- -;-:----- -- 0 2 .. . .. . . . .. . .. . ..... . . .. . . . -0.15 I--I-------- -- ----------------I I -------- - -0-2----------- -------------- -------- ---- ------------ -- -. .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 -0-6 --- --- - 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''' -------- 0-2 --- ------------- ------, -------- --- ------ 0.2 ............... .------------- -------------- -------- ----------- -- ............ -0-2 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-------------- ~ ~ ~t---------------L---:-------------------------- 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 minimal-maximum 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 -------------- . ----------- -- --- -- --------------- -- -- -------- --------------_ 0-08 -------------- ------ -------- --------------- --------------- --------S- ----- --.-------.---- -0.06 -------------- f------------- ------------------------------ ------------- ;- -------------- 0-04 _--.---------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 manipulability-based 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 torque-wise than compared to the quasi-dynamic 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, nickel-cadmiun, lithium-ion). 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 via-points. 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 |

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PAGE 1 MANIPULABILITY BASED PATH AN D TRAJECTORY PLANNING FOR CLIMBING MODE OF A HYBRID MOBILITY ROBOT By JAIME JOS BESTARD A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006 PAGE 2 Copyright 2006 by Jaime Jos Bestard PAGE 3 This work is dedicated to my parents. PAGE 4 iv ACKNOWLEDGMENTS The author profoundly thanks his advisor, Doctor Gloria J. Wiens. The author additionally extends his appr eciation 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, Gust avo Roman, Kaveh Albekord, Jessica Bronson, Nick Martinson, Takashi Hiramatsu, Sharan abasaweshwara Asundi, Sharath Prodduturi, Jean-Francois Kamath, and Daniel Jones, for their support. Finally, the author would like to express hi s appreciation and thanks to Natasha M. Elejalde, Joaqun A. Bestard, Juan P. Bestard, Carolina C. Bestard, Juan M. Fernndez, Rolanda M. Gmez, Mireya Llense, Enrique ta Perez, Maria S. Valdez, Martn R. Rosales, Roberto and Nancy Cachinero, Anshley Sardias, and Yadnaloy Acosta. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xiii 1 INTRODUCTION.........................................................................................................1 1.1 Types of Climbing Mechanisms and Their Algorithms....................................3 1.1.1 Climbing Inside Tubular Structures.......................................................5 1.1.2 Climbing Outside Tubular Structures....................................................7 1.1.3 Climbing About Structural Nodes.........................................................8 1.1.4 Climbing Smar t Structures.....................................................................9 1.2 Manipulability..................................................................................................11 1.3 Definitions and Terminology...........................................................................12 1.3.1 Path Planning.......................................................................................12 1.3.2 Trajectory 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 Dynamics......................................................................................19 2.3 Manipulability Measure...................................................................................24 2.3.1 Definition and Derivation of Mani pulability Ellipsoid and Kinematic Manipulability Measure.......................................................................25 2.3.2 Definition of the Dynamic Manipulability Measure............................26 2.4 Path Planning...................................................................................................28 2.5 Trajectory Generation......................................................................................29 3 PRELIMINARY CLIMBING OF 3DTP.....................................................................31 3.1 Design and Configuration of the 3DTP...........................................................31 3.2 Kinematics of the 3DTP...................................................................................33 3.3 Dynamics of the 3DTP.....................................................................................38 3.4 Climbing Methods...........................................................................................38 PAGE 6 vi 3.4.1 Docking/Undocking Maneuvers for the 3DTP....................................38 3.4.2 Flip Climbing Maneuver for 3DTP......................................................40 3.4.3 Side Climbing Maneuver for 3DTP.....................................................41 3.4.4 Rectilinear Climbing Maneuver for 3DTP..........................................43 3.5 Optimization via Exhaustive Search................................................................44 3.6 Flip Climbing Optimization via Minimal Torque Approach...........................46 3.7 Climbing Trajectories......................................................................................47 3.7.1 Minimal Torque Optimized Flip Trajectory........................................48 3.7.2 Preliminary Flip Climbing...................................................................53 3.7.3 Preliminary Side Climbing..................................................................56 3.7.4 Preliminary Rectilinear Climbing........................................................59 3.7.5 Further Comparison Results on Preliminary Climbing.......................62 3.8 Preliminary Observations.................................................................................63 4 OPTIMIZED CLIMBING OF 3DTP...........................................................................65 4.1 Docking/Undocking Optimization...................................................................65 4.2 Flip and Rectilinear Climbing Optimization....................................................66 4.3 Side Climbing Optimization............................................................................69 4.4 Trajectory Generation Using Ma nipulability Optimized Paths.......................70 4.4.1 Flip Climbing Method..........................................................................70 4.4.2 Side Climbing Method.........................................................................71 4.4.3 Rectilinear Climbing Method..............................................................73 4.5 Inverse Dynamics Simulation of 3DTP...........................................................74 4.6 Observations....................................................................................................81 5 CONCLUSIONS..........................................................................................................86 A MATLAB FUNCTIONS AND SCRIPTS...................................................................89 A.1 Recursive Lagrangian Dynami cs for Serial Manipulators...............................89 A.2 Inverse Tangent................................................................................................91 B JACOBIAN AND TRANSFORMATION MATRIX ELEMENTS FOR 3DTP........92 B.1 Jacobian Matrix Elements of 3DTP.................................................................92 B.2 Transformation Matrices Elements of 3DTP...................................................94 C TRAJECTORY GENERATION WITHOU T SEGMENT LINEARIZATION..........96 D ADDITIONAL TABLES...........................................................................................101 LIST OF REFERENCES.................................................................................................116 BIOGRAPHICAL SKETCH...........................................................................................119 PAGE 7 vii LIST OF TABLES Table page 2.1 Mechanism parameters for 3DTP.............................................................................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 pa rameters for 3DTP simulations.....................39 3.4 Flip maneuver boundary configurat ions 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 approa ch optimal configurations for 3DTP..............47 3.9 Effects of segment linearization on jo int maximum torques and maximum net power of different 3DTP manipulabil ity path optimized trajectories.......................48 3.10 Summary of preliminary simulation results for 3DTP climbing..............................64 4.1 Optimal manipulability conf igurations of 3DTP at post-undocking and pre-docking steps.......................................................................................................................... .66 4.2 Optimal manipulability confi gurations of 3DTP at select intermediate steps for the flip climbing maneuver.............................................................................................67 4.3 Optimal manipulability confi gurations of 3DTP at select intermediate steps for the rectilinear climbing method......................................................................................68 4.4 Optimal manipulability confi guration of 3DTP at interm ediate step for the side climbing maneuver....................................................................................................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 research.....................................................................................................................83 PAGE 8 viii D.1 Preliminary flip climbing joint torques for 3DTP...................................................101 D.2 Preliminary side climbing joint torques for 3DTP..................................................103 D.3 Preliminary rectilinear cl imbing joint torques for 3DTP........................................106 D.4 Optimized flip climbi ng 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 PAGE 9 ix LIST OF FIGURES Figure page 1.1 Alicia3 robot over outdoor concrete wall....................................................................2 1.2 Mechanical adaptations of an S-G platform...............................................................2 1.3 X-33 concept vehicle boarded by 3DMP robots.........................................................3 1.4 Climbing parallel robot (CPR) sequence of postures evading a structural node........5 1.5 Hyper-redundant robot climbing via bracing..............................................................6 1.6 Experimental results of the application of the kinematics control algorithm to the climbing parallel robot (CPR) prototype....................................................................7 1.7 Base modular climbing robot......................................................................................8 1.8 CPR robot.................................................................................................................. ..9 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 robots link and joint axes kinematic parameters....................................20 3.1 Different hybrid mobility kinematical designs.........................................................32 3.2 Close-loop mechanism parameters...........................................................................34 3.3 Configuration of end-effector 2 for 3DTP robot.......................................................39 3.4 Flip climbing maneuver............................................................................................41 3.5 Side climbing maneuver...........................................................................................42 3.6 Rectilinear climbing maneuver.................................................................................45 3.7 Minimal torque optimized flip traj ectory of 3DTP robot: joint angles.....................49 PAGE 10 x 3.8 Minimal torque optimized flip trajec tory of 3DTP robot: joint velocities................49 3.9 Minimal torque optimized flip trajecto ry of 3DTP robot: joint accelerations..........50 3.10 Minimal torque optimized flip trajectory of 3DTP robot: join t gravity effects........50 3.11 Minimal torque optimized flip trajec tory of 3DTP robot: joint torques...................51 3.12 Minimal torque optimized flip traj ectory of 3DTP robot: joint power.....................51 3.13 Minimal torque optimized flip tr ajectory of 3DTP robot: net power.......................52 3.14 Minimal torque optimized flip traj ectory 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 3D TP 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 3D TP 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 climbi ng 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 climbi ng of 3DTP robot: joint power..................................62 PAGE 11 xi 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 fo r 3DTP robot: jo int velocities...................................71 4.3 Path optimized flip method fo r 3DTP robot: joint accelerations..............................71 4.4 Path optimized side method for 3DTP robot: joint angles........................................72 4.5 Path optimized side method fo r 3DTP robot: jo int velocities...................................72 4.6 Path optimized side method fo r 3DTP robot: joint accelerations.............................73 4.7 Path optimized rectilinear met hod 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 fo r 3DTP robot: joint torques.......................................75 4.12 Path optimized side method for 3DTP robot: gravitational load effects..................76 4.13 Path optimized side method fo r 3DTP robot: joint torques......................................76 4.14 Path optimized rectilinear method for 3DTP robot: gravitational load effects.........77 4.15 Path optimized rectilinear met hod for 3DTP robot: joint torques............................77 4.16 Path optimized 3DTP robot cl imbing 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 me thod 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 angles........................................................................................................................96 C.2 Path optimized side climbing of 3DTP robot without segment linearization: joint angles........................................................................................................................97 PAGE 12 xii C.3 Path optimized rectilinea r climbing of 3DTP robot w ithout 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 torques.......................................................................................................................98 C.6 Path optimized rectilinea r climbing of 3DTP robot w ithout segment linearization: joint torques..............................................................................................................98 C.7 Path optimized flip climbing of 3DTP robot without segment linearization: joint power.........................................................................................................................99 C.8 Path optimized side climbing of 3DTP robot without segment linearization: joint power.........................................................................................................................99 C.9 Path optimized rectilinea r climbing of 3DTP robot w ithout segment linearization: joint power..............................................................................................................100 C.10 Path optimized climbing of 3DTP ro bot, no segment linearization: net power......100 PAGE 13 xiii Abstract of Thesis Presen ted 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 AN D TRAJECTORY PLANNING FOR CLIMBING MODE OF A HYBRID MOBILITY ROBOT By Jaime Jos Bestard December 2006 Chair: Gloria J. Wiens Major: Mechanical Engineering Inspection robots used in a wide range of applications requi re the ability to efficiently climb while carrying the necessary loads which may include sensors, special tooling, and batteries. Therefor e, it is essential to deve lop power efficient climbing algorithms for the overall design and anal ysis of autonomous multi-agent robotic systems. The vast majority of such robots are currently able to climb using only one method with no alternate or hybrid methods av ailable, in part due to the geometric constraints of the mechanism. Hybrid m obility robots consist of wheeled locomotion when traversing horizontal terrains with m odest grades and roughness. Upon encounter of vertical terrains, two hybrid r obots join to reconfigure in to a climbing mode on a smart structure. To this avail a six degree-o f-freedom mechanism was developed for the simulation of different methods of climbing while maintain ing constant geometric and inertia properties, thus providing grounds fo r comparison. This mechanism is capable of PAGE 14 xiv transforming into the different climbing conf igurations for the hybrid mobility robot developed in prior research. In this thesis, the research focuses on deve lopment of efficient path and trajectory planning algorithms for the climbing mode of the hybrid mobility robot. Three different climbing methods were compared on the basi s of their resulting maximum torques and instantaneous power requirements. These quant ities 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 via-points along a nominal trajectory. Piece-wise trajectories were then generated using quintic splines between the via-points of optimal configurations. The effect of the number of via-points se lected on the resulting torque-minimal trajectories was investigate d. Limitations on the optimality were found to be heavily weighted by the trajectory planning portion of the algorithms In spite of this, the manipulability-based approach was shown to be effective in generating paths and trajectories that exhibit overall lo w torque and power requirements. Of the three climbing methods, the recti linear 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 geomet ric 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 fu ture design optimization of climbing hybrid mobility robotic systems. PAGE 15 1 CHAPTER 1 INTRODUCTION The thrust of this thesis is to devel op path planning and trajectory generation techniques that optimize the climbing capabi lities of hybrid mobility multi-agent robotic systems. A hybrid mobility multi-agent robotic sy stem consists of robots within a team of robots that each have two or more loco motion types (e.g., wheeled and climbing) or acquire multiple locomotion types by reconfi guring 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 ground-based and space environments. Typical operating environments for c limbing mechanisms vary extensively throughout the planet. In the ci vilian environment, applications for climbing mechanisms range from the cleaning of high rise buildi ngs, 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 severa l maneuvers and applications in the cosmos. An example of this is the maneuvering a bout a Resident Space Object (RSO), e.g. satellite, on the space st ation and/or on space trusses, to be serviced in space where several robots must cooperate, and optimal trajectories about th e 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. PAGE 16 2 Figure 1.1 Alicia3 robot over outdoor concrete wall [Lon06]. Figure 1.2 Mechanical adaptations of an S-G platform for (a) climbing the exterior of tubes; (b) an open universal joint; an d (c) climbing the interior of tubes [Ara06]. An ideal application for climbing robots fo r space applications is for ground-based 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 illustrate d in Figure 1.3. Within each area the 3DMP PAGE 17 3 robots would separate and insp ect 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 robots climbing trajectory in which each end of the climbing robot would dock securely and undock as it climbed. Two joined 3DMP robots Figure 1.3 X-33 concept vehicle boarded by 3DMP robots [Cle03] 1.1 Types of Climbing Mechan isms and Their Algorithms Climbing algorithms have been developed for different situations and resource capabilities. For climbing inside tubular struct ures, there exist parallel mechanisms based in the Stewart-Gough platform [A ra06] 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 structur es can be used to climb outside tubular PAGE 18 4 structures with slight modifications to their design or configuration. Parallel mechanisms may also be used for climbing outside any other structure, incl uding a smart structure [Cle03], as demonstrated in related resear ch [Ara06]. However, the use of serial configurations is more common for climbi ng 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, Rip00]. All these mechanisms have the same purpose; however, they differ in approach to achieve it and in their overall performance. A subclass therefor e 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 si milar to a slinky, which is labeled as flip climbing in this thesis. And the final group cl imbs in a similar fashion to quadrupeds like geckos, where the climbing portion revolves abou t 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 define d as a corner in a st ructure, or a location at which the climbing path changes directi on abruptly [Sal05]. Th e performance of the climbing robot changes completely at this poi nt, 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. Ther efore, a category of climbing can also be PAGE 19 5 defined as the climbing about nodes. For example, Figure 1.4 illustrates a climbing parallel robot (CPR) following a se quence of moves around a corner node. Figure 1.4 Climbing parallel robot (CPR) seque nce of postures evading a structural node [Sal05] In both the serial and parallel configurati ons different climbing tasks can classify further the design of a robot, mainly the clim bing inside and outsid e tubular structures, climbing on smart structures, and climbing a bout structural nodes. Additionally, the climbing method used, further defi nes 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 a pplications 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 explor ation of oil and gas pipelines. Investigations on the usage of paralle l mechanisms to this avail are ongoing [Ara06]. However, the question remains: How e fficient can these parallel mechanisms be in different aspects, especially energy-wi se, 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, paralle l machines of the Stewart-Gough kind are PAGE 20 6 additionally out of range for the climbing application of r econfigurable inspection multiagents as in previous research [Cle03]. Th ese machines are difficult to implement in a reconfigurable fashion. Serial mechanisms for the same applic ation provide a different scope to the aforementioned questions. Regarding the question of efficiency, this thesis focuses on the development of new path optimization algor ithms for the climbi ng mode of a hybrid mobility robot. To this ava il, several methods will be explored where only serial mechanism configurations will be simulate d. Serial mechanisms for this purpose may also provide favorable answers to the ot her relevant questions regarding climbing [Kot97]. Climbing via bracing [Gre05] is an inte resting 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 show n in Figure 1.5, deviates this topic from the main methods to be investigate d. That is, to conform to the focus of this thesis, the hyper-redundant r obot 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. Figure 1.5 Hyper-redundant robot climbing via bracing [Gre05] PAGE 21 7 1.1.2 Climbing Outside Tubular Structures Climbing outside of tubular structures is a category that ma y 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 wi ring in potentially harmful situations. Figure 1.6 Experimental results of the appli cation of the kinematics control algorithm to the climbing parallel robo t (CPR) prototype [Alm03] Serial mechanisms have also been developed for climbi ng outside tubular structures. There have been many achieve ments and designs in this group, and consideration to modular and reconfigurab le designs have been given (Figure 1.7) [Rip00]. Climbing outside a t ubular structure can be done in all three climbing mode subclasses. The problem arises on the selecti on of the most efficient of these climbing algorithm subclasses. This dilemma exists be cause of the need of adding gripping and climbing motors [Pac97, Yan97]. Therefore, the topic of climbing outside tubular structures with serial mechanisms, reconfigur able in cases, will benefit from the methods presented in this thesis where efficient algorithms for climbing are developed. PAGE 22 8 Figure 1.7 Base modular climbing robot: (a) Si ngle robot (b) Coupling of two modules of climbing robot to form new capabilitie s of negotiating non-straight vertical motion [Rip00] 1.1.3 Climbing About Structural Nodes Climbing about structural nodes is a topi c that is of utter importance to any climbing robot; however, this topic is not in the scope of this inve stigation and is only used to consider and explain most situa tions encountered by a climbing robot. Most structures climbed will contain structural node s of some sort. Nodes may be evaded using path planning algorithms; however, their pres ence affects the simple climbing algorithm on a vertical surface. The most explicit descri ptions of this issue occur in designs of parallel mechanisms as the CPR [Sal05]. As shown in research with Stewart-Gough platform based mechanisms, these machines though excellent in certain aspect s of climbing, will lack the capability structurally of maneuvering about a corner node, as climbing a wall that ends into a flat roof or any kind of roof for that matter wit hout the aid of additiona l appendages. Parallel mechanisms developed to this avail become mo re 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 90 configurations, as shown in Figure 1.2 and PAGE 23 9 Figure 1.8. Thus, simply consider ing joint limitations, serial mechanisms are more adept structurally for the node kind of obstacles. Henceforth, serial mechanisms are defin itely superior in the matter of climbing about structural nodes. Another example is th at even inside a room where the walls and ceiling need to be inspected, the serial mech anism 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. Figure 1.8 CPR robot: (a) Posture where it is necessary to achieve 90 between both plates of Stewart-Gough platform (b) Modified joints [Sal05] 1.1.4 Climbing Smart Structures Climbing smart structures is the main scen ario of research for this thesis. These structures offer a favorable level of indepe ndence regarding the development of climbing PAGE 24 10 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 mechan ism, 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 compos ed of docking points, where a mechanism is in charge of detecting an end of the clim bing robot and clamping it or docking it to the smart structure. When the docking is comple ted, the other end of the robot is freed autonomously by the climbing structure, and th e 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] PAGE 25 11 Figure 1.10 Isometric and front view of the docking mechanism [Cle03] 1.2 Manipulability Manipulability is a measure that determin es the ease of changing the position and orientation of the end-effector in the sense of the required jo int torques being lower, thus easier to maneuver the end-e ffector. The manipulability el lipsoid, or vice versa, the manipulating-force ellipsoid [Yos 90], is a visual description of manipulability and is in turn inversely proportional to the manipulab ility measure. The ma nipulability ellipsoid delineates via its principal axes the direc tions in which a maximum manipulability and corresponding minimal manipula ting force are required to ge nerate motion. Non obstante, these concepts are devoid of dynamic consid erations and are further developed in [Yos90] into the dynamic-manipulab ility 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 PAGE 26 12 trajectories generated while ca pturing the dynamics of the system per the tasks to be achieved. Path planning and trajectory ge neration algorithms can be modified by following the direction in which a minimal ma nipulating 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 dete rmine 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 ge nerated 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 pat h. These reference traj ectories are usually provided in joint space as polynom ial 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 climbers 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 simp ly 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 limitati ons are dominated by the torque, mobility and maximum power required, thus the sizing of th e actuators is critical. To address these issues, a comparative study on different climbing strategies was conducted. These PAGE 27 13 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 suboptimally on robot manipulability metrics and minimal torque. The effectiveness of these algorithms is evaluated in terms of the maximum torque, maximu m power required, and robustness to inertia/payload variations. In addition, th e effects of implementation variations, such as number of intermediate via-points and boundary conditions used for the trajectory planning and the curve fitting selection (quintic, cubic, linear), were investigated. Hybrid mobility inspecti on mechanisms have previ ously been designed [Chi94, Mer05], and path planning and obstacle ev asion algorithms were developed [Cle03]. However, climbing ability for these mechanisms is in a primitive stage. Regardless of the previous design, though mainly using the geom etry, several climbing algorithms will be analyzed and ranked accordingly. Additio nally, 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 chap ter covers the definition of the Jacobian matrix, to be used finally in the chapter fo r the definition of the manipulability measure. The necessary concepts of Lagrangian dynami cs are also explained and an approach avoiding the necessity of symbolic computa tions is also presented. The concept of PAGE 28 14 manipulability is also introduced and its quantif ication is partly derived. Finally, path and trajectory planning concepts are introduced. Chapter 3 contains the development of th e hybrid mobility climbing robot 3DTP as a design and as it regards its geometric conf iguration. The basic solution for the closed loop form of the climbing robot is also intr oduced 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 undocki ng maneuvers. Additionally, in chapter 3 the preliminary paths and trajectories are develope d and tested, leading to initial conclusions regarding the behavior of each method. Chapter 4 tests the optimization usin g the concept of manipulability and observations are drawn regarding improveme nts in performance using this method. Additionally, chapter 4 explor es using static or quasi-dynamic methods [Cle03] for finding preliminary path via-points and optim izing them using manipulability concepts for flip climbing. Chapter 4 also delves in to the different methods considering motor loads and the effects that will have in carry ing 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. PAGE 29 15 CHAPTER 2 HYBRID MOBILITY ROBOT CLIMBING MECHANISM In this thesis, the hybrid m obility robot has two basic modes of operation, as a pair of wheeled locomotion robots and as a si ngle climbing robot. In todays research community, wheeled robots have become the platforms of choice for not only developing path planning for inspection, reconnaissan ce, and surveying, but for testing control algorithms involving multi-agent 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 ha s the ability to climb. In prior research [Cle03], an initial path planning algorithm for a hybrid mobility robot was developed based on a quasi-dynami c approach. In this chapte r, a generalized mechanism is defined which captures the climbing kine matics of the work of Clerc [Cle03] and others. In addition, the generalized mechan ism 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 su pporting 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) denot es a robot capable of climbing out of the plane of wheeled mobility. The 3DTPs rec onfigured workspace is defined by kinematics of a six-jointed serial mechanism and that of a closed-loop mechanism depending on the various climbing configurations, refer to Figure 2.1. PAGE 30 16 As opposed to the 3DMP (Three-Dimensional 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 stru cture consists of a series of automated ports located along the robots climbing trajectory in which each end of the climbing robot would dock securely and undock as it climbed. Figure 2.1 The 3DTP robot: (a) extended, free e nd (serial mechanism); and (b) docked to a smart structure after performing a c limbing step (closed-loop mechanism) 2.1 Fundamental Kinematics: The Jacobian Matrix The use of the Jacobian matrix provides a relationship between the velocities of the end-effector 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 T 0T0T eevp is used when expressing the ve locity of the e nd-effector, where 0 e p (a) (b) J oint 1 J oint 2 J oint 3 J oint 4 J oint 5 J oint 6 PAGE 31 17 refers to the absolute lin ear velocity component and 0 e refers to the absolute angular velocity component. The relation between velocities is given as vqq J (2.1) The subscript e denotes end effector and the superscr ipt 0 denotes the vector is expressed in terms of the inertial frame 0, and q is the vector of joint ve locities. Using the DenavitHartenberg notation, the angular and linear velocities of the e nd-effector with respect to the ground reference frame can be expressed as T 000 1 1001nn i ejjii jijpqp RR (2.2) T 00 1001n ejj jqR (2.3) Where 0 jR is the rotation matrix describing the orientation of frame j relative to frame 0 The T denotes the transpose of denotes the cross-product of two vectors, n is the number of links in the system, and the jq denotes the velocity of the jth joint. The 1 i i p is the position of i+1 frames origin in the ith reference frame where for n+1 frames origin is also the point on the end effector for which the end eff ector velocities are defined Considering the following definitions, T 00 001jjz R (2.4) 0000 ,1,11 00 n ij ejiiejjj ij ej p ppp pp RR (2.5) The generalized expression for the Jacobian matrix becomes PAGE 32 18000000 1,12,2, 000 11 eenen nzpzpzp zzz J (2.6) The vector 0jz represents the unit vector of joint axis j whilst 0 ejp refers to the vector from a point on joint axis j to the end-effector, nominally taken as a vector from the origin of the jth frame to the end effector. Additionally, considering the homogeneous transformation matrix from the ground reference frame to joint j 0000 0 0001jjjj j x yzp T (2.7) Where 0 jp is the jth frames origin relative to the fixed frame 0. The 0j x 0jy, and 0jz, are the unit vectors of the jth frame affixed to link j, determined according to DenavitHartenberg notation [Cra98]. Furthermore, 0011 12 j jj TTTT (2.8) And 1 1111 1 1111cossin0 sincoscoscossinsin sinsincossincoscos 0001jjj jjjjjjj j j jjjjjjja S S T (2.9) The j is the joint angle for revolute joint j The 1 j is the orientation of joint j axis relative to j -1 axis about the 1j X axis. The jS is the distance between j X and 1j X axes along j Z axis. The 1 ja is the shortest distance between j Z and 1 j Z axis, forming the 1 j X axis. PAGE 33 19 2.2 Lagrangian Dynamics To demonstrate the effectiveness of each climbing strategy and path/trajectory planning algorithm of the later chapters, th e equations of motion of the 3DTP robot are introduced. These were obtained in a genera lized fashion using a recursive Lagrange formulation. In robotics literature [Yos90], this method has been integrated with the Denavit-Hartenberg definitions such that diffe rentiations with respect to joint variables can be achieved via matrix multiplications. The Lagrangian function is LKP (2.10) Where K denotes the kinetic energy and P the potential energy of the system. By definition, Lagranges equations of motion are expressed as i iiidKKP Q dtqqq (2.11) 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 give n by equation (2.8), which is labeled the homogeneous transform from the 0 to the jth reference frame. Additionally, each recursive transform between links is defi ned by equation (2.9). In equation (2.9), j and jS are the jth link joint angle and o ffset respectively, while 1j and 1ja are the twist angle and link length, resp ectively. In the 3DTP mech anism all the joint angles j define the joint variables, since all joints are revolut e. Refer to Figure 2.2 for a visual description of the joint variables. The overall model parameters are provided in Table 2.1, the PAGE 34 20 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), 1ia Twist angles, 1i Joint offsets (m), iS Joint angles, i 1 0 90 01LL Variable 2 0 -90 0 Variable 3 2L 0 0 Variable 4 3L 0 0 Variable 5 4L 0 0 Variable 6 0 90 56LL Variable 7 0 0 0 0 L0 L1 L6 L5 J1J2J3J4J5J6L2L3L4 Z0 Z1X1 Z2 X2 Z3 X3 Z4 X4 Z5X5 Z6 X6 ZeXe X0 Figure 2.2 The 3DTP robots link and joint axes kinematic parameters Next, consider a point on link i, this point is labeled as ir with respect to the reference frame i The location of this point with respect to the ground, 0 reference frame, is given by PAGE 35 21 r ri i T0 0 (2.12) Relative to frame i ir is constant and 0idrdt Therefore, 00 0 1i i i j j jdr rqr dtdq T (2.13) Considering equation (2.13) and 0T000Ttrrrrr where tr denotes the trace of a matrix, then 00T 0T0T 11trii ii ii jk jk jkrrrrqq qq TT (2.14) Referencing to equations (2.10) through (2. 14), it is possible to compute the kinetic energy iK and the potential energy iP of link i. The former is found as follows 00T 0T0 link link 1111 tr 22ii ii iiijk ii jk jkKdKrrdvqq qq TT H (2.15) Where iH refers to the pseudo iner tia matrix [Yos90] given by T link 2 2 2ii i i ixxiyyizz ixyixziCMix ixxiyyizz ixyiyziCMiy ixxiyyizz ixziyziCMiz iCMixiCMiyiCMizirrdv III HHmr III HHmr III HHmr mrmrmrm H (2.16) To express the relation between the pseudo iner tia matrix and the inertia tensor consider ixxixyixz i iixyiyyiyz ixziyzizz I II I II I II I (2.17) PAGE 36 22 Where ixx I iyy I and izz I are the principal inertias of link i about its body-fixed frame and the remaining off-diagonal terms ar e the products of inertia for link i defined about the body-frame i axes. Then the following relations hold 22ixxixxiCMiyCMizIImrr (2.18) ixyixyiCMixCMiyHImrr (2.19) The same notational form is followed for iyy I izz I iyzH ixzH Additionally, the remaining parameters are Ti CMiCMixCMiyCMizrrrr (2.20) Where i CMir refers to the vector from the origin of link i to its center of mass in the coordinates of the link. The potential energy, iP, of link i is computed as follows T0i iiiCMiPmgr T (2.21) Considering T0xyzgggg as the gravitational a cceleration in the ground, 0, reference frame. In this fashion the Lagr angian, as expanded from equation (2.11) becomes 1 n ii iLKP (2.22) Substituting the previous equation into equati on (2.11) and rearranging, one obtains the dynamic equations for e ach joint torque as PAGE 37 23 00T 1 0 200T T 11tr trnk kk ikj kij ji nkkn j j kk kjmjCMj kijmji jmiiq qq qqmgr qqqq TT H T TT H (2.23) The generalized equation of mo tion is of the vector form qqVqqGq M (2.24) Where qM refers to an nn inertia matrix that can be further defined as 00T max,trn kk ijk kij jiM qq TT H (2.25) Additionally, Vqq is an n-dimensional vector th at contains the centrifugal and Coriolis forces and its ith element is expressed as 200T 11max,,trnnn kk ikjm jmkijm jmiVqq qqq TT H (2.26) The Gq vector contains the load due to gravity and is given by 0 n j Tj ijCMj ji iGmgr q T (2.27) Furthermore, the matrices 0 ijq T and 2020 ijkikjqqqq TT can be found using the relationship 1 1 i i i ii iq T T (2.28) Which comes from the fact that the transformation matrix is a function of the generalized coordinate. The parameter i, is defined as follows PAGE 38 240100 1000 for a revolute joint 0000 0000 0000 0000 for a prismatic joint 0001 0000i (2.29) The parameter i provides derivative definitions as shown 0 0j i j ji jq T T T (2.30) 0 20, ,max,jk jjkki i jkikj j ki qq T T T T 0 (2.31) Thus, these equations avoid the need fo r 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 closed-loop mechanism configurat ion, the additional constraint s 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 a nd design of any robot manipulator is the facility of changing position and orientation of the end-effector [Yos90]. A quantitative measure for this facility has been developed from the kinematics and dynamics viewpoint. In this thesis, the kinematic ma nipulability measure will be used for the purpose of path-planning, whereas the use of the dynamic manipulability measure will be left for future research. PAGE 39 25 2.3.1 Definition and Derivation of Ma nipulability Ellipsoid and Kinematic Manipulability Measure The relationship between the end-effector velocity v and the joint velocity q was previously given in equation (2.1). From th is relationship and per reference [Yos90], the definition and procedure to obtain the manipulability meas ure and its properties is provided below. First, consider the set of all end-effector velocities possible by the joint velocities in such a fash ion that the norm of q satisfies 222 121nqqqq (2.32) The set provides an ellipsoid. From the re lationship between the e nd-effector velocity v and joint velocities q given in equation (2.1), the ellipsoids major axis is the direction where the end-effector can move at high sp eeds (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 in dicator of the overall speed at which the end-effector can move. This definiti on is essential to the future path-planning process. The ellipsoid, since it represen ts 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 manipulab ility measure [Yos90] for a certain configuration and labeled as w. It is formulated as PAGE 40 26 Tdetkwqq JJ (2.33) However, when the degrees of freedom of the manipulator is equivalent to the number of joints, nm (refer to Section 2.2.1), the measure further reduces to detkwq J (2.34) The manipulability measure is a property th at also relates to the distance of the manipulator configuration from being singular. Generally th e manipulability measure is greater than zero, 0kw and is only zero, 0kw iff rankqm J (2.35) 2.3.2 Definition of the Dy namic 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 high-speed and high-precision 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 dynami cs as given by (1.24). Differentiating the relationship between joint a nd end-effector velocities given by (1.1) yields vqqqq JJ (2.36) The second term of the equation can be in terpreted as the virt ual acceleration and redefined as ,raqqqq J (2.37) From (2.24) and (2.36) PAGE 41 27 1,rrvaVqqGqaIJJJMMJ (2.38) Where J is the pseudo-inverse matrix of J. Further introducing the vectors ,rVqqGqa MJ (2.39) rvva IJJ (2.40) Equation (2.38) can be rewritten as 1v JM (2.41) Similar to the definition of the KME, th e DME is constrained by the inequality 1 The ellipsoid is described in Euclidean space as T T1Tvv JMMJ (2.42) The dynamic manipulability measure (DMM) is defined similarly to the KMM 1 TTdetdw JMMJ (2.43) Once again, similarly to the KMM, th e DMM is further reduced when the manipulator is not redundant, nm as given by det det detd kw w J M M (2.44) The denominator of (2.44) contains the eff ects of the dynamics of the manipulator while the numerator refers to the effects of th e kinematics on the mani pulability. The physical interpretations regarding singul ar configurations remain. PAGE 42 28 2.4 Path Planning Path planning provides a geometric de scription of the motion of a given manipulator [Spo05]. However, it does not provide any dyna mic 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 mani pulator must achieve a certain configuration for docking and undocking. For these maneuvers, the manipulator behaves entirely as a closed-loop 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 four-bar closed loop mechanism. The path in between the boundaries of th e 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 de sired 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 preliminar y path as shown in [Zhe96] 1lm C d CJWdsW msds (2.45) Where dms is the manipulability (dynamic or kinematic), at the given point and lW and mW are, respectively, weights deciding if the cost function to be minimized must adhere to the preliminary path or to the manipulability based path more. PAGE 43 29 The cost function provided beforehand is also computationally intensive for the 3DTP, since it was developed and demonstr ated 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 de fining endpoints to the desired path and at those endpoints obtaining the optimal manipulabil ity configuration, which in flip and side climbing are endpoints, but in the case of r ectilinear climbing are mainly vi a-points. From there it is only a problem of generating a convenient trajectory and d eciding which climbing method is more appropriate. Objectively, such minimi zation was done in previous research [Cle03] in the form of ,optimalminmax, 1,2,,6jij iqqti (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 ,optimalmaxjjqqw (2.47) Such optimizations are all further explai ned 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 via-points, on the path. For the case of the 3DTP, the via-points are the initial and final poi nts for each climbing step, including undocking and docking. Using cubic polynomial trajectories w ill provide discontinuities along the acceleration and as such are us eless. Using quintic polynomials is then a logical choice for planning the trajectory, suggested by fu rther research [Spo03, Atk78]. Therefore, if possible, it will be useful to use quin tic splines along the computed paths. PAGE 44 30 Trajectory generation is the intermediate step connecting the via-points and a simulation over time. As such, it is critical to use the proper fit for the via-points and to constrain it accordingly, since otherwise, the trajectory generator is free to choose trajectories that are well outsi de of the geometrical and physical capabil ities of the manipulator. With the knowledge of the differe nt 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 ge nerate a comparison database for optimized paths. PAGE 45 31 CHAPTER 3 PRELIMINARY CLIMBING OF THREE DIME NSIONAL TEST PLATFORM (3DTP) ROBOT The previous chapter provided the th eoretical background necessary for the development of a climbing algorithm exhibiting torque-minimal paths. In this chapter, three different climbing methodol ogies are explored as candida te methods of climbing for which optimal paths will be generated leading to correspond ing trajectories. The general characteristics of each climbing method ar e quantified for down-selecting each method used for further algorithm development. 3.1 Design and Configuration As previously stated, the 3DTP was designe d 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 diffe rent climbing methods. The 3DTP has a sixjoint configuration where the first and last jo int axes are perpendicular to the other four joint axes, which are in turn all parallel. Wh en two 3DMPs are joined they form a fourjoint serial manipulator with all joint axes pa rallel (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 robot s, 3DTP, 3DRP, and 3DMP demonstrating common features and differences. PAGE 46 32 Figure 3.1 Different hybrid mobility kinemati cal designs: (a) 3DTP on smart structure, (b) 3DRP joint axes and offsets [Cle 03], (c) 3DMP joint axes and offsets [Cle03] For a numerical analysis, the 3DTP is m odeled out of acrylic, which features the properties, 3.8 GPa E (Youngs modulus), 0.36 v (Poissons ratio), and 3kg 1190 m (density). For 3DTP kinematics, th e length parameters are specified as (a) (b) (c) S1 S2 S3 S4 S5 S6 PAGE 47 3300.05 m L 10.05 m L 20.15 m L 30.075 m L 40.15 m L 50.05 m L 60.05 m L 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 i Link lengths (m), 1 ia Twist angles, 1 i Joint offsets (m), iS Joint angles, i 1 0 90 0.1 Variable 2 0 -90 0 Variable 3 0.15 0 0 Variable 4 0.075 0 0 Variable 5 0.15 0 0 variable 6 0 90 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 (kg-m2) i Center of mass (m), ,CMir Mass (kg), im, x xi I ,yyi I ,zzi I 1 T 2002.34*10 2.64e-2 1.02e-51.01e-5 1.62e-6 2 T0.07500 7.17e-2 5.56e-61.13e-4 1.12e-4 3 T0.037500 4.31e-2 2.81e-63.10e-5 3.10e-5 4 T0.07500 7.17e-2 5.56e-61.13e-4 1.12e-4 5 T 202.34*100 2.64e-2 1.02e-51.62e-6 1.01e-5 6 T 2002.34*10 1.71e-2 3.45e-63.45e-6 8.32e-7 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 sequenc es with an arbitrary path between two docking ports. The arbitrary path will be optimized using a manipulability based PAGE 48 34 exhaustive search algorithm. For docking a nd undocking sequences, the first and last joints are set as 1690 throughout the motion sequence. The remaining joints can be found by considering the mechanism as a closed-loop kinematic chain and having four revolute joints parallel to each ot her. to this avail the kinemati cs reduces to the analysis of a four-bar mechanism, illustrated in Figure 3.2. Furthermore, the constraint 2345180 must be satisfied given 2290, 323 and 434 Refer to Figures 2.2 and 3.2. Figure 3.2 Close-loop mechanism parameters PAGE 49 35 Next, consider the closed loop equations 0122334456coscoscos0lllll (3.1) 223344sinsinsin0lllh (3.2) Setting one variable (such as 3 ) as a known parameter, the following derivations show that an instantaneous closed-f orm solution can be obtained for 2 and 4 given 01l and 56l 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 2244560133coscoscoslllll (3.3) 224433sinsinsinllhl (3.4) Define 3560133cos xlll and 333sin yhl Then solving for 2cos and 2sin 44 2 2cos cos xl l (3.5) 44 2 2sin sin yl l (3.6) Squaring and adding the previous two equations 222 4444 22 22 2 22cossin cossin1 xyllxy l (3.7) Therefore, 2222 42 44 4cossin 2 x yll xy l (3.8) Defining 2222 42 3 42 x yll z l And considering that PAGE 50 36 22 3444cossincosarctan y zxyxy x (3.9) One can easily solve for 4 as 43 22arccosarctan zy x xy (3.10) Then referring to equations (3.5) and (3.6) 2 23 2sin arctan cos (3.11) After obtaining 2 and 4 as a function of 3 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 f easible configurations and co rresponding endpoints to the climbing robots paths. This provides a simp lification 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 0000 1,16,6 00 16eezpzp zz J (3.12) The Jacobian matrix provides a basis for fu rther optimization of climbing methods using the manipulability measure. Every time the J acobian matrix is square, the definition of manipulability given in equati on (2.34) applies. However, for certain climbing methods the 3DTP behaves as a four revolute joint ma nipulator, and as such there is additional redundancy. During such climbing sequences the Jacobian must be reduced for PAGE 51 37 meaningful manipulability measures and in th at 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 r ectilinear climbing, the Jacobian is reduced to the following 22232425 32333435 42434445JJJJ JJJJ JJJJ reducedJ (3.13) Which reduced equation (2.1) to 2 3 4 5 xq y q z q q reducedJ (3.14) This Jacobian contains only the components affecting motion in the y z, and x directions due to the 2nd, 3rd, 4th, and 5th joints. On the other hand, the Jacobian for side climbing is reduced to 1112131415 3132333435 5152535455JJJJJ JJJJJ JJJJJ reducedJ (3.15) Which once again reduced equation (2.1) to 1 2 3 4 5 yq x q zq q q reducedJ (3.16) PAGE 52 38 Due to the motion in the x y and y 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 th e 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 op timized climbing using manipulability. 3.4 Climbing Methods This section will explore the three clim bing 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 op timized using a quasi-dynamic method, thus not considering the concept of manipula bility [Cle03]. The side method was also proposed in the previous resear ch; 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 optim ization 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 closed-loop mechanism as shown in Section 3.2. For every climbi ng maneuver, both 3DTP end-effectors 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 PAGE 53 39 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 pa rameters for 3DTP robot simulations Docking Undocking End-effector Parameters Initial Final Initial Final 0L 0.05 m 0.05 m 0.05 m 0.05 m 1, free end 0L 0.05 m 0.025 m0.025 m 0.05 m 6L 0.05 m 0.05 m 0.05 m 0.05 m 2, grounded end 6L 0.025 m0.025 m0.025 m 0.025 m 0L 0.05 m 0.05 m 0.05 m 0.05 m 1, free end 0L 0.025 m0.025 m0.025 m 0.025 m 6L 0.05 m 0.05 m 0.05 m 0.05 m 2, grounded end 6L 0.05 m 0.025 m0.025 m 0.05 m L0' L0 L6' L6 h L0' L0 L6' L6 h (a)(b) Undocking Docking 1 grounded 2 grounded 1 grounded 2 free Figure 3.3 Configuration of end-effector 2 for 3DTP robot (a) docked and (b) undocked PAGE 54 40 The 0L and 6L parameters refer to the distance from the first or last joint to the smart structure being climbed, respectively. In this fashion, the joint para meters 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 comp aring 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 sele ction was made to minimize the number of variables that could cause an inconsistent evaluation and subsequent conclusion of the comparative results, especially when consid ering the rectilinear climbing maneuver. This maneuver performs several docking maneuve rs 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 previo usly demonstrated for the 3DMP robot [Cle03], flip climbing is a me thod of climbing for which th e manipulator undocks one of its ends and follows a path by flipping to ar rive at the next desired docking point. The method can be further visualized in Figure 3.4. The flip maneuver is bounded by the cl osed-loop mechanis m configurations illustrated by Figure 3.4 (a) and (c). In th ese 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. PAGE 55 41 Figure 3.4 Flip climbing maneuver (a) E nd-effector 2 post-undocked (b) Sample sequence of flip steps (c) End-effector 2 pre-docked Table 3.4 Flip maneuver boundary conf igurations for numerical example Joint angles End-effector 2 docked (initial) End-effector 2 postundocked(Fig. 3.4 (a)) End-effector 2 predocked (Fig. 3.4 (c)) End-effector 2 docked (final) 1 90 90 90 90 2 -138.59 -132.65 -47.35 -41.41 3 -41.410 -47.35 47.35 41.41 4 -41.410 -34.67 34.67 41.41 5 41.410 34.67 -214.67 -221.41 6 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 disp lacement as for the flip method. Alternate PAGE 56 42 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. Figure 3.5 Side climbing maneuver (a) E nd-effector 2 post-undocked (b) Sample sequence of side steps (side view ) (c) End-effector 2 pre-docked A visualization of the side climbing met hod is shown in Figure 3.5, where Figure 3.5 (b) demonstrates the 90 to -90 rotati on 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 num erical analysis. Note, for the preliminary investigations of the side climbing method, the configuration of all inner joints is arbitrarily held constant. Upon introduci ng an optimization algorithm such as the manipulability based approach, these joint conditions are bound to change. PAGE 57 43 Table 3.5 Side maneuver boundary configurations Joint angles End-effector 2 docked (initial) End-effector 2 postundocked (Fig. 3.5 (a)) End-effector 2 predocked (Fig. 3.5 (c)) End-effector 2 docked (final) 1 90 90 -90 -90 2 -138.59 -132.65 -132.65 -138.59 3 -41.41 -47.35 -47.35 -41.41 4 -41.41 -34.67 -34.67 -41.41 5 -48.59 -55.33 -55.33 -48.59 6 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 ag ain, 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 step s (g) through (k)). In the simulated cases, the end-effector is raised half the initial distance between the two docked end-effectors (Figure 3.6 (b)). End-effector 2 then docks (Figure 3.6 (c)), fo llowed by a release of end-effector 1 (Figure 3.6 (d)). The robot continues to climb by rais ing end-effector 1 the user defined height once again (Figure 3.6 (e)). End-effector 1 doc ks 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 onl y two sequences of the climbing maneuvers. PAGE 58 44 In addition to the initial post-undock and final pre-dock configurations shown in Figures 3.6 (a) and (k), respectively; the rectilinear climbing yi elds additional boundary configurations, listed in Table 3.6. Thes e configurations provide boundaries for all climbing steps and optimizations. Table 3.6 Rectilinear maneuver boundary configurations for 3DTP robot Joint angles End-effector 2 predocked (Fig. 3.6 (b) and (h)) End-effector 2 docked (Fig. 3.6 (c) and (i)) End-effector 1 postundocked (Fig. 3.6 (d) and (j)) End-effector 1 pre-docked (Fig. 3.6 (e) and (k)) 1 90 90 90 90 2 -86.84 -104.48 -123.71 -145.33 3 -93.16 -75.52 -56.29 -34.67 4 -56.29 -75.52 -93.16 -47.35 5 56.29 75.52 93.16 47.35 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 th e robots workspace, that is, searching the entire range of moti on for each joint combination in the form of ,min,max iiiqqq where 1 jj iiiqqq 0 ,min iiqq ,max n iiqq In this fashion, solutions can be refined by decreasing the step size iq for finding different joint values at the cost PAGE 59 45 Figure 3.6 Rectilinear climbing maneuver (a ) End-effector 2 post-undocked (b) throu gh (j) Sample sequence of rectilinear steps including intermediate dockings (k) End-effector 1 pre-docked PAGE 60 46 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 HS-645MG, 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 postundocking position to find an optimal configur ation using the closed-loop form of the mechanism, varying 3 Secondly, the same method was used to find the optimal configuration at pre-docking. Afterwards, thes e two configurations set boundaries for an intermediate step which involved performing an exhaustive search in the serial mode of the mechanism varying 3 4 and 5 to determine the climbing configuration yielding minimal (max joint torques) at the midpoint of joint motion 2 Table 3.7 Actual mechanism inertia para meters for 3DTP robot including motors Inertia tensor (kg-m2) i Center of mass (m), CMirMass (kg), im, x xi I yyi I zzi I 1 T 2002.34*10 8.14e-2 3.37e-53.37e-5 1.25e-5 2 T0.07500 0.182 2.73e-57.53e-4 7.53e-4 3 T0.037500 4.31e-2 2.81e-63.10e-5 3.10e-5 4 T0.07500 0.182 2.73e-57.53e-4 7.53e-4 5 T 202.34*100 8.14e-2 3.37e-53.37e-5 1.25e-5 6 T 2002.34*10 1.71e-2 3.45e-63.45e-6 8.32e-7 PAGE 61 47 Table 3.8 Flip climbing minimal torque approa ch optimal configurations for 3DTP robot Joint variable End-effector 1 postundocked (Fig. 3.4 (a)) Flip step 1 (Fig. 3.4 (b)) End-effector 1 pre-docked (Fig. 3.4 (c)) 1q 90 90 90 2q -132.16 -90 -47.84 3q -92.84 77.37 92.84 4q 32.23 26.86 -32.23 5q 12.77 154.36 167.23 6q 0 0 0 3.7 Climbing Trajectories The preliminary joint trajec tories 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 co nfigurations defined in previous sections and intermediate steps as in the case of the minimal torque optimized flip method, the joint trajectories are generate d using quintic splines. It is important to note that using quintic spline fits between these given conf igurations will provid e continuity in the position, velocity, and finite a ccelerations of the resulting tr ajectories. However, if the difference in via-points is large, the results can be physically limiti ng since the joints will experience a wavy maneuver through undesired c onfigurations, such as going through the smart structure or through manipulator links as can be seen in A ppendix C, Figures C.1 through C.3. To this avail in this thesis, more via-points are obtained in between the ones optimized by simply using linear interpolati on between the boundary configurations. This method is further named segment linearization since segments of data are interpolated linearly. The method of segment linearization re duces the joint power required to perform the maneuver, however, in some cases this is at the cost of increasing joint torques as PAGE 62 48 seen in Table 3.9 with a combination of resu lts 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 manipulabil ity path optimized trajectories Climbing methodNo segment linearizationSegment linearization Flip 0.95179 1.0641 Side 1.091 1.0137 max (Nm) Rectilinear 0.46016 0.40237 Flip 0.83289 0.26146 Side 1.9589 0.1274 net,maxP (W) Rectilinear 0.18787 0.12837 As it can be seen, without segm ent linearization, in addition to the trajectories obtained being impossible to perform due to physical c onstraints, the maximum net power is quite large. When segment linearization is app lied, though not consta ntly, joint torques decrease slightly, while net pow er decreases considerably in all methods and the resulting trajectories are realistic as will be seen in ensuing chapters. Afterwards, this denser amount of via-points is interpolated usi ng 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 ma ximum overall power required, the effect of gravity (for comparison with joint tor ques and with the quasi-dynamic 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. PAGE 63 49 Figure 3.7 Minimal torque optimized flip trajectory of 3DTP robot: joint angles Figure 3.8 Minimal torque optimized flip trajectory of 3DTP robot: joint velocities PAGE 64 50 Figure 3.9 Minimal torque optimized flip tr ajectory of 3DTP robot: joint accelerations The joint torques were computed using the inverse dynamics as mentioned beforehand. The manipulability measure was al so computed along these joint trajectories. Additional measures for comparison were also computed from these trajectories and are shown in the following set of figures. Figure 3.10 Minimal torque optimized flip trajectory of 3DTP robot: joint gravity effects PAGE 65 51 Figure 3.11 Minimal torque optimized flip trajectory of 3DTP robot: joint torques Figure 3.12 Minimal torque optimized flip trajectory of 3DTP robot: joint power PAGE 66 52 Figure 3.13 Minimal torque optimized flip trajectory of 3D TP robot: net power 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 select ed 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 ad dition, the joint torques at the specific instant where the maximum occurs did not decrea se due to the fact that there will always PAGE 67 53 be a need to flip the arm. Furthermore, the manipulator configuration can only be realistically contracted up to its geometri c 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 cl imbing 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 traject ories for these preliminary simulations are shown in the following figures, followed by co mputed joint torques and other useful measures of the flip climbing method. Figure 3.15 Preliminary flip clim bing of 3DTP robot: joint angles PAGE 68 54 Figure 3.16 Preliminary flip climbing of 3DTP robot: joint velocities Figure 3.17 Preliminary flip climbing of 3DTP robot: joint accelerations PAGE 69 55 Figure 3.18 Preliminary flip climbing of 3DTP robot: joint gravity effects Figure 3.19 Preliminary flip clim bing of 3DTP robot: joint torques PAGE 70 56 Figure 3.20 Preliminary flip clim bing of 3DTP robot: joint power 3.7.3 Preliminary Side Climbing In this section, simulation results are pr esented for the preliminary side climbing method. Figure 3.21 Preliminary side clim bing of 3DTP robot: joint angles PAGE 71 57 Figure 3.22 Preliminary side climbi ng of 3DTP robot: joint velocities Figure 3.23 Preliminary side climbing of 3DTP robot: joint accelerations PAGE 72 58 Figure 3.24 Preliminary side climbing of 3DTP robot: joint gravity effects Figure 3.25 Preliminary side clim bing of 3DTP robot: joint torques PAGE 73 59 Figure 3.26 Preliminary side clim bing of 3DTP robot: joint power 3.7.4 Preliminary Rectilinear Climbing This section presents the trajectories a nd simulation results of the 3DTP climbing via the rectilinear method. Figure 3.27 Preliminary rectilinear cl imbing of 3DTP robot: joint angles PAGE 74 60 Figure 3.28 Preliminary rectilinear clim bing of 3DTP robot: joint velocities Figure 3.29 Preliminary rectilinear clim bing of 3DTP robot: joint accelerations PAGE 75 61 Figure 3.30 Preliminary rectilinear climbi ng of 3DTP robot: joint gravity effects Figure 3.31 Preliminary rectilinear cl imbing of 3DTP robot: joint torques PAGE 76 62 Figure 3.32 Preliminary rectilinear climbing of 3DTP robot: joint power 3.7.5 Further Comparison Resu lts on Preliminary Climbing For the trajectories presented in the above sections, the following plots are used to further delineate the comparison of the diffe rent climbing algorithms. Additional data was obtained regarding the maximum inst antaneous power for different climbing methods as shown in Figure 3.33. Figure 3.33 Preliminary climbi ng of 3DTP robot: net power PAGE 77 63 In this plot it can be clearly seen that th e 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 c limbing methods is shown in Figure 3.34. Figure 3.34 Preliminary climbi ng 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 summa ry, 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 le ss 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. Compar ed to the torque optimization it can be seen that all methods but preliminary flip climbing perform better overall. PAGE 78 64 Table 3.10 Summary of preliminary simu lation results for 3DTP robot climbing Method max (Nm)maxP (W) net,maxP (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 pr ocess in the ability of the robot to be autonomous and completely able to roam a nd inspect by itself, for if maximum joint torque is well below the maximum provide d 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 batter ies to run the robot the same amount of time. Overall, the rectilinear method is the met hod of choice for climbing a smart structure, though it may require larger batteries. The side method is feasible bu t cannot carry a large load if it can carry a lo ad at all. The flip method is the l east realistic of al l. In the next chapter, the optimization of all these methods using the manipulability measure will be explored to investigate whether or not the a bove conclusions of this chapter will remain valid upon application of th e optimization algorithms. PAGE 79 65 CHAPTER 4 OPTIMIZED CLIMBING OF THREE DI MENSIONAL 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 [C le03], where quasi-dynamic exhaustive searches were done to find optimal maximum torque configurations throughout the workspace of a 3DMP. Add itionally, via-points selected from the resulting min-max 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 closed-loop anal ysis, as presented in Chapter 3, the maximum manipulability configuration (q when maxww ) defines optimal docking/undocking configurations. Docking/undo cking optimization was done using an exhaustive search as introduced in Chapter 3 considering 3180180, where 3323, per the previous chapter, and 31 In addition, the feasible solution space is reduced by the introduction of the phys ical constraint that the mechanism cannot pass through the smart structure being clim bed. Considering the steps throughout this method, the resulting constrained optimal configurations for pre-docking and postundocking are shown in Table 4.1. PAGE 80 66 Table 4.1 Optimal manipulability configurat ions of 3DTP robot at post-undocking and pre-docking steps Method and description Joint variables Post-undocking (0.3 m h ) Pre-docking (0.3 m h ) 1q 90 90 2q -134.21 -45.79 3q -40.79 40.79 4q -41.42 41.42 5q 36.42 143.58 Flip climbing and side climbing cases, endeffector 2 free 6q 0 0 Post-undocking (0.3 mh ) Pre-docking (0.15 mh ) 1q 90 90 2q -134.21 -94.97 3q -40.79 -76.03 4q -41.42 -74.20 5q 36.42 65.20 Rectilinear climbing case, end-effector 2 free 6q 0 0 Post-undocking (0.15 m h ) Pre-docking (0.3 m h ) 1q 90 90 2q -114.80 -143.58 3q -74.20 -41.42 4q -76.03 -40.79 5q 85.03 45.79 Rectilinear climbing case, end-effector 1 free 6q 0 0 4.2 Flip and Rectilinear Climbing Optimization Flip and rectilinear climbing share a co mmon workspace where the Jacobian uses only its y z, and x 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 three-by-four matrix. PAGE 81 67 In order to optimize for flip climbing, the joint angles where searched in the following ranges, 2134.2145.79, 340.7936040.79, 441.4236041.42, and 5143.58360143.58 with the steps 222.103, 35, 420, and 540. The first step ensures that 2 stays within the range provided by post-undock and pre-dock confi gurations provided in Table 4.1, while obtaining optimal configurations at three intermediate 2 The remaining steps provide for full rotations of the corresponding joint variab les in order to maximize manipulability performing the exhaustive sear ch. 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 r obot at select intermediate steps for the flip climbing maneuver Joint variable1 22qq 2 22qq 3 22qq 1q 90 90 90 2q -134.21-90 -67.90 3q -40.79 -44.21 -44.21 4q -41.42 -58.58 -58.58 5q 36.42 103.5863.58 6q 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, 2134.2194.97, 376.0340.79, 474.2041.42, and 536.4265.20 with the steps 219.62, 30.49, 41.82, and 53.20. These steps provide a desired rang e of operation for the manipulator PAGE 82 68 throughout the rectilinear clim bing maneuver, obtaining one ve ry 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, 2143.58114.80, 374.2041.42, 476.0340.79, and 545.7985.03 with the steps 214.39, 30.46, 41.96, and 54.36. The optimal configurations repeat again as shown in Fi gure 3.6. Table 4.3 shows th e two optimal intermediate configurations found. Table 4.3 Optimal manipulability configurations of 3DTP r obot at select intermediate steps for the rectilinear climbing method Description Intermediate climbing step between Fig. 3.6 (a) and (b) Intermediate climbing step between Fig. 3.6 (c) and (d) Joint variable 1 22qq 1 22qq 1q 90 90 2q -114.59 -129.19 3q -51.07 -46.43 4q -50.53 -52.54 5q 62.00 80.67 6q 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 post-undocked position to the next pre-docking position in both mane uvers. Additionally, it can be easily observed that in the flip climbing maneuver, while performi ng the exhaustive search, the joints are searched through a whole revolu tion in order for the optimal manipulability configuration to be found. On the other hand, for the rec tilinear climbing maneuver, the joints are only PAGE 83 69 searched in the range of motion provided by boundary post-undocking and pre-docking optimized configurations. This reduced c onfiguration 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 sm art 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 y and joints 1 through 5 or 2 through 6, depending on the free and grounded end-effector (in this cas e, 1 through 5, because end-effector 1 is simulated as grounded). In side climbing the jo int angles are optimized in the ranges of 19090, 2134.21134.21360, 340.7940.79360, 441.4241.42360, and 536.42336036.423. The steps for these ranges are 145, 215, 315, 430, and 330. 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 variable1 11qq 1q 0 2q -224.21 3q 34.21 4q 48.58 5q 96.42 6q 0 PAGE 84 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 ab sent in between optimized via-points are populated by linear interpolation. This is follo wed by utilizing quintic spline interpolation to guarantee continuity in positions, veloci ties and accelerations between the optimized and the additional segment via-points. The re sults obtained in this fashion are shown graphically in the subsequent sections. 4.4.1 Flip Climbing Method The trajectory obtained for the flip met hod and the ensuing simulations are shown in this section. As it will be seen, the results lack the symmetry of preliminary simulations. Figure 4.1 Path optimized flip me thod for 3DTP robot: joint angles PAGE 85 71 Figure 4.2 Path optimized flip met hod for 3DTP robot: joint velocities Figure 4.3 Path optimized flip met hod for 3DTP robot: joint accelerations 4.4.2 Side Climbing Method The side climbing method traj ectories generated from the optimal path previously obtained are shown in this section. PAGE 86 72 Figure 4.4 Path optimized side me thod for 3DTP robot: joint angles Figure 4.5 Path optimized side met hod for 3DTP robot: joint velocities PAGE 87 73 Figure 4.6 Path optimized side met hod for 3DTP robot: joint accelerations 4.4.3 Rectilinear Climbing Method This section depicts the trajectories ge nerated via the optim al configurations obtained for the recti linear climbing method. Figure 4.7 Path optimized rectilinear method for 3DTP robot: joint angles PAGE 88 74 Figure 4.8 Path optimized rectilinear method for 3DTP robot: joint velocities Figure 4.9 Path optimized rectilinear me thod for 3DTP robot: joint accelerations 4.5 Inverse Dynamics Simulation As in the previous chapter, once the traject ories 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 Lagrangi an approach is used to obtain the equations PAGE 89 75 of motion from which the joint torques are calculated along the resu lting trajectories at each instant. The results of such simula tions are shown in the plots following. Figure 4.10 Path optimized flip method fo r 3DTP robot: gravitat ional load effects Figure 4.11 Path optimized flip me thod for 3DTP robot: joint torques PAGE 90 76 Figure 4.12 Path optimized side method fo r 3DTP robot: gravitat ional load effects Figure 4.13 Path optimized side me thod for 3DTP robot: joint torques PAGE 91 77 Figure 4.14 Path optimized rectilinear method for 3DTP robot: gravitational load effects Figure 4.15 Path optimized rectilinea r method for 3DTP robot: joint torques PAGE 92 78 Figure 4.16 Path optimized 3DTP r obot climbing met hods: manipulability As it can be seen from the previous set of results, throughout all climbing methods gravity has the most effect on joint torques. Th is 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 optimi zation, 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 minimal-maximum torques. Such performance i ndicates 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. PAGE 93 79 Figure 4.17 Path optimized flip method for 3DTP robot: joint power Figure 4.18 Path optimized side method for 3DTP robot: joint power PAGE 94 80 Figure 4.19 Path optimized rectilinea r method for 3DTP robot: joint power Figure 4.20 Path optimized climbing methods for 3DTP robot: net power Comparing all results with thos e of Chapter 3, it can be obser ved that after the path was optimized, the overall power requirement d ecreased 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 PAGE 95 81 combination of the joint configurations resulted in the 3DTP acquiring a flipped configuration. Overall; however, the ma ximum values for each method may have increased for all methods except for the rect ilinear which saw a si gnificant reduction in maximum power required. 4.6 Observations From the path optimized results, it can be seen that manipulability-based optimization did not consistently yield im proved torque characteristics and maximum power results throughout the generated trajectorie s. 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 provi des a summary of the results obtained. Table 4.5 Summary of preliminary, torque, and path optimized results for 3DTP robot Climbing method max (Nm)maxP (W)net,maxP (W) Optimization Flip 1.1059 0.35917 0.74262 Side Rectilinear Minimal torque Flip 1.1551 0.0614180.11075 Side 0.86389 0.0969280.096928 Rectilinear 0.34148 0.12857 0.36309 No optimization Flip 1.0641 0.11335 0.26146 Side 1.0137 0.11374 0.1274 Rectilinear 0.40237 0.0734 0.12837 Maximum manipulability Of all the climbing methods, the flippi ng method had less constraints on joint motion over which optimization could occur. As such Figures 4.16 and 3.34 indicate trajectories with greater range s of high manipulability values Side climbing on the other hand, the manipulability exhibi ted decreased values. As it can be seen from these results, of all the climbing methods, by using mani pulability only flippi ng had a large enough range to be optimized properly. Additiona lly, considering the initial manipulability PAGE 96 82 measures from the preliminary simulations flip climbing had a larger range for manipulability. This facilitated the optimiza tion. 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 0 w thus increasing joint torques. Rectilinear climbing was largely opt imized and in fact, the small optimization achieved increased the joint torques. Regard ing resulting power conditions, the maximum power required to achieve the maneuver decr eased considerably for the rectilinear climbing method while for the other methods in creases 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 compar ison are normalized by using the flip method optimized as presented in previous research as a baseline for comparison as shown %ir i r (4.1) %ir i rPP P P (4.2) Where r and rP 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, indicati ng an increase in performance in most cases. First observe that flip climbing was preliminarily worse torque-wise than compared to the quasi-dynamic optimizati on. Regarding power requirements, flip climbing was actually much better prelimin ary than the baseline. After being path PAGE 97 83 optimized, flip climbing performed better than its static optimizat ion baseline. Side climbing was actually considerab ly a better method than flip climbing torque and powerwise. 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 ove rall the better met hod of the three. Preliminarily, rectilinear climbing performed much better than the other two methods. After being path optimized, rec tilinear climbing still outperfor med its counterparts, in all indices. Table 4.6 Performance of different climbing optimizations for 3DTP robot as compared to previous research Climbing method max% max%Pnet,max%POptimization Flip 4.45 -82.9 -85.1 Side -21.9 -73.0 -86.9 Rectilinear -69.1 -64.2 -51.1 No optimization Flip -3.78 -68.4 -64.8 Side -8.34 -68.3 -82.8 Rectilinear -63.6 -79.6 -82.7 Maximum manipulability 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 simu lation. 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 intermedia te method in the sense that it outperforms flip climbing preliminarily and while statically optimized when regarding torque. Its power requirements are also si gnificantly improved as opposed to flip climbing. After optimized, side climbing is actually not a f easible method in part due to the segment linearization as can be observed in Table 4.5. Rectilinear climbi ng is once again by far PAGE 98 84 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 ot her robots designed for the purpose of inspection and hybrid mobility, th e capability of being autono mous is the most important concern. The loading capacity of these robots increases if they use rectilinear climbing as the preferred method when on a ve rtical surface. This is an indicator of requiring smaller batteries for a comparative time span of opera tion or having the abil ity to operate longer with the same size of batteries as those us ed if climbing in other methods. Lower power requirements as prescribed by Tables 4.5 and 4.6 are such indicator s. This relationship between power required and operational time comes from the relationships between power required, voltage, and discharge cu rrent defined by Peukerts law and the definition of the volt, where the former is kCIt (4.3) And the latter is defined as the poten tial 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 pow er and discharge time is given by CV P t (4.4) The capacity to weight ratio depends on th e kind of battery and behaves linearly for all types, (e.g. alkaline dry cells, nickel-c admiun, lithium-ion). This means that as the PAGE 99 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 opera tion. Therefore, as previously observed and discussed the rectilinear met hod will be the most useful for autonomous operations, for carrying additional loads as sensors, and fo r having the longest operational time if its battery is chosen to be of higher capacity. PAGE 100 86 CHAPTER 5 CONCLUSIONS This research presents an alternative approach to the optim ization of climbing methods using the concept of manipulabi lity and proposing different methods of climbing. The system developed for this re search 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 fo r smart structure climbing. Additionally, the concept of manipulability was used to optim ize the path being climbed at via-points. Trajectories were generated using quintic splines throughout the path in order to apply initial conditions for velocity and accelerati on while maintaining continuity to the second derivative. The three methods proposed, flip, side, a nd 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 th e 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 consum ption 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 PAGE 101 87 much more efficient one, but due to the fact that the 3DTP robot uses six motors, one in each joint, and that instead all climbing met hods use only four motors independently. If the unused motors are removed it is obvious that the 3DTP robot is th en lighter, therefore the load in the remaining motors decreases si gnificantly. In light of this, the rectilinear and flip climbing methods could be used in terchangeably, since their configurations consist of the same four moto rs, thus, reaping the benefits of both methods as needed. through creating hybrid climbing algorithms. Fo r example, the previously developed 3DMPs need to reconfigure for climbing along a vertical smart structure. To initiate the climbing mode, the rectilinear algorithm is not feasible. Therefore, perhaps a combination of side climbing and flip climbing may prove adequate for taking the 3DMPs to a vertical smart structure climbing configuration. The climbing methods provide three solid de finitions for approaches to climbing. However, further research must be done on the development of a robust climbing algorithm able to select the appropriate solution of the inverse kinematics while considering structural constr aints in the design, such as joint ranges and so forth. Additionally, experimental te sts regarding the robustness of the different climbing methods must also be performed, since th e rectilinear climbing algorithm may indeed prove to be more efficient when a load is being carried on the thir d and fifth links. The packaging of code for ease in generating via-poi nts, trajectories, time ranges, and so forth is also work in progress. As to future research, experimental validation must be performed on all three algorithms. Additiona l effects such as joint friction, link flexibilities, joint imperfections, and payload and inertia variations needs to be explored. PAGE 102 88 While the definition of the dynamic mani pulability measure was introduced, its implementation within the path optimization al gorithm has been left for future research exploring in greater depth th e inertia effects and faster climbing rates. With greater understanding of these effects on the path and trajectory planning, this work should impact design optimization of hybrid robots yielding optimal path, trajectory planning and control. PAGE 103 89 APPENDIX A MATLAB FUNCTIONS AND SCRIPTS The functions and scripts provided in this ap pendix are meant as useful snippets of code for verification purpos es or the use of readers in their own designs and optimizations. In the electronic copy of this docum ent, the users will be able to select the desired code, copy, and paste into MATLA B and perform the desired operations immediately. A.1 Recursive Lagrangian Dynamics for Serial Manipulators function [M,h,g]=EOM(q,qd,qdd,parameters) %% Numerical equations of motion via Lagrangian dynamics % q: nx1 vector of joint variables % qd: nx1 vector of joint velocities % qdd: nx1 vector of joint accelerations % parameters: structure containing the following variables: % joints: nx1 vector containing joint type (R=0, P=1) % a: (n+1)x1 vector containing link lengths % alpha: (n+1)x1 vector containing twist angles % d: (n+1)x1 vector containing joint offsets % theta: (n+1)x1 vector containing joint angles % mass: nx1 vector containing moving link masses % inertia: nx1 cell array containing moving link moments of inertia % shat: nx1 cell array containing moving link center of mass locations % gtilde: 4x1 array containing in its first three elements the % gravitational constant and 0 in the last element % Refer to Yoshikawa's Foundations of Robotics for further clarification on % the parameters necessary for Lagrangian dynamics joints=parameters.joints; a=parameters.a;alpha=parameters.alpha;d=parameters.d;theta=parameters.t heta; mass=parameters.mass;inertia=parameters.inertia;shat=parameters.shat; gtilde=parameters.gtilde; n=length(joints); for i=1:n if (joints(i)==0) theta(i)=q(i); elseif (joints(i)==1) d(i)=q(i); PAGE 104 90 end end for i=1:n+2 T{i,i}=(eye(4)); end for i=1:n+1 T{i,i+1}=tra(a(i),alpha(i),d(i),theta(i)); end for i=1:n for j=i+2:n+2 T{i,j}=T{i,j-1}*T{j-1,j}; end end for i=2:n+2 for j=1:i-1 T{i,j}=(zeros(4)); end end for i=1:n Ihat{i}=[ inertia{i}(1,1)+mass(i)*((shat{i}(2))^2+(shat{i}(3))^2),inertia{i}(1,2)+mass(i)*((shat{i}(1))*(shat{i}(2))),inertia{i}(1,3)+mass(i)*((shat{i}(1))*(shat{i}(3))); inertia{i}(2,1)+mass(i)*((shat{i}(2))*(shat{i}(1))),inertia{i}(2,2)+mas s(i)*((shat{i}(1))^2+(shat{i}(3))^2),inertia{i}(2,3)+mass(i)*((shat{i}(2))*(shat{i}(3))); -inertia{i}(3,1)+mass(i)*((shat{i}(3))*(shat{i}(1))),inertia{i}(3,2)+mass(i)*((shat{i}(3))*(shat{i}(2))),inertia{i}(3,3)+mas s(i)*((shat{i}(1))^2+(shat{i}(2))^2) ]; Hhat{i}=[ (-(Ihat{i}(1,1))+(Ihat{i}(2,2))+(Ihat{i}(3,3)))/2,Ihat{i}(1,2),-Ihat{i}(1,3),mass(i)*shat{i}(1); -Ihat{i}(2,1),((Ihat{i}(1,1))(Ihat{i}(2,2))+(Ihat{i}(3,3)))/2,-Ihat{i}(2,3),mass(i)*shat{i}(2); -Ihat{i}(3,1),-Ihat{i}(3,2),((Ihat{i}(1,1))+(Ihat{i}(2,2))(Ihat{i}(3,3)))/2,mass(i)*shat{i}(3); mass(i)*shat{i}(1),mass(i)*shat{i}(2),mass(i)*shat{i}(3),mass(i) ]; end for i=1:n if (joints(i)==0) DELTA{i}=([ 0,-1,0,0; 1,0,0,0; 0,0,0,0; 0,0,0,0 ]); elseif (joints(i)==1) DELTA{i}=([ PAGE 105 91 0,0,0,0; 0,0,0,0; 0,0,0,1; 0,0,0,0 ]); end end for i=1:n for j=1:n d0Tdq{i,j}=T{1,j+1}*DELTA{j}*T{j+1,i+1}; for k=1:n d20Tdqdq{i,j,k}=T{1,j+1}*DELTA{j}*T{j+1,k+1}*DELTA{k}*T{k+1,i+1}; end end end M=(zeros(n)); h=(zeros(n,1)); g=(zeros(n,1)); for i=1:n for j=1:n for k=max([i,j]):n M(i,j)=M(i,j)+trace(d0Tdq{k,j}*Hhat{k}*transpose(d0Tdq{k,i})); end for m=1:n for k=max([i,j,m]):n h(i)=h(i)+trace(d20Tdqdq{k,j,m}*Hhat{k}*transpose(d0Tdq{k,i}))*qd(j)*qd (m); end end g(i)=g(i)-mass(j)*transpose(gtilde)*d0Tdq{j,i}*[shat{j};1]; end end A.2 Inverse Tangent function theta=myatan2(y,x) %% Inverse tangent % This inverse tangent function obtains the four quadrant inverse tangent % and works with symbolic elements, as opposed to the default atan2 % function provided by MATLAB theta=-i*log((x+i*y)./sqrt(x.^2+y.^2)); PAGE 106 92 APPENDIX B JACOBIAN AND TRANSFORMATION MATRIX ELEMENTS FOR 3DTP The contents of this appendix provide th e reader with values for the Jacobian matrix and transformation matrices for the 3DTP. B.1 Jacobian Matrix Elements of 3DTP Robot The Jacobian matrix for the 3DTP is of the form 111213141516 212223242526 126 313233343536 126 414243444546 515253545556 616263646566 vvvJJJJJJ JJJJJJ JJJJJJ JJJ JJJJJJ JJJ JJJJJJ JJJJJJ J (B.1) Specifically, for the 3DTP, first consider the following variables 1 1, 25 25ii i ii ii i (B.2) Where 212 212 (B.3) Then 214161525354551626360JJJJJJJJJJ (B.4) 511J (B.5) PAGE 107 93 222 333 11 444 5655sinsin sinsin 1 2 sinsin coscosL L J L LL (B.6) 222 333 31 444 5655coscos coscos 1 2 coscos sinsin L L J L LL (B.7) 222 333 12 444 5655sinsin sinsin 1 2 sinsin coscos L L J L LL (B.8) 22221331 4415651coscos cossin JLL LLL (B.9) 222 333 32 444 5655coscos coscos 1 2 coscos sinsin L L J L LL (B.10) 424344451sin JJJJ (B.11) 626364651cos JJJJ (B.12) 333 13444 5655sinsin 1 sinsin 2 coscos L JL LL (B.13) 22331441 5651coscos sin JLL LL (B.14) PAGE 108 94 333 33444 5655coscos 1 coscos 2 sinsin L JL LL (B.15) 444 14 5655sinsin 1 2 coscos L J LL (B.16) 244415651cossin JLLL (B.17) 444 34 5655coscos 1 2 sinsin L J LL (B.18) 1556551 coscos 2 JLL (B.19) 255651sin JLL (B.20) 3556551 sinsin 2 JLL (B.21) 46551 sinsin 2 J (B.22) 5651cos J (B.23) 66551 cossin 2 J (B.24) B.2 Transformation Matrices Elements of 3DTP Robot The transformation matrices from referen ce frame to reference frame are given in this section. 11 01 0 1 11cossin00 001 sincos00 0001 LL T (B.25) PAGE 109 95 22 1 2 22cossin00 0010 sincos00 0001 T (B.26) 332 33 2 3cossin0 sincos00 0010 0001 L T (B.27) 443 44 3 4cossin0 sincos00 0010 0001 L T (B.28) 554 55 4 5cossin0 sincos00 0010 0001 L T (B.29) 66 5 5 6 66cossin00 001 sincos00 0001 L T (B.30) 6 61000 0100 001 0001eL T (B.31) PAGE 110 96 APPENDIX C TRAJECTORY GENERATION WITHOUT SEGMENT LINEARIZATION This appendix demonstrates to the read er the results obtai ned without segment linearization for trajectory generation. Figure C.1 Path optimized flip climbing of 3DTP robot without segment linearization: joint angles PAGE 111 97 Figure C.2 Path optimized side climbing of 3DTP robot without se gment linearization: joint angles Figure C.3 Path optimized rectilinear cl imbing of 3DTP robo t without segment linearization: joint angles Figure C.4 Path optimized flip climbing of 3DTP robot without segment linearization: joint torques PAGE 112 98 Figure C.5 Path optimized side climbing of 3DTP robot without se gment linearization: joint torques Figure C.6 Path optimized rectilinear cl imbing of 3DTP robo t without segment linearization: joint torques PAGE 113 99 Figure C.7 Path optimized flip climbing of 3DTP robot without segment linearization: joint power Figure C.8 Path optimized side climbing of 3DTP robot without se gment linearization: joint power PAGE 114 100 Figure C.9 Path optimized rectilinear cl imbing of 3DTP robo t without segment linearization: joint power Figure C.10 Path optimized climbing of 3DTP robot without segmen t linearization: net power PAGE 115 101 APPENDIX D ADDITIONAL TABLES The following appendix provides the reader with joint torques for different climbing operations. Table D.1 Preliminary flip climbi ng joint torques for 3DTP robot Time (s) 1 (Nm) 2 (Nm)3 (Nm) 4 (Nm) 5 (Nm) 6 (Nm) 0.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 0.30000 0.00000 0.19003 -0.22461-0.22349-0.04530 0.00000 0.60000 0.00000 0.21684 -0.21405-0.21418-0.04526 0.00000 0.90000 0.00000 0.23638 -0.20848-0.20807-0.04559 0.00000 1.20000 0.00000 0.26149 -0.19074-0.19590-0.04512 0.00000 1.50000 0.00000 0.28626 -0.17279-0.18525-0.04500 0.00000 1.80000 0.00000 0.31190 -0.15346-0.17338-0.04455 0.00000 2.10000 0.00000 0.33783 -0.13374-0.16109-0.04391 0.00000 2.40000 0.00000 0.36408 -0.11359-0.14833-0.04306 0.00000 2.70000 0.00000 0.39058 -0.09306-0.13516-0.04202 0.00000 3.00000 0.00000 0.41731 -0.07218-0.12159-0.04079 0.00000 3.30000 0.00000 0.44422 -0.05100-0.10766-0.03938 0.00000 3.60000 0.00000 0.47127 -0.02955-0.09341-0.03779 0.00000 3.90000 0.00000 0.49840 -0.00789-0.07886-0.03603 0.00000 4.20000 0.00000 0.52559 0.01395 -0.06406-0.03410 0.00000 4.50000 0.00000 0.55277 0.03592 -0.04904-0.03202 0.00000 4.80000 0.00000 0.57991 0.05798 -0.03383-0.02979 0.00000 5.10000 0.00000 0.60695 0.08008 -0.01848-0.02743 0.00000 5.40000 0.00000 0.63386 0.10217 -0.00303-0.02495 0.00000 5.70000 0.00000 0.66057 0.12421 0.01249 -0.02235 0.00000 6.00000 0.00000 0.68705 0.14614 0.02803 -0.01965 0.00000 6.30000 0.00000 0.71324 0.16793 0.04356 -0.01686 0.00000 6.60000 0.00000 0.73910 0.18952 0.05904 -0.01400 0.00000 6.90000 0.00000 0.76457 0.21087 0.07442 -0.01107 0.00000 7.20000 0.00000 0.78962 0.23192 0.08966 -0.00809 0.00000 7.50000 0.00000 0.81418 0.25264 0.10473 -0.00508 0.00000 7.80000 0.00000 0.83822 0.27298 0.11958 -0.00204 0.00000 8.10000 0.00000 0.86168 0.29289 0.13417 0.00101 0.00000 8.40000 0.00000 0.88453 0.31233 0.14848 0.00405 0.00000 8.70000 0.00000 0.90671 0.33124 0.16245 0.00707 0.00000 9.00000 0.00000 0.92818 0.34960 0.17605 0.01006 0.00000 PAGE 116 102 9.30000 0.00000 0.94891 0.36735 0.18924 0.01301 0.00000 9.60000 0.00000 0.96884 0.38446 0.20199 0.01590 0.00000 9.90000 0.00000 0.98794 0.40088 0.21427 0.01871 0.00000 10.20000 0.00000 1.00617 0.41659 0.22604 0.02144 0.00000 10.50000 0.00000 1.02349 0.43153 0.23726 0.02408 0.00000 10.80000 0.00000 1.03987 0.44569 0.24791 0.02660 0.00000 11.10000 0.00000 1.05527 0.45902 0.25796 0.02900 0.00000 11.40000 0.00000 1.06966 0.47149 0.26738 0.03128 0.00000 11.70000 0.00000 1.08302 0.48307 0.27615 0.03341 0.00000 12.00000 0.00000 1.09531 0.49375 0.28423 0.03538 0.00000 12.30000 0.00000 1.10651 0.50348 0.29162 0.03720 0.00000 12.60000 0.00000 1.11659 0.51225 0.29828 0.03885 0.00000 12.90000 0.00000 1.12553 0.52004 0.30421 0.04033 0.00000 13.20000 0.00000 1.13332 0.52683 0.30938 0.04162 0.00000 13.50000 0.00000 1.13994 0.53260 0.31377 0.04272 0.00000 13.80000 0.00000 1.14538 0.53734 0.31739 0.04363 0.00000 14.10000 0.00000 1.14962 0.54104 0.32021 0.04434 0.00000 14.40000 0.00000 1.15265 0.54369 0.32223 0.04485 0.00000 14.70000 0.00000 1.15447 0.54528 0.32345 0.04516 0.00000 15.00000 0.00000 1.15507 0.54581 0.32385 0.04526 0.00000 15.30000 0.00000 1.15446 0.54528 0.32345 0.04516 0.00000 15.60000 0.00000 1.15263 0.54369 0.32223 0.04486 0.00000 15.90000 0.00000 1.14959 0.54104 0.32021 0.04435 0.00000 16.20000 0.00000 1.14535 0.53734 0.31739 0.04363 0.00000 16.50000 0.00000 1.13991 0.53260 0.31378 0.04273 0.00000 16.80000 0.00000 1.13328 0.52683 0.30938 0.04162 0.00000 17.10000 0.00000 1.12548 0.52004 0.30422 0.04033 0.00000 17.40000 0.00000 1.11653 0.51225 0.29829 0.03886 0.00000 17.70000 0.00000 1.10644 0.50348 0.29163 0.03721 0.00000 18.00000 0.00000 1.09523 0.49375 0.28425 0.03539 0.00000 18.30000 0.00000 1.08294 0.48307 0.27616 0.03342 0.00000 18.60000 0.00000 1.06958 0.47149 0.26740 0.03129 0.00000 18.90000 0.00000 1.05518 0.45902 0.25798 0.02901 0.00000 19.20000 0.00000 1.03977 0.44569 0.24793 0.02661 0.00000 19.50000 0.00000 1.02338 0.43153 0.23728 0.02409 0.00000 19.80000 0.00000 1.00605 0.41659 0.22605 0.02146 0.00000 20.10000 0.00000 0.98782 0.40088 0.21429 0.01873 0.00000 20.40000 0.00000 0.96871 0.38446 0.20202 0.01591 0.00000 20.70000 0.00000 0.94877 0.36735 0.18926 0.01303 0.00000 21.00000 0.00000 0.92804 0.34960 0.17607 0.01008 0.00000 21.30000 0.00000 0.90656 0.33124 0.16247 0.00709 0.00000 21.60000 0.00000 0.88437 0.31232 0.14850 0.00407 0.00000 21.90000 0.00000 0.86152 0.29289 0.13420 0.00102 0.00000 22.20000 0.00000 0.83805 0.27298 0.11960 -0.00202 0.00000 PAGE 117 103 22.50000 0.00000 0.81401 0.25264 0.10475 -0.00506 0.00000 22.80000 0.00000 0.78944 0.23192 0.08969 -0.00807 0.00000 23.10000 0.00000 0.76440 0.21086 0.07444 -0.01105 0.00000 23.40000 0.00000 0.73892 0.18952 0.05907 -0.01398 0.00000 23.70000 0.00000 0.71306 0.16792 0.04359 -0.01684 0.00000 24.00000 0.00000 0.68686 0.14614 0.02806 -0.01963 0.00000 24.30000 0.00000 0.66038 0.12420 0.01252 -0.02233 0.00000 24.60000 0.00000 0.63366 0.10217 -0.00300-0.02492 0.00000 24.90000 0.00000 0.60675 0.08007 -0.01845-0.02741 0.00000 25.20000 0.00000 0.57971 0.05798 -0.03380-0.02977 0.00000 25.50000 0.00000 0.55256 0.03592 -0.04900-0.03199 0.00000 25.80000 0.00000 0.52538 0.01395 -0.06402-0.03407 0.00000 26.10000 0.00000 0.49819 -0.00789-0.07883-0.03600 0.00000 26.40000 0.00000 0.47105 -0.02956-0.09337-0.03776 0.00000 26.70000 0.00000 0.44401 -0.05100-0.10763-0.03935 0.00000 27.00000 0.00000 0.41710 -0.07219-0.12156-0.04077 0.00000 27.30000 0.00000 0.39037 -0.09306-0.13512-0.04200 0.00000 27.60000 0.00000 0.36386 -0.11360-0.14830-0.04304 0.00000 27.90000 0.00000 0.33762 -0.13375-0.16105-0.04388 0.00000 28.20000 0.00000 0.31167 -0.15348-0.17335-0.04452 0.00000 28.50000 0.00000 0.28611 -0.17268-0.18511-0.04496 0.00000 28.80000 0.00000 0.26043 -0.19228-0.19707-0.04532 0.00000 29.10000 0.00000 0.23839 -0.20435-0.20476-0.04494 0.00000 29.40000 0.00000 0.21571 -0.21462-0.21449-0.04526 0.00000 29.70000 0.00000 0.19988 -0.21948-0.22060-0.04523 0.00000 30.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 Table D.2 Preliminary side climbi ng joint torques for 3DTP robot Time (s) 1 (Nm) 2 (Nm) 3 (Nm) 4 (Nm) 5 (Nm) 6 (Nm) 0.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 0.30000 -0.00246 0.18820 -0.22529-0.22408-0.04530 0.00000 0.60000 -0.00466 0.21340 -0.21538-0.21553-0.04528 0.00000 0.90000 -0.01043 0.22964 -0.20965-0.20917-0.04527 0.00000 1.20000 -0.01264 0.24992 -0.20106-0.20231-0.04529 0.00000 1.50000 -0.04892 0.24364 -0.20368-0.20359-0.04521 0.00000 1.80000 -0.07740 0.24346 -0.20297-0.20298-0.04510 0.00000 2.10000 -0.10635 0.24255 -0.20226-0.20226-0.04494 0.00000 2.40000 -0.13514 0.24141 -0.20130-0.20130-0.04472 0.00000 2.70000 -0.16378 0.23998 -0.20011-0.20011-0.04446 0.00000 3.00000 -0.19223 0.23829 -0.19870-0.19870-0.04415 0.00000 3.30000 -0.22047 0.23632 -0.19706-0.19706-0.04378 0.00000 3.60000 -0.24845 0.23409 -0.19520-0.19520-0.04337 0.00000 3.90000 -0.27616 0.23159 -0.19312-0.19312-0.04291 0.00000 PAGE 118 104 4.20000 -0.30355 0.22883 -0.19082-0.19082-0.04240 0.00000 4.50000 -0.33060 0.22581 -0.18830-0.18830-0.04184 0.00000 4.80000 -0.35727 0.22254 -0.18557-0.18557-0.04123 0.00000 5.10000 -0.38354 0.21901 -0.18263-0.18263-0.04058 0.00000 5.40000 -0.40937 0.21524 -0.17948-0.17948-0.03988 0.00000 5.70000 -0.43474 0.21122 -0.17613-0.17613-0.03913 0.00000 6.00000 -0.45962 0.20696 -0.17258-0.17258-0.03834 0.00000 6.30000 -0.48397 0.20247 -0.16883-0.16883-0.03751 0.00000 6.60000 -0.50778 0.19775 -0.16490-0.16490-0.03664 0.00000 6.90000 -0.53101 0.19280 -0.16077-0.16077-0.03572 0.00000 7.20000 -0.55364 0.18763 -0.15647-0.15647-0.03476 0.00000 7.50000 -0.57565 0.18226 -0.15199-0.15199-0.03377 0.00000 7.80000 -0.59700 0.17667 -0.14733-0.14733-0.03273 0.00000 8.10000 -0.61768 0.17089 -0.14251-0.14251-0.03166 0.00000 8.40000 -0.63765 0.16491 -0.13753-0.13753-0.03056 0.00000 8.70000 -0.65690 0.15875 -0.13239-0.13239-0.02941 0.00000 9.00000 -0.67541 0.15241 -0.12710-0.12710-0.02824 0.00000 9.30000 -0.69316 0.14589 -0.12167-0.12167-0.02703 0.00000 9.60000 -0.71012 0.13921 -0.11610-0.11610-0.02579 0.00000 9.90000 -0.72627 0.13237 -0.11040-0.11040-0.02453 0.00000 10.20000 -0.74160 0.12538 -0.10457-0.10457-0.02323 0.00000 10.50000 -0.75609 0.11825 -0.09863-0.09863-0.02191 0.00000 10.80000 -0.76973 0.11099 -0.09257-0.09257-0.02057 0.00000 11.10000 -0.78249 0.10360 -0.08641-0.08641-0.01920 0.00000 11.40000 -0.79437 0.09609 -0.08015-0.08015-0.01781 0.00000 11.70000 -0.80534 0.08847 -0.07380-0.07380-0.01640 0.00000 12.00000 -0.81541 0.08075 -0.06736-0.06736-0.01497 0.00000 12.30000 -0.82455 0.07295 -0.06086-0.06086-0.01352 0.00000 12.60000 -0.83275 0.06506 -0.05428-0.05428-0.01206 0.00000 12.90000 -0.84002 0.05709 -0.04764-0.04764-0.01058 0.00000 13.20000 -0.84633 0.04906 -0.04094-0.04094-0.00910 0.00000 13.50000 -0.85168 0.04098 -0.03420-0.03420-0.00760 0.00000 13.80000 -0.85607 0.03285 -0.02742-0.02742-0.00609 0.00000 14.10000 -0.85949 0.02468 -0.02061-0.02061-0.00458 0.00000 14.40000 -0.86193 0.01648 -0.01378-0.01378-0.00306 0.00000 14.70000 -0.86340 0.00827 -0.00693-0.00693-0.00154 0.00000 15.00000 -0.86389 0.00004 -0.00008-0.00008-0.00002 0.00000 15.30000 -0.86340 -0.008180.00678 0.00678 0.00151 0.00000 15.60000 -0.86193 -0.016390.01363 0.01363 0.00303 0.00000 15.90000 -0.85949 -0.024590.02046 0.02046 0.00455 0.00000 16.20000 -0.85607 -0.032760.02727 0.02727 0.00606 0.00000 16.50000 -0.85168 -0.040890.03405 0.03405 0.00756 0.00000 16.80000 -0.84633 -0.048970.04079 0.04079 0.00906 0.00000 17.10000 -0.84002 -0.057000.04748 0.04748 0.01055 0.00000 PAGE 119 105 17.40000 -0.83275 -0.064970.05412 0.05412 0.01202 0.00000 17.70000 -0.82455 -0.072860.06070 0.06070 0.01349 0.00000 18.00000 -0.81541 -0.080670.06721 0.06721 0.01493 0.00000 18.30000 -0.80534 -0.088380.07364 0.07364 0.01636 0.00000 18.60000 -0.79437 -0.096000.07999 0.07999 0.01777 0.00000 18.90000 -0.78249 -0.103510.08625 0.08625 0.01916 0.00000 19.20000 -0.76973 -0.110900.09242 0.09242 0.02053 0.00000 19.50000 -0.75609 -0.118160.09847 0.09847 0.02188 0.00000 19.80000 -0.74160 -0.125290.10442 0.10442 0.02320 0.00000 20.10000 -0.72627 -0.132280.11024 0.11024 0.02449 0.00000 20.40000 -0.71012 -0.139120.11594 0.11594 0.02576 0.00000 20.70000 -0.69316 -0.145800.12151 0.12151 0.02700 0.00000 21.00000 -0.67541 -0.152320.12695 0.12695 0.02820 0.00000 21.30000 -0.65690 -0.158660.13224 0.13224 0.02938 0.00000 21.60000 -0.63765 -0.164830.13737 0.13737 0.03052 0.00000 21.90000 -0.61768 -0.170800.14236 0.14236 0.03163 0.00000 22.20000 -0.59700 -0.176590.14718 0.14718 0.03270 0.00000 22.50000 -0.57565 -0.182170.15183 0.15183 0.03373 0.00000 22.80000 -0.55364 -0.187550.15632 0.15632 0.03473 0.00000 23.10000 -0.53101 -0.192710.16062 0.16062 0.03569 0.00000 23.40000 -0.50778 -0.197660.16474 0.16474 0.03660 0.00000 23.70000 -0.48397 -0.202380.16868 0.16868 0.03748 0.00000 24.00000 -0.45962 -0.206870.17243 0.17243 0.03831 0.00000 24.30000 -0.43474 -0.211130.17598 0.17598 0.03910 0.00000 24.60000 -0.40937 -0.215150.17933 0.17933 0.03984 0.00000 24.90000 -0.38354 -0.218920.18248 0.18248 0.04054 0.00000 25.20000 -0.35727 -0.222450.18542 0.18542 0.04119 0.00000 25.50000 -0.33060 -0.225730.18815 0.18815 0.04180 0.00000 25.80000 -0.30355 -0.228750.19066 0.19066 0.04236 0.00000 26.10000 -0.27616 -0.231510.19296 0.19296 0.04287 0.00000 26.40000 -0.24845 -0.234000.19505 0.19505 0.04333 0.00000 26.70000 -0.22047 -0.236240.19691 0.19691 0.04375 0.00000 27.00000 -0.19223 -0.238200.19855 0.19855 0.04411 0.00000 27.30000 -0.16378 -0.239900.19996 0.19996 0.04443 0.00000 27.60000 -0.13514 -0.241320.20115 0.20115 0.04469 0.00000 27.90000 -0.10635 -0.242470.20210 0.20210 0.04490 0.00000 28.20000 -0.07742 -0.243360.20283 0.20283 0.04506 0.00000 28.50000 -0.04867 -0.243760.20342 0.20338 0.04517 0.00000 28.80000 -0.01648 -0.246660.20244 0.20298 0.04525 0.00000 29.10000 -0.00753 -0.231130.20935 0.20795 0.04522 0.00000 29.40000 0.00054 -0.216220.21427 0.21426 0.04524 0.00000 29.70000 -0.00004 -0.199830.21951 0.22062 0.04523 0.00000 30.00000 0.00000 -0.173450.22953 0.22953 0.04526 0.00000 PAGE 120 106 Table D.3 Preliminary rectilinear cl imbing joint torques for 3DTP robot Time (s) 1 (Nm) 2 (Nm)3 (Nm) 4 (Nm) 5 (Nm) 6 (Nm) 0.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 0.30000 0.00000 0.19006 -0.22458-0.22346-0.04530 0.00000 0.60000 0.00000 0.21659 -0.21442-0.21448-0.04531 0.00000 0.90000 0.00000 0.23990 -0.20322-0.20394-0.04499 0.00000 1.20000 0.00000 0.25217 -0.20827-0.20798-0.04541 0.00000 1.50000 0.00000 0.26550 -0.21154-0.21155-0.04529 0.00000 1.80000 0.00000 0.27694 -0.21622-0.21620-0.04529 0.00000 2.10000 0.00000 0.28759 -0.22070-0.22069-0.04529 0.00000 2.40000 0.00000 0.29734 -0.22512-0.22511-0.04529 0.00000 2.70000 0.00000 0.30618 -0.22946-0.22945-0.04529 0.00000 3.00000 0.00000 0.31407 -0.23373-0.23371-0.04529 0.00000 3.30000 0.00000 0.32101 -0.23791-0.23789-0.04529 0.00000 3.60000 0.00000 0.32696 -0.24201-0.24199-0.04529 0.00000 3.90000 0.00000 0.33192 -0.24603-0.24601-0.04529 0.00000 4.20000 0.00000 0.33586 -0.24996-0.24994-0.04529 0.00000 4.50000 0.00000 0.33878 -0.25381-0.25379-0.04529 0.00000 4.80000 0.00000 0.34065 -0.25757-0.25755-0.04529 0.00000 5.10000 0.00000 0.34148 -0.26123-0.26121-0.04528 0.00000 5.40000 0.00000 0.34126 -0.26481-0.26479-0.04528 0.00000 5.70000 0.00000 0.33995 -0.26829-0.26827-0.04528 0.00000 6.00000 0.00000 0.33790 -0.27171-0.27172-0.04530 0.00000 6.30000 0.00000 0.33095 -0.27439-0.27405-0.04508 0.00000 6.60000 0.00000 0.33583 -0.28410-0.28505-0.04591 0.00000 6.90000 0.00000 0.31212 -0.29577-0.29584-0.04533 0.00000 7.20000 0.00000 0.29542 -0.30703-0.30721-0.04538 0.00000 7.50000 0.00000 0.27375 -0.31177-0.31180-0.04481 0.00000 7.80000 0.00000 0.25181 -0.32082-0.32097-0.04538 0.00000 8.10000 0.00000 0.22522 -0.32326-0.32352-0.04531 0.00000 8.40000 0.00000 0.18799 -0.32867-0.32756-0.04585 0.00000 8.70000 0.00000 0.18119 -0.32195-0.32257-0.04507 0.00000 9.00000 0.00000 0.17097 -0.32393-0.32393-0.04528 0.00000 9.30000 0.00000 0.16429 -0.32329-0.32335-0.04527 0.00000 9.60000 0.00000 0.15766 -0.32230-0.32235-0.04527 0.00000 9.90000 0.00000 0.15137 -0.32077-0.32082-0.04527 0.00000 10.20000 0.00000 0.14540 -0.31871-0.31876-0.04527 0.00000 10.50000 0.00000 0.13976 -0.31613-0.31618-0.04527 0.00000 10.80000 0.00000 0.13444 -0.31304-0.31309-0.04527 0.00000 11.10000 0.00000 0.12944 -0.30944-0.30949-0.04527 0.00000 11.40000 0.00000 0.12475 -0.30534-0.30538-0.04527 0.00000 11.70000 0.00000 0.12038 -0.30074-0.30079-0.04528 0.00000 PAGE 121 107 12.00000 0.00000 0.11631 -0.29565-0.29570-0.04528 0.00000 12.30000 0.00000 0.11255 -0.29009-0.29014-0.04528 0.00000 12.60000 0.00000 0.10908 -0.28407-0.28411-0.04528 0.00000 12.90000 0.00000 0.10589 -0.27759-0.27763-0.04528 0.00000 13.20000 0.00000 0.10300 -0.27067-0.27071-0.04528 0.00000 13.50000 0.00000 0.10018 -0.26334-0.26334-0.04527 0.00000 13.80000 0.00000 0.10032 -0.25522-0.25581-0.04539 0.00000 14.10000 0.00000 0.09818 -0.24927-0.24787-0.04500 0.00000 14.40000 0.00000 0.12809 -0.24217-0.24228-0.04531 0.00000 14.70000 0.00000 0.15074 -0.23605-0.23606-0.04529 0.00000 15.00000 0.00000 0.17250 -0.22890-0.22872-0.04510 0.00000 15.30000 0.00000 0.19509 -0.22210-0.22213-0.04529 0.00000 15.60000 0.00000 0.21619 -0.21462-0.21458-0.04531 0.00000 15.90000 0.00000 0.23993 -0.20320-0.20393-0.04499 0.00000 16.20000 0.00000 0.25216 -0.20827-0.20798-0.04541 0.00000 16.50000 0.00000 0.26550 -0.21154-0.21155-0.04529 0.00000 16.80000 0.00000 0.27694 -0.21622-0.21620-0.04529 0.00000 17.10000 0.00000 0.28759 -0.22070-0.22069-0.04529 0.00000 17.40000 0.00000 0.29734 -0.22512-0.22511-0.04529 0.00000 17.70000 0.00000 0.30618 -0.22946-0.22945-0.04529 0.00000 18.00000 0.00000 0.31407 -0.23373-0.23371-0.04529 0.00000 18.30000 0.00000 0.32101 -0.23791-0.23789-0.04529 0.00000 18.60000 0.00000 0.32696 -0.24201-0.24199-0.04529 0.00000 18.90000 0.00000 0.33192 -0.24603-0.24601-0.04529 0.00000 19.20000 0.00000 0.33586 -0.24996-0.24994-0.04529 0.00000 19.50000 0.00000 0.33878 -0.25381-0.25379-0.04529 0.00000 19.80000 0.00000 0.34065 -0.25757-0.25755-0.04529 0.00000 20.10000 0.00000 0.34148 -0.26123-0.26121-0.04528 0.00000 20.40000 0.00000 0.34126 -0.26481-0.26479-0.04528 0.00000 20.70000 0.00000 0.33995 -0.26829-0.26827-0.04528 0.00000 21.00000 0.00000 0.33790 -0.27171-0.27172-0.04530 0.00000 21.30000 0.00000 0.33095 -0.27439-0.27405-0.04508 0.00000 21.60000 0.00000 0.33583 -0.28410-0.28505-0.04591 0.00000 21.90000 0.00000 0.31212 -0.29577-0.29584-0.04533 0.00000 22.20000 0.00000 0.29542 -0.30703-0.30721-0.04538 0.00000 22.50000 0.00000 0.27375 -0.31177-0.31180-0.04481 0.00000 22.80000 0.00000 0.25181 -0.32082-0.32097-0.04538 0.00000 23.10000 0.00000 0.22522 -0.32326-0.32352-0.04531 0.00000 23.40000 0.00000 0.18799 -0.32867-0.32756-0.04585 0.00000 23.70000 0.00000 0.18119 -0.32195-0.32257-0.04507 0.00000 24.00000 0.00000 0.17097 -0.32393-0.32393-0.04528 0.00000 24.30000 0.00000 0.16429 -0.32329-0.32335-0.04527 0.00000 24.60000 0.00000 0.15766 -0.32230-0.32235-0.04527 0.00000 24.90000 0.00000 0.15137 -0.32077-0.32082-0.04527 0.00000 PAGE 122 108 25.20000 0.00000 0.14540 -0.31871-0.31876-0.04527 0.00000 25.50000 0.00000 0.13976 -0.31613-0.31618-0.04527 0.00000 25.80000 0.00000 0.13444 -0.31304-0.31309-0.04527 0.00000 26.10000 0.00000 0.12944 -0.30944-0.30949-0.04527 0.00000 26.40000 0.00000 0.12475 -0.30534-0.30538-0.04527 0.00000 26.70000 0.00000 0.12038 -0.30074-0.30079-0.04528 0.00000 27.00000 0.00000 0.11631 -0.29565-0.29570-0.04528 0.00000 27.30000 0.00000 0.11255 -0.29009-0.29014-0.04528 0.00000 27.60000 0.00000 0.10908 -0.28407-0.28411-0.04528 0.00000 27.90000 0.00000 0.10589 -0.27759-0.27763-0.04528 0.00000 28.20000 0.00000 0.10300 -0.27067-0.27071-0.04528 0.00000 28.50000 0.00000 0.10018 -0.26334-0.26334-0.04527 0.00000 28.80000 0.00000 0.10032 -0.25522-0.25581-0.04539 0.00000 29.10000 0.00000 0.09821 -0.24925-0.24786-0.04500 0.00000 29.40000 0.00000 0.12766 -0.24236-0.24238-0.04531 0.00000 29.70000 0.00000 0.15598 -0.23365-0.23485-0.04530 0.00000 30.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 Table D.4 Optimized flip climbing joint torques for 3DTP robot Time (s) 1 (Nm) 2 (Nm)3 (Nm) 4 (Nm) 5 (Nm) 6 (Nm) 0.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 0.30000 0.00000 0.19018 -0.22081-0.22541-0.04528 0.00000 0.60000 0.00000 0.21696 -0.20671-0.21836-0.04526 0.00000 0.90000 0.00000 0.24136 -0.19365-0.21188-0.04541 0.00000 1.20000 0.00000 0.25126 -0.18961-0.21072-0.04515 0.00000 1.50000 0.00000 0.26312 -0.18525-0.20979-0.04486 0.00000 1.80000 0.00000 0.27377 -0.18132-0.20891-0.04421 0.00000 2.10000 0.00000 0.28468 -0.17705-0.20772-0.04327 0.00000 2.40000 0.00000 0.29574 -0.17250-0.20624-0.04205 0.00000 2.70000 0.00000 0.30694 -0.16768-0.20449-0.04055 0.00000 3.00000 0.00000 0.31828 -0.16260-0.20246-0.03878 0.00000 3.30000 0.00000 0.32972 -0.15728-0.20018-0.03676 0.00000 3.60000 0.00000 0.34128 -0.15171-0.19766-0.03449 0.00000 3.90000 0.00000 0.35291 -0.14593-0.19491-0.03199 0.00000 4.20000 0.00000 0.36461 -0.13995-0.19195-0.02929 0.00000 4.50000 0.00000 0.37635 -0.13378-0.18879-0.02639 0.00000 4.80000 0.00000 0.38812 -0.12745-0.18546-0.02331 0.00000 5.10000 0.00000 0.39989 -0.12098-0.18198-0.02008 0.00000 5.40000 0.00000 0.41164 -0.11439-0.17836-0.01672 0.00000 5.70000 0.00000 0.42335 -0.10770-0.17463-0.01325 0.00000 6.00000 0.00000 0.43498 -0.10093-0.17082-0.00969 0.00000 6.30000 0.00000 0.44652 -0.09411-0.16694-0.00607 0.00000 6.60000 0.00000 0.45793 -0.08727-0.16302-0.00241 0.00000 PAGE 123 109 6.90000 0.00000 0.46920 -0.08042-0.159080.00127 0.00000 7.20000 0.00000 0.48029 -0.07360-0.155150.00494 0.00000 7.50000 0.00000 0.49120 -0.06683-0.151260.00858 0.00000 7.80000 0.00000 0.50164 -0.06014-0.147380.01210 0.00000 8.10000 0.00000 0.51283 -0.05347-0.143890.01448 0.00000 8.40000 0.00000 0.52247 -0.04691-0.140540.01373 0.00000 8.70000 0.00000 0.53261 -0.04032-0.137270.01316 0.00000 9.00000 0.00000 0.54256 -0.03371-0.133960.01258 0.00000 9.30000 0.00000 0.55237 -0.02710-0.130600.01199 0.00000 9.60000 0.00000 0.56203 -0.02048-0.127210.01141 0.00000 9.90000 0.00000 0.57153 -0.01385-0.123780.01082 0.00000 10.20000 0.00000 0.58088 -0.00721-0.120320.01023 0.00000 10.50000 0.00000 0.59007 -0.00057-0.116830.00964 0.00000 10.80000 0.00000 0.59910 0.00606 -0.113300.00905 0.00000 11.10000 0.00000 0.60797 0.01270 -0.109740.00845 0.00000 11.40000 0.00000 0.61667 0.01934 -0.106150.00786 0.00000 11.70000 0.00000 0.62520 0.02597 -0.102530.00726 0.00000 12.00000 0.00000 0.63357 0.03259 -0.098880.00666 0.00000 12.30000 0.00000 0.64176 0.03921 -0.095200.00606 0.00000 12.60000 0.00000 0.64978 0.04581 -0.091500.00546 0.00000 12.90000 0.00000 0.65762 0.05241 -0.087770.00486 0.00000 13.20000 0.00000 0.66528 0.05899 -0.084010.00425 0.00000 13.50000 0.00000 0.67276 0.06555 -0.080240.00365 0.00000 13.80000 0.00000 0.68005 0.07210 -0.076440.00305 0.00000 14.10000 0.00000 0.68716 0.07863 -0.072620.00244 0.00000 14.40000 0.00000 0.69408 0.08513 -0.068770.00184 0.00000 14.70000 0.00000 0.70082 0.09161 -0.064920.00123 0.00000 15.00000 0.00000 0.70736 0.09807 -0.061040.00062 0.00000 15.30000 0.00000 0.71370 0.10450 -0.057140.00002 0.00000 15.60000 0.00000 0.71986 0.11091 -0.05323-0.00059 0.00000 15.90000 0.00000 0.72581 0.11728 -0.04931-0.00119 0.00000 16.20000 0.00000 0.73157 0.12362 -0.04537-0.00180 0.00000 16.50000 0.00000 0.73713 0.12992 -0.04142-0.00240 0.00000 16.80000 0.00000 0.74248 0.13619 -0.03746-0.00301 0.00000 17.10000 0.00000 0.74764 0.14243 -0.03349-0.00361 0.00000 17.40000 0.00000 0.75258 0.14862 -0.02951-0.00422 0.00000 17.70000 0.00000 0.75732 0.15477 -0.02552-0.00482 0.00000 18.00000 0.00000 0.76186 0.16088 -0.02153-0.00542 0.00000 18.30000 0.00000 0.76618 0.16695 -0.01753-0.00602 0.00000 18.60000 0.00000 0.77030 0.17297 -0.01353-0.00662 0.00000 18.90000 0.00000 0.77420 0.17894 -0.00952-0.00722 0.00000 19.20000 0.00000 0.77789 0.18486 -0.00551-0.00782 0.00000 19.50000 0.00000 0.78137 0.19073 -0.00150-0.00842 0.00000 19.80000 0.00000 0.78464 0.19655 0.00251 -0.00901 0.00000 PAGE 124 110 20.10000 0.00000 0.78769 0.20231 0.00651 -0.00960 0.00000 20.40000 0.00000 0.79052 0.20802 0.01052 -0.01020 0.00000 20.70000 0.00000 0.79313 0.21366 0.01452 -0.01078 0.00000 21.00000 0.00000 0.79554 0.21926 0.01852 -0.01137 0.00000 21.30000 0.00000 0.79764 0.22471 0.02245 -0.01197 0.00000 21.60000 0.00000 0.80049 0.23112 0.02716 -0.01243 0.00000 21.90000 0.00000 0.78997 0.22343 0.02101 -0.01463 0.00000 22.20000 0.00000 0.84408 0.28117 0.06983 -0.00648 0.00000 22.50000 0.00000 0.89043 0.33136 0.11610 0.00257 0.00000 22.80000 0.00000 0.93792 0.38295 0.16387 0.01220 0.00000 23.10000 0.00000 0.97913 0.42847 0.20708 0.02121 0.00000 23.40000 0.00000 1.01336 0.46718 0.24490 0.02925 0.00000 23.70000 0.00000 1.03948 0.49795 0.27621 0.03594 0.00000 24.00000 0.00000 1.05660 0.51989 0.30011 0.04098 0.00000 24.30000 0.00000 1.06405 0.53233 0.31592 0.04413 0.00000 24.60000 0.00000 1.06142 0.53485 0.32319 0.04526 0.00000 24.90000 0.00000 1.04853 0.52726 0.32172 0.04431 0.00000 25.20000 0.00000 1.02549 0.50968 0.31157 0.04133 0.00000 25.50000 0.00000 0.99267 0.48246 0.29305 0.03644 0.00000 25.80000 0.00000 0.95066 0.44620 0.26671 0.02989 0.00000 26.10000 0.00000 0.90033 0.40175 0.23333 0.02196 0.00000 26.40000 0.00000 0.84271 0.35014 0.19388 0.01302 0.00000 26.70000 0.00000 0.77903 0.29261 0.14949 0.00349 0.00000 27.00000 0.00000 0.71066 0.23050 0.10145 -0.00619 0.00000 27.30000 0.00000 0.63906 0.16527 0.05112 -0.01559 0.00000 27.60000 0.00000 0.56574 0.09843 -0.00009-0.02427 0.00000 27.90000 0.00000 0.49221 0.03150 -0.05076-0.03183 0.00000 28.20000 0.00000 0.41993 -0.03407-0.09954-0.03793 0.00000 28.50000 0.00000 0.35084 -0.09642-0.14479-0.04224 0.00000 28.80000 0.00000 0.28020 -0.15976-0.18941-0.04514 0.00000 29.10000 0.00000 0.24990 -0.18406-0.20432-0.04413 0.00000 29.40000 0.00000 0.21553 -0.20772-0.21903-0.04534 0.00000 29.70000 0.00000 0.20023 -0.21552-0.22257-0.04524 0.00000 30.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 Table D.5 Optimized side climbi ng joint torques for 3DTP robot Time (s) 1 (Nm) 2 (Nm) 3 (Nm) 4 (Nm) 5 (Nm) 6 (Nm) 0.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 0.30000 0.00004 0.19021 -0.22081-0.22541-0.04528 0.00000 0.60000 -0.00054 0.21658 -0.20674-0.21831-0.04524 0.00000 0.90000 0.00758 0.24683 -0.19313-0.21257-0.04572 0.00000 1.20000 -0.02304 0.23585 -0.18887-0.20660-0.04510 0.00000 1.50000 -0.05071 0.22968 -0.18235-0.19968-0.04509 0.00000 PAGE 125 111 1.80000 -0.08318 0.21909 -0.17644-0.19242-0.04471 0.00000 2.10000 -0.11654 0.20842 -0.17004-0.18474-0.04419 0.00000 2.40000 -0.15096 0.19736 -0.16329-0.17669-0.04351 0.00000 2.70000 -0.18632 0.18599 -0.15620-0.16830-0.04268 0.00000 3.00000 -0.22247 0.17437 -0.14881-0.15961-0.04169 0.00000 3.30000 -0.25929 0.16254 -0.14114-0.15065-0.04057 0.00000 3.60000 -0.29663 0.15056 -0.13325-0.14147-0.03931 0.00000 3.90000 -0.33436 0.13848 -0.12515-0.13211-0.03792 0.00000 4.20000 -0.37234 0.12635 -0.11690-0.12261-0.03642 0.00000 4.50000 -0.41041 0.11423 -0.10853-0.11301-0.03481 0.00000 4.80000 -0.44845 0.10217 -0.10008-0.10336-0.03311 0.00000 5.10000 -0.48629 0.09023 -0.09158-0.09370-0.03132 0.00000 5.40000 -0.52381 0.07845 -0.08309-0.08407-0.02946 0.00000 5.70000 -0.56084 0.06689 -0.07464-0.07452-0.02754 0.00000 6.00000 -0.59725 0.05560 -0.06626-0.06510-0.02557 0.00000 6.30000 -0.63291 0.04462 -0.05801-0.05584-0.02357 0.00000 6.60000 -0.66766 0.03401 -0.04992-0.04679-0.02155 0.00000 6.90000 -0.70138 0.02380 -0.04203-0.03799-0.01953 0.00000 7.20000 -0.73393 0.01405 -0.03437-0.02949-0.01751 0.00000 7.50000 -0.76520 0.00480 -0.02700-0.02132-0.01551 0.00000 7.80000 -0.79505 -0.00393-0.01993-0.01352-0.01355 0.00000 8.10000 -0.82338 -0.01207-0.01320-0.00614-0.01164 0.00000 8.40000 -0.85007 -0.01962-0.006860.00080 -0.00978 0.00000 8.70000 -0.87503 -0.02652-0.000920.00726 -0.00799 0.00000 9.00000 -0.89815 -0.032760.00458 0.01321 -0.00629 0.00000 9.30000 -0.91935 -0.038300.00961 0.01862 -0.00468 0.00000 9.60000 -0.93855 -0.043130.01416 0.02346 -0.00318 0.00000 9.90000 -0.95568 -0.047220.01819 0.02771 -0.00179 0.00000 10.20000 -0.97067 -0.050550.02170 0.03135 -0.00052 0.00000 10.50000 -0.98347 -0.053120.02465 0.03436 0.00061 0.00000 10.80000 -0.99404 -0.054910.02704 0.03672 0.00162 0.00000 11.10000 -1.00234 -0.055910.02886 0.03842 0.00247 0.00000 11.40000 -1.00835 -0.056120.03009 0.03945 0.00318 0.00000 11.70000 -1.01204 -0.055550.03074 0.03980 0.00374 0.00000 12.00000 -1.01341 -0.054180.03079 0.03947 0.00415 0.00000 12.30000 -1.01246 -0.052040.03025 0.03846 0.00440 0.00000 12.60000 -1.00922 -0.049120.02911 0.03678 0.00450 0.00000 12.90000 -1.00369 -0.045440.02740 0.03441 0.00444 0.00000 13.20000 -0.99591 -0.041020.02511 0.03139 0.00423 0.00000 13.50000 -0.98592 -0.035870.02226 0.02771 0.00387 0.00000 13.80000 -0.97378 -0.030020.01885 0.02340 0.00337 0.00000 14.10000 -0.95953 -0.023500.01492 0.01847 0.00273 0.00000 14.40000 -0.94326 -0.016250.01045 0.01289 0.00194 0.00000 14.70000 -0.92501 -0.009230.00584 0.00731 0.00115 0.00000 PAGE 126 112 15.00000 -0.90512 0.02012 -0.00931-0.01456-0.00292 0.00000 15.30000 -0.92543 0.00754 -0.00497-0.00601-0.00090 0.00000 15.60000 -0.94364 0.01613 -0.01024-0.01265-0.00191 0.00000 15.90000 -0.95991 0.02325 -0.01467-0.01815-0.00268 0.00000 16.20000 -0.97414 0.02979 -0.01861-0.02310-0.00332 0.00000 16.50000 -0.98628 0.03565 -0.02202-0.02742-0.00383 0.00000 16.80000 -0.99625 0.04080 -0.02488-0.03111-0.00419 0.00000 17.10000 -1.00402 0.04523 -0.02718-0.03414-0.00441 0.00000 17.40000 -1.00953 0.04892 -0.02891-0.03652-0.00447 0.00000 17.70000 -1.01276 0.05185 -0.03005-0.03822-0.00437 0.00000 18.00000 -1.01369 0.05400 -0.03061-0.03924-0.00413 0.00000 18.30000 -1.01229 0.05538 -0.03057-0.03959-0.00372 0.00000 18.60000 -1.00858 0.05597 -0.02994-0.03925-0.00317 0.00000 18.90000 -1.00256 0.05577 -0.02872-0.03824-0.00246 0.00000 19.20000 -0.99424 0.05478 -0.02692-0.03655-0.00161 0.00000 19.50000 -0.98365 0.05300 -0.02454-0.03421-0.00061 0.00000 19.80000 -0.97082 0.05045 -0.02160-0.031220.00052 0.00000 20.10000 -0.95580 0.04713 -0.01811-0.027600.00178 0.00000 20.40000 -0.93865 0.04305 -0.01410-0.023370.00317 0.00000 20.70000 -0.91943 0.03824 -0.00957-0.018550.00466 0.00000 21.00000 -0.89820 0.03271 -0.00455-0.013160.00627 0.00000 21.30000 -0.87506 0.02649 0.00093 -0.007230.00796 0.00000 21.60000 -0.85008 0.01960 0.00686 -0.000790.00975 0.00000 21.90000 -0.82336 0.01207 0.01318 0.00613 0.01160 0.00000 22.20000 -0.79501 0.00394 0.01989 0.01350 0.01351 0.00000 22.50000 -0.76513 -0.004770.02694 0.02127 0.01547 0.00000 22.80000 -0.73384 -0.014010.03431 0.02942 0.01746 0.00000 23.10000 -0.70126 -0.023740.04195 0.03791 0.01947 0.00000 23.40000 -0.66752 -0.033930.04982 0.04669 0.02149 0.00000 23.70000 -0.63274 -0.044530.05790 0.05572 0.02351 0.00000 24.00000 -0.59707 -0.055500.06614 0.06496 0.02551 0.00000 24.30000 -0.56063 -0.066780.07449 0.07437 0.02747 0.00000 24.60000 -0.52358 -0.078330.08293 0.08390 0.02938 0.00000 24.90000 -0.48605 -0.090090.09142 0.09352 0.03124 0.00000 25.20000 -0.44818 -0.102020.09990 0.10316 0.03303 0.00000 25.50000 -0.41013 -0.114070.10834 0.11280 0.03473 0.00000 25.80000 -0.37204 -0.126180.11670 0.12239 0.03634 0.00000 26.10000 -0.33405 -0.138300.12494 0.13188 0.03784 0.00000 26.40000 -0.29630 -0.150370.13303 0.14123 0.03923 0.00000 26.70000 -0.25895 -0.162340.14091 0.15039 0.04049 0.00000 27.00000 -0.22212 -0.174160.14857 0.15934 0.04161 0.00000 27.30000 -0.18596 -0.185780.15596 0.16802 0.04259 0.00000 27.60000 -0.15059 -0.197140.16304 0.17641 0.04343 0.00000 27.90000 -0.11616 -0.208190.16979 0.18445 0.04411 0.00000 PAGE 127 113 28.20000 -0.08276 -0.218910.17617 0.19212 0.04463 0.00000 28.50000 -0.05082 -0.228900.18224 0.19941 0.04498 0.00000 28.80000 -0.01665 -0.242400.18686 0.20615 0.04536 0.00000 29.10000 -0.00758 -0.229710.19728 0.21266 0.04481 0.00000 29.40000 0.00054 -0.216970.20677 0.21844 0.04529 0.00000 29.70000 -0.00004 -0.200110.21560 0.22262 0.04524 0.00000 30.00000 0.00000 -0.173450.22953 0.22953 0.04526 0.00000 Table D.6 Optimized rectilinear clim bing joint torques for 3DTP robot Time (s) 1 (Nm) 2 (Nm)3 (Nm) 4 (Nm) 5 (Nm) 6 (Nm) 0.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 0.30000 0.00000 0.19017 -0.22082-0.22541-0.04528 0.00000 0.60000 0.00000 0.21715 -0.20668-0.21838-0.04528 0.00000 0.90000 0.00000 0.23870 -0.19408-0.21170-0.04522 0.00000 1.20000 0.00000 0.25840 -0.18915-0.21104-0.04528 0.00000 1.50000 0.00000 0.27787 -0.18452-0.21041-0.04513 0.00000 1.80000 0.00000 0.29661 -0.18030-0.21010-0.04489 0.00000 2.10000 0.00000 0.31479 -0.17595-0.20965-0.04455 0.00000 2.40000 0.00000 0.33238 -0.17150-0.20910-0.04410 0.00000 2.70000 0.00000 0.34936 -0.16696-0.20845-0.04354 0.00000 3.00000 0.00000 0.36573 -0.16232-0.20767-0.04287 0.00000 3.30000 0.00000 0.38133 -0.15771-0.20689-0.04212 0.00000 3.60000 0.00000 0.39801 -0.15117-0.20463-0.04105 0.00000 3.90000 0.00000 0.40237 -0.15642-0.21076-0.04106 0.00000 4.20000 0.00000 0.39452 -0.17310-0.22481-0.04213 0.00000 4.50000 0.00000 0.38821 -0.18729-0.23684-0.04288 0.00000 4.80000 0.00000 0.38167 -0.20091-0.24825-0.04355 0.00000 5.10000 0.00000 0.37506 -0.21379-0.25894-0.04411 0.00000 5.40000 0.00000 0.36838 -0.22592-0.26886-0.04456 0.00000 5.70000 0.00000 0.36164 -0.23726-0.27797-0.04490 0.00000 6.00000 0.00000 0.35512 -0.24786-0.28641-0.04516 0.00000 6.30000 0.00000 0.34572 -0.25644-0.29214-0.04496 0.00000 6.60000 0.00000 0.34306 -0.27207-0.30470-0.04607 0.00000 6.90000 0.00000 0.31668 -0.28495-0.30573-0.04523 0.00000 7.20000 0.00000 0.29644 -0.30062-0.31106-0.04530 0.00000 7.50000 0.00000 0.27407 -0.31334-0.31325-0.04500 0.00000 7.80000 0.00000 0.25222 -0.32888-0.31838-0.04530 0.00000 8.10000 0.00000 0.22896 -0.34110-0.32033-0.04522 0.00000 8.40000 0.00000 0.19796 -0.36169-0.32821-0.04617 0.00000 8.70000 0.00000 0.19873 -0.34919-0.31917-0.04489 0.00000 9.00000 0.00000 0.19767 -0.34346-0.31811-0.04507 0.00000 9.30000 0.00000 0.20066 -0.33256-0.31291-0.04466 0.00000 9.60000 0.00000 0.20450 -0.32045-0.30644-0.04412 0.00000 PAGE 128 114 9.90000 0.00000 0.20943 -0.30685-0.29848-0.04341 0.00000 10.20000 0.00000 0.21540 -0.29182-0.28911-0.04253 0.00000 10.50000 0.00000 0.22238 -0.27541-0.27834-0.04148 0.00000 10.80000 0.00000 0.23015 -0.25782-0.26637-0.04030 0.00000 11.10000 0.00000 0.24126 -0.23659-0.25139-0.03866 0.00000 11.40000 0.00000 0.23813 -0.22897-0.24453-0.03865 0.00000 11.70000 0.00000 0.22039 -0.23570-0.24669-0.04029 0.00000 12.00000 0.00000 0.20499 -0.23986-0.24693-0.04147 0.00000 12.30000 0.00000 0.18925 -0.24403-0.24714-0.04252 0.00000 12.60000 0.00000 0.17336 -0.24802-0.24716-0.04340 0.00000 12.90000 0.00000 0.15732 -0.25184-0.24702-0.04411 0.00000 13.20000 0.00000 0.14118 -0.25547-0.24669-0.04466 0.00000 13.50000 0.00000 0.12455 -0.25909-0.24627-0.04504 0.00000 13.80000 0.00000 0.11261 -0.26042-0.24469-0.04521 0.00000 14.10000 0.00000 0.09437 -0.26567-0.24573-0.04532 0.00000 14.40000 0.00000 0.12879 -0.25026-0.23882-0.04527 0.00000 14.70000 0.00000 0.15079 -0.24019-0.23437-0.04528 0.00000 15.00000 0.00000 0.17268 -0.22905-0.22891-0.04513 0.00000 15.30000 0.00000 0.19520 -0.21826-0.22407-0.04528 0.00000 15.60000 0.00000 0.21675 -0.20688-0.21848-0.04528 0.00000 15.90000 0.00000 0.23873 -0.19407-0.21169-0.04522 0.00000 16.20000 0.00000 0.25840 -0.18915-0.21104-0.04528 0.00000 16.50000 0.00000 0.27787 -0.18452-0.21041-0.04513 0.00000 16.80000 0.00000 0.29661 -0.18030-0.21010-0.04489 0.00000 17.10000 0.00000 0.31479 -0.17595-0.20965-0.04455 0.00000 17.40000 0.00000 0.33238 -0.17150-0.20910-0.04410 0.00000 17.70000 0.00000 0.34936 -0.16696-0.20845-0.04354 0.00000 18.00000 0.00000 0.36573 -0.16232-0.20767-0.04287 0.00000 18.30000 0.00000 0.38133 -0.15771-0.20689-0.04212 0.00000 18.60000 0.00000 0.39801 -0.15117-0.20463-0.04105 0.00000 18.90000 0.00000 0.40237 -0.15642-0.21076-0.04106 0.00000 19.20000 0.00000 0.39452 -0.17310-0.22481-0.04213 0.00000 19.50000 0.00000 0.38821 -0.18729-0.23684-0.04288 0.00000 19.80000 0.00000 0.38167 -0.20091-0.24825-0.04355 0.00000 20.10000 0.00000 0.37506 -0.21379-0.25894-0.04411 0.00000 20.40000 0.00000 0.36838 -0.22592-0.26886-0.04456 0.00000 20.70000 0.00000 0.36164 -0.23726-0.27797-0.04490 0.00000 21.00000 0.00000 0.35512 -0.24786-0.28641-0.04516 0.00000 21.30000 0.00000 0.34572 -0.25644-0.29214-0.04496 0.00000 21.60000 0.00000 0.34306 -0.27207-0.30470-0.04607 0.00000 21.90000 0.00000 0.31668 -0.28495-0.30573-0.04523 0.00000 22.20000 0.00000 0.29644 -0.30062-0.31106-0.04530 0.00000 22.50000 0.00000 0.27407 -0.31334-0.31325-0.04500 0.00000 22.80000 0.00000 0.25222 -0.32888-0.31838-0.04530 0.00000 PAGE 129 115 23.10000 0.00000 0.22896 -0.34110-0.32033-0.04522 0.00000 23.40000 0.00000 0.19796 -0.36169-0.32821-0.04617 0.00000 23.70000 0.00000 0.19873 -0.34919-0.31917-0.04489 0.00000 24.00000 0.00000 0.19767 -0.34346-0.31811-0.04507 0.00000 24.30000 0.00000 0.20066 -0.33256-0.31291-0.04466 0.00000 24.60000 0.00000 0.20450 -0.32045-0.30644-0.04412 0.00000 24.90000 0.00000 0.20943 -0.30685-0.29848-0.04341 0.00000 25.20000 0.00000 0.21540 -0.29182-0.28911-0.04253 0.00000 25.50000 0.00000 0.22238 -0.27541-0.27834-0.04148 0.00000 25.80000 0.00000 0.23015 -0.25782-0.26637-0.04030 0.00000 26.10000 0.00000 0.24126 -0.23659-0.25139-0.03866 0.00000 26.40000 0.00000 0.23813 -0.22897-0.24453-0.03865 0.00000 26.70000 0.00000 0.22039 -0.23570-0.24669-0.04029 0.00000 27.00000 0.00000 0.20499 -0.23986-0.24693-0.04147 0.00000 27.30000 0.00000 0.18925 -0.24403-0.24714-0.04252 0.00000 27.60000 0.00000 0.17336 -0.24802-0.24716-0.04340 0.00000 27.90000 0.00000 0.15732 -0.25184-0.24702-0.04411 0.00000 28.20000 0.00000 0.14118 -0.25547-0.24669-0.04466 0.00000 28.50000 0.00000 0.12455 -0.25909-0.24627-0.04504 0.00000 28.80000 0.00000 0.11261 -0.26043-0.24469-0.04521 0.00000 29.10000 0.00000 0.09440 -0.26565-0.24573-0.04532 0.00000 29.40000 0.00000 0.12838 -0.25047-0.23892-0.04527 0.00000 29.70000 0.00000 0.15592 -0.23770-0.23311-0.04528 0.00000 30.00000 0.00000 0.17345 -0.22953-0.22953-0.04526 0.00000 PAGE 130 116 LIST OF REFERENCES Alm03 Almonacid, M, R J. Saltarn, R Arac il, and O Reinoso. "Motion Planning of a Climbing Parallel Robot." IEEE Tran sactions on Robotic s and Automation 19 (2003): 485-489. Ara06 Aracil, Rafael, Roque J. Saltaren, and Oscar Reinoso. "A Climbing Parallel Robot: a Robot to Climb Along Tubular and Metallic Structures." IEEE Robotics & Automation Magazine 13 (2006): 16-22. Asa86 Asada, Haruhiko, and J.-J. E. Slotine. Robot Analysis and Control New York, NY: J. Wiley, 1986. Atk78 Atkinson, Kendall E. An In troduction to Numerical Analysis New York: Wiley, 1978. But03 Butler, Zack, and Daniela Rus. Di stributed Locomotion Algorithms for SelfReconfigurable Robots Operating on Rough Terrain International Symposium on Computational Intelligence in Ro botics and Automation, 16 July 2003, Institute of Electrical & Electronics Engineers (I EEE). Kobe, Japan: IEEE, 2003. Pages 880-885. Volume 2. Chi94 Chirikjian, Gregory S. 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A Robotic Dynamic Manipulati on System with Trajectory Planning and Control 1996, Institute of Electrical a nd Electronics Engineers. IEEE, 1996. ns00 nsal, Cem, and Pradeep K. Khosla. Mechatronic Design of a Modular SelfReconfigurable Robotic System International Conf erence on Robotics & Automation, Apr. 2000, Institute of Electrical & Electronics Engineers (IEEE). San Francisco, CA: IEEE, 2000. Pages 1742-1747. Volume 2. PAGE 133 119 BIOGRAPHICAL SKETCH Jaime Jos Bestard was born in Santa Clar a, Villa Clara, Cuba, on the second day of October, 1982. He attended the Univers ity of Florida from August 2000 to May 2004 and has obtained a Bachelor of Science degree in Mechanical Engineering. He joined the graduate program in the Mechanical and Aerospace Engineering Department at the University of Florida in August 2004. He has worked in the generation of drivers for the interaction between the Space Automation and Manufacturing Mechanisms Laboratory (SAMM) parallel kinematic mechanism a nd the MATLAB and Simulink environments for ease of use and rapid development and te sting of control algorithms. He has also worked in the development of hardware /software interacti on interfaces for the Heterogeneous Expert Robots for onOrbit Servicing (HEROS) project. |