design and construction four-bay variable-...

Design and Construction of a Four-Bay Variable- Geomet ry-Truss

Manipulator Arm

by

S tephen Oliver Oikawa

A thesis submitted in partial fulfillment of the requirements for the Degree of Master of Applied Science,

at the Ins t i t ute for Aerospace S t udies, Univenity of Toronto

01995 Stephen Oliver Oikawa

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Abstract

The purpose of this thesis was to design and constnict a Variable-Geometry-Truss Manipulator (VGTM), known as Tmssarm Mark II, based upon Tmssarm Mark 1, the VGTM built at the University of Toronto Institute for Aerospace Studies (UTIAS). VGTM's have been the focus of robotic research as the design promises to provide a large stiffness for a relatively smdl maas. Stacking multiple VGTM's produces a multibay VGTM, a structure with high redundancy and long reach. To date most VGTM designs have remained in the world of the cornputer, with assumptions being made about exact geometry, mass, and joint play. The construction of a new four-bay trussann worked as a design excercise for the multibay VGTM concept. Trussarrn Mark II represents a major step forward in the development of the VGTM concept as lessons learned in its construction and testing bring us one step closer to a working productive robot.

Acknowledgements

This thesis and the work it represents would not have been possible without the help and support of the many skilled penons in the Space Robotics Laboratory. Thierry Cherpillod's technical skills were invaluable, dong with his sage advice. Regina Lee's determination and persistence provided kinematics software and vision, bringing the Mark II to life. James Crawford patiently taught me everything 1 know about machining and machine tools. Dynacon's belief in the tmssarm concept was essential to acquire funding for its construction. Last, but not least, 1 thank Dr Hughes for supervising this thesis and placing faith in my skills as an engineer.

On o persona1 note, 1 would take this opportunity to thank my mother and father who often never receive the appreciation they so deserve from their forever busy student son.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Trussum 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Tmssarm Mark 1 3

. . . . . . . . . . . . . . . . . . . . 1.3 Why Build Another Trussarm? 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Thesis Outline 5

. . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Conceptual Design 5

. . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Detailed Design 5

. . . . . . . . . . . . . . . . . . . . . 1.4.3 Evaluation and Testing 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Conclusions 6

2 Conceptual Design 7

. . . . . . . . . . . . . . . . . . 2.1 Design Review of Trussarm Mark 1 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Scale 7

. . . . . . . . . . . . . . . . 2.1.2 Custom Machined Components 8

. . . . . . . . . . . . . . . . . . . . . 2.1.3 Consistency of Design 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Specifications 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Workspace 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Payioad 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Speed 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Accuracy 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mark II is Born 11

3 Detailed Design 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Actuator Design 14

. . . . . . . . . . . . . . . . . . 3.1.1 Preliminary Actuator Design 14

. . . . . . . 3.1.2 Actuator Loads and the Differential Kinematics 16

. . . . . . . . . . . . 3.1.3 Drive Motor and Geax Ratio Selection 21

. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Passive Structure Design 21

. . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mass and Budget Goals 22

4 Evaluation and Testing 23

. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Workspace Evaluation 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Speed Evaluation 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Payload Evduation 26

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Accuracy Evaluation 26

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Kinematic Testing 27

5 Conclusions 29

. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Future Improvements 29

. . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Future Research Work 30

A FORTRAN Jacobian Solver

B Alternative Jacobian Solver

C Jacobian M a t h Analysis 49

. . . . . . . . . . . . . . . . . . . . . . . C.1 The Jacobiao of Trussarm 49

C.1.1 The inverse Kinernatics Algoritkm . . . . . . . . . . . . . . 50

. . . . . . . . . . . . . . . . . . . . . C.1.2 Member Length Method 52

. . . . . . . . . . . . . . . . . C . 1.3 Comparing the Two Met hods 55

D Drafting. and Parts List 57

E Manufacturer Specifications 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1 DC Motor 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Lead Screws 73

E.3 3/32" Hardened Steel Pin . . . . . . . . . . . . . . . . . . . . . . . 73

. . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4 Clamping Collars 73

. . . . . . . . . . . . . . . . . . . . . . . . . . . . E.5 Set Screw Collars 73

F Workspace

List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 One-Bay Trussarm 2

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Four-Bay Trussamn 3

. . . . . . . . . . . . . . . . . . . . 2.1 Initial Workspace Specification LO

. . . . . . . . . . . . . . . . . . 2.2 One Bay Workspace Requirements 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Actuator Plane 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Passive Structure 16

. . . . . . . . . . . . . . . . . . . . 3.3 Actuator Workspace Clearance 17

. . . . . . . . . . . . . . . . . . . . . . 3.4 Actuator Load Calculations 19

. . . . . . . . . . . . . . . . . 4.1 Standard Pick and Place Operation 25

. . . . . . . . . . . . . . . 4.2 Plotting Performed by Trussarm Mark 1 27

. . . . . . . . . . . . . . 4.3 Plotting Perforrned by Trussarm Mark II 28

. . . . . . . . . . . . . . . . . . . . . C.l Node Numbering of Trussarm 51

. . . . . . . . . . . . . . . . . . . C.2 Member Numbering of T m s s m 52

. . . . . . . . . . . . . . . . . . . . . . . . . . . D.l Trussarm Mark II 58

. . . . . . . . . . . . . . . . . . . . D.2 Assembly Drawing of Actuator 59

. . . . . . . . . . . . . . . D.3 Assembly Drawing of Passive Structure

. . . . . . . . . . . . . . . D.4 Passive Structure's Degrees of Freedom

. . . . . . . . . . . . . . . . . . D.5 Actuator Extension and Retraction

. . . . . . . . . . . . . . . . . . D.6 Offsets Superimposed on Structure

. . . . . . . . . . . . . . . . . . . . . D.7 Offsets and Member Lengths

. . . . . . . . . . . . . . . . . . . . . . . . . D.8 Actuator Numbering

. . . . . . . . . . . . . . . . . . . . . . . . . . D.9 Experimental Setup

. . . . . . . . . . . . . . . . . . . . F.l Gantry and Coordinate System

. . . . . . . . . . . . . F.2 Fully Retracted Reach X.Y. Z = O. O. 1.58 m

. . . . . . . . . . . . . F.3 Fully Extended Reach X.Y. Z = 0. O. 2.14 rn

F.4 Extended Negative X-axis Reach X.Y. Z = -1.64. 0. 0.49 m . . . . . F.5 Retracted Positive X-axis Reach X.Y. Z = 1.45. 0. 0.38 m . . . . . .

F.6 Negative X-axis LI-shape X.Y. Z = -1.08. O. -0.06 rn . . . . . . . . F.7 Positive X-axis u-shape X.Y. Z = 1 .07. 0. -0.23 m . . . . . . . . .

F.8 Negative X-axis S-cume X.Y. Z = -1.15. 0. 1.19 m . . . . . . . . . . F.9 Positive X-axis S-cvme X. Y Z = 1.33. 0. 0.99 m . . . . . . . . . . .

F.10 Extended Reach with Negative X-axis Hook X.Y. Z = -0.57,0, 1.62 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F.11 Retracted Reach with Positive X-axis Hook X.Y. Z = 0.68, 0. 1.34 rn

F.12 Straight Swing in Negative X-axis X.Y. Z = -1.34. 0. 1.50 m . . . . F.13 S traight Swing in Positive X-axis X.Y. Z = 1.46. 0. 1.45 m . . . . .

... V l l l

List of Tables

3.1 Configurations with Highest Actuator Loads. Actuator Length in . . . . . . . . . . . . . . . Centimetres With Respect to Baselength 20

. . . . . . . . . . . . . . . 3.2 Highest Actuator Loads for Bottom Bay 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.l Cables. Part 1 67

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Cables. Part 2 68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 Cables. Part 3 69

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 Parts List 70

. . . . . . . . . . . . . . . . . . . . . . . . . . D.5 Parts List Continued 71

. . . . . . . . . . . . . . . . . . . . . . . . . . E.l Motor Specifications 72

F.1 Actuator Positions in Centimetres With Respect to Baselength . . . 76

Chapter 1

Introduction

The design of any robot is an exercise in compromise. A compromise must always be drawn between speed, accuracy, payload, and workspace. For exarnple, four degree-of-freedom (DOF) SCARA type robots sacrifice workspace for speed and accuracy. Large welding robots sacrifice speed for accuracy, payload, and workspace. The designer must choose which parameter is rnost important, and then sacrifice others to attain desired performance in the choosen parameter. Only if the designer does something unconventional and creative can he hope to avoid the usual compromises.

VGTM's are an example of an unconventional and creative design which attempts to achieve a high level of accuracy, without sacrificing payload and workspace. The truss nature of these structures is very stiff for a relatively small rnass. This pro- vides the accuracy of movement without a heavy robot, thus maintainhg payload and workspace.

Onginally conceived as a design alternative to the Space Shuttle Remote Manip- ulator System (SSRMS or Canadam) [l], more terrestrial applications have been examined in recent years. The strength and stiffness of the design lends itself to applications requiring a large payload and workspace, without compromising accwacy.

Figure 1.1: One-Bay Trussarm

1.1 Trussarm

Trussarm' is an example of a VGTM. Best described as a double-octahedral tniss, a trussarm is pictured in Figure 1.1.

By varying the length of the actuators, the top plane can be moved with respect to the bottom plane. Figure 1.1 depicts a one-bay trussarm. A one-bay trussarm h a three DOF, and hence a very limited workspace. However, several bays can be stacked to form a multibay Trussarm as seen in Figure 1.2.

In theory, a multibay tnissarm of this design would have a very large workspace, provide good accuracy due to its stiff nature, and maintain a large payload capacity.

'The term "trussarm" refers to the general concept of a double-octahedral VGTM, while "Itussarm Mark I" refers to the one-bay robot previously built. "ri.ussarrn Mark II" is the subject of this thesis, and will be referred to as "Mark IIn for brevity.

CHAPTER 1 . INTRODUCTION

Figure 1.2: Four- Bay Trussarm

In order to demonstrate the concept, a prototype one-bay trussam was built by the Space Robotics Laboratory at UTIAS. The one-bay structure stands over one metre t d , has a mass of 41 kg, and is identical in design to Figure 1.1.

In order to demonstrate cornputer control of Trussarm Mark 1 two demonstration programs were developed. The first has bccome known as the typewriter demonstration in which Tmssarm Mark 1 plots user selected letters on a wipe board. This demonstration, including the kinematic theory and code implemented, is best documented by Joseph Sallmen's thesis [2]. It should be noted that Roger Hertz made invaluable contributions to the refinement of the kinematic algorithms.

The second demonstration was developed by David Fenske [3]. Using a PAMI- Virtek Vision System, Fenske demonstrated the ability of Trussarm Mark 1 to directionally tradc a target.

CHAPTER 1. INTRODUCTION 4

With both these demonstrations, the ability to control a one-bay t r u s s m in real time was confirmed. The design of the Mark 1 has stood the test of time with repeated use. However, moving a camera or magic marker can hardly be considered as taxing tasks for a 41 kg robot.

Why Build Anot her Trussarm?

With the success of Tmssarm Mark 1, the construction of a multibay trussarm would seem to be the next logical step. The workspace of the Mark 1 was severely Limi ted, and obviously incapable of demonstrat ing the advantages of multibay design~.

The reasons for not irnmediately building a four-bay trussarm are varied. To begin, the cost of duplicating the Mark I was prohibitive. Second, there were concerns with the design. As a one-bay, or even a two-bay structure, the Mark 1 has no problems. As a 124 kg four-bay structure standing over four metres tall, however, real concerns were raised about it supporting its own weight [4]. Finally, such a structure would not fit inside the Space Robotics Laboratory. These problems put a four-bay design on hold.

Despite this, work continued on the multibay concept. Naccarato and Hughes [5] showed that reference shape curves could resolve the kinematic redundancies of a multibay trussarm. Finite element rnodels prepared by Roger Hertz evaluated static loads2. A dynamic and control mode1 done by Carroll, Sincarsin, and D'Eleuterio [6] generated no concerns. A four-bay structure was, according to extensive research, a feasible concept.

With static loads, dynamic loads, and control issues having al1 been examined, the logical step was to design and construct a multibay trussarm. Such an exercise would serve multiple purposes. First, it would serve as a full scale real-life demonstration of the multibay tnissarm concept. If a picture is worth a thousand words, an animated Life size structure should be worth a million. Second, practi- cal elements of the concept, such as joint design, machining, assembly, and wiring, would d be brought to the forefront. Third, it would serve as a kinematic test bed for two- bay kinemat ics, reference shape curve kinemat ics, and obstacle avoidance algorthms with a real structure. Although ail have been tested in simulation, none have been tested on a real robot with real amplifiers and real proximity sensors.

lSorne but not all of this data cm be found in [4].

CHAPTER 1. INTRODUCTION

1.4 Thesis Outline

Wit h funding available through the Industnal Trussarm Project (ITP) , the deci- sion was made to design and construct a multibay trussarm. This thesis describes the design and constmction of Trussarm Mark II, a four-bay trussarm. An extensive part of this thesis was the development of differential kinernatics for trussarm. Much of this work was done prior to funding provided by the ITP. Though or&$- nally studied for other reasons, it was critical as a tool to evaluate actuator loads. Therefore, the differential kinematics of trussarm is described at length. A sum- mary for each chapter is provided below.

1.4.1 Conceptual Design

This chapter begins the sarne way the ITP began, with a full review of Trussarm Mark 1. A review of this first-generation trussam is presented. The second part of this chapter specifies the workspace, payload, speed, accuracy, and cost constraints of the ITP. Within this €rame work, the detailed design work began.

1.4.2 Detailed Design

This chapter documents the detailed design of al1 elernents of the Mark II. Dis- cussion begins with the actuator design. Vital to their design was the accurate calculation of the actuator loads. To this end, differential kinematics was used to evduate these loads quick1y and easily. The analysis of the differential kinematics was by no means trivial, and thus the discussion is included here. Finally, some discussion is made of the many minor design points.

1.4.3 Evaluat ion and Testing

This chapter evaiuates the Mark II. It describes some of the probiems discovered once the Mark II was operational. Possible fixes for these problems me presented. How well workspace, speed, payload, and accuracy requirements were met is also documented.

CHAPTER 1. INTRODUCTION

1.4.4 Conclusions

This chopter discusses future improvements to the Mark II. The Mark II is by no means a perfect structure and improvements that should be made first are pointed out. This chapter ends this thesis by suggesting possible research projects for which the facility is idedy suited.

Chapter 2

Concept ual Design

The ITP provided funding for a new robotic facility based upon trussarrn technology. For reasons discussed in the Introduction, it was decided that a new multibay trussarm should be designed.

2.1 Design Review of Trussarm Mark 1

Those who forget history are destined to repeot it.

Trussaxm Mark 1 has been in many ways a success. But the design, Iike many first attempts, hm shortcomings. A review of the Mark 1's design was done in order to avoid repeating mistakes.

2.1.1 Scale

Does sire rnake a différence?

For reasons that have never been documented, Trussarm Mark 1 was built with 1 m long passiuel members. Although sirnplifying the kinematic mathernatics, no reason for this choice has been discovered. Tnissamn Mark I is a fairly large robot

- - - - - - - -

'Members of h e d length are referred to as passive.

7

CHAPTER 2. CONCE3PTUA.L DESIGN 8

standing over 1 metre tall, and having a mass of 41 kg. A four-bay trussarm of this scde would not fit in the Space Robotics Labonitory as the ceiling is too low. The total mass of the structure would exceed 100 kg, making it hard to secure the base without permanently bolting it to the concrete floor (like the Daisy facility). Because of its size, Trussaxm Mark 1 is difficult to transport outside the lab for dernonstrations. A four-bay trussarm of this scde would be impossible.

More than anything, the scale of Trussarm Mark I rnakes it extremely expensive to build more bays. The cost of the alurninum increases as a function of the scale cubed. The price of parts such as beaings, motors, and bal1 screws ail tend to increase proportionally with scaie. Al1 parts which must be customed machined become more expensive as the labour involved usudly increases with scale for any given part.

The key point here is that when building prototype robots which are not task specific, the designer is free to choose scale. Scale affects bot h cost and the pract ical nature of the facility. Thus it should be chosen with extreme care.

2.1.2 Custom Machined Components

KISS, Keep It Simple Stupid.

Custom machined parts on Trussarm Mark 1, though quite functional and very robust, are nonetheless rather complex. The cost of machining the parts for Trussarm Mark I is not documented anywhere. Based upon experience in getting parts machined for this thesis, they no doubt cost a s m d fortune.

For the Mark II, all elements would need to be kept as simple as possible. The key benefit would be Lower construction cost. Other benefits such as easier assembly and maintenance would anse from a simple design.

2.1.3 Consistency of Design

A chain is only as strong as its weakest link.

In his report, h g [4] noted that Trussarm Mark 1 would not be able to support an additional three bays. In fact this statement is not entirely tme. The structure is more than up to the task, except for the method chosen to secure the duminum

members to the hinges. The members are secured using steel pins. Analysis showed that the steel pins would rip through the soft aluminum with the weight of three more bays.

Another weak point of the design is the actuator hinges. The actuators are connected to each other off the centre line of the ball screws. Under heavy loads, this produces strong bending moment on the ball screws. Manufacturen of ball screws and lead screws expressly stâte that such mountings should be avoided. Though not a real concern for a one-bay trussami as the loads are light, a multibay trussarrn severely loads the act uators.

At 41 kg, there is more than enough material to do the job right, but small lapses in judgement produce weak points, ruining the potentia.1 of this design for more bays.

2.2 Specificat ions

Before design work began, specificatioos for payload, speed, accuracy, and workspace were required. The desire was to produce a robot that accentuated trussarm's strong points, while not costing an enormous amount to build. Wit h this in rnind, the following specifications were drawn up.

2.2.1 Workspace

The workspace of a robot is defined as the set of kinematically possible configurations. In order to best demonstrate the workspace of trussaxm, a multibay stmcture with the ability to form both a ü-shape and an S-curve would be ideal. See Figure 2.1. These configurations demonstrate well obstacle avoidance capabilities, and the dexterity of trussaan.

In t e m of total reach, one metre was thought large enough to be simple to build, and yet s m d enough to be dordable. Retraction ability2 is a function of the extension ratio3 of the actuaton. It was hoped Tmssarm Mark II would have a minimum reach 50% of its maximum reach. Meeting this goal would depend upon the actuator design.

'One of trussarm's unique abilities is to deploy and retract dong a straight path. 3E~ension ratio is defined as maximum length divided by mininum length.

CHAPTER 2. CONCEPTUAL DESIGN

U-Shau and S-Curve Confiaurations

Figure 2.1 : initial Workspace Specificatioo

2.2.2 Payload

A payload capacity of 3 kg was deemed to be a reasonable goal. This payload would be in line with the capabilities of the CRS A465 robot and other industrial robots of similar physical size.

2.2.3 Speed

The speed of robots is a most difficult thing to define. The maximum speed of a l l robots changes depending on where the end effector is located, and in what direction the end effector is moving. For the Mark II, an end effector speed of a 15 cm/s was chosen. However, at this time it was unclear whether this speed was a realistic goal.


2.2.4 Accuracy

Accuracy of robots is easier to define than speed, but impossible to measure without sophisticated calibration equipment. In addition, the accuracy of a robot is highly dependent upon the quality of machining, and precision cornponents such a bearings. Though the technology exists, and precision components are available off- t he-shelf, t hey are very expensive, and beyond the budget of t his project . Since it was going to be near impossible to rneasure accuracy, and prohibitively expensive to attain it, no accuracy goals were set.

2.3 Mark II is Born

With specifications laid out above, the conceptuai design could be stasted.

In order to achieve the workspace goals a minimum of four bays would be required. Each bay would need to be able to tilt at least 45' and retract by 50%. See Figure

This would enable four bays to form Il-shape and S-cuwe configurations, and obviously retract 50%. Since four bays were deemed necessary, and a 1 rn total reach specified, the height of each bay would be approximately 25 cm. It became quickly apparent that such a design would require actuators that would be very hard to build due to their small size. The individual components for the actuators would be difficult and expensive to obtain. Hence, the total reach was increased to the 2 m range. The minimum reach goal would then be 1 m.

DC electric motor driven Iead screws were selected as the basis for linear actuation. Pneumatic and hydraulic actuators were quickly discounted due to cost, and lack of experience in the lab with these systems. In addition, DC electric motor driven lead screws, dong with appropriate encoden, would be compatible the existing CRS C500.

Due to budget contraints it was decided to use the CRS C500 controller in the lab. Since the (3500 is an 8-axis controller, only two of the four bays would be controllable. Four bay control would have to wait for a new controiler. Aowever, it was felt important to constnict dl four bays at once. This was due to the nature of machining costs. The main cost is in the setup. Once the tools have been set up, parts can be run off relatively quicHy. For this reason, ordering enough parts for four bays at once would be Iess expensive than ordering parts for two bays twice.


Figure 2.2: One Bay Workspace Requirements

The above reasoning also meant that each bay would be identical. This would reduce the number of unique parts and thus reduce costs. Ideally, the upper bays could be made with smaileï and lighter components since they carry less deadweight. This was deemed as being too costly for this project.

With control available for only two bays, power would be needed br only six actuators. Trussarm Mark I's existing amplifiers were deemed suit able. Three new amplifers would be purchaaed to provide a total of six.

A mass budget of 3 kg per bay was set, but would later turn out to be mattainable without resorting to very expensive materiah. Aowever it would serve as a ball park figure with which to begin calculating actuator loads.

A budget of $1500 per bay was set as a target goal. This price would inciude the cost of the motors, lead screws, gearing, bearings and bushings, custom machining, and materials. Amplifiers and other electronics were budgetted separately as they were mostly the responsibility of the electronics technologist, Thierry CherpilIod.

CHAPTER 2. CONCEPTUAL DESIGN 13

With the above outline, detailed design work began by October 1994. A fully functional two-bay trussarm, with two bays attached but not powered, would be operational by April 1995.

Chapter 3

Det ailed Design

This chapter discusses the detailed design. This chapter, like the design process of the Mark II, is split into two sections. The first deals with the design of the actuator plane pictured in Figure 3.1. The second deds with the design of the passive structure seen in Figure 3.2.

3.1 Actuator Design

The actuators are by far the most critical component of any trussarm design. This is due to the fact the actuators serve the dual purposes of actuation and support. Industriai linear actuators were looked at and disregarded for two reasons. First, their cost was prohibitive. Second, most do not provide the proper position information for robotic controilers. Hence, the actuators were designed in-house.

3.1.1 Preliminary Actuator Design

The linear actuators built for Mark II differ from those on Mark 1. First and foremost is the method of mounting the actuators. The actuators on Mark 1 are mounted with an offset fiom their centre Line. This means the actuator must cany bending moments. In general, this is a very bad idea. When a lead screw, or b d screw, is loaded in this fashion, it will bend, causing binding between the nut and screw. For a rnuitibay structure Kke Mark II, with relative loads much higher, keeping the lead screw in pure compression or tension is critical.

CHAPTER 3. DETMLED DESIGN

Figure 3.1: Actuator Plane

Figure D.2 shows the split yoke, Part #13, which connects the actuators. Mounted dong the centre line of the lead screw, large bending moments are avoided. Unfor- tunately a compromise is made in terrns of extension ability. In order to provide clearance as the angle closes between adjacent actuaton, as seen in Figure 3.3, the minimum length is inneased, effectively decreasing the extension ratio. For this reason, the extension ratio of Mark II is much less than Mark 1. This factor limits the retractability.

The split yoke proved to be easy to machine. The reader is encouraged to compare this piece with the complicated curved clamp used on Mark 1 (see Figure 1.1). This design is simple enough that off-the-shelf linear actuators could be retrofitted in the future with little or no modification to the existing structure.

Findy, timing belts were used to transfer power from the motor shaft to the lead screw. This has many advantages over the w o m gear used on Trussarm Mark 1. First, it was cheaper. Second, different gear ratios could be selected by interchanging pulleys with no modification to the mounting structure. This

CHAPTER 3. DETAILED DESIGN

Figure 3.2: Passive Structure

allowed lower gearing on the lower bays with more load, and higher gearing on the upper bays with less load. Third, worm gears (or any gear train) require very precise dignrnent. With timing belts, the belts act as a flexible transmission systern which will absorb any misalignment with ease. Fourth, timing belts are used in precision plotters such as the lab's CALCOMP, and industriai robots such as the lab's CRS A465. This disrnissed concerns raised about backlash or slop in this type of transmission. Findy, the pulleys are easily interchangable ailowing a vaxiety of gear ratios depending on whether the lab wished to maximize speed or pay load for different experiment S.

3.1.2 Actuator Loads and the DifEerential Kinematics

Finding the actuator loads was the first cntical step. Since al1 actuaton were to be identical, the task was sirnplified to identifying which actuator experienced the greatest load, and evaluate that load. AU actuators would then be built to handle

Figure 3.3: Actuator Workspace Clearance

this load. Intuition, and finite element models done by Roger Hertz, indicated t hat the lowestl bay actuators would have the greatest loads. Thus the actuators were built to handle the worst case load of the bottom bay.

A finite element model (FEM) could calculate the loads at the actuators. However, a FEM would be time consuming to set up. In addition, the best software package for the job was IDEAS, which is installed at the Engineering Computing Facility (ECF). The CPU time cost, and the learning curve required to produce the model were prohibitive.

Instead, differential kinematics was used to find the actuator loads. The differential kinematics of a robot relate differential movements of the actuators to differential movernents of the end effector. Differential kinematics ultimately involve the determination of a robot's Jacobian matrix. In robotics, the Jacobian matrix J is

where r = [rl rz ~3 ... r,,,IT

'Note that the term lowest or éotiom bay will alway refer to the bay cloaest to the base.


is a vector of the coordinates of the end effector and

is a vector of the joint variables. Since a single bay is a three DOF device, J is a 3 x 3 matrix. From the above definitions, the Jacobian is simply a matrix containing the partial derivatives of the rate of change of the ri with respect to a11 qi. This implies that the forward kinematics, the behaviour of r as a function of q is known. For trussarm, no closed-form solution of even the one-bay is known. Hence, the Jacobian matrix cannot be analytically expressed. However, the inverse kinematics of trussarm is known, and can be expressed in a closed-form. This allows us to form an analytical expression of the inverse Jacobian for trussarm. Thus we have

where, in accordance with the definition in Equation 3.1,

a(l, &Il

This expression is in no way trivial. Appendix C contains a more in-depth look into how the Jacobian is formulated for trussam geometries.

Now with J, the principle of virtual work can be used to show that

T = J=C (3-6)

where f is a vector of the generalized forces at the end effector, and r is a vector of the actuator forces [7]. Hence, given f, one can find T .

Of course this only works for a one-bay trussarm. But the inverse kinematics allow an arbitrary tool vector? Thus, to find the static load on the lowest three actuators,

2The tool vector points fiom the centroid of the top plane to the location of an arbitrary tool.

CHAPTER 3. D E T ' L E D DESIGN

centre of mass single bay

centre of mass of upper three bays

vector

drawing not to scale

Figure 3.4: Actuator Load Calculations

the upper three bays are put in an arbitrary configuration. Next, knowing the mass of each bay, and assuming that the centre of mass of each bay is Located at the midpoint of the vector drawn from the centroid of the bottom plane to the centroid of the top plane, the centre of mass of the top three bays can be found. A tool vector can then be defined to the centre of mass. A mass equal to the mass of the upper three bays can be placed at the tip of the tool vector. Next, defining a gravity vector produces f, the load acting at the tool tip. Finally, finding J with the tool vector, and using Equation 3.6, T is found. This process is pictured in Figure 3.4.

As with most design processes, the problem of which cornes first, the chicken or the egg, is always faced. The design of the actuators could not start without the the mass of the structure being known. But, the mass of the structure could not be found without the design of the actuators. Thus, the bays were assumed to have a mass of 3 kg each. A 3 kg payload was located at the top of the fourth bay so payload goals would be met.

Table 3.1: Configurations wit h Highest Act uator Loads, Act uator Lengt h in Cen- timetres With Respect to Baselengt h

Table 3.2: Highest Actuator Loads for Bottom Bay

F.13 1 +7.6 1 -7.6 1 -7.6 1 -7.6 1 -7.6 1 -7.6 ( -7.6 1 -7.6 1 -7.6 1 -7.6 1 -7.6 1 -7.6 1

11 +7.6 -7.6 -7.6

Different configurations were tried. Parameters such as the lengths of passive members, maximum actuator length, and minimum actuator length, were varied to find a workable design. Designs were run through forward kinematics to ensure

9 +7.6 -7.6 -7.6

12 +7.6 -7.6 -7.6

Actuator F .6 F.7 F.12 F.13

workspace requirements were being met. Eventually a workable design was settled

10 -7.6 +7.6 -7.6

5 f7.6 -7.6 -7.6

Actuator F.6 F.7 F.12

2 -405 N +202 N -423 N +213 N

1 3.345 N -910 N +362 N -959 N

upon. Setting passive members around 28 cm in length, and using actuators with a 30 cm minimum length and 46 cm maximum length, most workspace require-

2 +7.6 -7.6 f7.6

1 -7.6 +7.6 -7.6

3 -405 N +201 N -423 N +213 N

ments would be met. In addition, actuators loads rernained under 1000 N. Widely available 1/4"-20 lead screws would just be able to handle this load3.

6 f7.6 -7.6 -7.6

Table 3.1 lists the four actuator configurations that gave the highest predicted loads. Actuator positions are given in centimet res with respect to baselengt h. These configurations are pictured in Appendix F. The loads are contained in Table 3.2. Two observations of note. Loads are highest when the upper bays are fully extended out since the centre of gavity is furthest from the base. Second, loads are slightly higher when the lowest bay tilts by extending only one actuator as in F.13, as opposed to F.12 where tilt is achieved by extending two actuators.

3 +7.6 -7.6 +7.6

It should be noted that this process could be repeated up the structure for each bay. The gavity vector would be constantly changing as the base of the upper bays is not fixed, but with some programming the loads on al1 twelve actuators couid have been calculated.

7 -7.6 $7.6 -7.6

4 -7.6 +7.6 -7.6

3See Appendk E for manufacturer specifications.

8 +7.6 -7.6 -7.6


3.1.3 Drive Motor and Gear Ratio Selection

DC electric motors corne in wide variety of sizes, power, and price. Chosing the right one was a tricky task.

The maximum actuator load was predicted to be around 1000 N. With a quarter inch lead screw having a rated efficieny of 0.42 oz-in per 1 lb, 225 oz-in would be the maximum torque required. A DC motor capable of this amount of torque would be to large. DC motors with a s t a l l torque of 50 oz-in were available and were reasonably priced, and sized. Using a gear ratio of 4.4:1, 50 oz-in motors would be suitable to the task. Twelve such motors with built in encoders were puchased.

3.2 Passive Structure Design

The passive stmcture can be seen in Figure 3.2. The passive structure was designed to fit around the actuator plane. The design needed to meet t hree critical items. First, the buckiing loads of the passive members needed to be more than adequate. Second the joints needed to have a minimum arnount of play. Third, the design needed to be simple enough that machining costs would be minimal.

Extruded aluminurn tubing provided more than enough strength for passzue members #22. For the lengths used, this provided a buckling load 8200 N, far exceeding the maximum actuator loads. Not surprisingly, the FEM by Roger Hertz showed that passive member loads were of sirnila magnitude to the actuator loads. Thus with a smail leap of faith, the 318" aluminurn was assumed to be more than adequate.

For #23 passive members 1/4" drill rod provided a precise and hardened surface about which rotation was possible. This provided the so called petaki of the passive stmcture with one DOF.

Securing the ends of the passive members was performed using split clamps. Capa- ble of withstanding axial loads of 2600 N4, this system was chosen for its simplicity of construction, strengt h, and ease of assembly.

Findy, simpliSing the design was by far the hardest goal to achieve, and required the moat creativity. Literdy hundreds of rough p e n d sketches were examineci

'See Appendix E.

CHAPTER 3. DETALZIED DESIGN 22

before the final design was achieved. This design employs a single profde with a 120" bend (see Figure D.3). This simplified rnachining irnmensely as the machine shop needed only to make a single profile, and then cut and drill the individual pieces. Aluminum was used for Parts #18,19, and 20.

The passive structure is connected to the actuator plane by t h small yokes, Part #14. The srnall yokes, secured by shoulder socket screws provide the three DOF needed at the junctwe between the passive structure and the actuator plane. The small yokes use a 3/32" hardened steel pin to rotate about. Despite their small size, these pins have a 500 lb double shear strength. This is more than adequate (see Appendix E).

Ideally Mwk II should be a perfect truss. The reality is members do not intersect at single points. Figures D.6 and D.7 show the offsets required for Mark II to be mechanically possible. These offsets are required by the kinematics. Note the lengths of the passive members are both around 28 cm in length as desired. Making both passive members equal length was not possible with the simple design of the joints.

3.3 Mass and Budget Goals

The mass and budget goals of the project were known to be exceeded before construction even began (see Table D.5). The cost per bay was around $1800. This exceeded the original target of $1500 per bay by 20%. The bays would also end up with a mass of 4.6 kg each, exceeding the original god of 3 kg by over 50%.

The budget was able to hande the s m d cost overrun. The mass budget was another issue. There was some concern that the actuators may have trouble reaching some of the high-load configurations. However, not much could be done to reduce weight wit hout increasing costs even more.

Chapter 4

Evaluat ion and Test ing

Once completed, Mark II was tested. Workspace goals were met for the rnost part. Speed was reasonable, but not anywhere near originally set goals. The desired payload capacity was eàsily met. Finally, the stiffness of the structure was good, but the slop in the joint assernbly proved unacceptable. Despite this, the prototype robot is capable of perforrning many experimental tasks. Much was lemed by this exercise, and more can be learned with continued use of the Mark II.

4.1 Workspace Evaluat ion

The kinematicdy possible workspace is documented in Appendix F. This is the workspace that would be achievable with al1 four bays actuated, and actuators capable of their full extension.

The reality of the situation is that only two bays on Mark II are actuated at this point. In addition, testing revealed that actuators on the lowest bay are overstressed and cannot fully extend in al configurations. It was realized during the design process that the bottom three actuators would be operating at around 120% of their maximum compression load specifications (see Appendix E). It was hoped these specifications were overly conservative. This appears not to be the case.

When attempting to achieve the configuration in Figure F. 13, the extending actuator protests loudly. It is believed that the actuator will extend fully, but not

CHAPTER 4. EVAL UATION AND TESTING 24

without causing severe Wear to the lead screw and its threads. The lead screw is not in danger of collapse. The compression load is causing the lead screw to slightly buckle. This causes the threads between the lead screw nut and lead screw to bind (Parts #5 and #4). Binding is also likely occurring between the lead screw and its bearing support (Parts #4 and #7).

Right now the actuator is only capable of 90% of its extension, or a length of $6 cm with respect to baselength ( s e Figure D.5) while moving to configuration F.13. Therefore, not much work space is lost. The second bay does not suffer these problems as the load is much reduced. If the third and fourth bays were powered, no problems would be expected at a l l with their actuators as the Ioads are even less.

Two solutions may fix this problem. The first is simple and inexpeosive. Replace the lead screw with one having a custom machined end to accept a radial bearing replacing Part #7, a bronze bushing. This will eliminate one source of binding. In order to preveot the lead screw itself from buckling, support will need to be provided at the opposite end, or a fixed-supported bearing support. This is a much more expensive proposition as the inside of the Thompson shafting will have to be hished smooth.

The retraction of Trussarm Mark II is 73% of its maximum reach, instead of 50%. This is due to the actuator design, which has an extension ratio of 150%. Actuators with a 168% extension ratio would be needed to realize 50% retraction, al1 other dimensions rernaining the same. An actuator with that type of extension would have to be telescoping in order to remain rigid when fully extended. Such a design would be very expensive to build.

4.2 Speed Evaluat ion

The original speed goal was 15 cm/s. The maximum speed obsenred out of the two bays is 1.9 c m / s . This speed was detennined by moving from a fully retracted configuration, to a fidly extended one. This motion resdts in a 29 cm displacement of the end effector along the z-axis. The motion required a total of 15 S. With two more bays actuated, the total motion would have b e n 58 cm, with the time remaining at 15 S. Hence four bays should be able to reach end effector speeds of 3.8 cmfs.

A couple of notes should be made at this point. The doaded speed of the DC

CHAPTER 4. EVAL UATrON AND TESTING

Figure 4.1: Standard Pick and Place Operation

motors is 6500 RPM. The C500 was instructed to limit the maximum speed of the motors to 2925 RPM. When speeds higher than 2925 RPM itre cornmanded, the 12 Voit amplifiers fail to provide enough power. Heoce the factor most lirniting speed at this point is the amplifiers. The DC motors were not overheating, and according to their specifications in Appendix E, they should be able to provide more speed.

Second, the observed speed is in fact an average speed. The actuators require some time to r m p up to 2925 RPM. Hence the maximum speed is in fact slightly higher.

Speed of course varies with trajectory. A standard test in robotics is the 12 inch pick and place seen in Figure 4.1. Note nothing is ever really picked up, it is a simple test of speed over a standard trajectory. Though this test is meant to compare high-speed assembly robots, its the closest thing to a standard speed test the fieid of robotics has ever invented, and so is used here. Tmssamn Mark II is capable of a 26 s pick and place. This is nowhere near the 1.2 s of the CRS A465. Then again the tnissann concept was never meant for high-speed assembly. This if anything should underline the problem with attaining high speeds with such a structure. A hydradic trussarm would be required to achieve speeds of similar sized serial m s .

For now, the speed of is limited by the amplifiers. The drive train could handle more power if larger amplifiers became available. With better amplifiers and four bays, end effector speeds wouid increase to around 5 cm/s.

CHAPTER 4. EVALUATION AND TESTING

4.3 Payload Evaluat ion

The original payload goal was 3 kg. The highest loaded carried by Mark II is 3.3 kg. With 3.3 kg bolted to the top plane of the fourth bay, Mark II easily moved £rom position F.3 to F.2. This up and down motion was achieved with maximum motor speeds of 2275 RPM before the 12 Volt amplifiera gave out. With 2.3 kg bolted to the top plane of the fourth bay, al1 of the workspace achievable with no load was realized, albei t at slower speeds.

Mark II bas a total mass of 18.4 kg and can easily c a n y 3.3 kg. The CRS A465 robot has a mass of 31 kg and has a maximum payload of 3 kg. The A465 h a rnuch higher speed, but Mark II possesses far geater reach. Hence, in terms of payload, Mark II performs admirably.

4.4 Accuracy Evaluat ion

Testing the accuracy of a robot can be a difficult and expensive task. For this t hesis, accuracy was only qualitatively evaluated. The easiest way to test accuracy was to place a wipe board underneath Mark II, and with a marker acting as a tool, evaluate its plotting skills. A sirnilar task was set for T r u s s m Mark 1 with its typewriter demonstration. Figure 4.2 contains a picture of the plotting skills of Trussarm Mark 1 [2]. Figure 4.3 contains a picture of a plot perfomed by Trussarm Muk II. Neither plot is to scale. The letters are in redity 4 cm tail in Figure 4.2 and 5 cm ta11 in Figure 4.3.

The plot is not as precise as originally hoped. Some wiggle in the pen can be seen. However, it should be noted that Mark II was plotting on a board over 1.6 m fiom its base. For an 18.4 kg robot this is impressive. The CRS A465 robot has a maximum reach of 0.711 m and a m a s of 31 kg. In addition, much of the wiggle is caused by flexibilty in the mounting of the baseplate to the gantry. This mounting is being replaced.

On a more critical note, the slop in the joint assembly connecting the actuator plane with the passive structure is unacceptable. It is apparent upon close inspection that Part #13 in the bottom bay is too weak. This split yoke is supposed to clamp tightly around the Part #2, the large aluminum tube forming the actuator. During testing, the split yoke has obviously opened up by about 1 mm. This results in end effector play at the fourth bay of around 2.5 cm. The split yokes in the other

CHAFTER 4. EVAL UATION AND TESTING

UTIAS Figure 4.2: Plot ting Performed by Tnissarm Mark I

three bays seem to be holding firm.

Until the baseplate is properly mounted, the slop is removed, and problems in the kinematics have been worked out (see 4.5) a final judgement on the potential accuracy of is hard to make.

4.5 Kinemat ic Test ing

In order to control Mark II in real time, forward and inverse kinematics needed to be programmed into the C500 controller. This task was performed by Regina Lee (81. Since only two bays were controllable, two-boy kinematics developed by Hughes and Nacarrato were implemented [5].

Unfortunately, some problems were soon noted with this algorithm. The two- bay inverse kinematics determine only position, pitch, and roll, or five degrees of Ereedom. The sixth degree is used to optimize the shape of the two bays'. This means the yaw angle of the tool vector can drift. While not a problem if the tool vector is located at the centroid of the top plane of the second bay, it becomes apparent when the tool vector is located away fiom this point.

'A cost function penaliaes deviations €tom the centre of the worhpace [5].

CHAPTER 4. EVALUATION AND TESTING

UTIAS Figure 4.3: Plotting Performed by T r u s s m Mark II

In the case of the plotting routines, the pen was mounted at the vertex of the top plane of the fowt h bay. Drift in the yaw angle is believed to be responsible for the slight enlarging of letters in Figure 4.3 from left to right.

Other simple tests such as pure x-suis or y-amrs moves of the peu dong the wipe board reveded pointing errors too large to simply be slop in the structure. Errors as large as 5 cm in position were discovered while attempting to move the pen 30 cm. These errors were minimized in the plotting routine used in Figure 4.3 by performing many small moves of less than 1 cm.

Other tests were performed with the C500 alone. Executing the forward kinematics on a set of actuator lengths produced a cartesian location of the end effector. When the inverse kinematics was run on the cartesian point, the same set of actuator lengths should result. With the tool vector set to zero, this was the case. With a nonzero tool vector, this wacr not the case.

Confidence is very high that this is not a programming bug. A more detailed examination of this problem can be found in Regina Lee's thesis [SI.

Chapter 5

Conclusions

The construction of Trussarm Mark II has been successfully completed. Though only two bays are actuated, the knowledge and tools are al1 in place for the actuation of al1 four. Mark II has been successfully integrated with the CRS C500 controiler. Three s m d dernonstration progams have been run on the C500. The first is the plotting routine discussed previously. Second, the PAMI-Virtek Vision System used by Fenske [3] has been integrated and cas perform target tracking. Last, a program to demonstrate the workspace has been written. These three pro- g a m s prove that a multibay t r u s s m can be controlled in real time and perforrn useful tasks. They have dso brought to light problerns with the two-bay kinematics never known before.

This exercise has forced manufacturing issues, such as joint design, to corne to the forefront. The cost of building a robot should not be underestimated. The CRS A465 costs over $40,000 to purchase. Prototypes, due to their one-of-a-kind nature, always cost an order of magnitude more than production models. Mark II by cornparison was assembled on a shoe string budget. However, with some creativity and carefid attention to detail, the construction of a useful faciiity was accomplished.

5.1 Fut m e Improvements

Tmssarm Mark II has demonstrated, for the fist tirne, that a working multibay trussarm with a usefd workspace, speed, and payload capacity can be built. The

CHAPTEX 5. CONCL USIONS 30

two most glaring problems with are overly stressed bot tom bay actuators, and slop in the joints. Both of these problems can be fked with off-the-shelf parts. With around another $500 per bay, the shortcornings discussed should be pretty much eliminated. Some suggestions for how to go about this are discussed in Chapter 4. Experimental fixes could be attempted on only one bay. Once a satisfactory fix is found, it could be implemented on the other three.

5.2 Future Research Work

Trussarm Mark II has much to offer once al1 four bays are actuated, even if problems with overloaded actuators and slop are never adàressed. Implementat ion of two-bay kinematics has revealed problems with the algorithm never realized before, despite extensive computer testing. O d y once a real task, on a real robot was performed did problems with the optimization method become apparent. Multi- bay reference curve kinematics [5] has never been implemented on a real robot performing real tasks. Implement ion will reveal any unforeseen difficult ies. T hus, Mark II should be used to confirm the workings of the reference curve kinematics.

Wi t h the successful implementation of multibay kinemat ics, development of obstacle avoidance technologies can begin. The reference curve algorithm can be used to avoid obstacles. In computer simulation, the obstacle's position and velocity are given with respect to inertial space. On a red robot such as Mark 11, obstacle detection will be performed by proximity sensors mounted to Mark II itself. Where will these sensors be mounted? What type of sensors should be used? Can these senson be used to calculate velocity? Can t hey accurately detect the direction in which the obstacle is approaching fiom? Obstacle detection will be relative and not absolute. In order to know whether the robot is approaching the obstacle or the obstacle is approaching the robot, the position and velocity of each sensor with respect to inertial space must be known. Assuming a minimum of three sensors per bay (since each bay has three distinct sides), this means the location of 12 sensors plus one end effector must be caldated in real time. 1s this numerically realistic given the complexity of the kinematics? What happens when a trussarm is in a U'shape configuration like in Figure F.7 and proximity sensors ultimately begin detecting the lower bays of itself? Clearly Mark II has a future as a vehicle for tes ting redundancy resolution and obstable avoidance schemes.

Finally, Thissann Mark II is a working advertisement for the trussazm concept. Its impact is fax greater than an animated simulation on a smail computer screen.

CHAPTER 5. CONCL USIONS 31

The faciiity as a whole (robot, amplifiers, controllet, software) has proven to be very robust so far ônd has given less headaches than Trussarm Mark 1 many a times. In the role of a living billboard, Mark II may prove to be the most valuable.

Appendix A

FORTRAN Jacobian Solver

Program: Jacob.too1.f Purpose: To find numercial values of the Jacobian of

one bay Trussarm. Input: vector(3) - vector from base to end effector Output : j (3.3) - Jacobian

act (3) - length of the three actuators

subroutine jacob(vector,lo,deltaJtool,j,act)

real mag, L02 t e a i j (3,3) , act (3)

real m(91, n(3), e (9) , r, 02, Dl, common m(91, n(3) , e(9) , r, D2, Di,

C C Def ine ail COMMON parameters bef ore

C Hot e the D2 ueed by jac0b.f is double the D2 used by fbS1,subs.f

sngl(delta(1) )

APPENDIX A. FORTRAN JACOBIAN SOLVER

C C Define e and m vectors as per Frank N. C

mag = 1.0/2.0/3.0**0.5 * LOI e(1) = m(1) / mag 4 2 ) = m(2) / mag e(3) = 0.0

C C Define the tool and the r.y.2 location of the end effector. C


Output magnitude o f act0, act3, and acte

Output Jacobian matrix

end

APPENDE A. FORTRAN JACOBIAN SOLVER

Function returns the value of t h i .

real function thi( k, x, y, z )

real m ( 9 ) , 1431, 4 9 1 , r, D2, Dl, L O I , xt, yt, zt real gamma, c, el, 92, e3 comon m(91, n(31, dg). r, D2, D l , Loi, x t , yt, zt

t h i = ( > gamma*((r-xt) **2 + (y-yt)**2 + (z+zt) **2) + > a*(x-xt) + pt*(y-yt) - zt*(z+zt) - > c* (el* (1-xt)+eZ*(y-yt) +e3*(z+a) ) - Dl*(z+zt) ) > / > ( > r* ( (z+zt) **2 + (ei*(x-xt) +e2* (y-yt) +e3* (z+zt) ) **2 ) **O. 5 )

end

Function returns the value of beta.

real function beta( k , x, y, z )

APPENDLX A. FORTRAN JACOBIAN SOLVER

beta = ( ei*(x-rt) + e2*(y-yt) + e3*(z+zt) ) / (z+zt)

return

end

Function retums the value of theta.

real function thetac k , x, y, z )

return

end

Function returns the value of dthi-dx.

real function dthi,dx( k, x, y, z )


tO = (0. XEO*DZ/ ((x-xt) **2+(y-yt ) **2+(z+zt) **2) **O .5EO*(2*x-2*xt) + >(0.5EO-O.SEO*D2/((x-xt)**2+(y-yt)**2+(z+zt)**2)**O.SEO)*(2*x-2*xt) >+n-c*e1) /s/( (z+zt) **2+(el*(x-xt) +e2* (y-yt)+e3*(z+zt) ) **2)**O. SEO- >((O.SEO-O.5EO*D2/((x-xt)**2+(y-f)**2+(z+zt)**2)**0.SEO)*((x-xt)** >2*(y-@)**2+(z+zt)**2)+xt* (x-xt) +@*(y-yt) -zt*(z+zt) -c* (e i * (x-n) + >e2*(y-yt)+e3*(z+zt) ) -Di*(z+zt) ) /r/((z+zt)**2+(ei* (x-xt)+e2*(y-yt) + >e3* (z+zt) )**SI * * O . iSEl* (el* (x-xt) +e2*(y-yt)+e3*(z+zt) )*el

end

C Function r e t m s the value of dthi-dy.

real function dthi,dy( k, x , y, z )

realm(9), n(3), e(9), r , D2, Di, LOI, x t , yt, zt real tO, el, e2, 83, c common m(9), n(3), d9), r, D2, Dl, LOI, xt, yt, zt

tO - (0.25EO*D2/( (x-xt) **2+(y-yt) **2+(z+zt) **2) **O. SEO* (2*y-Z*yt) + >(O~5EO-O.5EO*D2/((rxt)**2+(y-~t)**2+(z+n)**2)**O.SEO)*(2*y-2*pt) >+yt-c*e2) /r/((z+zt) **2+ (el* (x-xt) +a* (y-yt)+e3*(z+zt) ) **2) **O. SEO- >((O.SEO-O.5EO*D2/((x-xt)+*2+(y-~)**2+(z+zt)**2)**0.5EO)*((x-n)** >2+(y-yt) **2+(z+zt)**2)+xt* (x-.t)+yt*(y-yt) -zt*(z+zt)-c*(ei*(x-a)+ >e2* (p-yt)+e3*(Z+Zt) )-Di* (z+zt) ) /r/ ((z+zt) **?+ (eî*(x-xt 1 +&*(y-yt) + >e3*(z+zt) ) **2) **O. lSEl*(el*(x-xt) +eZ*(y-yt) +e3*(z+zt) )*e2


return

end

Function returns the value of dthi-dz.

real function dthi_dz( k. x, y, z )

t0 = (O.2SEO*D2/ ( (x-xt) **2+(y-yt) **2+(z+zt) **2) * *O . SEO* (2*2+2*2t) + >(O. SEO-O . SEO*DZ/ ( (x-xt) **2+(y-yt) **2+(z+zt) **2) * *O. SEO) * ( 2 * ~ + 2 * ~ t ) >-zt-c*e3-Dl) /r/((z+zt) **2+(el* (x-xt)+e2* (y-yt)+e3*(z+zt) )*a?) **O .5 >EO-0. SEO* ((0.5EO-0. SEO*D2/ ((x-rt) **2+(y-yt) **2+(z+zt) **2) **O. SEO) * >((~-xt)**2+(y-fl)**2+(z+zt)**2)+xt*(x-xt)+pt*(y-yt)-zt*(~+~t)-~*(e >i* (x-xt )+eZ*(y-yt) +e3* (z+zt) ) -Dl*(z+zt) ) /t/ ((z+zt) **2+(el* (x-xt ) +e >Z*(y-yt)+&* (z+zt) ) **2) **O. lSEl* (2*2+2*zt+2* (el*(x-xt)+e2* (y-yt) +e >3* (2+2t) > *83)

return

Function returns the value of dbeta-dx.

reaï function dbeta-dx( k, x, y, z


real m(9 ) , n(3). e(S), r, D2, Dl, LOI, xt, y t , zt real tO, e l , 02, 83 common dg), n(31, e(9). r, D2, Dl, LOI, x t , yt, z t

return

end

Function returns the value of dbeta-dy .

real function dbeta-dy ( k, x, y, z )

reaï m ( 9 ) , n(3), e(9). s, D2, Dl, L01, x t , yt, z t real tO, e l , 02, 83 comnon dg), n(3). e(9), r, D2, Dl, LOI, x t , y t . zt

return

end

Function returris the value of dbeta-dz.

APPENDE A. FORTRAN JACOBIAN SOLVEIR

real function dbeta,dz( k , x, y, z )

real m ( 9 ) . n(3), 4 9 1 , r , D2, Dl, LOI, x t , y t , z real tO, e l , 02, e3 comnon m ( 9 ) , n(31, e (9 ) , r, D2, Dl, LOI, x t , @,

t O = e3/ (z+zt) -(el* (x-xt) +e?*(y-yt) +e3* (z+zt) ) / (z+zt) **2

dbeta-dz = t O

retum

end

C Function returns t h e value of dtheta-dx.

real function dtheta,dx( k, x, y , z )

real m(9). n(3) , e (9 ) , r , D2. Dl, LOI, x t , yt, zt commonm(9). n(3). e(9), r, D2, Dl, Loi, n, yt, z t

end

C Function tetuma the value of dtheta-dy.

real function dtheta-dy( k, x, y, z 1

APPENDIX A. FORTRAN JACOBIAN SOLWR 41

return

end

Function returns the value of dtheta-dz .

real fuaction dtheta,dz( k, x, y, z )

reaï m(9), n(3), 4 9 1 , r , D2, Dl, Loi, xt , y t , zt common m(9), n(3), e(9), r, D2, Dis LOI, x t , y t , zt

end

Function returno the value of dR,dx[j] , ahere j=1,2,3.

real function dR,dx( j, k, x, y, z

return

end


Function returns the value of dR,dy [j] . where j=1,2.3.

real function dR-dy( j , k, x , y. z )

return

end

Function returns the value of dR_dz[j]. where j=1,2,3.

reaï function dl,&( j , k, x , y. z )

real m ( 9 ) , n(3). e(9). r. D2, Dl. Loi, x t , y t , zt conmion m ( 9 ) , n (3) , e(9), r , D2, D l , Loi, x t , yt, zt

return

end

Funct ion returns the value of R C f , k] , uhere j=1,2,3 and k=0,3,6

k=O refers t o vector R7 k-3 ref ers t a vector R 8 k=6 refers t o vector R9

real function Rj( j, L. x, y, z 1


return

end

C Function returns the value of Act [j .k] where j=1,2.3 and k=0,3,6

k = O refers to R7-R8 k = 3 refers to R8-R9 k = 6 refers t o R9-R7

real functionActj( j. k, x, y , z )

if ( k .eq. 6 ) then Actj = Rj(j, k, x, y , z ) J 0, x, Y, 2)

return else

Actj = j k x, y , z ) - Rj(j, k+3, x. y. 2)

retum endif

end

C Function returns the magnitude of Act N where j=0,3,6

real function ActHagj ( k, x, y, z )

real m ( 9 ) . n(3). e(9). r, D2, Dl, L01, a, gt,


end

C Calculates the dx temm of the Jacobian matrix

real function dActMagj-dx( k. x, y, z )

real m ( 9 ) , n(31, e(9) , r , D2, Di, Loi, x t , pt, zt comon m ( 9 ) , n(3), 4 9 ) . r, D2, Dl, LOI, xt, yt, zt

elae

end

C C Calculatea the dy terma of the Jacobian matrix


real function dActHagj,dy ( k, x, y, z )

real m ( 9 ) , n(3). e(91, r. D2, Dl, LOI, xt, yt, zt common m ( 9 ) . n(3) . dg), r, D2. Dl, LOI, x t , yt, zt

end

C C Calculates the dz terms of the Jacobian matrix C C

real function dActMagj,dz( k, x, y, z )


end

Appendix B

Alternative Jacobian Solver

C

c subrout ine calculates the j acobian matrix relating c the actuator velocitiea to the nodal velocities C

n=ntmem-nbmem do 10 i= l ,n

do 20 j=l,a jac(i , j)=O.dO

20 continue 10 continue

if (il .gt. nbase) then

jac(ii, jZ+j)=rnode(iZ, j)-mode(i1, j) continue

do 60 j = i , 3 jac(ii , jZ+j)=niode(i2, j)-rnode(i1, j)

continue

end if

continue

********************************** invert ' j acobiad matrix **********************************

do 70 j=l,ntact j ltlistact (j ) j j = j 1-nbrnem do 80 i4.n

hmat(i, j)=jacinv(i,jj)*actlen(j) continue

continue

return end

Appendix C

Jacobian Mat rix Analysis

Before the Indust rial Tmssarm Pro ject (ITP) started, extensive work had been done on trussarm differential kinematics. This was done for three main reasons. First, it had never been done before. Second, the differential kinematics promised to hel p charac terize and compare trussarm witb other serial robots using methods developed by Yoshikawa [9]. Third, Cartesian controllers' and their applicability to tnissarm had becorne an area of personal researdi.

When the ITP started, most of this work had been done, and could be immediately used to find actuator loads instead of deveioping a Cartesian controiler. Though this t hesis bas nothing to do with Cartesian controilers, the mathematics of finding the Jacobian does, and hence is dotnunented in detail here.

C.l The Jacobian of Trussarm

The use of the Jacobiaa matrix J for control of robotic systems in real time implies that J c m be cdculated in real time as wd. If J requires too mu& computation time, the control algorithm which may seem very good in theory, will not work in practice as the algorithm will be unable to keep up with rapid changes of state.

Typically, in robotics, the Jacobian of a robot is expressed as

'In robotics, Cartesian controllers refer to controllus which apply gains to errors in the end efktor reference fiame, as opposed to errors in joint coordinates. They al1 rely upon the ciifFecentid kinernatics being known.

where e = [el e2 e3 ... e,jT

is a vector of the coordinates of the end effector and

is a vector of the joint variables. For trussarm the vector q = [ql qz q3]T where qi is the length of the each of the actuators. The vector e = [el ez e31T represents the Cartesian location of the end effector. The end effector in this case refers to a point fixed with respect to the top plane of trussarm.

Finding the Jacobian for trussarm is not as straightforward as with other manipulators. For example, the pardel geometry of trussarm produces a forward kinematics that has no closed-form solution. Different methods of cdculating the Jacobian for trussarm have been proposed in recent years. However, very little has been done to compare their individual meri ts for cont rol purposes.

The method described irnmediately below was used in this thesis. This method, referred to as the Inverse Kinernatics Algorithm, uses knowledge of the kinematics to find J. Another method of finding the Jacobian, originally proposed by Hughes [IO] will be referred to as the Member Length Method. These two methods are described in more detail below.

C. 1.1 The Inverse Kinematics Algo~ithm

From the above definitions, the Jacobian is simply a matrix containing the partial derivatives of the rate of change of the ei with respect to ail qi- This implies that the forward kinematics, the behavior of e as a function of q is known. For trussarm, no closed-form solution of even the one-bay is known. Forward kinematics are performed by using a root finding algorithm to solve for e. Aence, the Jacobian m a t h cannot be analyticdy expressed. However, the inverse kinematics of trussarm is known, and can be expressed in a closed-form. By analytically differentiating the inverse kinematics, an analytical expression of the inverse Jacobian for trussarm can be found. T hus, we have

top plm.

Figure C. 1: Node Numbering of Trussarm

where in amordance with Equation C.1

The analyticd expression of J(q)-l is by no means trivial. The inverse kinematics begin by expressing the length of the actuators as

where

and k = n + 6

The vector r, is a vector fiom the centroid of the bottom plane to the node n. The above numbering convention can seen in Figure C.l and Figure C.2.

Eaeh vector r, can be expressed as a fanction of a single petal angle fl,. The ptal angle is defined in Figure C.1. Note, trnssarm has three petal angles, one for each node

Figure C.2: Member Numbering of Trussarm

n E (4,5,6). Each of the 0, can be expressed as a function of e, the vector of the end effector. The exact function of 0, wiil not be developed here, suffice it to say that it involves two inverse trigonomettic functions and some basic geometry (see [2] for det ails).

As one can imagine, the computation required to find di nine terms of J is qui te large. The FORTRAN subroutine to perform this task is documented in Appendix A.

C.1.2 Member Length Method

The method origindy developed by Hughes 1101 and employed by Sincarsin, D'Eleuterio, and Carroll [6] differs immensely from the method described above.

The Mernber Length Method defines a matrix

(C. IO)

where each ri is a vector describing the Cartesian location of nodes 4 through 9 with respect to the centroid of the bottom plane. See Figure C.1. Note that these are the

APPENDDC C. JACOBIAN MATRlX ANALYSTS 53

&ed nodes, since nodes 1 through 3 are fixed with respect to inertid space.

A matrix J', different fkom J is dehed as

J' =

where

Two more matrices need to be

-r& oT oT oT oT T r,, r oT oT oT * oT oT oT of -r4,6 * oT oT oT oT

I'i,j = ri - rj

defined. The matrix L

L = diag(14, f5, 16,

(CA 1)

(C.12)

(C. 13)

is a diagonal 18 by 18 matrix with each f i being the length of member i. Note that only the unfued members form matrix L. Findy,

It cm be shown that, J'iT = ~i (C.15)

The proof of Equation C.15 cornes fiom differentiating the length of every member with respect to time. For example,

APPENDLX C. JACOBIAN MATRIX ANALYSIS 54

Since ri is inertidy fuced, ri = 0, and substituting Equation C.12

which is the equation for the first term in C.15. Similarly, the seventh term in C.15 is derived from

which c a n be written as T , - [iolto = r4,5 (r4 - 9)

Now, C.15 can be written as i~ = [(J')-IL] i

Let H = [(Y)-'LI. Note that H is a Jacobian matrix of sort. It relates the rate of change of lengt h of all unfixed members to nodal velou t ies in Cartesian space. Since only (Ito, lii, fi*), the actuator mernbers, have variable lengt h, C .21 reduces down to

where H' is an 18 by 3 matrix of columns 7-9 of ($)-le

Since the end effector is fixed with respect to the top plane of trussarm, then

where E is a 3 by 18 matrix Jacobian relating end effector velocity with the velouty of ail 18 unfixed nodes of tnissarm.

Note t hat O O

0 0 112

APPENDIX C. JACOBIAN MATRIX ANALYSE

which is the same J as produced by the Invene Kinematics Algorithm. Note also that this method, with the srnail exception of E, required no knowledge of the kinematics. Only the constraint that the distance between certain pairs of nodes must equal the length of a specific member at all times was required; thus, the term Mernber Length Method. The computer code to perform the Memkr Length Methad can be found in Appendix B.

C.1.3 Comparing the Two Methods

The question of which method is better for controlling or simulating trussarm is not simple. Both methods have theh merits and their drawbacks.

The Inverse Kinematics Algorithm produces an enormous amount of computer code. The reader is encouraged to briefly look at Appendices A and B to compare the size of the code. However, the Inwrse Kinematics Algorithm requires oniy e as an input. Trussarm's encoders only provide actuator lengt h data. Hence, forward kinematics mus t be run to find e given q.

The Member Length Method code is very compact and simple. However, rT, the location of al16 moving nodes of trussam, is required as input. Since t mssarm's encoders provide only the actuator lengths, forward kinematics will need to be run as wd. However, the forward kinematics routine wiU need to calculate rT, as opposed to just e.

Another cornparison can be made regarding the caledation of inverse matrices. The Inverse Kinematics Algotithm outputs a 3 by 3 matrix equal to J-l. This matrix must be inverted. An invertible 3 by 3 matrix can always be inverted using a simple analytical scheme. It is straightforward and requires few computations.

The Member Length Method requires the inversion of the 18 by 18 matrix J'. This matrix must be inverted by Gaussian elimination or equivalent method. The efficiency of this algorithm will largely determine the speed at which the Mernber Length Methal will operate. It should be noted that Gaussian eümination is an 0 ( n 3 ) operation, where n is the dimension of the matrix. Thus, depending upon the eficiency of the matrix inversion aigorithm, J' will require over 600 times more compntation to invert than J.

A common thread can be drawn fiom the above comments. Using the knowledge of the inverse kinematics reduces the input stream fiom the 18 terms of rT, to the 3 terms of e; and, the size of the matrix inversion problem is reduced fiom o(18~) = Q(5832) to 0(3~) = O(27). Reducing the size of the input stream reduces the amount of forward kinematics as well, which both methods rely on since encoders only provide q. The question is whether the computational cost of finding the derivative of the inverse kinematics is greater than the compntational cost of additional forward kinematics and

APPENDlX C. JACOBLAN 1MATR.K ANALYSE

matrk inversion routines.

Appendix D

Drafting, and Parts List

This Appendix contains many of the AutoCAD drawings that were done during the design. AutoCAD was an invaluable tool in estimating the mass of parts, and checking clearances of the moving components. AU drawings were done in AutoCAD Version 12 using the AME Modeller. Figures D.2 and D.3, and Tables D.4 and D.5, nearly contain enough information build a new bay. Figure D.8 and Tables D.1 through D.3 provide a l l the winng information required to connect Mark II to the C500, or any other controller.

It should be noted that some figures do not show ail parts. For example Figure D.1 is missing the motor mounts, motors, and etc. These drawings were simplified to reduce rendering times on the Pentium 90. Even this powerfd PC required nearly an hour to produce some drawings. Note that Figures D.2 and D.3 are complete in ail respects.

Findy, without the Pentium 90, all the drawings in this Appendix, and much of the design assistance provided by AutoCAD, would have been impossible.

APPENDIX D. DRAFTING, AND PARTS LIST

Figure D.1: Trussarm Mark II

APPENDIX D. D W T I N G , AND PARTS LIST

Figure D.2: Assembly Drawing of Actuator


Figure D .3: Assembly Drawing of Passive S tmcture


Figure D .4: Passive Structure's Degrees of Freedom


MINIMUM LENGTH

BASELENGTH

LENGTH = 38.8 * 7.6

Figure D .5: Actuator Extension and Ret raction

APPENDlX D. DRAFTING, AND PARTS LIST

Figure D.6: Offaet s Superimposed on Structure

APPENDlX D. DRAFTING, AND PARTS LIST

Figure D.7: Offsets and Member Lengths


Figure D.8: Actuator Numbering

I

I

I

II' U3MOd

APPENDLX D. DRAFTING, AND PARTS LIST

CIRCUIT BOARD

8

CIRCUIT PIN I C W CdNNtCTûR

1 1 BUCK 1 6 1 ROBOTARM 1 24 1 GND WHITE 1 15 Vcc

BUCK 5 ROBOT ARM 6 1 8' RED 4 2 1 B

BUCK 3 ROBOT ARM 5 1 A'

YEUOW 2 1 1 A

BROWN LIMIT SWITCH CONNECTOR H 19 HOME - 1

RED 43 + S v & r

RED M f R A NIA ORANGE

ACTüAl'ûR 1 lWSTEDPAiR 1 CIRCUITPIN PIN Ii 1 DESCRIPTION 1 2 BUCK 6

GREEN 1 1

RED 5 BUE 4

r

RED 3

WHITE 2 BLACK LlMlT SWITCH

BROWN

RED EXTRA

ROBOT ARM 25 GND 16 Vcc

ROBOT ARM 10 2 0' 4 2 6

ROBOT ARM 9 2 A' 3 2A

CONNECTOR #1 23 HOME - 2 43 +5vdc

NIA

1 GREEN 1 A C V I I O R J TYVISTED P H CIRCUIT PIN C M CONNECTOR PiY U 1 OESCRIPiWN

3 GREEN e ROBOT ARM JO GND

WHITE 1 17 Vcc

RED S ROBOT ARM 30 3 Ba YULOW 4 21 3 B

Table D.1: Cables, Part 1

APPENDIX D. DRAFTLNG, AND PARTS LIST

BLACK 3 ROBOT ARM 29 3 A'

ORANGE 2 20 3 A

BLACK LIMIT SWITCH CONNECTOR #î 27 HOME - 3

BUE 43 + 5 vdc

GREEN MTRA N A BLUE

BUCK RoeoTARM

GND WITE -- - ---

BUCK 5 ROBOT ARM 4C) 4 B' RED 4 23 4 8

W C K 3 ROBOT ARM 38 4 A' VULOW 2 22 4 A

BROWN 1 LYlT SWITCH 1 CONNECTOR #l 31 HOME - 4 RED 1 43 1 * I v d c

ORANGE RED 1 - 1 " I I

BUCK = I GND

GREEN

RED 5 ROBOT ARM 37 5 B' BUE 4 49 SB

- -

BUCK LIMIT SWTCH CONNECTOR #l 35 HOME - 5 BROWN 43 + 5 *

RED EXTRA NIA GREEN


APPENDIX D. DRAFTING, AAND PARTS LIST

1 ORANGE

1 BUCK

3 ROBOT ARM 34 6 A'

2 19 6 A

LIMIT SWlTCH CONNECTOR # 1 39 HOME - 0 43 + S v & - I " I I - . - I I

I - - - AMp- coooco~~-bnO~ 1 3 A 1 ORANGE GALIL-A CONNECTOR #1 1 Vc~m -1

U C K 44 GND I

6 2 W H E GALIL-8 CONNECTOR #1 3 V W ~ -2

c ,

F

3 GAUL-C

THIERRY - X

U C K BWE

4

D


CONNECTOR #1

CONNECTOR Il BLACK

RED

l

E

6

44 5

44 2

6

GNO V C O ~ -3

GND V C O ~ -4

BLACK VEUOW BLACK

BROWN

1 THIERRY - Y 1 CONNECTOR 1ii

THIERRY - Z

44 4

GND V W ~ -5

CONNECTOR #l

44 6

GND V C O ~ d


Table D.4: Parts List

1 2 3 4 5 6 7 g 9 10 11

3 3 3 3 3 18 15 24 3 3 3

I linear bearing large Al tube tubular bearing shaft 1/4-20 lead screw, 10.5" bng 1/4-20 supernut 1/4 'Sût~~fe~&mp 114" bushing, 318" long 1/4'thtu6tbearing smaîl pulley large pulley timing belt

Canada Bea!rig ldeal MeWs BGS B e d f l ~ s Stock Drive Stock Drive

canada 8Win~r Rotofre cision RotoReasion

Canada Bearing

12 13

motormount 3 large spîit yoke + steel plug 3

Thompson Stetphen Oikawa / IRM

f hornmn Acme Acme

k t 0 n mat

14 15 i 6 17 18 19 20 21 22 23 24

26

NA NA

NA NA

A122026 NA NA

He1 1 SA-25201 1 HB20BN-25201

SC25

IRM Technologies IRM Technobgies

PA3û440FûûO 3M060060

Canada Beating Canada Bearinq

Nordex Nordex

NA B46-2

EPCIC2-4 DM472

NA NA NA NA NA NA

AQB-A208 DeS-Al-24 DBS-Al-22

Bastan Boston Giear

AOB-Al-9 AFC-A2-25

IRM Technob~ies Boston Glear

Nordex Clifion

IRM Technob~ies lRM Technobaies IRM Technob~ies Stephen Oikawa Stephen Oikawa Stephen Oikawa

Nordex Nordex Nordex

NA Canada Bearing Rata Pleasi on

S e m Proâucts NA NA NA

ldeal Metals C o ~ & B m s

Brafasco Roto Preasio n RotoReasion Roto Reasion

hinge assembly 6

&&in @ar

1/4"kishing,1/4"long 3/32" dowel pin DCmotor active-passiva connecter passive plane connecter passive hinge connecter jead saew beering bbck Al passive members Steel passive members flanged 114' bushing bng s houlder scrm short shoulder screw

PA301 ODFûûO

6 6 3 6 3 12 3 12 6 6 6 6

APPENDrX D. DRAFTING, AND PARTS LIST

I . r 1

Table D.5: Parts List Continued

$0.44 $0.94 $3.82

1

$7.85 $1 1.75

I

$4.25 $20.00

-

$67.a I

$38.00 $0.52 $0.50

$35.00 $23.00 $23.00 $23.00

r

$2.00 $2.00 $2.00 $0.94 $3.!30 $3.04

$22.07 $1 5.00 $62.50

-press lit into large Al tube - in house mhining, e û f t - some IRM machining required

100 21 0 300

5 4 5 5

50 5

----

25 40 5 5 2

170 27

- --

42 17 30 33 31 5 5 5

- 10 tooth, 1/8" shaft - 44 W h , 1/4" shait - 6ô tooth beit - aistom machined - aistom machined - araorn rnachined - for A-? hinge - 112" long pin - surplus motor + encoder - custom ki l t

- - - -

0 i5~sto~buiit - aistom kilt - in buse mschining - 38" exüuded Ai tubiiy), 0.049" W.T. - 114" drill rod - 114" IO^ - 0.751 5" shoulder, 1 û-32 threaâ - 0.501 5 shoulder, 10-32 threaâ

1 2

$66.21 $45.00

300 630

$7.92 $1 4.1 0 $91.68 $23.55 $25.25 $1 2.75 $60.00

- - -- -

$201 .O0 $228.00

$3.1 2 $3 .O0

S1oS.00 $1 38.00 $69.00 $276.00

$6.00 $24.06 $1 2.00 $5.64

$21 .O0 $1 8.24

$1 87 -50 900l 3

90 60

120 15

150 15

- - - - 7s

120 30 30

6 7 8 9 10 11 12

- -

13 14 15

12 510

- -- 162 126

16 7

17

~ 18 19

204 90

20 21

396 186 30 30 30

22 23 24 25 26

Appendix E

Manufacturer Specificat ions

The DC motors' strongest suit is their price. At $35 each, complete with encoders, these motors could not be beat. Their inexpensive cost was due to the fact t hey were surplus motors, not new motors. New motors with similar specifications were priced at anywhere from $200 each for AstroFlight motors, to over $800 each for motors from MicroMo.

The main disadvantage of purchasing surplus motors is the fact these motors wiLl be h a d , if not impossible, to replace if one is damaged. Second, the motors are often not as advertised. For example, the encoders were mpposed to have 200 pulses/rev. After much debugging of computer software, and electronics it was discovered to great surprise these encoders output 192 puises/rev. Though a simple software fuc compensated for this, it underscores the risks involved in purçhasing surplus motors in the future.

Maximum Terminal Voltage D C Resis t ance

No Load Speed at 40 V Torque Sensitivi ty

Motor Inertia Weight Encoder

40 V DC 6 OHMS

6500 RPM 8 oz-in/amp

2.6 x 10-~ oz-in /sec2 5.2 oz

192 pdses/revolution

Table E.l: Motor Specifications

APPENDIX E. MANUFACTURER SPEClFICATIONS

E.2 Lead Screws

A 114"-20 lead screw is 114" in diameter, and has 20 threads per inch. The mechanical advantage of the screw is not the best design parameter to reiy upon as the screws typicdy have efficiencies of 50%, or less. Good rnanofacturers rate lead screws by combining efficiency and gear reduction into a single number, oz-in of torque per 1 lb of lift. For a 1/4"-20 lead screw, 0.42 oz-in per 1 lb is typicd.

Besides ensuring sufficient mechanical advantage, the screws must be able to withstand the maximum compression load of 1000 N. The 1/4"-20 lead screws are rated to 200 lbs (880 N) of compression, with a fixed-free bea,ring support, and 5 inches length. This is the Achilles heal of Mark II. The maximum load exceeds the rated load. With a fixed- supported bearing support the Iead screw could handle much more load. However, such a support system is difficult to build. This is why industrial actuators can cost upwards of $1000 each.

E.3 3/32" Hardened Steel Pin

The 3/32" harden pins are rated at 500 lbs double shear strength. These pins ultimately support the entire weight of ail bays located above them. The total mass of the structure is approximately 18.5 kg, or 40.5 lbs. Each pin alone can support the entire structure. Examining the design, a minimum of three pins aiways supports the weight of the structure above it. Hence, these pins were felt strong enongh for the task.

E.4 Clamping Collars

Boston Gear produces damping colIars for 318" shafts which use a single 6-32 machine serew for clamping. Constructed of aluminum, with a width of only 5/16", these collas are rated to hold 600 lbs axidy (2600 N). Parts #18, 19, and 20 have a width of 3/Bf'(1/16" more), are made of aluminum, and use 6-32 machine screws for damping. It is a safe bet these parts will hold similar loads.

E.5 Set Screw Collars

Set screw coilars, Part #6, were used to prevent the lead screw fiom moving axially with respect to the bearing block (see Figure D.2). These collars must be able to withstand

APPElNDIX E. MANUFACTURER SPECWICATIONS 74

1000 N of axial load. Set screw collas are not designed for this. Ideaiiy, a damping colla, like the ones described above, should have been used. However, such collars cost on the order of $10 each, where as set screw collars cost a mere $0.50 each. The solution was to drill and pin the set screw coilas permanently to the lead screws. The 1/16" roll pins have a double shear strength of 425 Lbs, or 1870 N, double the axially load experienced by the lead screws. The roll pins cost less than $0.05 each, and hence was the most economical solution.

Appendix F

Wor kspace

This Appendix contains figures describing the workspace of Trussarrn Mark II. The workspace is very complicated, with many redundant ways to place the end effector in a given location and orientation. Hence, there is no simple way to describe it. It is hoped these figures provide a giimpse of what is possible with multibay tnissarms such as Mark II. The gantry depicted is the gantry origindy used to move the Doisy faulity into the Space Robotics Labomtoy. AU fi y e s in this Appendix ore to scale. The view is 30" d o m from the x-y plane, and aligned with the y-ais . Table F.l gives the actuator positions for d the figures to foiiow. The position wit h respect to the base of the centroid of the top plane of the fourth bay is provided in each figure

Act uator F.2 F.3 F.4 F.5 F.6 F.7 F.8 F .9

F.10 F . l l F.12 F.13

Table F.1: Actuator Positions in Centimetres With Respect to Baselengt h

Figure F.l: Gantry and Coordinate System

Figure F.2: Fully Retracted Reach X,Y,Z = 0, 0, 1.58 m

Figure F.3: Fdy Extended Reach X,Y,Z = O, 0, 2.14 m

Figure F.4: Extended Negative X-axis Reach X,Y,Z = -1.64, 0, 0.49 m

Figure F.5: Retracted Positive X-axis Reach X7Y7Z = 1.45, 0, 0.38 m

Figure F.6: Negative X-axis LT-shape X,Y,Z = - 1.08, 0, -0.06 m

Figure F.7: Positive X-axis &shape X,Y,Z = 1.07, 0, -0.23 m

Figure F.8: Negative X-axis S-cume X,Y,Z = -1.15, 0, 1.19 m

Figure F.9: Positive X-axis S-curve X,Y,Z = 1.33, O, 0.99 m

Figure F.10: Extended Reach with Negative X-axis Hook X,Y,Z = -0.57, 0, 1.62 m

Figure F.11: Retracted Reach with Positive X-auis Hook X,Y,Z = 0.68, 0, 1.34 m

APPENDIX F. WORKSPACE

Figure F.12: Straight Swing in Negative X-axis X,Y,Z = -1 34 , 0, 1.50 m

Figure F.13: Straight Swing in Positive X-axis X,Y,Z = 1.46, 0, 1.45 m

Bibliography

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