This paper analyzes the variations in joint angles of a two link planar manipulator used for welding applications. The inverse kinematics equations are used to calculate the joint angles (θ1 and θ2) of the manipulator for a given position of the end effector moving linearly between points A and B. A program is written in Fortran 90 to calculate the joint angles and link positions at multiple step points along the welding seam, as solving the inverse kinematics equations by hand for each point would be complicated. The program uses homogeneous transformations and geometric relationships based on the Denavit-Hartenberger representation to determine the manipulator configuration corresponding to each end effector position.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Lesson 12: Linear Approximations and Differentials (handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Lesson 12: Linear Approximations and Differentials (handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Business Proposal Presentation in Purple Monochrome Corporate Style.pptxJOHN35307
Trajectory planning is moving from point A to point B while avoiding collisions over time. This can be computed in both discrete and continuous methods. Trajectory planning is a major area in robotics as it gives way to autonomous vehicles.
Trajectory planning is sometimes referred to as motion planning and erroneously as path planning. Trajectory planning is distinct from path planning in that it is parametrized by time. Essentially trajectory planning encompasses path planning in addition to planning how to move based on velocity, time, and kinematics.
Problem Constraints
Holonomicity
Holonomicity is the relationship between the controllable degrees of freedom of the robot and the total degrees of freedom of the robot. If the number of controllable degrees of freedom are greater than or equal to the total degrees of freedom a robot is said to be holonomic. By using a holonomic robot many movements are much easier to make and return to a past pose is much easier.
A car would be non-holonomic, as it has no way to move laterally. This makes certain movements, such as parallel parking, difficult. An example of a holonomic vehicle would be one using mecanum wheels, such as the new Segway RMP.[1]
Dynamic Environments
In dynamic environments, such as the real world, many possible collision objects are not stationary. This makes trajectory planning more difficult as time is constantly changing and objects are moving. A robot cannot simply move backward in time as it might simply back away from a stationary collision. In addition to this many choices are completely irreversible due to terrain, such as moving off of a cliff.
Concepts
Concepts of Trajectory Planning
Trajectory planning gives a path from a starting configuration S to a goal configuration G avoiding collisions in a 2D or 3D space.
A configuration is the pose of a robot describing its position. Configuration Space C, is the set of all configurations. For instance, in two dimensions a robot's configuration would be described by coordinates (x, y) and angle θ. Whereas in three dimensions a robot's configuration would be described by coordinates (x, y, z) and angles (α, β, γ).
Free space Cfree is the set of all configurations that are collision-free. Computing the shape of Cfree is not efficient, however, computing if a given configuration is a collision free is by simply using kinematics and collision detection from sensors.
Target space is a linear subspace of free space which we want robot go there. In global motion planning, target space is observable by robot's sensors. However, in local motion planning, robot cannot observe the target space in some states. To solve problem, robot assume several virtual target space which is located in observable area (around robot). The virtual target space is called sub-goal.[2]
Planning Algorithms
Artificial Potential Field
Example of a Potential Field
Artificial Potential Field Planning places values over the map with the goal having the low
Synchronverters: Inverters that Mimic Synchronous GeneratorsQing-Chang Zhong
Inverters are made mathematically equivalent to conventional synchronous generators, which considerably facilitates the integration of renewable energy and distributed generation into smart grids.
Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Ro...James Smith
Using ideas developed in detail in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016, this document solves one of the special cases of the famous Problem of Apollonius. A new Appendix presents alternative solutions.
See also:
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Business Proposal Presentation in Purple Monochrome Corporate Style.pptxJOHN35307
Trajectory planning is moving from point A to point B while avoiding collisions over time. This can be computed in both discrete and continuous methods. Trajectory planning is a major area in robotics as it gives way to autonomous vehicles.
Trajectory planning is sometimes referred to as motion planning and erroneously as path planning. Trajectory planning is distinct from path planning in that it is parametrized by time. Essentially trajectory planning encompasses path planning in addition to planning how to move based on velocity, time, and kinematics.
Problem Constraints
Holonomicity
Holonomicity is the relationship between the controllable degrees of freedom of the robot and the total degrees of freedom of the robot. If the number of controllable degrees of freedom are greater than or equal to the total degrees of freedom a robot is said to be holonomic. By using a holonomic robot many movements are much easier to make and return to a past pose is much easier.
A car would be non-holonomic, as it has no way to move laterally. This makes certain movements, such as parallel parking, difficult. An example of a holonomic vehicle would be one using mecanum wheels, such as the new Segway RMP.[1]
Dynamic Environments
In dynamic environments, such as the real world, many possible collision objects are not stationary. This makes trajectory planning more difficult as time is constantly changing and objects are moving. A robot cannot simply move backward in time as it might simply back away from a stationary collision. In addition to this many choices are completely irreversible due to terrain, such as moving off of a cliff.
Concepts
Concepts of Trajectory Planning
Trajectory planning gives a path from a starting configuration S to a goal configuration G avoiding collisions in a 2D or 3D space.
A configuration is the pose of a robot describing its position. Configuration Space C, is the set of all configurations. For instance, in two dimensions a robot's configuration would be described by coordinates (x, y) and angle θ. Whereas in three dimensions a robot's configuration would be described by coordinates (x, y, z) and angles (α, β, γ).
Free space Cfree is the set of all configurations that are collision-free. Computing the shape of Cfree is not efficient, however, computing if a given configuration is a collision free is by simply using kinematics and collision detection from sensors.
Target space is a linear subspace of free space which we want robot go there. In global motion planning, target space is observable by robot's sensors. However, in local motion planning, robot cannot observe the target space in some states. To solve problem, robot assume several virtual target space which is located in observable area (around robot). The virtual target space is called sub-goal.[2]
Planning Algorithms
Artificial Potential Field
Example of a Potential Field
Artificial Potential Field Planning places values over the map with the goal having the low
Synchronverters: Inverters that Mimic Synchronous GeneratorsQing-Chang Zhong
Inverters are made mathematically equivalent to conventional synchronous generators, which considerably facilitates the integration of renewable energy and distributed generation into smart grids.
Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Ro...James Smith
Using ideas developed in detail in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016, this document solves one of the special cases of the famous Problem of Apollonius. A new Appendix presents alternative solutions.
See also:
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Full-RAG: A modern architecture for hyper-personalizationZilliz
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!SOFTTECHHUB
As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdfMalak Abu Hammad
Discover how MongoDB Atlas and vector search technology can revolutionize your application's search capabilities. This comprehensive presentation covers:
* What is Vector Search?
* Importance and benefits of vector search
* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
* Best practices and optimization strategies
Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
2. 356 Welding
y
y x2
y2
A B
s a
o2 n
l2 θ2 y1 l2 θ2 z2
x1
y0 l1 o1
l1
θ1 z1
o θ1
o o0 x0 x
x
Fig. 1. Linear movement of the end effector from A to B z0
Fig. 4. Coordinate frames attached to two link planar manipulator
y
end-effector or tool frame. The origin of the tool frame is
l -l
1 2 most often placed symmetrically between the fingers of the
l +l
1 2 gripper.6 The unit vectors along the xn , yn and zn axes are
labelled as n, s and a, respectively. According to the D-
H representation, the transformation matrix in general form
x can be written as below:
Cosθi −Cosβi Sinθi Sinαi Sinθi ai Cosθi
Sinθi Cosαi Sinθi −Sinαi Cosθi ai Sinθi
Ai =
di
Fig. 2. The workspace of the two link planar manipulator
i−1 0 Sinαi Cosαi
0 0 0 1
A(x1 , y1 ) C x 0 , y 0 B(x 2 , y 2 ) (4)
m n where i = 1, 2. Equation (4) describes the transformation
matrix between two coordinate frames i and (i − 1) attached
to links i and i − 1 which are connected by a rotational joint.
Fig. 3. Dividing the welding seam into step points In the above equation θi is the angle, αi is the twist angle,
ai is the length and di is the distance between the links. The
of a given step point may calculated. Consider point C joint variables of the two link planar manipulator are given
in Fig. 3; x0 and y0 coordinates are calculated by using in Table I.
geometric equalities as below: According to (4), the transformation matrices A1 and A2 ,
0 1
which relates to the first and second coordinate frames, can
[CA] m be written as follows:
= =k (1)
[CB] n
c1 −s1 0 l1 c1
then, x0 and y0 are written as follow: s1 c1 0 l1 s1
A0 =
1
0 0 1 1
x1 + kx2 0 0 0 1
x0 = (2)
1+k c2 −s2 0 l2 c2
s2 c2 0 l2 s2
A1 =
2
1
and (5)
0 0 1
y1 + ky2 0 0 0 1
y0 = (3)
1+k
In Fig. 4 coordinate frames of the two link planar manipulator Table I. The joints variables.
is given. In this figure, oxyz, ox1 y1 z1 and ox2 y2 z2 coordinate
Link (i) di ai αi θi
frames are placed on the manipulator links according to
1 0 l1 0 θ1
the Denavit Hartenberger (D-H) representation.5 The final 2 0 l2 0 θ2
coordinate system on xn yn zn is commonly referred to as the
3. Welding 357
where c1 = cosθ1 , s1 = sinθ1 , c2 = cosθ2 and s2 = sinθ2 .
y
The homogeneous transformation matrix is,
(Px0, Py0)
T = A2 = A1 A2
0 0 1 (6)
c12 −s12 0 l1 c1 + l2 c12 l2
l1 c1 + l2 c12 r
s12 c12 0 θ2
A2 =
0 (7)
0 0 1 1
0 0 0 1 ρ
l1
In the above equation, c12 = c1 c2 − s1 s2 = cos(θ1 + θ2 ) and θ1
s12 = s1 c2 + c1 s2 = sin(θ1 + θ2 ). The matrix A2 relates the
0
end-effector coordinates to the base coordinate frame. From x
Equations (6) and (7), the position of the end-effector in the
base coordinate system can be obtained. From (6), A2 can be
1
Fig. 5. The manipulator joint angles
written as follows:
−1
A2 = A1
1 0 T (8) where,
where px0 = r cosρ
¯
(13)
py0 = sinρ
¯
c1 s1 0 −l1 r= px0 + py0
¯2 ¯2 (14)
−1 −s1 c1 0 0
A1 = A0 =
1 0 and ρ = tan−1 ( py0 ). By using (13) and the equality
, ¯
p
0 1 0 0 ¯ x0
0 0 0 1 cosθ1 cosρ + sinθ1 sinρ = cos(θ1 − ρ), the Equation (12) is
rewritten as below:
and 2l1 r cos(θ1 − ρ) = px0 + py0 + l1 − l2
¯2 ¯2 2 2
(15)
¯ The above equation can be rewritten by using (14) as;
nx0 sx0 ax0 Px0
ny0 sy0 ay0 ¯
Py0
T =
Pz0
.
nz0 sz0 az0 ¯ 2l1 px0 + py0 cos(θ1 − ρ) = px0 + py0 + l1 − l2
¯2 ¯2 ¯2 ¯2 2 2
(16)
0 0 0 1
Then, Equation (8) can be rewritten as follows
¯
c1 s1 0 −l1 nx0 sx0 ax0 Px0
−s1 c1 0 0 ny0 sy0 ay0 ¯
Py0
A0 T =
1 0 0 1 0 nz0 sz0 az0 Pz0
¯
0 0 0 1 0 0 0 1
c1 nx0 + s1 ny0 c1 sx0 + s1 sy0 c1 ax0 + s1 ay0 c1 px0 + s1 py0 − 11
¯ ¯
(9)
0 −s1 nx0 + c1 ny0 s1 sx0 + c1 sy0 s1 ax0 + c1 ay0 s1 px0 + c1 py0
¯ ¯
A1 = A1 T
2
nz0 sz0 az0 ¯
pz0
0 0 0 1
The corresponding entries in (5) and (9) are set equal, From (16),
therefore the following equations can be written:
px0 + py0 + l1 − l2
¯2 ¯2 2 2
c1 px0 + s1 py0 − l1 = l2 c2
¯ ¯ (10) cos(θ1 − ρ) = (17)
2l1 px0 + py0
¯2 ¯2
−s1 px0 + c1 py0 = l2 s2
¯ ¯ (11)
and
The first and second joint angles, θ1 and θ2 in Fig. 5 can be
2
calculated from Equations (10) and (11). Taking the square px0 + py0 + l1 − l2
¯2 ¯2 2 2
of (10) and (11) and adding them side by side the following cos (θ1 − ρ) =
2
(18)
equation results 4l1 px0 + py0
2
¯2 ¯2
can be written. By using the equality cos2 (θ1 − ρ) +
(2l1 px0 )c1 + (2l1 py0 )s1 =
¯ ¯ ¯2
px0 + ¯2
py0 + 2
l1 − 2
l2 (12) sin2 (θ1 − ρ) = 1 and Equation (18), the following equation
4. 358 Welding
Table II. The joint variables of the manipulator which are used in 8
the program. 6
The length of the first link l1 = 10 cm 4
The length of the second link l2 = 9 cm 2
Time (Second)
x Coordinate (starting point) 7 x 0
y Coordinate (starting point) 5 -2 0 5 10 15 20 25 30 35
x Coordinate (ending point) −7
-4
y Coordinate (ending point) 2
Step Number (n) 50 -6
Total Loggin Time 30 sn -8
follows Fig. 6. The position of the end-effector on x-axis
2 6
px0 + py0 + l1 − l2
¯2 ¯2 2 2 y
sin(θ1 − ρ) = 1− (19) 5
4l1 px0 + py0
2
¯2 ¯2
4
From (17) and (19), 3
2
px0 +py0 +l1 −l2
¯2 ¯2 2 2
√ 1
2l1 px0 +py0
¯2 ¯2
tan(θ1 − ρ) =
2 0
px0 +py0 +l1 −l2
¯2 ¯2 2 2
1− 0 5 10 15 20 25 30 35
4l1 px0 +py0
2
¯2 ¯2 Time (Second)
2
4l1 px0 + py0 − px0 + py0 + l1 − l2
2
¯2 ¯2 ¯2 ¯2 2 2
Fig. 7. The position of the end-effector on y-axis
tan(θ1 − ρ) =
¯2
px0 + ¯2
py0 + 2
l1 − 2
l2
(20) 6 y
can be written. Then above equation is; 5
2
4
4l1 px0 + py0 − px0 + py0 + l1 − l2
2
¯2 ¯2 ¯2 ¯2 2 2
−1
θ1 = ρ + tan 3
px0 + py0 + l1 − l2
¯2 ¯2 2 2
2
(21) 1
x
0
The second joint angle, θ2 can be solved by dividing (11) to -10 -5 0 5 10
(10) as below:
Fig. 8. The position of the end-effector on xoy coordinate frame
−1 −px0 s1 + py0 c1
¯ ¯
θ2 = tan + kπ (22)
px0 c1 + py0 s1 − l1
¯ ¯ Time (Second)
Teta 1 0
The program is written acccording to variables which are (Gradient) 0 10 20 30 40
-20
given in Table II and Equations (21), (22). The algorithm of
the program is given in detail in Appendix I. -40
As it is shown in Table II, the program is run for 30 seconds
and 50 step points. After running the program data in Table III -60
are obtained.
-80
IV. RESULTS AND DISCUSSION -100
The position of the end-effector on the x-axis is shown in
Fig. 6. There is a linear relationship between time and posi- -120
tion. After the middle of the sampling time (after 15 seconds),
-140
the end-effector is moving along the negative part of the x-
axis in order to reach to final point set in the program. -160
Similarly, the variation in the end effector position on the
y-axis is shown in Fig. 7. As it is shown in this figure, the start- Fig. 9. The variations in the first joint angle, θ1
ing and ending points of the end-effector matches the values
that are given in the program which is rather desirable. It The variations in the first joint angle, θ1 are shown in Fig. 9.
follows from the above two figures the position of the end In this figure there is a linear relationship between time and
effector on the xoy coordinate frame is shown in Fig. 8. the joint angle until 10 sn. After this point, linearity no longer
5. Welding 359
Table III. The joint variables of the manipulator which are derived from the program.
Step Time θ1 (Gradient) θ2 (Gradient) ω1 (rad/sn) ω2 (rad/sn)
0 0 −2,816486 126,473 0,00 0,00
1 0,6 −4,252487 128,2504 −0,0418 0,0517
2 1,2 −5,715397 130,0042 −0,0426 0,051
3 1,8 −7,208878 131,7343 −0,0434 0,0503
4 2,4 −8,736984 133,4404 −0,0445 0,0496
5 3 −10,30421 135,122 −0,0456 0,0489
6 3,6 −11,91557 136,7784 −0,0469 0,0482
7 4,2 −13,57665 138,4088 −0,0483 0,0474
8 4,8 −15,2937 140,0119 −0,0499 0,0466
9 5,4 −17,07373 141,5863 −0,0518 0,0458
10 6 −18,92464 143,1301 −0,0538 0,0449
11 6,6 −20,85528 144,6412 −0,0562 0,044
12 7,2 −22,8756 146,117 −0,0588 0,0429
13 7,8 −24,99669 147,5543 −0,0617 0,0418
14 8,4 −27,23099 148,9493 −0,065 0,0406
15 9 −29,59219 150,2976 −0,0687 0,0392
16 9,6 −32,09529 151,5941 −0,0728 0,0377
17 10,2 −34,75644 152,8328 −0,0774 0,036
18 10,8 −37,59261 154,0066 −0,0825 0,0341
19 11,4 −40,62097 155,1076 −0,0881 0,032
20 12 −43,85803 156,127 −0,0942 0,0297
21 12,6 −47,31818 157,0547 −0,101 0,027
22 13,2 −51,01178 157,8804 −0,107 0,024
23 13,8 −54,94283 158,593 −0,114 0,0207
24 14,4 −59,10625 159,1815 −0,121 0,0171
25 15 −63,48522 159,6359 −0,127 0,0132
26 15,6 −68,04936 159,9473 −0,133 0,00906
27 16,2 −72,75414 160,1095 −0,137 0,00472
28 16,8 −77,54247 160,1188 −0,139 0,000272
29 17,4 −82,34827 159,9752 −0,14 −0,00418
30 18 −87,10191 159,6816 −0,138 −0,00854
31 18,6 −91,73608 159,2443 −0,135 −0,0127
32 19,2 −96,19121 158,6716 −0,13 −0,0167
33 19,8 −100,4192 157,9736 −0,123 −0,0203
34 20,4 −104,3854 157,1611 −0,115 −0,0236
35 21 −108,0681 156,2452 −0,107 −0,0266
36 21,6 −111,4578 155,2365 −0,0986 −0,0293
37 22,2 −114,5544 154,1449 −0,0901 −0,0318
38 22,8 −117,365 152,9796 −0,0818 −0,0339
39 23,4 −119,9021 151,7484 −0,0738 −0,0358
40 24 −122,1811 150,4586 −0,0663 −0,0375
41 24,6 −124,219 149,1163 −0,0593 −0,039
42 25,2 −126,0335 147,7267 −0,0528 −0,0404
43 25,8 −127,6423 146,2944 −0,0468 −0,0417
44 26,4 −129,0621 144,8232 −0,0413 −0,0428
45 27 −130,3088 143,3162 −0,0363 −0,0438
46 27,6 −131,3971 141,7762 −0,0317 −0,0448
47 28,2 −132,3404 140,2054 −0,0274 −0,0457
48 28,8 −133,1511 138,6058 −0,0236 −0,0465
49 29,4 −133,8403 136,9787 −0,02 −0,0473
50 30 −134,4179 135,3254 −0,0168 −0,0481
exist and the joint angle is varying rapidly. Similar behaviour of the end-effector, the variation in the second joint angle
can be seen in the angular velocity of the first joint angle, as is rather nonlinear, but still there are no sudden increases or
shown in Fig. 10. There, after 1.5 seconds the variations in decreases, therefore it assumes a gradual form. As a result, the
the joint angle velocity are rapid and nonlinear. angular velocity of the second joint angle is varying linearly
Similarly, the variation in the second joint angle, θ2 is for about 2 seconds, as shown in Fig. 12. After that time the
shown in Fig. 11. In order to reach the required position changes are rather nonlinear, but they are still gradual.
6. 360 Welding
Time (Second) kinematic equations together with some geometric equalities.
0
The joint angles variations and angular velocities are shown
Velocity 1 0 10 20 3 0
in Fig. 6–12. The results derived in this work are completely
(rad/sn) -0,02
theoretical.
-0,04 The program results may give us an idea how the real robot
may behave under same condition. Obviously, the real robot
-0,06
may not give same responses. This is due to the fact that
-0,08 in this analyses the mass of the end-effector and links, the
friction forces in the joints are not considered.
-0,1 The variation in the first joint angle, θ1 is rapid and
-0,12
nonlinear are shown in Fig. 9. This could cause problems, in
real life applications due to the fact that under this condition
-0,14 the inertial forces could become very effective; hence, this
could lead to instability.
-0,16
In welding application, the linear movements of the end-
effector is crucial in order to have a homogeneous (in terms
Fig. 10. The changes of the first joint angle velocity, ω1
of materials) welding seam. For that reason, the stability and
linear movement of the end-effector is very important. In this
180 case, the stability of the manipulator may be assured by using
Teta 2 160 one of the adaptive control strategies which can adapt itself
(Gradient) to changes in the system and working condition.7
140
One could consider modelling the manipulator dynamic-
120 ally by using one of the dynamic system modelling packages
100
program, like simulink. The joint angles and position of the
links may be found by using this mathematical model of
80
the system. Later on, a prototype of the real manipulator
60 may be built, and the two theoretical results (the inverse
40
kinematic analyses results and the dynamic model results)
may be compared with real robot responses in order to see
20
which method is of an effective and practical in use. It is also
0 possible to find which method gives better understanding of
0 5 10 15 20 25 30 35 the real robot.
Time (Second)
Fig. 11. The variations in the second joint angle, θ2
References
1. J. C. Colson and N. D. Perreira, “Kinematic Arrangements Used
0,06
in Industrial Robots”, Proc. 13th International Symposium on
Industrial Robots (1983). pp. 28–37.
0,04 2. J. M. Hollerbach and S. Gideon, “Wrist-Partitioned Inverse
Kinematic Accelerations and Manipulator Dynamics”, Int. J.
0,02 Robotics Res. 4, 61–76 (1983).
Time (Second) 3. A. J. Koivo, Fundamentals for Control of Robotic Manipulators,
Velocity 2
(rad/sn)
0 (John Wiley and Sons, 1989).
0 5 10 15 20 25 30 35 4. A. A. Goldenberg, B. Benhabib and R. G. Fenton, “A Complete
-0,02 Generalized Solution to the Inverse Kinematics of Robots”,
IEEE Journal of Robotics and Automation, RA-1, No. 1, 14–20
-0,04 (March 1985).
5. M. W. Spong and M. Vidyasagar, Robot Dynamics and Control
-0,06
(John Wiley and Sons, 1989).
6. R. P. Paul, B. E. Shimano and G. Mayer, “Kinematic Control
Fig. 12. The changes of the second joint angle velocity, ω2 Equations for Simple Manipulators” IEEE Trans. Systems,
Man., and Cybernetics SMC-11, No. 6, 339–455 (1981).
7. S. Bulut, “The robustness of Reduced Order Minimal Controller
V. CONCLUSIONS Synthesis Control in Plastic Region”, Proc. Instn Mech. Engrs,
In this paper, the joint angles and angular velocities of the Part I, Journal of Systems and Control Engineering 214, 185–
two link planar manipulator are derived by using inverse 195 (1999).
7. Welding 361
APPENDIX I
The calculation program which is written in Fotran 90
START
ul1, ul2, bax, bay
bix, biy, adim, time
baku = (bax**2 + bay**2)**0.5
biku = (bix**2 + biy**2)**0.5
top1 = ul1 + ul2
fark1 = ABS(ul1 – ul2)
time1 = time/adim
H
top1 < baku
E
H
baku < fark1
E
H
top1 < biku
E
H
biku < fark1
E
H
bay < 0
E
H
biy < 0
E
m=0
A B C
8. 362 Welding
A B C
Create “results”
Data file
If the given
coordinates are
outside of the
I = 0,adim working space then
enter button“1” for
the new
coordinates
E
i = adim
H
bulx = bix
buly = biy n = adim-m
Vgir
saykat = m/n E
Vgir = 1
bulx = (bax + saykat*bix)/(1 + saykat) H
buly = (bax + saykat*biy)/(1 + saykat)
teta1 = atan(buly/bulx) +
atan(((44*ul1**2*(bulx**2 + buly**2) –
(bulx**2 + buly**2 + ul1**2 – ul2**2)**0.5)/
(bulx**2 + buly**2 + ul1**2 – ul2**2))
teta2 = atan((bulx*sin(teta1) + buly*cos(teta1)/
(bulx*cos(teta1) + buly*sin(teta1) – ul1))
E
i=0
H
t1fark = teta1 – teta1s
t2fark = teta2 – teta2s
teta1s = teta1
teta2s = teta2
F D E
9. Welding 363
F D E
hız1 = t1fark/time1
hız2 = t2fark/time1
time t = time1*i
I
Write all data accordinly into teta1s,
teta2s, hız1, hız2 and timet data files
Close the
“results” data file
STOP
Table IV. The description of notations that are in the Fortran 90 program.
adim: the total step number n: the total step number along [CB]
baku: the distance between o0 , origin of the base saykat: the coefficient k, k = m /n
coordinate frame and the starting point
bax: x coordinate of the starting point teta1: the first joint angle, θ1
bay: y coordinate of the starting point teta2: the second joint angle, θ2
biku: the distance between o0 , origin of the base teta1s = teta1 − t1fark
coordinate frame and the ending point teta2s = teta2 − t2fark
bix: x coordinate of the ending point t1fark: θ1 (t + 1) − θ 1 (t)
biy: y coordinate of the ending point t2fark: θ2 (t + 1) − θ 2 (t)
bulx: x coordinate of the every step point t: time
buly: y coordinate of the every step point time: one cycle of time
fark1: the subtractive length of the first and second link time1: one step of time
timet: the total time
hız1: the angular velocity of the first joint angle, θ1 top1: the total length of the first and second link
hız2: the angular velocity of the second joint angle, θ2 ul1: the length of the first link
m: the total step number along [AC] ul2: the length of the second link