353 research outputs found
Fast Manipulability Maximization Using Continuous-Time Trajectory Optimization
A significant challenge in manipulation motion planning is to ensure agility
in the face of unpredictable changes during task execution. This requires the
identification and possible modification of suitable joint-space trajectories,
since the joint velocities required to achieve a specific endeffector motion
vary with manipulator configuration. For a given manipulator configuration, the
joint space-to-task space velocity mapping is characterized by a quantity known
as the manipulability index. In contrast to previous control-based approaches,
we examine the maximization of manipulability during planning as a way of
achieving adaptable and safe joint space-to-task space motion mappings in
various scenarios. By representing the manipulator trajectory as a
continuous-time Gaussian process (GP), we are able to leverage recent advances
in trajectory optimization to maximize the manipulability index during
trajectory generation. Moreover, the sparsity of our chosen representation
reduces the typically large computational cost associated with maximizing
manipulability when additional constraints exist. Results from simulation
studies and experiments with a real manipulator demonstrate increases in
manipulability, while maintaining smooth trajectories with more dexterous (and
therefore more agile) arm configurations.Comment: In Proceedings of the IEEE International Conference on Intelligent
Robots and Systems (IROS'19), Macau, China, Nov. 4-8, 201
Handling robot constraints within a Set-Based Multi-Task Priority Inverse Kinematics Framework
Set-Based Multi-Task Priority is a recent framework to handle inverse
kinematics for redundant structures. Both equality tasks, i.e., control
objectives to be driven to a desired value, and set-bases tasks, i.e., control
objectives to be satisfied with a set/range of values can be addressed in a
rigorous manner within a priority framework. In addition, optimization tasks,
driven by the gradient of a proper function, may be considered as well, usually
as lower priority tasks. In this paper the proper design of the tasks, their
priority and the use of a Set-Based Multi-Task Priority framework is proposed
in order to handle several constraints simultaneously in real-time. It is shown
that safety related tasks such as, e.g., joint limits or kinematic singularity,
may be properly handled by consider them both at an higher priority as
set-based task and at a lower within a proper optimization functional.
Experimental results on a 7DOF Jaco$^2
Self-motion control of kinematically redundant robot manipulators
Thesis (Master)--Izmir Institute of Technology, Mechanical Engineering, Izmir, 2012Includes bibliographical references (leaves: 88-92)Text in English; Abstract: Turkish and Englishxvi,92 leavesRedundancy in general provides space for optimization in robotics. Redundancy can be defined as sensor/actuator redundancy or kinematic redundancy. The redundancy considered in this thesis is the kinematic redundancy where the total degrees-of-freedom of the robot is more than the total degrees-of-freedom required for the task to be executed. This provides infinite number of solutions to perform the same task, thus, various subtasks can be carried out during the main-task execution. This work utilizes the property of self-motion for kinematically redundant robot manipulators by designing the general subtask controller that controls the joint motion in the null-space of the Jacobian matrix. The general subtask controller is implemented for various subtasks in this thesis. Minimizing the total joint motion, singularity avoidance, posture optimization for static impact force objectives, which include maximizing/minimizing the static impact force magnitude, and static and moving obstacle (point to point) collision avoidance are the subtasks considered in this thesis. New control architecture is developed to accomplish both the main-task and the previously mentioned subtasks. In this architecture, objective function for each subtask is formed. Then, the gradient of the objective function is used in the subtask controller to execute subtask objective while tracking a given end-effector trajectory. The tracking of the end-effector is called main-task. The SCHUNK LWA4-Arm robot arm with seven degrees-of-freedom is developed first in SolidWorksรยฎ as a computer-aided-design (CAD) model. Then, the CAD model is converted to MATLABรยฎ Simulink model using SimMechanics CAD translator to be used in the simulation tests of the controller. Kinematics and dynamics equations of the robot are derived to be used in the controllers. Simulation test results are presented for the kinematically redundant robot manipulator operating in 3D space carrying out the main-task and the selected subtasks for this study. The simulation test results indicate that the developed controllerรขโฌโขs performance is successful for all the main-task and subtask objectives
์์ ํ ์ฌ๊ตฌ์ฑ ๋ก๋ด ์์คํ : ์ค๊ณ, ํ๋ก๊ทธ๋๋ฐ ๋ฐ ๋ฐ์ํ ๊ฒฝ๋ก๊ณํ
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ) -- ์์ธ๋ํ๊ต ๋ํ์ : ๊ณต๊ณผ๋ํ ๊ธฐ๊ณํญ๊ณต๊ณตํ๋ถ, 2020. 8. ๋ฐ์ข
์ฐ.The next generation of robots are being asked to work in close proximity to humans. At the same time, the robot should have the ability to change its topology to flexibly cope with various tasks. To satisfy these two requirements, we propose a novel modular reconfi gurable robot and accompanying software architecture, together with real-time motion planning algorithms to allow for safe operation in unstructured dynamic environments with humans.
Two of the key innovations behind our modular manipulator design are a genderless connector and multi-dof modules. By making the modules connectable regardless of the input/output directions, a genderless connector increases the number of possible connections. The developed genderless connector can transmit as
much load as necessary to an industrial robot. In designing two-dof modules, an offset between two joints is imposed to improve the overall integration and the safety of the modules.
To cope with the complexity in modeling due to the genderless connector and multi-dof modules, a programming architecture for modular robots is proposed. The key feature of the proposed architecture is that it efficiently represents connections of multi-dof modules only with connections between modules, while existing architectures should explicitly represent all connections between links and joints. The data structure of the proposed architecture contains properties of tree-structured multi-dof modules with intra-module relations. Using the data structure and connection relations between modules, kinematic/dynamic parameters of connected modules can be obtained through forward recursion.
For safe operation of modular robots, real-time robust collision avoidance algorithms for kinematic singularities are proposed. The main idea behind the algorithms is generating control inputs that increase the directional manipulability of the robot to the object direction by reducing directional safety measures. While
existing directional safety measures show undesirable behaviors in the vicinity of the kinematic singularities, the proposed geometric safety measure generates stable control inputs in the entire joint space. By adding the preparatory input from the geometric safety measure to the repulsive input, a hierarchical collision avoidance
algorithm that is robust to kinematic singularity is implemented.
To mathematically guarantee the safety of the robot, another collision avoidance algorithm using the invariance control framework with velocity-dependent safety constraints is proposed. When the object approached the robot from a singular direction, the safety constraints are not satis ed in the initial state of the
robot and the safety cannot be guaranteed using the invariance control. By proposing a control algorithm that quickly decreases the preparatory constraints below thresholds, the robot re-enters the constraint set and avoids collisions using the invariance control framework.
The modularity and safety of the developed reconfi gurable robot is validated using a set of simulations and hardware experiments. The kinematic/dynamic model of the assembled robot is obtained in real-time and used to accurately control the robot. Due to the safe design of modules with o sets and the high-level safety
functions with collision avoidance algorithms, the developed recon figurable robot has a broader safe workspace and wider ranger of safe operation speed than those of cooperative robots.๋ค์ ์ธ๋์ ๋ก๋ด์ ์ฌ๋๊ณผ ๊ฐ๊น์ด์์ ํ์
ํ ์ ์๋ ๊ธฐ๋ฅ์ ๊ฐ์ ธ์ผํ๋ค. ๊ทธ์ ๋์์, ๋ก๋ด์ ๋ค์ํ๊ฒ ๋ณํ๋ ์์
์ ๋ํด ์ ์ฐํ๊ฒ ๋์ฒํ ์ ์๋๋ก ์์ ์ ๊ตฌ์กฐ๋ฅผ ๋ฐ๊พธ๋ ๊ธฐ๋ฅ์ ๊ฐ์ ธ์ผ ํ๋ค. ์ด๋ฌํ ๋ ๊ฐ์ง ์๊ตฌ์กฐ๊ฑด์ ๋ง์กฑ์ํค๊ธฐ ์ํด, ๋ณธ ๋
ผ๋ฌธ์์๋ ์๋ก์ด ๋ชจ๋๋ผ ๋ก๋ด ์์คํ
๊ณผ ํ๋ก๊ทธ๋๋ฐ ์ํคํ
์ณ๋ฅผ ์ ์ํ๊ณ , ์ฌ๋์ด ์กด์ฌํ๋ ๋์ ํ๊ฒฝ์์ ์์ ํ ๋ก๋ด์ ์ด์ฉ์ ์ํ ์ค์ํ ๊ฒฝ๋ก ๊ณํ ์๊ณ ๋ฆฌ์ฆ์ ์ ์ํ๋ค.
๊ฐ๋ฐ๋ ๋ชจ๋๋ผ ๋ก๋ด์ ๋ ๊ฐ์ง ํต์ฌ์ ์ธ ํ์ ์ฑ์ ๋ฌด์ฑ๋ณ ์ปค๋ฅํฐ์ ๋ค์์ ๋ ๋ชจ๋์์ ์ฐพ์ ์ ์๋ค. ์
๋ ฅ/์ถ๋ ฅ ๋ฐฉํฅ์ ์๊ด ์์ด ๋ชจ๋์ด ์ฐ๊ฒฐ๋ ์ ์๋๋ก ํจ์ผ๋ก์จ, ๋ฌด์ฑ๋ณ ์ปค๋ฅํฐ๋ ๊ฒฐํฉ ๊ฐ๋ฅํ ๊ฒฝ์ฐ์ ์๋ฅผ ๋๋ฆด ์ ์๋ค. ๊ฐ๋ฐ๋ ๋ฌด์ฑ๋ณ ์ปค๋ฅํฐ๋ ์ฐ์
์ฉ ๋ก๋ด์์ ์๊ตฌ๋๋ ์ถฉ๋ถํ ๋ถํ๋ฅผ ๊ฒฌ๋ ์ ์๋๋ก ์ค๊ณ๋์๋ค. 2 ์์ ๋ ๋ชจ๋์ ์ค๊ณ์์ ๋ ์ถ ์ฌ์ด์ ์คํ์
์ ๊ฐ์ง๋๋ก ํจ์ผ๋ก์จ ์ ์ฒด์ ์ธ ์์ฑ๋ ๋ฐ ์์ ๋๋ฅผ ์ฆ๊ฐ์์ผฐ๋ค.
๋ฌด์ฑ๋ณ ์ปค๋ฅํฐ์ ๋ค์์ ๋ ๋ชจ๋๋ก ์ธํ ๋ชจ๋ธ๋ง์ ๋ณต์ก์ฑ์ ๋์ํ๊ธฐ ์ํด, ์ผ๋ฐ์ ์ธ ๋ชจ๋๋ผ ๋ก๋ด์ ์ํ ์ํํธ์จ์ด ์ํคํ
์ณ๋ฅผ ์ ์ํ์๋ค. ๊ธฐ์กด ๋ชจ๋๋ผ ๋ก๋ด์ ์ฐ๊ฒฐ์ ๋ํ๋ด๋ ๋ฐฉ๋ฒ์ด ๋ชจ๋ ๋งํฌ์ ์กฐ์ธํธ ์ฌ์ด์ ์ฐ๊ฒฐ ๊ด๊ณ๋ฅผ ๋ณ๋๋ก ๋ํ๋ด์ผํ๋ ๊ฒ๊ณผ ๋ค๋ฅด๊ฒ, ์ ์๋ ์ํคํ
์ณ๋ ๋ชจ๋๋ค ์ฌ์ด์ ์ฐ๊ฒฐ๊ด๊ณ๋ง์ ๋ํ๋์ผ๋ก์จ ํจ์จ์ ์ธ ๋ค์์ ๋ ๋ชจ๋์ ์ฐ๊ฒฐ๊ด๊ณ๋ฅผ ๋ํ๋ผ ์ ์๋ค๋ ๊ฒ์ ํน์ง์ผ๋ก ํ๋ค. ์ด๋ฅผ ์ํด ํธ๋ฆฌ ๊ตฌ์กฐ๋ฅผ ๊ฐ์ง๋ ์ผ๋ฐ์ ์ธ ๋ค์์ ๋ ๋ชจ๋์ ์ฑ์ง์ ๋ํ๋ด๋ ๋ฐ์ดํฐ ๊ตฌ์กฐ๋ฅผ ์ ์ํ์๋ค. ๋ชจ๋๋ค ์ฌ์ด์ ์ฐ๊ฒฐ๊ด๊ณ ๋ฐ ๋ฐ์ดํฐ ๊ตฌ์กฐ๋ฅผ ์ด์ฉํ์ฌ, ์ ํํ ๊ธฐ๊ตฌํ/๋์ญํ ๋ชจ๋ธ ํ๋ผ๋ฏธํฐ๋ฅผ ์ป์ด๋ด๋ ์๋ฐฉํฅ ์ฌ๊ท ์๊ณ ๋ฆฌ์ฆ์ ๊ตฌํํ์๋ค.
๋ชจ๋๋ผ ๋ก๋ด์ ์์ ํ ์ด์ฉ์ ์ํด, ๊ธฐ๊ตฌํ์ ํน์ด์ ์ ๊ฐ๊ฑดํ ์ค์๊ฐ ์ถฉ๋ํํผ ์๊ณ ๋ฆฌ์ฆ์ ์ ์ํ์๋ค. ๋ฐฉํฅ์ฑ ์์ ๋๋ฅผ ์ค์ด๋ ๋ฐฉํฅ์ ์ ์ด ์
๋ ฅ์ ์์ฑํ์ฌ ๋ฌผ์ฒด ๋ฐฉํฅ์ผ๋ก์ ๋ก๋ด ๋ฐฉํฅ์ฑ ๋งค๋ํฐ๋ฌ๋น๋ฆฌํฐ๋ฅผ ์ฆ๊ฐ์ํค๋ ๊ฒ์ด ์ ์ํ ์๊ณ ๋ฆฌ์ฆ์ ํต์ฌ์ด๋ค. ๊ธฐ์กด์ ๋ฐฉํฅ์ฑ ์์ ๋๊ฐ ๊ธฐ๊ตฌํ์ ํน์ด์ ๊ทผ์ฒ์์ ์ํ์ง ์๋ ์ฑ์ง์ ๊ฐ์ง๋ ๊ฒ๊ณผ๋ ๋ฐ๋๋ก, ์ ์ํ ๊ธฐํํ์ ์์ ๋๋ ์ ์ฒด ์กฐ์ธํธ ๊ณต๊ฐ์์ ์์ ์ ์ธ ์ ์ด ์
๋ ฅ์ ์์ฑํ๋ค. ์ด ๊ธฐํํ์ ์์ ๋๋ฅผ ์ด์ฉํ์ฌ, ๊ธฐ๊ตฌํ์ ํน์ด์ ์ ๊ฐ๊ฑดํ ๊ณ์ธต์ ์ถฉ๋ํํผ ์๊ณ ๋ฆฌ์ฆ์ ๊ตฌํํ์๋ค.
์ํ์ ์ผ๋ก ๋ก๋ด์ ์์ ๋๋ฅผ ๋ณด์ฅํ๊ธฐ ์ํด, ์๋์๋์ ์ข
์์ ์ธ ์์ ์ ์ฝ์กฐ๊ฑด์ ๊ฐ์ง๋ ๋ถ๋ณ ์ ์ด ํ๋ ์์ํฌ์ ์ด์ฉํ์ฌ ๋ ํ๋์ ์ถฉ๋ ํํผ ์๊ณ ๋ฆฌ์ฆ์ ์ ์ํ์๋ค. ๋ฌผ์ฒด๊ฐ ํน์ด์ ๋ฐฉํฅ์ผ๋ก๋ถํฐ ๋ก๋ด์ ์ ๊ทผํ ๋, ๋ก๋ด์ ์ด๊ธฐ ์ํ์์ ์์ ์ ์ฝ์กฐ๊ฑด์ ๋ง์กฑ์ํค์ง ๋ชปํ๊ฒ ๋์ด ๋ถ๋ณ์ ์ด๋ฅผ ์ ์ฉํ ์ ์๊ฒ ๋๋ค. ์ค๋น ์ ์ฝ์กฐ๊ฑด์ ๋น ๋ฅด๊ฒ ์๊ณ์ ์๋๋ก ๊ฐ์์ํค๋ ์๊ณ ๋ฆฌ์ฆ์ ์ ์ฉํจ์ผ๋ก์จ, ๋ก๋ด์ ์ ์ฝ์กฐ๊ฑด ์งํฉ์ ๋ค์ ๋ค์ด๊ฐ๊ณ ๋ถ๋ณ ์ ์ด ๋ฐฉ๋ฒ์ ์ด์ฉํ์ฌ ์ถฉ๋์ ํํผํ ์ ์๊ฒ ๋๋ค.
๊ฐ๋ฐ๋ ์ฌ๊ตฌ์ฑ ๋ก๋ด์ ๋ชจ๋๋ผ๋ฆฌํฐ์ ์์ ๋๋ ์ผ๋ จ์ ์๋ฎฌ๋ ์ด์
๊ณผ ํ๋์จ์ด ์คํ์ ํตํด ๊ฒ์ฆ๋์๋ค. ์ค์๊ฐ์ผ๋ก ์กฐ๋ฆฝ๋ ๋ก๋ด์ ๊ธฐ๊ตฌํ/๋์ญํ ๋ชจ๋ธ์ ์ป์ด๋ด ์ ๋ฐ ์ ์ด์ ์ฌ์ฉํ์๋ค. ์์ ํ ๋ชจ๋ ๋์์ธ๊ณผ ์ถฉ๋ ํํผ ๋ฑ์ ๊ณ ์ฐจ์ ์์ ๊ธฐ๋ฅ์ ํตํ์ฌ, ๊ฐ๋ฐ๋ ์ฌ๊ตฌ์ฑ ๋ก๋ด์ ๊ธฐ์กด ํ๋๋ก๋ด๋ณด๋ค ๋์ ์์ ํ ์์
๊ณต๊ฐ๊ณผ ์์
์๋๋ฅผ ๊ฐ์ง๋ค.1 Introduction 1
1.1 Modularity and Recon gurability 1
1.2 Safe Interaction 4
1.3 Contributions of This Thesis 9
1.3.1 A Recon gurable Modular Robot System with Bidirectional Modules 9
1.3.2 A Modular Robot Software Programming Architecture 10
1.3.3 Anticipatory Collision Avoidance Planning 11
1.4 Organization of This Thesis 14
2 Design and Prototyping of the ModMan 17
2.1 Genderless Connector 18
2.2 Modules for ModMan 21
2.2.1 Joint Modules 21
2.2.2 Link and Gripper Modules 25
2.3 Experiments 26
2.3.1 System Setup 26
2.3.2 Repeatability Comparison with Non-recon gurable Robot Manipulators 28
2.3.3 E ect of the O set in Two-dof Modules 30
2.4 Conclusion 32
3 A Programming Architecture for Modular Recon gurable Robots 33
3.1 Data Structure for Multi-dof Joint Modules 34
3.2 Automatic Kinematic Modeling 37
3.3 Automatic Dynamic Modeling 40
3.4 Flexibility in Manipulator 42
3.5 Experiments 45
3.5.1 System Setup 46
3.5.2 Recon gurability 46
3.5.3 Pick-and-Place with Vision Sensors 48
3.6 Conclusion 49
4 A Preparatory Safety Measure for Robust Collision Avoidance 51
4.1 Preliminaries on Manipulability and Safety 52
4.2 Analysis on Reected Mass 56
4.3 Manipulability Control on S+(1;m) 60
4.3.1 Geometry of the Group of Positive Semi-de nite Matrices 60
4.3.2 Rank-One Manipulability Control 63
4.4 Collision Avoidance with Preparatory Action 65
4.4.1 Repulsive and Preparatory Potential Functions 65
4.4.2 Hierarchical Control and Task Relaxation 67
4.5 Experiments 70
4.5.1 Manipulability Control 71
4.5.2 Collision Avoidance 75
4.6 Conclusion 82
5 Collision Avoidance with Velocity-Dependent Constraints 85
5.1 Input-Output Linearization 87
5.2 Invariance Control 89
5.3 Velocity-Dependent Constraints for Robot Safety 90
5.3.1 Velocity-Dependent Repulsive Constraints 90
5.3.2 Preparatory Constraints 92
5.3.3 Corrective Control for Dangerous Initial State 93
5.4 Experiment 95
5.5 Conclusion 98
6 Conclusion 101
6.1 Overview of This Thesis 101
6.2 Future Work 104
Appendix A Appendix 107
A.1 Preliminaries on Graph Theory 107
A.2 Lie-Theoretic Formulations of Robot Kinematics and Dynamics 108
A.3 Derivatives of Eigenvectors and Eigenvalues 110
A.4 Proof of Proposition Proposition 4.1 111
A.5 Proof of Triangle Inequality When p = 1 114
A.6 Detailed Conditions for a Danger Field 115
Bibliography 117
Abstract 127Docto
Learning Singularity Avoidance
With the increase in complexity of robotic systems and the rise in non-expert
users, it can be assumed that task constraints are not explicitly known. In
tasks where avoiding singularity is critical to its success, this paper
provides an approach, especially for non-expert users, for the system to learn
the constraints contained in a set of demonstrations, such that they can be
used to optimise an autonomous controller to avoid singularity, without having
to explicitly know the task constraints. The proposed approach avoids
singularity, and thereby unpredictable behaviour when carrying out a task, by
maximising the learnt manipulability throughout the motion of the constrained
system, and is not limited to kinematic systems. Its benefits are demonstrated
through comparisons with other control policies which show that the constrained
manipulability of a system learnt through demonstration can be used to avoid
singularities in cases where these other policies would fail. In the absence of
the systems manipulability subject to a tasks constraints, the proposed
approach can be used instead to infer these with results showing errors less
than 10^-5 in 3DOF simulated systems as well as 10^-2 using a 7DOF real world
robotic system
Safety-related Tasks within the Set-Based Task-Priority Inverse Kinematics Framework
In this paper we present a framework that allows the motion control of a
robotic arm automatically handling different kinds of safety-related tasks. The
developed controller is based on a Task-Priority Inverse Kinematics algorithm
that allows the manipulator's motion while respecting constraints defined
either in the joint or in the operational space in the form of equality-based
or set-based tasks. This gives the possibility to define, among the others,
tasks as joint-limits, obstacle avoidance or limiting the workspace in the
operational space. Additionally, an algorithm for the real-time computation of
the minimum distance between the manipulator and other objects in the
environment using depth measurements has been implemented, effectively allowing
obstacle avoidance tasks. Experiments with a Jaco manipulator, operating in
an environment where an RGB-D sensor is used for the obstacles detection, show
the effectiveness of the developed system
Singularity Avoidance with Application to Online Trajectory Optimization for Serial Manipulators
This work proposes a novel singularity avoidance approach for real-time
trajectory optimization based on known singular configurations. The focus of
this work lies on analyzing kinematically singular configurations for three
robots with different kinematic structures, i.e., the Comau Racer 7-1.4, the
KUKA LBR iiwa R820, and the Franka Emika Panda, and exploiting these
configurations in form of tailored potential functions for singularity
avoidance. Monte Carlo simulations of the proposed method and the commonly used
manipulability maximization approach are performed for comparison. The
numerical results show that the average computing time can be reduced and
shorter trajectories in both time and path length are obtained with the
proposed approachComment: 8 pages, 2 figures, Accepted for publication at IFAC World Congress
202
Design, Control and Motion Planning for a Novel Modular Extendable Robotic Manipulator
This dissertation discusses an implementation of a design, control and motion planning for a novel extendable modular redundant robotic manipulator in space constraints, which robots may encounter for completing required tasks in small and constrained environment.
The design intent is to facilitate the movement of the proposed robotic manipulator in constrained environments, such as rubble piles. The proposed robotic manipulator with multi Degree of Freedom (m-DOF) links is capable of elongating by 25% of its nominal length. In this context, a design optimization problem with multiple objectives is also considered. In order to identify the benefits of the proposed design strategy, the reachable workspace of the proposed manipulator is compared with that of the Jet Propulsion Laboratory (JPL) serpentine robot. The simulation results show that the proposed manipulator has a relatively efficient reachable workspace, needed in constrained environments. The singularity and manipulability of the designed manipulator are investigated. In this study, we investigate the number of links that produces the optimal design architecture of the proposed robotic manipulator. The total number of links decided by a design optimization can be useful distinction in practice.
Also, we have considered a novel robust bio-inspired Sliding Mode Control (SMC) to achieve favorable tracking performance for a class of robotic manipulators with uncertainties. To eliminate the chattering problem of the conventional sliding mode control, we apply the Brain Emotional Learning Based Intelligent Control (BELBIC) to adaptively adjust the control input law in sliding mode control. The on-line computed parameters achieve favorable system robustness in process of parameter uncertainties and external disturbances. The simulation results demonstrate that our control strategy is effective in tracking high speed trajectories with less chattering, as compared to the conventional sliding mode control. The learning process of BLS is shown to enhance the performance of a new robust controller.
Lastly, we consider the potential field methodology to generate a desired trajectory in small and constrained environments. Also, Obstacle Collision Avoidance (OCA) is applied to obtain an inverse kinematic solution of a redundant robotic manipulator
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