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Hassan Thesis Presentation - Dec16

A mobile parallel robot used for pipeline inspection is analyzed. The robot consists of single modules resembling a snake. The study determines the robot's singularity-free workspace and derives optimal geometric parameters. When passing through sharp corners in the pipe, the robot encounters singular configurations that disrupt mobility. To overcome this, prismatic joints are added to the arms, allowing continuous singularity-free motion across corners through optimal control strategies like path following control and gradient ascent to maximize arm length. Simulation results demonstrate the robot can navigate corners without getting stuck, leading to a continuous singularity-free workspace.

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KINEMATICS AND OPTIMAL CONTROL OF A MOBILE PARALLEL ROBOT FOR INSPECTION OF PIPE-LIKE ENVIRONMENTS Hassan Sarfraz The Ottawa-Carleton Institute for Electrical and Computer Engineering 1
Snake-like Pipeline Inspection Robot Explorer 10/14 [Photograph]. Retrieved 11 December, 2013, from Pipetel technologies Inc. http://www.pipetelone.com/explorer_10-14.html 2
Problem Statement Goal: To maximize the reachable workspace 3   0  0
Contribution Analysis of a single module of a snake-like pipeline inspection robot.  Study of Workspace and Singularities  Determination of Optimal Geometry  Optimal Control in a Geometrically Singular pipe Explorer 10/14 [Photograph]. Retrieved 11 December, 2013, from Pipetel technologies Inc. http://www.pipetelone.com/explorer_10-14.html 4
Single Module: a Mobile Parallel Robot qJxJ qx   2121 ,,, ssq     ,, GG yxx  5
Single Module: a Mobile Parallel Robot                                                    cossinsin00 sincoscos00 cossin 2 sin 2 1 10 cossin 2 cos 2 1 01 21 21 1 1 wll wll l w ah l w ah Jx                                                    2 22 1 11 21 2 22 1 11 21 1 11 1 1 11 1 sinsin coscos 00sin 00cos ds sdy ds sdy ll ds sdx ds sdx ll ds sdy l ds sdx l J PP PP P P q     qJxJ qx   2121 ,,, ssq    ,, GG yxx  6
Singular Configurations Serial Singularity                    222222 111111 sin'cos' sin'cos' sxsy sxsy PP PP   0det T qq JJ Active joints motion resulting in no motion in the end-effector 7 Applied to this robot Singularity occurs when at least one arm is perpendicular to pipe wall
Singular Configurations Parallel Singularity   0det T xx JJ             ,4,2,0, 0 0sinsin2cos12 21 2121 22 ii w wllw   Motion in the end-effector is admitted for motionless active joints 8 Practically, the above condition is not possible Analytical expression defining parallel singularity
Singularity-free Workspace, Гsf Introduction to four pipe-like structures 9
Singularity-free Workspace, Гsf Discretization Method 1. Forming a Grid on test area, Гref 10
Singularity-free Workspace, Гsf Discretization Method 1. Forming a Grid on surface area, Гref 2. Direct Search Algorithm, Inverse Kinematic Sol. 11
Singularity-free Workspace, Гsf Discretization Method 1. Forming a Grid on surface area, Гref 2. Direct Search Algorithm, Inverse Kinematic Sol. 3. Collision Avoidance Algorithm 12
10 1 max min   KCI KCI    1. Forming a Grid on surface area, Гref 2. Direct Search Algorithm, Inverse Kinematic Sol. 3. Collision Avoidance Algorithm 4. Proximity to singularity  Kinematic Conditioning Index Singularity-free Workspace, Гsf Discretization Method 13
Proximity to singularity in a Straight pipe  0  5  5 15  15 Singularity-free Workspace, Гsf 14
 0 10  10  20  20 Proximity to singularity in 135° elbow Singularity-free Workspace, Гsf 15
Optimization of Geometric ParametersOptimization Problem Formulation   sf ref sf ahwl     ofContinuity ConstraintContact ConstraintAvoidanceCollision ConstraintAvoidanceySingularit:tosubjected ,,,FMaximize 16
PWl 26.0 PWw 5.0 PWh PWa 25.0 Optimization of Geometric Parameters Constrained Optimization in a Straight Pipe Initial Design Parameters   0 15 17
PWw 5.0 PWh  Optimization of Geometric Parameters Constrained Optimization in a Straight Pipe PWl 58.0 PWa 96.0 Converged Design Parameters   0 15 18
Constrained Optimization in a Straight Pipe Converged Design Parameters,l PWa 96.0 Cost function v.s. θ for values of l Average Cost function v.s. l PWw 5.0 PWh  Optimization of Geometric Parameters 19
PWw 5.0 PWh  Constrained Optimization in 135° elbow PWl 75.0 PWa 91.0 Converged Design Parameters   0 Optimization of Geometric Parameters 20
Critical Mobility Scenario • Collision • Singular Configuration • Discontinuity in Гsf Optimal Control in a Geometrically Singular pipe 21
• Prismatic Joints on the arms • Additional degrees of freedom to overcome singularities at the corner Modified Parallel Mobile Robot Optimal Control in a Geometrically Singular pipe 22
Modified Parallel Mobile Robot (continued) Optimal Control in a Geometrically Singular pipe                                                                2 22 1 11 2211 2 22 1 11 2211 1 11 11 1 11 11 sinsin coscos 00sin 00cos ds sdy ds sdy ldld ds sdx ds sdx ldld ds sdy ld ds sdx ld J PP PP P P q     qJxJ qx   2121 ,,, ssq   21,,,, ddyxx GG                                                                              212211 212211 111 111 coscoscossinsin00 sinsinsincoscos00 0coscossin 2 sin 2 1 10 0sincossin 2 cos 2 1 01 wldld wldld ld w ah ld w ah Jx 23
1. Forward motion using Path-Following Control with proportional term • Xg, Yg, θ 2. Optimal arm length using gradient ascent Path Following and Optimal Trajectories Optimal Control in a Geometrically Singular pipe 24
Discrete-time Simulation Results Optimal Control in a Geometrically Singular pipe 25
Prismatic Arm length vs. Path AbscissaSingularity measure vs. Path Abscissa Performance Evaluation Optimal Control in a Geometrically Singular pipe 26
• Continuous Singularity-free Workspace using Mobile Parallel Robot with prismatic arms • Discontinuity in Singularity- free Workspace using Mobile Parallel Robot with rigid arms vs. Comparison of Singularity-free Workspace Optimal Control in a Geometrically Singular pipe 27
Summary and Conclusion  Singular configurations  Singularity-free workspace  Optimization of Geometric Parameters  Mobile robot with discontinuous workspace when crossing a sharp corner.  Formulated and simulated a kinematical model to navigate singularity-free across a corner  An Optimal control strategy used to maximize a performance index and deal with collisions  Proposed Solution leads to continuous singularity- free workspace. 28
Publication Journal Paper  Lounis Douadi, Davide Spinello, Wail Gueaieb and Hassan Sarfraz. “Planar kinematics analysis of a snake-like robot”. Robotica. doi:10.1017/S026357471300091X. Conference Paper  Davide Spinello, Hassan Sarfraz, Wail Gueaieb, Lounis Douadi, “Critical Maneuvers of an Autonomous Parallel Robot in a Confined Environment”, In Proceedings of the International Conference on Mechanical Engineering and Mechatronics (ICMEM), 8 pp. Paper no. 196, 2013. 29
Thank you for your time Any questions or comments? 30
 Contact Constraint  Continuity of Гsf   PWwl 2 Optimization of Geometric MethodOptimization Problem Formulation (continued) 31
Step 1: Step 2: ]0[l ]0[w ]0[h ]0[a         0001 ,,,Fmax ahwll ldl  ]1[l Optimization of Geometric MethodOptimization Technique: Parametric Variation 32
Step 1: Step 2: ]0[l ]0[w ]0[h ]0[a         0001 ,,,Fmax ahwlw wdw  ]1[l ]1[w Optimization of Geometric MethodOptimization Technique: Parametric Variation 33
Step 1: Step 2: ]0[l ]0[w ]0[h ]0[a         ahwla ada ,,,Fmax 0001   ]1[l ]1[w ]1[h ]1[a Optimization of Geometric MethodOptimization Technique: Parametric Variation 34
Step 1: Step 2: Step 3: Repeat the above process ]0[l ]0[w ]0[h ]0[a ]1[l ]1[w ]1[h ]1[a       qq 1 ][ql ][qw ][qh ][qa ][q  Optimization of Geometric MethodOptimization Technique: Parametric Variation 35
PWl 5.0 PWw 5.0 PWh PWa 5.0 Optimization of Geometric MethodConstrained Optimization in a Straight Pipe Initial Design Parameters   0 36
PWw 5.0 PWh  Optimization of Geometric MethodConstrained Optimization in a Straight Pipe PWl 58.0 PWa 96.0 Converged Design Parameters   0 37
0w 0h Optimization of Geometric MethodUnconstrained Optimization in a Straight Pipe PWl  PWa  Converged Design Parameters   0 38

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Hassan Thesis Presentation - Dec16

  • 1.
    KINEMATICS AND OPTIMAL CONTROLOF A MOBILE PARALLEL ROBOT FOR INSPECTION OF PIPE-LIKE ENVIRONMENTS Hassan Sarfraz The Ottawa-Carleton Institute for Electrical and Computer Engineering 1
  • 2.
    Snake-like Pipeline Inspection Robot Explorer10/14 [Photograph]. Retrieved 11 December, 2013, from Pipetel technologies Inc. http://www.pipetelone.com/explorer_10-14.html 2
  • 3.
    Problem Statement Goal: Tomaximize the reachable workspace 3   0  0
  • 4.
    Contribution Analysis of asingle module of a snake-like pipeline inspection robot.  Study of Workspace and Singularities  Determination of Optimal Geometry  Optimal Control in a Geometrically Singular pipe Explorer 10/14 [Photograph]. Retrieved 11 December, 2013, from Pipetel technologies Inc. http://www.pipetelone.com/explorer_10-14.html 4
  • 5.
    Single Module: aMobile Parallel Robot qJxJ qx   2121 ,,, ssq     ,, GG yxx  5
  • 6.
    Single Module: aMobile Parallel Robot                                                    cossinsin00 sincoscos00 cossin 2 sin 2 1 10 cossin 2 cos 2 1 01 21 21 1 1 wll wll l w ah l w ah Jx                                                    2 22 1 11 21 2 22 1 11 21 1 11 1 1 11 1 sinsin coscos 00sin 00cos ds sdy ds sdy ll ds sdx ds sdx ll ds sdy l ds sdx l J PP PP P P q     qJxJ qx   2121 ,,, ssq    ,, GG yxx  6
  • 7.
    Singular Configurations Serial Singularity                   222222 111111 sin'cos' sin'cos' sxsy sxsy PP PP   0det T qq JJ Active joints motion resulting in no motion in the end-effector 7 Applied to this robot Singularity occurs when at least one arm is perpendicular to pipe wall
  • 8.
    Singular Configurations Parallel Singularity  0det T xx JJ             ,4,2,0, 0 0sinsin2cos12 21 2121 22 ii w wllw   Motion in the end-effector is admitted for motionless active joints 8 Practically, the above condition is not possible Analytical expression defining parallel singularity
  • 9.
    Singularity-free Workspace, Гsf Introductionto four pipe-like structures 9
  • 10.
    Singularity-free Workspace, Гsf DiscretizationMethod 1. Forming a Grid on test area, Гref 10
  • 11.
    Singularity-free Workspace, Гsf DiscretizationMethod 1. Forming a Grid on surface area, Гref 2. Direct Search Algorithm, Inverse Kinematic Sol. 11
  • 12.
    Singularity-free Workspace, Гsf DiscretizationMethod 1. Forming a Grid on surface area, Гref 2. Direct Search Algorithm, Inverse Kinematic Sol. 3. Collision Avoidance Algorithm 12
  • 13.
    10 1 max min   KCI KCI    1. Forming aGrid on surface area, Гref 2. Direct Search Algorithm, Inverse Kinematic Sol. 3. Collision Avoidance Algorithm 4. Proximity to singularity  Kinematic Conditioning Index Singularity-free Workspace, Гsf Discretization Method 13
  • 14.
    Proximity to singularityin a Straight pipe  0  5  5 15  15 Singularity-free Workspace, Гsf 14
  • 15.
     0 10  10 20  20 Proximity to singularity in 135° elbow Singularity-free Workspace, Гsf 15
  • 16.
    Optimization of Geometric ParametersOptimizationProblem Formulation   sf ref sf ahwl     ofContinuity ConstraintContact ConstraintAvoidanceCollision ConstraintAvoidanceySingularit:tosubjected ,,,FMaximize 16
  • 17.
    PWl 26.0 PWw5.0 PWh PWa 25.0 Optimization of Geometric Parameters Constrained Optimization in a Straight Pipe Initial Design Parameters   0 15 17
  • 18.
    PWw 5.0 PWh  Optimizationof Geometric Parameters Constrained Optimization in a Straight Pipe PWl 58.0 PWa 96.0 Converged Design Parameters   0 15 18
  • 19.
    Constrained Optimization ina Straight Pipe Converged Design Parameters,l PWa 96.0 Cost function v.s. θ for values of l Average Cost function v.s. l PWw 5.0 PWh  Optimization of Geometric Parameters 19
  • 20.
    PWw 5.0 PWh  ConstrainedOptimization in 135° elbow PWl 75.0 PWa 91.0 Converged Design Parameters   0 Optimization of Geometric Parameters 20
  • 21.
    Critical Mobility Scenario •Collision • Singular Configuration • Discontinuity in Гsf Optimal Control in a Geometrically Singular pipe 21
  • 22.
    • Prismatic Jointson the arms • Additional degrees of freedom to overcome singularities at the corner Modified Parallel Mobile Robot Optimal Control in a Geometrically Singular pipe 22
  • 23.
    Modified Parallel MobileRobot (continued) Optimal Control in a Geometrically Singular pipe                                                                2 22 1 11 2211 2 22 1 11 2211 1 11 11 1 11 11 sinsin coscos 00sin 00cos ds sdy ds sdy ldld ds sdx ds sdx ldld ds sdy ld ds sdx ld J PP PP P P q     qJxJ qx   2121 ,,, ssq   21,,,, ddyxx GG                                                                              212211 212211 111 111 coscoscossinsin00 sinsinsincoscos00 0coscossin 2 sin 2 1 10 0sincossin 2 cos 2 1 01 wldld wldld ld w ah ld w ah Jx 23
  • 24.
    1. Forward motionusing Path-Following Control with proportional term • Xg, Yg, θ 2. Optimal arm length using gradient ascent Path Following and Optimal Trajectories Optimal Control in a Geometrically Singular pipe 24
  • 25.
    Discrete-time Simulation Results OptimalControl in a Geometrically Singular pipe 25
  • 26.
    Prismatic Arm lengthvs. Path AbscissaSingularity measure vs. Path Abscissa Performance Evaluation Optimal Control in a Geometrically Singular pipe 26
  • 27.
    • Continuous Singularity-free Workspaceusing Mobile Parallel Robot with prismatic arms • Discontinuity in Singularity- free Workspace using Mobile Parallel Robot with rigid arms vs. Comparison of Singularity-free Workspace Optimal Control in a Geometrically Singular pipe 27
  • 28.
    Summary and Conclusion Singular configurations  Singularity-free workspace  Optimization of Geometric Parameters  Mobile robot with discontinuous workspace when crossing a sharp corner.  Formulated and simulated a kinematical model to navigate singularity-free across a corner  An Optimal control strategy used to maximize a performance index and deal with collisions  Proposed Solution leads to continuous singularity- free workspace. 28
  • 29.
    Publication Journal Paper  LounisDouadi, Davide Spinello, Wail Gueaieb and Hassan Sarfraz. “Planar kinematics analysis of a snake-like robot”. Robotica. doi:10.1017/S026357471300091X. Conference Paper  Davide Spinello, Hassan Sarfraz, Wail Gueaieb, Lounis Douadi, “Critical Maneuvers of an Autonomous Parallel Robot in a Confined Environment”, In Proceedings of the International Conference on Mechanical Engineering and Mechatronics (ICMEM), 8 pp. Paper no. 196, 2013. 29
  • 30.
    Thank you foryour time Any questions or comments? 30
  • 31.
     Contact Constraint Continuity of Гsf   PWwl 2 Optimization of Geometric MethodOptimization Problem Formulation (continued) 31
  • 32.
    Step 1: Step 2: ]0[l]0[w ]0[h ]0[a         0001 ,,,Fmax ahwll ldl  ]1[l Optimization of Geometric MethodOptimization Technique: Parametric Variation 32
  • 33.
    Step 1: Step 2: ]0[l]0[w ]0[h ]0[a         0001 ,,,Fmax ahwlw wdw  ]1[l ]1[w Optimization of Geometric MethodOptimization Technique: Parametric Variation 33
  • 34.
    Step 1: Step 2: ]0[l]0[w ]0[h ]0[a         ahwla ada ,,,Fmax 0001   ]1[l ]1[w ]1[h ]1[a Optimization of Geometric MethodOptimization Technique: Parametric Variation 34
  • 35.
    Step 1: Step 2: Step3: Repeat the above process ]0[l ]0[w ]0[h ]0[a ]1[l ]1[w ]1[h ]1[a       qq 1 ][ql ][qw ][qh ][qa ][q  Optimization of Geometric MethodOptimization Technique: Parametric Variation 35
  • 36.
    PWl 5.0 PWw5.0 PWh PWa 5.0 Optimization of Geometric MethodConstrained Optimization in a Straight Pipe Initial Design Parameters   0 36
  • 37.
    PWw 5.0 PWh  Optimizationof Geometric MethodConstrained Optimization in a Straight Pipe PWl 58.0 PWa 96.0 Converged Design Parameters   0 37
  • 38.
    0w 0h Optimization of Geometric MethodUnconstrainedOptimization in a Straight Pipe PWl  PWa  Converged Design Parameters   0 38

Editor's Notes

  • #2 My name is Hassan Sarfraz. My thesis is on ‘Kinematics and Optimal Control of a Mobile parallel robot for inspection of pipe-like Environments’ .
  • #3 Energy Resources are an essential part of every individual’s life. Population and economic growth is deriving the demand of energy which depends on a reliable supply of energy such as natural gas, diesel, petroleum and other products made from crude oil. These resources are mainly delivered from large networks of pipelines. These pipelines are susceptible to corrosion and cracking and therefore requires regular maintenance and inspection. A flexible multi-body mechanism or a wheeled snake-like robot exhibits excellent mobility in such confined environments. The modules which are connected using passive revolute joints. Where each module is a parallel mechanism on a mobile platform with 3DOF. This project was funded through a collaboration with an industrial partner that owns a robotic device for nondestructive inspection in gas pipelines. The device is currently operated manually, and this project is the first step towards the next generation to build an autonomous system for navigation pipe-like environment with suitable sensing.
  • #4 The goal of this thesis is Given a confined environment Analyze the kinematics of the robot And maximize the reachable workspace
  • #5  In this project, the emphasis is on a single module that is a parallel robot on a moving platform My contribution includes the kinematic analysis in which I study the workspace and singularity of the mobile parallel robot in various pipe structures. The optimization of geometric parameters of the system to maximize the Singularity-free workspace. And an optimal control to generate singularity free trajectories when crossing a geometric singularity such as a sharp 90 elbow is proposed.
  • #6  This is a 3DOF mechanism which comprises of a rigid block with two rigid arms attached to the side of the body and in-contact with the walls. The mechanism is able to position itself using four active joints located at H1, H2, P1, P2. The forward motion of the robot is achieved by two active wheels in contact with the walls at P1 and P2 which is modeled here as rolling without slipping. The parameters which defines the geometry of the robot are L w h and position of the pinning point of arms. The velocity kinematics equations can be represented in the form of matrix. (J q + J x =0) The end-effector velocity vector is composed of 3 states, defining the positional and orientation velocities of the robot . The joint velocity vector is composed of 4 states, defining the angular and positional velocities of the arms and the wheels respectively. The analytical expressions for the serial and parallel Jacobians have been derived and they depend on the kinematic parameters of the system.
  • #7  The analytical expressions for the serial and parallel Jacobians have been derived and they depend on the kinematic parameters of the system.
  • #8 Two types of singularity exist in a parallel mechanism. A serial singularity exists when a motion in active joints result is no motion in the end-effector. Theoretically these are the equations which describes the serial singularity and these figures depicts the singular configuration of the robot.
  • #9 The second type of singularity is a parallel singularity which occurs when a motion in the end-effector exist even when the active joints are motionless. Parallel singularity exist when width of the payload is equal to zero and both arm overlap each other. Due to the kinematical constrains imposed on the robot a parallel singularity can never exist. Therefore only serial singularity is considered in our analysis and results.
  • #10  Four types of pipe-like structures are used as a test bench to determine the singularity-free workspace of the robot. The pipe structure has a constant width and the curvature of elbow pipes are inspired by industrial standards.
  • #11  The workspace is computed using a Discretization method. The test area is divided into multiple cells and a reference workspace region is formed. A direct search algorithm is performed on each cell which solves the robot’s inverse kinematic equation. Once a solution exist, a collision avoidance algorithm is run which examines the contact and intersection of all four corners of the module and the rigid arms of the robot with the inner boundaries. The final step of this analysis is to make sure the robot does not encounter a singular configuration at a given cell position. Kinematic conditioning index is used to quantify the proximity to a singularity.
  • #12  The workspace is computed using a Discretization method. The test area is divided into multiple cells and a reference workspace region is formed. A direct search algorithm is performed on each cell which solves the robot’s inverse kinematic equation. Once a solution exist, a collision avoidance algorithm is run which examines the contact and intersection of all four corners of the module and the rigid arms of the robot with the inner boundaries. The final step of this analysis is to make sure the robot does not encounter a singular configuration at a given cell position. Kinematic conditioning index is used to quantify the proximity to a singularity.
  • #13  The workspace is computed using a Discretization method. The test area is divided into multiple cells and a reference workspace region is formed. A direct search algorithm is performed on each cell which solves the robot’s inverse kinematic equation. Once a solution exist, a collision avoidance algorithm is run which examines the contact and intersection of all four corners of the module and the rigid arms of the robot with the inner boundaries. The final step of this analysis is to make sure the robot does not encounter a singular configuration at a given cell position. Kinematic conditioning index is used to quantify the proximity to a singularity.
  • #14  The workspace is computed using a Discretization method. The test area is divided into multiple cells and a reference workspace region is formed. A direct search algorithm is performed on each cell which solves the robot’s inverse kinematic equation. Once a solution exist, a collision avoidance algorithm is run which examines the contact and intersection of all four corners of the module and the rigid arms of the robot with the inner boundaries. The final step of this analysis is to make sure the robot does not encounter a singular configuration at a given cell position. Kinematic conditioning index is used to quantify the proximity to a singularity.
  • #15 The results of singularity-free workspace in a straight pipe is shown The workspace region is symmetric and resembles the boundary line of the straight pipe. The width of the workspace decreases as the robot is oriented from -15 to 15 The color coded map shows the variation of KCI on singularity-free workspace region. The center line depict the farthest configuration from singularity.
  • #16 Same consideration apply here in a 135 elbow pipe
  • #17 A constraint optimization of the geometric parameters is performed. The objective is to maximize the singularity-free workspace subjected to the following constraints Singularity Avoidance Collision Avoidance The contact constraint between the wheel and the boundary. And a resulting workspace which is not disjoint
  • #18 During the constraint optimization …the payload area denoted by w and h is kept constant and only the length of the arms and their position on the body is optimized. The singularity-free workspace of the robot is presented when the robot is oriented 0 degree and 15 degrees about the center line of the pipe.
  • #19 The converged optimal design parameters lead to a larger singularity-free workspace area. And The results are shown below. These converged values are independent of the initial parameters selected during the optimization process.
  • #20 These plots shows the converged value of parameter L. The plot on the left shows the value of the cost function versus the orientation angle of the robot for various values of L. The plot is symmetric about the orientation angle 0 due to the robot’s symmetric design. Moreover the cost function value reduces as the orientation angle diverts from 0 due to possible collision and singularity avoidance. The plot on the right , shows the average cost function versus parameter L .. Where the value of l=0.58 Wp maximizes the cost function .
  • #21 Similar analysis performed on a 135 degree elbow pipe gives the following converged values. These values are represented in figure below and a singularity-free workspace using these parameters is also shown on right
  • #22 When operating autonomously in a pipe-like environment, it is important to obtain a continuous path without singularities. A typically challenging maneuver is the one related to a sharp 90 degree angled elbow as illustrated in left Figure. A mobility analysis on this type of pipe-like structure shows a discontinuity in the workspace as shown in the figure to the right. The workspace is discontinuous due to the change in the orientation which in turn implies the constraint violation on the left arm. Furthermore, rolling without slipping constraint is also violated when the left arm collides to the pipe. The left figure shows a robot with one of the configuration leading to discontinuity in the workspace.
  • #23 We considered to tackle the critical scenario by augmenting the kinematics and introducing prismatic joints on the two arms of the robot. This modification adds additional two degrees of freedom on the robot, where the arms will be able to dynamically change their lengths and avoid collision between the wall as well as singular configurations.
  • #24 The velocity kinematics equations can be represented in the form of matrix. (J q + J x =0) The end-effector velocity vector is composed of 5 states, defining the positional and orientation velocities of the robot as well as the velocities of the prismatic joints. The joint velocity vector is composed of 4 states, defining the angular and positional velocities of the arms and the wheels respectively. The analytical expressions for the serial and parallel Jacobians have been derived and they depend on the kinematic parameters of the system.
  • #25  The forward motion of the robot inside the pipe is illustrated by path following control acting on center of mass (G) and orientation of the module. The optimal arm lengths are obtained by updating the state according to the gradient ascent which maximizes the singularity measure det(JxTJx).
  • #26 A simulation has been performed using Matlab which shows the path-following of the module and optimal control of the arm lengths. [Show the video] Here the robot is to follow the dotted center-line of the pipe and successfully pass through the sharp 90 degree elbow while avoiding collisions and singularities. A collision avoidance algorithm refrains all four corners of the module to come in contact with the inner wall of the pipe. Most importantly, it also avoids collision between the arms and the wall during the turn. Initially , the robot’s module starts following the line with constant speed and the arms quickly adjust themselves for optimality. As the robot centers itself in the pipe, the arms of the robot extracts to their maximum lengths. This indicates that as the length of the arms increases the robot moves far from a singular configuration. As the robot crosses the elbow the arms are dynamically adjusted to avoid singularities as well as collision between the left arm and the wall. In practice the speed of the robot should be reduced in this area for the robot to make adjustments accordingly.
  • #27 The figure on the left shows how the length of arms d1 and d2 vary as the robot follows the centerline of the pipe. These parameters were set to have an upper bound of 0.55 normalized with respect to the width of the pipe, to reflect the behavior of a prismatic joint. When the robot does not encounter any turn, the prismatic joint length tends to increase to the maximum as it leads the robot as far as possible from singular configuration. The figure on the right shows the value of the determinant of serial jacobian which is bounded between 0 and 1, while following the desired path. A sudden drop in the value of the index occurs when the robot tries to cross the elbow. It is noted that at this configuration the two arms are close to being normal to the pipe wall causing the robot to be closer to a singular configuration. The fluctuation after the 300th iteration is due to the length of the left arm being adjusted by the collision avoidance algorithm in order for the left wheel to remain in contact with the inner wall to maintain its forward motion.
  • #28 This slide shows a comparison of singularity-free workspace of mobile robot with rigid arms and with prismatic arms. It can be seen that with the augmented kinematics (prismatic arms) we are able to cross the elbow successfully and achieve a continuous singularity-free workspace.
  • #29  In summary, I have presented to you the singular configuration and singularity free workspace of the pipe-like structures. I have also optimized the geometric parameter which maximizes the singularity-free workspace. Moreover I have presented to you a critical scenario for a mobile robot which leads to a discontinuity in its workspace. We formulated and simulated a kinematic model to navigate singularity-free across the corner. A path following controller along with optimal control to maximize a performance index was proposed which measures the distance from singularity. Moreover a Collision Avoidance Algorithm was implemented to avoid collisions of the arms and the body of the robot with the inner wall of the pipe. And therefore the proposed solution lead to a continuous singularity-free workspace. Thank you for your time
  • #30 The following publications are produced during the work on this thesis
  • #37 Show initial design parameters Constrained because w and h are constants due to a need for required hardware area Show the workspace at theta=0
  • #38 Present the Converged values and respective workspace (theta=0) Current initial design parameters were very good estimate -The workspace looks the same at theta=0 ONLY - The fact that the converged values effect maximizes the workspace at overall when theta changes from -45 to 45 degrees.
  • #39 Present the Converged values and respective workspace (theta=0) Current initial design parameters were very good estimate -The workspace looks the same at theta=0 ONLY - The fact that the converged values effect maximizes the workspace at overall when theta changes from -45 to 45 degrees.