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Inverse Manipulator Kinematics

Inverse Manipulator Kinematics. Review. Direct Versus Inverse Kinematics. Direct (Forward) Kinematics Given: Joint angles and links geometry Compute: Position and orientation of the end effector relative to the base frame Inverse Kinematics

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Inverse Manipulator Kinematics

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  1. Inverse Manipulator Kinematics Review Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  2. Direct Versus Inverse Kinematics Direct (Forward) Kinematics Given:Joint angles and links geometry Compute: Position and orientation of the end effector relative to the base frame Inverse Kinematics Given:Position and orientation of the end effector relative to the base frame Compute: All possible sets ofjoint angles and links geometry which could be used to attain the given position and orientation of the end effetor Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  3. Central Topic - Inverse Manipulator Kinematics - Examples • Geometric Solution - Concept Decompose spatial geometry into several plane geometry Examples - RRR (3R) manipulators - Geometric Solution • Algebraic Solution - Concept Examples- PUMA 560 - Algebraic Solution Direct Kinematics Goal (Numeric values) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  4. Solvability • Existence of Solutions • Multiple Solutions • Method of solutions • Close form solution • Algebraic solution • Geometric solution • Numerical solutions Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  5. Solvability - Existence of Solution • For a solution to exist, must be in the workspace of the manipulator • Workspace - Definitions • Dexterous Workspace (DW): The subset of space in which the robot end effector can reach all orientation. • Reachable Workspace (RW): The subset of space in which the robot end effector can reach in at least 1 orientation • The Dexterous Workspace is a subset of the Reachable Workspace Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  6. Solvability - Existence of Solution - Workspace - 2R Example 1 - Reachable Workspace Dexterous Workspace Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  7. Solvability - Existence of Solution - Workspace - 2R Example 2 - Reachable Workspace NO Dexterous Workspace Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  8. Solvability - Existence of Solution - Workspace - 3RExample 3 - End Effector Rotation Reachable Workspace & Dexterous Workspace Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  9. Solvability - Multiple Solutions • Problem: The fact that a manipulator has multiple multiple solutions may cause problems because the system has to be able to choose one • Solution: Decision criteria • The closest (geometrically) - minimizing the amount that each joint is required to move • Note 1: input argument - present position of the manipulator • Note 2: Joint Weight - Moving small joints (wrist) instead of moving large joints (Shoulder & Elbow) • Obstacles exist in the workspace - avoiding collision Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  10. Solvability - Multiple Solutions • Multiple solutions are a common problem that can occur when solving inverse kinematics because the system has to be able to chose one • The number of solutions depends on the number of joints in the manipulator but is also a function of the links parameters • Example: The PUMA 560 can reach certain goals with 8 different (solutions) arm configurations • Four solutions are depicted • Four solutions are related to a “flipped” wrist Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  11. Solvability - Multiple Solutions - Number of Solutions • Task Definition - Position the end effector in a specific point in the plane (2D) • No. of DOF = No. of DOF of the task Number of solution: 2 (elbow up/down) • No. of DOF > No. of DOF of the task Number of solution: Self Motion - The robot can be moved without moving the the end effector from the goal Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  12. Solvability - Methods of Solutions • Solution (Inverse Kinematics)- A “solution” is the set of joint variables associated with an end effector’s desired position and orientation. • No general algorithms that lead to the solution of inverse kinematic equations. • Solution Strategies • Closed formSolutions - An analytic expression includes all solution sets. • Algebraic Solution - Trigonometric (Nonlinear) equations • Geometric Solution - Reduces the larger problem to a series of plane geometry problems. • Numerical Solutions - Iterative solutions will not be considered in this course. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  13. Solvability Robot - 6 DOF Single Series Chain Revolute & Prismatic Joints Non Real-Time Real-Time Numeric Solution Analytic Solution Close Form Solution Sufficient Condition Three adjacent axes (rotary or prismatic) must intersect Iterations Industrial Robots Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  14. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 1-7 Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  15. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 1 • Equation • Solution (Unique) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  16. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 2 • Equation • Solution • Two Solutions • Singularity at the Boundary Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  17. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 3 • Equation • Solution • Two Solutions apart • Singularity Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  18. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 4 • Equation Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  19. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 4 Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  20. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 4 • Solution • Two Solutions • For a solution to exist • No solution (outside of the workspace) • One solution (singularity) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  21. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 5 • Equation • Solution Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  22. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 6 • Equation • Solution • For an exiting solution (the determinant must be positive) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  23. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 7 • Transcendental equations are difficult to solve because they are a function of • Making the following substitutions yields an expression in terms of a single veritable , Using this substitutions, transcendental equation are converted into polynomial equation Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  24. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 7 • Transcendental equation • Substitute with the following equations • yields Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  25. Inverse Kinematics - Generalized Algebraic (Analytical) Solutions – Case 7 • Which is solved by the quadratic formula to be • Note • If u is complex there is no real solution to the original transcendental equation • If then Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  26. Inverse Manipulator Kinematics Geometric Approach Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  27. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 1/ • Given: • Manipulator Geometry • Goal Point Definition: The position of the wrist in space • Problem: What are the joint angles ( ) as a function of the goal (wrist position and orientation) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  28. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 2/ Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  29. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 3/ • The planar geometry - top view of the robot x3 yd L4 y0 L3 x2 z3 x0 xd z2 Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  30. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 4/ Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  31. L4 L3 r2 r1 L1+L2 zd z0 Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 5/ • The planar geometry - side view of the robot: • where Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  32. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 6/ • By Apply the law of cosines we get • Rearranging gives • and • Solving for we get • Where is defined above in terms of known parameters Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  33. L4 β L3 r2 α r1 Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 8/ • Finally we need to solve for • where Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  34. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 9/ • Based on the law of cosines we can solve for Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  35. Inverse Kinematics - 3D RRR (3R) - Geometric Solution - 10/ • Summary Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  36. Inverse Manipulator Kinematics Analytical Approach Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  37. Central Topic - Inverse Manipulator Kinematics - Examples • Algebraic Solution - Concept Examples- PUMA 560 - Algebraic Solution Direct Kinematics Goal (Numeric values) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  38. Solvability - PUMA 560 Given : PUMA 560 - 6 DOF, Solve: Total Number of Equations: 12 Independent Equations: 3 - Rotation Matrix 3 - Position Vector Type of Equations: Non-linear Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  39. Central Topic - Inverse Manipulator Kinematics - Examples • Geometric Solution - Concept Decompose spatial geometry into several plane geometry Examples - Planar RRR (3R) manipulators - Geometric Solution • Algebraic Solution - Concept Examples- PUMA 560 - Algebraic Solution Direct Kinematics Goal (Numeric values) Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  40. Inverse Kinematics - PUMA 560 - Algebraic Solution - 1/ • Given: • Direct Kinematics: The homogenous transformation from the base to the wrist • Goal Point Definition: The position and orientation of the wrist in space Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  41. Inverse Kinematics - PUMA 560 - Algebraic Solution - 2/ • Problem: What are the joint angles ( ) as a function of the wrist position and orientation ( or when is given as numeric values) Direct Kinematics Goal Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  42. Inverse Kinematics - PUMA 560 - Algebraic Solution - 3/ • Solution (General Technique): Multiplying each side of the direct kinematics equation by a an inverse transformation matrix for separating out variables in search of solvable equation • Put the dependence on on the left hand side of the equation by multiplying the direct kinematics eq. with gives Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  43. Inverse Kinematics - PUMA 560 - Algebraic Solution - 4/ • Put the dependence on on the left hand side of the equation by multiplying the direct kinematics eq. with gives Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  44. Inverse Kinematics - PUMA 560 - Algebraic Solution - 5/ Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  45. Inverse Kinematics - PUMA 560 - Algebraic Solution - 6/ Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  46. Inverse Kinematics - PUMA 560 - Algebraic Solution - 7/ • Solution for • Equating the (2,4) elements from both sides of the equation we have • To solve the equation of this form we make the trigonometric substitution Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  47. Inverse Kinematics - PUMA 560 - Algebraic Solution - 8/ • Substituting with we obtain • Using the difference of angles formula Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  48. Inverse Kinematics - PUMA 560 - Algebraic Solution - 9/ • Based on • and so • The solution for may be written • Note: we have found two possible solutions for corresponding to the +/- sign Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  49. Inverse Kinematics - PUMA 560 - Algebraic Solution - 10/ • Solution for • Equating the (1,4) element and (3,4) element • We obtain Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  50. Inverse Kinematics - PUMA 560 - Algebraic Solution - 11/ • If we square the following equations and add the resulting equations Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

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