Jacobian Matrix in Robotics

Jacobian Matrix in Robotics
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Jacobian matrix in robotics is crucial for differentiating the forward kinematics equations. It involves methods such as explicit derivation, iterative techniques, and recursive equations. The matrix columns map angular velocity contributions to linear and angular velocities of the end effector along various axes. Jacob Rosen, an instructor at UCLA, provides detailed explanations and insights into this essential concept.

  • Robotics
  • Jacobian Matrix
  • Forward Kinematics
  • Differentiation
  • Mechanical Engineering
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Uploaded on Mar 05, 2025 | 12 Views

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  1. Jacobian Explicit Method - Differentiation the Forward Kinematics Eqs. (Method No. 1) Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  2. Jacobian Matrix - Derivation Methods Explicit Method Iterative Methods Recursive Equations Velocity Propagation Base to EE (Method 2) Force/Torque Propagation EE to Base (Method 3) Differentiation the Forward Kinematics Eqs. (Method 1) Jacobian Matrix Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  3. Jacobian Explicit Form Overview 0 v ( ) = = 0 X J 0 0 v 1 x ( ) / 0 J v 2 y d 0 v = 3 3 z 0 x ( ) 0 J y 0 z N Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  4. Jacobian Matrix The meaning of each column (e.g. the first column) of the Jacobian matrix: J x 11 1 ( ) J J y 21 2 J z 31 = J 41 x ( ) J J 51 y J 61 z N The first column maps the contribution of the angular velocity of the first joint to the linear and angular velocities of the end effector along all the axis (x,y,z) Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  5. Jacobian Explicit Method Technique Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  6. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  7. Jacobian Explicit Method Rational Intuitive Explanation Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  8. Jacobian Matrix The meaning of each column of the Jacobian matrix: ( ) J J x 1 i 1 J y 2 i 2 J z 3 i = ( ) J J 4 i x J 5 i y J 6 i z N The i th column maps the contribution of the angular velocity of the i th joint to the linear and angular velocities of the end effector along all the axis (x,y,z) Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  9. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  10. Jacobian Explicit Form Linear Velocity Two Cases J The i th column of the Jacobian can be generated by holding all the joints fixed but the i th and actuating the i th at a unite velocity Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  11. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z i k R 6 i i i 0 i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  12. J Jacobian Explicit Form Angular Velocity ( ) J ( ) J 0 1 0 1 0 1 0 2 Z Z Z Z 1 n n n n 2 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  13. J Jacobian Explicit Form Angular Velocity Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  14. Jacobian Explicit Method Rational Geometrical / Analytical Explanation Based on Velocity Propagation Method Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  15. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  16. J Jacobian Explicit Form Linear Velocity f f f = + + + 1 1 1 y x x x 1 1 2 6 x x x = ( , , , , , ) y f x x x x x x 1 2 6 1 1 1 2 3 4 5 6 f f f = ( , , , , , ) y f x x x x x x = + + + 2 2 2 y x x x 2 2 1 2 3 4 5 6 2 1 2 6 x x x 1 2 6 = ( , , , , , ) y f x x x x x x 6 6 1 2 3 4 5 6 f f f = + + + 6 6 6 y x x x 6 1 2 6 x x x 1 2 6 F Y = (X ) F = Y X X F = = ) ( Y X J X X X Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  17. J Jacobian Explicit Form Linear Velocity f f f = + + + 1 1 1 y x x x 1 1 2 6 x x x 1 2 6 f f f 0 P = + + + 2 2 2 y x x x P = i = 0 q 6 q 2 1 2 6 x x x i d 6 i i 1 2 6 i i i f f f = + + + 6 6 6 y x x x 6 1 2 6 x x x 0 P 1 2 6 = 6 J vi i Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  18. Jacobian Explicit Form Linear Velocity Derivative J nP 0 The linear velocity of the end effector is . By the chain rule for differentiation 0 n P P = 0 n q n i q i i iq Where is the generalized notation for both the angle (revolute joint) and displacement (prismatic joint) = i d i q i Thus the i th column of which denoted as is given by ( ) J J i 0 P n q i = = 0 1 j q iq This expression is just the linear velocity of the end effector that would result if and all the others Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  19. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  20. Jacobian Matrix The meaning of each column of the Jacobian matrix: ( ) J J x 1 i 1 J y 2 i 2 J z 3 i = ( ) J J 4 i x J 5 i y J 6 i z N The i th column maps the contribution of the angular velocity of the i th joint to the linear and angular velocities of the end effector along all the axis (x,y,z) Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  21. Jacobian Explicit Form Linear Velocity Two Cases J The i th column of the Jacobian can be generated by holding all the joints fixed but the i th and actuating the i th at a unite velocity Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  22. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  23. Jacobian Explicit Form Linear Velocity Case 1 Revolute Joint J Freeze all DOF Except the i th DOF (Revolute ) = v r = 0 0 0 ( ) Z P P J 6 i i i vi i 0 * * * P 0 i 0 * * Z P 6 x x ix 0 * * * P 0 i 0 * * Z P = 0 6 6 y T = 0 y ix T 0 i * * * P 0 i 0 * * Z P 6 z z ix 0 0 0 1 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  24. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  25. Jacobian Explicit Form Linear Velocity Case 2- Prismatic Joint J All the joints are fixed except a single prismatic joint. The i th prismatic joint generates pure translation of the end effector The direction of the translation is parallel to the axis i Z 0?? 0?? 0?? 0 0 1 ? 0? 0? 0??= ?? ? = ?? = ?? ? ? ? 0?? 0?? 0?? 0 ? 0? = ? ? 0 0 1 ? Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  26. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  27. J Jacobian Explicit Form Angular Velocity 0 + + = + 1 1 i i i i 0 R + 1 i i + 1 i 0 0 For i=0 = + 1 1 0 0 0 R 1 0 1 1 0 0 0 For i=1 = + = + 2 2 1 1 2 1 0 0 0 R R 2 1 2 1 2 0 2 1 2 0 0 0 0 0 0 = + = + + = + + 3 3 2 2 3 2 2 1 3 1 3 2 0 0 0 0 0 0 0 R R R R R For i=2 3 2 3 1 2 3 1 2 3 3 1 2 3 1 2 3 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  28. J Jacobian Explicit Form Angular Velocity I 0 0 0 0 0 = + = + + + 1 n n 1 n n 1 n 2 n 3 n n 0 0 0 0 0 R R R R R For i=n-1 1 1 2 3 n n n n n 3 1 2 3 n 0 Multiply both side of the equations by R n 0 0 0 0 0 = + = + + + + 0 0 1 0 0 0 0 n n 1 n n 1 n 2 n 3 n n 0 0 0 0 0 R R R R R R R R R R R 1 1 2 3 n n n n n n n n n n n 3 1 2 3 n Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  29. J Jacobian Explicit Form Angular Velocity 0 0 0 0 0 = + = + + + + 0 0 1 0 0 0 0 n n 1 n n 1 n 2 n 3 n n 0 0 0 0 0 R R R R R R R R R R R 1 1 2 3 n n n n n n n n n n n 3 1 2 3 n 0 0 0 0 = = + + + + 0 0 1 0 2 0 3 0 0 0 0 0 R R R R 1 2 3 n n n 1 2 3 n 0 n n n = i = i = i k i i = = = 0 0 0 0 0 R R Z n i i i i i i 1 1 1 i Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  30. J Jacobian Explicit Form Angular Velocity ( ) J ( ) J 0 1 0 1 0 1 0 2 Z Z Z Z 1 n n n n 2 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  31. J Jacobian Explicit Form Angular Velocity Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  32. 3R Example Explicit Methods Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  33. Angular and Linear Velocities - 3R Robot - Example For the manipulator shown in the figure, compute the angular and linear velocity of the tool frame relative to the base frame expressed in the tool frame (that is, calculate and ). 4 4v 4 4 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  34. Angular and Linear Velocities - 3R Robot - Example Frame attachment Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  35. Angular and Linear Velocities - 3R Robot - Example DH Parameters i 1 2 3 4 i 0 90 0 0 i 1 0 id 0 0 0 0 ia 0 L1 L2 L3 1 1 2 3 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  36. Angular and Linear Velocities - 3R Robot - Example From the DH parameter table, we can specify the homogeneous transform matrix for each adjacent link pair: ?1 ?1 0 0 ?2 0 ?2 0 ?3 ?3 0 0 1 0 0 0 ?1 ?1 0 0 ?2 0 ?2 0 ?3 ?3 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ??? ??? ?????? 1 ?????? 1 0 0 ?? 1 0? = ?????? 1 ?????? 1 0 ??? 1 ??? 1 0 ??? 1?? ??? 1? 1 1 ? 1? = ? 0 ?1 0 0 1 1 0 0 0 0 1 0 ?3 0 0 1 1? = 2 ?2 0 0 1 2? = 3 0 0 1 0 3? = 4

  37. Jacobian Explicit Form i Revolute Joint Prismatic Joint ( ) J 0 P 0 P 0 i 0 * * Z P 6 6 x ix OR id 0 i 0 * * Z P i = 0 y ix T OR i 0 i 0 * * Z P z ix 0 i 0 0 ( ) Z P P 0 0 0 1 0 Z 6 i i 0 * * * P ( ) 6 x J 0 * * * P = 0 6 6 y T 0 = * * * P 0 0 = Z 1 6 z i 0 0 0 1 Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

  38. Angular and Linear Velocities - 3R Robot - Example The homogeneous transform matrix from frame 0 to each one of the joints (1,2,3,4) ?1 ?1 0 0 ?1 ?1 0 0 0 0 1 0 0 0 0 1 0? = 1 ?1 ?1 0 0 ?1 ?1 0 0 0 0 1 0 0 0 0 1 ?2 0 ?2 0 ?2 0 ?2 0 0 ?1 0 0 1 ?1?2 ?1?2 ?2 0 ?1?1 ?1?2 ?2 0 ?1 ?1 0 0 ?1?1 ?1?1 0 1 1 0 0 0? =1 0?2 1? = = 2 ?1?23 ?1?23 ?23 0 ?1?23 ?1?23 ?23 0 ?1 ?1 0 0 ?1(?1 + ?2?2) ?1(?1 + ?2?2) ?2?2 1 ?1 ?1 0 0 ?1 ?1 0 0 0 0 1 0 0 0 0 1 ?2 0 ?2 0 ?2 0 ?2 0 0 ?1 0 0 1 ?3 ?3 0 0 ?3 ?3 0 0 0 0 1 0 ?2 0 0 1 1 0 0 0? =1 0?2 1?3 2? = = 3 ?1?23 ?1?23 ?23 0 ?1?23 ?1?23 ?23 0 ?1 ?1 0 0 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 1 ?1 ?1 0 0 ?1 ?1 0 0 0 0 1 0 0 0 0 1 ?2 0 ?2 0 ?2 0 ?2 0 0 ?1 0 0 1 ?3 ?3 0 0 ?3 ?3 0 0 0 0 1 0 ?2 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 ?3 0 0 1 1 0 0 0? =1 0?2 1?3 2?4 3? = = 4

  39. Angular and Linear Velocities - 3R Robot - Example Expressing the columns of ?? . Since all the joints are revolute joints i.e. ? = 1 and the last frame is frame 4 the generic expression for the columns of ?? is ? ?? 0? ( 0??) for ? = 1,2,3,4 0?4 ??? ? = 1 ? 0 ? 0 ? 1 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ?1(?1 + ?3?23 + ?2?2 ?1(?1 + ?3?23 + ?2?2 0 0 0 0 0 0 1 = 0? 0?4 0?1 = = 1 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ??? ? = 2 ? ? ? 0 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ?1 (?3?23 + ?2?2) ?1(?3?23 + ?2?2) ?3?23 + ?2?2 ?1 ?1 0 ?1?1 ?1?1 0 0? 0?4 0?2 = = ?1 ?1 = 2 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ??? ? = 3 ? ? ? ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ?1(?1 + ?2?2) ?1(?1 + ?2?2) ?2?2 ?1 ?1 0 ?3?1?23 ?3?1?23 ?3?23 0? 0?4 0?3 = = ?1(?1 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?2?2 = 3 ?3?23 + ?2?2 ??? ? = 4 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ?1(?1 + ?3?23 + ?2?2) ?1(?1 + ?3?23 + ?2?2) ?3?23 + ?2?2 ?1 ?1 0 ? ? ? 0 0 0 0 0 0? 0?4 0?4 = = = ?1 0 ?1 0 4

  40. Angular and Linear Velocities - 3R Robot - Example ?? Expressing the columns of ?? . Since all the joints are revolute joints i.e. ? = 1 and the last frame is frame 4 the generic expression for the columns of ?? is ? 0? for ? = 1,2,3,4 0 0 1 0? = ??? ? = 1 1 ?1 ?1 0 0? = ??? ? = 2 2 ?1 ?1 0 0? = ??? ? = 3 3 ?1 ?1 0 0? = ??? ? = 4 4

  41. Angular and Linear Velocities - 3R Robot - Example Compiling the results into a single matrix ?1(?1 + ?3?23 + ?2?2 ?1(?1 + ?3?23 + ?2?2 0 0 0 1 ?1 (?3?23 + ?2?2) ?1(?3?23 + ?2?2) ?3?23 + ?2?2 ?1 ?1 0 ?3?1?23 ?3?1?23 ?3?23 ?1 ?1 0 ? =

  42. Angular and Linear Velocities - 3R Robot - Example With the end results mapping the joint space velocities to task space velocities ? ? ? ?? ?? ?? ?1(?1 + ?3?23 + ?2?2 ?1(?1 + ?3?23 + ?2?2 0 0 0 1 ?1 (?3?23 + ?2?2) ?1(?3?23 + ?2?2) ?3?23 + ?2?2 ?1 ?1 0 ?3?1?23 ?3?1?23 ?3?23 ?1 ?1 0 ?1 ?2 ?3 = Note 1- Size The size of the matrix is 6x3. Can we control all six velocities of the end effector with only three DOF (Hint: The dimension of the Jacobian matrix will have to be reduced) Note 2 Frame of Reference The explicit method produce an expression of the Jacobian matrix in frame zero

  43. Jacobian Methods of Derivation & the Corresponding Reference Frame Summary Method Jacobian Matrix Reference Frame Transformation to Base Frame (Frame 0) Explicit (Diff. the Forward Kinematic Eq.) None 0 J N Iterative Velocity Eq. Transform Method 1: = 0 0 N v R v N N N = R 0 0 N N N N NJ Transform Method 2: N 0 0 R 0 ( ) ( ) = 0 N N J J N N 0 R N Iterative Force Eq. Transpose = N N T N T [ ] J J N NJ T N Transform 0 0 R 0 ( ) ( ) = 0 N N J J N N 0 R N Instructor: Jacob Rosen Advanced Robotic - Department of Mechanical & Aerospace Engineering - UCLA

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