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Chapter 3 Manipulator Kinematics




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Chapter 3 Manipulator Kinematics




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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.1 Forward kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.2 Inverse Kinematics
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Chapter 3 Manipulator Kinematics

               3.3 Manipulator Jacobian
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2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 03 Manipulator Kinematics

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  1. 1. Chapter 3 Manipulator Kinematics 1 Chapter Lecture Notes for Manipulator Kinematics A Geometrical Introduction to Forward kinematics Robotics and Manipulation Inverse Kinematics Manipulator Richard Murray and Zexiang Li and Shankar S. Sastry Jacobian CRC Press Redundant Manipulators Reference Zexiang Li and Yuanqing Wu ECE, Hong Kong University of Science & Technology July ,
  2. 2. Chapter 3 Manipulator Kinematics 2 Chapter Manipulator Kinematics Chapter Manipulator Kinematics Forward kinematics Forward kinematics Inverse Kinematics Inverse Kinematics Manipulator Jacobian Manipulator Jacobian Redundant Manipulators Reference Redundant Manipulators Reference
  3. 3. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 3 Chapter Manipulator Kinematics Forward kinematics Inverse Play/Pause Stop Kinematics (a) Adept Cobra i600 (SCARA) Manipulator Figure 3.1 Jacobian Redundant Manipulators Reference
  4. 4. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 3 Chapter Manipulator Kinematics Forward kinematics Inverse Play/Pause Stop Kinematics (a) Adept Cobra i600 (SCARA) Manipulator Figure 3.1 Jacobian Redundant Lower Pair Joints: Manipulators revolute joint S ↦ SO( ) prismatic joint R ↦ T( ) R T Reference Li Li+ Li Li+
  5. 5. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 3 Chapter Manipulator Kinematics Forward kinematics Inverse Play/Pause Stop Kinematics (a) Adept Cobra i600 (SCARA) (b) Forward kinematics of SCARA Manipulator Figure 3.1 Jacobian Redundant Lower Pair Joints: Manipulators revolute joint S ↦ SO( ) prismatic joint R ↦ T( ) R T Reference Li Li+ Li Li+ Forward kinematics: L base: stationary
  6. 6. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 3 Chapter Manipulator Kinematics Forward kinematics Inverse Play/Pause Stop Kinematics (a) Adept Cobra i600 (SCARA) (b) Forward kinematics of SCARA Manipulator Figure 3.1 Jacobian Redundant Lower Pair Joints: Manipulators revolute joint S ↦ SO( ) prismatic joint R ↦ T( ) R T Reference Li Li+ Li Li+ Forward kinematics: L joint L base: stationary Link : first movable link
  7. 7. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 3 Chapter Manipulator Kinematics Forward kinematics Inverse Play/Pause Stop Kinematics (a) Adept Cobra i600 (SCARA) (b) Forward kinematics of SCARA Manipulator Figure 3.1 Jacobian Redundant Lower Pair Joints: Manipulators revolute joint S ↦ SO( ) prismatic joint R ↦ T( ) R T Reference Li Li+ Li Li+ Forward kinematics: L joint L joint n Ln base: stationary Link : first movable link Link n: end-effector
  8. 8. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 4 ◻ Joint space: Revolute joint: S , θ i ∈ S or θ i ∈ [−π, π] Chapter Prismatic joint: R Q ∶ S × ⋯ × S ×R × ⋯ × R Manipulator Kinematics Joint space: Forward kinematics no. of R joint no. of P joint Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  9. 9. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 4 ◻ Joint space: Revolute joint: S , θ i ∈ S or θ i ∈ [−π, π] Chapter Prismatic joint: R Q ∶ S × ⋯ × S ×R × ⋯ × R Manipulator Kinematics Joint space: Forward kinematics no. of R joint no. of P joint Q ∶S ×S ×S ×R Inverse Adept Q = Γ ∶S ×⋯×S Kinematics Manipulator Elbow Jacobian Redundant Manipulators Reference
  10. 10. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 4 ◻ Joint space: Revolute joint: S , θ i ∈ S or θ i ∈ [−π, π] Chapter Prismatic joint: R Q ∶ S × ⋯ × S ×R × ⋯ × R Manipulator Kinematics Joint space: Forward kinematics no. of R joint no. of P joint Q ∶S ×S ×S ×R Inverse Adept Q = Γ ∶S ×⋯×S Kinematics Manipulator Elbow Jacobian Redundant θ = ( , ,..., ) ∈ Q Manipulators Reference (nominal) joint config: gst ( ) ∈ SE( ) Reference Reference (nominal) end-effector config:
  11. 11. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 4 ◻ Joint space: Revolute joint: S , θ i ∈ S or θ i ∈ [−π, π] Chapter Prismatic joint: R Q ∶ S × ⋯ × S ×R × ⋯ × R Manipulator Kinematics Joint space: Forward kinematics no. of R joint no. of P joint Q ∶S ×S ×S ×R Inverse Adept Q = Γ ∶S ×⋯×S Kinematics Manipulator Elbow Jacobian Redundant θ = ( , ,..., ) ∈ Q Manipulators Reference (nominal) joint config: gst ( ) ∈ SE( ) Reference Reference (nominal) end-effector config: Arbitrary configuration gst (θ): gst ∶ θ ∈ Q ↦ gst (θ) ∈ SE( )
  12. 12. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 5 ◻ Classical Approach: Chapter gst (θ , θ ) = gst (θ ) ⋅ gl l ⋅ gl t Manipulator Kinematics Disadvantage: A coordinate frame needed for each link Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  13. 13. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 5 ◻ Classical Approach: Chapter gst (θ , θ ) = gst (θ ) ⋅ gl l ⋅ gl t Manipulator Kinematics Disadvantage: A coordinate frame needed for each link ◻ The product of exponentials formula: Forward kinematics Inverse Kinematics Consider Fig 3.2. Manipulator Jacobian θ2 Redundant Manipulators θ1 Reference L2 T L1 S Figure 3.2: A two degree of freedom manipulator (Continues next slide)
  14. 14. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 6 Step 1: Rotating about ω by θ ξ = −ωω× q Chapter Manipulator Kinematics gst (θ ) = e ξ θ ⋅ gst ( ) ˆ Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  15. 15. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 6 Step 1: Rotating about ω by θ ξ = −ωω× q Chapter Manipulator Kinematics gst (θ ) = e ξ θ ⋅ gst ( ) ˆ Forward kinematics Inverse Step 2: Rotating about ω by θ ξ = −ωω× q Kinematics gst (θ , θ ) = e ξ θ ⋅ e ξ θ ⋅ gst ( ) Manipulator Jacobian ˆ ˆ Redundant Manipulators o set Reference θ ∶ ( , ) ↦ ( , θ ) ↦ (θ , θ ) (Continues next slide)
  16. 16. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 7 What if another route is taken? Chapter θ ∶ ( , ) ↦ (θ , ) ↦ (θ , θ ) Manipulator Kinematics Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  17. 17. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 7 What if another route is taken? Chapter θ ∶ ( , ) ↦ (θ , ) ↦ (θ , θ ) Manipulator Kinematics Forward kinematics Step 1: Rotating about ω by θ ξ = −ωω× q Inverse Kinematics gst (θ ) = e ξ θ ⋅ gst ( ) Manipulator ˆ Jacobian Redundant Manipulators Reference Step 2: Rotating about ω′ by θ Let e ξ θ = R p ˆ ω′ = R ⋅ ω q′ = p + R ⋅ q (Continues next slide)
  18. 18. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 8 −ω′ × q′ −R ω RT (p + R q ) ξ′ = = ˆ Chapter Manipulator Kinematics ω′ Rω Forward = R pRˆ −ω × q = Ad ˆ ⋅ ξ ⇒ ξ ′ = eξ θ ⋅ ξ ⋅ e− ξ θ ˆ ˆ ˆ ˆ kinematics R ω eξ θ Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  19. 19. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 8 −ω′ × q′ −R ω RT (p + R q ) ξ′ = = ˆ Chapter Manipulator Kinematics ω′ Rω Forward = R pRˆ −ω × q = Ad ˆ ⋅ ξ ⇒ ξ ′ = eξ θ ⋅ ξ ⋅ e− ξ θ ˆ ˆ ˆ ˆ kinematics R ω eξ θ Inverse Kinematics Manipulator gst (θ , θ ) = eξ ⋅ eξ θ ⋅ gst ( ) Jacobian ˆ′ θ ˆ Redundant = ee ⋅ eξ θ ⋅ gst ( ) ˆ ˆ θ Manipulators ξ θ ⋅ ξ θ ⋅e− ξ ˆ ˆ Reference = eξ θ ⋅ eξ ⋅ e− ξ θ ⋅ eξ θ ⋅ gst ( ) ˆ ˆθ ˆ ˆ = eξ θ ⋅ eξ ⋅ gst ( ) ˆ ˆθ Independent of the route taken
  20. 20. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 9 ◻ Procedure for forward kinematic map: Chapter Identify a nominal configuration: Manipulator Θ = (θ , . . . , θ n ) = , gst ( ) ≜ gst (θ , . . . , θ n ) Kinematics Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  21. 21. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 9 ◻ Procedure for forward kinematic map: Chapter Identify a nominal configuration: Manipulator Θ = (θ , . . . , θ n ) = , gst ( ) ≜ gst (θ , . . . , θ n ) Kinematics Forward kinematics Inverse Kinematics Simplification of forward kinematics mapping: ξi = −ωωi qi Manipulator × Jacobian Revolute joint: Redundant Manipulators Choose qi s.t. ξi is simple. vi ξi = Reference Prismatic joint:
  22. 22. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 9 ◻ Procedure for forward kinematic map: Chapter Identify a nominal configuration: Manipulator Θ = (θ , . . . , θ n ) = , gst ( ) ≜ gst (θ , . . . , θ n ) Kinematics Forward kinematics Inverse Kinematics Simplification of forward kinematics mapping: ξi = −ωωi qi Manipulator × Jacobian Revolute joint: Redundant Manipulators Choose qi s.t. ξi is simple. vi ξi = Reference Prismatic joint: Write gst (θ) = e ξ θ . . . e ξn θ n ⋅ gst ( ) (product of exponential ˆ ˆ mapping)
  23. 23. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 10 Example: SCARA manipulator θ3 ⎡ ⎤ ⎢ I ⎥ Chapter θ2 ⎢ l +l ⎥ Manipulator gst ( ) = ⎢ θ1 l2 ⎥ θ4 Kinematics ⎢ ⎥ l1 l ⎣ ⎦ Forward kinematics z T ω =ω =ω = l0 q1 S q2 q3 Inverse y Kinematics x Manipulator Jacobian Figure 3.3 Redundant Manipulators Reference
  24. 24. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 10 Example: SCARA manipulator θ3 ⎡ ⎤ ⎢ I ⎥ Chapter θ2 ⎢ l +l ⎥ Manipulator gst ( ) = ⎢ θ1 l2 ⎥ θ4 Kinematics ⎢ ⎥ l1 l ⎣ ⎦ Forward kinematics z T ω =ω =ω = l0 q1 S q2 q3 Inverse y Kinematics x Manipulator Jacobian Figure 3.3 q = ,q = ,q = l +l Redundant Manipulators l Reference
  25. 25. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 10 Example: SCARA manipulator θ3 ⎡ ⎤ ⎢ I ⎥ Chapter θ2 ⎢ l +l ⎥ Manipulator gst ( ) = ⎢ θ1 l2 ⎥ θ4 Kinematics ⎢ ⎥ l1 l ⎣ ⎦ Forward kinematics z T ω =ω =ω = l0 q1 S q2 q3 Inverse y Kinematics x Manipulator Jacobian Figure 3.3 q = ,q = ,q = l +l Redundant Manipulators l Reference ⎡ ⎤ ⎡ l ⎤ ⎡ l +l ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⇒ξ =⎢ ⎥, ξ = ⎢ ⎥, ξ = ⎢ ⎥, ξ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (Continues next slide)
  26. 26. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 11 gst (θ) = eξ θ ⋅ eξ ⋅ eξ ⋅ eξ ⋅ gst ( ) = R(θ) p(θ) ˆ ˆ θ ˆθ ˆ θ Chapter Manipulator Kinematics Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  27. 27. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 11 gst (θ) = eξ θ ⋅ eξ ⋅ eξ ⋅ eξ ⋅ gst ( ) = R(θ) p(θ) ˆ ˆ θ ˆθ ˆ θ Chapter ⎡ c −s ⎤ ⎡ ls ⎤ ⎥ ξ θ ⎢ c −s Manipulator ⎢ l s ⎥, =⎢ s ⎥, e ˆ = ⎢ s ⎥ Kinematics ⎢ ⎥ ⎢ ⎥ e ˆ ξθ c c ⎢ ⎥ ⎢ ⎥ Forward ⎣ ⎦ ⎣ ⎦ kinematics ⎡ (l s )s ⎤ ⎢ c −s ⎥ ˆ ⎡ ⎤ Inverse ⎢ s (l s )v ⎥, eξ θ = ⎢ I ⎥ Kinematics =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ˆ c ⎢ ⎥ ξ θ e ⎢ ⎥ Manipulator ⎢ ⎥ θ ⎣ ⎦ Jacobian Redundant ⎣ ⎦ Manipulators Reference
  28. 28. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 11 gst (θ) = eξ θ ⋅ eξ θ ⋅ eξ θ ⋅ eξ θ ⋅ gst ( ) = R(θ) p(θ) ˆ ˆ ˆ ˆ Chapter ⎡ c −s ⎤ ⎡ ls ⎤ ⎥ ξ θ ⎢ c −s Manipulator ⎢ s l s ⎥, ⎢ ⎥, e ˆ = ⎢ s ⎥ Kinematics =⎢ c ⎥ ⎢ c ⎥ ˆ ξθ e ⎢ ⎥ ⎢ ⎥ Forward ⎣ ⎦ ⎣ ⎦ kinematics ⎡ (l s )s ⎤ ⎢ c −s ⎥ ˆ ⎡ ⎤ Inverse ⎢ s (l s )v ⎥, eξ θ = ⎢ I ⎥ Kinematics =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ˆ c ⎢ ⎥ ξ θ e ⎢ ⎥ Manipulator ⎢ ⎥ θ ⎣ ⎦ Jacobian ⎣ ⎦ ⎡ −s −l s − l s ⎤ Redundant ⎢ c ⎥ ⎢ l c +l c ⎥ Manipulators gst (θ) = ⎢ s c ⎥ ⎢ l +θ ⎥ Reference ⎢ ⎥ ⎣ ⎦ in which, c = cos(θ + θ + θ ) and c = cos(θ + θ ).
  29. 29. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 12 Example: Elbow manipulator Chapter Manipulator ξ1 ξ4 ξ2 ξ3 ξ5 ⎡ ⎤ Kinematics ⎢ I l +l ⎥ l1 l2 ⎢ ⎥ Forward q1 q2 gst ( ) = ⎢ ξ6 ⎥ kinematics T ⎢ l ⎥ ⎣ ⎦ Inverse Kinematics l0 Manipulator S Jacobian Redundant Figure 3.4 ⎡ ⎤ ⎡ ⎤ Manipulators ⎢ − × ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Reference ⎢ ⎥ ⎢ ⎥ ξ =⎢ l ⎥=⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (Continues next slide)
  30. 30. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 13 ⎡ − ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ − × ⎥ ⎢ −l ⎥ ⎢ −l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ l ⎥ ξ =⎢ − l ⎥ = ⎢ − ⎥, ξ = ⎢ − ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Chapter ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Manipulator ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Kinematics ⎡ l +l ⎤ ⎡ ⎤ ⎡ −l ⎤ ⎢ ⎥ ⎢ −l ⎥ ⎢ ⎥ Forward ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ kinematics ⎢ ⎥ ⎢ l +l ⎥ ⎢ ⎥ ξ =⎢ ⎥, ξ = ⎢ − ⎥, ξ = ⎢ ⎥ Inverse ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Kinematics ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Manipulator Jacobian Redundant → gst (θ , . . . θ ) = eξ θ . . . eξ ⋅ gst ( ) = Manipulators ˆ ˆ θ R(θ) p(θ) Reference ⎡ −s (l c + l c ) ⎤ ⎢ ⎥ p(θ) = ⎢ c (l c + l c ) ⎥ , R(θ) = [rij ] ⎢ l −l s −l s ⎥ ⎣ ⎦ (Continues next slide)
  31. 31. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 14 in which, r = c (c c − s c s ) + s (s s c + s c c s + c s s ) Chapter r = −c (s c c + c s ) + s s s r = c (−c s s − (c c s + c s )s ) + (c c − c s s )s Manipulator Kinematics r = c (c s + c c s ) − (c c s + (c c c − s s )s )s Forward kinematics r = c (c c c − s s ) − c s s Inverse Kinematics Manipulator Jacobian r = c (c c s + (c c c − s s )s ) + (c s + c c s )s Redundant r = −(c s s ) − (c c − c s s )s r = −(c c s ) − c s Manipulators Reference r = c (c c − c s s ) − s s s
  32. 32. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 14 in which, r = c (c c − s c s ) + s (s s c + s c c s + c s s ) Chapter r = −c (s c c + c s ) + s s s r = c (−c s s − (c c s + c s )s ) + (c c − c s s )s Manipulator Kinematics r = c (c s + c c s ) − (c c s + (c c c − s s )s )s Forward kinematics r = c (c c c − s s ) − c s s Inverse Kinematics Manipulator Jacobian r = c (c c s + (c c c − s s )s ) + (c s + c c s )s Redundant r = −(c s s ) − (c c − c s s )s r = −(c c s ) − c s Manipulators Reference r = c (c c − c s s ) − s s s Simplify forward Kinematics Map: Choose base frame or ref. Config. s.t. gst ( ) = I
  33. 33. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 15 ◻ Manipulator Workspace: Chapter Manipulator W = {gst (θ) ∀θ ∈ Q} ⊂ SE( ) Kinematics Forward Reachable Workspace: kinematics Inverse Kinematics WR = {p(θ) ∀θ ∈ Q} ⊂ R Manipulator Jacobian Redundant Manipulators Reference
  34. 34. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 15 ◻ Manipulator Workspace: Chapter Manipulator W = {gst (θ) ∀θ ∈ Q} ⊂ SE( ) Kinematics Forward Reachable Workspace: kinematics Inverse Kinematics WR = {p(θ) ∀θ ∈ Q} ⊂ R Manipulator Jacobian Redundant Dextrous Workspace: Manipulators WD = {p ∈ R ∀R ∈ SO( ), ∃θ, gst (θ) = (p, R)} Reference
  35. 35. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 16 Example: A planar serial 3-bar linkage Chapter (a) Workspace calculation: θ3 g = (x, y, ϕ) θ2 Manipulator l2 l3 x =l c +l c +l c Kinematics y =l s +l s +l s θ1 l1 Forward kinematics ϕ=θ +θ +θ (a) (b) Inverse Kinematics (b) Construction of Workspace: Manipulator Jacobian (c) Reachable Workspace: l1 l2 l3 2l3 2l3 Redundant (d) Dextrous Workspace: Manipulators (c) (d) Reference Figure 3.5
  36. 36. Chapter 3 Manipulator Kinematics 3.1 Forward kinematics 16 Example: A planar serial 3-bar linkage Chapter (a) Workspace calculation: θ3 g = (x, y, ϕ) θ2 Manipulator l2 l3 x =l c +l c +l c Kinematics y =l s +l s +l s θ1 l1 Forward kinematics ϕ=θ +θ +θ (a) (b) Inverse Kinematics (b) Construction of Workspace: Manipulator Jacobian (c) Reachable Workspace: l1 l2 l3 2l3 2l3 Redundant (d) Dextrous Workspace: Manipulators (c) (d) Reference Figure 3.5 ◻ R manipulator with maximum workspace (Paden): Elbow manipulator and its kinematics inverse. † End of Section †
  37. 37. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 17 Definition: Inverse kinematics Given g ∈ SE( ), find θ ∈ Q s.t. gst (θ) = g, where gst ∶ Q ↦ SE( ) Chapter Manipulator Kinematics Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  38. 38. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 17 Definition: Inverse kinematics Given g ∈ SE( ), find θ ∈ Q s.t. gst (θ) = g, where gst ∶ Q ↦ SE( ) Chapter Manipulator Kinematics Forward kinematics Example: A planar example Inverse Kinematics (x, y) (x, y) Manipulator l2 Jacobian θ2 θ2 r Redundant y α Manipulators l1 β Reference θ1 φ θ1 B x (a) (b) Figure 3.6 x = l cos θ + l cos (θ + θ ) y = l sin θ + l sin (θ + θ ) Given (x, y), solve for (θ , θ ).
  39. 39. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 18 Review: Chapter Polar Coordinates: (r, ϕ), r = x +y Manipulator Kinematics Forward Law of cosines: θ = π ± α, α = cos− kinematics l +l −r Inverse Flip solution: π + α Kinematics ll Manipulator r +l −l Jacobian Redundant Manipulators θ = atan (y, x) ± β, β = cos− lr Reference
  40. 40. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 18 Review: Chapter Polar Coordinates: (r, ϕ), r = x +y Manipulator Kinematics Forward Law of cosines: θ = π ± α, α = cos− kinematics l +l −r Inverse Flip solution: π + α Kinematics ll Manipulator r +l −l Jacobian Redundant Manipulators θ = atan (y, x) ± β, β = cos− lr Reference Hight Lights: Subproblems Each has zero, one or two solutions!
  41. 41. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 19 ◻ Paden-Kahan Subproblems: Chapter Subproblem 1: Rotation about a single axis Manipulator Kinematics Let ξ be a zero-pitch twist, with unit magnitude and two points p, q ∈ R . Find θ s.t. eξθ p = q Forward kinematics ˆ Inverse Kinematics Manipulator p Jacobian u Redundant u′ Manipulators r θ ξ θ Reference v′ v q (a) (b) Figure 3.6
  42. 42. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 19 ◻ Paden-Kahan Subproblems: Chapter Subproblem 1: Rotation about a single axis Manipulator Kinematics Let ξ be a zero-pitch twist, with unit magnitude and two points p, q ∈ R . Find θ s.t. eξθ p = q Forward kinematics ˆ Inverse Kinematics Manipulator p Jacobian u Redundant u′ Manipulators r θ ξ θ Reference v′ v q (a) (b) Figure 3.6 Solution: Let r ∈ l ξ , define u = p − r, v = q − r, eξθ r = r ˆ (Continues next slide)
  43. 43. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 20 Moreover, Chapter Manipulator ⇒ eξθ p = q ⇒ eξθ (p − r) = q − r ⇒ ˆ ˆ eωθ ˆ ∗ u = v Kinematics u v w u=w v Forward ⇒ eωθ u = v kinematics T T ˆ Inverse Kinematics u = v Manipulator Jacobian p Redundant Manipulators u u′ Reference r θ ξ θ v′ v q (a) (b) Figure 3.6 (Continues next slide)
  44. 44. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 21 u′ = (I − ωωT )u, v′ = (I − ωωT )v u′ = v′ Chapter Manipulator ωT u = ωT v Kinematics The solution exists only if Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  45. 45. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 21 u′ = (I − ωωT )u, v′ = (I − ωωT )v u′ = v′ Chapter Manipulator ωT u = ωT v Kinematics The solution exists only if Forward kinematics If u′ ≠ , then Inverse Kinematics Manipulator u′ × v′ = ω sin θ u′ v′ Jacobian Redundant Manipulators u′ ⋅ v′ = cos θ u′ v′ Reference ⇒ θ = atan (ωT (u′ × v′ ), u′T v′ )
  46. 46. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 21 u′ = (I − ωωT )u, v′ = (I − ωωT )v u′ = v′ Chapter Manipulator ωT u = ωT v Kinematics The solution exists only if Forward kinematics If u′ ≠ , then Inverse Kinematics Manipulator u′ × v′ = ω sin θ u′ v′ Jacobian Redundant Manipulators u′ ⋅ v′ = cos θ u′ v′ Reference ⇒ θ = atan (ωT (u′ × v′ ), u′T v′ ) If u′ = , ⇒ Infinite number of solutions!
  47. 47. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 22 Subproblem 2: Rotation about two subsequent axes Chapter Let ξ and ξ be two zero-pitch, unit magnitude twists, with intersecting axes, and p, q ∈ R . find θ and θ s.t. Manipulator Kinematics eξ θ eξ θ p = q. Forward ˆ ˆ kinematics Inverse Kinematics p ξ2 Manipulator Jacobian θ2 Redundant Manipulators c Reference r θ1 ξ1 q Figure 3.6 (Continues next slide)
  48. 48. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 23 Solution: If two axes of ξ and ξ coincide, then we get: Subproblem 1: θ + θ = θ Chapter Manipulator Kinematics Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  49. 49. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 23 Solution: If two axes of ξ and ξ coincide, then we get: Subproblem 1: θ + θ = θ Chapter Manipulator If the two axes are not parallel, ω × ω ≠ , then, let c satisfy: p = c = e− ξ θ q Kinematics ˆθ ˆ Forward eξ kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  50. 50. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 23 Solution: If two axes of ξ and ξ coincide, then we get: Subproblem 1: θ + θ = θ Chapter Manipulator If the two axes are not parallel, ω × ω ≠ , then, let c satisfy: p = c = e− ξ θ q Kinematics ˆθ ˆ Forward eξ Set r ∈ l ξ ∩ l ξ kinematics Inverse p − r = c − r = e− ξ θ (q − r), ⇒ eω u = z = e−ω θ v Kinematics ˆθ ˆ Manipulator eξ ˆ θ ˆ Jacobian u z v ω u=ω z Redundant = z = v Manipulators T T ωT v = ωT z Reference ⇒ , u
  51. 51. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 23 Solution: If two axes of ξ and ξ coincide, then we get: Subproblem 1: θ + θ = θ Chapter Manipulator If the two axes are not parallel, ω × ω ≠ , then, let c satisfy: p = c = e− ξ θ q Kinematics ˆθ ˆ Forward eξ Set r ∈ l ξ ∩ l ξ kinematics Inverse p − r = c − r = e− ξ θ (q − r), ⇒ eω u = z = e−ω θ v Kinematics ˆθ ˆ Manipulator eξ ˆ θ ˆ Jacobian u z v ω u=ω z Redundant = z = v Manipulators T T ωT v = ωT z Reference ⇒ , u As ω , ω and ω × ω are linearly independent, z = αω + βω + γ(ω × ω ) ⇒ z = α + β + αβωT ω + γ ω × ω (Continues next slide)
  52. 52. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 24 ⎧ ⎪α ⎪ = (ωT ω )ωT u−ωT v ωT u = αωT ω + β ⎪ ⇒⎨ (ωT ω ) − ω v = α + βω ω ⎪β ⎪ = ⎪ Chapter T T (ωT ω )ωT v−ωT u ⎩ Manipulator Kinematics (ωT ω ) − Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  53. 53. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 24 ⎧ ⎪α ⎪ = (ωT ω )ωT u−ωT v ωT u = αωT ω + β ⎪ ⇒⎨ (ωT ω ) − ω v = α + βω ω ⎪β ⎪ = ⎪ Chapter T T (ωT ω )ωT v−ωT u ⎩ Manipulator Kinematics (ωT ω ) − − α − β − αβωT ω Forward = u ⇒γ = (∗) kinematics u ω ×ω Inverse z Kinematics Manipulator (∗) has zero, one or two solution(s): Jacobian eξ θ p = c Redundant ˆ Manipulators Given z ⇒ c ⇒ eξ θ p = c ˆ Reference for θ and θ Two solutions when the two circles intersect.
  54. 54. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 24 ⎧ ⎪α ⎪ = (ωT ω )ωT u−ωT v ωT u = αωT ω + β ⎪ ⇒⎨ (ωT ω ) − ω v = α + βω ω ⎪β ⎪ = ⎪ Chapter T T (ωT ω )ωT v−ωT u ⎩ Manipulator Kinematics (ωT ω ) − − α − β − αβωT ω Forward = u ⇒γ = (∗) kinematics u ω ×ω Inverse z Kinematics Manipulator (∗) has zero, one or two solution(s): Jacobian eξ θ p = c Redundant ˆ Manipulators Given z ⇒ c ⇒ eξ θ p = c ˆ Reference for θ and θ Two solutions when the two circles intersect. One solution when they are tangent
  55. 55. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 24 ⎧ ⎪α ⎪ = (ωT ω )ωT u−ωT v ωT u = αωT ω + β ⎪ ⇒⎨ (ωT ω ) − ω v = α + βω ω ⎪β ⎪ = ⎪ Chapter T T (ωT ω )ωT v−ωT u ⎩ Manipulator Kinematics (ωT ω ) − − α − β − αβωT ω Forward = u ⇒γ = (∗) kinematics u ω ×ω Inverse z Kinematics Manipulator (∗) has zero, one or two solution(s): Jacobian eξ θ p = c Redundant ˆ Manipulators Given z ⇒ c ⇒ eξ θ p = c ˆ Reference for θ and θ Two solutions when the two circles intersect. One solution when they are tangent Zero solution when they do not intersect
  56. 56. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 25 Subproblem 3: Rotation to a given point Chapter Given a zero-pitch twist ξ, with unit magnitude and p, q ∈ R , find θ s.t. q − eξθ p = δ Manipulator Kinematics ˆ Forward kinematics Inverse Kinematics p Manipulator u u′ Jacobian r θ ξ θ0 θ Redundant Manipulators ωθ ′ e u Reference v v′ δ δ′ q ω T (p−q) Figure 3.7
  57. 57. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 25 Subproblem 3: Rotation to a given point Chapter Given a zero-pitch twist ξ, with unit magnitude and p, q ∈ R , find θ s.t. q − eξθ p = δ Manipulator Kinematics ˆ Forward Define: u = p − r, v = q − r, v − eωθ u ˆ =δ kinematics Inverse Kinematics p u = u − ωω u ′ Manipulator u u′ T Jacobian r θ v′ = v − ωωT v ξ θ0 θ Redundant Manipulators ωθ ′ e u Reference v v′ δ δ′ q ω T (p−q) Figure 3.7 ⇒ u = u′ + ωωT u, v = v′ + ωωT v, δ ′ = δ − ωT (p − q) (Continues next slide)
  58. 58. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 26 Chapter Manipulator (v′ + ωωT v) − eωθ (u′ + ωωT u) ˆ =δ ⇒ v − e u + ωωT (v − u) =δ Kinematics ′ ωθ ′ ˆ Forward kinematics Inverse ωωT (q−p) Kinematics Manipulator Jacobian Redundant Manipulators Reference
  59. 59. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 26 Chapter Manipulator (v′ + ωωT v) − eωθ (u′ + ωωT u) ˆ =δ ⇒ v − e u + ωωT (v − u) =δ Kinematics ′ ωθ ′ ˆ Forward kinematics Inverse ωωT (q−p) Kinematics Manipulator v′ − eωθ u′ = δ − ωT (p − q) = δ′ , Jacobian ˆ Redundant θ = atan (ωT (u′ × v′ ), u′T v′ ), Manipulators ϕ = θ − θ ⇒ u′ + v′ − u′ ⋅ v′ cos ϕ = δ ′ , Reference u′ + v′ − δ ′ θ = θ ± cos− (∗) u′ ⋅ v′ Zero, one or two solutions!
  60. 60. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 27 ◻ Solving inverse kinematics using Chapter subproblems: Manipulator Kinematics Technique 1: Eliminate the dependence on a joint eξθ p = p, if p ∈ l ξ . Given eξ θ eξ θ eξ θ = g, select p ∈ l ξ , p ∉ l ξ Forward ˆ ˆ ˆ ˆ kinematics Inverse or l ξ , then: gp = eξ θ eξ Kinematics ˆ ˆθ Manipulator p Jacobian Redundant Manipulators Reference
  61. 61. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 27 ◻ Solving inverse kinematics using Chapter subproblems: Manipulator Kinematics Technique 1: Eliminate the dependence on a joint eξθ p = p, if p ∈ l ξ . Given eξ θ eξ θ eξ θ = g, select p ∈ l ξ , p ∉ l ξ Forward ˆ ˆ ˆ ˆ kinematics Inverse or l ξ , then: gp = eξ θ eξ Kinematics ˆ ˆθ Manipulator p Jacobian Redundant Manipulators Technique 2: subtract a common point Reference = g, q ∈ l ξ ∩ l ξ ⇒ eξ θ eξ p − q = gp − q ⇒ ˆ ˆθ ˆθ ˆ ˆθ ˆθ eξ θ eξ eξ ˆ ˆ eξ p − q = gp − q ˆθ eξ
  62. 62. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 28 Example: Elbow manipulator Chapter Manipulator ξ1 ξ4 Kinematics ξ2 ξ3 ξ5 Forward l1 l2 kinematics pb ξ6 qw Inverse Kinematics Manipulator l0 Jacobian S Redundant Manipulators Reference Figure 3.7 gst (θ) = eξ θ eξ gst ( ) = gd ˆ ˆθ ˆθ ˆ θ ˆθ ˆ θ eξ eξ eξ eξ = gd ⋅ gst ( ) = g − ˆ ˆ θ ˆθ ˆ θ ˆθ ˆ θ ⇒ eξ θ eξ eξ eξ eξ eξ (Continues next slide)
  63. 63. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 29 Step 1: Solve for θ qω = g ⋅ qω Chapter ˆ ˆ θ Manipulator Let eξ θ . . . eξ Kinematics qω = g ⋅ qω Forward ˆ ˆθ ˆθ kinematics ⇒ eξ θ eξ eξ Inverse Kinematics Subtract pb from g qω : Manipulator (eξ qω − pb ) = g qω − pb Jacobian ˆ ˆθ ˆθ Redundant eξ θ eξ qω − pb ≜ δ ← Subproblem 3 Manipulators ˆθ Reference ⇒ eξ
  64. 64. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 29 Step 1: Solve for θ qω = g ⋅ qω Chapter ˆ ˆ θ Manipulator Let eξ θ . . . eξ Kinematics qω = g ⋅ qω Forward ˆ ˆθ ˆθ kinematics ⇒ eξ θ eξ eξ Inverse Kinematics Subtract pb from g qω : Manipulator (eξ qω − pb ) = g qω − pb Jacobian ˆ ˆθ ˆθ Redundant eξ θ eξ qω − pb ≜ δ ← Subproblem 3 Manipulators ˆθ Reference ⇒ eξ Step 2: Given θ , solve for θ , θ (eξ qω ) = g qω , Subproblem 2 ⇒ θ , θ ˆ ˆθ ˆθ eξ θ eξ (Continues next slide)
  65. 65. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 30 Step 3: Given θ , θ , θ , solve θ , θ = e− ξ Chapter e− ξ e− ξ θ g ˆ θ ˆθ ˆ θ ˆθ ˆθ ˆ Manipulator Kinematics eξ eξ eξ Forward g let p ∈ l ξ , p ∉ l ξ or l ξ , e eξ p = g p, kinematics ˆ ξ θ ˆθ Inverse Kinematics Subproblem 2 ⇒ θ and θ . Manipulator Jacobian Redundant Manipulators Reference
  66. 66. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 30 Step 3: Given θ , θ , θ , solve θ , θ = e− ξ Chapter e− ξ e− ξ θ g ˆ θ ˆθ ˆ θ ˆθ ˆθ ˆ Manipulator Kinematics eξ eξ eξ Forward g let p ∈ l ξ , p ∉ l ξ or l ξ , e eξ p = g p, kinematics ˆ ξ θ ˆθ Inverse Kinematics Subproblem 2 ⇒ θ and θ . Step 4: Given (θ , . . . , θ ), solve for θ Manipulator Jacobian Redundant eξ θ = (eξ θ . . . eξ θ )− ⋅ g ≜ g Manipulators ˆ ˆ ˆ Let p ∉ l ξ ⇒ eξ θ p = g ⋅ p = q ⇐ Subproblem 1 Reference ˆ Maximum of solutions:
  67. 67. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 31 Example: Inverse Kinematics of SCARA ⎡ ⎤ θ3 ⎢ l +l ⎥ Chapter gst ( ) = ⎢ ⎥ θ2 ⎢ ⎥ Manipulator ⎢ ⎥ Kinematics l2 l ⎣ ⎦ θ1 θ4 Forward l1 gst (θ) = e ξ θ . . . e ξ θ gst ( ) kinematics ˆ ˆ ⎡ cϕ −sϕ ⎤ Inverse z ⎢ ⎥ Kinematics l0 T x ⎢ ⎥ q1 S q2 q3 = ⎢ sϕ cϕ ⎥ ≜ gd y Manipulator y ⎢ ⎥ x ⎢ ⎥ Jacobian z ⎣ ⎦ Redundant Figure 3.8 Manipulators ⎡ ⎤ −l s − l s ⎢ ⎥ Reference p=⎢ ⎥ ⇒ p(θ) = x ⎢ ⎥ l c +l c = y ⇒θ =z−l ⎢ ⎥ l +θ z ⎣ ⎦ = gd gst ( )e− ξ ≜g ˆ ˆ θ ˆ θ ˆ θ eξ θ eξ eξ − (Continues next slide)
  68. 68. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 32 Let p ∈ l ξ , q ∈ l ξ ⇒ e ξ θ e ξ p = g p, ˆ ˆ θ Chapter e ξ θ (e ξ p − q) = g p − q , Manipulator ˆ ˆ θ Kinematics p − q = δ ← Subproblem 3 to get θ Forward ˆ ξ θ kinematics e Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  69. 69. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 32 Let p ∈ l ξ , q ∈ l ξ ⇒ e ξ θ e ξ p = g p, ˆ ˆ θ Chapter e ξ θ (e ξ p − q) = g p − q , Manipulator ˆ ˆ θ Kinematics p − q = δ ← Subproblem 3 to get θ Forward ˆ ξ θ kinematics e Inverse ⇒ e ξ θ (e ξ p) = g p ⇒ θ ← Subproblem 1 to get θ Kinematics ˆ ˆ θ Manipulator = e− ξ e− ξ θ gd gst ( )e− ξ ≜g Jacobian ˆ ˆ θ ˆ ˆ θ ξ θ − Redundant ⇒e p = g p, p ∉ l ξ ← Subproblem 1 to get θ Manipulators ˆ ξ θ e Reference There are a maximum of two solutions! † End of Section †
  70. 70. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 33 Example: ABB IRB4400 θ x Chapter θ l l Manipulator z Kinematics l T θ θ y Forward kinematics z y l Inverse l S θ Kinematics θ q x Manipulator Jacobian Redundant Manipulators Reference − ω = , ω = −ω = −ω = ,ω = ω = l l l +l q = ,q = ,q = , pw ∶= q = q = q = l l +l (Continues next slide)
  71. 71. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 34 ⎡ l +l +l ⎤ ⎢ − ⎥ qi × ω i gst ( ) = ⎢ ⎢ l +l ⎥ , ξi = ⎥ ⎢ ⎥ ωi Chapter ⎣ ⎦ gst (θ) = e ξ θ e ξ gst ( ) ∶= gd Manipulator ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ Kinematics eξ eξ eξ eξ Forward kinematics Inverse Kinematics Manipulator Jacobian Redundant Manipulators Reference
  72. 72. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 34 ⎡ l +l +l ⎤ ⎢ − ⎥ qi × ω i gst ( ) = ⎢ ⎢ l +l ⎥ , ξi = ⎥ ⎢ ⎥ ωi Chapter ⎣ ⎦ gst (θ) = e ξ θ e ξ gst ( ) ∶= gd Manipulator ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ Kinematics eξ eξ eξ eξ Forward pw = gd pw =∶ q ⇒ e ξ pw = e− ξ θ q kinematics ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ Inverse eξ θ eξ eξ eξ Kinematics qx ⇒ =[ ] ⋅ e− ξ θ q = cos θ qy − sin θ qx , q = ˆ Manipulator qy Jacobian qz Redundant ⇒ θ = tan− (qy qx ) Manipulators Reference
  73. 73. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 34 ⎡ l +l +l ⎤ ⎢ − ⎥ qi × ω i gst ( ) = ⎢ ⎢ l +l ⎥ , ξi = ⎥ ⎢ ⎥ ωi Chapter ⎣ ⎦ gst (θ) = e ξ θ e ξ gst ( ) ∶= gd Manipulator ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ Kinematics eξ eξ eξ eξ Forward pw = gd pw =∶ q ⇒ e ξ pw = e− ξ θ q kinematics ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ Inverse eξ θ eξ eξ eξ Kinematics qx ⇒ =[ ] ⋅ e− ξ θ q = cos θ qy − sin θ qx , q = ˆ Manipulator qy Jacobian qz Redundant ⇒ θ = tan− (qy qx ) Manipulators pw − q = e− ξ θ q − q =∶ δ ← Subproblem 3 to get θ ˆ θ ˆ Reference eξ (e pw ) = e ˆ ξ θ ˆ ξ θ ˆ −ξ θ e q ← Subproblem 1 to get θ
  74. 74. Chapter 3 Manipulator Kinematics 3.2 Inverse Kinematics 34 ⎡ l +l +l ⎤ ⎢ − ⎥ qi × ω i gst ( ) = ⎢ ⎢ l +l ⎥ , ξi = ⎥ ⎢ ⎥ ωi Chapter ⎣ ⎦ gst (θ) = e ξ θ e ξ gst ( ) ∶= gd Manipulator ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ Kinematics eξ eξ eξ eξ Forward pw = gd pw =∶ q ⇒ e ξ pw = e− ξ θ q kinematics ˆ ˆ θ ˆ θ ˆ θ ˆ θ ˆ Inverse eξ θ eξ eξ eξ Kinematics qx ⇒ =[ ] ⋅ e− ξ θ q = cos θ qy − sin θ qx , q = ˆ Manipulator qy Jacobian qz Redundant ⇒ θ = tan− (qy qx ) Manipulators pw − q = e− ξ θ q − q =∶ δ ← Subproblem 3 to get θ ˆ θ ˆ Reference eξ (e pw ) = e ˆ ξ θ ˆ ξ θ ˆ −ξ θ e q ← Subproblem 1 to get θ = e− ξ e− ξ θ gd gst ( ) =∶ g ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ ˆ eξ eξ eξ e− ξ − Use subproblem 1,2 to solve for θ , θ , θ † End of Section †
  75. 75. Chapter 3 Manipulator Kinematics 3.3 Manipulator Jacobian 35 Given gst ∶ Q → SE( ), θ(t) = (θ (t) . . . θ n (t))T → eξ θ . . . eξn θ n gst ( ) ˆ ˆ Chapter and θ(t) = (θ (t) . . . θ n (t))T , Manipulator Kinematics ˙ ˙ ˙ Forward kinematics Inverse What is the velocity of the tool frame? Kinematics Manipulator Jacobian Redundant Manipulators Reference
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