305-572A Mechanics of Robotics Systems I by 81jnAQ

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									      Dept. Of Mechanical Engineering


      MECH572A
Introduction To Robotics


       Lecture 7
                                Review
• Basic Robotic Kinematic Problems
      Direct (forward) Kinematics
      Inverse Kinematics
• DH Notation
                                  Zi-
                                  1

                                                i-                 i
                         bi-1                   1          Zi
                 Oi-                   ai-1
                 1

                                      Xi-                       i
                                i-1 1 Oi bi
                                                      ai
                                                i
                       i-1                                               Oi+
                                               Xi                        1
                                                                                Xi+1
                                                                                i+1

               Revolute joints                                               Zi+1
                       Review
• Transformation Between Neighboring Links
    Fi to Fi+1


       Orientation:




         Position:
                          Review
• Forward Kinematics




     Known joint angles            End Effector Position + Orientation
                   Inverse Kinematics

• Overview
  - Problem description:
    Known EE position and orientation, find joint angles (inverse process)
     Direct Kinematics Problem (DKP) -> Solution unique
     Inverse Kinematics Problem (IKP) -> May have multiple solutions,
                   not always solvable (Kinematic Invertibility)
  - Equations in IKP are usually highly nonlinear, analytically solvable
     (closed form solution available) only for certain types of
     manipulators, examples:
          PUMA (6R decoupled)
          Stanford Arm (5R-1P)
          Canadarm 2 (7R with 3 parallel pitch joint axes)
     other types of manipulator rely on numerical methods for solution
               Inverse Kinematics

• Overview (cont'd)
  - PUMA – 6R decoupled (Arm + Wrist)
                            Inverse Kinematics

• Overview (cont'd)
  - Canada Arm 2 – 7R (Off-pitch Joints + Pitch joints)

  3 parallel pitch joints
  4 off-pitch joints
                 Inverse Kinematics

• Overview (cont'd)
  Scope of this course – Decoupled manipulators
  - Have Special architecture that allows the decoupling of
    position problem from orientation problem. e.g. PUMA
  - Analytical IKP solution available
                 Inverse Kinematics

• 6-R Decoupled Manipulator




          Arm (Position)
                                 Wrist (Orientation)

                                 C: wrist centre
                Inverse Kinematics
• 6-R Decoupled Manipulator
   – Position Problem




                                     Recall ai =Qibi -
                                     eq(4.3d)
                    Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
     EE position vector p, wrist centre position vector c




                                                            c expressed in
                                                            terms of p and Q
                    Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)

    From eq. (4.18c)
    position problem can be
    decoupled from
    orientation problem.
    Three equations in
    (4.17) for three
    unknowns 1, 2, and 3
                     Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
     Solve the equation:




            2              3                   1
     Linear transformation of vector in 3 to vector in 1
     Norm of the vector is invariant, i.e., || VLHS|| = || VRHS||
                  eliminate 2
                     Inverse Kinematics
• 6-R Decoupled Manipulator
   – Position Problem (cont'd) - coefficients of eq. (4.19a)




       The 3rd scalar equation of (4.17) does not contain 2 and thus leads to
                      Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
     Two equations (4.19a) and (4.20a) linear in c1, s1, c3, s3, solve c1, s1 in
     terms of c3, s3 :




   If       0, then we have a singularity. To be discussed later

        (4.21a) & (4.21b):   c12  s12  1




              1 is eliminated
                     Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)




                                                    Tan-half-angle-
                                                    identities
   – Use trigonometric identities to treat (4.22)
                    Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)




     Solving (4.22):

     Substitute c3 and s3 to (4.21a) & (4.21b) to determinefour different
     solutions of 1
                     Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
     To get 2, use the first two equations of (4.17), which yield,
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)




     If          there is a Singularity, which is to be discussed next
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
     Discussion on solutions


                           e1 // e2   1 = 0
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
                           e1 intersects with e2
                  Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
     PUMA Robot – A special case




             Four solutions for same
             given wrist centre
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)
                              Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd)




                                                                   Z2
                                                 Z1                      C
     The first two components of
     [c]2 vanish


    a 1  Q 1 c2  c1  c                                 Z3
    Q1 c2  c  a 1
                                                              a2    a3
    c2  Q   T
                   (c  a 1 )  Q c  b 1
                                 T
                                                 a1
                                                      [c]1


       C lies on Z2 axis                    X1               Y1
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd) – Example




   From figure




   Compute the coefficients
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd) –Example




     The quadric equation in 3


     solve the equation
                   Inverse Kinematics

• 6-R Decoupled Manipulator
   – Position Problem (cont'd) –Example
     Compute 1




     Compute 2




     The remaining roots are computed likewise:
                    Inverse Kinematics

• 6-R Decoupled Manipulator
   – Orientation Problem
    The EE orientation in terms
    of Q along with 1, 2, and
    3 are known data. 4, 5
    and 6 are to be computed
                   Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
   – Orientation problem (cont'd)
     Geometric relationship yields




                                 2 roots – if radical > 0
             Solution for 4     1 root – if radical = 0
                                 No root – if radical < 0
                  Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
   – Orientation problem (cont'd)
     Workspace of spherical wrist:




     The workspace description
                   Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
   – Orientation problem
     Recall
     Assume




     Equating the first two elements of the 3rd column (independent of
     6)



                    Solve for 5
                   Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
   – Orientation problem

     Take the first column of both side
                                          Recall
                                          Q6= [p6, q6, u6]




      where




                      Solve for 6
                    Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
   – Orientation problem summary
     2 sets of solution:  {4, 5}1, 6
                          {4, 5}2, 6
                   Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
   – Overall Solution of IKP

       Arm (Position)          Wrist (Orientation)   Total
           4                          2               8

								
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