VIEWS: 6 PAGES: 34 POSTED ON: 9/1/2012
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 determinefour 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 c2 c1 c Z3 Q1 c2 c a 1 a2 a3 c2 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