McGill Dept of Mechanical Engineering MECH572A Introduction To robotics Lecture 7
MECH572A Introduction To Robotics Lecture 7 Dept. Of Mechanical Engineering
Review Basic robotic Kinematic problems Direct(forward) Kinematics Inⅴ erse Kinematics DH Notation Z b KZi C-1 X Blok X Revolute joints Z
Review • Basic Robotic Kinematic Problems Direct (forward) Kinematics Inverse Kinematics • DH Notation Oi- 1 Oi Oi+ 1 Zi- 1 Zi Zi+1 i-1 i i+1 Xi- 1 Xi Xi+1 Revolute joints bi-1 bi i ai-1 ai i i- 1 i-1
Review Transformation Between Neighboring Links Fi to Fil cosei -isin ei Hi sin e, Orientation: Qi= sin ]i Mi cos @ li cos d 0 入2 Position ai Sin e osa三sinc
Review • Transformation Between Neighboring Links Fi to Fi+1 Orientation: Position:
Review Forward Kinematics e 2 g$ es ON 94 03 PfE, 1, a) Known joint angles End effector position Orientation Q1Q2Q3Q4Q3 Q6=Q a1+Q1(a2+Q2a+Q2Q3a4+Q2Q3Q4a5+Q2Q3Q4Q5a6)=p
Review • Forward Kinematics Known joint angles End Effector Position + Orientation
nⅴ erse Kinematics Overview Problem description Known ee position and orientation find joint angles (inverse process) Direct Kinematics Problem(DKP)-> Solution unique Inverse Kinematics Problem(lkp)-> May have multiple solutions not al ways solvable(Kinematic Invertibility) Equations in IKP are usually highly nonlinear, analytically sol vable (closed form solution available)only for certain types of manipulators, examples PUMA(6R decoupled) Stanford Arm(5R-IP) Canadarm 2(7R with 3 parallel pitch joint axes) other types of manipulator rely on numerical methods for solution
Inverse Kinematics • Overview - Problem description: Known EE position and orientation, find joint angles (inverse process) Direct KinematicsProblem(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(contd) PUMa-6R decoupled(Arm+ Wrist) onsite b A、H4 Y7 FIGURE 43. Coordinate frames of a puma robot
Inverse Kinematics • Overview (cont'd) - PUMA – 6R decoupled (Arm + Wrist)
Inverse Kinematics Overview(contd) Canada Arm 2-7R(Off-pitch Joints Pitch joints) 3 parallel pitch joints off-pitch Joints
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(contd) 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 • 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
nⅴ erse Kinematics 6-R Decoupled manipulator de 0 03 e P(a,g, z) Arm(Position) Wrist (Orientation) C. wrist centre
Inverse Kinematics • 6-R Decoupled Manipulator Arm (Position) Wrist (Orientation) C: wrist centre
nⅴ erse Kinematics 6-R Decoupled manipulator Position problem a+Qia2+Q1Q2a3+Q1Q2Q3a4=c (4.16) a2+Q2a+Q2Q3a4=Qf(c-al Recall ai=Qibi Q2(b2+Q3b3+Q3Q4b4)=Q1c-b1 eq(4.3d 0 a4三QAb4三0≡be→Q2Q4b4≡b2Qe≡b4ul Q2(b2+Q3b3 +b4u3)=Q1c-bI (417)
Inverse Kinematics • 6-R Decoupled Manipulator – Position Problem Recall ai =Qibi - eq(4.3d)
