McGill Dept Of Mechanical Engineering MECH572A Introduction To robotics Lecture s
MECH572A Introduction To Robotics Lecture 8 Dept. Of Mechanical Engineering
Review Robot Kinematics Geometric Analysis x=f(0) f Differential analysis 00 Forward(direct)vs Inverse Kinematics problem Inverse Kinematics Problem(IKP Problem description Known QEE and pee, Seek 01,.On Possibility of Analytical(closed form) solution depends on the architecture of the manipulator 6R Decoupled manipulator(e.g, PUMA) Position problem- position ofC(wrist centre) Orientation problem-eE orientation
Review • Robot Kinematics Geometric Analysis Differential analysis Forward (direct) vs. Inverse Kinematics problem • Inverse Kinematics Problem (IKP) - Problem description: Known QEE and pEE, Seek 1, … n - Possibility of Analytical (closed form) solution depends on the architecture of the manipulator - 6-R Decoupled manipulator (e.g., PUMA) Position problem – position of C (wrist centre) Orientation problem – EE orientation θ θ f x = x = f(θ)
Review IKP-6-R Decoupled manipulator Solution process overview Arm(position) Wrist(orientation) 1+91a2+Q1 a3+q1Q q3a=c 01.02.0 04.05.66 Eliminate e2 Special geometrv in Elimination Equ's in 01, 03 wrist axis efe =d △1≠0 Eliminate e1 Quadratic equ inτ4(64) Quartic equ in t3(03) Radical≥0 0 Solution △2≠0 Max Number of solution 4 Max Number of solution 2
Review • IKP – 6-R Decoupled Manipulator Solution process overview: Arm (position) Wrist (orientation) 1, 2, 3 Equ's in 1, 3 Quartic equ in 3 (3) Eliminate 2 1 0 Eliminate 1 3 1 2 2 0 Max Number of Solution: 4 Elimination Solution 4, 5, 6 Quadratic equ in 4 (4) 4 5 6 Radical 0 Max Number of Solution: 2 Special geometry in wrist axis
Manipulator Kinematics Velocity(Differential) Analysis e 0 2=1e1+62 445 Wn=6e1+b2e2+…+bnen
Manipulator Kinematics • Velocity (Differential) Analysis
Manipulator Kinematics Velocity analysis Angular velocity ofee 61e1+02 anen=∑h Position ofee p=a1+a2t.t an p=a+A2+…an (4.47) ai=wi x ai, i=l, 2. (4.48 p=be1xa1+(61e1+b2e2)×a2+ (4.49) e1+2e2+…+bnen)×a e1x(a1+a2+…+an)+b2e2×(a2+a3+…+an) +…+6 en x a
Manipulator Kinematics • Velocity Analysis Angular velocity of EE Position of EE