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
Manipulator Kinematics Velocity Analysis(cont'd) Define r≡a+a+1++an (4.50) Position vector from Oi to p p=∑lexr A≡[ee2…·en] (4.51a) B=[e1 xri exr2… en xIn] (4.51b) b=[A12…b A0, P=B Recall twist J6=t (4.53)
Manipulator Kinematics • Velocity Analysis (cont'd) Define Let Position vector from Oi to P Recall twist
Manipulator Kinematics Velocity Analysis(cont'd Jacobian matrix A B (4.54a) el x e x r2 ener 犹t inear transformation between joint rates 60 and Cartesian rates(ee) ith column (revolute joints) The Plucker array of ith axis w.r.t e:x r point p of ee
Manipulator Kinematics • Velocity Analysis (cont'd) Jacobian matrix: ith column (revolute joints) Linear transformation between joint rates and Cartesian rates (EE) The Plücker array of ith axis w.r.t point P of EE
Manipulator kinematics Velocity Analysis(cont'd) Prismatic joint x aite =1e1+e2+…+b-e-1+b11e+1+…+nen p=el xr1+02e2 xr2+.+Bi-lej-1 xri-1+bie i +62计+1ei+1Xr计1+…+nen×an b=[1b2…b-1bb1…b The ith column of jacobian matrix 0
Manipulator Kinematics • Velocity Analysis (cont'd) Prismatic joint: The ith column of Jacobian matrix
Manipulator Kinematics Velocity Analysis(cont'd For 6 joint manipulator, J is a 66 square matrix 6=J-t (4.57) Solve equations using Gauss-elimination (Lu decomposition)algorithm J=LU 0 11 0 00 12 Ly= t U0 Compute y Compute (Forward substitution) (Backward substitution)
Manipulator Kinematics • Velocity Analysis (cont'd) For 6 joint manipulator, J is a 66 square matrix Solve equations using Gauss-elimination (LU decomposition) algorithm Compute y (Forward substitution) Compute (Backward substitution)
Manipulator kinematics Velocity Analysis(cont'd Transformation of jacobian matrix In general, Jacobian can be defined wrt different points. For decoupled manipulators Wrist Centre to= Point p at ee tp Recall twist transformation P-c 1 Two point a 10 and b on ee JB=UJA A-B 1 Property det(JB)=det(JA)
Manipulator Kinematics • Velocity Analysis (cont'd) Transformation of Jacobian matrix In general, Jacobian can be defined wrt different points. For decoupled manipulators: Recall twist transformation Property: Wrist Centre Point P at EE Two point A and B on EE
