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