Kinematics and the Robot class

import screws

ur5 = screws.robots.ur5()                          # M, screw axes, inertias from the textbook's URDF
T = ur5.fk([0.3, -1.2, 0.8, -0.4, 1.1, 0.2])         # a reachable, non-singular pose
result = ur5.ik(T, theta0=[0.1, -1.4, 0.1, 0.1, 1.4, 0.1])   # result.theta, result.converged, result.history

The free functions are the reference implementation and read as the book does: screws.exp6(screws.vec_to_se3(S * theta)) is \(e^{[\mathcal{S}]\theta}\), screws.fk_space(M, S, theta) is the space form of the product of exponentials, and screws.FKinSpace is the very same function under MR's name.

Screw-axis lists are 6xn, one axis per column, as MR writes them. Nothing guesses the orientation (a 6x6 array is ambiguous for a six-axis arm), so hand entry goes through a sequence of 6-vectors: screws.Robot.from_screw_axes(M, [S1, S2, ...]).

Two UR5s

screws.robots.ur5() is built from the URDF the textbook prints in section 4.2, whose lengths are the manufacturer's to a tenth of a millimetre (89.159, 135.85, 425, 119.7, 392.25, 93, 94.65 and 82.3 mm) and which carries the link inertias. screws.robots.ur5(source="textbook") is the rounded table of MR Figure 4.6 (\(W_1 = 109\), \(W_2 = 82\), \(L_1 = 425\), \(L_2 = 392\), \(H_1 = 89\), \(H_2 = 95\) mm) with no inertias. Use the first to match a simulator or a real arm; use the second to reproduce the book's worked examples digit for digit. The two agree in every axis direction and differ by under a millimetre in position.