Question A. Consider a serial chain with two helical joints with the same pitch, h, and distinct axes through the same point, O. Describe the screws of all possible instantaneous motions of the end-effector.
Question B. Consider the set, H, composed of all wrenches of arbitrary finite pitch with axis (line of application) Oz.
B1. Is H a vector subspace of se(3)*? Prove your answer by showing either that H is closed under addition or that it is not. B2. Find the dimension and a basis of span(H). Prove your answers.
Question C. Let A and B be subspaces of a vector space V (not necessarily finite-dimensional). For any 𝐶 ⊂ 𝑉, 𝐶⊥ ⊂ 𝑉∗ is its annihilator.
C1. Prove that (𝐴 ∩ 𝐵)⊥ ⊃ 𝐴⊥ + 𝐵⊥.
C2. Prove that (𝐴 ∩ 𝐵)⊥ = 𝐴⊥ + 𝐵⊥ if it is given that the following three lemmas are true:
L1. dim(A + B) = dim(A) + dim(B) – dim(A B)
L2. (𝐴 + 𝐵)⊥ = 𝐴⊥ ∩ 𝐵⊥ L3. dim(𝐴⊥) = dim(V) – dim(A) for any subspace A.
Question D.
D1.Two screws of finite pitches p and –p are reciprocal. Describe precisely the conditions for the relative location of the two axes. Prove your answer.
D2. A screw with pitch 0.5 m is on the Oy axis. Consider the set of all screws satisfying the following conditions: (1) and are reciprocal; (2) the pitch of is 0.5 m; (3) the common normal between the axes is in the Oxz plane; (4) the angle between the axes is /4. Show that the set of points with coordinates (x, y, z) on the axes of all screws is a quadric surface. Provide the equation Q(x, y, z) = 0 for this surface and identify its type.
Question E. A twist system, T, is spanned by two planar pencils of pure rotations, each with concurrent axes. The planes of the pencils are horizontal and distinct. The two points of concurrence define a vertical line.
E1. Find the dimension of T.
E2. Find a basis of the reciprocal wrench system T⊥.
E3. Describe all screws in T⊥. Note. Prove your answers and illustrate them with clear drawings.
Question F. The subspace H of se(3) is spanned by all the twists of the same finite non-zero pitch, 0 < h1 < ∞, with vertical axes (i.e., axes parallel to the Oz coordinate direction).
F1. Find a basis of the reciprocal wrench system H⊥.
F2. Describe all elements of H⊥.
Note. Prove your answers and illustrate them with clear drawings.
