A - Summary of Useful Formulas

Summary

Chapter 2

• dof = (sum of freedoms of bodies) − (number of independent configuration constraints)

• Gr¨ubler's formula is an expression of the above formula for mechanisms with N links (including ground) and J joints, where joint i has fi degrees of freedom and m = 3 for planar mechanisms or m = 6 for spatial mechanisms:

• Pfaffian velocity constraints take the form A(θ)θ = 0.

Chapter 4

• The product of exponentials formula for a serial chain manipulator is

where M is the frame of the end-effector in the space frame when the manipulator is at its home position, Si is the spatial twist when joint i rotates (or translates) at unit speed while all other joints are at their zero position, and Bi is the body twist of the end-effector frame when joint i moves at unit speed and all other joints are at their zero positions.

Chapter 5

• For a manipulator end-effector configuration written in coordinates x, the forward kinematics is x = f(θ), and the differential kinematics is given by x˙ = (∂f/∂θ)θ˙ = J(θ)θ˙, where J(θ) is the manipulator Jacobian.

• Written using twists, the relation is V∗ = J∗(θ)θ˙, where ∗ is either s (for a space Jacobian) or b (for abody Jacobian). The columns Jsi, i = 2, … , n, of the space Jacobian are

with Js1 = S1, and the columns Jbi, i = 1, … , n−1, of the body Jacobian are

with Jbn = Bn. The spatial twist caused by joint-i motion is altered only by the configurations of joints inboard from joint i (i.e., between the joint and the space frame), while the body twist caused by joint i is altered only by the configurations of joints outboard from joint i (i.e., between the joint and the body frame).

The two Jacobians are related by