Track 4. Screw Theory for Robotics

Robots are commonly modeled as a multi-body system, that is, a set of rigid bodies connected by joints. For instance, the mechanical structure of today's humanoid robots is a kinematic tree of cylindrical joints, with the root of the tree at the (rigid body corresponding to the) waist of the robot, and one branch for each limb, left leg, right leg, etc. Motors located in each joint produce torques, which in turn generate a chain of forces between the rigid bodies of the kinematic chain until end-effectors, hands or feet. If the end-effector is free (like a hand in the air), it will perform a pure motion. If it is in contact (like a foot firmly planted on the ground), it will not move directly, but the interaction with the environment will produce contact forces that in turn move the location of the humanoid in space via the Newton-Euler equations of motion. This phenomenon is central to locomotion, and it can be studied, like all rigid-body motions, using the framework of screw theory.

Screws

The motion of any rigid body is fully described by a mathematical object called a screw, also known as spatial vectors (there may be a subtle difference between these two concepts but I don't understand it for now). A screw sO=(r,mO)sO=(r,mO) is given by:

•           its resultant rr, a vector, and

•           its moment mOmO, a vector field over the Euclidean space E3E3.

The resultant rr is the same everywhere, but the moment mOmO depends on the point O∈E3O∈E3 where it is taken. However, the moment field has a particular structure: from mOmO and rr, the moment at any other point P∈E3P∈E3 is given by the Varignon formula:

mP = mO+PO−→−×r.mP = mO+PO→×r.

Although the coordinate vector sPsP of a screw depends on the point PP where it is taken, the screw itself does not depend on the choice of PP as a consequence of this formula. There is therefore a distinction to make between the screw itself and its coordinate vector at a given point. A common convention is to denote screws with hats s^s^ and their coordinates with point subscripts sOsO.