Rigid Body Motion

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Chapter 2 - Rigid Body Motion

Next: Manipulator Kinematics

A rigid motion of an object is a motion which preserves distance between points. The study of robot kinematics, dynamics, and control has at its heart the study of the motion of rigid objects. In this chapter, we provide a description of rigid body motion using the tools of linear algebra and screw theory.

Chapter Summary

The following are the key concepts covered in this chapter:

1. Rotational motion of a rigid body is represented by an element of the special orthogonal group
   

   Other parameterizations of SO(3) include fixed and Euler angle sets, and unit quaternions.

2. The configuration of a rigid body is represented as an element . An element may also be viewed as a mapping which preserves distances and angles between points. In homogeneous coordinates, we write
   

   The same representation can also be used for a rigid body transformation between two coordinate frames.

3. Rigid body transformations can be represented as the exponentials of twists:
   

   The twist coordinates of are .

4. A twist is associated with a screw motion having attributes
   

   Conversely, given a screw we can write the associated twist. Two special cases are pure rotation about an axis by an amount and pure translation along an axis :

   
5. The velocity of a rigid motion can be specified in two ways. The spatial velocity,
   

   is a twist which gives the velocity of the rigid body as measured by an observer at the origin of the reference frame. The body velocity,

   

   is the velocity of the object in the instantaneous body frame. These velocities are related by the adjoint transformation

   

   which maps . To transform velocities between coordinate frames, we use the relations

   

   where is the spatial velocity of coordinate frame relative to frame and is the body velocity.

6. Wrenches are represented as a force, moment pair
   

   If is a coordinate frame attached to a rigid body, then we write for a wrench applied at the origin of , with and specified with respect to the coordinate frame. If is a second coordinate frame, then we can write as an equivalent wrench applied at :

   

   For a rigid body with configuration , is called the spatial wrench and is called the body wrench.

7. A wrench is associated with a screw having attributes
   

   Conversely, given a screw we can write the associated wrench.

8. A wrench and a twist are reciprocal if . Two screws and are reciprocal if the twist about and the wrench along are reciprocal. The reciprocal product between two screws is given by
   

   where represents the twist associated with the screw . Two screws are reciprocal if the reciprocal product between the screws is zero.

9. A system of screws describes the vector space of all linear combinations of the screws . A reciprocal screw system is the set of all screws which are reciprocal to . The dimensions of a screw system and its reciprocal system sum to 6 (in ).

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