Robot Modeling and Control. Mark W. Spong

and are coordinate vectors that represent the location of this point in space with respect to coordinate frames o₀x₀y₀ and o₁x₁y₁, respectively. When no confusion can arise, we may simply refer to these coordinate frames as frame 0 and frame 1, respectively.

      Since the origin of a coordinate frame is also a point in space, we can assign coordinates that represent the position of the origin of one coordinate frame with respect to another. In Figure 2.1, for example, we may have

      Thus, o⁰₁ specifies the coordinates of the point o₁ relative to frame 0 and o¹₀ specifies the coordinates of the point 0₀ relative to frame 1. In cases where there is only a single coordinate frame, or in which the reference frame is obvious, we will often omit the superscript. This is a slight abuse of notation, and the reader is advised to bear in mind the difference between the geometric entity called p and any particular coordinate vector that is assigned to represent p. The former is independent of the choice of coordinate frames, while the latter obviously depends on the choice of coordinate frames.

      While a point corresponds to a specific location in space, a vector specifies a direction and a magnitude. Vectors can be used, for example, to represent displacements or forces. Therefore, while the point p is not equivalent to the vector v₁, the displacement from the origin o₀ to the point p is given by the vector v₁. In this text, we will use the term vector to refer to what are sometimes called free vectors, that is, vectors that are not constrained to be located at a particular point in space. Under this convention, it is clear that points and vectors are not equivalent, since points refer to specific locations in space, but a free vector can be moved to any location in space. Thus, two vectors are equal if they have the same direction and the same magnitude.

      When assigning coordinates to vectors, we use the same notational convention that we used when assigning coordinates to points. Thus, v₁ and v₂ are geometric entities that are invariant with respect to the choice of coordinate frames, but the representation by coordinates of these vectors depends directly on the choice of reference coordinate frame. In the example of Figure 2.1, we would obtain

In order to perform algebraic manipulations using coordinates, it is essential that all coordinate vectors be defined with respect to the same coordinate frame. In the case of free vectors, it is enough that they be defined with respect to “parallel” coordinate frames, that is, frames whose respective coordinate axes are parallel, since only their magnitude and direction are specified and not their absolute locations in space.

      Using this convention, an expression of the form , where and are as in Figure 2.1, is not defined since the frames o₀x₀y₀ and o₁x₁y₁ are not parallel. Thus, we see a clear need not only for a representation system that allows points to be expressed with respect to various coordinate frames, but also for a mechanism that allows us to transform the coordinates of points from one coordinate frame to another. Such coordinate transformations are the topic for much of the remainder of this chapter.

      2.2 Representing Rotations

      In order to represent the relative position and orientation of one rigid body with respect to another, we attach coordinate frames to each body, and then specify the geometric relationship between these coordinate frames. In Section 2.1 we saw how one can represent the position of the origin of one frame with respect to another frame. In this section, we address the problem of describing the orientation of one coordinate frame relative to another frame. We begin with the case of rotations in the plane, and then generalize our results to the case of rotations in a three-dimensional space.

      2.2.1 Rotation in the Plane

Figure 2.2 shows two coordinate frames, with frame o₁x₁y₁ obtained by rotating frame o₀x₀y₀ by an angle θ. Perhaps the most obvious way to represent the relative orientation of these two frames is merely to specify the angle of rotation θ. There are two immediate disadvantages to such a representation. First, there is a discontinuity in the mapping from relative orientation to the value of θ in a neighborhood of θ = 0. In particular, for θ = 2π − ε, small changes in orientation can produce large changes in the value of θ, for example, a rotation by ε causes θ to “wrap around” to zero. Second, this choice of representation does not scale well to the three-dimensional case.

Figure 2.2 Coordinate frame o₁x₁y₁ is oriented at an angle θ with respect to o₀x₀y₀.

      A slightly less obvious way to specify the orientation is to specify the coordinate vectors for the axes of frame o₁x₁y₁ with respect to coordinate frame o₀x₀y₀:

in which and are the coordinates in frame o₀x₀y₀ of unit vectors x₁ and y₁, respectively.¹ A matrix in this form is called a rotation matrix. Rotation matrices have a number of special properties that we will discuss below.

      In the two-dimensional case, it is straightforward to compute the entries of this matrix. As illustrated in Figure 2.2,

      which gives

(2.1)

      Note that we have continued to use the notational convention of allowing the superscript to denote the reference frame. Thus, is a matrix whose column vectors are the coordinates of the unit vectors along the axes of frame o₁x₁y₁ expressed relative to frame o₀x₀y₀.