Qualitative Conguration Space

In the last section we showed how to represent in c-space the behavior of all of the components in^SketchIT's domain. We also demonstrated that we can use c-space to reason about behavior: we can compute the overall behavior of a device directly from the c-space description of its parts. Hence, c-space has many of the properties we require of a behavioral representation.

To facilitate synthesizing new designs, we require a behavioral representation with one other essential property: it must generalize the design by abstracting away the implementation of the behaviors. As we mentioned at the outset, c-space does not adequately generalize the design, and hence in this section we extend c-space to produce a new representation that does. We call this new representation qualitative conguration space (qualitative c-space or qc-space).

To illustrate, we use the rotor-slider device from Figure 2-1. The cs-curve in Figure 2-2 represents the interaction between the face on the rotor and the face on the slider. Notice that if we extend the slider's face farther upward, we obtain the same cs-curve. Hence the particular cs-curve in Figure 2-2 represents many pairs of faces (we discussed just two of the possibilities) and in doing so abstracts away the particular geometry of the rotor and slider. However, the cs-curve is still too specic for our purposes.

To see why, we examine the c-space trajectory that describes the motion of the rotor and slider (Figure 2-6). As the trajectory moves vertically through the cs-plane, it is deected to the right by the cs-curve. In physical terms, the slider pushes the rotor counterclockwise. The essential behavior|the slider pushing the rotor counterclockwise|is the physical manifestation of the cs-curve deecting the trajectory to the right. Hence, any cs-curve that deects the trajectory to the right will produce the same qualitative behavior (i.e., the slider pushing the rotor counter-clockwise).

Consequently, to generalize the design far enough for our purposes, we want to identify a class of curves that displace the trajectory to the right. There are many curve shapes that would do this, but if we generalize only to the class of mono-tonic curves, we make tractable the task of computing trajectories through multi-dimensional c-spaces (see Chapter 4).² Hence, we generalize to monotonic curves.

In addition to its shape, a single cs-curve has two other properties that we can vary, the location of each end point. In this example there is only one constraint on the locations of the end points: one end point must be on either side of the initial vertical segment of the trajectory (otherwise the \particle" will miss the curve, i.e., the slider will miss the rotor). Hence, we generalize to monotonic curves that have one end point on either side of the initial segment of the trajectory.

Figure 2-10 shows our generalization of the c-space of the rotor and slider. We

2A multi-dimensional c-space occurs when a device is composed of many xed-axis components.

2.3. QUALITATIVECONFIGURATION SPACE 49 represent the end points of the generalized cs-curve and the starting point of the tra-jectory with landmarks. Landmarks are quantities with constraints on their relative ordering, but which do not have absolute values. In Figure 2-10 for example, A, B, and C are landmarks on the ^UR axis; all we know about their values is that A ^<B

< C (which ensures that one end point of the curve is one either side of the initial segment of the trajectory). The generalized cs-curve is a monotonic curve with pos-itive slope. (All of this assumes that blocked space is on the appropriate side of the curve as shown.) Any particular (i.e., numeric) cs-curve whose end points satisfy the conditions expressed by the landmark orderings, and whose shape is monotonic with positive slope, will produce the same qualitative behavior as the original.

Figure 2-10 is an example of what we call qualitative conguration space (qc-space). Qc-space, which is summarized in Figure 2-11, is ^SketchIT's primary rep-resentation of behavior. It inherits all of the desirable properties of c-space that we described in Section 2.2, and in addition adequately generalizes the design.

U_R US

A B C

D E F

Figure 2-10: The generalization of the c-space for the rotor and slider. The cs-curve represents any monotonic cs-curve, but is shown as a straight line for the sake of convenience. The values A through F are landmark values rather than specic numerical values. The landmark values are constrained to have the ordering shown.

In c-space the interaction between a pair of faces is a cs-curve, but in qc-space the interaction is a qcs-curve (i.e., a qualitative conguration space curve). A qcs-curve represents a family of monotonic cs-curves that all have the same qualitative slope.

Qualitative slope is the obvious notion of describing curves as horizontal, vertical,

Qualitative Conguration Space

The interaction between a pair of faces is represented with a qcs-curve.

Spring neutral positions, actuator motion limits, and interactions with xed surfaces are represented by innite boundaries.

Qcs-curves are monotonic and have a qualitative slope.

Landmark values denote the end points of qcs-curves and the axis crossings of innite boundaries.

The relative locations of qcs-curves and innite boundaries are encoded by the ordering of the landmark values.

Figure 2-11: The Qualitative Conguration Space Representation.

Figure 2-12: There are eight types of qcs-curves. For convenience, the diagonal curves are drawn as straight lines, but they represent any diagonal monotonic curve.

positively sloped, or negatively sloped. We often used the term diagonal slope to refer to both positively and negatively sloped curves. When the four qualitative slopes are combined with the two possible locations of blocked space, the result is the eight qualitatively dierent cs-curves in Figure 2-12.

In qc-space, the locations of qcs-curves and innite boundaries are dened by landmark values: A pair of landmark values denes the location of each end point of a nite qcs-curve; one landmark value denes the axis crossing of each innite boundary. The ordering of the landmark values encodes the relative locations of the qcs-curves and innite boundaries.

Implicit in the qc-space representation is the assumption that each interaction is