POLÍTICAS CURRICULARES

In the literature, the control based on maximizing the difference of taus between two features has been shown to align a vehicle parallel to the beacons. The minimum difference, is clearly achieved when the vehicle is perpendicular to the line segment connecting these two points of interest (Figure 3·1a). The maximum difference, is therefore achieved when the vehicle is parallel to this line segment (Figure 3·1b).

(a) (b)

Figure 3·1: Time-to-transit comparison between points O1 and O2

with: (a) minimized difference, and (b) maximized difference.

In order to achieve this goal, two separate methods can be utilized, both based on the same control law. The first is a continuous method developed in (Kong et al., 2013) which is summarized as follows.

Theorem 1. (Kong et al., 2013) Consider point features O1, O2 located respectively

at (x1, y1) and (x2, y2). Let τj(t) be the time-to-transit associated with feature Oj for

j = 1, 2. Suppose the initial orientation, θ0, of the vehicle is such that τ2 > τ1 (which

implies that cosθ(x2−x1) + sinθ(y2−y1) > 0). Further assume that the vehicle travels

at constant speed v = 1. Then for any k > 0, the steering control

u = u(t) = k[τ₂0(θ(t)) − τ₁0(θ(t))], (3.5) where τ_j0(θ) = δτj

δθ will asymptotically align the vehicle with the line segment directed

from O1 to O2.

For the proof and further details, refer to (Kong et al., 2013). This law appears to be the best choice for control when steering around corners at the end of corridor segments. It is assumed that as the end of the corridor is approached, visual features on the outer wall of the next corridor segment will become visible. These features can be used to align the vehicle with the new segment. Assume, without loss of

generality, that the new corridor segment will involve a left-hand turn. To understand the geometry of the control law, assume that we have chosen the global coordinates such that the motion of the vehicle along the first (old) corridor segment is in the vertical (positive y) direction in the coordinate system. The motion along the new segment will then be in the negative x-direction in the chosen coordinate system. Also, assume both corridor segments are one unit wide, and that the vehicle should begin its turn at the instant it passes the convex (inner) corner between the new and old segments. There are two features on the outer wall segment at the corner, O1 and

O2. The tau difference maximizing control law of (Kong et al., 2013) that will align

the vehicle parallel to the vector between features at (x1, y1) and (x2, y2) (where the

second is to the left of the first) is given explicitly by

u(t) = k(−sinθ(x2− x1) + cosθ(y2 − y1). (3.6)

Again without loss of generality, we assume that the vehicle begins its turn at the global frame point (x, y) = (1, −1) and that the feature (global) coordinates are on the global x-axis at (x1, 0) and (x2, 0). Assume for the left turn (ccw) in question,

(x2 − x1) = −1. Then the steering law is u(t) = ksinθ. Recalling that the vehicle

kinematics are given by 3.1, solving for θ(t) explicitly is easily done. It satisfies the differential equation

˙

θ(t) = ksinθ(t), θ(0) = π

2 (3.7)

This can be explicitly integrated to yield θ(t) = 2ArcCot((e−kt)). This in turn allows explicit integration

˙

which yields

y(t) = 2ArcT an((e

kt₎₎

k − 1 (3.9)

From this equation, we have the following result.

Proposition 1 : Using the tau difference maximizing steering law u(t) = ksinθ, the vehicle will turn as to avoid colliding with the outside wall of the new corridor provided that k > π (Figure 3·2).

Proof. This is easily proved using the explicit form of 3.9. Using the formulation in (Kong et al., 2013), it is easy to see that both θ(t) and y(t) are monotonically increasing functions of t, with θ(t) → π. Recalling that lims→infArcT an(s) = π/2 we

see that limt→infy(t) = π/k −1, and this will be less than zero if and only if k > π.

Remark 1: Conceptually, this describes steering using tau difference maximizing. In our actual implementation, there is no direct access to the global coordinate vari- ables, therefore placing all reliance on visual cues acquired by the imaging system. Nevertheless, what is shown is that provided there is enough steering authority to react appropriately to the perceived values of the taus, it is possible to negotiate the turn on to the new corridor segment.

Remark 2: A further word of caution is that the exact form of the steering law will depend on the features that are being sensed. Because the robot vehicle will be proceeding down the new corridor segment, it will lose sight of the features as it passes them. Hence, when the above asymptotic result serves to justify the approach, in actual implementation, the features being used in the feedback will need to be continually updated.

The discrete control utilizes only the sign of Equation 3.5 and the determined value for Ω.

Figure 3·2: The trajectory of the vehicle using the tau maximizing control law and feature beacons located at (x1, y1 = (-3,0) and (x2, y2)

= (-4,0), with k = 2.

The value of δθ is equivalent to ˙θ, or the control for the previous motion. For this law to work, it is imperative that the previous control was not 0, or 3.5 cannot be computed. This will cause a chain effect if a turning rate of 0 is returned for the aforementioned case. In a real implementation, the previous values of δθ must be tracked. In order to avoid the division by zero problem, previous controls should only be recorded when they are not equal to zero.

Figure 3·3 shows a robot’s current tau difference between two features (O1 and

O2) as ρ, and the desired tau difference as ρ0.

Figure 3·3: The current tau difference ρ is not maximal. ρ0 represents the maximum difference.

3.3

Special Considerations for Egocentric, Non-Holonomic