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Figure 4.5 (a) Optimal braking strategy requires at most one switch of control. (b) The corresponding time velocity relation.
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starts with the initial values x = rv and 0 V0 < Vmax , the system will rst move, with control p = pmax , along parabola I to parabola I I (Figure 4.5a), x(V ) = V 2 V02 r 2pmax
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and then, with control p = pmax , toward the origin, along parabola I I , x(V ) = V2 2pmax
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The optimal time tb of braking is a function of the initial velocity V0 , radius of vision rv , and the control limit pmax , tb (V0 ) = 2V02 + 4pmax rv V0 pmax
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Function tb (V0 ) has a minimum at V0 = Vmax = 2pmax rv , which is exactly the upper bound on the velocity given by (4.1); it is decreasing on the interval V0 [0, Vmax ] and increasing when V0 > Vmax (see Figure 4.5b). For the interval V0 [0, Vmax ], which is of interest to us, the above analysis leads to a somewhat counterintuitive conclusion: Proposition 1. For the initial velocity V0 in the range V0 [0, Vmax ], the time necessary for stopping at the boundary of the sensing range is a monotonically decreasing function of V0 , with its minimum at V0 = Vmax . Notice that this result (see also Figure 4.5) leaves a comfortable margin of safety: Even if at the moment when the robot sees an obstacle on its way it moves with
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the maximum velocity, it can still stop safely before it reaches the obstacle. If the robot s velocity is below the maximum, it has more control options for braking, including even one of speeding up before actual braking. Assume, for example, that we want the robot to stop in minimum time at the sensing range boundary (the origin in Figure 4.5a); consider two initial positions: (i) x = rv , V = V01 and (ii) x = rv , V = V02 ; V02 > V01 . Then, according to Proposition 1, in case (i) this time is bigger than in case (ii). Note that because of the discrete control it is the permitted maximum velocity, Vp max , that is to be substituted into (4.4) to obtain the minimum time. (More details on the braking procedure can be found in Ref. 99). 4.2.5 Dynamics and Collision Avoidance The analysis in this section consists of two parts. First we incorporate the control constraints into the model of our mobile robot and develop a transformation from the moving path coordinate frame to the world frame (see Section 4.2.1). Then the Maximum Turn Strategy is produced, an incremental decision-making mechanism that determines forces p and q at each step.
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Transformation from Path Frame to World Frame. The remainder of this section refers to the time interval [ti , ti+1 ), and so index i can be dropped. Let (x, y) R2 be the robot s position in the world frame, and let be the (slope) angle between the velocity vector V = (Vx , Vy ) = (x, y) and x axis of the world frame (see Figure 4.2). The planning process involves computation of controls u = (p, q), which for every step de nes the velocity vector and eventually the path, x = (x, y), as a function of time. The normalized equations of motion are
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x = p cos q sin y = p sin + q cos The angle between vector V and x axis of the world frame is found as arctan Vy , Vx 0 Vx = arctan Vy + , Vx < 0 Vx To nd the transformation from path frame to the world frame (x, y), present the velocity in the path frame as V = V t. Angle is de ned as the angle between t and the positive direction of x axis. Given that control forces p and q act along the t and n directions, respectively, the equations of motion with respect to the path frame are V =p = q/V
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