Page 177 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

152 ACCOUNTING FOR BODY DYNAMICS: THE JOGGER’S PROBLEM

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 =−r v , V = V 0 1
2
1
2
and (ii) x =−r v , V = V ; V >V . Then, according to Proposition 1, in case
0 0 0
(i) this time is bigger than in case (ii). Note that because of the discrete control it
is the permitted maximum velocity, V p 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.
Transformation from Path Frame to World Frame. The remainder of this
section refers to the time interval [t i ,t i+1 ), and so index i can be dropped.
2
Let (x, y) ∈ R be the robot’s position in the world frame, and let θ be the
(slope) angle between the velocity vector V = (V x ,V y ) = (˙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 defines the velocity vector and even-
tually the path, x = (x, y), as a function of time. The normalized equations of
motion are

¨ x = p cos θ − q sin θ
¨ y = p sin θ + q cos θ
The angle θ between vector V and x axis of the world frame is found as



V y
 arctan , V x ≥ 0

V x
θ =

V y
 arctan + π, V x < 0

V x
To find the transformation from path frame to the world frame (x, y),present the
velocity in the path frame as V = V t. Angle θ is defined 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
θ = q/V
˙