Page 185 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

160 ACCOUNTING FOR BODY DYNAMICS: THE JOGGER’S PROBLEM

4.3.1 The Model
To a large extent the model that we will use in this section is similar to the model
used by the Maximum Turn Strategy above. There are also some differences. For
convenience we hence give a complete model description here.
2
As before, the scene is two-dimensional physical space W ≡ (x, y) ⊂ ;it
may include a finite set of locally finite static obstacles O ∈ W. Each obstacle
O k ∈ O is a simple closed curve of arbitrary shape and of finite length, such that
a straight line will cross it in only a finite number of points. Obstacles do not
touch each other; if they do, they are considered one obstacle.
The robot is a point mass,ofmass m. Its vision sensor allows it to detect any
obstacles and the distance to them within its sensing range (radius of vision)—a
disk D(C i ,r v ) of radius r v centered at its current location C i .Atmoment t i ,the
robot’s input information includes its current velocity vector V i and coordinates
of C i and of target location T .
The robot’s means to control its motion are two components of the accel-
eration vector u = f/m = (p, q),where m is the robot mass and f the force
applied. Controls u come from a set u(·) ∈ U of measurable, piecewise continu-
2
ous bounded functions in , U ={u(·) = (p(·), q(·))/p ∈ [−p max ,p max ], q ∈
[−q max ,q max ]}. By taking mass m = 1, we can refer to components p and q as
control forces, each within a fixed range |p|≤ p max , |q|≤ q max ; p max ,q max > 0.
Force p controls the forward (or backward when braking) motion; its positive
direction coincides with the robot’s velocity vector V.Force q, the steering con-
trol, is perpendicular to p, forming a right pair of vectors (Figure 4.8). There is
no friction: For example, given velocity V, the control values p = q = 0 will
result in a constant-velocity straight-line motion along the vector V.
Without loss of generality, assume that no external forces except p and q act
on the system. Note that with this assumption our model and approach can still
handle other external forces and constraints using, for example, the technique
suggested in Ref. 95, whereby various dynamic constraints such as curvature,
engine force, sliding, and velocity appear in the inequality describing the limi-
tations on the components of acceleration. The set of such inequalities defines a
convex region of the ¨x ¨y space. In our case the control forces act within the inter-
section of the box [−p max ,p max ] × [−q max ,q max ], with the half-planes defined
by those inequalities.
The task is to move in W from point S (start) to point T (target) (see
Figure 4.1). The control of robot motion is done in steps i, i = 0, 1, 2,... .Each
step i takes time δt = t i+1 − t i = const; the path length within time interval
δt depends on the robot velocity V i .Steps i and i + 1 start at times t i and
t i+1 , respectively; C 0 = S. Control forces u(·) = (p, q) ∈ U are constant within
the step.
We define three coordinate systems (follow Figure 4.8):

• The world frame, (x, y), is fixed at point S.
• The primary path frame, (t, n), is a moving (inertial) coordinate frame. Its
origin is attached to the robot; axis t is aligned with the current velocity