Page 191 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

166 ACCOUNTING FOR BODY DYNAMICS: THE JOGGER’S PROBLEM

Since p and q are constant over the time interval t ∈ [t i ,t i+1 ), the solution for
V(t) and θ(t) within the interval becomes

V(t) = pt + V 0
q log(1 + tp/V i )
θ(t) = θ 0 + (4.13)
p
where θ 0 and V 0 are constants of integration and are equal to the values of θ(t i )
and V(t i ), respectively. By parameterizing the path by the value and direction
of the velocity vector, the path can be mapped into the world frame (x, y) using
the vector integral equation

t i+1

r(t) = V · t · dt (4.14)
t i
Here r(t) = (x(t), y(t)),and t is a unit vector of direction V, with the projections
t = (cos(θ), sin(θ)) onto the world frame (x, y). After integrating Eq. (4.14),
obtain the set of solutions

2p cos θ(t) + q sin θ(t) 2
x(t) = V (t) + A
2
4p + q 2
q cos θ(t) − 2p sin θ(t) 2
y(t) =− V (t) + B (4.15)
4p + q 2
2
where terms A and B are
2
V 0 (2 p cos(θ 0 ) + q sin(θ 0 ))
A = x 0 −
2
4 p + q 2
2
V 0 (q cos(θ 0 ) − 2 p sin(θ 0 ))
B = y 0 +
2
4 p + q 2
and V(t) and θ(t) are given by (4.13).
Equations (4.15) describe a spiral curve. Note two special cases: When p = 0
and q = 0, Eqs. (4.15) describe a straight-line motion under the force along the
vector of velocity; when p = 0and q = 0, the force acts perpendicular to the
2
vector of velocity, and Eqs. (4.15) produce a circle of radius V /|q| centered at
0
the point (A, B).
4.3.4 Canonical Solution
Given the current position C i = (x i ,y i ), the intermediate target T i , and the veloc-
ity vector V i = (˙x i , ˙y i ), the canonical solution presents a path that, assuming no
obstacles, would bring the robot from C i to T i with zero velocity and in minimum
time. The L ∞ -norm assumption allows us to decouple the bounds on accelera-
tions in ξ and η directions, and thus treat the two-dimensional problem as a set