Page 190 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

MINIMUM TIME STRATEGY 165

the path, (x(t), y(t)), as a function of time. Taking mass m = 1, the equations
of motion become

¨ x = p cos θ − q sin θ
¨ y = p sin θ + q cos θ

The angle θ between vector V = (V x ,V y ) and x axis of the world frame is
found as



V y
 arctan , V x ≥ 0

V x
θ =
 arctan V y + π, V x < 0

V x
The transformations between the world frame and secondary path frame, from
(x, y) to (ξ, η) and from (ξ, η) to (x, y), are given by
ξ x − x T

η = R y − y T (4.11)
and
x ξ x T

y = R η + y T (4.12)

where

cos θ sin θ
R =
− sin θ cos θ
R is the transpose matrix of the rotation matrix between the frames (ξ, η) and

(x, y),and (x T ,y T ) are the coordinates of the (intermediate) target in the world
frame (x, y).
To define the transformations between the world frame (x, y) and the primary
path frame (t, n), write the velocity in the primary path frame as V = V t.To
find the time derivative of vector V with respect to the world frame (x, y), note
that the time derivative of vector t in the primary path frame (see Section 4.3.1)
is not equal to zero. It can be defined as the cross product of angular velocity
ω = θb of the primary path frame and vector t itself: ˙ t = ω × t, where angle θ
˙
is between the unit vector t and the positive direction of x axis. Given that the
control forces p and q act along the t and n directions, respectively, the equations
of motion with respect to the primary path frame are

V = p
˙
˙
θ = q/V