Page 178 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

MAXIMUM TURN STRATEGY 153

Since the control forces are constant over time interval [t i ,t i+1 ), within this
interval the solution for V(t) and θ(t) becomes

V(t) = V 0 + pt
tp
q log(1 + )
V i
θ(t) = θ 0 + (4.5)
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 onto the world frame using the
vector integral equation

t i+1
r(t) = V · dt (4.6)
t i
Here r(t) = (x(t), y(t)) and V = (V · cos(θ), V · sin(θ)) are the projections of
vector V onto the world frame (x, y). After integrating Eq. (4.6), we obtain a set
of solutions in the form

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 (4.7)
2
4p + q 2
where terms A and B are
2
V 0 (2p cos(θ 0 ) + q sin(θ 0 ))
A = x 0 −
2
4p + q 2
2
V 0 (q cos(θ 0 ) − 2p sin(θ 0 ))
B = y 0 +
2
4p + q 2
Equations (4.7) are directly defined by the control variables p and q; V(t) and
θ(t) thereinare givenbyEq. (4.5).
In general, Eqs. (4.7) describe a spiral curve. Note two special cases: When
p = 0,q = 0, Eqs. (4.7) describe a straight-line motion along the vector of veloc-
2
ity; when p = 0and q = 0, Eqs. (4.7) produce a circle of radius V /|q| centered
0
at the point (A, B).
Selection of Control Forces. We now turn to the control law that guides the
selection of forces p and q at each step i, for the time interval [t i ,t i+1 ).To
ensure a reasonably fast convergence to the intermediate target T i , those forces
are chosen such as to align, as fast as possible, the direction of the robot’s