  This article examines a symbolic numerical approach to optimize a vehicle's
track for autonomous driving and collision avoidance. The new approach uses the
classical cost function definition incorporating the essential aspects of the
dynamic state of the vehicle as position, orientation, time sampling, and
constraints on slip angles of tires. The optimization processes minimize the
cost function and simultaneously determine the optimal track by varying
steering and breaking amplitudes. The current velocity of the vehicle is
limited to a maximal velocity, thus, allowing a stable search of the optimal
track. The parametric definition of obstacles generates a flexible environment
for low and high speed simulations. The minimal number of influential
optimization variables guarantees a stable and direct generation of optimal
results. By the current new approach to control a vehicle on an optimal track,
we are able to autonomously move the vehicle on an arbitrary track approximated
by low order polynomials. The optimization approach is also able to deal with a
variety of different obstacles and the corresponding optimal smooth obstacle
path. The computations demonstrate the effective control of a four wheel
vehicle in normal operation and exceptional obstacle avoidance with
continuously differentiable obstacle avoidance tracks. Simulation tests are
done using vehicle's velocities of 3m/s, 6m/s, 7.6m/s, 10m/s, 12m/s, and 18m/s.
At higher vehicle's velocities, a mathematical-only approach is not sufficient
and a mechanical intervention for tires is needed as a complimentary part to
control the slip angle. The results shows that the cost function reached a
considerably high average convergence-to-zero rate success in most of the
tested scenarios.
