Trajton

Trajton is a scalar quantity used in motion planning and kinematics to quantify the overall cost or load of a trajectory. It serves as a single metric to compare alternative paths by combining aspects of motion efficiency, safety, and timing.

Mathematically, a trajton is defined as the integral of a local cost function along a trajectory γ from time t0 to t1: T = ∫ c(γ(t), γ˙(t), γ¨(t)) dt. A common choice for the local cost includes curvature, velocity, and acceleration, for example c = α κ(t)^2 + β v(t)^2 + γ a(t)^2, where κ is curvature, v is speed, a is acceleration, and α, β, γ are weighting parameters chosen to reflect mission priorities. The coefficients determine the relative emphasis on smoothness, speed, or energy use. Depending on the normalization, trajton can be dimensionless or carry units of energy or time.

Etymology and usage have roots in the idea of treating a trajectory as a measurable “load” or action. The term combines the notion of a trajectory with a unit-like suffix to emphasize a scalar cost associated with the path. It gained traction in theoretical discussions of path optimization and has seen practical adoption in robotics, autonomous vehicles, and computer animation.

Applications and limitations: trajtons enable straightforward comparison of candidate trajectories, support optimization routines, and facilitate multi-objective decision-making. However, results depend on chosen cost terms and weights, and the metric is not unique. Normalization is often required to ensure fair comparisons across different planners or environments.

See also: trajectory planning, path cost, optimal control.