Trajectory Criticality Index (TCI)#

Description#

The ac{TCI} metric models criticality using an optimization problem [Junietz2018]. The task is to find a minimum difficulty value, i.e. how demanding even the easiest option for the vehicle will be under a set of physical and regulatory constraints. For example, if the constraint is to avoid obstacles, then driving straight towards an obstacle and being only a few seconds away requires a large change in steering angle and acceleration to satisfy the constraint of collision avoidance.

Here, the possible set of vehicle actions are not only constrained by physically possible behavior; it additionally shall adhere to a mathematically modeled set of requirements. Said requirements are based on the necessary longitudinal (\(a_x\)) and lateral acceleration (\(a_y\)) to avoid collisions as well as the margin (‘reserve’) for corrections in speed (\(R_x\)) and course angle (\(R_y\)). Since both \(R_x\) and \(R_y\) are dependent on \(a_x\) and \(a_y\), it suffices to minimize the combined function w.r.t. \(a_x\) and \(a_y\). The requirements include concepts such as holding a safe following distance and maximizing distance to obstacles.

Assuming the vehicle behaves according to Kamm’s circle, acs{TCI} for a scene $S$ with an ego vehicle $A_1$ reads as .. math:

\mathit{TCI}(A_1,S,t,t_H) = \min_{a_x, a_y} \sum_{\tilde{t}=t}^{t+t_H} w_x R_x(\tilde{t}) + w_y R_y^2(\tilde{t}) + \frac{w_{\mathit{ax}} a_x^2(\tilde{t}) + w_{\mathit{ay}} a_y^2(\tilde{t})}{(\mu_\mathit{max}g)^2}

where \(t_H\) is the prediction horizon, \(a_x\) and \(a_y\) the longitudinal and lateral accelerations, \(\mu_\mathit{max}\) the maximum coefficient of friction, \(g\) the gravitational constant, \(w\) weights, and \(R_x\) and \(R_y\) the longitudinal and lateral margin for angle corrections:

\[\begin{split}R_x(t) = \frac{\max(0, x(t) - r_x(t))}{d_x(t)},\\ R_y^2(t) = \frac{(y(t) - r_y(t))^2v(t-t_s)}{d_y^2(t)v_\mathit{max}}.\end{split}\]

Here, \(x(t)\), \(y(t)\) is the position, \(t_s\) the discrete time step size, \(v_\mathit{max}\) the maximum velocity, \(r_x(t)\) the reference for a following distance (set to \(2\text{s} \cdot v(t)\)), \(r_y\) the position with the maximum lateral distance to all obstacles in \(S\), \(d_x(t)\), \(d_y(t)\) the maximum longitudinal and lateral deviations from \(r_x\), \(r_y\).

Properties#

Run-time capability#

Yes, but not designed for active trajectory control

Target values#

None found

Subject type#

Road vehicles (esp. automated)

Scenario type#

Mostly highways

Inputs#

Velocities \(v_i\), positions \(p_i\), following distance \(r_x\), position maximizing lateral distances \(r_y\), \(\mu_\mathit{max}\) maximum coefficient of friction

Output scale#

\([0, \infty]\), number, ratio scale

Reliability#

High, under the assumption of a reliable encoding of safety-critical factors in constraints

Validity#

High, as demonstrated by an expert-based assessment of the metric’s results in four scenarios, but also found to be dependent on the cost function and constraints [Junietz2018]

Sensitivity#

High, no false negative identified in initial expert-based validation [Junietz2018], but also depends on cost function and constraints

Specificity#

High, no false positive identified in initial expert-based validation [Junietz2018], but also depends on cost function and constraints

Prediction model#

Time window#

Unbound, but depends on computational power and choice of cost function and constraints

Time mode#

Branching time