Required Longitudinal Acceleration (\({a}_{\mathit{long,req}}\))¶

Description¶

For two actors \(A_1\), \(A_2\) at time \(t\), \({a}_{\mathit{long,req}}\) measures the maximum longitudinal backward acceleration required, on average, by actor \(A_1\) to avoid a collision in the future. It can be formalized as

\[{a}_{\mathit{long,req}}(A_1, A_2, t) = \max \{ a_{1,\mathit{long}} \le 0 ~|~\forall \, \tilde{t} \ge 0: d(p_1(t+\tilde{t}),p_2(t+\tilde{t})) > 0\}\,.\]

The \({a}_{\mathit{long,req}}\) can be adapted for the situation where the acceleration of \(A_1\) needs to be positive in order to avoid a collision by taking the minimum \(a_{1,\mathit{long}} \ge 0\) instead. An interesting special case, cf. [Jansson2005], is exhibited when constant acceleration of the actors is assumed, resulting in

\[{a}_{\mathit{long,req}}(A_1, A_2, t) = \min\Big(a_{2,\mathit{long}} + \frac{(v_{1,\mathit{long}}(t)-v_{2,\mathit{long}}(t))^2}{2d(p_1(t),p_2(t))}, 0\Big)\,.\]

For constant acceleration, the concept of \({a}_{\mathit{long,req}}\) is also known under the term Deceleration Rate To Avoid Crash (DRAC) [Archer2005]. Similarly, the \({a}_{\mathit{lat,req}}\) metric [Jansson2005] is defined as the minimal absolute lateral acceleration required for a steering maneuver to evade collision.

Properties¶

Run-time capability¶

Yes

Target values¶

-6 m/s² [Stellet2016] (AEB), \(\mu \cdot g\) [Jeppsson2018] (point of no return), -3.4 m/s² [Huber2020] (scenario classification), \([-8,-4]\) m/s², dependent on speed [Benmimoun2011] (incident detection), -5 m/s² [UNECE157] (Requirement on emergency maneuver deployment in ALKS)

Subject type¶

Road vehicles (automated and human)

Scenario type¶

Intersecting predicted paths for a significant time span in the scenario

Inputs¶

\(v_i, a_i, p_i\) for \(i \in \{1,2\}\) assuming the constant acceleration motion model

Output scale¶

\((-\infty, \infty)\), acceleration (m/s²), ratio scale

Reliability¶

High, under the assumption that the non-collision condition can be reliably predicted

Validity¶

High, but only longitudinal evasion considered, knowledge on vehicle capabilities necessary for interpretation [Zheng2019]

Sensitivity¶

High, as most critical situations between two actors impose a high required acceleration at some point, more sensitive than CPI [Guido2011]

Specificity¶

Medium, as there exists situations with intersecting paths of actors, but planned trajectory is deviating (e.g. turning maneuvers)