nits:
this could be heavy.
I think there are better ways to make it faster, such as extracting the closest object, the closest point, etc.

pseudo code

backward_objects = extraceBackwardObjects(lanelets, ego_pose, objectse); // extract objects whose arc length is less than ego's one
closest_backward_object = getClosestObject(backward_objects); // get object whose arc length is max
closet_object_corner_point = getClosestCornerPoint(closest_backward_object); // get cornaner whose arc length is max
most_backward_ego_corner_point = getMostBackwardEgoCornerPoint(ego_pose, vehicle_footprint); // get corner whose arc length is min
distance = most_backward_ego_corner_point - closet_object_corner_point; // get arc length between two points

Thanks!
I will consider improving the efficiency

In my current implementation, distance a is calculated but distance b should be calculated, correct?
If so, the method of extracting objects for back distance against back object should be changed. In this sketch, only object [2] should be extracted.

And for caring lateral collision implemented with the following implementetion. objects[0,1,2] should be extracted 🤔

      (utils::checkCollisionBetweenFootprintAndObjects(
        local_vehicle_footprint, *backed_pose, back_stop_objects_in_pull_out_lanes,
        parameters_->objects_collision_check_margins.back())))

I will consider improving the efficiency

thanks! but it's ok if it takes time to improve because the check target objects may be not so many

In my current implementation, distance a is calculated but distance b should be calculated, correct?

no, I think distance a should be calculated,

And for caring lateral collision implemented with the following implementetion. objects[0,1,2] should be extracted 🤔

I agree

how about creating polygon along the start_pose_candidates_path?

And extracts closest object overlaps with the polygon

a: objects_collision_check_margins.back()
b: back_objects_collision_check_margin

It's heavier but checking lateral distance of object pose is not considering the size of the object?

looks good!
the attention area is along the start_pose_candidates_path not just nagative x-coords direction, right?

which pattern did you mean?

and, better to check just side to ego?

(sorry for my mistake, this pseudo code did not consider footprint)

#6149 (comment)

    const auto obj_arc_coods = getArcCoordinates(lanelets, object.pose.position);
    if(std::abs(ego_arc_coords.distance -obj_arc_coods.distance) > lateral_threshold){
      continue;
    }

not just check object.pose.position but need to check 4 footprint points by like std::any