In this chapter we first develop a constraint tightening strategy that uses the logistic function and rotation approach to conservatively approximate the analytical solution for the fundamental dimension of the geometric collision area according to the specified confidence threshold and the uncertainty distribution. We then implement the proposed constraint tightening method to multi-vehicle grouping frame work and solve a stochastic MPC based motion planning problem for collision avoidance and traffic interaction. In the numerical experiment, we compare the approaches with and without using the constraint tightening method as well as vehicle grouping method. The statistic results shows that the specified confidence threshold for collision avoidance can be always satisfied when applying the proposed constraint tightening method. In addition, the vehicle grouping approach can help to exclude the undesired local minimum for the non-convex MPC based motion planning problem with a warm start strategy based numerical solver. Therefore, it can facilitate the motion planning of autonomous vehicle. In the future, we will work on finding a way to adjust the conservatism of the constraint tightening approach while also to ensure a safety.
CONCLUSIONS AND FUTURE WORK
1. Conclusion
The dissertation focus on predictive motion planning and control for autonomous driving with the consideration of 1) selection and planning of multiple maneuvers, 2) handling the stochastic uncertainties from state observation and 3) accommodating the dynamic environment with multiple object vehicles.
To deal with the above considerations, firstly, a hybrid predictive trajectory guidance framework is proposed that models the maneuvers of ego vehicle (tracking a specific reference speed on a specific lane) with particle description under predictive control as a hybrid control system. The multiple maneuver options provide the autonomous vehicle with flexibility and robustness in accommodating the complex traffic scenarios with proper behaviors comparing to the case with only one maneuver options such like collision avoidance.
Then the predictive reference speed assignment is combined with each optional reference lane in the configuration of the optimization of maneuver selection problem, thus realizing the predictive maneuver planning. Relaxation method is then applied to transform the mixed integer programming problem into nonlinear programming problem for real time application. Comparing with the hybrid predictive trajectory guidance framework, the predictive maneuver planning have better solutions in maneuver selection as the predictive motion of the surrounding traffic are considered and the switches among the optional maneuvers at each step of predictive horizon are optimized. However, this framework suffers from the local minimum do to the non-convex problem formulation. Therefore, a
forced maneuver selection (e.g. lane change) due to the reference speed adjustment is used to help the optimization problem jump out of the undesired local minimum. In addition, the obstacle filtering algorithm with vehicle grouping strategy is also used to effectively exclude the undesired local minimum via generating a more conservative collision field for the group of object vehicles with close proximity in positions.
When stochastic uncertainties are induced, a probabilistic evolution model is applied to track the structure, state and boundary of the object vehicle group. The probabilistic collision/closeness criteria is derived between any two individual object vehicles with non-negligible geometric size and motion uncertainties based on their state estimation via Bayesian tracking. Based on the closeness evaluation, a density-based method is applied to group/cluster the IOVs without a prior guess about the number of groups. The Monte Carlo Simulation with large samples (1000~100000) is used to demonstrate the correctness of the derived integral expression in evaluating the probabilistic closeness. In addition, while computationally more efficient approaches of closeness evaluation that ignore geometric sizes exist, they could not resolve group attributes and are not applicable for road vehicles.
Based on the probabilistic closeness/collision evaluation, a constraint tightening strategy is developed via deriving the approximated analytical solution for the fundamental dimension of the collision area with a specified confidence threshold and the estimated uncertainty distribution. The constraint tightening method is also extended to multi-vehicle grouping framework and implemented in solving the predictive maneuver planning problem with probabilistic collision avoidance constraint. The performance of the
constraint tightening strategy is demonstrated via statistic study based on the results of 100 Monte Carlo simulation samples, which shows that the specified confidence threshold can be always satisfied. Also, the vehicle group behaviors like merge and split clearly indicates the driving intention change of the object vehicles and therefore affects the maneuver plan of the autonomous vehicle.
2. Future work
In the future, the following problems need further exploration:
The feasibility issue of the predictive maneuver planning with observation uncertainties. As the proposed predictive maneuver planning and control framework is based on the formulation of an MPC problem with various constraints. The successful implementation of the work is highly depends on the feasible solution found by the solver. However, in reality, due to the complexity of the dynamic environment (e.g. unexpected object vehicle maneuvers) and the uncertain condition of the sensing device (sensing range/view limitations, delays, faults and clutter), the feasible solutions might not be always found. Therefore, a feasibility study and perhaps some reactive approached to avoid an inevitable collision condition are still open for research.
The accurate and efficient motion/interaction prediction of the detected object
vehicles and vehicle groups. As the predictive maneuver/motion planning method require the future information of the surrounding traffic environment, the accurate, efficient prediction of the intention, motion of the other traffic participants are very important. Furthermore, the maneuver of ego vehicle will
have influence on the other object vehicles (ego vehicles or human driven vehicles), thus the interaction between them needs to be considered and predicted to improve the performance of the maneuver/motion planning framework.
Conservatism of the constraint tightening approach. While with applying the
constraint tightening approach, the specified confidence threshold is satisfied, the distribution of the collision probability are too conservative comparing to the confidence threshold, especially when the size of the geometric shape of the object vehicle is large(e.g. a vehicle group case). Therefore, a safe way to tune the conservatism of the constraint tightening approach is open for research.
APPENDIX
Sufficient Condition for Non-Overlapping of Two Ellipses with Parallel Axles
Use the parametric equation to describe the position of the ellipse defined in equation (4.3), (4.4), we obtain: 1 1 1 1 1 1 1 1 cos sin E E x x a y y b (A.1) 2 2 2 2 2 2 2 2 cos sin E E x x a y y b (A.2)
Considering the externally tangential condition of the two ellipses defined in equation (A.1),(A.2), as shown in Figure A-1, the position of the intersection between the two ellipses at the comment tangent should satisfied:
1 1 2 2 2 1 cot cot b b a a (A.3)
Assume ellipse 1 is fixed, combining equation (A.1)-(A.2), the algebraic equation for the center of ellipse 2 that externally tangential to the ellipse 1 can be derived by:
2 2 1 2 2 2 2 2 2 1 1 1 2 2 1 2 1 1 1 2 2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 2 2 1 cos cos sin sin cos sin a b x x a a b a b a b y y b a b a b (1) which is bounded by 2 1 1 1 2 1 1 2 1 1 2 1 2 2 1 1 2 2 2 1 1 2 cos min[ , ] sin min[ , ] a b x x a a b a b a b y y b a b a b (2)
Thus the sufficient condition for the split of ellipse 2 from ellipse 1 can be defined by: 2 2 2 1 2 2 2 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 min[ , ] min[ , ] x x y y a b a b a b a b a b a b a b (3)
Similarly, if ellipse 2 is fixed, the sufficient condition for the split of ellipse 1 from ellipse 2 can be defined by:
1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 2 2 1 1 min[ , ] min[ , ] x x y y a b a b a b a b a b a b a b (4)
Therefore, simultaneously satisfying equation (A.6) and (A.7) guarantees the non- overlap of the two ellipses with parallel axles.
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