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Fastest motion planning for an unmanned vehicle in the presence of accelerating obstacles

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Abstract

Estimating the fastest trajectory is one of the main challenges in autonomous vehicle research. It is fundamental that the vehicle determines its path not only to minimize travel time, but to arrive at the destination safely by avoiding any obstacles that may be in collision route. In this paper, we consider estimating the trajectory and acceleration functions of the trip simultaneously with an optimization objective function. By approximating the trajectory and acceleration function with B-splines, we transform an infinite-dimensional problem into a finite-dimensional one. Obstacle avoidance and kinematic constraints are carried out with the addition of a penalization function that penalizes trajectories and acceleration functions that do not satisfy the vehicles’ constraints or that are in a collision route with other obstacles. Our approach is designed to model observations of the obstacles that contain measurement errors, which incorporates the realistic stochasticity of radars and sensors. We show that, as the number of observations increases, the estimated optimization function converges to the optimal one where the obstacles’ positions are known. Moreover, we show that the estimated optimization function has a minimizer and that its minimizers converge to the minimizers of the optimization function involving the true threat zones.

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References

  • Atta D, Subudhi B (2013) Decentralized formation control of multiple autonomous underwater vehicles. Int J Robot Autom 28:303–310

    Google Scholar 

  • Awerbuch B, Betke M, Rivest RL, Singh M (1999) Piecemeal graph exploration by a mobile robot. Inf Comput 152:155–172

    MathSciNet  MATH  Google Scholar 

  • Barraquand J, Latombe J (1993) Nonholonomic multibody mobile robots: controllability and motion planning in the presence of obstacles. Algorithmica 10:121–155

    MathSciNet  MATH  Google Scholar 

  • Belkouche F (2009) Reactive path planning in a dynamic environment. IEEE Trans Robot 25:902–911

    Google Scholar 

  • Bodin P, Villemoes LF, Wahlberg B (2000) Selection of best orthonormal rational basis. SIAM J Control Optim 38(4):995–1032

    MathSciNet  MATH  Google Scholar 

  • Borenstein J, Koren Y (1989) Real-time obstacle avoidance for fast mobile robots. IEEE Trans Syst Man Cybern 19:1179–1187

    Google Scholar 

  • Borenstein J, Koren Y (1991) The vector field histogram-fast obstacle avoidance for mobile robots. IEEE Trans Robot Autom 7:278–288

    Google Scholar 

  • Borrelli F, Falcone P, Keviczky T, Asgari J, Hrovat D (2005) Mpc-based approach to active steering for autonomous vehicle systems. Int J Veh Auton Syst 3:265–291

    Google Scholar 

  • Bouthemy P (1989) A maximum likelihood framework for determining moving edges. IEEE Trans Pattern Anal Mach Intell 11:499–511

    Google Scholar 

  • Brooks RA (1982) Solving the find path problem by representing free space as generalized cones. Massachusetts Institute Technology, A.I Memo No 674

  • Chiang H-T, Malone N, Lesser K, Oishi M, Tapia L (2015a) Aggressive moving obstacle avoidance using a stochastic reachable set based potential field. In: Springer tracts in advanced robotics book series, Algorithmic foundations of robotics XI, pp 73–89

  • Chiang H-T, Malone N, Lesser K, Oishi M, Tapia L (2015b) Path-guided artificial potential fields with stochastic reachable sets for motion planning in highly dynamic environments. In: IEEE international conference on robotics and automation

  • Chiang H-T, Rackley N, Tapia L (2015c) Stochastic ensemble simulation motion planning in stochastic dynamic environments

  • Coutinho WP, Battarra M, Fliege J (2018) The unmanned aerial vehicle routing and trajectory optimisation problem, a taxonomic review. Comput Ind Eng 120:116–128

    Google Scholar 

  • de Boor C (1978) A practical guide to splines, applied mathematical sciences. Springer, New York

    MATH  Google Scholar 

  • de Ponte Muller F (2017) Survey on ranging sensors and cooperative techniques for relative positioning of vehicles. Sensors (Basel) 17:271

    Google Scholar 

  • DeVore R, Petrova G, Temlyakov V (2003) Best basis selection for approximation in l p. Found Comput Math 3(2):161–185

    MathSciNet  MATH  Google Scholar 

  • Dias R, Garcia N, Zambom AZ (2010) A penalized nonparametric method for nonlinear constrained optimization based on noisy data. Comput Optim Appl 45:521–541

    MathSciNet  MATH  Google Scholar 

  • Dias R, Garcia N, Zambom AZ (2012) Monte Carlo algorithm for trajectory optimization based on Markovian readings. Comput Optim Appl 51:305–321

    MathSciNet  MATH  Google Scholar 

  • Donald B (1984) Motion planning with six degrees of freedom. Massachusetts Institute Technology Artificial Intelligence Laboratory, Technical report AIM-791

  • Elnagar A, Gupta K (1998) Motion prediction of moving objects based on auto regressive model. IEEE Trans Syst Man Cybern 28:803–810

    Google Scholar 

  • Ferguson D, Stentz A (2006) Using interpolation to improve path planning: the field d* algorithm. J Field Robot 23:79–101

    MATH  Google Scholar 

  • Fiorini P, Shiller Z (1993) Motion planning in dynamic environments using the relative velocity paradigm. In: Proceedings of IEEE international conference on robotics and automation, pp 560–565

  • Fujimura K, Samet H (1989) A hierarchical strategy for path planning among moving obstacles. IEEE Trans Robot Autom 5:61–69

    Google Scholar 

  • Galceran E, Carreras M (2013) A survey on coverage path planning for robotics. Robot Auton Syst 61:1258–1276

    Google Scholar 

  • Giyanani A, Bierbooms W, van Bussel G (2015) Lidar uncertainty and beam averaging correction. Adv Sci Res 12:85–89

    Google Scholar 

  • Gong C, Tully S, Kantor G, Choset H (2012) Multi-agent deterministic graph mapping via robot rendezvous. In: IEEE international conference on robotics and automation

  • Gonzalez J, Dornbush A, Likhachev M (2012) Using state dominance for path planning in dynamic environments with moving obstacles. In: Proceedings of the IEEE international conference on robotics and automation, pp 4009–4015

  • Goulden T, Hopkinson C (2013) Quantification of LiDAR measurement uncertainty through propagation of errors due to sensor sub-systems and terrain morphology. In: AGU fall meeting abstracts

  • Hsu D, Kindel R, Latombe J-C, Rock S (2002) Randomized kinodynamic motion planning with moving obstacles. Int J Robot Res 21:233–255

    MATH  Google Scholar 

  • Hsu D, Latombe J, Motwani R (1997) Path planning in expansive configuration spaces. In: Proceedings of IEEE international conference on robotics and automation, pp 2719–2726

  • Huang C, Shi S, Wang X, Chung W (2005) Parallel force/position controls for robot manipulators with uncertain kinematics. Int J Robot Autom 20:158–168

    Google Scholar 

  • Kala R, Shukla A, Tiwari R (2012) Robot path planning using dynamic programming with accelerating nodes. Paladyn 3:23–24

    Google Scholar 

  • Kambhampati S, Davis LS (1986) Multiresolution path planning for mobile robots. Int J Robot Res 5:90–98

    Google Scholar 

  • Karlin S (1973) Some variational problems on certain Sobolev spaces and perfect splines. Bull Am Math Soc 79:124–128

    MathSciNet  MATH  Google Scholar 

  • Kavraki L, Svestka P, Latombe J, Overmars M (1996) Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans Robot Autom 12:566–580

    Google Scholar 

  • Koenig S, Likhachev M (2002) D* lite. In: Proceedings of the AAAI conference of artificial intelligence, pp 476–483

  • Krishnan J, Rajeev U, Jayabalan J, Sheela D (2017) Optimal motion planning based on path length minimisation. Robot Auton Syst 94:245–263

    Google Scholar 

  • Kruse T, Pandey AK, Alami R, Kirsch A (2013) Human-aware robot navigation: a survey. Robot Auton Syst 61:1726–1743

    Google Scholar 

  • Kuwata Y, Teo J, Fiore G, Karaman S, Frazzoli E, How JP (2009) Real-time motion planning with applications to autonomous urban driving. IEEE Trans Control Syst Technol 17:1105–1118

    Google Scholar 

  • Lachout P, Liebscher E, Vogel S (2005) Strong convergence of estimators as \(\epsilon _n\)-minimisers of optimization problems. Ann Inst Stat Math 57:291–313

    MATH  Google Scholar 

  • LaValle S (1998) Rapidly-exploring random trees: a new tool for path planning. Technical Report 98-11, Computer Science Dept., Iowa State University

  • Lindemann S, LaValle S (2009) Simple and efficient algorithms for computing smooth, collision-free feedback laws over given cell decompositions. Int J Robot Res 28:600–621

    Google Scholar 

  • Luus R (2007) Choosing grid points in solving singular optimal control problems by iterative dynamic programming. In: Proceedings of the 10th IASTED international conference on intelligent systems and control, p 592

  • Macrina G, Di Puglia Pugliese L, Guerriero F, Laporte G (2020) Drone-aided routing: a literature review. Transport Res Part C Emerg Technol 120:102762

    Google Scholar 

  • Mahdavian M, Shariat-Panahi M, Yousefi-Koma A, Ghasemi-Toudeshki A (2015) Optimal trajectory generation for energy consumption minimization and moving obstacle avoidance of a 4dof robot arm. In: 3rd RSI international conference on robotics and mechatronics, pp 353–358

  • Malone N, Lesser K, Oishi M, Tapia L (2014) Stochastic reachability based motion planning for multiple moving obstacle avoidance. In: Proceedings of the 17th international conference on hybrid systems: computation and control (New York, NY, USA), HSCC ’14. ACM, pp 51–60

  • Matveev AS, Wang C, Savki AV (2012) Real-time navigation of mobile robots in problems of border patrolling and avoiding collisions with moving and deforming obstacles. Robot Auton Syst 60:769–788

    Google Scholar 

  • Milford MJ (2008) Robot navigation from nature: simultaneous localisation, mapping, and path planning based on hippocampal models. Springer Tracts in Advanced Robotics, Berlin, p 41

  • Mitteta M-A, Nouiraa H, Roynarda X, Goulettea F, Deschauda J-E (2016) Experimental assessment of the quanergy m8 lidar sensor. In: The international archives of the photogrammetry, remote sensing and spatial information sciences, XXIII ISPRS Congress, Prague XLI-B5, pp 12–19

  • Nakhaei A, Lamiraux F (2008) A framework for planning motions in stochastic maps. In: 2008 10th international conference on control, automation, robotics and vision, pp 1959–1964

  • Nanao M, Ohtsuka T (2010) Nonlinear model predictive control for vehicle collision avoidance using c/gmres algorithm. In: IEEE international conference on control applications, Yokohama

  • Naranjo JE, Gonzalez C, Garcia R, Pedro T (2008) Lane-change fuzzy control in autonomous vehicles for the overtaking maneuver. IEEE Trans Intell Transport Syst 9:438–450

    Google Scholar 

  • Nash A, Daniel K, Koenig S, Felner A (2007) Theta*: any-angle path planning on grids. In: Proceedings of the national conference on artificial intelligence, pp 1177–1183

  • Ni J, Wu W, Shen J, Fan X (2014) An improved vff approach for robot path planning in unknown and dynamic environments. In: Computational Intelligence Approaches to Robotics, Automation, and Control, vol 2014, Article ID 461237. https://doi.org/10.1155/2014/461237

  • Pandey A, Parhi DR (2017) Optimum path planning of mobile robot in unknown static and dynamic environments using fuzzy-wind driven optimization algorithm. Def Technol 13:47–58

    Google Scholar 

  • Park J, Kim D, Yoon Y, Kim HJ, Yi K (2009) Obstacle avoidance of autonomous vehicles based on model predictive control. Proc IMechE Part D J Automob Eng 223:1499–1516

    Google Scholar 

  • Park J, Kim HJ (2016) Generation of locally optimal trajectories against moving obstacles using gaussian sampling. In: 16th international conference on control, automation and systems, pp 273–277

  • Petres C, Pailhas Y, Patron P, Petillot Y, Evans J, Lane D (2007) Path planning for autonomous underwater vehicles. IEEE Trans Robot 23:331–341

    Google Scholar 

  • Qiang H, Yokoi K, Kajita S, Kaneko K, Arai H, Koyachi N, Tanie K (2001) Planning walking patterns for a biped robot. IEEE Trans Robot Autom 17:280–289

    Google Scholar 

  • Reif U (1997) Orthogonality of cardinal b-splines in weighted Sobolev spaces. SIAM J Math Anal 28:1258–1263

    MathSciNet  MATH  Google Scholar 

  • Reif U (1997) Uniform b-spline approximation in Sobolev spaces. Numer Algorithms 15:1–14

    MathSciNet  MATH  Google Scholar 

  • Rubagotti M, Vedova M, Ferrara A (2011) Time-optimal sliding mode control of a mobile robot in a dynamic environment. IET Control Theory Appl 5:1916–1924

    MathSciNet  Google Scholar 

  • Sahraei A, Manzuri MT, Razvan MR, Tajfard M, Khoshbakht S (2007) Real-time trajectory generation for mobile robots. In: Proceedings of the 10th congress of the Italian association for artificial intelligence on AI*IA, artificial intelligence and human-oriented computing, Rome

  • Sen PK, Singer JM (1993) Large sample methods in statistics. an introduction with applications. Chapman & Hall, New York

  • Sgorbissa A, Zaccaria R (2012) Planning and obstacle avoidance in mobile robotics. Robot Auton Syst 60:628–638

    Google Scholar 

  • Shiller Z, Large F, Sekhavat S (2001) Motion planning in dynamic environments: obstacles moving along arbitrary trajectories. In: Proceedings of IEEE international conference on robotics and automation

  • Silverman B (1986) Density estimation for statistics and data analysis. Chapman and Hall, London

    MATH  Google Scholar 

  • Stentz A (1994) Optimal and efficient path planning for partially-known environments. In: Proceedings of the 1994 IEEE international conference on robotics and automation, pp 3310 – 3317

  • Takano G, Obayashi M, Uto K (2015) Path planning for autonomous car to avoid moving obstacles by steering using tangent-arc-tangent-arc-tangent model. In: IEEE conference on control applications, pp 1702–1709

  • Tazir M, Azouaoui O, Hazerchi M, Brahimi M (2015) Mobile robot path planning for complex dynamic environments. In: International conference on advanced robotics, pp 200–206

  • Tiwari A, Chandra H, Yadegar J, Wang J (2007) Constructing optimal cyclic tours for planar exploration and obstacle avoidance: a graph theory approach. Advances in variable structure and sliding mode control. Springer, Berlin

    MATH  Google Scholar 

  • Toit NED, Burdick JW (2011) Robot motion planning in dynamic, uncertain environments. IEEE Trans Robot 28:101–115

    Google Scholar 

  • van den Berg J, Lin M, Manocha D (2008) Reciprocal velocity obstacles for real-time multi-agent navigation. In: Proceedings of IEEE international conference on robotics and automation, pp 1928–1935

  • Vidakovic B (1999) Statistical modeling by wavelets. Wiley series in probability and statistics: applied probability and statistics. Wiley-Interscience, New York

  • Volos CK, Kyprianidis M, Stouboulos N (2012) A chaotic path planning generator for autonomous mobile robots. Robot Auton Syst 60:651–656

    Google Scholar 

  • Wooden DT (2006) Graph-based path planning for mobile robots. Dissertation—Georgia Institute of Technology

  • Yang K, Sukkarieh S (2008) Real-time continuous curvature path planning of UAVs in cluttered environments. In: Proceedings of the 5th international symposium on mechatronics and its applications, pp 1–6

  • Ye C, Yung NHC, Wang D (2003) A fuzzy controller with supervised learning assisted reinforcement learning algorithm for obstacle avoidance. IEEE Trans Syst Man Cybern B 33:17–27

    Google Scholar 

  • Yoon Y, Shin J, Kim HJ, Park Y, Sastry S (2009) Model-predictive active steering and obstacle avoidance for autonomous ground vehicles. Control Eng Pract 17:741–750

    Google Scholar 

  • Yu J, LaValle SM (2016) Optimal multirobot path planning on graphs: complete algorithms and effective heuristics. IEEE Trans Robot 32:1163–1177

    Google Scholar 

  • Zambom AZ, Seguin B, Zhao F (2018) Robot path planning in a dynamic environment with stochastic measurements. J Glob Optim. https://doi.org/10.1007/s10898-018-0704-4

    Article  MATH  Google Scholar 

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Correspondence to Adriano Zanin Zambom.

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Appendices

Appendix

Proof of Lemma 3

Proof

We need to show that for all \({\varvec{\omega }}_0 =({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\in D_0(v_0)\subset {\mathbb {R}}^{K_1}\times {\mathbb {R}}^{K_2}\)

$$\begin{aligned} \lim _{{\varvec{\omega }}\rightarrow {\varvec{\omega }}_0 }\sup _{t\ge T_0}d(\mathbf{p}_t({\varvec{\omega }}_0 ),\mathbf{p}_t({\varvec{\omega }}))=0. \end{aligned}$$
(18)

By performing a change of variables, (7), which is used to implicitly define \(p_{t1}({\varvec{\omega }})\), can be rewritten in the form

$$\begin{aligned} \int _0^{p_{t1}({\varvec{\omega }})} G(u,{\varvec{\theta }})\, \mathrm{d}u=\int _0^{t-T_0} F(s,{\varvec{\omega }})\, \mathrm{d}s, \end{aligned}$$
(19)

where

$$\begin{aligned} G(u,{\varvec{\theta }})&=\sqrt{1+({\varvec{\theta }}^\top \mathbf{B}'(u))^2},\\ F(s,{\varvec{\omega }})&=v_0+\tau ({\varvec{\omega }})\int _0^{s/\tau ({\varvec{\omega }})} {\varvec{\beta }}^\top \mathbf{C}(u)\, \mathrm{d}u. \end{aligned}$$

Notice that G is continuous and \(G(u,{\varvec{\theta }})\ge 1\). Moreover, since \(\tau \) is continuous, so is F.

Since the functions \(B_k\) are only defined on [0, b] and the functions \(C_k\) are only defined on [0, 1], \(G(u,{\varvec{\theta }})\) is only defined for \(0\le u\le b\) and \(F(s,{\varvec{\omega }})\) is only defined for \(0 \le s\le \tau ({\varvec{\omega }})\). For the purposes of this proof, let us extend the B-spline basis functions \(B_k\) and \(C_k\), so that they are zero past their original domains of definition. Doing so means that \(G(u,{\varvec{\theta }})\) will be defined for all \(u\ge 0\) and \(F(s,{\varvec{\omega }})\) will be defined for all \(s\ge 0\). Moreover, this will result in \(F(s,{\varvec{\omega }})\) being constant for \(s\ge \tau ({\varvec{\omega }})\). Notice that \(F(\tau ({\varvec{\omega }}),{\varvec{\omega }})\) is related to the horizontal speed of the vehicle at the time it reaches the destination. Since the trajectory of the vehicle is determined by a function, it must approach the destination from the left. Hence, it follows that \(F(\tau ({\varvec{\omega }}),{\varvec{\omega }})\ge 0\). Putting this together with the fact that \(F(s,{\varvec{\omega }})\) is constant for \(s\ge \tau ({\varvec{\omega }})\), we obtain

$$\begin{aligned} F(s,{\varvec{\omega }})\ge 0,\quad \text {for all}\ s\ge \tau ({\varvec{\omega }}). \end{aligned}$$
(20)

These extensions allow us to define \(q_t({\varvec{\omega }})\) implicitly through

$$\begin{aligned} \int _0^{q_t({\varvec{\omega }})} G(u,{\varvec{\theta }})\, du=\int _0^{t-T_0} F(s,{\varvec{\omega }})\, ds \end{aligned}$$
(21)

for any \(t\ge T_0\). For \(t\in [T_0,T_0+\tau ({\varvec{\omega }})]\), \(q_t({\varvec{\omega }})=p_{t1}({\varvec{\omega }})\); however, for \(t\ge T_0+\tau ({\varvec{\omega }})\), by (20) and (21), \(q_t({\varvec{\omega }})\ge p_{1t}({\varvec{\omega }})\). We will begin by showing that for any \(M>T_0\)

$$\begin{aligned} \lim _{{\varvec{\omega }}\rightarrow {\varvec{\omega }}_0 }\sup _{t\in [T_0,M]}|q_t({\varvec{\omega }}_0 )-q_t({\varvec{\omega }})|=0. \end{aligned}$$
(22)

Fix \(t\ge T_0\) and \(({\varvec{\theta }},{\varvec{\beta }}),({\varvec{\theta }}_0,{\varvec{\beta }}_0)\in D_0(v_0)\). Under the assumption that \(q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\ge q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }})\), since \(G\ge 1\), by (21), we have

$$\begin{aligned} |q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }})|&=q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }})\\&\le \int _{q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }})}^{q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )} G(u,{\varvec{\theta }}_0 )\, \mathrm{d}u\\&\le \int _0^{t-T_0}|F(s,{\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-F(s,{\varvec{\theta }}_0 ,{\varvec{\beta }})|\, \mathrm{d}s. \end{aligned}$$

A similar argument shows that this inequality also holds if \(q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )< q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }})\). Similarly, under the assumption that \(q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\ge q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 )\), we have

$$\begin{aligned} |q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 )|&=q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 ) \le \int _{q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 )}^{q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )} G(u,{\varvec{\theta }}_0 )\, \mathrm{d}u\\&=\int _0^{t-T_0} F(s,{\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\, \mathrm{d}s-\int _0^{q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 )}G(u,{\varvec{\theta }}_0 )\, \mathrm{d}u\\&=\int _0^{t-T_0} F(s,{\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-F(s,{\varvec{\theta }},{\varvec{\beta }}_0 )\, \mathrm{d}s \\&\quad +\int _0^{q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 )}G(u,{\varvec{\theta }})-G(u,{\varvec{\theta }}_0 )\, \mathrm{d}u\\&\le \int _0^{t-T_0} |F(s,{\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )-F(s,{\varvec{\theta }},{\varvec{\beta }}_0 )|\, \mathrm{d}s\\&\quad +q_t({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\sup _{u\in [0,q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )]}|G(u,{\varvec{\theta }})-G(u,{\varvec{\theta }}_0 )|. \end{aligned}$$

A similar argument shows that this inequality also holds if \(q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )< q_{t}({\varvec{\theta }},{\varvec{\beta }}_0 )\).

The above inequalities show that to establish (22), it suffices to show that

$$\begin{aligned} \lim _{{\varvec{\theta }}\rightarrow {\varvec{\theta }}_0 }\sup _{t\in [T_0,M]}&q_t({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\sup _{u\in [0,q_{t}({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )]}|G(u,{\varvec{\theta }})-G(u,{\varvec{\theta }}_0 )|=0, \end{aligned}$$
(23)
$$\begin{aligned} \lim _{{\varvec{\omega }}\rightarrow {\varvec{\omega }}_0 } \sup _{t\in [T_0,M]}&\int _0^{t-T_0} |F(s,{\varvec{\omega }}_0 )-F(s,{\varvec{\omega }})|\, \mathrm{d}s=0. \end{aligned}$$
(24)

To obtain (23), notice that

$$\begin{aligned} |G(u,{\varvec{\theta }})-G(u,{\varvec{\theta }}_0 )|&\le \big | |{\varvec{\theta }}^\top \mathbf{B}(u)|-|{\varvec{\theta }}_0 ^\top \mathbf{B}(u)|\big |\\&\le |({\varvec{\theta }}-{\varvec{\theta }}_0 )^\top \mathbf{B}(u)|\\&\le |{\varvec{\theta }}-{\varvec{\theta }}_0 | M_2, \end{aligned}$$

where \(M_2=\sup _{u\ge 0 } |\mathbf{B}(u)|\). The result (23) then immediately follows from the fact that the function \(t\mapsto q_t({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\) is continuous. To obtain (24), write F in the form

$$\begin{aligned} F(s,{\varvec{\omega }})=v_0+\int _0^{se({\varvec{\omega }})} {\varvec{\eta }}({\varvec{\omega }})^\top \mathbf{C}(u)\, \mathrm{d}u, \end{aligned}$$

where

$$\begin{aligned} e({\varvec{\omega }})=1/\tau ({\varvec{\omega }})\qquad \text {and}\qquad {\varvec{\eta }}({\varvec{\omega }})=\tau ({\varvec{\omega }}){\varvec{\beta }}\end{aligned}$$

are continuous functions of \({\varvec{\omega }}\). For all \({\varvec{\omega }}\), we have

$$\begin{aligned} |F(s,{\varvec{\omega }}_0 )-F(s,{\varvec{\omega }})|&\le \Bigg |\int _{s e({\varvec{\omega }})}^{s e({\varvec{\omega }}_0 )} {\varvec{\eta }}({\varvec{\omega }}_0 )^\top \mathbf{C}(u)\, \mathrm{d}u\Bigg |\\&\qquad +\Bigg |\int _0^{s e({\varvec{\omega }})} ({\varvec{\eta }}({\varvec{\omega }}_0 )-{\varvec{\eta }}({\varvec{\omega }}))^\top \mathbf{C}(u)\, \mathrm{d}u\Bigg |\\&\le s |{\varvec{\eta }}({\varvec{\omega }}_0 )| |e({\varvec{\omega }}_0 )-e({\varvec{\omega }})| M_3 + s|e({\varvec{\omega }})| |{\varvec{\eta }}({\varvec{\omega }}_0 )-{\varvec{\eta }}({\varvec{\omega }})| M_3\\&=sM_3\Bigg [ |{\varvec{\eta }}({\varvec{\omega }}_0 )||e({\varvec{\omega }}_0 )-e({\varvec{\omega }})| + |e({\varvec{\omega }})||{\varvec{\eta }}({\varvec{\omega }}_0 )-{\varvec{\eta }}({\varvec{\omega }})|\Bigg ], \end{aligned}$$

where \(M_3=\sup _{u\ge 0 } |\mathbf{C}(u)|\). It follows that:

$$\begin{aligned}&\sup _{t\in [T_0,M]} \int _0^{t-T_0} \left| F(s,{\varvec{\omega }}_0 )-F(s,{\varvec{\omega }})\right| \, \mathrm{d}s\nonumber \\&\quad \le \frac{(M-T_0)^2M_3}{2} \Big [ |{\varvec{\eta }}({\varvec{\omega }}_0 )||e({\varvec{\omega }}_0 )-e({\varvec{\omega }})| + |e({\varvec{\omega }})||{\varvec{\eta }}({\varvec{\omega }}_0 )-{\varvec{\eta }}({\varvec{\omega }})|\Big ], \end{aligned}$$
(25)

and so, (24) holds, since e and \({\varvec{\eta }}\) are continuous.

Since \(p_{t1}({\varvec{\omega }})=q_t({\varvec{\omega }})\) for \(t\in [T_0,T_0+\tau ({\varvec{\omega }})]\) and \(p_{t1}({\varvec{\omega }})\le q_t({\varvec{\omega }})\) for \(t>T_0+\tau ({\varvec{\omega }})\), it can be shown that

$$\begin{aligned} |p_{1t}({\varvec{\omega }}_0 )-p_{t1}({\varvec{\omega }})|\le |q_t({\varvec{\omega }}_0 )-q_t({\varvec{\omega }})|, \end{aligned}$$

and so, from (22), we have

$$\begin{aligned} \lim _{{\varvec{\omega }}\rightarrow {\varvec{\omega }}_0 }\sup _{t\in [T_0,M]}|p_{t1}({\varvec{\omega }}_0 )-p_{t1}({\varvec{\omega }})|=0 \end{aligned}$$

for all \(M>T_0\). Since \(\tau \) is continuous, there is a \(M>0\), such that for all \({\varvec{\omega }}\in D_0(v_0)\) satisfying \(|{\varvec{\omega }}-{\varvec{\omega }}_0 |<1\) we have \(T_0+\tau ({\varvec{\omega }})\le M\). Using this fact and the fact that \(p_{t1}({\varvec{\omega }})\) is constant for \(t\ge T_0+\tau ({\varvec{\omega }})\), it follows that:

$$\begin{aligned} \lim _{{\varvec{\omega }}\rightarrow {\varvec{\omega }}_0 }\sup _{t\ge T_0}|p_{t1}({\varvec{\omega }}_0 )-p_{t1}({\varvec{\omega }})|=\lim _{{\varvec{\omega }}\rightarrow {\varvec{\omega }}_0 }\sup _{t\in [T_0,M]}|p_{t1}({\varvec{\omega }}_0 )-p_{t1}({\varvec{\omega }})|=0. \end{aligned}$$
(26)

To finish the proof, we must now establish a result of the form (26) with the first component of \(\mathbf{p}_t\) replaced by the second component [see (6)]. However, this follows from the fact that the B-spline basis functions \(B_k\) are continuously differentiable and that \(p_{t1}\) satisfies (26).\(\square \)

Proof of Lemma 4

Proof

Fix \({\varvec{\omega }}_0 =({\varvec{\theta }}_0 ,{\varvec{\beta }}_0 )\in D_0(v_0)\). Since \(\tau \) is continuous, find \(\delta >0\), such that if \({\varvec{\omega }}\in D_0(v_0)\) and \(|{\varvec{\omega }}-{\varvec{\omega }}_0|<\delta \), then \(0<|\tau ({\varvec{\omega }})-\tau ({\varvec{\omega }}_0)|<\tau ({\varvec{\omega }}_0)/2\). Now, consider \({\varvec{\omega }}=({\varvec{\theta }},{\varvec{\beta }})\in D_0(v_0)\) with \(|{\varvec{\omega }}_0 -{\varvec{\omega }}|<\delta \). By the triangle inequality and (4), we have for \(t\in [T_0,\infty )\)

$$\begin{aligned}&|a_{{\varvec{\omega }}}(t)-a_{{\varvec{\omega }}_0 }(t)|\le \sum _{k=1}^{K_2} (|{\varvec{\beta }}_{\circ }|+1) \Bigg |C_k\Big ( \frac{t-T_0}{\tau ({\varvec{\omega }})}\Big ) - C_k\Big (\frac{t-T_0}{\tau ({\varvec{\omega }}_0 )}\Big )\Bigg | \nonumber \\&\quad +\,|\beta _k-\beta _{\circ k}| \Bigg |C_k\Bigg (\frac{t-T_0}{\tau ({\varvec{\omega }}_0 )}\Bigg )\Bigg |, \end{aligned}$$
(27)

where here we use a constant extension of \(C_k\) past the interval [0, 1] to ensure that \(a_{\varvec{\omega }}(t)\) is constant past the interval \([T_0,T_0+\tau ({\varvec{\omega }})]\). We will investigate three different regimes for t. If \(t\ge T_0+\max \{\tau ({\varvec{\omega }}),\tau ({\varvec{\omega }}_0)\}\), then

$$\begin{aligned} \Bigg |C_k\Bigg ( \frac{t-T_0}{\tau ({\varvec{\omega }})}\Bigg ) - C_k\Bigg (\frac{t-T_0}{\tau ({\varvec{\omega }}_0 )}\Bigg )\Bigg |=|C_k(1)-C_k(1)|=0. \end{aligned}$$

Now, assume that \(t\le T_0+\min \{\tau ({\varvec{\omega }}),\tau ({\varvec{\omega }}_0)\}\). Since the B-splines \(C_k\) are smooth functions, there is a constant \(M_4>0\), such that \(|C'_k(s)|\le M_4\) for all \(s\in [0,1]\) and all \(k=1,\dots ,K_2\). Thus, it follows from the mean value theorem that:

$$\begin{aligned} \Bigg |C_k\Bigg ( \frac{t-T_0}{\tau ({\varvec{\omega }})}\Bigg ) - C_k\Bigg (\frac{t-T_0}{\tau ({\varvec{\omega }}_0 )}\Bigg )\Bigg |\le {\frac{M_4}{\tau ({\varvec{\omega }}_0)} |\tau ({\varvec{\omega }})-\tau ({\varvec{\omega }}_0 )|.} \end{aligned}$$

For the final case, we consider t between \(T_0+\tau ({\varvec{\omega }}_0)\) and \(T_0+\tau ({\varvec{\omega }})\). Here, we assume that \(\tau ({\varvec{\omega }}_0)\le \tau ({\varvec{\omega }})\). The case where \(\tau ({\varvec{\omega }})\le \tau ({\varvec{\omega }}_0)\) is treated in a similar manner. Since \(T_0+\tau ({\varvec{\omega }}_0)\le t\le T_0+\tau ({\varvec{\omega }})\), we can use the mean value theorem again to find that

$$\begin{aligned} \Bigg |C_k\Bigg ( \frac{t-T_0}{\tau ({\varvec{\omega }})}\Bigg ) - C_k\Bigg (\frac{t-T_0}{\tau ({\varvec{\omega }}_0 )}\Bigg )\Bigg |&=\Bigg |C_k\Bigg ( \frac{t-T_0}{\tau ({\varvec{\omega }})}\Bigg ) - C_k(1)\Bigg |\\&\le M_4 \Bigg | \frac{t-T_0}{\tau ({\varvec{\omega }})}-1\Bigg |\\&\le {\frac{M_4}{\tau ({\varvec{\omega }})} |\tau ({\varvec{\omega }}_0)-\tau ({\varvec{\omega }})|.} \end{aligned}$$

Putting these cases together, we find that there is a constant \(M_5\), such that for all \(t\ge T_0\)

$$\begin{aligned} \Bigg |C_k\Bigg ( \frac{t-T_0}{\tau ({\varvec{\omega }})}\Bigg ) - C_k\Bigg (\frac{t-T_0}{\tau ({\varvec{\omega }}_0 )}\Bigg )\Bigg |\le M_5 |\tau ({\varvec{\omega }})-\tau ({\varvec{\omega }}_0 )|. \end{aligned}$$

Since the \(C_k\) are bounded functions, by possibly enlarging \(M_5\), we may assume that \(|C_k(s)|\le M_5\) for all \(s\in [0,1]\) and all \(k=1,\ldots ,K_2\). Putting these facts together yields

$$\begin{aligned} |a_{{\varvec{\omega }}}(t)-a_{{\varvec{\omega }}_0 }(t)|\le K_2M_5\Big [ (|{\varvec{\beta }}_{\circ }|+1)|\tau ({\varvec{\omega }})-\tau ({\varvec{\omega }}_0 )| +|{\varvec{\beta }}-{\varvec{\beta }}_{\circ }| \Big ] \end{aligned}$$

for all \(t\in [T_0,\infty )\). Since \(\tau \) is continuous, it follows that \(a_{{\varvec{\omega }}}(t)\) is equicontinuous in t.

The equicontinuity in t of \(\int _0^t a_{{\varvec{\omega }}}(u)du\) follows from the equicontinuity in t of \(a_{{\varvec{\omega }}}(t)\). \(\square \)

1.1 Derivation of the maximum-likelihood estimators

The partial derivatives of \(L({\varvec{\eta }}_0^\ell , \mathbf{V}^\ell , \mathbf{W}^\ell |\mathbf {N}^\ell )\) with respect to the parameters \({\varvec{\eta }}_0^\ell , \mathbf{V}^\ell \), and \(\mathbf{W}^\ell \) are

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{ \partial l}{ \partial {\varvec{\eta }}_0^\ell } = - \frac{1}{2} \sum _{t=1}^T \left( -2\varSigma ^{-1}\mathbf {N}_t + 2\varSigma ^{-1}{\varvec{\eta }}_0^\ell + 2t\varSigma ^{-1}\mathbf{V}^\ell + t^2 \varSigma ^{-1} \mathbf{W}^\ell \right) ,\\ \frac{ \partial l}{ \partial \mathbf{V}^\ell } = - \frac{1}{2} \sum _{t=1}^T \left( -2t\varSigma ^{-1}\mathbf {N}_t + 2t\varSigma ^{-1}{\varvec{\eta }}_0^\ell + 2t^2\varSigma ^{-1}\mathbf{V}^\ell + t^3\varSigma ^{-1}\mathbf{W}^\ell \right) , \\ \frac{ \partial l}{ \partial \mathbf{W}^\ell } = - \frac{1}{2} \sum _{t=1}^T \left( -t^2\varSigma ^{-1}\mathbf {N}_t + t^2\varSigma ^{-1}{\varvec{\eta }}_0^\ell + t^3\varSigma ^{-1}\mathbf{V}^\ell + \frac{t^4}{2}\varSigma ^{-1}\mathbf{W}^\ell \right) . \end{array}\right. } \end{aligned}$$

Equating these to zero, we obtain the following system of equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\varvec{\eta }}_0^\ell = \frac{1}{T} \sum _{t=1}^T \mathbf {N}_t - \frac{c_1}{T} \mathbf{V}^\ell - \frac{c_2}{2T} \mathbf{W}^\ell , \\ \mathbf{V}^\ell = \frac{1}{c_2} \sum _{t=1}^T t \mathbf {N}_t - \frac{c_1}{c_2} {\varvec{\eta }}_0^\ell - \frac{c_3}{2c_2} \mathbf{W}^\ell , \\ \mathbf{W}^\ell = \frac{2}{c_4} \sum _{t=1}^T t^2 \mathbf {N}_t - \frac{2c_2}{c_4} {\varvec{\eta }}_0^\ell - \frac{2c_3}{c_4} \mathbf{V}^\ell , \end{array}\right. } \end{aligned}$$

where \(c_j = \sum _{t=1}^T t^j\). Substituting \({\varvec{\eta }}_0^\ell \) from the first equation into the second equation, and the resulting \({\widehat{\mathbf{V}}}^\ell \) into the third equation, we obtain

$$\begin{aligned} {\widehat{\mathbf{W}}}^\ell&= \frac{ 2\sum _{t=1}^T t^2\mathbf {N}_t^\ell - 2c_2 {\bar{\mathbf {N}}}_T^\ell + \Bigg (\frac{2c_1c_2}{T}- 2c_3\Bigg )\Bigg (\frac{ \sum _{t=1}^T t\mathbf {N}_t^\ell - c_1{\bar{\mathbf {N}}}_T^\ell }{c_2 - \frac{c_1^2}{T}}\Bigg )}{c_4 - \frac{c_2^2}{T} - \frac{\left( \frac{c_1c_2}{T} - c_3\right) ^2 }{\left( c_2 - \frac{c_1^2}{T}\right) }},\\ {\widehat{\mathbf{V}}}^\ell&= \frac{ \sum _{t=1}^T t\mathbf {N}_t^\ell - c_1{\bar{\mathbf {N}}}_T^\ell }{c_2 - \frac{c_1^2}{T}}\\&\quad + \Bigg (\frac{\frac{c_1c_2}{2T} - \frac{c_3}{2}}{c_2 - \frac{c_1^2}{T}}\Bigg )\frac{ 2\sum _{t=1}^T t^2\mathbf {N}_t^\ell - 2c_2 {\bar{\mathbf {N}}}_T^\ell + \Bigg (\frac{2c_1c_2}{T}- 2c_3\Bigg )\Bigg (\frac{ \sum _{t=1}^T t\mathbf {N}_t^\ell - c_1{\bar{\mathbf {N}}}_T^\ell }{c_2 - \frac{c_1^2}{T}}\Bigg )}{c_4 - \frac{c_2^2}{T} - \frac{\left( \frac{c_1c_2}{T} - c_3\right) ^2 }{\left( c_2 - \frac{c_1^2}{T}\right) }},\\{\hat{{\varvec{\eta }}}}_0^\ell&= {\bar{\mathbf {N}}}_T^\ell - \frac{c_1}{T} {\widehat{\mathbf{V}}}^\ell - \frac{c_2}{2T} {\widehat{\mathbf{W}}}^\ell , \end{aligned}$$

which in sequence define the maximum-likelihood estimators.

Derivation of Var\(({\hat{{\varvec{\eta }}}}_t^\ell )\)

After a lengthy calculation, one can write

$$\begin{aligned} \mathrm{Var}({\hat{{\varvec{\eta }}}}_t^\ell )&= \mathrm{Var}({\hat{{\varvec{\eta }}}}_0^\ell + t{\widehat{\mathbf{V}}}^\ell + \frac{t^2}{2}{\widehat{\mathbf{W}}}^\ell )\\&= Var\Bigg \{\sum _{k=1}^T \mathbf {N}_k\Bigg [t^2\Bigg (\frac{k^2}{d} - \frac{c_2}{dT} + \frac{ak}{db} - \frac{ac_1}{dbT}\Bigg )\\&\quad + t\Bigg (\frac{k}{b} - \frac{c_1}{Tb} + \frac{ak^2}{db} - \frac{ac_2}{dbT} + \frac{a^2k}{db^2} - \frac{a^2c_1}{Tdb^2}\Bigg )\\&\quad + \frac{1}{T} - \frac{c_1k}{Tb} + \frac{c_1^2}{T^2b} - \frac{c_1ak^2}{Tdb} + \frac{c_1c_2a}{T^2db} \\&\quad - \frac{c_1a^2k}{Tdb^2} + \frac{c_1^2a^2}{T^2db^2} - \frac{c_2k^2}{Td} + \frac{c_2^2}{T^2d} - \frac{c_2ak}{Tdb} + \frac{c_1c_2a}{T^2db}\Bigg ]\Bigg \}\\&= d_t\varSigma ^\ell , \end{aligned}$$

where \(a = c_1c_2/T - c_3\), \(b = c_2-c_1^2/T\), \(d = c_4-c_2^2/T - a^2/b\), and

$$\begin{aligned} d_t&= \sum _{k=1}^T \Bigg [t^2\Bigg (\frac{k^2}{d} - \frac{c_2}{dT} + \frac{ak}{db} - \frac{ac_1}{dbT}\Bigg )\\&\qquad + t\Bigg (\frac{k}{b} - \frac{c_1}{Tb} + \frac{ak^2}{db} - \frac{ac_2}{dbT} + \frac{a^2k}{db^2} - \frac{a^2c_1}{Tdb^2}\Bigg )\\&\qquad + \frac{1}{T} - \frac{c_1k}{Tb} + \frac{c_1^2}{T^2b} - \frac{c_1ak^2}{Tdb} + \frac{c_1c_2a}{T^2db} - \frac{c_1a^2k}{Tdb^2}\\&\qquad + \frac{c_1^2a^2}{T^2db^2} - \frac{c_2k^2}{Td} + \frac{c_2^2}{T^2d} - \frac{c_2ak}{Tdb} + \frac{c_1c_2a}{T^2db}\Bigg ]^2\\&= \sum _{k=1}^T \Bigg [k^2\Bigg (\frac{t^2}{d} + \frac{ta}{db} - \frac{c_1a}{Tdb} - \frac{c_2}{Td}\Bigg )\\&\qquad + k\Bigg ( \frac{at^2}{db} + \frac{t}{b} + \frac{a^2t}{db^2} - \frac{c_1}{Tb} - \frac{c_1a^2}{Tdb^2} - \frac{c_2a}{Tdb}\Bigg )\\&\qquad - \frac{c_2t^2}{dT} - \frac{ac_1t^2}{dbT} - \frac{c_1t}{Tb} - \frac{ac_2t}{dbT} - \frac{a^2c_1t}{Tdb^2} + \frac{1}{T}\\&\qquad + \frac{c_1^2}{T^2b} + \frac{c_1c_2a}{T^2db} + \frac{c_1^2a^2}{T^2db^2} + \frac{c_2^2}{T^2d} + \frac{c_1c_2a}{T^2db}\Bigg ]^2\\&= c_4\Bigg (\frac{t^2}{d} + \frac{ta}{db} - \frac{c_1a}{Tdb} - \frac{c_2}{Td}\Bigg )^2\\&\qquad + c_2\Bigg ( \frac{at^2}{db} + \frac{t}{b} + \frac{a^2t}{db^2} - \frac{c_1}{Tb} - \frac{c_1a^2}{Tdb^2} - \frac{c_2a}{Tdb}\Bigg )^2\\&\qquad \times \Bigg (- \frac{c_2t^2}{dT} - \frac{ac_1t^2}{dbT} - \frac{c_1t}{Tb} - \frac{ac_2t}{dbT} - \frac{a^2c_1t}{Tdb^2} + \frac{1}{T}\\&\qquad + \frac{c_1^2}{T^2b} + \frac{c_1c_2a}{T^2db} + \frac{c_1^2a^2}{T^2db^2} + \frac{c_2^2}{T^2d} + \frac{c_1c_2a}{T^2db}\Bigg )^2\\&\qquad + 2c_3\Bigg (\frac{t^2}{d} + \frac{ta}{db} - \frac{c_1a}{Tdb} - \frac{c_2}{Td}\Bigg )\Bigg ( \frac{at^2}{db} + \frac{t}{b} + \frac{a^2t}{db^2} - \frac{c_1}{Tb} - \frac{c_1a^2}{Tdb^2} - \frac{c_2a}{Tdb}\Bigg )\\&\qquad + 2c_2\Bigg (\frac{t^2}{d} + \frac{ta}{db} - \frac{c_1a}{Tdb} - \frac{c_2}{Td}\Bigg )\Bigg (- \frac{c_2t^2}{dT} - \frac{ac_1t^2}{dbT} - \frac{c_1t}{Tb} - \frac{ac_2t}{dbT} - \frac{a^2c_1t}{Tdb^2}\\&\qquad + \frac{1}{T} + \frac{c_1^2}{T^2b} + \frac{c_1c_2a}{T^2db} + \frac{c_1^2a^2}{T^2db^2} + \frac{c_2^2}{T^2d} + \frac{c_1c_2a}{T^2db}\Bigg )\\&\qquad + 2c_1\Bigg ( \frac{at^2}{db} + \frac{t}{b} + \frac{a^2t}{db^2} - \frac{c_1}{Tb} - \frac{c_1a^2}{Tdb^2} - \frac{c_2a}{Tdb}\Bigg )\Bigg (- \frac{c_2t^2}{dT} - \frac{ac_1t^2}{dbT} - \frac{c_1t}{Tb}\\&\qquad - \frac{ac_2t}{dbT} - \frac{a^2c_1t}{Tdb^2}+ \frac{1}{T} + \frac{c_1^2}{T^2b} + \frac{c_1c_2a}{T^2db} + \frac{c_1^2a^2}{T^2db^2} + \frac{c_2^2}{T^2d} + \frac{c_1c_2a}{T^2db}\Bigg ). \end{aligned}$$

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Zambom, A.Z., Seguin, B. Fastest motion planning for an unmanned vehicle in the presence of accelerating obstacles. Comp. Appl. Math. 40, 119 (2021). https://doi.org/10.1007/s40314-021-01511-9

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