Mobile Robots Perception & Navigation Part 10 doc

Unified Dynamics-based Motion Planning Algorithm for Autonomous Underwater Vehicle-Manipulator Systems (UVMS) 351 We have also plotted the simulation results for surge-sway motion, power requirement and energy consumption of the UVMS in case of CMP method (in the left column) and that of in case of proposed UDMP method (in the right column) in Fig. 26. Top two plots in this figure show the profile of the surge- sway movements of the vehicle in the said two methods. In case of the CMP method, the vehicle changes the motion sharply and moves more as compared to the motion generated from the UDMP method. It may so happen that, in practice, this type of sharp and fast movements may be beyond the capability of the heavy dynamic subsystem and consequently large errors in trajectory tracking will occur. Additionally, this may cause saturation of the thrusters and the actuators resulting in degradation in performance. Moreover, the vehicle will experience large velocity and acceleration in CMP method that result in higher power requirement and energy consumption, as we observe it in next two sets of plots in Fig. 26. Thus, this investigation reveals that our proposed Unified Dynamics-Based Motion Planning method is very promising for autonomous operation of dynamic system composed of several subsystems having variable dynamic responses. 0 5 10 6 6.5 7 7.5 8 8.5 surge pos [m] 0 5 10 1.5 2 2.5 3 sway pos [m] 0 5 10 -0.5 0 0.5 heave pos [m] 0 5 10 -20 0 20 rollpos [deg] 0 5 10 -60 -40 -20 0 pitch pos [deg] 0 5 10 -60 -40 -20 0 yaw pos [deg] 0 5 10 0 50 100 joint 1 pos [ de g] time [sec] 0 5 10 -100 -50 0 joint 2 pos [ de g] time [sec] 0 5 10 -100 -50 0 joint 3 pos [ de g] time [sec] Fig. 24. Joint-space trajectories: Unified Dynamics-based Motion Planning method (solid lines) and Conventional Motion Planning method (dashed lines). 352 Mobile Robots, Perception & Navigation 0 5 10 -500 0 500 thrust 1 [N ] sat uratio n lim it sat uratio n lim it A 0 5 10 -500 0 500 thrust 2 [N ] sat uratio n lim it sat uratio n lim it 0 5 10 -500 0 500 thrust 3 [N ] saturation limit saturation limit 0 5 10 -500 0 500 thrust 4 [N] sat uratio n lim it sat uratio n lim it 0 5 10 -200 0 200 thrust 5 [N] sat uratio n lim it sat uratio n lim it B 0 5 10 -200 0 200 time [sec] thrust 6 [N] saturation limit saturation limit 0 5 10 -200 0 200 thrust 7 [N] sat uratio n lim it sat uratio n lim it 0 5 10 -200 0 200 thrust 8 [N] sat uratio n lim it sat uratio n lim it 0 5 10 -200 0 200 time [sec] torque 1 [N.m] sat uratio n lim it sat uratio n lim it 0 5 10 -100 0 100 time [sec] torque 2 [N.m] sat uratio n lim it sat uratio n lim it 0 5 10 -50 0 50 time [sec] torque 3 [N.m] saturation limit saturation limit Fig. 25. Thruster and actuator forces and torques of the UVMS. Unified Dynamics-based Motion Planning method (solid lines) and Conventional Motion Planning method (dashed lines). Thruster faults are marked by “A” (Thruster 1) and “B” (Thruster 5). 6 6.5 7 7.5 8 8.5 1.5 2 2.5 vehicle sway pos [m] vehicle surge pos [m] 0 2 4 6 8 10 0 1 2 total power [HP] time [sec] 0 2 4 6 8 10 0 2000 4000 time [sec] total energy [J] 6 6.5 7 7.5 8 8.5 1.5 2 2.5 vehicle sway pos [m] vehicle surge pos [m] 0 2 4 6 8 10 0 1 2 total power [HP] time [sec] 0 2 4 6 8 10 0 2000 4000 time [sec] total energy [J] Fig. 26. Surge-sway motion of the vehicle, power requirement and energy consumption of the UVMS. Results from Conventional Motion Planning method are in the left and that of Unified Dynamics-based Motion Planning method are in the right. Conventional Method Unified D y namics-based Method Unified Dynamics-based Motion Planning Algorithm for Autonomous Underwater Vehicle-Manipulator Systems (UVMS) 353 8. Conclusions We have proposed a new unified dynamics-based motion planning algorithm that can generate both kinematically admissible and dynamically feasible joint-space trajectories for systems composed of heterogeneous dynamics. We have then extended this algorithm for an autonomous underwater vehicle-manipulator system, where the dynamic response of the vehicle is much slower than that of the manipulator. We have also exploited the kinemetic redundancy to accommodate the thruster/actuator faults and saturation and also to minimize hydrodynamic drag. We have incorporated thruster dynamics when modeling the UVMS. Although, some researchers have exploited kinematic redundancy for optimizing various criteria, but those work have mainly addressed to problems with land-based robotics or space-robotics. Hardly any motion planning algorithm has been developed for autonomous underwater vehicle- manipulator system. In this research, work we have formulated a new unified motion planning algorithm for a heterogeneous underwater robotic system that has a vastly different dynamic bandwidth. The results from computer simulation demonstrate the effectiveness of the proposed method. It shows that the proposed algorithm not only improves the trajectory tracking performance but also significantly reduce the energy consumption and the power requirements for the operation of an autonomous UVMS. We have not presented results from Case II (Total Decomposition) because of the length of the paper. However, these results are comparable to the conventional motion planning approach. In future, instead of Fourier decomposition, one can try to use wavelet approach to decompose the task-space trajectory into system’s sub-component compatible segments. There are a few drawbacks of this paper as well. We used a model-based control technique to evaluate our planning algorithm. However, the underwater environment is uncertain and we need to use adaptive control techniques in future. Although the fault-tolerant control algorithm has been experimentally verified, the other proposed algorithms need to be validated by experiments. 9. References Klein, C.A. & C.H. Huang, C.H. (1983). Review of pseudoinverse control for use with kinematically redundant manipulators, IEEE Transactions on Systems, Man and Cybernetics, Vol. SMC-13, No. 3, pp. 245-250. Paul, R. (1979). Manipulator Cartesian path planning, IEEE Transactions on Systems, Man and Cybernetics, Vol. SMC-9, No. 11, pp. 702-711. Zhou, Z.L. & Nguyen, C.C. (1997). Globally optimal trajectory planning for redundant manipulators using state space augmentation method, Journal of Intelligent and Robotic Systems, Vol. 19, No. 1, pp. 105-117. Siciliano, B. (1993). 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Dynamic robot manipulator trajectory planning for obstacle avoidance, Journal of Mechanics Research Communications, Vol. 26, No. 2, pp. 139-144, 1999. Faiz, N. & Agrawal, S.K. (2000). Trajectory planning of robots with dynamics and inequalities, Proceedings of IEEE International Conference on Robotics and Automation, pp. 3977-3983, San Francisco, CA, April 2000. Macfarlane, S. & Croft, E.A. (2003). Jerk-bounded manipulator trajectory planning: design for real-time applications, IEEE Transactions on Robotics and Automation, Vol. 19, No. 1, pp. 42-52, February 2003. Brock, O. & Khatib, O. (1999). High-speed navigation using the global dynamic window approach, Proceedings of the IEEE International Conference on Robotics and Automation, pp. 341-346, Detroit, MI, May 1999. Huang, Q.; Tanie, K. & Sugano, S. (2000). Coordinated motion planning for a mobile manipulator considering stability and manipulation, International Journal of Robotics Research, Vol. 19, No. 8, pp. 732-742, August 2000. Yamamoto, Y. & Fukuda, S. (2002). Trajectory planning of multiple mobile manipulators with collision avoidance capability, Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3565-3570, Washington D.C., May 2002. Yamashita, A.; Arai, T.; Jun, O. & Asama, H. (2003). Motion planning of multiple mobile robots for cooperative manipulation and transportation, IEEE Transactions on Robotics and Automation, Vol. 19, No. 2, pp. 223-237, April 2003. Yoerger, D.R. & Slotine, J.J.E. (1985). Robust trajectory control of underwater vehicles, IEEE Journal of Oceanic Engineering, Vol. 10, No. 4, pp. 462-470, October 1985. Unified Dynamics-based Motion Planning Algorithm for Autonomous Underwater Vehicle-Manipulator Systems (UVMS) 355 Spangelo, I. & Egeland, O. (1994). Trajectory planning and collision avoidance for underwater vehicle using optimal control, IEEE Journal of Oceanic Engineering, Vol. 19, No. 4, pp. 502-511, 1994. Kawano, K. & Ura, T. (2002). Motion planning algorithm for nonholonomic autonomous underwater vehicle in disturbance using reinforced learning and teaching method,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 4032-4038, Washington D.C., May 2002. Sarkar, N. & Podder, T.K. (2001). Coordinated motion planning and control of underwater vehicle-manipulator systems subject to drag optimization,” IEEE Journal of Oceanic Engineering, Vol. 26, No. 2, pp. 228-239, 2001. Fu, K.S. Gonzalez, R.C. & Lee, C.S.G. (1988). Robotics: Control, Sensing, Vision, and Intelligence, McGraw-Hill International Edition, 1988. Craig, J.J. (1989). Introduction to Robotics: Mechanics and Control, Addison-Wesley Publishing Company, 1989. Kane, T.R. & Lavinson, D.A. (1985). Dynamics: Theory and Applications, McGraw-Hill Book Co., 1985. Fossen, T.I. (1984). Guidance and Control of Ocean Vehicles, John Willey & Sons Ltd., 1994. Podder, T,K. (2000). 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A sufficient condition for full linearization via dynamic state feedback, Proceedings of IEEE Conference on Decision and Control, pp. 203-208, Athens, Greece, December 1986. Yun, X. (1988). Dynamic state feedback control of constrained robot manipulators, Proceedings of IEEE Conference on Decision and Control, pp. 622-626, Austin, Texas, December 1988. Ben-Israel, A. & Greville, T.N.E. (1974). Generalized Inverse: Theory and Applications, New York, Wiley, 1974. Yoerger, D.R.; Cooke, J. & Slotine, J.J.E. (1990). The influence of thruster dynamics on underwater vehicle behavior and their incorporation into control system design, IEEE Journal of Oceanic Engineering, Vol. 15, No. 3, pp. 167- 178, 1990. Healey, A.J.; Rock, S.M.; Cody, S. Miles, D. & Brown, J.P. (1995). Toward an improved understanding of thruster dynamics for underwater vehicles, IEEE Journal of Oceanic Engineering, Vol. 20, No. 4, pp. 354-361, 1995. 356 Mobile Robots, Perception & Navigation Whitcomb, L.L. & Yoerger, D.R. (1999). Development, comparison, and preliminary validation of non-linear dynamic thruster models, IEEE Journal of Oceanic Engineering, Vol. 24, No. 4, pp. 481-493, 1999. 16 Optimal Velocity Planning of Wheeled Mobile Robots on Specific Paths in Static and Dynamic Environments María Prado Dept. Mechanical Engineering, University of Malaga Spain 1. Introduction The control system of a mobile robot generally comprises two different modules: a trajectory planner and a trajectory tracking controller, although some researchers have proposed algorithms that integrate both tasks. To completely solve the trajectory planning problem is to define an open-loop path and its velocity profile from an initial to a final posture, while avoiding any potential obstacles. In time-optimal planning of a wheeled mobile robot (WMR), the problem is solved by defining control inputs for the wheels that minimize navigation time from the origin to the target posture. This goal implies two tasks, which can be carried out simultaneously or sequentially: path-planning (PP), which involves the computation of the shortest feasible path; and velocity-planning (VP), which involves the computation of the fastest feasible velocity profile for the entire domain of the path. Several approaches have been developed to perform both tasks. The most widely used approaches are free configuration-time space based methods, (Reinstein & Pin, 1994), but these algorithms are computationally expensive, even when one is only dealing with PP or VP separately. To reduce the computational cost, researchers have recently published methods which do not require computing the C-space obstacles (Wang et al., 2004), as well as methods that search for a probabilistic road map (LaValle & Kuffner, 2001). Some other approaches that use intelligent computing-based methods have also been presented, such as those that use artificial potential fields-based methods (Liu & Wu, 2001), fuzzy logic (Takeshi, 1994), genetic algorithms (Nerchaou, 1998) or neural networks (Zalama et al., 1995). In order to find an optimal and feasible solution for the two problems, mechanical, kinematic and dynamic characteristics of the WMR that limit its motion must be taken into account, as well as other environmental, task-related and operational issues. These constraints can be summarized by upper boundary functions of the velocity, acceleration and deceleration of the WMR. In general, the functions are not constant, nor are they even continuous. They are therefore nonintegrable constraints, and the time optimal planning is a nonholonomic problem. A significant number of nonholonomic constraints, which include not only mechanical and kinematic but also dynamic characteristics of the WMR, are difficult to deal with when PP and VP are approached simultaneously. The vast majority of existing algorithms consider 358 Mobile Robots, Perception & Navigation only kinematic constraints or some dynamic conditions derived from simplified models of the WMR and/or its environment. But the resulting trajectory may be unexecutable, or tracked by the robot with high spatial and temporal errors. However, when PP and VP are approached sequentially, the difficulty of both problems is significantly reduced. Such approaches make it possible to include more complex constraints for the WMR’s velocity and acceleration, especially with regards to its kinematic and dynamic characteristics. To our knowledge, the first references addressing VP with kinematic and dynamic constraints for WMR is (O’Dunlaing, 1987). This paper, like a number of other algorithms to solve the VP stage, is based on constant maximum values for robot velocity and acceleration, set to arbitrary constants which are unrelated to the mechanical characteristics of the system. More recent works seek to find more efficient bounds for these operating variables, but never in a global way and always based on simplified dynamic robot models. (Weiguo et al., 1999) propose a velocity profile planner for WMRs on flat and homogeneous terrains, where velocity and acceleration are limited only by the outer motor torques and by the absolute slippage of the vehicle on the ground. (Choi & Kim, 2001) develop another planner where velocity and acceleration are constrained by dynamic characteristics related to the performance of the robot's electric motors and its battery's power. (Guarino Lo Bianco & Romano, 2005) present a VP algorithm for specific paths that generate a continuous velocity and acceleration profile, both into safety regions limited by upper boundary functions not described in the paper. The method involves an optimization procedure that has a significant computational cost. Some other limitations have been studied, mainly within the framework of projects for planetary exploration. (Shiller, 1999) deals with some dynamic constraints: sliding restrictions, understood as the avoidance of absolute vehicle slippage, tip-over and loss of wheel-ground contact constraints, which are important issues when dealing with irregular outdoor terrains. The author works with a very simplified robot model, neglecting sideslip and assuming pure rolling, so wheel deformations and microslippages which can cause important tracking errors are not quantified. (Cheriff, 1999) also proposes a set of kinematic and dynamic constraints over the robot’s path, dealing specifically with 3D irregular and non-homogeneous grounds. The resulting trajectory planner directly incorporates a complete dynamic WMR model, considering non-linear motions and specifically accounting for wheel-ground interactions, which makes it necessary to run complex algorithms that significantly increase computational cost. (Lepetic et al., 2003) present a VP method that considers dynamic constraints by bounding the acceleration by the maximum wheel-ground adherence capacity. This maximum is computed as a function of a constant friction coefficient for every posture and of the weight borne by the wheel. Load transfer due to lateral forces is considered to calculate the weight on the wheel, but only as a constant maximum value, derived from a simplified model of the WMR, that reduces the lateral maximum acceleration to the same value for every posture. The VP method published by (Krishna et al., 2006) builds a trajectory continuous in space and velocity, which incorporates environment and sensory constraints by setting a maximum velocity for the entire path of the robot that is decreased when an obstacle is detected within its visibility circle. The velocity constraint is computed as a function of the position and velocity of the obstacle and of a maximum acceleration or deceleration value of the WMR, established as constant values for every posture. This chapter deals with time-optimal planning of WMRs when navigating on specific spatial paths, i.e., when the PP is previously concluded. First, the computation of the upper Optimal Velocity Planning of Wheeled Mobile Robots on Specific Paths in Static and Dynamic Environments 359 boundary functions of its velocity, acceleration and deceleration are described. Then a method for time-optimal planning is proposed, the main goals of which are: - To fully exploit velocity, acceleration and deceleration constraints, avoiding the planning of velocities or accelerations that lead to dangerous motions. - To plan a feasible trajectory, with continuous velocity and deceleration - To bound the jerk of the WMR - To be of low computational cost. The method firstly deals with velocity planning in static environments and then presents an algorithm to modify the resulting trajectory to avoid moving obstacles. Special attention is paid to the efficiency of the second algorithm, an advantage which makes it highly useful for local and/or reactive control systems. 2. Problem definition Problem 1: Given a WMR’s path, computed to navigate in a static and known environment, plan the fastest, feasible and safe trajectory, considering the constraints imposed by the mechanical configuration, kinematics and dynamics of the robot and by environmental and task-related issues. Problem 2: Modify the trajectory quickly and locally to avoid moving obstacles. A generalized posture of a WMR, parameterizing by the path length, s, can be defined by the vector [] T į(s)ș(s),Y(s),X(s),q(s) = . [X(s), Y(s)] is the position and θ(s) the orientation of the WMR’s guide point on a global frame (Z coordinate is constant by assuming navigation is on flat ground). δ(s) is a function kinematically related to the curvature of the trajectory, Nj(s); specifically, it is a function of the steer angles of the wheels of WMRs with steering wheels or a function of the difference between the angular velocities at the traction wheels for WMRs with differential drive. The path, P(s), can be defined by a continuous series of generalized postures from the initial posture, q 0 , to the final one, q f. Therefore, if S is the total length of the path: {} [] f qSPqPSsqsP =∨=ℜ→= )()0(;,0:)()( 0 4 (1) To transform P(s) into a trajectory, a velocity function must be generated for the entire path domain. It must be defined in positive real space (if the WMR is only required to move forward, as is the usual case) and planned to make the robot start from a standstill and arrive at the final posture also with null velocity. That is: {} [] 0)(0)0(;,0:)()( =∨=ℜ→= + SVVSsvsV (2) Additional conditions are strongly required of V(s) to obtain a feasible trajectory: 1. Continuity, since the kinematics of WMR make it impossible to develop other types of maneuvers. 2. Confinement into a safety region of the space-velocity plane (s×v), upper limited by a boundary function of the velocity, V Lim (s). 3. Confinement of its first derivative with respect to time, acceleration or deceleration, into a safety region of the space-acceleration plane (s×a), upper limited by a boundary function of the acceleration, a Lim (s), and lower limited by the negative value of a boundary function of deceleration d Lim (s). 360 Mobile Robots, Perception & Navigation 4. Continuity of acceleration or deceleration: this condition ensures that the jerk of the robot, the second derivative of its velocity, is finite, so that the robot’s movements are smooth. High jerk is not recommended for WMRs for a number of reasons: it causes the robot to shake significantly and thus complicates on-board tasks; it makes tracking control more difficult, since wheel microslippage increases and wheel behavior becomes less linear (Wong, 2001); and it increases the error of on- board sensor systems. 5. Additionally, low computational cost is beneficial for the generation of the velocity profile. This goal is especially pursued when solving problem 2, for the purpose of possibly incorporating the algorithm into local controls or reactive planners, to adjust the trajectory in the presence of new unexpected obstacles that appear in the visibility area of the robot’s sensorial systems (Krishna et al., 2006). 3. Velocity constraints This section deals with constructive characteristics, kinematic configuration and the dynamic behaviour of a WMR, as well as operational matters, in order to identify the constraints that influence the maximum velocity of a WMR’s guide point. For all the constraints detailed in the following subsections, an upper boundary function of velocity, parametrized by s, can be generated. The function is built by assigning the lowest upper bound of all the velocity constraints to each posture: { } [] S,0s/V V,Vmin)s(V nLim2Lim1LimLim ⊂= (3) This chapter addresses the case of a WMR guided by steering wheels; in the case of WMRs with differential drive, the approach will be similar and therefore the constraints can easily be deduced under the same considerations. 3.1. Construction constraints Thermal and mechanical characteristics of motors and batteries impose maximum rotational velocities on the tractive and steering servomotors, ω tm max and ω sm max , respectively (Choi & Kim 2001). Thus, if Ǐ t is the reduction ratio of the drive-train and R the wheel’s radius, the maximum linear velocity of driven wheels on the ground is: R max tmt max tw v ωξ= (4) Further, if Ǐ s is the reduction ratio of the steering-train, the maximum velocity of variation of the steering angle, i.e. the maximum steering gain, is: max sms max s G ωξ= (5) 3.2. Kinematic constraints With regards to kinematic linkages between the driven wheels and the guide point, if d tw max is the position vector on the ground of the most distant driven wheel with respect to the guide point, an upper bound for the WMR’s velocity is given by: [...]... Planning of Wheeled Mobile Robots on Specific Paths in Static and Dynamic Environments 377 s(m) Y(m) 8 P7 P6 10 80 P8 P5 60 6 40 P9 P4 4 P3 P10 P2 P11 20 2 P1 P0 0 2 P13 P12 4 6 8 10 12 14 X(m) 0 10 20 30 40 50 60 t(s) 6 7 8 10 Fig 4 Planned and tracked trajectory with maximum velocity 1.7m/s s(m) Y P7 P6 10 80 8 P8 P5 60 6 P4 40 P9 4 P3 P10 20 2 P2 P1 P0 0 P11 2 4 P12 6 8 P13 10 12 14 _ Planned;... Vol 45, N0 34, 199- 210, 2003, ISSN 0921-8890 Liu, G.Y & Wu, C.J.(2001) A discrete method for time optimal motion planning a class Optimal Velocity Planning of Wheeled Mobile Robots on Specific Paths in Static and Dynamic Environments 381 of mobile robots, Journal of Intelligent and Robotic Systems, Vol.32,No.1, 7592, ISSN 09 2102 96 Muñoz, V.F (1995) Trajectory planning for mobile robots, Ph.D Thesis,... Huitang, C & Peng-Yung, W (1999) Optimal Motion Planning for a Wheeled Mobile Robot, Proceedings of IEEE International Conference on Robotics and Automation, pp 41-46, ISBN 0-7803-5180-0 , Michigan, USA, June 1999, IEEE, New York, USA 382 Mobile Robots, Perception & Navigation Wong, J.Y (2001), Theory of Ground Vehicles, John Wiley & Sons, ISBN 0-471-35461-9, New York, USA Zalama, E.; Gaudiano, P & Lopez-Coronado,... modeling of mobile robots Application to the trajectory planning, Ph.D Thesis, University of Málaga, Spain Prado, M.; Simón, A & Ezquerro, F (2002) Velocity, acceleration and deceleration bounds for a time-optimal planner of a wheeled mobile robot, Robotica, Vol 2, N0 2, 181193, ISSN 0263-5747 Prado, M.; Simon, A Carabias, E.; Perez, A & Ezquerro, F (2003) Optimal velocity planning of wheeled mobile robots. .. path: 1 TE 2 (~) = E 2 (~)d~ s s s s 0 s (10) If the planned path, P(s) in (1) is particularized for stationary manoeuvres, i.e with constant velocity and curvature, the WMR’s planned position in the same world reference frame can ~ be expressed as a function of s as: s s [X(~), Y(~ )] = sin( 2π~ ) 1 s , (1 − cos( 2π~ ) s κ κ (11) 362 Mobile Robots, Perception & Navigation The actual tracked trajectory,... (Leonard & Durrant-Whyte, 1992) (Ko et al., 1996) On the other hand, in the global method, the local map-making and the matching processes are avoidable and the self-localization is computationally efficient and fast (Leonard & Durrante-Whyte, 1991) (Hernandez et al., 2003) A global ultrasonic system presented in this chapter is a kind of a pseudo-lite system with a 384 Mobile Robots, Perception & Navigation. .. For autonomous navigation in workspace, a mobile robot should be able to figure out where it is and what direction it moves towards, which is called the self-localization (Singh & Keller, 1991) The self-localization capability is the most basic requirement for mobile robots, since it is the basis of the on-line trajectory planning and control The trajectory error of a dead-reckoning navigation, which... υk represents the ωk radius of rotation The position vector and the heading angle of the mobile robot are augmented so as to become P = x, y, θ t , which is referred to as the robot posture The bold and normal symbols represent the vector and the scalar variables, respectively 386 Mobile Robots, Perception & Navigation As a consequence of (1) and (2), the state equation for the ultrasonic receivers... Operational constraints The need to fit and synchronise the robot's motion with its environment, whether static or dynamic, makes operational constraints necessary Fig 1 Tip-over of the WMR 364 Mobile Robots, Perception & Navigation 3.4.1 Maximum velocity to prevent collisions V is limited by a value that ensures the WMR will come to a complete stop at a distance greater than a safety distance from any obstacle,... acceleration constraint causes the planned trajectory to take 3.7s (26.2% of the total time) to reach the velocity boundary function Low errors are found again: spatial errors are always t(s) 378 Mobile Robots, Perception & Navigation lower than 0.80m and temporal errors are lower than 2s for a trajectory of 14.2 s The velocity and acceleration profiles of the trajectory of Fig 5, along with their boundary functions . () () () () ° ° ¯ ° ° ®  >− − =σ ≤=σ σ∩σ=σ c ii c ii i 2 i c i c i i 1 i 2 i 1 ii tt tt tt A t'' tt t t A t'' ;t''t''t'' (46) By integrating. Planning method (dashed lines). 352 Mobile Robots, Perception & Navigation 0 5 10 -500 0 500 thrust 1 [N ] sat uratio n lim it sat uratio n lim it A 0 5 10 -500 0 500 thrust 2 [N ] sat uratio. responses. 0 5 10 6 6.5 7 7.5 8 8.5 surge pos [m] 0 5 10 1.5 2 2.5 3 sway pos [m] 0 5 10 -0.5 0 0.5 heave pos [m] 0 5 10 -20 0 20 rollpos [deg] 0 5 10 -60 -40 -20 0 pitch pos [deg] 0 5 10 -60 -40 -20 0 yaw