A Unified Dynamics-Based Motion Planning Algorithm for Autonomous
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Robotica (2004) volume 22, pp. 117–128. © 2004 Cambridge University Press DOI: 10.1017/S0263574703005368 Printed in the United Kingdom A unified dynamics-based motion planning algorithm for autonomous underwater vehicle-manipulator systems (UVMS) Tarun Kanti Podder* and Nilanjan Sarkar† (Received in Final Form: May 19, 2003) SUMMARY space trajectory is not unique. It admits infinite number of A new unified motion planning algorithm for autonomous joint-space solutions for a given task-space trajectory. Underwater Vehicle-Manipulator Systems (UVMS) has However, there are various mathematical tools such as been presented in this paper. Commonly, a UVMS consists Moore-Penrose Generalized Inverse, which map the desired of two sub-systems, a vehicle and a manipulator, having Cartesian trajectory into the corresponding joint-space vastly different dynamic responses. The proposed algorithm trajectory for a kinematically redundant system. Research- considers the variability in dynamic bandwidth of the ers have developed various trajectory planning methods for complex UVMS system and generates not only kine- redundant systems.1,3–5,9 A kinematic approach of motion matically admissible but also dynamically feasible reference planning has been reported in references [1–5]. Zhou and trajectories. Additionally, this motion planning algorithm Nguyen3 formulated optimal joint-space trajectories for exploits the inherent kinematic redundancy of the whole kinematically redundant manipulators by applying Pon- system and provides reference trajectories that accom- tryagin’s Maximum Principle. Siciliano4 has proposed modates other important criteria such as thruster/actuator an inverse kinematic approach for motion planning of a faults and saturations, and also minimizes hydrodynamic redundant spacecraft-manipulator system. Antonelli and drag. Effectiveness of the proposed unified motion planning Chiaverini5 have used a pseudoinverse method for task- algorithm has been verified by extensive computer simula- priority redundancy resolution for an autonomous tion. The results are quite promising. Underwater Vehicle-Manipulator System (UVMS) using a kinematic approach. Several researchers,6–15 on the other hand, have con- KEYWORDS: Underwater vehicle-manipulator system; Under- sidered the dynamics of the system for trajectory planning. water robots; Dynamics-based motion planning; Heterogeneous Vukobratovic and Kircanski6 proposed an inverse problem dynamic system; Trajectory planning. solution to generate nominal joint-space trajectory consider- ing the dynamics of the system. Bobrow7 presented the 1. INTRODUCTION Cartesian path of the manipulator with a B-spline polyno- In robotics, trajectory planning is one of the most challeng- mial and then optimized the total path traversal time ing problems.1 Traditionally, trajectory planning problem is satisfying the dynamic equations of motion. Shiller and formulated as a kinematic problem and therefore the Dubowsky8 presented a time-optimal motion planning dynamics of the robotic system is neglected.2 Although the method considering the dynamics of the system. Shin and kinematic approach to the trajectory planning has yielded McKay9 proposed a dynamic programming approach to some very successful results, they are essentially incomplete minimize the cost of moving a robotic manipulator. as the planner does not consider the system’s dynamics Recently, Hirakawa and Kawamura10 have proposed a while generating the reference trajectory. As a result, the method to solve trajectory generation problem for redundant reference trajectory may be kinematically admissible but robot manipulators using the variational approach with a B- may not be dynamically feasible. spline function to minimize the consumed electrical energy. Researchers, in the past several years, have developed Saramago and Steffen11 have formulated off-line joint-space various trajectory planning methods for robotic systems trajectories to optimize traveling time and minimize considering different kinematic and dynamic criteria such as mechanical energy of the actuators using spline functions. obstacle avoidance, singularity avoidance, time minimiza- Zhu et al.12 have formulated real-time collision free tion, torque optimization, energy optimization, and other trajectory by minimizing an energy function. Faiz and objective functions.3–21 A robotic system that has more than Agrawal13 have proposed a trajectory planning scheme that 6 dofs (degrees-of-freedom) is termed as a kinematically explicitly satisfy the dynamic equations and the inequality redundant system. For a kinematically redundant system, constraints prescribed in terms of joint variables. Recently, the mapping between task-space trajectory and the joint- Macfarlane and Croft14 have developed and implemented a jerk-bounded trajectory for an industrial robot using concat- * Monterey Bay Aquarium Research Institute, 7700 Sandholdt enated quintic polynomials. Motion planning of land-based Road, Moss Landing, California, CA 95039 (USA) mobile robotic systems has been reported in references E-mail: [email protected] 15 † Department of Mechanical Engineering, Vanderbilt University, [15–18]. Brock and Khatib have proposed a global Nashville, Tennessee, TN 37235 (USA) dynamic window approach that combines planning and real- E-mail: [email protected] time obstacle avoidance algorithms to generate motion for 118 Underwater vehicle mobile robots. Huang et al.16 have presented a coordinated such reference trajectories that either the UVMS may not be motion planning approach for a mobile manipulator con- able to track them or while tracking, it may consume sidering system stability and manipulation. Yamamoto and exorbitant amount of energy which is extremely precious Fukuda17 formulated trajectories considering kinematic and for autonomous operation in an oceanic environment. dynamic manipulability measures for two mobile robots Additionally, in this research, we exploit the kinematic carrying a common object while avoiding a collision by redundancy of the UVMS to formulate a unified dynam- changing their configuration dynamically. Recently, Yama- ically feasible reference trajectory generation algorithm that shita et al.18 have proposed a motion planning method for can accommodate thruster/actuator faults, saturations, and multiple mobile robots for cooperative transportation of a provide a minimum drag trajectory for a given task. All large object in a 3D environment. To reduce the computa- these performance criteria are very important for autono- tional burden, they have divided the motion planner into a mous underwater operation. They provide a fault-tolerant global path planner and a local manipulation planner then and reduced energy consuming autonomous operation they have designed it and integrated it. All the previously framework. mentioned researches6–18 are performed for either space robotic or land-based robotic systems. On the other hand, 2. THEORETICAL DEVELOPMENT very few works on motion/trajectory planning of underwater robotic systems have been reported.19–21 Yoerger and Slo- tine19 formulated a robust trajectory control approach for 2.1. UVMS dynamics underwater robotic vehicles. Spangelo and Egeland20 devel- For convenience, we commonly use two reference frames to oped an energy-optimum trajectory for underwater vehicles describe underwater robotic systems. These two frames are by optimizing a performance index consisting of a weighted namely the earth-fixed frame (denoted by XYZ) and the combination of energy and time consumption by the system. body-fixed frame (denoted by XvYvZv), as shown in Figure 1. Recently, Kawano and Ura21 have proposed a motion planning algorithm for nonholonomic autonomous under- The dynamic equations of motion of a UVMS can be water vehicle in disturbance using reinforcement learning expressed as follows: 22 (Q-learning) and teaching method. Sarkar and Podder have Mb(qm )˙w+Cb(qm , w)w+Db(qm , w)w+Gb(q)= b (1) presented a coordinated motion planning algorithm for a where the subscript ‘b’ denotes the corresponding parame- UVMS to minimize the hydrodynamic drag. Note that ters in the body-fixed frames of the vehicle and the UVMS always implies an autonomous UVMS in this ᑬ(6+n)ϫ (6+n) manipulator. Mb(qm ) is the inertia matrix paper. ᑬ(6+n)ϫ (6+n) including the added mass and Cb(qm ,w) is the However, the majority of the trajectory planning methods centrifugal and Coriolis matrix including terms due to added available in the literature that considered the dynamics of ᑬ(6+n)ϫ (6+n) mass. Db(qm ,w) is the drag matrix, the system are formulated for land-based robots. They have G(q) ᑬ(6+n) is the vector of restoring forces and either optimized some objective functions related to trajec- ᑬ(6+n) b is the vector of forces and moments acting on tory planning satisfying dynamic equations or optimized T the UVMS. The displacement vector q=[qv,qm ] , where energy functions. Moreover, for the land-based robotic q =[q ,....,q]T, and q =q ,....,q ]T ·q,q and q system, the dynamics of the system is either homogeneous v 1 6 m 7 6+n 1 2 3 6–18 are the linear (surge, sway, and heave) displacements of the or very close to homogeneous. On the other hand, most vehicle along X, Y, and Z axes, respectively, expressed in of the trajectory planning methods that have been developed the earth-fixed frame. q4, q5 and q6 are the angular (roll, for space and underwater robotic systems use the pseu- pitch, and yaw) displacements of the vehicle about X, Y and doinverse