Randy L Haupt and Sue Ellen Haupt (2004), Practical Genetic. Algorithms. USA: Wiley ..... Chotiprayankul, Liu. D.K, Wang. D and Dissanayake (2011). Collision.
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Optimal Trajectory Planning for Autonomous Robots -A Review 1 J. Jeevamalar and 2 Dr. S. Ramabalan
I
Research Scholar and Assistant Professor 2 Principal and Professor
1,2 Department of Mechanical Engineering, E.G.S. Pillay Engineering College, Nagapattinam. 1 J.Jeevamalar @ gmail.com 2 cadsrb @ gmail.com
Abstract Optimal Trajectory planning is very important to the operation of robot manipulators. Its main aim is to generate the trajectory from initial to goal that satisfies some objectives, like minimization of Time interval, Acceleration, Joint Jerk, Torque, Vibration, mechanical energy consumption of Actuator and obstacle, collision avoidance criteria by satisfYing the manipulator's kinematic and dynamic constraints. In this review, discussion of optimization techniques to find the optimal trajectory planning either in Cartesian space or Joint space is presented. Optimal trajectory selection approaches such as kinematics and dynamics techniques with various constraints are presented and explained. Although the kinematics approach is simple and straight forward, it will experience some problems in implementation because of lack of Inertia and torque constraints. The application of Algorithms to find the optimal trajectory of manipulators in the obstacle avoidance is more important for getting better result of manipulators. -
Index Terms
Optimal Trajectory, Obstacle avoidance, Genetic Algorithms, Kinematic and Dynamic constraints.
of interpolating functions which meet the imposed kinematic and dynamic constraints. Apart from the particular strategy adopted, the motion laws generated by the trajectory planner must fulfill the constraints set a priori on the maximum values of the generalized joint torques, and must be such that no mechanical resonance mode is excited. This can be achieved by forcing the trajectory planner to generate smooth trajectories: in particular, it would be desirable to obtain trajectories with continuous joint accelerations, so that the absolute value of the jerk keeps bounded. Limiting the jerk is very important, because high jerk values can wear out the robot structure, and heavily excite its resonance frequencies. Vibrations induced by non-smooth trajectories can damage the robot actuators, and introduce large errors while the robot is performing tasks such as trajectory tracking. [1]. The objective of the paper is to investigate the different optimization techniques for trajectory selection of manipulators.
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I. INTRODUCTION OF TRAJECTORY PLANNING
The trajectory planning problem is a fundamental one in Robotics. It may be formulated thus: defme a sequential motion law along a given geometric path, such as certain requirements set on the trajectory properties are fulfilled. Hence, the aim of trajectory planning is to generate the reference inputs for the control system of the manipulator, in order to be able to execute the motion. The inputs of any trajectory planning algorithm are: the geometric path, the kinematic and dynamic constraints; and the output are the trajectory of the joints, expressed as a time sequence of position, velocity and acceleration values. Normally, the geometric path is specified in the operating space, i.e. with reference to the end effector of the robot, because both the task to perform and the obstacles to avoid can be more naturally described in this space. However, the trajectory is normally planned in the joint space of the robot, after a kinematic inversion of the given geometric path has been done. The joint trajectories are then obtained by means
II. OPTIMIZATION TECHNIQUES
Optimization is the process of adjusting the inputs to or characteristics of a device, mathematical process, or experiment to find the minimum or maximum output or result. The input consists of variables; the process or function is known as the cost function, objective function, or fitness function; and the output is the cost or fitness. If the process is an experiment, then the variables are physical inputs to the experiment. III. MINIMUM SEEKlING ALGORITHS
Searching the cost surface for the minimum cost lies at the heart of all optimization routines. Usually a cost surface has many peaks, valleys, and ridges. An optimization algorithm works much like a hiker trying to fmd the minimum altitude. The following are some of the minimum seeking algorithms [2]. A. Exhaustive Search
ISBN: 978-81-909042-2-3 ©2012 IEEE
IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012 This Exhaustive search will not get stuck in local mmuna and work for either continuous or discontinuous parameters. It is only practical for a small number of parameters in a limited search space since it takes an extremely long time to reach global minima [3]. B. Analytical Optimization
An extremum is found by setting the first derivative of a cost function to zero or undefined and solving for the parameter value. If the second derivative is greater than zero, the extremum is a minimum and conversely, if the second derivative is less than zero, the extremum is a maximum. It quickly finds a single minimum, but requires a search scheme to find the global minimum. [3]. C.
Nedler-Mead Downhill Simplex Method
Linear programming concerns the minimization of a linear function of many variables subject to constraints that they are linear equations and equalities. It does not require calculation of the derivatives [3]. D. Optimization Based on Line Minimization
The largest category of optimization methods fall under the general title of successive line optimization method. An algorithm begins at some random point on the cost surface, chooses a direction to move, then moves in that direction until the cost function begins to increase. Next, the procedure is repeated in another direction [2]. IV. GENETIC ALGORITHMS
The genetic algorithm (GA) is an optimization and search technique based on the principles of genetics and natural selection and allows a population composed of many individuals to evolve under specified selection rules to a state that maximizes the "fitness". The advantages of a GA include that it optimizes with large number of continuous or discrete variables and well suited for parallel computers. V. MULTIPLE OBJECTIVE OPTIMIZATIONS
In many applications the cost function has multiple, often times conflicting, objectives. Two approaches of MOO are weighted cost functions and fmding the Pareto front. A. Weighted Cost Functions
The most straightforward approach to MOO is to weight each function and add them together. The problem with this method is determining appropriate values of weighting function. Different weights produce different costs for the same cost function. B. Pareto Optimization
In MOO there is usually no single solution that is optimum with respect to all objectives. Consequently there are a set of optimal solutions, known as Pareto-optimal solutions,
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non-inferior solutions, or effective solutions. The goal of MOO is to find as many of these solutions as possible. The multi-objective GA (MOGA) starts by finding all non-dominated chromosomes of a population and gives them a rank of one. These chromosomes are removed from the population. Next all the non-dominated chromosomes of this smaller population are found and assigned a rank of two. This process continues until all the chromosomes are assigned a rank. The largest rank will be less than or equal to the size of the population. A non-dominated sorting GA (NSGA) ranks chromosomes in the same manner as MOGA. The NSGA algorithm then calculates a uniqueness value. Distance may be calculated from the variable values or the associated costs. A newer version (NSGA-II) improves the NSGA algorithm by the way of reduces the computational complexity of the non dominated sorting, introduces elitism and replaces sharing with crowded-comparison to reduce computations and the need for a user-defined sharing parameter. VI. MINIMUM TIME TRAJECTORY
Minimum-time algorithms were the first trajectory planning techniques proposed in the scientific literature because they were tightly linked to the need of increasing the productivity in the industrial sector. The two approaches for optimal trajectory are summarized below, A. Kinematic Approach
Kinematic motion planning involves the planning of motion through space, while not violating some combination of position, velocity, acceleration, and jerk constraints. These kinematic constraints are often determined by actuator limits like torque and velocity. Abstract limits like torque can be converted into acceleration limits, based on position and velocity. As a result of the non-linear characteristics of the manipulator, kinematic limits vary over the workspace. A minimum time approach for obstacle avoidance and envelope protection of UAV by using Non-Linear Trajectory Generation. The reference is considered in Cartesian space and is expressed as a Sequential quadratic Programming [4]. A technique for generating the optimal time trajectory enables to take into kinematic constraints on the robot motion expressed as upper bounds on the absolute values of velocity, Acceleration and jerk constraints [5]. A new algorithm is to calculate the time optimal, 3rd order trajectories delivering the parameter of 7 cubic polynomials for computing the smooth set of points for Position, Velocity and Acceleration along a given path [6]. A fast algorithm is generated for smoothing collision free trajectories for PUMA manipulator and Honda ASIMO robot manipulator. The constraints like time, velocity, acceleration are taken into account [7]. A feed forward technique for minimum time motion control of an automatic guided vehicle on a G3 path along with Velocity, Acceleration, Jerk constraints. The optimal planning based on dynamics path inversion algorithm
ISBN: 978-81-909042-2-3 ©2012 IEEE
IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012 [8]. A new technique that combines the closed loop pseudo inverse method with genetic algorithm which adopts direct kinematics for optimizing the trajectory planning of redundant manipulators [9]. A multi objective Genetic Algorithm technique is proposed to address the trajectory planning of manipulator based on kinematic approach. The optimal trajectory is obtained indicate that obstacles in the workspace may interference in the Pareto front [lO]. A simultaneous algorithm has been applied to PUMA 560 robot and 4 operational parameters like execution time, computational time, distance travelled and number of configurations are computed and analyzed their cost function on the trajectory. The SQP technology is used for Cubic spline trajectory for solving optimization problem [11]. A new method proposes a soft parallel robot for ankle joint rehabilitation. Kinematic workspace analysis is carried out and the singularity criterion of the SPR's Jacobian matrix is used to define the feasible workspace. Results from simple GA and modified GA are compared and discussed [12]. A kinematic analysis is proposed to create the trajectory planning for a novel machine tool structure consisting of a six degree-of-freedom hexapod machine and a two-degree-offreedom rotary table [13]. An offline genetic algorithm searching is introduced for the optimal or suboptimal placement of a robot's base during work cell design to obtain a position and orientation path following task of a given 3D curved path and orientation, maximizing the manipulator's velocity performance [14]. A new technique that combines the closed-loop pseudo inverse method with genetic algorithms for finding the optimal trajectory. The results are compared with a genetic algorithm that adopts the direct kinematics [15]. A new method introduces a new inverse static kinematic calibration technique based on genetic programming, which is used to establish and identify model structure and parameters [16]. A new method allows the matching of the best anatomy of the metamorphic robot to a task in order to increase its kinematic performance during task execution. The comparison of the results is made capable, which strongly hints that metamorphic robots could indeed show far better performance, than fixed anatomy manipulators [17]. The jerk analysis of a 3-RRPS parallel manipulator to realize six degrees of freedom is approached by means of the theory of screws. The input/output equations of velocity, acceleration and jerk of the moving platform with respect to the fixed platform are obtained systematically by resorting to reciprocal-screw theory by the method of kinematic analysis and the hybrid algorithm is used to reduce the jerk [18]. A proposed approach was validated and evaluated by considering actuation limits and different assembly modes to optimization a force, for a planar 3-RRR parallel manipulator. The numerical results are compared with modified differential evolution (MDE) under taking the kinematic constraints [19]. A new method is introduced to increase the singularity avoidance ability of a three dimensional planar manipulator, by way of increasing its degrees of freedom. The manipulability measure values of both of proposed manipulators and PUMA arm have been calculated and
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analyzed, and the results of the illustrated for singularity avoidance under consideration of Kinematic constraints [20]. A model-based algorithm is proposed to obtain a solution to calculate the manipulator's pose accurately by comparing with Lowe's method. Besides the accuracy, the proposed algorithm also performs better in computation efficiency [21]. B. Dynamic Approach
Dynamic motion planning involves motion through space, while avoiding dynamic constraints. The dynamic constraints are usually in the form of torque and force limits. This method always requires some model of the manipulator dynamics. This model may be explicit with a numerical integration, or approximate with a discrete estimation. If position, velocity and acceleration are provided, the inverse dynamics of the manipulator may be used to find the instantaneous joint torques. If the position, velocity and torque are provided, the forward dynamic equations may be used to determine instantaneous acceleration. If a torque is applied for some time, then the forward dynamics equation must be integrated to find the effect of torque on position and velocity. If the position, velocity, and acceleration are applied, then the inverse dynamics may be integrated over some time period to find the path energy. A search algorithm to generate near minimum time trajectory with near optimal time motion planning under kinematic redundancy, Joint angle, Joint rate, Acceleration, Jerk, workspace, and actuator Torque limitations by forward kinematics and inverse dynamics [22]. A direct description method is applied for a 6DOF KUKA 361 robot system through a nonlinear change of variable, the time optimal trajectory planning is transferred into a convex optimal control problem with a single state that reduces finding the globally optimal trajectory to solve a 2nd order Cone Program by using robust numerical algorithm [23]. An efficient algorithm to obtain shorter motion times under the given near torque condition compared with theoretically and experimentally for 6 DOF Industrial Robot [24]. A global optimization approach based on hybrid genetic algorithm to obtain minimum time cubic-spline trajectory subjected to energy constraint was introduced for Parallel Platform Manipulator under kinematic and dynamic constraints by using the PSO optimization technique [25]. A new acceleration continuous procedure is added to the minimum time feed rate optimization algorithm to address Jerk constraints and remove discontinuities in the acceleration profile. The algorithms maintain computational efficiency and support the incorporation of a variety of Position, velocity, acceleration and Jerk constraints [26]. An algorithm is proposed to consider both execution time and the integral of squared Jerk for fast execution and smooth trajectory. The cubic and B-Spline techniques were considered, evaluated and experimentally validated for obtaining minimum time-jerk trajectory planning [27]. A genetic algorithm is presented to search the optimal joint inters knot parameters which includes joint angle and joint velocities of a space robotic manipulator in joint space in order to realize the maximum Jerk [28]. A new trajectory planning is introduced to suppress residual vibration
ISBN: 978-81-909042-2-3 ©2012 IEEE
IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012 in two link rigid flexible manipulators. The joint angle of the flexible link was expressed as a Cubic Spline function and PSO technique is implemented [29]. By using Cubic Spline and 5th order B-Splines trajectory to minimize the maximum absolute value of the jerk along the whole trajectory and to ensure the continuity of the position, velocity, acceleration values [30]. A novel method that uses 8 DOF polynomial functions to generate a smooth trajectory for parametric representation of a given path. A Genetic Algorithm is used to minimize the mechanical energy consumed in the robot manipulator which includes inverse kinematics and dynamics of a manipulator [31]. An offline trajectory planning algorithm that provides a class of parametric trajectories to the unactuated joint in order to reach desire static configurations of the system based on dynamic constraint. The Nedler-mead Simplex method fmin search is crude and prone to local minima [32]. A SQP algorithm is used to reduce the global vibration or residual vibration of flexible link by joint trajectory which is obtained by hybrid optimization technique [33]. A Genetic Algorithm is used to optimize the joint angles in a given search space for 3 armed planar manipulator that would contribute to a productive and quality way of material handling and processing [34]. The Genetic Algorithm is utilized to optimize the trajectory of free joints of a Crawling Giant in a robot in order to minimize the effort. B-Spline curves are used to convert into parametric optimization [35]. The performance of the trajectory planning and the fitness of the selection of the Interactive Robot controller model/parameters can be visually evaluated by the simulation and Cubic Spline trajectories are used for the simulation [36]. A new trajectory is optimized by minimizing the integral square of the 4th time derivative by adjusting the velocity and acceleration conditions to enable the maximum time compression by considering axis velocity, acceleration, and jerk constraints for 5-axis on-the-fly laser drilling [37]. A one-layer recurrent neural network is proposed for solving pseudo convex optimization problems subject to linear equality and bound constraints by comparing with the existing neural networks for optimization the proposed neural network is capable of solving more general pseudo convex optimization problems dynamic constraints [38]. A different level bi-criteria minimization method is proposed for eliminating joint-torque instability, joint-acceleration, and guarantees the final joint velocity of the motion to be near zero under consideration of dynamic constraints for PUMA 650 robot manipulator [39]. A novel method is suggested a framework for using an Orthogonal crossover in DE variants and proposed OXDE, a combination of DE and ox. The experiments is carried out to study OXDE and to demonstrate that the framework can also be used for improving the performance of other DE variants [40]. An experimental evaluation and validation is carried out of two minimum timejerk trajectory planning algorithms compared with a global minimum jerk and a "classic" spline algorithms namely a cubic splines based technique (named SPL3J) and a fifthorder B-splines based technique (named BSPL5J) [41]. A general framework is presented for the dynamic modeling and
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analyses of the legged robots using the concept of kinematic modules. For the proposed modular framework, the inverse and forward dynamics algorithms with inter and intra modular recursions has developed for the simulation of controlled legged robots are solved by Recursive algorithms [42].
VII. MINIMUM ENERGY TRAJECTORY
In some fields the time is not the most important, but the consumed energy is considered as the primal criteria. This can be the case where the amount of available energy is limited Planning the robot trajectory using energetic criteria provides several advantages. On one hand, it yields smooth trajectories resulting easier to track and reducing the stresses to the actuators and to the manipulator structure. A novel trajectory planning is generated by the dynamic behavior of the robot to reduce total time and torque for PMA 560 robot manipulator in different workspaces [43]. A smooth trajectory is generated for a given path to minimize the mechanical energy consumed in the manipulator. The results are compared with trajectory planning algorithm and exhaustive algorithm [44]. A genetic port based model for variable stiffness actuator is presented with velocity of designs which can be modeled and analyzed. A concept design of an energy efficient variable stiffness actuator is implemented [45]. A real coded Genetic Algorithm is developed to obtain the optimal trajectory for a minimum energy. A higher degree polynomial is used to compute minimum energy point to point trajectory which is carried out by using inverse dynamic analysis [46]. A Multi-Objective Genetic Algorithm has done for PKM to determine the optimal location of a given test path in order to minimize the electric energy used by the actuators, their maximal torque and the shaking forces subject to the kinematic, dynamic and geometric constraints [47]. A tree based algorithm is used to find a feasible trajectory between the start position and a distant goal to improve the computation time [48]. A new optimal control problem is introduced to find a intersection-free trajectory planning in which the dynamic equation of the mechanical system and the holonomic constraints are treated using special derivative multipliers [49]. An energy consumption model is developed to minimize dissipating energy for optimal feet forces distributions. By considering the kinematics and dynamic models of a realistic six-legged robot to analyze complex relationships among gait parameters and its power consumption [50]. VIII. COLLISION FREE TRAJECTORY
When dealing with autonomous robots one is always faced with the serious problem of collision avoidance, which is one of the main issues in applications for a wide variety of tasks in industry. When the required tasks cannot be carried out by a single robot, then multiple robots are used cooperatively. However, this may lead to a collision if they
ISBN: 978-81-909042-2-3 ©2012 IEEE
IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012 are not properly navigated. So the main task of path planning for robot manipulators is to find an optimal collision-free trajectory from an initial to a final configuration. The optimal solution to prevent a collision can be defined by the planned arrival time the total traveling distance or time and the time delay. A real time modification method of reference trajectories with collision avoidance for multiple robots [51]. A hybrid optimization technique that combines Genetic Algorithm and pattern Search techniques for obstacle avoidance under kinematic constraint [52]. A novel methodology is used to plan collision free paths for an n DOF manipulator in a 3D environment for facilitating intuitive robot programming [53]. A collision free trajectory is obtained and integrated with the continuation method, creating an algorithm is enabled the boundary mapping of collision free reachable workspaces. The reachable workspaces are calculated and compared with the results of manipulators [54]. A real time collision free trajectory planning method using the virtual force based approach. This method is able to drive the robot arm in a narrow environment without collisions [55]. A method is generated for designing optimal smoothing spline with equality and inequality constraints including cross coupled constraints. The B spline curve is used for optimal trajectory planning [56]. The end effector trajectory is subjected to consider equality and inequality constraints for optimal collision free control of manipulator with minimal energy [57]. A Constraint supervision algorithm is proposed in robotic systems to fulfill configuration and workspace constraints caused by robot mechanical limits, collision avoidance, and industrial security under using the Dynamic constraints [58]. A new controller approach is applied to redundant robot manipulators constrained by mobile obstacles to achieve a good trajectory tracking of the end effector even if the obstacles are fixed or mobile manipulator by using inverse kinematic constraints [59]. A Sensor-based motion planning is introduced for sensor calibration and 3D data segmentation to use it to automatically plan grasps and manipulation actions for a service robot which allows the robot to move within dynamic without collision [60].
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A review for optimal trajectory planning of autonomous robot has been illustrated in this paper. The techniques are based on the minimization of objective functions that takes into account constraints like minimum Time, Velocity, Energy, Jerk, Torque and vibrations are expressed by means of Kinematic and Dynamics constraints along the whole trajectory. Kinematics approaches give the results but when comes to reality inertia and torque constraints make it difficult to execute. Dynamical approaches prove to be more realistic in terms of fitting in torque constraints and joints physical limits. The algorithms like Genetic Algorithms are used for optimal trajectory.
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