Optimal Control Motion Planning

Optimal Control Motion Planning O. Hachour Abstract Motion planning is one o the important tasks in intelligent control o an autonomous mobile robot. An optimal ree path without collision is solicited in any design o movement o an autonomous mobile robot. he robots are compelling noor reasons o mobility but because o their autonomy, and so their ability to maintain a sense o position and to navigate without human intervention is paramount. o deal with optimal control concept and to present a real intelligent task, we propose an optimal control motion or autonomous mobile robot. he objective o optimization theory is to determine the control process that will cause a process to satisy the physical constraints and at the same time minimize (or maximize) the perormance measure (cost). he best response is gotten at the end where all boundary elements constraints are satisied. Our discussion here will be restricted to systems which are described by ordinary dierential equations (in state variable model). he theory developed is aimed to solve the problem o optimal control. hat means that, ind an admissible control which causes the systems to ollow an admissible trajectory that minimize the perormance. As case study o optimization theory, we have practiced the motion planning o an autonomous mobile robot where we have studied clearly the main steps o optimal control. his is a clariied model o optimal control o moving a robot where the elapsed time is the principal perormance to be evaluated. Keywords Intelligent system, optimal control, artiicial intelligence, decision. I. INRODUCION he history o autonomous mobile robotics research has largely been a story o closely supervised, isolated experiments on platorms which do not last long beyond the end o the experiment. here is no universally accepted deinition o the term robot. ypical deinitions encompass notion o mobility, programmability, and the use sensory eedback in determining subsequent behavior [4,5]. Autonomous robots which navigate without human interventions are required in robotic ields. In order to achieve tasks, autonomous robots have to be intelligent and should decide their own action. When the autonomous robot decides its action, it is necessary to plan optimally depending on their tasks. More, it is necessary to plan a collision ree path minimizing a cost such as time, energy and distance. When an autonomous robot moves rom a point to a target point in its given environment, it is necessary to plan an optimal or easible path avoiding obstacles in its way and answer to some criterion o autonomy requirements such as : thermal, energy, time, and saety or example [3,4]. A robotic system is an intelligent mobile machine capable o autonomous operations in structured and unstructured environment. It must be capable o sensing, thinking and acting. he mobile robot is an appropriate tool or investigating optional artiicial intelligence problems relating to world understanding and taking a suitable action, such as, planning missions, avoiding obstacles, and using data rom many sources. According to the Robotics Industries Association (RIA): A robot is a reprogrammable multiunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions or the perormance o a variety o tasks. his artiicial automaton produce the remarkable repetitive actions that may be tedious and hazardous health human risks as exposure to unsae materials like radioactive and high pressure in underwater applications. he automaton mobile robots are able to react and act in structured or unstructured environments as human being. hese machines are able o taking a suitable decision with appropriate best intelligence. Finally, the notion o simulating biological organism has a certain instinctive reproductive appeal and oers the possibility o satisying our curiosity as to how come to be as we are. Being able to interact and communicate with robots in the same way we interact with people as long been a goal o Artiicial Intelligence AI and robotic researches. However, much o the robotics research in the past has emphasized the goal o achieving autonomous agents. Classical artiicial intelligence systems presuppose that all knowledge is stored in a central database o logical assertions or other symbolic representation and that reasoning consist largely o searching and sequentially updating that database. While this model has been successul or disembodied reasoning system, it is problematic or robots. he objective o intelligent mobile robots is to improve machine autonomy. his improvement concerns three (03) essential aspects. First, robots must perorm eiciently some tasks like recognition, decision-making, and action which constitute the principal obstacle avoidance problems. hey must also reduce the operator load by using natural language and common sense knowledge in order to allow easier decision making. Finally, they must operate at a human level with adaptation and learning capacities [5,6,7,8]. he navigation o mobile robots in an environment where stationary obstacles, and other moving objects, requires the ISBN: 978--6804-5-5 36

existence o algorithms that are able to solve the path and motion planning problem o these robots so that collisions are avoided and the easible path is ound. On the other hand, a suitable control law has to be designed, in order or the mobile robot to execute the desired motion. he problem becomes more diicult when the parameters that describe the model and/or the workspace o the robot are not exactly known. However, the mobile robot is an appropriate tool or investigating optional artiicial intelligence problems relating to world understanding and taking a suitable action, such as, planning missions, avoiding obstacle, and inding data rom many sources. Many traditional working machines already used are going through changes to become remotely operated or even autonomous. echnology has made this easible by using advanced computer control systems. o detect all possible obstacles, the robot is supposed to have vision system (camera). o operate in certain dynamic environments, the use o two or more sensors can guarantee to deliver acceptably accurate inormation all o the time. hus the redundancy can be useul or autonomous systems as in the human sensory system When an autonomous robot moves rom a source point to a target point in its given environment it is necessary to plan an optimal or easible path avoiding obstruction in its way and answering to autonomy requirements such as: thermal, energy, Communication Management, Mechanical design, etc. he theory and practice o Intelligent Autonomous Robots IAR are currently among the most intensively studied and promising areas in computer science and engineering which will certainly play a primary goal role in uture. hese theories and applications provide a source linking all ields in which intelligent control plays a dominant role. echnology has made this easible by using advanced computer control systems. Also, the automotive industry has put much eort in developing perception and control systems to make the vehicle saer and easier to operate. o perorm all tasks in dierent environments, the vehicle must be characterized by more sever limits regarding mass volume, power consumption, autonomous reactions capabilities and design complexity. Particularly, or planetary operations sever constraints arise rom available energy and data transmission capacities, e.g., the vehicles are usually designed as autonomous units with: data transer via radio modems to rely stations ( satellite in orbit or ixed surace stations) and power rom solar arrays, batteries or radio-isotope thermo electric generators (or larger vehicles). Classical control design is generally a trial- and -error process in which various methods o analysis are used iteratively to determine the design parameters o an acceptable system. Acceptable system perormance is generally deined in terms o time and requency domain such as time arise, gain and phase margin, peak overshoot and bandwidth. Radically dierence perormance criteria must be satisied, however by complex, multiple-input, multiple outputs systems required to meet the demand s o modern technology. For example the design o a spacecrat attitude control system that minimizes uel expenditure is not amenable to solution by classical methods. A new and direct approach to the synthesis o these complex systems, called optimal control theory, has been made easible by the development o the digital computer. his paper deals with the intelligent navigation optimal control o autonomous mobile robot in an unknown environment. he objective is to determine the control models that will cause a process a robot to move reaching the best path. II. HE OPIMAL CONROL PROBLEM Optimal control needs three steps to be developed which are A. Mathematical Model It is the model that describes the process to be controlled. he mathematical model is considered as a nontrivial part o any control problem. it is just the uniied ramework which it has a strong motivation beore starting any control design. he objective is to obtain the simplest mathematical description that adequately predicts the response o the physical system to all anticipated inputs. B. Physical constraints Ater we have selected a mathematical model, the next step is to deine the statement o physical constraints on the state and control values. he selection ocus about admissibility concept. Admissibility is an important concept, because it reduces the range o values that can be assumed by the states and controls. Rather than consider all control histories and their trajectories to see which are the best (according to some criterion) we investigate those trajectories and controls that are admissible. C. Perormance measure (the cost) In order to evaluate the perormance o a system quantitatively; the designer selects a perormance measure. An optimal control is deined as one that minimizes (or maximizes) the perormance measure. In certain cases the problem statement may clearly indicate that to selecor a perormance measure. Whereas in other problems the selection is a subjective matter. Generally or any optimal control design the perormance o a system is evaluated by a measure o the orm. t J = h(x(t), t) + g(x(t),u(t), t)dt () t0 ISBN: 978--6804-5-5 37

Where and are the initial and inal time; h and g are scalar unctions. may be speciied or ree, depending on the problem statement. starting rom the initial state x( )=x 0 t,t, causes a and applying a control signal u(t); or t system to ollow some state trajectory o the system. he optimal control problem is to ind an admissible control U* which causes the system to ollow an admissible trajectory X*that minimizes or maximizes the perormance measure []. In this context, an optimal control may be non unique. A non-unique optimal control may complicate computational procedures, but they do allow the possibility o choosing among several controller conigurations. his is certainly helpul to the designer, because he can then consider other actors, such as cost, size, reliability, etc; which may not have been included in the perormance measure. III. HE PROPOSED OPIMAL CONROL MOION Consider a robot shown in igure is to be controlled to navigate rom initial point position S to the target position. o simpliy the model, let us approximate that the shape o the robot by a unit point mass, or by a material point in the beginning. At the end o conception we will take the size o this robot (just to be added to the material point or gravity center). his physical ormaacilitates a lot o tasks. he distance o this roborom initial position at time t is denoted by d(t). he unit point mass can be accelerated, deviated, decelerated, or keeping a constant velocity. he dierential equation is given by:.. d (t) = a(t) + b(t) () Where : a(t) the control vector o acceleration. b(t) the control vector o deceleration., are a weighting actors included to permit adjustment o the relative importance o the three terms in d(t). he states variables are: X (t) d ( t) X ( t ) d.. ( t ) he control variables are: C ( t) a( t) C ( t) b( t) We deine the physical constraints or the state variables as: X ( t 0 ) S (5) X ( t ) 0 (3) (4) X X ( t ( t 0 ) 0 ) 0 Since the robot starts rom rest (initial velocity is zero 0) and stops when reaching the target.we can call these constraints by admissible states. A history o state values in the interval [, ] is called a state trajectory as it is shown in the igure where we present an example o a single valued unction o time which is denoted by x. We assume the ollowing constraints or the control inputs: 0 a(t) max - max b(t) 0 Where max is the maximum o acceleration max is the maximum o deceleration hese constraints are called admissible controls. he constraints are also called boundary conditions limit the system parameters. his is certainly very useul we investigate only with those are admissible to reduce the range values and to satisy some autonomy requirements such as: time, energy, thermal, etc. A history o control input values during the interval [, ] is called a control history. he autonomy implies that the robot is capable o reacting to static obstacles and unpredictable dynamic events that may impede the successul execution o a task. o achieve this level o robustness, methods need to be developed to provide solutions to localization, map building, planning and control. he robots are compelling noor reasons o mobility but because o their autonomy, and so their ability to maintain a sense o position and to navigate without human intervention is paramount. For example, AGV (Autonomous Guided Vehicle) robots autonomously deliver parts between various assembly stations by ollowing special electrical guide wires using a custom sensor. Several autonomy requirements must be satisied to well perorm the tasks o autonomous mobile robots. Ater selecting the admissible trajectories (stets) and the admissible controls the next step is to evaluate the perormance o a system quantitatively measure. Selecting o perormance measure is the subjective matter, this is due i a nonunique existence o perormance. In such cases the designer may be required to try several perormance measures beore selecting one which yields what he considers to be optimal perormance. For each investigated optimal control, we estimate the best perormance measure. his is when a nonunique optimal control exits. Here we are seeking the absolute or global minimum o the perormance measure o J, not merely local minima. It may be helpul to visualize the optimization as shown in the igure3. U (),U (), U (3) are points at which J has local or (6) (7) ISBN: 978--6804-5-5 38

relative minima u () is the point where J has its goal or absolute minimum. For our case study: optimal motion planning o an autonomous mobile robots we investigate with the ollowing perormance measure: J t (8) his is because the short elapsed time is the main necessity to reach the target without collision.we want that our autonomous mobile robot reach the target as quickly as possible without damage taking into consideration all embedded actors o navigation while answering to the main criteria o autonomy requirements. Navigation is one o the most challenging competences required o a mobile robot. Success in navigation requires success at the our building blocks o navigation: perception, the robot must interpret its sensors to extract meaningul data; localization, the robot must determine its position in the environment; cognition, the robot must decide how to act to achieve its goals; and motion control(see the igure 4), he robot must modulate its motor outputs to achieve the desired trajectory. O these our components, localization plays the role to ound the exact local points o the robot. he robot has to ind a collision-ree trajectory between the starting coniguration and the goal coniguration in a static environment containing some obstacles. o this end, the robot needs the capability to build a map o the environment, which is essentially a repetitive process o moving to a new position, sensing the environment, updating the map, and planning subsequent motion. his is necessary to build the trajectory o sub positions the easible optimal trace line towards the target without collisions». When an autonomous robot moves rom a source position to a target position, it musind a easible connection between the source and the target. In other word: It is necessary to plan an optimal or easible path avoiding obstacles in its way and answer to some criterion o autonomy requirements such as: thermal, energy, time, and saety or example. o operate independently in unknown or partially known environments is the basic eature o an autonomous mobile robot. he autonomy implies that the robot is capable o reacting to static obstacles and unpredictable dynamic events that may impede the successul execution o a task. o achieve this level o robustness, methods need to be developed to provide solutions to localization, map building, planning and control. he development o such techniques or autonomous robot navigation is one o the major trends incurrent robotics research. Coniguration space obstacles have high artiicial potentials that decline gradually with distance rom the obstacle. At any instance, the robot calculates the derivative o the potential unction and descends the maximal downward gradient in an eort to reach the minimum at the goal position. his calculation quickly determines the motion to take next. For every coniguration space, there is an optimal number o samples that must be selected to construct a suicient approximation o coniguration space connectivity. Motion planning will requently reer to motions o a robot in a D or 3D world that contains obstacles. he robot could model an actual robot, or any other collection o moving bodies, such as humans or lexible molecules. A motion plan involves determining what motions are appropriate or the robot so that it reaches a goal state without colliding into obstacles. When the autonomous robot decides its action, it is necessary to plan optimally depending on their tasks and on the environments complexity. o illustrate a complete problem ormulation and to deal with principle o optimal control, let us now present the optimal control and trajectory or the autonomous mobile robot shown in the igure 6. We assume that the robot has enough uel available to reach the target. he robot is simulated in dierent environments. o relect the robot behavior acquired by learning in the explored environment and in new unvisited environments. he robot reacts in eicient and a satisactory manner in these environments. he coniguration o the environments changes by adding other shapes o static obstacles, in each situation the robot can navigate successully. he lexible proposed approach is able to achieve the main task without collisions or every developed or proposed environment. he algorithm is implemented in several static environments; whereby the environment is studied in a two dimensional coordinate system. he algorithm permits the robot to move rom the initial position to the desired position ollowing an optimal estimated trajectory. aking a suitable action and reacting at the appropriate way, the roboinds its sae way without collisions in eicient manner. he igure 5 shows one example o this training algorithm. Robot S Fig. A robot control problem ISBN: 978--6804-5-5 39

x(t) a*(t) x(t) max ime ½(+t) t Fig. the state x(t) and its value b*(t) (a) J Admissible Control U u () u (3) u () Fig.3 a representation o the optimization problem max ½( +t) (b) Mission commands x*(t) Cognition path planning Path Actuators Path execution Commnads x*(t) ½( +t) (c) t Acting Fig. 4 Motion control t S ½( +t) (d) Fig.6 the optimal control and trajectory or the proposed autonomous mobile robot Fig. 5 an optimal example navigation set-up ISBN: 978--6804-5-5 40

IV. CONCLUSION he theory and practice o Intelligent Autonomous Systems are currently among the most intensively studied and promising areas in computer science and engineering which will certainly play a primary goal role in uture. In this paper, we have presented a model o an optimal control optimal control motion or autonomous mobile robot. he objective o optimization theory is to determine the control process that will cause a process to satisy the physical constraints and at the same time minimize (or maximize) the perormance measure (cost). he best response is gotten at the end where all boundary elements constraints are satisied.. he objective o optimal control theory is to determine the control signals that will cause a process to satisy the physical constraints and at the same time minimize or maximize some perormance measure criterion. he theory developed is aimed to solve the problem o optimal control. hat means that, ind an admissible control which causes the systems to ollow an admissible trajectory that minimize the perormance. As case study o optimization theory, we have practiced the motion planning o an autonomous mobile robot where we have studied clearly the main steps o optimal control. his is a clariied model o optimal control o moving a robot where the elapsed time is the principal perormance to be evaluated.. his is a clariied model o optimal control o moving a robot where the elapsed time is the principal perormance to be evaluated. his proposed approach has made the robot able to achieve these tasks: avoiding obstacles, deciding, perception, and recognition and to attend the target which are the main actors to be realized o autonomy requirements. Hence; the results are promising or next uture work o this domain. Besides, the proposed approach can deal a wide number o environments. REFERENCES [] D. KIRK, Optimal control, Prentice Hall edition 988 [] R.F.Stengel, stochastic optimal control: theory and application, Jhon wiley & Sons edition,986. [3] L.R Medsker, hybrid intelligent systems, Kluwer Academic Publishers, 995. [4] H K. Schilling and C.Jungius : Mobile robots or planetary explorations, in Proceeding nd International Conerence IFAC Intelligent Autonomous Vehicles, Finland, 995, pp.0-0. [5] O.Hachour, Pth planning o autonomous mobile robot, International Journal O System Applications, Engineering & Developments, issue4, volume,008,pp.78-90. [6] O.Hachour, he proposed Fuzzy Logic Navigation approach o Autonomous Mobile robots in unknown environments, International journal o mathematical models and methods in applied sciences, Issue 3, Volume 3, 009, pp 04-8. [7] O.Hachour, the proposed hybrid intelligent system or path planning o Intelligent Autonomous Systems; International journal o mathematics and computers in simulation,issue 3, Volume 3, 009, Pages 33-45. [8] O. Hachour,, path planning o Autonomous Mobile Robot, International Journal o Systems Applications, Engineering & Development, Issue4, vol., 008, pp78-90. ISBN: 978--6804-5-5 4