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Leng-Feng Lee (llee3@eng.buffalo) Advisor : Dr. Venkat N. Krovi Mechanical and Aerospace Engineering Dept. State Univers

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## Leng-Feng Lee (llee3@eng.buffalo) Advisor : Dr. Venkat N. Krovi Mechanical and Aerospace Engineering Dept. State Univers

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**Decentralized Motion Planning within an Artificial Potential**Framework (APF) for Cooperative Payload Transport by Multi-Robot Collectives Leng-Feng Lee (llee3@eng.buffalo.edu) Advisor : Dr. Venkat N. Krovi Mechanical and Aerospace Engineering Dept. State University of New York at Buffalo**Agenda**• Motivation & System Modeling • Literature Survey & Research Issues • Local APF & limitations • Global APF-Navigation Function • Case Studies-Single robot with APF Part I • Dynamic Formulation-Group of Robots • Motion Planning-Three Approaches • Case Studies-Multi Robots with APF • Performance Evaluation of Three Approaches Part II • Conclusion & Future Work**Motivation**• Examples of Multi-robot groups: • Tasks are too complex; • Gain in overall performance; • Several simple, small-sized robot are easier, cheaper to built, than a single large powerful robot system; • Overall system can be more robust and reliable. • Group Cooperation in Nature: Armies of Ants Schools of Fish Flocks of Birds • How do we incorporate similar cooperation in artificial multi robot group? Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Motivation**• Example of Multi robot groups: Robots in formation • Cooperative payload transport Robots in group Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Motion Planning (MP) for Robot Collectives**• Definition: • The process of selecting a motion and the associated set of input forces and torques from the set of all possible motions and inputs while ensuring that all constraints are satisfied. • Why Motion Planning? • To realize all the functionalities for mobile robots, the fundamental problem is getting a robot to move from one location to another without colliding with obstacles. • MP for Robot Collective - • MP exist for individual robots such as manipulator, wheeled mobile robot (WMR), car-like robot, etc. • We want to examine extension of MP techniques to • Robot Collectives Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**Motion Planning Algorithm Classification**• Explicit Motion Planning: • Decompose MP problem into 3 tasks: • Path Planning, Trajectory Planning, & Robot Control; • Example: Road Map Method, Cell Decomposition, etc. • Implicit Motion Planning: • Trajectory and actuators input are not explicitly compute before the motion occur. • Artificial Potential Field (APF) Approach belongs to this category. • Combine Path Planning, Trajectory Planning, and Robot Control in a single framework. Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**Motion Planning (cont’)**• Artificial Potential Field (APF) Approach: • Obstacles generated a artificial Repulsive potential and goal generate an Attractive potential. • Motion plan generated when attractive potential drives the robot to the goal and repulsive potential repels the robot away from obstacles. • Combine Path Planning, Trajectory Planning, and Robot Control in one framework. Subclass of Implicit Motion Planning Algorithm Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**Research Issues**• Broad Challenges: • Extending APF approach for Multi-robot collectives. • Ensuring tight formations required for Cooperative Payload Transport application. • Specific Research Questions: • Which type of potential function is more suitable for MP for multi robot groups? • How can we use the APF framework to help maintain formation? and • How this framework be extended to realize the tight formation requirement for cooperative payload transport? Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**Research Issues (cont’)**• To answer these research questions: • Part I: • Study various APF & their limitations; • Determined a suitable APF as our test bed; • Create a GUI to design and visualize the potential field; • Case studies: MP for single robot using APF approach. • Part II: • E.O.M. for group of robots with formation constraints; • Solved the MP planning problem using three approaches; • Performance evaluation using various case studies. Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**Research Issues (cont’)**• Hierarchical difficulties in MP: (Dynamic Model) • Our results: • Multiple point-mass robots; • Sphere World; • Stationary Obstacles & Target. Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**System Modeling**• Individual level system models include: • Point Mass Robot; • Differentially Driven Nonholonomic Wheel Mobile Robot (NH-WMR). Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**System Modeling (cont’)**• Group level system model is formed using: • Point Mass Robot; • Differentially Driven Nonholonomic Wheel Mobile Robot. Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion**PART I: Artificial Potential Approach**• Examine: • Variants of APF & their limitations; • Navigation function ; • Single module formulations; • Simulation studies.**Local APF -background**• Artificial Potential Field Approach • Proposed by Khatib in early 80’s. • FIRAS Function. [Khatib, 1986] • Later, various kind of Potential Functions were proposed: • GPF Function. [Krogh, 1984] • Harmonic Potential Function. [Kim, 1991] • Superquadric Potential Function. [Khosla, 1988] • Navigation Function.[Koditschek, 1988] • Ge New Potential. [Ge, 2000] Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF Approach-Formulation**• Idea: • Goal generate an attractive potential well; • Obstacle generate repulsive potential hill; • Superimpose these two type of potentials give us the total potential of the workspace. Where: denote the total artificial potential field; denote the attractive potential field; and is the repulsive potential field. is the position of the robot. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Attractive potential**• Characteristics: • Affect every point on the configuration space; • Minimum at the goal. • The gradient must be continuous. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Attractive potential**• Example 1: Where: = Positive scaling factor = Euclidean distance between the robot and the target = Position of the target. = Position of the robot. is commonly used. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Attractive potential**• Example 2: Where: = Positive scaling factor For distance smaller than s, conical well. For distance larger than s, constant attractive force. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Repulsive potential**• Characteristics: • The potential should have spherical symmetry for large distance; • The potential contours near the surface should follow the surface contour; • The potential of an obstacle should have a limited range of influence; • The potential and the gradient of the potential must be continuous. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Repulsive potential**• Example 1 - FIRAS Function: Where: = Positive scaling factor = the shortest Euclidean distance between the robot from the obstacle surface Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Repulsive potential**• Example 2 - Superquadric Potential Function: • Approach Potential; • Avoidance Potential. • Avoid creation of local minima result from flat surface by creating a symmetry contour around the obstacle. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Repulsive potential**• Example 3 - Harmonic Potential Function: Attractive Potential Repulsive Potential • Superimpose of another harmonic potential is also a harmonic potential. • More complicated shape can be modeled using ‘panel method’. Detail Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Repulsive potential**• Example 4 - Ge New Function: Where: = Minimal Euclidean distance from robot to the target. • Modified from FIRAS function to solve the ‘Goal NonReachable for Obstacle Nearby’ -GNRON problem. • Ensures that the total potential will reach its global minimum, if and only if the robot reaches the target where Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Repulsive potential**• Potential Function with Velocities Information: • Some potential function include the velocities information of the robots, obstacles and target. • Example: Ge & Cui Potential [Dynamic obstacle & Target]. • Provide a APF for dynamic workspace. • Example: GPF Function. [Dynamic obstacles only]. • Can be used with our formulation for group of robots for motion planning in dynamic workspace. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF –Total Potential**• Total Potential of Workspace: • Superimpose different repulsive potential from obstacles and different attractive potential from the goal, we get the total potential for the workspace. • At any point of the workspace, the robot will reach the target byfollowing the negative gradient flow of the total potential. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF –Total Potential**• Example: FIRAS Function Rectangular Obstacle: Circular Obstacle: Radius 2 unit in height, 1 unit in width. Target : More Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF –Limitations**• Local Minimum - result from single obstacle 3D View Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF –Limitations**• Local Minimum - result from multiple obstacles Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF –Limitations**• Limitation - Target close to obstacle: Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Local APF -Limitations**• Some other limitations include: • No passage between closely spaced obstacle. • Non optimal path. • Implementation related limitations. • Oscillation in the presence of obstacle; • Oscillation in narrow passages; • Infinite torque is not possible. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Global APF – Navigation Function**[ Proposed by: Rimon & Koditschek] • Properties: • Guarantee to provide a global minimum at target. • Bounded maximum potential. Let be a robot free configuration space, and let be a goal point in the interior of , A map is a Navigation Function if it is: . function. , that is, at least a 1. Smooth on . 2. Polar at ,i.e., has a unique minimum at on the path-connected component of containing , i.e., uniformly maximal on the boundary of 3. Admissible on 4. A Morse Function Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Feature: Tunable by a single parameter :**Navigation Function • Navigation Function of a sphere world : Where: Detail is the implicit form of bounding sphere. is the implicit form of obstacle geometric Eq. Number of obstacles Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Navigation Function**• Example - Navigation Function of a sphere world : Where: Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Navigation Function -Constructions**• At low value of , local minima may exist: Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Navigation Function – MATLAB GUI**• A GUI to properly select a value: Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Formulation & Simulation**• Idea: • We want the robot to follow the negative gradient flow of the workspace potential field; • Analogy to a ball rolling down to the lowest point in a given potential. • Thus the gradient information will serve as the input to the robot system. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Formulation**• Formulation – Single point-mass robot: Kinematic Model: Dynamic Model: is a positive diagonal scaling matrix is the gradient of the potential field is dissipative term added to stabilize the system Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Formulation**• Formulation – Nonholonomic Wheeled Mobile Robot (NH-WMR): Kinematic Model: is the projected gradient onto the direction of forward velocity. is the proportional to the angular error between the gradient and robot direction. the desired x-direction velocity. desired y-direction velocity. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Formulation**• Formulation – Group robot without formation constraints: Generalize position: -number of point-mass robot Kinematic Model: Dyanamic Model: Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Simulations**• Simulation 1 – Single robot with single obstacle: Detail Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Simulations**• Simulation 2 – Single robot with two obstacles: Detail Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Simulations**• Simulation 3 – Single NH-WMR with four obstacles: Detail More Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Simulations**• Simulation 4 – Group robots without formation constraint: Detail Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**APF Approach – Simulations**• Simulation 5 – Group robots without formation constraint: Detail Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**PART II: Group Robots Dynamic Formulation**• Include: • Dynamic Formulation for Group of Robots with Formation; • Solved the E.O.M using three Methods; • Simulation Studies; • Performance evaluation of each Methods.**Group Robots Dynamic Formulation**• Approaches for formation maintenance: • Formation Paradigm • Leader-follower [Desai et. al., 2001] • Virtual structures [Lewis and Tan, 1997] • Virtual leaders [Leonard and Fiorelli, 2001], [Lawton, Beard et al., 2003] • Our Approaches: • View as a constrained mechanical system. • Formation constraints – holonomic constraints added to a unconstrained dynamic system. • Motion planning now can be treated as a forward dynamic simulation of a constrained mechanical system. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Group Robots Dynamic Formulation**• The dynamic of group of robot can be formulated using Lagrange Equation by: (1) is the n-dimensional vector of generalized coordinates is the n-dimensional vector of generalized velocities is the n-dimensional vector of generalized velocities is the n-dimensional vector of external forces is the vector of input forces, which is is the Jacobian matrix Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Group Robots Dynamic Formulation**• The Lagrange Equation can be solved using following three methods: • Method I: Direct Lagrange Multiplier Elimination Approach. • Explicitly computing the Lagrange multiplier by a projection into the constrained force space. • Method II: Penalty Formulation Approach. • Approximating the Lagrange multiplier using artificial compliance elements such as virtual springs and dampers. • Method III: Constraints Manifold Projection Based Approach • By projecting the equations of motion onto the tangent space of the constraint manifold in a variety of ways to obtain constraint-reaction free equations of motions. Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Group Robots Dynamic Formulation**• Method I: Direct Lagrange Multiplier Elimination Approach: • The direct Lagrange multiplier elimination is a centralized approach where the Lagrange multiplier is explicitly calculated to ensure formation constraints are not violated. (2) The resulting Dynamic Equation can be expressed as: (3) Detail Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion**Group Robots Dynamic Formulation**• Method II: Penalty Formulation Approach: • The holonomic constraints are relaxed and replaced by linear/non-linear spring with dampers. • Here, the Lagrange multipliers are explicitly approximated as the force of a virtual spring or damper based on the extent of the constraint violation and assumed spring stiffness and damping constant. This can be expressed as: Resulting Dynamic Equation: (4) Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion