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Leng-Feng Lee (llee3@eng.buffalo) Advisor : Dr. Venkat N. Krovi Mechanical and Aerospace Engineering Dept. State University of New York at BuffaloPowerPoint Presentation

Leng-Feng Lee (llee3@eng.buffalo) Advisor : Dr. Venkat N. Krovi Mechanical and Aerospace Engineering Dept. State University of New York at Buffalo

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Leng-Feng Lee (llee3@eng.buffalo) Advisor : Dr. Venkat N. Krovi Mechanical and Aerospace Engineering Dept. State University of New York at Buffalo

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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

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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

- 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

- 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

- Example of Multi robot groups:

Robots in formation

- Cooperative payload transport

Robots in group

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- 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

- 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

- 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

- 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

- 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

- 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

- Individual level system models include:
- Point Mass Robot;
- Differentially Driven Nonholonomic Wheel Mobile Robot (NH-WMR).

Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion

- Group level system model is formed using:
- Point Mass Robot;
- Differentially Driven Nonholonomic Wheel Mobile Robot.

Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion

- Examine:
- Variants of APF & their limitations;
- Navigation function ;
- Single module formulations;
- Simulation studies.

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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 Minimum - result from single obstacle

3D View

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Local Minimum - result from multiple obstacles

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Limitation - Target close to obstacle:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- 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

[ 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 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

- Example - Navigation Function of a sphere world :

Where:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- At low value of , local minima may exist:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- A GUI to properly select a value:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- 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

- 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

- 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

- 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

- Simulation 1 – Single robot with single obstacle:

Detail

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Simulation 2 – Single robot with two obstacles:

Detail

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Simulation 3 – Single NH-WMR with four obstacles:

Detail

More

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Simulation 4 – Group robots without formation constraint:

Detail

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Simulation 5 – Group robots without formation constraint:

Detail

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Include:
- Dynamic Formulation for Group of Robots with Formation;
- Solved the E.O.M using three Methods;
- Simulation Studies;
- Performance evaluation of each Methods.

- 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]

- Formation Paradigm
- 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

- 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

- 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.

- Method I: Direct Lagrange Multiplier Elimination Approach.

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- 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

- 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

- Method III: Constraints Manifold Projection Based Approach:
- In this approach, the dynamic equation with constraint-reactions is projected into the tangent space (feasible motion subspace) to obtain the constraint free projected dynamics equations.

Is the independent velocities.

Thus, the resulting Dynamic Equation become:

(5)

Detail

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Baumgarte Stabilization:
- To prevent numerical drift in the simulation, we adopted Baumgarte stabilization method.
- Baumgarte stabilization method involves the creation of an artificial first or second order dynamical system which has the algebraic position-level constraint as its attractive equilibrium configuration.

For example, the holonomic constraint of Eq.(1) is replaced with:

Where the solution of the above equation is :

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Three point-mass robots forming a triangular shape:

The governing Equation can be written as:

where:

We will use this model to perform various case studies.

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Performance Evaluation – Formation Error:

Formation Error:

is the total formation error;

is the actual Euclidean distance between robot

and robot

and robot

is the desired Euclidean distance between robot

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 1 – Three robots in formation, without obstacle:

Method I:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 1 – Three robots in formation, without obstacle:

Method II:

Decentralized Formulation:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 1 – Three robots in formation, without obstacle:

Method III:

Partial Decentralized Formulation:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 1 – Formation Error from three methods:

Method I

Method II

Method III

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 1 – Formation Error & Effect of Ks

Method II:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 1 – Formation Error & Effect of

Method II:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 2 – Three robots in Formation, one obstacle

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 2 – Three robots in formation, one obstacle:

Method I

Method III

Method II

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 2 – Formation Error from three methods:

Method I

Method II

Method III

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 2 – Formation Error & Effect of Ks &

Method II

Method III

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 2 – Three robots in Formation with Expansion.

Each sides change from 2 units to 4 units in 4 seconds:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 3 – Three robots in Formation with Expansion.

Method I

Method III

Method II

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 3 – Formation Error & Effect of Ks &

Method II

Method III

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 4 – Three robots in Formation with Shape Change.

Constraint between robot A & B change from 2 units to 4 units in 4 seconds:

Note: Method I cannot perform this task because when three robots in a straight line, the inverse of the Jacobian matrix become singular.

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 4 – Three robots in Formation with Shape Change.

Method II

Method III

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Case Study 4 – Formation Error & Effect of Ks &

Method II:

Method III:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- General Characteristics – Formation Accuracy

The average total formation error for each method :

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- General Characteristics – Computational Time

The average total Computational Time (sec) for each method :

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- General Characteristics – Decentralize formulation capability

The decentralize formulation capability for each method :

Centralized Decentralized

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- General Characteristics – Formation relatedconcerns:
- The Jacobian matrix in Method I and Method III can become singular in some specific position.
- Method II has no such limitations.

In Summary:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Evaluation of various potential functions.
- Development of a GUI to generate navigation function.
- Develop the group motion planning problem as a forward dynamic simulation problem;
- Evaluation of three different method in solving motion planning problem for a group of robots in formation.
- Critical evaluation of the performance by the three approaches.

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

- Provide a way to avoid Jacobian matrix become singular.
- Incorporate nonholonomic constraints in the formulation.
- Implement a more efficient gradient finding method by utilizing the available information from each robot.
- Implement the algorithm in a decentralized computation manner.

Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion

Thank You!

Acknowledgments:

Dr. V. Krovi, Dr. T. Singh & Dr. J. L. Crassidis