Petri Net Modeling for Robotic Assembly and Trajectory Planning

Brenan J. McCarragher

IEEE Transaction on Industrial Electronics, Vol. 41, No. 6, December 1994

Summary

                   A new approach for task-level control of robotic assembly is proposed considering 3 components: a process monitor (for tracking the assembly process), a discrete event controller (for determining the velocity commands that maintained desired contacts) and a novel means of trajectory planning which incorporates the system’s ability to track and control the process. Assembly is treat as a discrete even dynamic system modeled with Petri nets. A Petri net is composed of four parts: a set of places P, a set of transitions T, an input function and an output function O. The Input function is mapping from places to transitions, while the output function is doing the opposite. P is denoted as P=P^S+P^C (respectively state places of current place and place conditions of external inputs that enable transitions to occur). A state place is defined by a contact pair (Φ,φ) for respectively edges and surfaces. The occurrence of a transition is to be considered a discrete event, the input function activates the places for a given transition (when conditions are then met, the transition is said “place enabled”); the output function defines the contact pair resulting from a discrete event. Inhibitor arc leave control place p_i to reach transition t_j, so that t_j is enabled only if place p_i does not contain tokens (not inhibited transition are called “control enabled”). A token in a state place indicates contact between edge and surface, in this way the Petri net is controlled. Input and output function can be expressed through the use respectively of two matrices: N^- and N⁺, when the element i,j for given matrix is equal to one then i place is required as input of transition j; N=N⁺-N^- is the composite change matrix, indicating the overall change in the marking of the net.

The matrix can be used to determine the result of firing transition t_j (next-marking γ_d ∈ R^p), where t_j is a vector equal to 0 except in jth component so that γ_d=γ + N(t_j), this equation is used to define the marking that result from the execution of the Pertri Net.

·       Model

Process Monitoring is the task of determining the current status of the assembly process and it is important in regard of reliability and preventing failures. A strain gage sensor is attached to the workpiece to check forces, a quality approach of signal understanding is used in order to reduce the noise, ensure a faster system and avoid estimate several unrequired quantities. The decidability problem is based on the criterion that templates can’t have similar force readings. Events are measured in terms of change in force, therefore of change in acceleration, which denotes of course a change in speed. According to basic kinematic equation introduced at page 635, it is possible to define mainly a state of Gain Contact (for which distance between the edge and the surface for a peg-in-hole problem must decrease), Loss Contact (for which the distance must increase) and lastly a state for which an event may not occur. The equation and disequalities may bring to more than once solution, the optima is obtained as: J=max[min(a_iq’], where a is the velocity coefficient vector describing the constraint represented by place p_i. and q is velocity of generic coordinates. A further constrain must be introduced to impose upper boundary conditions: x’²+y’²+(lα’)²=1 where α’ is rotational speed linearized with l, the distance the of contact to the origin. The Net may be used also for error recovery and of course for trajectory planning according to page 637 algorithm, considering decidability, directability and ability weights obtained through the path length.

Key Concepts

Petri Net

Key Results

Experiments performed in a two-peg-in-hole problem have been performed showing success in fast and slow speed, with some problem in final placing, having some non feasible solutions (the transition is not directable). The system was anyways able to recover from mismatching at different speeds.

A hidden Markov model-based assembly contact recognition system

H.Y.K. Lau

Mechatronics, No.13, 2003

Summary

                  Robotic assembly is a defined as a classical problem in industrial robotics where a rigid robot with accurate actuators performed sequences of predetermined operations in an assigned workspace through a position-based controller. Limitations during assembly tasks appear to be due to fixtures, features and tolerances, in order to overcome these limitations force-based control strategies have been proposed, such as: spring damper, hybrid force/position control, impendence control and so on. The fundamental issue in robotic assembly is the ability to recognize assembly states and this should be addressed before the deployment of appropriate control strategies. The proposed system (HMM-based contact state recognized) can be considered a high-level feedback control system for perceiving the environment in terms of symbolic expressions. With Fundamental Contact (FC) the author intends to describe contact formations that may consist of zero, one or more pairs between the work piece and the environment in which it is involved (it is a primitive for of contact formation involving a single pair of contact features). A contact is defined as a result of geometric arrangement, contact based control strategy for robot assembly is based on the pattern of force/torque which is formed between a work piece and the manipulator, so that the d.o.f. is reduced at least by one. Different contact states are then defined: 1) Configuration of an object (c, is the set of kinematics parameters that locates an object for a given set of joint angles); 2) Configuration Space (C, the Rⁿ space which collects different c); 3) Contact Feature ( F≜{f,e,v}, where f is the face, e the edge and v the vertex of the object); 4) Fundamental Contact (FC ordered pair of contact feature; 5) Contact State (CS is a set of FC between to polyhedra).

In robotic assembly Contact Recognition is defined as the process using a contact sensing techniques to obtain geometrical information of contact with corresponding symbolic interpretations. Symbolic interpretations of an assembly provide high-level knowledge of assembly operations to robot programmers, but some limitations in the models (Petri-nets, ANN, Rule-Based contact analysis and further on) have been encountered due to complexity. HMM, already used successfully for speech recognition, is proposed considering assembly states interrelated and occurrence of contact formation between a work piece and the mating parts normally distributed.

·       Model

A HMM is composed of two stochastic processes, a basic Markov chain and an observable stochastic process. HMM are defined with a triplet λ=(A,B,π) (A is the transition matrix for the transition probabilities, B the matrix of probabilities for the observables and π is the initial state of the distribuition). HMM can be used to generate the observation sequence, mainly observation probabilities can be evaluated, the most probable sequence, given the observation sequence, can be also obtained and the probability of a sequence given an observable for a certain triplet can be maximized to find the HMM performance. The architecture of the HMM-based contact state recognition works in the following manner: defining the force torque signals, process them, map them and transfer the data to the HMM, after which the model with maximum probability must be selected in order to finally classify the state. The method, which for its final implementation uses forward and backward algorithm, Baum Welch algorithm (Appendix A),ALBG-VQ algorithm and symbol mapping algorithm (Appendix B), has been tested to investigate sensitivity, performance of recognizing 2D and 3D contact formations and the ability to classify sequence of contact formation during peg-in-hole tasks.

Key Concepts

HMM, assembly contact recognition

Key Results

The system is insensitive to the number of states and small amount of training data is required. Superiority of HMM-based system has been shown compared with other traditional methods already mentioned above.