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École Polytechnique de Montreal
Mechanical Engineering Department
Dissertation Proposal
Multi-Objective Trajectory Planning by Augmented Lagrangian
and Neuro-Fuzzy Techniques for Robotic Manipulators
Amar Khoukhi
Dissertation Advisers:
Luc Baron and Marek Balazinski
November 2006
1
Abstract
This research considers the multi-objective dynamic trajectory planning (TP) for robotic manipulators. In
the first part, the offline planning problem is formulated in a variation calculus framework. It is based on the
dynamic model of the robot including actuator and friction models. Optimised criteria are robot travelling
time and electrical and kinetic energy, and a measure of manipulability in order to avoid kinematic
singularities. The optimisation process is done while satisfying several constraints such as link length
limitations, and passing through imposed poses. The augmented lagrangian with decoupling is used to solve
the resulting constrained non linear and non convex optimal control problem. A first implementation on a
parallel kinematic machine case study gave good results - compared to only kinematic based control and
approaches based on penalty methods - in time and energy minimisation and constraints satisfaction.
The second part of this thesis considers the application of neuro-fuzzy techniques to adaptive motion
planning of redundant manipulators. Three approaches are considered. The first approach implements a
feed-forward neural network to learn the redundant inverse kinematics. This is followed by pre-processing
steps, involving a minimum time trajectory parameterisation in the joint space, satisfying starting and
ending conditions and limits on joint angles, rates, accelerations, and jerks. The corresponding torques are
computed using inverse dynamics. These torques are then projected into the admissible domain. A
subtractive clustering procedure is performed to initialise membership function parameters of a neuro-fuzzy
network, which is then trained and optimised. The learning performances of the neuro-inverse kinematics
and neuro-fuzzy system are shown to give very good results in learning robot inverse kinematics as well as
fitting the constrained near minimum-time trajectory. The second approach is similar to the first one, except
the first neural network is replaced by a neuro-fuzzy system. It is observed that the learning performances of
this approach are better than the first one. In the third approach, the redundant inverse kinematics is
included in the offline pre-processing steps, where the augmented lagrangian technique developed in the
first part of the thesis is applied to construct as many as needed trajectories covering the workspace and
satisfying the constraints. From the outcomes of such a realistically-model-based offline planned
trajectories, a data-driven neuro-fuzzy system is built and optimized to concisely represent the robot
dynamic multi-objective motion. Once the neuro-fuzzy controller structure (rules number and premise and
consequence membership functions) parameters are identified and optimized, it is used to achieve online
planning with reasonable real-time complexity. The preliminary results of this approach show better
performance as compared to the existing ones, mainly in redundancy resolution, and time and energy
minimization.
Keywords: Robot Manipulators, Parallel Kinematic Machines (PKM), Constrained Optimal Control,
Multi-objective Trajectory Planning, Augmented Lagrangian, Decoupling, Redundancy Resolution, Neuro-
Fuzzy Inverse Kinematics, Data-driven Neuro-Fuzzy Systems.
2
Nomenclature
B : Reference frame attached to the center of mass of the base
A : Reference frame attached to the center of mass of the end-effector (EE)
, : iiA iB th attachment point of leg i on body A and B
Tzyx ] [ =p : Position vector of the origin of A relative to B in B
Tzyx ] [ ...
=.p : Velocity vector of the origin of A in B
: Position and orientation of A in B = 1x TTE ] [ ψθϕpq =
: Cartesian and angular velocity of end-effector =2x TTT ] [ ..ωp=q
T] [ 21 xxx = : Continuous-time state of the manipulator
=kx Tkk ] [ 21 xx : Discrete-time state of the manipulator
)(tτ : Cartesian space force/torques wrench
: Vector of electric currents Tiii ] ,... , ,[ 621=i
Tlll ] ,... , ,[ 621=l : Vector of the link lengths
J : Jacobian matrix of the manipulator
)(qM j , : Inertia matrix expressed in joint and Cartesian space )(qM c
, : Coriolis and centrifugal force/torque in joint and Cartesian space ),(.qqN j ),(
.qqN c
)(qG j , : Gravity force in joint and Cartesian space )(qGc
M , aaM : Actuator inertia matrix and its component
V , aaV : Actuator viscous damping coefficient matrix and its component
K , aaK : Actuator gain matrix and its component
K : Control law gain matrix
mτ : Joint torque vector produced by the DC motors
3
λ : Lagrange multipliers (or co-states) associated to state variables
)( σρ , : Lagrange multipliers associated to inequality and equality constraints
, : Generic function defining inequality and equality constraints g s
) ,( Sg μμ : Penalty coefficients associated to inequality and equality constraints
N : Total number of discretizations of the trajectory
*1** , , ηηw : Cost minimization, equality and inequality constraints optimal tolerances
t1 , , ηηttw : Cost minimization, equality and inequality constraints feasible tolerances
List of Abbreviations
PKM : Parallel Kinematic Machines
IKP : Inverse Kinematic Problem
FK : Forward Kinematic
TP : Trajectory Planning
DOG : Degree Of Freedom
IDM : Inverse Dynamic Model
AL : Augmented Lagrangian
ALDP : Augmented Lagrangian with Decoupling and Projection
NetIK : Net Inverse Kinematics
ANFIS : Adaptive Neuro-Fuzzy Inference System
NeFIK : Neuro-Fuzzy Inverse Kinematics
NeFOTC : Neuro-Fuzzy Optimal-Time Controller
NeFuMOP : Neuro-Fuzzy Multi-Objective Planning
4
List of Figures
Figure 1. Robot Off-line Programming Framework ……………………………………………..............10
Figure 2. Example of a serial assembly robot, M16iB Series by Fanuc Robotics ……………………….10
Figure 3. Example of a parallel robot, F200iB by Fanuc Robotics ………………………………………10
Figure 4. Parallel Robot Flight Simulator: Series 500 by CAE…………………………………………..14
Figure 5. Illustration of EE passage through imposed poses……………………………………………...18
Figure 6. A schematic representation of a planar parallel manipulator……………………...……………24
Figure 7. Kinematic simulation results……………………………………………………………………25
Figure 8. Simulation results with the augmented Lagrangian (Minimum Time-Energy) ………………..25
Figure 9. Geometry of a 3 DOF serial planar robot…………………………………………………….…28
Figure 10. Neuro-Fuzzy Inverse Kinematic learning through Forward kinematic (FK).,…………….….28
Figure 11. Performance of NeFIK, Root mean square error………………………………………………28
Figure 12. Neuro-Fuzzy Optimal Time Controller learning through the Inverse Dynamics (ID)…………28
Figure 13. Overall architecture and function of the first neuro-fuzzy near optimal time trajectory planning
system …………………………………………………………………………………………………....30
Figure 14. Architecture of the Tsukamoto fuzzy inference system with one output noted z……………..30
Figure 15. Overall architecture and function of the neuro-fuzzy muli-objective planning system..……....31
List of Tables
Table 1. Limits of the workspace, actuator torques and sampling periods. ……………………................24
Table 2. Flowchart diagram describing milestones and progress of the thesis project …………………...32
5
Table of contents Abstract…...………………………………………………..…………………………………….2
Nomenclature…………………………………………………………………………………….3
List of Abbreviations…………………………………………………………………………….4
List of Figures……………………………………………………………………………………5
List of Tables……………………………………………………………………………………..5
Table of Contents……………………………………………………………………………..….6
1. Introduction and basic terminology…………………………………………………………7
2. Research objectives …………………………………………………………………………8
2. 1. Research motivation………………………………………………………………….8
2. 2. Scope of the Thesis…………………………….....................................................…..9
2. 3. Original contributions...……………………….....................................................….10
3. Literature survey………………………………………..……………………………………10
3. 1. Offline trajectory planning of robotic manipulators…..…………………….……..10
3. 2. Online trajectory planning of robotic manipulators………… ………..……...........13
4. Proposed research..……………………………………………………………………...........15
4. 1. Offline trajectory planning; Augmented Lagrangian with decoupling ……………..15
4. 1. 1. Problem statement. ………………………………………………………............15
4. 1. 2. Problem resolution ……………………...……………………..……...................18
4. 1. 3. Augmented Lagrangian with decoupling…………………….……………………19
4. 1. 4. Implementation on a planar parallel manipulator…………………………………23
4. 2. Online trajectory planning, A hierarchical data-driven neuro-fuzzy system…….…....24
4. 2. 1. Introduction………………………………………………………..………..........24
4. 2. 2. First approach: Neuro and neuro-fuzzy near optimal time planning.. ………….27
4. 2. 3. Second approach: Neuro-Fuzzy near optimal time planning..… ……………….28
4. 2. 4. Third approach: Neuro-Fuzzy multi-objective planning.....……………………..28
5. Progress to date and milestones………….………………….. …………...............................30
6. Conclusions ………………………………….. …….................................................................30
7. Acknowledgment……………………………………………………………………………..32
8. References …………………………………………………………………………….............33
Appendix: Summary and list of contributions……….. ………………………...........……….37
6
1. Introduction and basic terminology
Robot Trajectory Planning (TP) can take place in two different ways: on-line or off-line. On-line
TP is meant for real time robot motion planning and requires lower time consumption algorithms
compared to off-line planning. The latter takes place on a computer and sees the robot, task, and
workspace as a unique entity. TP might be defined through different levels of increasing
complexity. The first one is the path planning, which consists to find the shortest path to move the
robot end-effector (EE) from a starting to a target pose (positions and orientations). This case had
been classically identified as the piano-mover problem and had received a great interest from
robotics and computational geometry communities. The second level includes the of addition
velocity and acceleration information, while avoiding singular poses and obstacles. This yields to
the trajectory planning problem. When the forces and energy minimization are involved, one gets
higher performance task execution at a price of higher complexity [1-3]. Three basic related issues
to robot TP are robot kinematics and dynamics, task, and workspace. The kinematics deals with the
relationships between the EE and actuated joint motions and the mapping between their respective
spaces. Dynamics comes into play when the torques and forces causing the motion are considered.
The workspace defines the area or the set of poses that can be reached by the EE without
singularity, link interference, or collision with obstacles. A task might be defined by specific
starting, target and imposed intermediate poses [1-4].
In this research, two sub-problems are considered. First, a new offline TP is developed for
parallel manipulators. It is based on the dynamic model as well as workspace and task
requirements. Optimized criteria include time and energy necessary for a task execution, as well as
a measure of manipulability for singularity avoidance. Computational optimization techniques are
explored to solve this problem on a parallel manipulator case study. As for the second sub-
problem, the online trajectory planning; conventional optimal control techniques are too time-
consuming. This complexity related mainly to strong non linear and coupling dynamics and
kinematic transformation between EE and actuated joint spaces, makes them impractical for online
implementation. In this research, data-driven neuro-fuzzy systems are considered to develop an
online multi-objective trajectory planning with reasonable real-time complexity.
7
2. Research objectives
2. 1. Research motivations
In today’s fast paced control and automation world, high performance in flexibility, agility, cost
reducing, quick and accurate planning are increasingly in demand. To be competitive, companies
utilize tools which quickly provide reliable and accurate information about an application without
the expensive and time consuming set up of a physical prototype. Robotics is an archetype
engineering field where such tools are a must [9]. Indeed, a great force of robots is their flexibility
and ability to rearrange for new tasks. Utilization of robot's flexibility presupposes effective
programming. The ability to program offline and simulate a robot trajectory prior to production
might help eliminating the potential for collisions between robots, parts, tools, and fixtures, and
saving inevitably large amounts of non-productive time if manual programming were used. In fact,
several demanding tasks in manufacturing such as assembly, pick and place, wilding and screwing
are repetitive and there exist room for enhancing productivity while decreasing production costs.
Henceforth, if it is possible to capture the essential components that govern Robot - Task -
Workspace interactions in a mathematical model, predictions regarding the outcome of the
experiment could be made using a computer, instead of a physical robot [9]. Moreover, this enables
the user to identify the influence of some single parameters upon overall performance, something
that can not be done with a real robot (because there are never two truly identical situations in the
real world!). Therefore, robot off-line programming may be encapsulated in the two basic levels
(Fig. 1). The modeling and approach level, and the simulation and testing level. In the first level
one develops robot kinematics and dynamics, including actuators models, task model, consisting in
starting, ending and intermediate poses and workspace model including obstacles, tools, humans,
or other robots to avoid or to cooperate with. A feasible approach could be satisfactory if required
performances are not too tight. Otherwise, one must get an optimal approach, and then test how is
it robust to changes of the dynamic parameters [10-13]. By including robot dynamics and forces
optimization, one can set TP as an optimal control problem and solve it using advanced non linear
programming techniques. For serial robots, this approach has been proved as highly effective.
Now, one would like to know how about its generalization to parallel kinematic structures.
Because of the differences between serial and parallel robots geometric, kineto-static and dynamic
characteristics [14-17], the nature of TP problems and their requirements are also different. Figures
2 and 3 show typical examples of industrial serial and parallel-robots. For serial manipulators, the
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objective is to move the EE from a starting to an ending pose such that the path remains within the
workspace and there is no collision with obstacles at any point. In the case of parallel manipulators,
in addition, the associated question of practical importance is avoiding singularities. Unlike with
serial robots, several types of singularity had been identified for parallel manipulators and each has
its own impact on their degeneracy conditions [18-26]. Moreover, it had been reported recently that
a full consideration of PKMs dynamics will result in reducing the cycle times and energy
consumption while enhancing process accuracy. In fact, PKMs can be advantageously applied to
industrial production processes, if their superior dynamic capabilities can fully be exploited [27,
28]. Developing efficient multi-criteria TP systems is a step ahead towards the achievement of
such objectives.
2. 1. 2. Research motivations in other fields
Trajectory planning has always been a major and dynamic research area in the wider robotics
community: Mobile and car-like robots, walking and humanoid robots, and serial and parallel
kinematic manipulators. The application areas are even larger: From traditional manufacturing, to
rescue and hazardous environments, to space and undersea robotics. Effective multi-objective
motion planning and optimization systems under several constraints will undoubtedly enhance
performance of these machines at both levels: offline and online planning. It goes without saying
that trajectory planning covers more general areas of applications such as motor cars, aircrafts,
ships, railroads,…etc.
2. 2. Scope of the Thesis
This thesis studies the motion planning for manipulator robots. The question we ask ourselves in
this research is the following: Is there a feasible framework to formalize the multi-objective TP
problem for manipulator robots, while optimizing time and energy consumption, resolving
redundancy, avoiding singularities and satisfying several constraints, related to actuator and
workspace limitations and task requirements? Then how is it possible to use the outcomes of such
an offline TP to build an intelligent system for real-time online planning with a reasonable
implementation complexity? Moreover, the following assumptions are made throughout the thesis:
The robot basis, moving platform, and struts along the legs are considered as rigid bodies.
In section 3, a literature survey on serial and parallel robot trajectory planning is given. Section 4
looks at various caveats in optimal TP, and develops the proposed offline planning scheme. Then,
9
three neuro-fuzzy approaches are introduced for multi-objective online TP. These approaches are
implemented on a case study of three degrees of freedom planar serial manipulator. Achieved
works and milestones for the ongoing work are given in section 5. Section 6 concludes this report.
An appendix is provided, containing a summary of contributions and copies of realized
publications.
2. 3. Original contributions
A variational calculus formulation of the trajectory planning problem for parallel kinematic
machines is proposed. The cost functional relates a weighted time-energy and singularity
avoidance criteria, along with several non linear and non convex equality and inequality constraints
on both state and control inputs. The sampling periods and actuator torques are taken as control
variables to be optimized during the trajectory planning procedure.
Implementing a discrete augmented Lagrangian with decoupling technique to solve the
resulting non linear and non convex constrained optimal control problem.
Design of two hierarchical data-driven neuro-fuzzy controllers for online near minimum-
time trajectory planning of redundant manipulators. One implements a feedforward neuro-inverse
kinematics, whereas the second uses a neuro-fuzzy inverse kinematic network. Both NF controllers
aim is to concisely represent the robot dynamic behavior based on an optimal offline trajectory
planner outcomes.
A third data-driven neuro-fuzzy controller is built for online multi-objective trajectory
planning. It is characterized by a desired starting and ending points and a trajectory in Cartesian
space as inputs to give the optimal time-energy singularity-free outputs in terms of sampling
periods and actuator torques. In this approach, the redundant inverse kinematics is included in
offline pre-processing steps, where the augmented lagrangian technique with decoupling and
projection is applied to generate as many as needed multi-objective trajectories covering the
workspace, on which to build the data-driven neuro-fuzzy network.
3. Literature survey
3. 1. Offline trajectory planning of robotic manipulators
Over the last two decades, an increasing number of researchers have been working on
computational methods to generate optimal control for general manipulator robots for both offline
and online programming. Robot motion planning had been essentially considered from two
10
Fig. 1. Overall Robot Off-line Programming Framework
Fig. 2. Example of a serial robot Fig. 3. Example of a parallel robot
Assembly robot, M16iB Series by Fanuc Robotics F200iB by Fanuc Robotics
different points of view. From computational kinematics and CAD standpoints, it consists merely
to assimilate the robot, workspace, and environment to that of a Windows application using a
dedicated CAD/CAM graphics-based interactive simulation system, with menus, toolbars and
icons and implements advanced 3D modeling, drawing, and simulation tools to obtain as accurate
positioning results as possible in 3D space. Examples of such software packages are CATIA-
Robotics, IGRID, RobotMaster, and ROBCAD [29-33]. From control systems standpoint, the
problem consists in finding the optimal torques sequence to achieve the displacement of the robot
from a starting to an ending poses, while optimizing a cost function.
Our project pertains to the second point of view. Within this class, there are two main categories
of methods for optimal TP. For both, the focus was mainly on time optimality [34-42]. This is due
to the obvious reasons of production increase. The first class decomposes the problem into two
steps. First, the optimal path is computed according to a parameterized form, such as B-Spline,
11
Bezier, cubic spline or polynomial. Then, a minimum time control problem which is independent
of the degrees of freedom (DOF) of the robot is obtained. This class includes works reported in
[34-42]. In [41, 42], it was shown that the unconstrained time-optimal trajectory was either bang-
bang or bang-singular-bang. For path-constrained time-optimal trajectory, it was shown in [35, 36]
that the optimal parameterized path was also bang-singular-bang, and in [10] it was proved that all
optimal paths must be saturated in at least one actuator at all times. The main drawback of these
approaches is the hard switching between minimal and maximal acceleration on the trajectory
causing inadmissible wear of drives and robot mechanics. The standard approach to deal with this
problem is to use a linear segment with parabolic blends – type trajectory with carefully selected
switching points, or to approximate the time-optimal motion with splines [12]. In the same
streamline, it has been shown, that an intuitive solution of this problem, eventually based on
Pontryaging maximum principle, can be found [36].
The second class of algorithms does not go through the intermediate geometric path computation,
but rather computes a collision free-path that is also optimal with respect to given performance
criteria. Works falling in this category are provided in [43, 44]. In [44], the authors propose to
include the drive dynamics in the trajectory generation using a cost functional, which weights cycle
time as well as drive energy in order to achieve a ‘‘softer’’ trajectory characteristics. Another
approach is to design smooth trajectories that approximate optimal motion ── yet do not
inherently require the high torque rates associated with the switching points ── by considering
constrained multi-objective criteria, based on torques and time optimal motion. This trajectory is
designed such that several inherent constraints associated to high robot-task-workspace
performance requirements are satisfied. This approach had been considered to solve a constrained
non linear and non convex optimal TP for serial robots [11], by applying the augmented
Lagranigan technique [45, 46].
Regarding closed loop parallel kinematic structures, the trajectory planning is quite different
from that of serial structures. This is due to major basic differences between the two types of
machines [14, 15]. Unlike with serial manipulators for whom the EE is connected to the ground
through a single chain, in parallel manipulators the EE is connected at more than one point with a
number of chains, each connecting it to the ground in parallel. Figure 4 shows a PKM example of
flight simulator. For these machines, the major complexity issue in the dynamic control and
trajectory planning is the strong non linear mapping between task space represented by the moving
12
EE poses and joint space represented by the actuated joints. The singularity-free constraint is
another or subsequent basic issue. The architecture-dependent performance associated with strong
coupled non linear dynamics makes harder the trajectory planning and control system design for
PKMs as compared to serial machines. The overwhelm criteria considered hitherto for PKM-TP
are essentially kinematics and design-based. These include singularity avoidance and dexterity
optimization [18-22]. The reason for this is the fact that conditions for which jacobian’s
determinant is zero or infinity are difficult to find analytically for a general PKM, since an
analytical expression for the determinant of the Jacobian matrix is not available. Several studies
had been dealt with the PKM-TP problem [18-22, 28, 47-48]. For instance, in [18], the authors had
developed a clustering scheme to isolate and avoid singularities and obstacles to parallel robots
path planning. In [19] the authors developed a scheme for avoiding singularities of a Gough-
Stewart platform by restructuring a pre-planned path in the vicinity of a singularity. In [20], an
algorithm is reported for planning a well-conditioned path between two end-points for Gough-
Stewart platform manipulators. In [21], it is shown that a singularity-free pose change is possible
for parallel manipulators. However, a starting point for reaching high performance is the inclusion
of PKM dynamics. A variational approach is reported in [22] for planning a singularity-free
minimum-energy path between two end-points for Gough-Stewart platforms. This approach is
based on a penalty optimization method. Penalty methods, however, have several drawbacks [49-
50]. Another basic issue recently identified for PKM-TP and practical use of PKMs in industry (in
a machining process, for example) is that for a prescribed tool path in the workspace, the control
system should guarantee the prescribed task completion within the workspace, for a given set up of
the platform (i.e. for which limitations on actuator lengths and physical dimensions are not
violated). This problem has been recently considered in [47 and 48], where design methodologies
involving workspace limitations and actuator forces optimization, using optimization techniques
had been reported. In this context, the first part of the present research constitutes an extension of
the mentioned literature.
3. 2. Online trajectory planning of robotic manipulators
For online trajectory planning of manipulator robots, the primary weakness of conventional
robot’s TP methods is the massive amount of computer time needed to obtain a solution. As robot
tasks are demanding in precision and diversity, these tasks execution might be difficult to achieve
because of strong non linearity and coupling of the dynamic parameters, such as Coriolis and
13
centrifugal wrenches, inertia parameters, and friction forces. Implementation of pure time-optimal
motion with torque limits might result in non satisfactory results. For instance, direct tracking of a
time-optimal trajectory leads to joint vibrations and overshoot of the nominal torque limits. For
serial robots, in [38] a secondary path velocity controller was used to modify online a nominal
minimum time velocity profile, which allows using the available torque range for path tracking. In
[39] the authors developed a complex computed torque-based controller to attempt to track the
constrained minimum time trajectory, while limiting actuator chatter. However, such rather open
loop controllers are not necessarily a practical choice for industrial robots, which typically utilize
closed loop controllers. To overcome these difficulties, several researchers have been using soft
computing techniques. For a start, genetic algorithms are increasingly applied to motion planning
and dynamic control of robotic systems [51- 53]. Neural networks and fuzzy control constitute
perhaps the most applied artificial intelligence fields in control systems of industrial manipulators
and mobile robots. Beside, these artificial intelligence techniques have added a new dimension to
many engineering fields to study, and have been effectively applied to everything from voice and
image recognition to toasters and automobile transmissions [54-57]. Neural networks are best
known for their learning and generalization potential. They are robust and fault tolerant. However,
because of their black box nature, neural networks lack interpretability and making use of
linguistic knowledge capabilities, which constitute the power of fuzzy logic. Fuzzy systems are
successful in using human skills and thinking processes within a machine [56]. They are easy to
understand and interpret because they are transparent. They have the ability to mimic human
thinking, by acquiring their knowledge from an expert who encodes his knowledge in a series of
IF/THEN rules. But the problem arises when a system has many inputs and outputs, obtaining a
rule base for such a large system is difficult [57]. In robotics literature, there is a large amount of
published works using neuro-fuzzy techniques. In these approaches, several objectives had been
considered. Path tracking, pose attainability, and minimum torque control were proposed [6-8]. In
this proposal, data-driven neuro-fuzzy techniques are used to build a multi-objective online robot
TP system, under actuator, workspace and task constraints. Such a multi-objective trajectory
planner was not considered before according to our best knowledge.
14
A. Constraints modeling:
Some of these constraints are defined in joint space, while others are defined in the task space, as
below:
Singularity avoidance: This is of primary importance for a successful PKM-TP system.
While PKM’s Jacobian allows motion transformation between joint and Cartesian spaces, it is also
the force transformation matrix from actuated joints to the moving EE. At poses where this matrix
is singular, i.e. when its determinant is either zero or infinity, the robot becomes uncontrollable.
This corresponds either to unreachable poses due to limitations of angular displacements, or to
uncontrollable displacements because all active links are locked [23-28]. A common kinematic
performance index related to singularity avoidance is the manipulability measure [23].
Accordingly, the following singularity avoidance function is defined as
))()(det(
1 )( 11
1kk
kxJxJ
xT
=ϖ (4)
Link intermediate length limits:
, withMaxkMin lll << N,...,,k 2 0 = , and (5) )( xl MaxMax Θ=
Torques limits: Non violation of control torques nominal values is another TP issue. That,
the leg forces demand at a given point on the trajectory must be continuously checked for possible
violation of preset limits. To avoid actuator saturation and singular poses, as soon as any actuator
crosses its limit, the optimal planning procedure has to determine an alternate actuation leading to
an alternate path on which the actuator forces would be constrained within the limits. The robot
torques is assumed to belong to a compact and bounded set C N6 ℜ⊂ , expressed as:
C (6) { }10 , :such that , maxmin6 −=<<ℜ∈= ,...,NkkNk ττττ
Sampling period limits: The sampling periods are defined such that the overall robot
traveling time is , where is the traveling time between two successive discrete poses k
and , . If the total traveling time for robot EE to move from a starting to an ending
point is too small, there may be no admissible solution to the optimal control problem, since the
torque constraints bound indirectly the path traversal time [12, 39]. Conversely, the sampling
period must be smaller than the system smallest time constant in order to prevent the system
∑−
==
1
0
N
kkhT kh
1 +k 1,...,0 −= Nk
kh
16
from being uncontrollable between two control times. In this research a tradeoff is made through
variation of sampling periods within an admissible domain H defined as:
H (7) { }maxmin :such that , hhhh kk <<ℜ∈= +
Task requirements: In a typical industrial robotic manipulator set-up, the user or task-planner
provides the trajectory-planner with a series of poses, in Cartesian or joint space, through which
the manipulator should travel. These constraints are quantified by a set of L pre-defined poses (pl,
Rl) through which the moving EE must pass (Fig. 5), where pl refers to the lth imposed Cartesian
position of the EE, and Rl to its orientation. This set includes initial and final poses. These
constraints represent equality constraints and are written as:
0 )(lPassTh1 =−−= lP
l Tx pps and 0 )( )(2 =−= lT
ll epsx RRs vect (8) Ll ,...,1 =
Where (p, ) describes the current computed pose of the EE, while vect(.) is the axial vector of its
3 3 matrix argument, measures the passage tolerance between p and p
R
×lPassThPT l , and measures
the absolute value of the angle of rotation between
leps
lRR and .
The above inequality constraints are written in the following simplified forms:
, , , (9) )( )( Min1 xlx Θg −= Max2 )( )( lxx −=Θg τττ ) ( Min3 −=g Max4 ) ( τττ −=g
For the sake of development simplicity, all inequality constraints noted
as regardless if they depend only on state, or on control variables or on both. 41 0, ) , ( ,...,jhj =≤τg x,
Hence, we turn up with J = 24 inequality constraints, 2L equality constraints (imposed
passages), and 12 equality constraints representing state dynamics equations.
B. Performance Index
Broadly speaking, it is possible to optimize any cost function that has a physical sense.
According to task and performance targets, the cost function is given a special definition. The
performance index considered in this thesis relates time and energy consumption, and a measure of
manipulability. For energy criterion, both electric and kinetic energies are considered.
17
Way-configuration Z z y x
X Y Passage tolerance
xs
xT
Fig. 5. Illustration of EE passage through imposed poses (positions and orientations)
_____ optimal path, ---- feasible path, xs Starting pose, , xT Target pose, For time criterion, the number of sampling periods is guessed from an initial feasible kinematic-
based solution. Then the sampling periods and the actuator torques are both taken as control
variables. In its continuous-time the performance index is defined as:
h τ
∫⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡ +++=
∈
Η∈
T TT
T
t
tdtttttt
ttt
E0
122
,0
))(( )()(21 1 )()(
)(
Min xQxxU ϖδιτττ C
(10)
4. 1. 2. Problem resolution
In its continuous-time version, the constrained optimal control problem can be stated as follows:
Among all admissible control sequences ( )ht ),(τ ∈C Η× , that allow the robot to move from an
initial state to a final state , find those which minimize the cost function : St xx =)( 0 Tft xx )( = E
(11) )(
Min
,0
EtttT
C∈
Η∈
τ
subject to constraints (3), (5), (6), (7), and (8)
The corresponding discrete-time problem consists of finding the optimal sequences
and , allowing the robot to move from an initial state to a final state ,
while minimizing the cost :
) ,..., ,( 21 Nτττ
),..., ,( 21 Nhhh Sxx 0 = TN xx =
dE
18
(12) ⎭⎬⎫
⎩⎨⎧ +++= ∑
=∈Η∈
N
kkk
Tkk
Tkkd hE
h 11221
)]( [Min
xQxxU ϖδιττ
Cτ
subject to : 10 ), , ,(1 ,..., N- k hf kkkdkk
==+ τxx
0 ) , ,( ≤kkkj hτxg 10 ,41 , ,..., N- k ,..,j ==
L, ..., ii 210 )( k == , xs ,..., N k 0 , = In the next section, the resulting constrained, nonlinear and non convex optimal trajectory planning
problem is solved using augmented lagrangian optimization technique.
4. 1. 3. Augmented Lagrangian with decoupling
A. Augmented Lagrangian
In the course of solving the constrained non-linear optimal control problem (10), the Augmented
Lagrangean (AL) function is used to transform this problem into a non-constrained one, where the
degree of penalty for violating the constraints is regulated by penalty parameters. This method
relies basically on quadratic penalty methods, but reduces the possibility of ill conditioning of sub-
problems that are generated with penalization by introducing explicit Lagrange multipliers
estimates at each step into the function to be minimized, which fastens convergence iterates.
Furthermore, while the ordinary Lagrangean is generally non convex (in the presence of non
convex constraints as for the case at hand), AL might be convexified to some extent with a
judicious choice of the penalty coefficients.
Thus, a rough sketch of the algorithm is: Given a feasible initial trajectory, control inputs, initial
Lagrange multipliers and penalty parameters, solve within an inner loop for the AL function
minimization with respect to control variables for a suitable tolerance. Depending on the outcomes,
the penalty coefficients and Lagrange multipliers are adjusted by testing the norms of all equality
and inequality constraints against the tolerances. If the constraints are satisfied within the given
penalty and tolerances rang, then the current value of penalty is doing a good job of maintaining
near-feasibility of iterates. The multipliers are updated without decreasing penalty. If the
constraints are violated with respect to tolerances, then we decrease penalty to ensure that the next
sub-problem will place more emphasis on reducing constraints violations. In both cases, tolerances
are decreased to force the subsequent primal iterates to be increasingly accurate solutions of the
primal problem. For more details on theoretical and algorithmic aspects for the augmented
Lagrangean derivation, the reader is referred to references [45, 46, 49, 50, 58 and 59].
19
The AL function transforming the constrained optimal control problem to an unconstrained one is:
= ) , , , , ,( σρλτμ hL x [ ] )( 1
1221 k
N
kkk
Tkk
Tk h∑
=+++ xQxxU ϖδιττ { }+−+∑
−
=++
1
011 ),,((
N
kkkkdk
Tk h
kτλ xfx
(13) +⎢⎣⎡ Ψ∑ ∑∑
−
= =
−
= ))( ,(
1
0
2
1
1
1
N
kk
li
ik
i
L
lk
Sh xsσμ +⎥
⎦
⎤Φ∑
= )) , ,( ,(
J
1kkkj
jk
jh
gτρμ xg ∑
=Ψ
2
1))( ,(
SiN
Li
iNNh xsσμ
Where is defined by the discrete state equation (3) at the sampling time k, is the
sampling total number, designates the adjoint (or co-state) obtained from the adjunct
equations, are Lagrange multipliers, associated to inequality and equality constraints,
are penalty coefficients. Penalty functions
),,( kkkd hk
τxf N
N 12 R∈λ
σ ρ,
, Sg μμ Ψμ andΦμ adopted here combine penalty and
dual methods. This allows relaxation of the inequality constraints as soon as they are satisfied.
These penalty functions are defined by:
bbaba TS
S)2 ( ) ,( μ+=Ψμ and { }22
) ,0(Max 21 ) ,( ababa −+=Φ g
ggμ
μμ (14)
The Karush-Kuhn-Tucker first order optimality necessary conditions state that for a
trajectory , to be optimal solution to the problem, there must exists some positive
Lagrange multipliers , unrestricted sign multipliers , and finite positive penalty
coefficients such that for :
kkk h ,,τx ,...,Nk 0 =
) ,( kk ρλ kσ
) ,( Sg μμ ,...,Nk 0 =
0 =∂∂
k
Lxμ , 0 =
∂∂
k
Lτμ , 0 =
∂∂
khLμ , 0 =
∂∂
k
Lλμ , 0 =
∂∂
k
Lρμ , 0 =
∂∂
k
Lσμ , 0 ) ( =hgT
k ,x,τρ , , and 0 )( =xsTkσ
0 ) ( ≤h,g τx, (15)
The development of these conditions enables us to derive the iterative formulas to solve the
optimal control problem by adjusting control variables, Lagrange multipliers as well as penalty
coefficients. With these conditions satisfied, the search line for the optimal solution will be
directed towards the feasible (constraints satisfied), and usable (objective function minimized)
domain of . However, in equation (3), contains the inverse of the total inertia
matrix of the EE, struts and actuators as well as their Coriolis and centrifugal
wrenches . These might take several pages long to display for a PKM plant. In
) ,( ** hτ
) ,( hτ ) , ,( kkkd hk
τxf
)(1
xM−−c
) ,( 21 xxN c−
20
developing the first necessary optimality conditions and computing the co-states , one has to
determine the inverse of the mentioned inertia matrix and its derivatives with respect to state
variables. This results in an intractable complexity even by using symbolic calculation.
kλ
B. Augmented Lagrangian with Decoupling:
The major computational difficulty mentioned beforehand can not be solved by performing with
the original non linear coupled formulation. Instead, it is tackled using a linear-decoupled
formulation similar to that used in decoupled non linear control systems [60].
Theorem:
Under the invertibility condition of the inertia matrix, the control law defined in the Cartesian
space as
(16) )( ) ,( )( 12211 xGxxxNvxMu ccc−−−
++=
allows the robot to have a linear and decoupled behavior with a dynamic equation:
(17) vx 2 =&
Where v is an auxiliary input
Proof:
It follows simply by substituting the proposed control law (16) into the dynamic model (2).
One gets
vxMxxM )( )( 121 cc−−
=&
Since is invertible, it follows that )(xM c−
vx 2 =&
This brings the robot to have the decoupled and linear behavior described by the following linear
dynamic equation written in discrete form as:
(18) ) , , ( )( 1 kkkD
dkkkdkk hhk
vxfvBxFx =+=+
21
with ) , , ( kkkD
d hk
vxf k
k
kk
k
h
hh vIIxIO
II 2
66
662
6666
6666
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎥⎦
⎤⎢⎣⎡=
×
×
××
××
Notice that this formulation reduces drastically the computations, by alleviating us the
calculation at each iteration of the inertia matrix inverse and its derivatives with respect to state
variables, which in turn results in ease calculation of the co-states. The non-linearity is however
transferred to the objective function.
The decoupled formulation transforms the discrete optimal time-energy control problem to
finding the optimal sequences of sampling periods and acceleration inputs ,
, allowing the robot to move from an initial state to a final state , while
minimizing the cost function , expressed for the decoupled problem as:
) ,..., ,( 21 Nhhh
)),..., ,(( 21 Nvvv Sxx = 0 TN xx =
dE
⎪⎩
⎪⎨⎧
⎢⎢⎣
⎡
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ ++⎥⎦
⎤++= ∑−
=
−−−−−−
∈∈
1
01221112211 )( ) ,( )()( ) ,( )( Min
N
kkckkkckkc
T
kckkkckkcv
Dd
kh
E xGxxxNvxMUxGxxxNvxMΗV
(19) )( 1221
⎪⎭
⎪⎬⎫
⎥⎥
⎦
⎤+++ kkk
T
k hS xQxx ϖι δ
subject to : 10 ), , ,(1 ,..., N- k hf kkkD
dkk
==+ νxx
and satisfying the above mentioned constraints, which mainly remain the same, except for actuator
torques’, which become:
(20) MaxkckkkckkcMin ττ 12211 )() ,( )( <++<−−−
xGxxxNvxM
Henceforth, inequality constraints and can be re-written as: 3g 4g
(21) ⎥⎦
⎤⎢⎣
⎡ ++=−−−
)( ) ,( )( - ) ,( 12211Min3 kckkkckkckkD xGxxxNvxMvxg τ
(22) Max122114 )( ) ,( )( ) ,( τ−⎥⎦
⎤⎢⎣
⎡ ++=−−−
kckkkckkckkD xGxxxNvxMvxg
The augmented Lagrangian associated to the decoupled formulation (after removing bars and c
indexes for writing simplicity) is:
22
= ) , , , , ,(D σρμ λvx hL [ ][[{∑−=
++1
012211 )( ) ,( )(
N
k
Tkkkkkk UxGxxxNvxM [ ]])() ,( )( 12211 kkkkkk xGxxxNvxM ++
] }++++ )( 1221 kkkTk hxQxx ϖ δι { }∑
−
=++ −
1
011 ) , ,( (
N
kkkk
Ddk
Tk h
kvxfxλ
(23)
where the function is defined by the state equation (16) at time k, is the total
sampling number, other parameters appearing in (19) are defined above.
+⎢⎣⎡ Ψ+∑ ∑∑
−
= =
−
= ))( ,(
1
0
2
1
1
1
N
kk
Dli
ik
i
L
lkh xs
Sσμ ∑
=+⎥
⎦
⎤Φ
J
1 )) ,( ,(
jkk
Dj
jk vxg
gρμ ∑
=Ψ
2
1))( ,(
iN
DLi
iNNh xs
Sσμ
),,( kkkD
d hk
vxf N
Again, the development of the first order Karush-Kuhn-Tucker optimality necessary conditions
require that for , to be solution to the problem (19), there must exist some
positive Lagrange multipliers , unrestricted sign multipliers , and finite positive penalty
coefficients such that conditions (15) hold for the decoupled problem. The co-states kλ
and control inputs are then determined from the necessary optimality conditions, and
because of the strong non linearity of the system at hand, a numeric solution is adopted to solve
this optimal control problem through implementation of the augmented lagrangian. More
development details are provided in [17].
kkk h , , vx ,...,Nk 0 =
) ,( kk ρλ kσ
) ,( sg μμμ=
kk h ,v
4. 1. 4. Implementation on a planar parallel manipulator:
The proposed approach is implemented on a 2 degrees of freedom parallel manipulator reported
in [61] and shown in Fig. 6. The program was coded in Matlab [62]. The following numeric values
are used: The EE mass is =200.0 kg , that of each leg is = 570.5 , and that of the slider
is =100 . The platform radius is
pm lm kg
sm kg mr 750.0= , mR 2030.1 = and the strut length . Table. 1
shows the limits of the workspace, actuator torques and sampling periods. For the augmented
Lagrangian, the following parameter values had been
taken , , , , , , , ,
mL 9725.1 =
0.5=sω 0.5=sη 4.0== ηααw 0.25 1=γ =0w 2100 10 −==ηη 1.2 2=γ =*w 5*
1* 10 −==ηη 4.0== ηββw 01.0=υ ,
. The initial Lagrange multipliers components are set to zero. The singularity weight
is
3.0=−υ 00 , σρ
1 =δ . The maximum value for , the minimum value for . The scenario
consists of a straight line trajectory from an initial Cartesian state position , to a
52Max 01 =δ -52
Min 01 =δ
7.0 0 −=x 1.0 0 −=y
23
final position , (in meters). The initial and final linear and angular velocities are
equal to zero. The maximum velocity is 0.2m/sec and maximum acceleration of 2 m/sec
7.0 =Tx 6.1 −=Ty
2. The
maximum travelling time for this trajectory is 10 sec. In the presented simulations, the focus is on
time-energy constrained trajectory planning by the augmented Lagrangian, more kinematic related
performance evaluation and design for a similar case study might be found in [37, 39]. Typically,
four simulation objectives are considered: 1) Compare robot trajectories for different values of the
weights ; 2) Asses the effects of the dynamic parameters changes on the augmented
Lagrangian sensitivity and on the behavior of the PKM; 3) At which precision versus time
consumption, the augmented Lagrangian achieves passage satisfaction through imposed poses; 4)
To what extent the number of inner and outer iterations of AL impacts PKM performance versus
CPU Time. Figures 7 and 8 show the simulation results for both initial kinematic and augmented
Lagrangian solutions. In part (a) of these figures, the first plots from the top show the displacement
along x-axis of the platform point of operation. The second plots show the displacement along y-
axis of the platform point of operation. The third shows the instantaneous values of consumed time
to achieve the trajectory. In part (b), the first and second plots from the top show instantaneous
variations of joint torques. While the third ones, they show the instantaneous values of the
consumed energy. One notes that although the initial solution is kinematically feasible, when the
corresponding torques is computed considering the dynamic model and forces, one gets shortly
torque values outside the admissible domain resulting in high values for the energy cost. With the
augmented Lagrangian, however, the variations of the energy consumption are increasing smoothly
and monotonously. More details on this implementation and simulation results are provided in [17].
, 1 and, ιQU
Table 1. Limits of the workspace, actuator torques and sampling periods.
Parameter x-coordinate (m) y-coordinate (m) (Nm) (Nm) h (sec) 1τ 2τ
max min
0.8 -0.8
-0.720 -1.720
550 -550
700 -700
0.7 0.01
4. 2. Online trajectory planning, a hierarchical data-driven neuro-fuzzy system
4. 2. 1. Introduction
The second part of this thesis is the multi-objective online TP for manipulator robots.
Implementation of pure conventional time-optimal motion planning with torque limits is
problematic. First, a direct tracking of a time-optimal trajectory leads to joint vibrations and
24
Fig. 6. A schematic representation of the planar parallel manipulator overshoot of nominal torque limits. Second, such motion controllers need a large computational
time making them simply out of use for real-time applications. Furthermore, according to
performance targets, the trajectory planner must balance between path speed and accuracy
demands of the user, as well as kinematic limitations of the robot, and time and energy
consumption. Finally, because of their open loop structures, these controllers are not necessarily a
practical choice for industrial robots, which typically utilize closed loop controllers.
This problem is handled in this thesis using data-driven neuro-fuzzy systems [63]. There exist
some published works on data-driven neuro-fuzzy networks for mobile robot motion planning [64].
In the proposed work, instead of gathering experimental information from a robot, our approach is
based on simulated data. The use of simulated data provides reliable training and comparison for
neuro-fuzzy networks. As a case study we chose to implement it on a redundant manipulator as it is
more challenging compared to non redundant robots. The basic idea of the proposed approach is
that from a purely but realistically-model-based offline TP outcomes, and once the neuro-fuzzy
structure (rules number and premise and consequence membership function parameters) is learnt
and identified, the neuro-fuzzy controller is used in a generalization phase to achieve online motion
planning with reasonable real-time complexity.
To this end, three approaches are considered. The first system is built upon three levels. At the
first level, a feedforward neural network named NetIK (for Neuro Inverse Kinematic) is used to
solve the inverse kinematics problem (IKP) of the redundant serial manipulators. The second level
is a set of pre-processing steps, where a minimum time trajectory parameterization satisfying
starting and ending targets attainability with zero joint rates, and boundary conditions on
25
Fig. 7. kinematic simulation results
Fig. 8. Simulation results with the augmented Lagrangian (Minimum Time-Energy)
accelerations and jerks is performed. The third level is a neuro-fuzzy system, named NeFOTC
(for Neuro-Fuzzy Optimal Time Controller). It is built upon training dataset generated from the
parameterized minimum time trajectory and robot Inverse Dynamics (ID) including friction and
actuator models, allowing the robot EE to move from a starting point S to a final point T. The
second approach is similar to the first one, except the feedforward neural network is replaced by a
neuro-fuzzy system. In the third approach, the inverse kinematic is included in the pre-processing
steps. It is solved through a variation calculus framework, applying the augmented lagrangian
methodology developed in the first part of the thesis. From the outcomes of such a purely but
realistically-model-based offline planned trajectory, the data-driven neuro-fuzzy system parameters
are learnt and optimised. Once the latter is identified by rules number and premise and
consequence membership function parameters, the trained neuro-fuzzy controller ─ which
concisely represent the dynamic multi-objective motion ─ of the robot is used to achieve online
programming with reasonable real-time complexity. A comparison is performed between the three
neuro-fuzzy approaches on a three degrees of freedom redundant serial planar manipulator.
26
4. 2. 2. First approach, A hierarchical neuro-fuzzy near optimal time planning
This system contains three levels briefly introduced below. More details are provided in [67].
A. Neuro Inverse Kinematics
In the first level, a feedforward neural network named NetIK (for Neuro Inverse Kinematic) is
used to solve the inverse kinematics problem (IKP). IKP function consists in transforming
Cartesian coordinates (given in the task space) into the corresponding joint coordinates (given in
the actuation joint space) of the manipulator. For a planar three DOF manipulator (Fig. 9), NetIK
gets the Cartesian coordinates of the robot EE defined within the task space as inputs to give
the corresponding joint variables as outputs. The forward kinematics (FK) equations act
as a supervisor. NetIK is a multi-layer feedforward neural network. It implements the Levenberg-
Marqurad version of Backpropagation algorithm to learn robot inverse kinematics.
) ,( yx
) , ,( 321 qqq
B. Pre-Processing
The second level is a set of pre-processing steps, where a minimum time trajectory
parameterization satisfying starting and ending targets attainability with zero joint rates, and
boundary conditions on accelerations and jerks is performed. The optimization is performed under
actuator dynamics constraints and workspace limitations. The corresponding torques are computed
using the inverse dynamic model, which in turn are projected into the torques admissible domain.
C. Hierarchical Neuro and Neuro-Fuzzy Near Optimal Time Trajectory Planning
A neuro-fuzzy controller, named NeFOTC (for Neuro-Fuzzy Optimal Time Controller) is built
upon training dataset generated from the parameterized minimum time trajectory and robot Inverse
Dynamics (ID) including friction and actuator models, allowing the robot EE to move from a
starting point S to a final point T (Fig. 12). After the training dataset constituted of as needed of
minimum time trajectories, built from the previous pre-processing steps, basic issues here are how
to partition the obtained dataset input/output entries for a better initialization of membership
function parameters at both premise and consequent parts, and what is the number of rules that
better fits the optimal dynamic planned trajectory. A subtractive clustering [65, 66] of input-output
dataset is first performed to better tailor the neuro-fuzzy system to the robot behaviour. Then a
Tsukamoto neuro-fuzzy inference is used to train an adaptive neuro-fuzzy network. The premise
27
parameters (antecedent membership functions parameters) and rule-consequences parameters are
learnt using an optimization procedure on the error function. The definition of fuzzy membership
functions used for fuzzification and defuzzification of input and output variables plays a significant
role in the ability of the neuro-fuzzy controller to learn and generalize. Figure 13 displays an
overall architecture and function of the proposed hierarchical neuro-fuzzy optimal time trajectory
planning system. Figure 14 displays the topology and architecture of NeFOTC. For both networks
(NefIK and NeFOTC), the Levenberg-Marquard version of the gradient back-propagation
algorithm is used to learn premise and consequent parameters of the fuzzy rules. This approach had
been implemented on a case study of redundant planar serial robot. Figure 15 depicts NeFOTC
learning performance (Root mean square error vs. Epochs). More implementation details and
simulation results are provided in [67, 68].
4. 2. 3. Second approach, Neuro-Fuzzy near optimal time planning
As mentioned above, the second approach is similar to the first one, except the feedforward neural
network is replaced by a neuro-fuzzy system. This approach allows substantial improvements in
the inverse kinematic solution as compared to the previous approach. This results in better solution
for the overall minimum time planning.
4. 2. 4. Third approach, Neuro-Fuzzy multi-objective planning:
In this approach, the system is built upon two levels. In the first level, an offline planning is first
formulated in a variation calculus framework considering singularity avoidance through kinematic
redundancy resolution, time and joint torques minimization and limits on joints, joint rates, and
joint torques. This allows building a multi-objective singularity-free offline TP, satisfying
workspace limitations, task requirements, and several constraints such as torques, position,
velocity, and acceleration boundary conditions. From the outcomes of this purely but realistically-
model-based offline TP, a Tsukamoto Neuro-Fuzzy inference network named NeFuMOP, (for
Neuro-Fuzzy Multi-Objective Planning) is designed to achieve the online motion planning of the
28
Fig. 9. Geometry of a 3 dof planar manipulator of joint angles 321 , ,,jq j =
Fig. 10 The use of the forward kinematic (FK) to the learning Fig. 11.Performance of NeFIK: Root mean square
Fig. 12. The use of the inverse dynamics (ID) to the learning of the
robot, in the same way as it was done at the third level for the first and second approaches
NetNetNet qqq 321 , ,
Error
NeFIKP
yx,FK
321 , , qqq
Of the Fuzzy Neuro- Inverse Kinematic (NeFIK) module error with respect to learning epochs
Neuro-Fuzzy Near Optimal Time Controller (NeFOTC) module
developed in the previous sub-sections. Once such a neuro-fuzzy structure (rules number and
premise and consequence membership function parameters) is identified, it is used to achieve
NeFOTCError
Adjustment
Desired inputs Computed outputs Target outputs kooo ) , ,( 321kqq ) ,(
. kddd ) , ,( 321 τττ
29
online planning. The purpose of this Neuro-Fuzzy network is to learn and fit the robot dynamic
behavior corresponding to the determined multi-objective planned trajectory. A comparison is
performed between both NFN approaches on a three degrees-of-freedom serial planar manipulator
under workspace limitations and task requirements. This approach is now being performed in an
ongoing work.
5. Progress to date and milestones:
As to date, basic achievements had been done for the first part of this research project. The
formulation, design and implementation of the augmented lagrangian algorithm with decoupling
have been performed for PKM offline trajectory planning. A new design for a data-driven neuro –
fuzzy network was realized based for a redundant robot multi-objective trajectory planning. This is
performed with satisfaction of velocity, acceleration, and jerk limitations. Two approaches had
been considered. The first one is a three-level approach, which consists in a neuro-fuzzy inverse
kinematic, followed by pre-processing steps, on which to build a second neuro-fuzzy network to
achieve online near minimum time planning. The second approach is a two-level system that
solves the kinematic redundancy within a first offline planning level.
The second level is a Tsukamoto neuro-fuzzy inference system to perform online trajectory
planning as done for the system of the previous section. This second approach is now being
implemented. The flowchart in Table 2 shows the milestones and progress of the thesis project.
6. Conclusion
A formulation and resolution approach has been considered to offline and online trajectory
planning problems of robot manipulators. The offline planning uses a variational calculus
framework, and is implemented on parallel kinematic machines, under robot kinematics, dynamics
and intrinsic and extrinsic constraints. While optimizing time and energy necessary to achieve the
trajectory, it allows satisfaction of several constraints related to robot, task and workspace
interactions. The robot dynamic model includes the moving platform, struts and actuators models.
The augmented Lagrangian algorithm with a decoupled and linearized formulation is used to solve
for the resulting non linear and non convex optimal control problem. This allows a computational
ease of the co-states and other variables necessary to perform the optimization process. Although it
30
Figure 13. Overall architecture and function of the first neuro-fuzzy near
optimal time trajectory planning system
Fig. 14. NeFOTC Topology and Architecture
31
Figure 15. Overall architecture and function of the neuro-fuzzy muli-objective planning system
is task and algorithm parameter settings dependent, the computational time is drastically reduced
when we use the decoupled and linearized formulation is used. A major advantage of this approach
is that one can introduce several criteria and constraints to satisfy, which are related to robot, task
or environment, like singularity, link interference and obstacles avoidance, by deriving the
corresponding constraint expressions and adding them naturally in the Lagrangian in order to have
them included in the trajectory planning system. The online trajectory planning system is a
hierarchical neuro-fuzzy system built from the outcomes of a prior offline planning. First
implementation results of this intelligent trajectory planning on a case study of a redundant serial
planar manipulator is encouraging. As recommendation for a possible extension of this work, it
consists to incorporate obstacle and link interference avoidance strategies and other specific task
requirements at both levels - offline and online - trajectory planning processes.
7. Acknowledgment
The author would like to gratefully thank the Natural Science and Engineering Council of Canada
(NSERC) for supporting this work under PhD scholarship grant ES D3-317622 and Strategic
Project grant STPGP-269579.
32
1st Year 2nd Year 3rd Year
Activity Status F04 W05 S05 F05 W06 S06 F06 W07 S07 Literature Survey √
- Robots offline & online trajectory planning √ - Existing approaches for serial & parallel robots √ - Advantages and drawbacks of existing approaches √ -Optimization techniques & Augmented Lagrangian √
1
- Neuro-Fuzzy techniques to online planning √ Offlline Trajectory Planning √
- Formulation of PKM offline trajectory planning
√
- Implementation and simulation on a case study √ - Writing & publication of 2 conference papers √
2
- Writing & submission of a journal paper √ Online Trajectory Planning
- Neuro and neuro-fuzzy minimum time planning √ - Writing & publication of a conference paper √ - Hierarchical neuro-fuzzy minimum time planning √ - Writing & submission of a journal paper √ - Neuro-fuzzy multi-objective planning ⊥
3
- Writing & submission of a journal paper ⊥ Dissertation Writing & Submission ! Dissertation Defence !
Table 2. Flowchart diagram describing milestones and progress of the thesis project (√) Work done, (⊥) Being performed, (!) To be done
8. References [1] J.C. Latombe, “Robot Motion Planning”, Kluwer Academic Publishers, Norwell, Mass. 1991.
[2] J.H. Reif, “Complexity of the Mover's Problem and Generalizations”, Proc. of the 20th IEEE Symposium on
Foundations of Computer Science, IEEE, New York, pp:421-427, 1979
[3] J. Canny, “The Complexity of Robot Motion Planning”, MIT Press, Cambridge, Mass. 1988.
[4] C. Ahrikencheikh and A. Seireg, “Optimized-Motion Planning”, John Wiley &Sons, 1994.
[5] J. K. Son, H. Y. Sik, P. C. Kug, “Neuro-Fuzzy Control of a Robot Manipulator for a Trajectory Design”, IEEE Int’l
Workshop on Robot and Human Communication, 1997.
[6] H. Shao, K. Nonami, T. Wojtara, R. Yuasa, S. Amano, D. Waterman, “Neuro-Fuzzy Position Control of Demining
Tele-Operation System Based on RNN Modeling”, Rob. and Computer Integrated Manuf., 22, pp: 25-32, 2006.
[7] Y. L. Sun, M. J. Er, “Hybrid Fuzzy Control of Robotics Systems”, IEEE Trans. on Fuzzy Sys. vol. 12, No. 6, p 755-
765, 2004.
[8] W. S. Lin, C. H. Tsai, and C. H. Wang, “Robust Neuro-Fuzzy Model-Following Control of Robot Manipulators”,
Proceedings of the IEEE Int’l Conference on Control Applications, Trieste, Italy, September, 1998.
[9] U. Nehmzow, “Mobile Robotics: A Practical Introduction”, Springer-Verlag, London, 2000.
[10] Y. Chen and A. A. Desrochers, “Structure of Minimum-Time Control Law for Robotic Manipulators with
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36
Appendix: Summary of the contributions
In this appendix, we enclose copies of publications achieved hitherto. These consist mainly in
three journal papers (two submitted, and one in preparation), and four conference papers (three
published and one submitted).
The first journal paper relates the proposed off-line trajectory planning using augmented
Lagrangian with decoupling technique, and is submitted for publication to Robotics and Computer
Integrated Manufacturing. Two conference communications as parts of this paper had been
published. One at CCTOMM 2006, held at Canadian Space Agency, Saint-Hubert QC, Canada,
May, 26-27 2005, the second at ROMANSY 2006, 16th CISM-IFFTOMM Symposium on Robot
Design, Dynamics and Control held at Warsaw, Poland, June, 20-24 2006. A third conference
paper is also achieved within the same topic.
The second journal paper describes the hierarchical neuro-fuzzy system developed to online
trajectory planning purpose. This paper is submitted for publication to Engineering Applications of
Artificial Intelligence. This has been extended to prepare a third journal paper. A conference
communication as part of this work had been published at NAFIPS’6 (North American Fuzzy
Information Processing Society), held at Concordia University, Montreal, QC, Canada, June 3-6,
2006. In summary, here is the list of publications realized as to date.
List of publications:
Book Chapter
[1] A. Khoukhi, L. Baron, M. Balazinski:“A Decoupled Approach to Time-Energy Trajectory
Planning of Parallel Kinematic Machines”, CISM Courses and Lectures No. 487, Proc. of 16th CISM-
IFFToMM Symposium, Robot Design, Dynamics and Control, T. Zielinska, C. Zielinski Eds, p 179-
186, Springer Wien N.Y, 2006.
Refereed Journal Articles
[2] A. Khoukhi, L. Baron, M. Balazinski: “Constrained Multi-Objective Trajectory Planning of
Parallel Kinematic Machines”, Submitted to Robotics and Computer Integrated Manufacturing, 2006
37
[3] A. Khoukhi, K. Demirli, L. Baron, M. Balazinski: “Hierarchical Neuro-Fuzzy Near Optimal
Time Trajectory Planning for Redundant Manipulators” , Submitted to Engineering Applications of
Artificial Intelligence, 2006
[4] A. Khoukhi, K. Demirli, L. Baron, M. Balazinski: “Data-Driven Neuro-Fuzzy Multi-Objective
Trajectory Planning of Manipulator Robots”, in preparation (2006)
Refereed Conference Publications
[5] A. Khoukhi, L. Baron, M. Balazinski: “Programmation Dynamique Time-Énergie Minimum des
Robots Parallèles par Lagrangien Augmenté”, CCToMM, Symposium on Mechanics, Machines and
Mechatronics Canadian Space Agency (CSA) , Saint Hubert, Canada, pp:147-148 , May 26-27, 2005
[6] A. Khoukhi, K. Demirli, L. Baron, M. Balazinski: “Neuro-Fuzzy Optimal Time Energy
Trajectory Planning of Manipulator Robots”, NAFIPS’6 (North American Fuzzy Information
Processing Society) Conference, Concordia University, Montreal, Canada, June 3-6, 2006.
[7] A. Khoukhi, L. Baron, M. Balazinski: “Multi-Objective Trajectory Planning for Redundant
Manipulators ”, Submitted to Fifth International Conference on Industrial Automation, Montréal,
CANADA, June 11-13, 2007
Other Contributions
[8] A. Khoukhi “Hierarchical Neuro-Fuzzy Optimal Time Energy Trajectory Planning for
Redundant Manipulators”, Seminar at CAE R. Frazer Elliott Laboratory, Mechanical Eng. Dept,
University of Montreal, April 27, 2006
[9] A. Khoukhi, M. Balazinski, L. Baron: Hybrid Fuzzy Systems for Traffic Light Control,
Presentation at AXOR Inc. October 4 2005, Research Project: Strategic-NSERC STPGP-26957
[10] A. Khoukhi, L. Baron, M. Balazinski: “Multi-Objective Off-Line Programming of Parallel
Kinematic Machines”, Technical Report, Mechanical Eng. Dept, Ecole Polytechnique of Montreal,
in preparation (2006).
38
