Modeling, Simulation and Control of Redundantly Actuated Parallel Manipulators

202 Pages·2007·9.02 MB·English

Universit´e catholique de Louvain
Facult´e des Sciences Appliqu´ees
Modeling, Simulation and Control of
Redundantly Actuated Parallel
Manipulators
Latchezar GANOVSKI
Mechanical Engineer
Ph.D. Thesis
Members of the Jury:
Prof. Paul Fisette (promotor) Universit´e catholique de Louvain
Prof. Jean-Claude Samin (promotor) Universit´e catholique de Louvain
Prof. Joris De Schutter Katholieke Universiteit Leuven
Prof. Pierre Duysinx Universit´e de Li`ege
Prof. Pierre-Yves Willems Universit´e catholique de Louvain
Prof. Gr´egoire Winckelmans Universit´e catholique de Louvain
Louvain-la-Neuve, Belgium
December 2007
I would like to express my deepest gratitude to:
Paul Fisette, for his friendship, for being of constant help and support to me,
for guiding perfectly my work on this thesis from the very beginning – with his
precious ideas, advice and remarks, his words of encouragement and motivation,
when I needed them the most,
Jean-Claude Samin, for so many fruitful discussions and enlightening remarks, as
well as for his valuable supervision of this work,
Pierre-Yves Willems, for being such a magnificent mentor to me, in the scientific
and every other sense of this term,
Paul Thibaut, another precious friend and colleague, with whom we spent many
hours together, tackling mechatronic design problems,
Thierry Daras, for his kind assistance with respect to some hardware implemen-
tation aspects of this thesis,
GuyCampion, JorisDeSchutterandBramDemeulenaere, fortheirhelpfuladvice
on some robot control issues,
All my past and present colleagues at the PRM Division of the UCL, where I
have been working for almost nine years, in a wonderful atmosphere of team spirit,
union and friendship,
Maxime Raison, Vincent Spinewine and Franc¸ois Nicolas, for being such good
office mates,
Paul Fisette and Jean-Claude Samin, once more, as well as Benoˆıt Raucent – for
landing me a helping hand in the edition of the text,
All my friends in Belgium and Bulgaria,
All of you, that I have probably forgot to thank here,
And, finally, my wife Evgenia and my stepdaughter Lora, for their love and sup-
port, for being my courage and my guiding light.
Thank you all from the bottom of my heart!
Latchezar, December 2007
Felix qui potuit rerum cognoscere causas.

i
List of symbols
Subscripts
n : corresponds to the number of degrees of freedom of the
multibody system;
k : correspondstothenumberofjointsofthemultibodysys-
tem;
u : stands for the independent coordinates of the multibody
system;
v : stands for the dependent coordinates of the multibody
system;
a : stands for the actuated (active) coordinates of the multi-
body system;
p : stands for the non-actuated (passive) coordinates of the
multibody system;
a : stands for the actuated (active) coordinates, correspond-
d
ing in number to the degrees of freedom of the multibody
system;
a : stands for the actuated (active) coordinates, correspond-
r
ing in number to the redundant actuators of the multi-
body system;
c : stands for the kinematic constraint equations;
m : stands for the parallel manipulator Jacobian;
Superscripts
n : stands for ”power n”;
d : stands for desired (reference) values of the controlled
variables;
Operators
a˙ : first time derivative of a;
a¨ : second time derivative of a;
a˜ : tilde matrix associated with a (3 1) array a;
×
a˜ : tilde tensor associated with a vector a;
Main symbols
X : set of end-effector (absolute) generalized coordinates of
the manipulator;
q : set of joint (relative) generalized coordinates of the ma-
nipulator;
J : Jacobian matrix;
u : sub-setofindependentgeneralizedcoordinatesofthepar-
allel manipulator;
ii
v : sub-set of dependent generalized coordinates of the par-
allel manipulator;
a : sub-setofactuated(active)generalizedcoordinatesofthe
multibody system;
p : sub-set of non-actuated (passive) generalized coordinates
of the multibody system;
W : end-effector ”wrench” - resultant vector of the external
torques and forces, applied on the end-effector;
τ : vectoroftorques/forcesappliedtothemanipulatorjoints;
n : Euclidean space of a dimension n;
ℜ
ǫn : ellipsoid, defined in the n-dimensional Euclidean space
n;
ℜ
ς : sphere of radius equal to 1 (unit sphere);
u
Ψ : surface (set of points), defined in the n-dimensional Eu-
clidean space n;
ℜ
U : left unitary matrix of the singular value decomposition
J = U Σ VT;
Σ : diagonalmatrix(withnonnegativeelementsindecreasing
order) of the singular value decomposition J = U Σ VT;
V : right unitary matrix of the singular value decomposition
J = U Σ VT;
λ : vector of Lagrange multipliers;
N : number of independent actuators (actuated joints);
act
d (d ) : number of degrees of actuation (force) redundancy;
AR FR
d : number of end-effector degrees of freedom that the par-
eff
allel manipulator;
d : number of degrees of freedom of the multibody system;
mbs
Nbody : number of bodies of the multibody system;
Njoint : number of joints of the multibody system;
O, ˆI : orthonormal inertial frame of origin O in the three-
{ { }}
dimensional Euclidian space, rigidly fixed to the base
(body 0) of the multibody system;
Xˆi : orthonormalreferenceframeinthethree-dimensionalEu-
{ }
clidian space, attached to body i;
Ri : (3 3) absolute rotation matrix of Xˆi with respect to
× { }
ˆI ;
{ }
Ri,h : (3 3) relative rotation matrix between frames Xˆi and
× { }
Xˆh ;
{ }
Ωi : relative angular velocity vector of body i with respect to
its parent h;
ωi : absolute angular velocity vector of body i;
iii
Oi, O′i : reference attachment points of joint i on the parent body
h and its child i, respectively;
zi : relative position vector −O−i−O→′i;
dik : positionvectorofthereferenceattachmentpointOk with
respect to O′i;
dik : extended position vector of Ok with respect to O′i;
z
Gi : center of mass of body i;
dii : position vector of the center of mass Gi with respect to
O′i;
dii : augmented position vector of Gi with respect to Oi;
z
xi : absolute position vector of the center of mass Gi;
mi : mass of body i;
Ii : inertia matrixof bodyi with respect to its center of mass
Gi;
Ii : inertia tensor of body i with respect to its center of mass
Gi;
g : gravity vector;
Fi : vector of the forces applied to body i by its parent h
through joint i;
Fi : resultantvectoroftheexternalforces(excludingthegrav-
ext
ity), applied to body i;
Fi : resultant vector of all the forces applied to body i;
tot
Li : vector of the torques applied to body i by its parent h
through joint i;
Li : resultant vector of the external torques, applied to body
ext
i;
Li : resultant vector of all the torques applied to body i;
tot
Qi : generalizedforceassociatedwiththegeneralizedjointco-
ordinate qi;
S : diagonal selection matrix of a hybrid controller;
d
S : diagonal selection matrix of a redundant hybrid
r
controller;
Abbreviations
MBS : multibody system;
d.o.f. : degrees of freedom;
RPPM : reconfigurable planar parallel manipulator;
RAPM : redundantly actuated parallel manipulator;
PD : proportional-derivative;
PDFF : proportional-derivative plus feed forward;
FCTT : feed-forward computed-torque.
iv
Contents
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Introduction 1
I Parallel Robots 5
1 Definitions and Classification 7
1.1 Some History About Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 General terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Manipulator architecture elements and characteristics . . . . . . . . 14
1.2.2 Parallel manipulator definitions . . . . . . . . . . . . . . . . . . . . . 16
2 Parallel Manipulator Configurations 19
2.1 Inverse kinematic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Direct kinematic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Maximal number of direct kinematic problem solutions. . . . . . . . 22
2.2.2 A possible solution from a given manipulator configuration . . . . . 22
3 Singularities 23
3.1 Problematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Singularity classification through kinematics . . . . . . . . . . . . . . . . . . 25
3.3 Velocity and force ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Velocity ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Force ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Singularity classification through velocity and force ellipsoids . . . . . . . . 32
4 Redundant actuation 35
4.1 Definitions and Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Redundant actuation classification . . . . . . . . . . . . . . . . . . . 39
4.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 Force redundancy applications . . . . . . . . . . . . . . . . . . . . . 40
4.2.2 Redundancy resolution schemes . . . . . . . . . . . . . . . . . . . . . 45
v
vi CONTENTS
II Multibody Formalism 49
5 Multibody formalism terminology 53
5.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Tree-like MBS topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Kinematics: basic notations and definitions . . . . . . . . . . . . . . . . . . 55
5.4 Dynamics: basic notations and definitions . . . . . . . . . . . . . . . . . . . 60
6 Multibody Kinematics 63
6.1 Forward sub-chain kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Closed-loop multibody systems . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.1 Closed loop cutting procedures . . . . . . . . . . . . . . . . . . . . . 68
6.2.2 Coordinate partitioning method . . . . . . . . . . . . . . . . . . . . 71
6.2.3 Closed-loop inverse kinematics . . . . . . . . . . . . . . . . . . . . . 74
7 Multibody Dynamics 77
7.1 Direct dynamics of tree-like multibody systems . . . . . . . . . . . . . . . . 77
7.2 Inverse dynamics of tree-like multibody systems . . . . . . . . . . . . . . . . 78
7.3 Newton-Euler recursive formalism . . . . . . . . . . . . . . . . . . . . . . . 79
7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3.2 Forward recursive kinematics . . . . . . . . . . . . . . . . . . . . . . 79
7.3.3 Backward recursive dynamics . . . . . . . . . . . . . . . . . . . . . . 81
7.4 Direct dynamics of closed-loop multibody systems . . . . . . . . . . . . . . 83
7.4.1 Plant dynamics reduction using the coordinate partitioning method 84
7.5 Inverse dynamics of closed-loop multibody systems . . . . . . . . . . . . . . 85
7.6 Dynamic model extension: joint friction forces. . . . . . . . . . . . . . . . . 86
7.7 Closed-loop multibody system dynamics computation . . . . . . . . . . . . 87
7.7.1 Direct dynamics numerically robust time integration . . . . . . . . . 88
III Redundant Actuation and Singularities 95
8 Piecewise and redundant actuation 97
8.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 Piecewise actuation approach . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.3 Redundant actuation of parallel manipulators . . . . . . . . . . . . . . . . . 101
8.3.1 Pseudoinverse-equivalent force redundancy solution . . . . . . . . . . 102
8.3.2 Minimal infinity-norm force redundancy solution . . . . . . . . . . . 105
8.4 Algorithmic computer implementation . . . . . . . . . . . . . . . . . . . . . 107
9 Control of redundant parallel robots 109
9.1 Introduction: robot control . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1.1 Control of redundantly actuated parallel manipulators: research
achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110