- Email: [email protected]
Modeling of Vehicles Using Robotics Formulation
Copyright COl IFAC Advances in Automotive Control. Karlsruhe. Gennany. 2001
MODELING OF VEHICLES USING ROBOTICS FORMULATION Eric Guillo _.1 Maxime Gautier-
- Institut de Recherche en Communications et Cybemitique de Nantes (IRCCyN) 1 rue de la Noe - B.P. 92101 - 44321 Nantes Cedex 3 - France eric. guillo/maxime. gautier@irccyn. ec-nantes.fr
Abstract: The aim of this paper is to give a general and unifying presentation of the modeling issues of the mobile machines, in order to simulate their behavior and to implement advanced control laws. Considering the structural diversity and complexity of those mobile machines, a systematic method of geometrical description based on the Modified Denavit-Hartenberg parameterization is proposed and applied to a civil engineering machine, the compactor. Copyright fS>2001IFAC Keywords: Mobile machines, robotics, geometric modeling, kinematic modeling, dynamic modeling.
1. INTRODUCTION
According to Dudzinski systematic classification (Dudzinski, 1989), there exists many kinds of steering mechanisms for mobile machines. They allow them to follow uneven rolling ground. Even if such mechanical structures are used for a long time , their modeling are essentially restricted to the kinematic and dynamic models taking into account the restriction to machine mobility induced by the kinematics constraints of pure rolling and non slipping along the motion. Real working conditions of motion do not necessarily satisfy such hypotheses. Consequently the dynamic model that explicitly takes into account forces between wheels and soil is needed.
Recent advances in the field of robotics, dynamic localization, computer vision and production practices have made possible the automation of mobile machines involved with transportation, mining and lumbering tasks, road construction and maintenance... The aim of this paper is to give a general and unifying presentation of the modeling issues of these mobile machines, in order to simulate their behavior and to implement advanced control laws. Several examples of derivation of kinematic and/or dynamic models for mobile machines are available in the literature for particular mechanical structures of machines ((Tilbury et al., 1994)). For instance, the usual approach of mobile robotics only considers systems made up of a rigid cart equipped with rigid wheels moving on a horizontal plane. Here, a more general point of view is adopted and the general class of earthmoving equipment is considered.
I
Considering the structural diversity and complexity of those mobile machines, a systematic method of geometrical description based on the Modified Denavit-Hartenberg parameterization is proposed in this paper. It amounts to consider a mobile machine as a robot manipulator with a tree structure where wheels and tools are terminal links. This original description allows to automatically calculate symbolic expressions of the geometrical, kinematic and dynamic models using symbolic softwares like SYMORO+ (for SYmbolic MOd-
Partially supported by the LCPC. France.
119
eling of RObots) ... It is also convenient to model mechanical systems with lumped elasticity. For instance, it allows to take suspensions into account in the modeling of a car. In these conditions, only small adaptations are needed to calculate the Inverse Dynamic Model of the system. The Geometrical, Kinematic and Dynamic Models that are of interest for understanding the behavior of mobile machines are developed in this paper.
Fig. 2. Articulated frame machine with floating rear axle. units linked together. Futhermore, an additional horizontal axle joint is generally added to earthmoving machines in order to ensure permanent ground contact on all four wheels (see Figure 2). The compactor (see Figure 3), the civil engineering machine that is studied all this paper long, is a typical illustration of this kind of steering system.
2. MOBILE MACHINES DESCRIPTION 2.1 Steering systems
III (a)
(b)
(c)
.J. ,
I
;e
(d)
Fig. 1. Schematic for typical steering systems in mobile machines: (a) Knuckle steering, (b) Skid steering, (c) Articulated-frame steering, (d) Combined steering.
Fig. 3. A typical compactor: Albaret VA12DV Combined steering is used with machines that require extraordinary versatility in application, for instance motor graders. The driver can always adapt the type of steering precisely to application requirements. Articulated-frame and knuckle steering increase manceuvrability in the most confined spaces and is particularily appropriate in curves and up against curbstones.
Figure 1 shows a schematic representation of typical steering systems for mobile wheeled machines. Knuckle steering is typical for machines with rigid frames. Earthmoving equipment with knuckle steering is distinguished according to single-axle steering, either as front-wheel or rear-wheel steering, four-wheel (coordinated) steering and crab steering. They correspond to the general class of vehicles that had been studied in the context of mobile robotics (see (Canudas de Wit et al., 1997)).
2.2 Wheel-soil interaction In the area of mobile robotics, authors generally assume that the mobile robots under study are made up of a rigid cart equipped with rigid wheels and that they are moving on an horizontal plane. In each case, it is assumed that the contact between the wheel and the ground is reduced to a single point of the plane. Furthermore, the interaction between the wheel and the ground is supposed to satisfy both conditions of pure rolling and non-slipping along the motion.
Most vehicles with skid steering are small with two or even three driven, but not steerable, axles. In principle, this steering process acts like a tracked vehicle one. The steering motions are controlled by using the difference in the circumferential speed of the wheels on each side of the vehicle. The overall centre of gravity, which is dependent on the loaded condition at any given time, plays an essential role in the determination of the turning circle. This kind of steering design provides one of the greatest manamvrability of all the steering systems described here.
Considering the general case of mobile machines, these hypotheses are too limited. Many types of interaction between the wheel and the ground can be met:
In comparison to traditional equipment with knuckle steering, machines with articulated-frame steering are more manceuvrable. The driving direction is changed by "collapsing" the entire machine, which usually consists of two rigid axle
• • • •
120
Rigid wheel over rigid soil, Rigid wheel over unprepared soil, Tire over rigid soil, Tire over unprepared soil.
of a tree structure of n = 7 rigid bodies Cj where Co is the base body and with the following body definitions (see Figure 4):
2.3 Description using robotics formulation The mechanical structure of a mobile machine defines a system ~ composed of a set of n rigid bodies Cj linked together by rotoid and prismatic joints.
• Cl , C 2 are two virtual bodies (Le. without mass and inertia) used to define the compactor position with respect to the frame Ho, • C3 , Cs are the front and rear chassis, • C4 , C 7 are the front and rear rolls, • C6 is a virtual body (used to define a second frame Rs attached to Cs) ·
By analogy with classical robot manipulator description (see (Canudas de Wit et al., 1997)), it is possible to consider a mobile machine as a tree structure of rigid bodies where wheels and tools are terminal bodies. Consequently, the Modified Denavit-Hartenberg (MDH) notations can be applied to the mobile machine to obtain its geometric parameters.
The system ~ is provided with a frame R j respectively attached to each of the (n + 1) bodies Cj . Let Rj be defined as Rj = (OJ';[j ' J!.i'~j) (see Figure 4).
Many definitions are needed to perform this systematic geometric description of mobile machines:
Classical tree structure description using the MDH notations (Khalil and Kleinfinger, 1986) applied to the system ~ defines the geometric parameters of the compactor (see Table 1) with respect to the position and orientation of the body Co. Table l. Geometric parameters of the compactor
• The definition of a reference frame for all studied motions of the mobile machine, • The definition of a reference body on the mobile machine. Then, let Rg be a Galilean reference frame attached to the ground so that Rg = (0g, ;[g' J!.g , ~g) (R g may be different from the base frame Ho of the mobile machine).
i 1
= a(j)
2
Let Cr be the reference body on the mobile machine. It corresponds to the body whose situation (Le. position and orientation) gives the mobile machine posture refered to the frame R g • According to the studied motion, the mobile machine posture is specified by a set of generalized coordinates. According to the MDH notations, it amounts to define a virtual carrying arm in the equivalent tree structure where each joint variable corresponds to a generalized coordinate of the posture vector.
3 4 5
6 7
Ct:i.
di
°i
ri.
7r/2 7r/2 7r/2 -7r/2
0 0 0 0
7r/2 7r/2
rl
03 04
0 0
-Ds -D6
Os
2 0
-7r/2
0
07
0 0 0 0 0
Uj
0
2 3 3 5 6
1 0 0 0
0
r2
In that case the homogeneous transform (1) of the frame R j with respect to Ri (where i = a(j) is the antecedent of j) is expressed as a function of the 4 following parameters:
• or
angle between ~i and ~j ' corresponding to a rotation about ;[i' • dj : distance from ~i to ~j along ;[i' • Br angle between ;[i and ;[j ' corresponding to a rotation about ~j' • r{ distance from ;[i to ;[j along ~j.
At last , to complete the geometric description of the mobile machine, the situation of wheels and tools (terminal bodies) are given with respect to the reference body C r using the MDH notations. In order to illustrate this purpose, the geometric description of the compactor is presented.
iSj
i
nj iaj
(1)
Ol x 3
2.4 The compactor example
where i Aj is the (3 x 3) rotation matrix which defines the orientation of the frame R j with respect to the frame R; and i P j is the origin of the frame R j expressed in the frame R; .
A planar motion of the compactor is considered. Let Rg be a Galilean reference frame attached to the rolling plan IT. In these conditions, a vector of three generalized coordinates is needed to specify the compactor posture. The front chassis is choosen to be the reference body Cr of the compactor, i.e. its situation gives the compactor posture.
Remark 1. Ilj = 1 means that the i-joint is actuated and Ilj = 0 that it is not. (1j specifies the type of the joint ((1j = 0 if rotational, (1j = 1 if translational, <>j = 2 if fixed) .
According to classical robot manipulator description (Canudas de Wit et al. , 1997), the compactor is considered as a mechanical system ~ composed
According to the previous description of the compactor, the vehicle motion is completely described
121
I
.4 [
I
i
i
__~ ____ _ -- ------ -4-i i
i
i
0'::::-7+---ch::'
i
(a) Bodies definition
(b) Equivalent tree structure
(c) Projection frames definition
Fig. 4. Geometric description of the compactor using the MDH notations. where V7.~ and Wj ,g respectively are the velocity of the point Oj and the rotation velocity of the body Cj with respect to the reference frame R g . ip~j is called base jacobian matrix of the vehicle.
by the vector q (see eq. ??) of six generalized coordinates: q = [rl r2 ()3 ()4 ()5 ()7) T. 3. GEOMETRIC MODEL
The expression of the body velocity is developped using the following recurrence relations applied from the base body Co to the considered terminal body.
Using the expression of homogeneous transform iTj of the frame R j with respect to ~ (see eq. 1) , the Direct Geometric Model (DGM) is developped . It gives bodies situation with respect to the reference frame Rg as a function of q, the vector of joint variables.
(4)
For example, equation 2 gives the situation of the rear chassis (C6 ) of the compactor with respect to the frame R g • The symbolic modeling software SYMORO+ (see (Khalil and Creusot, 1997)) had been used to determine the expression of equation 2 from the table 1 of the geometric parameters of the compactor.
For example, the application of these recurrence relations to determine the velocity of the rear roll (C7 ) of the compactor with respect to the frame Rg gives the expression 5 of the base jacobian matrix 6ip~7. The symbolic modeling software SYMORO+ had been used to determine the expression of equation 5 from the table 1 of the geometric parameters of the compactor.
(2)
6~~7 = .. , C(93 + 95) S(93 - S(93 + 95) C(93 [
4. KINEMATIC MODEL In the previous section, the geometric description of vehicles using robotics formulation is presented. From this description, the Direct Kinematic Model (DKM) of vehicles can be developped. It is composed of nt relations (see eq. 3) corresponding to the velocity of each of the nt vehicle terminal bodies (wheels, tools, .. . ).
+ 95) + 115)
- Ds S 9s 0 0 0] -(D6 + Ds C 9s) 0 - D6 0
~
~
~
~
~
~
o
0 0
0 1
0 0
0 1
0
o
(5)
1
4.1 Conditions of pure rolling and non-slipping Let be assumed that the contact between the rigid wheel and the ground is reduced to a single point B j of the rolling plan IT (see Figure 5) . The contact between the wheel and the ground is supposed to satisfy both conditions of pure rolling and non-slipping along the motion. This means that the velocity of the contact point B j is equal to zero and implies that the two components of this velocity, respectively parallel to the plan of the wheel and orthogonal to this plan (see eq. 6), are equal to zero.
The equation 3 gives the kinematic wrench of a vehicule terminal body Cj with respect to the reference frame R g .
(3)
122
o o
Zi
o
4= ®-"--_+-....X;,
1 - - (D6 D6
xj
(12)
+ D5C85)
Ds
- - 58 5 re
with the following definition for the vector 1] : • the component 1]1 corresponds to the linear . : 1]1 = V0 3 veIOC1ty 3,g ' X3, • the component 1]2 corresponds to the yaw rate : 1]2 = W3,g 'Z3 = rh.
Fig. 5. Contact between a rigid wheel and the ground reduced to a single point.
{ VZ~ 'Yi = 00 'Xi
=
(6)
Vj,~
5. DYNAMIC MODEL
The velocity of the contact point B j can be deduced from the jacobian matrix
°
The Inverse Dynamic Model (IDM) of a vehicle is written as following
M(q)ij+H(q,q) =lL+Qc
(13)
(7)
where:
• M(q) is the mass matrix of the system E. • H(q, q) is the vector of centrifugal, Coriolis
where
and gravity terms. • lL is a vector depending on the internal forces
Considering a vehicle equipped with nt wheels, its Direct Kinematic Model under constraints is given by the following expression
•
(8)
between the vehicle bodies: motor torques, friction, lumped elasticity. ~t is a vector depending on the external contact forces between the ground and the wheels.
for all wheels Cj of the vehicle.
5.1 Internal forces
With these notations, the constraints can be written in the general matrix form
The vector lL of internal forces is composed of three components:
K(q)q
=0
• The j-component lLj of the actuation vector lLa is written as
(9)
This means that whatever the type of vehicle, the velocity q is restricted to belong a distribution ~c defined as
q E ~c =
span{col(S(q))}
if /.Lj if /.Lj
(14)
where Uj is the motor torque on the j-joint. • The j-component lL{ of the friction vector lLt is written as
(10)
where the columns of the matrix S(q) form a basis of ker(K(q)). This is equivalent ot the following statement: for all time t, there exists a timevarying vector 1]( t) such that
q = S(q)1]
= 1, = O.
lL{
=-
F vj
qj -
F sj sign(q])
(15)
where F vj and F sj respectively are the viscous and striction friction parameters for the j-joint • The j-component lLj of the elastic forces vector lL e is written as
(11)
The relation 11 is called the Inverse Kinematic Model under constraints of the vehicle.
if elastic joint, if not.
For example, considering the case of the compactor, the Inverse Kinematic Model under constraints is written as
where kj, the stiffness of the j-joint.
123
(16)
5.2 Contact forces between wheel and the ground
6. CONCLUSION
Using the principle of virtual powers, the vector of external forces is developed to obtain
In this paper, the geometric, kinematic and dynamic models of vehicles using robotics formulation is presented. In order to describe the configuration of vehicles, a parameterization using the Denavit-Hartenberg notations is used. It amounts to consider a mobile machine as a robot manipulator with a tree structure where wheels and tools are terminal links. This original description allows to automatically calculate symbolic expressions of the geometrical, kinematic and dynamic models using symbolic softwares like SYMORO+ (for SYmbolic MOdeling of RObots) ... It is also convenient to model mechanical systems with lumped elasticity. To illustrate this purpose, the case of a civil engineering machine is studied.
(t
n:-c
i o· where 'lI' points out the wrench of the j resultant contact forces between the ground II j on the wheel Cj at point Dj projected in frame
Ri . 5.3 Linearity property of the inverse model The expression of kinematic and potential energies are linear in relation to a set of (np = 11) parameters, Xs . Consequently, the expression of the Inverse Dynamic Model is also linear in relation to the same set of parameters and then it is possible to write it as following
Y. with Ys
= Ds(q, q)Xs
ACKNOWLEDGEMENTS This paper presents research results managed in collaboration with the LCPC (Laboratoire Central des Ponts et Chaussees, France). The authors would like to thank Fran
(18)
= lLa.
Using this property of the Inverse dynamic Model, a Weighted Least Squares method of identification is proposed by (Gautier, 1997) to obtain the values of the dynamic parameters Xs and applied to the compactor.
7. REFERENCES Canudas de Wit, C., B. Siciliano and G. Bastin (1997). Theory of Robot Control. Chap. 1, pp. 4~29 . Springer-Verlag. London. Dudzinski, P.A. (1989). Design characteristics of steering systems for mobile wheeled earthmoving equipment. Journal of Terramechanics 6(1), 25~82 . Gautier, M. (1997). Dynamic identification of robots with power model. In: IEEE International Conference on Robotics and Automation. Albuquerque. pp. 1922~1927. Guillo, E. and M. Gautier (2000) . Dynamic modeling and identification of earthmoving engines without kinematic constraints: application to the compactor. In: Proceedings of IEEE International Conference on Robotics and A utomation. San Francisco, California. Khalil, W. and D. Creusot (1997). Symoro+: A system for the symbolic modelling of robots. Robotica 15, 153~ 161. Khalil, W. and J.-F. Kleinfinger (1986) . A new geometric notation for open and closed-loop robots. In: Proceedings of IEEE International Conference on Robotics and Automation. San Francisco, California. pp. 1174~ 1180. Tilbury, D., O. Sordalen, L. Bushnell and S. Sastry (1994). A multi-steering system: conversion into chained form using dynamic feedback. In: Preprints of the 4th IFAC Symposium on Robot Control. Capri, Italy. pp. 159164.
5.4 General expressions of the dynamic model The general expression of the Inverse Dynamic Model is given by the relation 13. According to the expression of the internal and external forces that take part in the model, there are two possible expression of this model: • The conditions of pure rolling and nonslipping are satisfied. Then, the Inverse Dynamic Model has the following expression
f
S(q)T M(q)S(q)i]
+ S(q)T H'(q,7)) =
S(q)T(JLa
l
...
+1[/ +JL e) (19) q = S(q)7)
• The conditions of pure rolling and nonslipping are not satisfied. Then, the Inverse Dynamic Model has the following expression
fl
M(q)ij
+ H(q,q) = La +0 +Qc nc = )"' 'L
i4>0jT
J
iTOj n ·-c ·
j ]
(20)
]
The Inverse Dynamic Model of the compactor without kinematic constraints is developped in (Guillo and Gautier, 2000) and used to identify its dynamic parameters along a straight line motion .
124
MODELING OF VEHICLES USING ROBOTICS FORMULATION Eric Guillo _.1 Maxime Gautier-
- Institut de Recherche en Communications et Cybemitique de Nantes (IRCCyN) 1 rue de la Noe - B.P. 92101 - 44321 Nantes Cedex 3 - France eric. guillo/maxime. gautier@irccyn. ec-nantes.fr
Abstract: The aim of this paper is to give a general and unifying presentation of the modeling issues of the mobile machines, in order to simulate their behavior and to implement advanced control laws. Considering the structural diversity and complexity of those mobile machines, a systematic method of geometrical description based on the Modified Denavit-Hartenberg parameterization is proposed and applied to a civil engineering machine, the compactor. Copyright fS>2001IFAC Keywords: Mobile machines, robotics, geometric modeling, kinematic modeling, dynamic modeling.
1. INTRODUCTION
According to Dudzinski systematic classification (Dudzinski, 1989), there exists many kinds of steering mechanisms for mobile machines. They allow them to follow uneven rolling ground. Even if such mechanical structures are used for a long time , their modeling are essentially restricted to the kinematic and dynamic models taking into account the restriction to machine mobility induced by the kinematics constraints of pure rolling and non slipping along the motion. Real working conditions of motion do not necessarily satisfy such hypotheses. Consequently the dynamic model that explicitly takes into account forces between wheels and soil is needed.
Recent advances in the field of robotics, dynamic localization, computer vision and production practices have made possible the automation of mobile machines involved with transportation, mining and lumbering tasks, road construction and maintenance... The aim of this paper is to give a general and unifying presentation of the modeling issues of these mobile machines, in order to simulate their behavior and to implement advanced control laws. Several examples of derivation of kinematic and/or dynamic models for mobile machines are available in the literature for particular mechanical structures of machines ((Tilbury et al., 1994)). For instance, the usual approach of mobile robotics only considers systems made up of a rigid cart equipped with rigid wheels moving on a horizontal plane. Here, a more general point of view is adopted and the general class of earthmoving equipment is considered.
I
Considering the structural diversity and complexity of those mobile machines, a systematic method of geometrical description based on the Modified Denavit-Hartenberg parameterization is proposed in this paper. It amounts to consider a mobile machine as a robot manipulator with a tree structure where wheels and tools are terminal links. This original description allows to automatically calculate symbolic expressions of the geometrical, kinematic and dynamic models using symbolic softwares like SYMORO+ (for SYmbolic MOd-
Partially supported by the LCPC. France.
119
eling of RObots) ... It is also convenient to model mechanical systems with lumped elasticity. For instance, it allows to take suspensions into account in the modeling of a car. In these conditions, only small adaptations are needed to calculate the Inverse Dynamic Model of the system. The Geometrical, Kinematic and Dynamic Models that are of interest for understanding the behavior of mobile machines are developed in this paper.
Fig. 2. Articulated frame machine with floating rear axle. units linked together. Futhermore, an additional horizontal axle joint is generally added to earthmoving machines in order to ensure permanent ground contact on all four wheels (see Figure 2). The compactor (see Figure 3), the civil engineering machine that is studied all this paper long, is a typical illustration of this kind of steering system.
2. MOBILE MACHINES DESCRIPTION 2.1 Steering systems
III (a)
(b)
(c)
.J. ,
I
;e
(d)
Fig. 1. Schematic for typical steering systems in mobile machines: (a) Knuckle steering, (b) Skid steering, (c) Articulated-frame steering, (d) Combined steering.
Fig. 3. A typical compactor: Albaret VA12DV Combined steering is used with machines that require extraordinary versatility in application, for instance motor graders. The driver can always adapt the type of steering precisely to application requirements. Articulated-frame and knuckle steering increase manceuvrability in the most confined spaces and is particularily appropriate in curves and up against curbstones.
Figure 1 shows a schematic representation of typical steering systems for mobile wheeled machines. Knuckle steering is typical for machines with rigid frames. Earthmoving equipment with knuckle steering is distinguished according to single-axle steering, either as front-wheel or rear-wheel steering, four-wheel (coordinated) steering and crab steering. They correspond to the general class of vehicles that had been studied in the context of mobile robotics (see (Canudas de Wit et al., 1997)).
2.2 Wheel-soil interaction In the area of mobile robotics, authors generally assume that the mobile robots under study are made up of a rigid cart equipped with rigid wheels and that they are moving on an horizontal plane. In each case, it is assumed that the contact between the wheel and the ground is reduced to a single point of the plane. Furthermore, the interaction between the wheel and the ground is supposed to satisfy both conditions of pure rolling and non-slipping along the motion.
Most vehicles with skid steering are small with two or even three driven, but not steerable, axles. In principle, this steering process acts like a tracked vehicle one. The steering motions are controlled by using the difference in the circumferential speed of the wheels on each side of the vehicle. The overall centre of gravity, which is dependent on the loaded condition at any given time, plays an essential role in the determination of the turning circle. This kind of steering design provides one of the greatest manamvrability of all the steering systems described here.
Considering the general case of mobile machines, these hypotheses are too limited. Many types of interaction between the wheel and the ground can be met:
In comparison to traditional equipment with knuckle steering, machines with articulated-frame steering are more manceuvrable. The driving direction is changed by "collapsing" the entire machine, which usually consists of two rigid axle
• • • •
120
Rigid wheel over rigid soil, Rigid wheel over unprepared soil, Tire over rigid soil, Tire over unprepared soil.
of a tree structure of n = 7 rigid bodies Cj where Co is the base body and with the following body definitions (see Figure 4):
2.3 Description using robotics formulation The mechanical structure of a mobile machine defines a system ~ composed of a set of n rigid bodies Cj linked together by rotoid and prismatic joints.
• Cl , C 2 are two virtual bodies (Le. without mass and inertia) used to define the compactor position with respect to the frame Ho, • C3 , Cs are the front and rear chassis, • C4 , C 7 are the front and rear rolls, • C6 is a virtual body (used to define a second frame Rs attached to Cs) ·
By analogy with classical robot manipulator description (see (Canudas de Wit et al., 1997)), it is possible to consider a mobile machine as a tree structure of rigid bodies where wheels and tools are terminal bodies. Consequently, the Modified Denavit-Hartenberg (MDH) notations can be applied to the mobile machine to obtain its geometric parameters.
The system ~ is provided with a frame R j respectively attached to each of the (n + 1) bodies Cj . Let Rj be defined as Rj = (OJ';[j ' J!.i'~j) (see Figure 4).
Many definitions are needed to perform this systematic geometric description of mobile machines:
Classical tree structure description using the MDH notations (Khalil and Kleinfinger, 1986) applied to the system ~ defines the geometric parameters of the compactor (see Table 1) with respect to the position and orientation of the body Co. Table l. Geometric parameters of the compactor
• The definition of a reference frame for all studied motions of the mobile machine, • The definition of a reference body on the mobile machine. Then, let Rg be a Galilean reference frame attached to the ground so that Rg = (0g, ;[g' J!.g , ~g) (R g may be different from the base frame Ho of the mobile machine).
i 1
= a(j)
2
Let Cr be the reference body on the mobile machine. It corresponds to the body whose situation (Le. position and orientation) gives the mobile machine posture refered to the frame R g • According to the studied motion, the mobile machine posture is specified by a set of generalized coordinates. According to the MDH notations, it amounts to define a virtual carrying arm in the equivalent tree structure where each joint variable corresponds to a generalized coordinate of the posture vector.
3 4 5
6 7
Ct:i.
di
°i
ri.
7r/2 7r/2 7r/2 -7r/2
0 0 0 0
7r/2 7r/2
rl
03 04
0 0
-Ds -D6
Os
2 0
-7r/2
0
07
0 0 0 0 0
Uj
0
2 3 3 5 6
1 0 0 0
0
r2
In that case the homogeneous transform (1) of the frame R j with respect to Ri (where i = a(j) is the antecedent of j) is expressed as a function of the 4 following parameters:
• or
angle between ~i and ~j ' corresponding to a rotation about ;[i' • dj : distance from ~i to ~j along ;[i' • Br angle between ;[i and ;[j ' corresponding to a rotation about ~j' • r{ distance from ;[i to ;[j along ~j.
At last , to complete the geometric description of the mobile machine, the situation of wheels and tools (terminal bodies) are given with respect to the reference body C r using the MDH notations. In order to illustrate this purpose, the geometric description of the compactor is presented.
iSj
i
nj iaj
(1)
Ol x 3
2.4 The compactor example
where i Aj is the (3 x 3) rotation matrix which defines the orientation of the frame R j with respect to the frame R; and i P j is the origin of the frame R j expressed in the frame R; .
A planar motion of the compactor is considered. Let Rg be a Galilean reference frame attached to the rolling plan IT. In these conditions, a vector of three generalized coordinates is needed to specify the compactor posture. The front chassis is choosen to be the reference body Cr of the compactor, i.e. its situation gives the compactor posture.
Remark 1. Ilj = 1 means that the i-joint is actuated and Ilj = 0 that it is not. (1j specifies the type of the joint ((1j = 0 if rotational, (1j = 1 if translational, <>j = 2 if fixed) .
According to classical robot manipulator description (Canudas de Wit et al. , 1997), the compactor is considered as a mechanical system ~ composed
According to the previous description of the compactor, the vehicle motion is completely described
121
I
.4 [
I
i
i
__~ ____ _ -- ------ -4-i i
i
i
0'::::-7+---ch::'
i
(a) Bodies definition
(b) Equivalent tree structure
(c) Projection frames definition
Fig. 4. Geometric description of the compactor using the MDH notations. where V7.~ and Wj ,g respectively are the velocity of the point Oj and the rotation velocity of the body Cj with respect to the reference frame R g . ip~j is called base jacobian matrix of the vehicle.
by the vector q (see eq. ??) of six generalized coordinates: q = [rl r2 ()3 ()4 ()5 ()7) T. 3. GEOMETRIC MODEL
The expression of the body velocity is developped using the following recurrence relations applied from the base body Co to the considered terminal body.
Using the expression of homogeneous transform iTj of the frame R j with respect to ~ (see eq. 1) , the Direct Geometric Model (DGM) is developped . It gives bodies situation with respect to the reference frame Rg as a function of q, the vector of joint variables.
(4)
For example, equation 2 gives the situation of the rear chassis (C6 ) of the compactor with respect to the frame R g • The symbolic modeling software SYMORO+ (see (Khalil and Creusot, 1997)) had been used to determine the expression of equation 2 from the table 1 of the geometric parameters of the compactor.
For example, the application of these recurrence relations to determine the velocity of the rear roll (C7 ) of the compactor with respect to the frame Rg gives the expression 5 of the base jacobian matrix 6ip~7. The symbolic modeling software SYMORO+ had been used to determine the expression of equation 5 from the table 1 of the geometric parameters of the compactor.
(2)
6~~7 = .. , C(93 + 95) S(93 - S(93 + 95) C(93 [
4. KINEMATIC MODEL In the previous section, the geometric description of vehicles using robotics formulation is presented. From this description, the Direct Kinematic Model (DKM) of vehicles can be developped. It is composed of nt relations (see eq. 3) corresponding to the velocity of each of the nt vehicle terminal bodies (wheels, tools, .. . ).
+ 95) + 115)
- Ds S 9s 0 0 0] -(D6 + Ds C 9s) 0 - D6 0
~
~
~
~
~
~
o
0 0
0 1
0 0
0 1
0
o
(5)
1
4.1 Conditions of pure rolling and non-slipping Let be assumed that the contact between the rigid wheel and the ground is reduced to a single point B j of the rolling plan IT (see Figure 5) . The contact between the wheel and the ground is supposed to satisfy both conditions of pure rolling and non-slipping along the motion. This means that the velocity of the contact point B j is equal to zero and implies that the two components of this velocity, respectively parallel to the plan of the wheel and orthogonal to this plan (see eq. 6), are equal to zero.
The equation 3 gives the kinematic wrench of a vehicule terminal body Cj with respect to the reference frame R g .
(3)
122
o o
Zi
o
4= ®-"--_+-....X;,
1 - - (D6 D6
xj
(12)
+ D5C85)
Ds
- - 58 5 re
with the following definition for the vector 1] : • the component 1]1 corresponds to the linear . : 1]1 = V0 3 veIOC1ty 3,g ' X3, • the component 1]2 corresponds to the yaw rate : 1]2 = W3,g 'Z3 = rh.
Fig. 5. Contact between a rigid wheel and the ground reduced to a single point.
{ VZ~ 'Yi = 00 'Xi
=
(6)
Vj,~
5. DYNAMIC MODEL
The velocity of the contact point B j can be deduced from the jacobian matrix
°
The Inverse Dynamic Model (IDM) of a vehicle is written as following
M(q)ij+H(q,q) =lL+Qc
(13)
(7)
where:
• M(q) is the mass matrix of the system E. • H(q, q) is the vector of centrifugal, Coriolis
where
and gravity terms. • lL is a vector depending on the internal forces
Considering a vehicle equipped with nt wheels, its Direct Kinematic Model under constraints is given by the following expression
•
(8)
between the vehicle bodies: motor torques, friction, lumped elasticity. ~t is a vector depending on the external contact forces between the ground and the wheels.
for all wheels Cj of the vehicle.
5.1 Internal forces
With these notations, the constraints can be written in the general matrix form
The vector lL of internal forces is composed of three components:
K(q)q
=0
• The j-component lLj of the actuation vector lLa is written as
(9)
This means that whatever the type of vehicle, the velocity q is restricted to belong a distribution ~c defined as
q E ~c =
span{col(S(q))}
if /.Lj if /.Lj
(14)
where Uj is the motor torque on the j-joint. • The j-component lL{ of the friction vector lLt is written as
(10)
where the columns of the matrix S(q) form a basis of ker(K(q)). This is equivalent ot the following statement: for all time t, there exists a timevarying vector 1]( t) such that
q = S(q)1]
= 1, = O.
lL{
=-
F vj
qj -
F sj sign(q])
(15)
where F vj and F sj respectively are the viscous and striction friction parameters for the j-joint • The j-component lLj of the elastic forces vector lL e is written as
(11)
The relation 11 is called the Inverse Kinematic Model under constraints of the vehicle.
if elastic joint, if not.
For example, considering the case of the compactor, the Inverse Kinematic Model under constraints is written as
where kj, the stiffness of the j-joint.
123
(16)
5.2 Contact forces between wheel and the ground
6. CONCLUSION
Using the principle of virtual powers, the vector of external forces is developed to obtain
In this paper, the geometric, kinematic and dynamic models of vehicles using robotics formulation is presented. In order to describe the configuration of vehicles, a parameterization using the Denavit-Hartenberg notations is used. It amounts to consider a mobile machine as a robot manipulator with a tree structure where wheels and tools are terminal links. This original description allows to automatically calculate symbolic expressions of the geometrical, kinematic and dynamic models using symbolic softwares like SYMORO+ (for SYmbolic MOdeling of RObots) ... It is also convenient to model mechanical systems with lumped elasticity. To illustrate this purpose, the case of a civil engineering machine is studied.
(t
n:-c
i o· where 'lI' points out the wrench of the j resultant contact forces between the ground II j on the wheel Cj at point Dj projected in frame
Ri . 5.3 Linearity property of the inverse model The expression of kinematic and potential energies are linear in relation to a set of (np = 11) parameters, Xs . Consequently, the expression of the Inverse Dynamic Model is also linear in relation to the same set of parameters and then it is possible to write it as following
Y. with Ys
= Ds(q, q)Xs
ACKNOWLEDGEMENTS This paper presents research results managed in collaboration with the LCPC (Laboratoire Central des Ponts et Chaussees, France). The authors would like to thank Fran
(18)
= lLa.
Using this property of the Inverse dynamic Model, a Weighted Least Squares method of identification is proposed by (Gautier, 1997) to obtain the values of the dynamic parameters Xs and applied to the compactor.
7. REFERENCES Canudas de Wit, C., B. Siciliano and G. Bastin (1997). Theory of Robot Control. Chap. 1, pp. 4~29 . Springer-Verlag. London. Dudzinski, P.A. (1989). Design characteristics of steering systems for mobile wheeled earthmoving equipment. Journal of Terramechanics 6(1), 25~82 . Gautier, M. (1997). Dynamic identification of robots with power model. In: IEEE International Conference on Robotics and Automation. Albuquerque. pp. 1922~1927. Guillo, E. and M. Gautier (2000) . Dynamic modeling and identification of earthmoving engines without kinematic constraints: application to the compactor. In: Proceedings of IEEE International Conference on Robotics and A utomation. San Francisco, California. Khalil, W. and D. Creusot (1997). Symoro+: A system for the symbolic modelling of robots. Robotica 15, 153~ 161. Khalil, W. and J.-F. Kleinfinger (1986) . A new geometric notation for open and closed-loop robots. In: Proceedings of IEEE International Conference on Robotics and Automation. San Francisco, California. pp. 1174~ 1180. Tilbury, D., O. Sordalen, L. Bushnell and S. Sastry (1994). A multi-steering system: conversion into chained form using dynamic feedback. In: Preprints of the 4th IFAC Symposium on Robot Control. Capri, Italy. pp. 159164.
5.4 General expressions of the dynamic model The general expression of the Inverse Dynamic Model is given by the relation 13. According to the expression of the internal and external forces that take part in the model, there are two possible expression of this model: • The conditions of pure rolling and nonslipping are satisfied. Then, the Inverse Dynamic Model has the following expression
f
S(q)T M(q)S(q)i]
+ S(q)T H'(q,7)) =
S(q)T(JLa
l
...
+1[/ +JL e) (19) q = S(q)7)
• The conditions of pure rolling and nonslipping are not satisfied. Then, the Inverse Dynamic Model has the following expression
fl
M(q)ij
+ H(q,q) = La +0 +Qc nc = )"' 'L
i4>0jT
J
iTOj n ·-c ·
j ]
(20)
]
The Inverse Dynamic Model of the compactor without kinematic constraints is developped in (Guillo and Gautier, 2000) and used to identify its dynamic parameters along a straight line motion .
124