Transcript Poster

Velocity Transformation Based Multi-Body
Approach for Vehicle Dynamics
Dr. Y.P. Daniel Chang
Weidong Zhang
Start
Abstract: An automobile is a complex close loop multi-body system, which includes
the chassis, front suspension, rear suspension, wheels etc. In such a close loop
topology, the chassis is modeled as a rigid body, which is linked to low arms, upper
arms, tire rods, and tie bars in SLA suspensions by kinematics joints. The dampers
and springs in suspensions are considered as velocity based and position based force
elements separately. The magic formula is adopted to describe tire’s slip behavior.
Based on the virtual power principle, the complete equations of motion including
Langrage multipliers are formulated. This formulation is based on the velocity and
acceleration vectors of the center of gravity and the angular velocity and acceleration
vectors of each element. Velocity transformation is adopted to build the relationships
between the dependent velocities the independent velocities, which consist of
translational velocities at the center of gravity and angular velocities of the base body
and at the kinematics joints. By using Runge-Kutta method, position variables for
base body translation and rotation at kinematics joint can be directly acquired. Euler
quaternion four parameters are solved from angular velocity and previous quaternion,
and they are used to orientate the base body and spherical joints. In terms of these
position variables, position and velocity of the rigid body are easily obtained
recursively at a specific time. After a series of computations, the vehicle dynamic
behavior time history is eventually reached. Velocity based multi-body approach
highly increase computational efficiency, and provide an effective solver for vehicle
dynamics and tire’s dynamics studies.
Vehicle Model
start time t, in which q, z i , zi are known
q , R1
Tire force Fx , Fy , Fz , and M z
q
Calculate force from damper and spring
1, q , q

S 1c1 from z
R1T MR1 , R1T Q, R1T MS1c1

 2q , c2 from q, q
 2q R1 , c 2   2q S 1c1
z1
Numerica l intergrati on z1
z1 (t  h)
Quaternion for base body
t>tfinal
End
Multi-body Vehicle Model
Orientation and Recursive Kinematics
Euler quaternion 4 parameters q  [s, v , v , v ]
orientation of the base body.
q ( t )  0 0.5( t )q( t )
x
0
y
, are used to describe the
z
0
The quaternion to the rotation matrics
1  2 v  2 v

A   2 v v  2sv
 2 v v  2sv

1
Vehicle model sketch

0
0
M



1

0
0
T
2T
q
1
q
2
q
q
Multi-Body Position Relation
x
x
  q  Q
     c 
   
   c 
1
A A A
1
k
1
 , , c corresponding to the constraint of the close constraint.
2
z
y
z
z
y
2 v v  2sv
1  2v  2v
2 v v  2sv
x
y
y
2
2
x
z
x
2 v v  2sv
2 v v  2sv
1  2v v
z
x
x
z
y
y
z
x
2
2
x
y
2
kj
j
Where A andAare the rotational matrics of body j and k respectively. A is the
rotational matrics between body k and j when a revolute joint is considered.
A  I~
u~
u (1  cos z )  ~
u sin z
kj
k
kj
j
j
kj
j
kj
q
The center of gravity in each body is evaluated here, each body has 6 DOF, three
transnational and three rotational corresponding to the global reference frame.
q  u , u , u ,  ,  ,  
i
xi
yi
zi
xi
yi
zi
The projection matrices R1 ,which defines the null space of  ,can be directly
obtained through velocity computations, without the need of forming and factoring
the Jacobian matrix . The velocities of a body b can be computed from the velocities
of base body, and relative velocities of kinematics Joint .
q1
 R 0
q  R z
q  R z  S c
Velocity Analysis
r  g    (r  g )
u    u
    u
  
r  r  d
g  r    (g  r )
j2
j
j
j
j
j2
j
k1
k
k1
k
j
kj
k1
k
j
j2
jk
k1
k
k
k1
Acceleration Analysis
1
q
r  r  d

 
 

r  g  
  (r  g )      (r  g ) 
1
1
1
1
1
1
k1
j2
k
1
j2
R MR
 R

  R
T
1
1
1
q
1
2
q
1
R MR
 R

T
1
2
q
1
1
R
0
0





2
q
2
y
2
 , , c corresponding to the constraint of the Open-chain system by cutting close joint.
1
2
Considering a pair of rigid bodies shown in the left figure, the orientation metrics is
represented by:
j
1
2
T
1T
1
q
R   z

0  

0  
T
2T
1
q
1
 R Q  R MS c 
 


c


S
c
 

  c  Sc 
 

T
T
1
1
1
1
1
2
q
R   z  R Q  R MS c 
 


0     c   S c 
T
2T
1
q
1
T
T
1
1
2
2
1
1
1
1
q
2
2
1
2
q
1
1
1
1
1
jk
j
kj
j
j2
j
j
j
j2
j
g  r  
  (g  r )    (  (g  r ))
k
k1
k
k
k1
k
k
k
k1