Kinematic and Dynamic model of a tracked Robot:
Kinematic and Dynamic Model:
The MRWS is a skid-steer type vehicle consisting of a chassis and two magnetic tracks. The magnetic
tracks provide tractive forces for climbing steel surfaces. Several assumptions are noted for this model:
1) planar climbing surface with uniform frictional properties, 2) the tracks are parallel, 3) the magnets
create a uniform normal force between the climbing surface and the tracks. Figure 1 provides a schematic
of the track-based skid steer platform. The inertial frame {I} is denoted with base coordinates (X, Y)
while the robot frame {R} attached to the robot at the center of mass (point C) is denoted with base
coordinates (x, y) with the z axis normal to the climbing surface, the x axis lying parallel to the tracks and
pointing in the longitudinal direction, and the y axis pointing in the lateral direction. The vector
components {XC, YC} represent the position of the robot in the inertial frame while  represents the
rotation. In this paper, upper case letters will indicate description in the inertial frame while lower case
letters will indicate the local robot frame where appropriate. The left and right tracks are actuated with
inputs l and r, the length of the track is given by l while the distance between the centerline of the tracks
is given by 2b.
x
y
Vc
C
Y
2b
l
X
FIGURE 1: MRWS SCHEMATIC
For purposes of this model, the system is represented as a multi-body system connected with kinematic
pairs as shown in Fig. 2 which consists of three primary bodies; the robot chassis, the contact portion of
the left track (CPl) and the contact portion of the right track (CPr). CPl represents the portion of the left
track that is in contact with the ground while CPr represents the portion of the right track that is in contact
with ground. The robot chassis represents the remainder of the robot system. The left and right contact
patches, CPl and CPr contact the climbing surface over the area bm x l and are connected to the robot
chassis through prismatic joints.
vl

CPl
vlat
vr
y x
bm
Y
vx
vy
2b
CPr
C
bm
l
X
.
FIGURE 2: Kinematic Model of MRWS that Considers Track Contact Portions as Independent
Bodies
The velocities of each of the three bodies are shown on fig. 2 with a common angular velocity () and
velocity in the local y direction (vlat) and local x direction velocities of vx, vl, and vr for the robot chassis,
left contact patch and right contact patch respectively leading to a description of the system as _dot =
[vx, vy, , vl, vr]T. The kinematic relationship between the relative motion along the prismatic joints and
the input drive sprocket rotations is given as,
dl   rl r
(1)
dr   lr r
(2)
and
where dl , dr represent the time rate of change of length in the left and right prismatic joints respectively,
l , r the angular velocities of the left and right tracks, l, r transmission correction coefficients and r
represents the pitch radius of the left and right track drive sprockets in a similar manner as used for
wheeled skid-steer robots in [Mandow et al.,]. ..The velocity of the robot chassis can be related to the
input velocities ( l , r ) and the contact patch velocities as,
vr   rr r  vl   ll r
2
v y  vlat
v    r  v    r
vx 

r
r

r
l
l
l
(3)
(4)

.
b
(5)
In the sense that the system is a mobile robot with two degrees of input motion, with lateral slip of the
tracks characterized by xic (the distance from c to the line of instant centers along x) (citation) and
longitudinal slip of the tracks characterized by the instant centers associated with CPl, CPr (ylic, yric)
(citation), three constraints are defined for the system as:
v y  xic  0
(6)
vl    ylic  b  0
vr    yric  b  0
Or
A  0
(7)
.
(8)
(9)
xlic
0 0
0 1

A  0 0   ylic  b  1 0
0 0  yric  b  0 1
with
(10)
where (6) defines the lateral slip, and (7), (8) define the longitudinal slip of the left and right tracks
respectively. Equations (3-8) can be combined to describe the motion of the robot chassis as a function of
the inputs and slip parameters as,
(10.5)
 vx 
 yric l rl
1
 
x  r
vy  
 ic l l
   yric  ylic    r
 z
 ll
 ylic r rr 
 
 xic r rr   l 

  r rr   r 
The equations of motion can be expressed in the robot frame as,
  Vq, q   Gq   Eτ  Q  AT λ
Mqq
(11)
Where
mc
0

M q    0

0
 0
0
0
0
mc  2mt
0
0
I c  2mt b
0
0
mt
0
0
0
0
2
0
0
0 
0

0
mt 
(12)
the mass matrix,
Vq, q   0
the vector of coriolis and centripetal accelerations,
  mc  2mt gs 
 m  2m gc 
c
t



G q   
0


0




0
the gravity vector,
1
 1
 r2 r2 
 0
0
 b b 

  
Eτ   r 2 r 2   l 

 1
 r 
0
 r2


1
 0

r2 

the input torque vector,
(13)
(14)
(15)
xlic
0 0  1 
0 1
 

A λ  A  0 0   ylic  b  1 0 2 

0 0  yric  b  0 1 
3 
The generalized constraint forces and
T
(16)
 l r 
2 
 2


0
 b

Q   2  l   r 
(17)
 l



2
 r



2
the generalized input forces. Equation (11) presents the equations of motion with track instant
center locations and generalized constraint forces exposed through the lagrange multipliers.
According to the constraint, the lagrange multipliers are defined as follows: 1 = sum of the
lateral friction forces from the left and right tracks, 2 the longitudinal friction in the left track
and l3 the longitudinal friction in the right track.
Equation (11) can be used in the following manner. If the slip kinematics are prescribed via
estimated instant center locations (See for example Mandow et. Al., others), or through assigned
slip ratios for the left and right tracks (see for example … and others), Equation 11 can define the
friction forces required to maintain the slip kinematics prescribed over the trajectory.
Alternatively, if a particular friction model is assumed and used to calculate the friction forces at
the surface contact, Eq. 11 can serve as a means to calculate the instant center locations of the
track portions and yield the parameters needed to complete the kinematic relationships in Eq.
(10.5). Note that this allows the kinematics to be approximated via the model and can account
for desired trajectory. This method will be demonstrated for a climbing mobile robot application
below.
Revisiting Eq. 11, the equations present equations of motion expose the track instant center
locations and generalized friction forces along with the robot chassis motion and two driving
torques. Assume that the robot chassis accelerating motion is prescribed as _dot = 0 and
vx_dot = 0. Combining these input acceleration terms and the differential of the constraints and
eliminating the input torques from Eq. 11 results in the following a set of differential equations
for the instant centers in state space form as,
mc  2mt
 

mt
0
  xic 
 0
  y   1  
  lic  2b  1  2
 1 2
mt   y ric 


1  mc  2mt gc
0   xic  

   32



3
3   ylic   

 mc  2mt g 2 s 
2
2
 3   yric   2  33  m  2m g 2 s 


c
t
2
2
(18)
Or
A  B  C ,
with
 xic 
 
   ylic 
y 
 ric 
And a solution,
A1

C  B 


(19)
(20)
(21)
Stationary values of the instant centers are found when  -> infinity or C  B   0 . This second
case,
C  B   0
(22)
can be written as a linear equations on the friction forces,


 1
 
0
0
 mc  2mt g cθ 
 x
 1 





y

b
y

b


ic
lic
ric
1

 2    mc  2mt g 2 sθ 
2b
2b
 2b
 
 m  2m g 2 sθ 
  ylic  b 
  yric  b   3 
c
t


  xic
1
 2b

2b
2b

Solving for the friction forces,
mc  2mt g cθ
 1  

  

2    mc  2mt g b sθ xic cθ   ylic  yric  4b 
   m  2m g b sθ x cθ   y  y  4b  
t
ic
lic
ric
 3  c

(23)
(24)
Equation 24 creates a condition on the friction forces for the instant center positions to be
stationary (fixed) in the robot frame. Equation 24 will be termed the stationary instant center
criteria. Equation 24 could serve multiple purposes. If the instant center locations (or
conversely track slip) were assumed in the model, Eq. 24 could provide a calculation of the
necessary friction forces to maintain that assumption. Conversely, if a friction model is
assumed, the instant center locations could be determined.
Equation 24 provides the fundamental relationship of skid-steer mobile robots that assume
slipping for motion, in that it relates the kinematics (instant center locations) and kinetics
(friction forces) directly (for an assumed accelerating state). This relationship is independent of
the friction model assumed for the system. An additional kinetic relationship for slipping of
SSMR is given as,
f
(25)
 vˆ
f
which defines the direction of friction vector as opposing the motion. From the friction forces
given in Eq. 24, with 1 = sum of the lateral friction forces from the left and right tracks, 2 the
longitudinal friction in the left track and 3 the longitudinal friction in the right track, it will be
assumed that lateral friction is equally distributed between the left and right track. Then, the
direction of the friction vector for the left and right tracks are defined as,
 1 

 22 
(26)
 1 

 23 
(27)
 l  tan 1 
 r  tan 1 
where l = direction of friction vector acting on the left track and r is the direction of the
friction vector acting on the right track. The direction of the velocity of the centroid of the left
and right track are similarly defined as,
 xic 

(28)
 l  3 2  tan 1 
  ylic  b  

xic 

  yric  b 
 r  3 2  tan 1 
(29)
Combining Eq’s 26-29 according to Eq. 25 yields,
 xic  
  
 
tan 1  1   tan 1 
2


2

y

b
 2
 lic

(30)
  
 xic  
 
(31)
tan 1  1   tan 1 
2
 23 
  yric  b  
And further implementing the angle addition identity and noting that the tangent of /2
approached 1/0 yields,
xic
1
(32)
1
22  ylic  b 
xic
1
(33)
 1.
23  y ric  b 
Equations (32)-(33) are combined with Eqs. (24) to eliminate the friction constraint forces and express the
stationary instant center criteria in terms on the instant centers themselves as,
(34)
cθ xic  ylic  yric  2b  2b sθ ylic  b  0
cθ xic  ylic  yric  6b  2b sθ yric  b  0
or
.
(35)
 xic
 b  ylic
tan 
(36)
3 xic
 b  y ric
.
tan 
(37)
for the left hand track and
for the right hand track.
Equations (36)-(37) describe the slip criteria as linear relationships between lateral and longitudinal slip
for the left and right track, under the assumptions that the rate of curvature of the path is constant and the
lateral friction is equal on the left and right tracks. These relationships are independent of the friction
model and can be used to selecting approximating kinematics for skid-steer mobile robot systems.
END HERE!!!
The velocity of an arbitrary point on CPl, A is given as,
v A  v C   z  rC  A  dl xˆ
(3)
where rC-A is a vector from point C to A as shown in Fig. 3. Similarly, the velocity of an arbitrary point
on CPr , B is given as,
v B  v C   z  rC  B  dr xˆ
(4)
where rC-B is a vector from point C to B as shown in Fig. 3. The velocity of points A and B can also be
described in terms of rotation about the instant centers associated with CPl, CPr respectively as,
v A   z  rICl  A
(5)
v B   z  rICr  B
(6)
where rICl-A is a vector from the instant center of CPl to A and rICl-B, is a vector from the instant center of
CPr to B as shown on Fig. 3. Note from Fig. 3 that the instant centers from all three bodies lie on a line
parallel to the y axis due to the Arnhold Kennedy theorem (CITATION?) and the fact that the axes of the
left and right prismatic joints in the kinematic model Fig. 2 align with the local x axis.
IC
ICl
x
y
rIcl-A
rICl
A
rC-A
rICr
C r
C-B
ICr
rIcl-B
B
Y
X
FIGURE 3: Kinematic Model of MRWS with Instant Centers Labeled
Combining equations 1, 3 and 5 for the left track, 2, 4, and 6 for the right track, and collecting terms
through the distributive nature of addition in the cross product yields




0  v C  z  rC  A  rICl  A  dl xˆ
0  v   r r
 d xˆ
C
or
z
C B
ICl  B
r
 ll rl xˆ  v C  z  rIC
 rr rr xˆ  v C  z

 r 
(7)
(8)
l
(9)
ICr
(10)
Where rICl = (xic, ylic)T, rICr = (xic, yric)T, (LABEL xic, yric, ylic on fig. 3) are vectors from the robot chassis
centroid (C) to the instant centers for CPl and CPr respectively and vC = (vx, vy)T is the velocity of the
robot at C. These equations are expanded into scalar components,
v x   z ylic   l rll 



 v y   z xic   0 
vx   z yric   r rrr 



 v y   z xic   0 
(11)
(12)
and then combined into the matrix expression,
 vx 
 
 
J1 vy   J 2  l 

 r 
 
 z
(13)
1 0  ylic 
J 1  1 0  yric 
0 1 xic 
(14)
where
 l rl
J 2   0
 0
0 
 r rr 
0 
(15)
vx 
l 
 
 v y   K eq   
 r 
 
 z
(13)
.
The direct kinematics of the tracked robot system is written in its usual form as,
where
K eq
 yric l rl
1
x  r
 J J2 
 yric  ylic   ic l l
  l rl
1
1
 ylic r rr 
 xic r rr  .
  r rr 
(14)
The portion of Keq that considers longitudinal translation and rotation only is given by Keq* as,
K *eq  (J 11 J 2 )* 
 yric l rl
1
 yric  ylic    l rl
 ylic r rr 
 r rr 
(15) ELIMINATE?
Given the generalized coordinate vector, q defined as,
q  X C
YC  l  r 
(16)
where q is based on coordinates specified in inertial frame, the kinematic relationships can be written as,
Aq  0
(17)
where

A  J 1 R IR
or
T
J 2

(18)
 c
A   c
 s
s
 ylic
  l rl
s
 yric
0
c
xic
0
0 
  r rr 
0 
(19)
where J1, J2 are given in Eqs. 14, 15 and R R is the rotation operation of  about the z axis and projects
the robot frame {R} onto the inertial frame {I}. From Eq. 17, it can be seen that q lies in the null space
of A.
I
Dynamics:
The method of Lagrange is applied to solve the dynamic equations for the MRWS. In this
model, the MRWS is represented by five primary massive components (i.e., components with a
non-negligible percentage of system mass). Table I below lists these components and their
velocity in the inertial frame while Fig. 5 labels these components.
Table I: Important MRWS components for the dynamics analysis
MRWS component
Mass labels
Robot chassis
mc, Ic
Approx.
percentage of
system mass*
20%
Left track module
mm,l, Im,l
20%
Left track with
adhering members
mt,l, It,l
20%
Right track module
mm,r, Im,r
20%
Right track with
adhering members
mt,r, It,r
20%
Velocity
 X 
Vc   c 

 Yc 
 X   z  d l s  bc 
Vm,l   c


 Yc   z d l c  bs  
 X   z  d tl s  bc 
Vt ,l   c


 Yc   z d tl c  bs  
 X   z  d r s  bc 
Vm,r   c


 Yc   z d r c  bs  
 X   z  d tr s  bc 
Vt ,r   c


 Yc   z d tr c  bs  
Notes:
 Mass percentage based on typical construction of MRWS system
 I* = Ic +( Im.l + It.l + Im.r + It,r) + (mm,l + mt,l + mm,r + mt,r)b2
What about I about the track axes?.
Fig. 5: CAD Model with components labeled.
From table I, parameters dl, dtl, dr, dtr represent the x coordinates of the center of mass (cm) in the
robot frame for the left track module, left track, right track module and right track respectively
while parameter b represents the y coordinate of the cm for each of the MRWS components.
This assumes that the center of mass for the left and right tracks and track modules lie on the
track centerline and are symmetric about the robot x axis.
The dynamic equations are derived using Lagrange techniques and written in the general form,
  Vq, q   Gq   Eτ  Q  AT λ
Mqq
()
with
 mc  2mm  mt 

0

M q    2mm d m  mt d t s

0


0
 2mm d  mt d t s
0
mc  2mm  mt 
 
2mm d m  mt d t c
0
2mm d  mt d t c


2 mm d  b  mt d  b
2
2
2
t
2
  I
0
*
0
0
0
It
0
0
0
0
0 
0  ()

0
I t 
the mass matrix,

 2mm d  mt d t  z2 c

 2mm d  mt d t  z2 s

V q, q    2mm d m  mt d t  X c  z c  Yc  z s

0


0

the vector of coriolis and centripetal accelerations,
0


m  2m  m g 
tm
t
 c


G q   
0


0




0
the gravity vector,
0 0 
0 0 

  
Eτ  0 0  l 

  r 
1
0


0 1
the input torque vector,
c
 s 
 c
 s
s
c   1 

 
A T λ   ylic  yric xic  2 



0
0 
3 
 r
 0
r
0 
the constraint matrix,
T
λ  1 2 3 

The vector of lagrange multipliers and









()
()
()
()
()
f l ,x  f r ,x




f l ,r  f r ,r


 f l , x  d tl s  bc   f l , y d tl c  bs  

 Ml  Mr 
Q  

  f r , x  d tr s  bc   f r , y d tr c  bs 



 f l , x r  f pl , x r


 f r , x r  f pr , x r


()
the vector non-conservative forces in the system (other than gravity forces and the inputs).
 ), two input torques () and three
In this model, there are five generalized accelerations ( q
lagrange multipliers (). In the usual solution method for a wheeled mobile robot, the kinematic
model is fully known according to the no slip, pure roll constraints consisting of two holonomic
and one non-holonomic constraint (CITATION HERE!). The holonomic constraints can be used
to reduce the generalized coordinate set (SEE FOR EXAMPLE A CITATION THAT DOES
THIS) or included in the lagrange multipliers (SEE FOR EXAMPLE A CITATION THAT
DOES THIS). Thus, two inputs would be defined, yielding four (or eight if holonomic
constraints are retained in equations of motion) unknowns to be solved from the three (five)
equations of motion and additional one (three) constraint equation(s). The solution defines the
motion and the corresponding forces at the wheel contact points through the lagrange multipliers.
In the proposed MRWS model, the kinematic model is based on qic = {xic, ylic, yric}T the vector
of parameters defining the instant center locations of the left and right track contact patches.
These parameters are contained implicitly in the kinematic, dynamic and friction force models.
The goal of the dynamic model is to determine these parameters. In order to solve this system,
two assumptions are made: 1) The friction forces can be determined directly from the kinematics
and an appropriate friction model, and 2) the lagrange multipliers are known independently of
the dynamic equations. The first assumption is based on the fact that for all motion other than
idealized straight-line motion, a state of slipping must occur between the contact patch bodies
(CPl, CPr) and ground. The second assumption notes that the lagrange multipliers enforce the
corresponding kinematic constraints, which define the instant center locations. Since all nonconservative forces have been accounted for and there are no kinematic pairs associated with
these constraints, the lagrange multipliers are known and equal to zero.
Thus, the system yields eight equations (equations of motion and constraints) and eight
unknowns (generalized coordinates and/or motor torques, instant center locations). The instant
center locations are contained implicitly in the friction model resulting in an iterative solution of
the dynamic model. The friction model and suggested solution for the dynamic equations are
provided in the sections below.
Friction model
This section will present a friction model that is appropriate for the MRWS
TRISTAN TO ENTER
The virtual work associated with the non-conservative forces acting on the system is derived
from three primary sources; input torques, friction between the left and right contact patches and
the climbing surface, and friction between the track and track module guides (i.e., friction in the
prismatic joint discussed in the kinematic modeling section);
W  τ l  θ l  τ r  θ r
 f l  rCP ,l  f r  rCP ,r  M CP.l  θ l  M CP.r  θ r
.
 f p ,l  r p ,l  f p.r  r p ,r
where l, l, are the input torques on the tracks, fl, fr are the resultant friction forces acting on the
centroid of the left and right contact patches ( rCP,l, rCP,r ), MCPl, MCP,r are the resultant moments
about the centroid of the left and right contact patches, and fp,l, fp,r are the friction forces in the
left and right track guides with virtual displacement rp,l, rp,r . This leads to the leads to the
generalized force associated with the jth generalized coordinate,
Q j  τ l   q j θ l  τ r   q j θ r
 
 
 X c   z  d tl s  bc   dl c 


 Y   d c  bs   d s   f r 
z
tl
l
 c

 d 
 d 
 f p ,l   q j  l   f p ,r   q j  r 
0
0
 fl 

q j

q j
 X c   z  d tr s  bc   dr c 
ω z


 Y   d c  bs   d s   M l  M r  
q j
z
tr
r
 c

This leads to a generalized force vector as:
f l ,x  f r ,x




f l ,r  f r ,r


 f l , x  d tl s  bc   f l , y d tl c  bs  

  M l  M r 
Q'  

  f r , x  d tr s  bc   f r , y d tr c  bs 



 l  f l , x r  f pl , x r


 r  f r , x r  f pr , x r


where the friction forces are described in the friction model in the section below.
Dynamic Model solution:
These are solved in an iterative fashion according to the flow-chart below.
Appendix: Detailed determination of the Equations of Motion via Lagrange
The kinetic energy of the MRWS is now defined as,

T  1 mc Vc  Vc  mm,l Vm,l  Vm,l  mm,r Vm,r  Vm,r  mt ,l Vt ,l  Vt ,l  mt ,r Vt ,r  Vt ,r  1  z  I z  l  I t ,ll  r  I t ,r
2
2
with velocities defined in table I and
 X c2 
vc  vc   2 

 Yc 
v ml  v ml  X c2  Yc2   z2 b 2  d l2  2 X c z bc  d l s   2Yc z  bs  d l c 
v  v  X 2  Y 2   2 b 2  d 2  2 X  bc  d s   2Y  bs  d c 

mr
mr
c
c


z




r

c
z
r
c
z
r
v tl  v tl  X  Y   b  d  2 X c z bc  d tl s   2Yc z  bs  d tl c 
v tr  v tr  X c2  Yc2   z2 b 2  d tr2  2 X c z bc  d tr s   2Yc z bs  d tr c 
2
c
2
c
2
z
2
2
tl
(1)
The potential energy of the climbing MRWS system is given as
V  mgYc
where the climbing surface is assumed to be vertical and the inertial Y axis is aligned with the
vertical, and m represent the sum of the MRWS component masses.
The system energy can be rewritten as,


T  1 mc X c2  Yc2
2
 1 mml X c2  Yc2   z2 b 2  d l2  2 X c z bc  d l s   2Yc z  bs  d l c 
2
 1 mmr X c2  Yc2   z2 b 2  d r2  2 X c z bc  d r s   2Yc z bs  d r c 
2
 1 mtl X c2  Yc2   z2 b 2  d tl2  2 X c z bc  d tl s   2Yc z  bs  d tl c 
2
 1 mtr X c2  Yc2   z2 b 2  d tr2  2 X c z bc  d tr s   2Yc z bs  d tr c 
2
2
2
 1 I * z2  I trr  I tll
2
V  mgYc


















Further, it will be assumed that the left and right track modules are similar in mass and
symmetrically located on the platform: mml = mmr = mm; mtl = mtr = mt; dl = dr = d; dtl = dtr =dt;
and Itl = Itr = It , yielding,




T  1 mc X c2  Yc2
2
 mm X c2  Yc2   z2 d 2  b 2  2 X c  z ds   2Yc  z dc 
 mt X c2  Yc2   z2 d t2  b 2  2 X c  z d t s   2Yc  z d t c 
2
2
 1 I * z2  I t r  l
2
V  mgYc .









Following the method of Lagrange, the spatial and time derivatives of the energy are taken with
respect to the 5x1 generalized coordinate vector q as,
For q1 = Xc:
T
0
X c
T
 X c mc  2mm  mt   2mm d  mt d t  z s
X c

d  T
 
2

  X c mc  2mm  mt   2mm d  mt d t   s   z c


X
dt 
c
V
0
X c
For q2 = Yc:
T
0
Yc
T
 Yc mc  2mm  mt   2mm d  mt d t  z c
Yc
d  T
 
2

  Yc mc  2mm  mt   2mm d  mt d t   c   s

dt  Yc 

V
Yc


 mg
For q3 = :
T
 2mm d  mt dt X c  c   Yc  s 

T
 2 mm d 2  b 2  mt d t2  b 2  z  2mm d m  mt d t   X c ds   Yc dc   I * z
 z
 
 







 

 

  
d  T 
2
2
2
2
*

  2 mm d  b  mt d t  b   2mm d m  mt d t   X c s  X c  z c  Yc c  Yc  z s  I 
dt   
V

0
For q4 = r:
T
0
 r
T
 I tr
r
d  T


  I tr

dt   r 
V
0
 r
For q5 = l:
T
 I tl
l
d  T


  I wl

dt   l 
V
0
 l
The three constraints discussed in the early model and kinematic section are incorporated into the
Lagrange derivation through the use of Lagrange multipliers. These constraints can be written in
the form:
 AT i ,q j i
where A is defined in the kinematic model.
The virtual work associated with the non-conservative forces acting on the system is derived
from three primary sources; input torques, friction between the left and right contact patches and
the climbing surface, and friction between the track and track module guides (i.e., friction in the
prismatic joint discussed in the kinematic modeling section);
W  τ l  θ l  τ r  θ r
 f l  rCP ,l  f r  rCP ,r  M CP.l  θ l  M CP.r  θ r
.
 f p ,l  r p ,l  f p.r  r p ,r
where l, l, are the input torques on the tracks, fl, fr are the resultant friction forces acting on the
centroid of the left and right contact patches ( rCP,l, rCP,r ), MCPl, MCP,r are the resultant moments
about the centroid of the left and right contact patches, and fp,l, fp,r are the friction forces in the
left and right track guides with virtual displacement rp,l, rp,r . This leads to the leads to the
generalized force associated with the jth generalized coordinate,
Q  τ   q θ  τ   q θ
j
l
j
 
l
r
j
 
r
 X c   z  d tl s  bc   dl c 

f 
q j 
   d c  bs   d s  r
Y
z
tl
l
 c

 d 
 d 
 f p ,l   q j  l   f p ,r   q j  r 
0
0
 fl 

This leads to a generalized force vector as:

 X c   z  d tr s  bc   dr c 

  M l  M r   ω z
q j 

q j


 Yc   z d tr c  bs   d r s 
f l ,x  f r ,x




f l ,r  f r ,r


 f l , x  d tl s  bc   f l , y d tl c  bs  

 Ml  Mr 
Q'  

  f r , x  d tr s  bc   f r , y d tr c  bs 



 l  f l , x r  f pl , x r


 r  f r , x r  f pr , x r


where the friction forces are described in the friction model in the section below.
Using equations x,xx, xxx, xxx …, the equations of motion are assembled according to equation
xx to yield,



Xc mc  2mm  mt   2mm d  mt dt   z ds  z2 dc  c1  c2  s3  f l , x  f r , x
Yc mc  2mm  mt   2mm d  mt dt  dc   2 ds  mc  2mm  mt g  s1  s2  c3  f l , y  f r , y
 


2 mm d  b  mt d  b
2
2
2
t
2
  2m
m
 

 
 
 s  X  c  Y c  Y  s  I *
d m  mt d t   X
c
c
z
c
c
z

 ylic1  yric2  xic 3  f l , x  d tl s  bc   f l , y d tl c  bs   f r , x  d tr s  bc   f r , y d tr c  bs 
I tl   l rl 1   l  f l , x r  f pl, x r
I tr   r rr 2   r  f r , x r  f pr, x r
or, as cast in matrix form,
  Vq, q   Gq   Eτ  Q  AT λ
Mqq
with
 mc  2mm  mt 

0

M   2mm d m  mt d t s

0


0
 2mm d  mt d t s
0
mc  2mm  mt 
2mm d m  mt d t c
 
0
2mm d  mt d t c


2 mm d  b  mt d  b
2
2
2
t
2
  I
0
*
0
0
0
It
0
0
0

 2mm d  mt d t  z2 c

 2mm d  mt d t  z2 s

V q, q    2mm d m  mt d t  X c  z c  Yc  z s

0


0











0
0 
0

0
I t 
0


m  2m  m g 
tm
t
 c


G q   
0


0




0
0 0 
0 0 


E  0 0 


1 0 
0 1
 s  c  c 
 c  s  s 


AT   d
b
b 


r
0 
 0
 0
0
r 
T
λ  1 2 3  .
f l ,x  f r ,x




f l ,r  f r ,r


 f l , x  d tl s  bc   f l , y d tl c  bs  

 Ml  Mr 
Q  

  f r , x  d tr s  bc   f r , y d tr c  bs 



 f l , x r  f pl , x r


 f r , x r  f pr , x r


Friction Model:
This section will develop a friction model that is appropriate for the MRWS
FOR FOR TRISTAN TO ENTER
Solving the Model
The MRWS dynamic model system gives a total of 8 equations: 5 equations of motion and an
additional three constraint equations (Eqs. X, xx). A typical solution would consider the
corresponding set of unknowns as the generalized coordinate vector (q double dot) and the vector
of lagrange multipliers. However, in the MRWS dynamics problem, the kinematics are based on
an additional set of parameters, the instant centers of the right and left contact patch (xic, ylic,
yric), fig. 3. The parameters are contained implicitly in the kinematic, dynamic and friction
force models. In order to solve this system, two assumptions are made:
1) The friction forces can be determined directly from the kinematics and the friction model
2) The Lagrange Multipliers are known independently of the dynamic equations
3)
4) Except in the ideal case of straight-line motion, a state of slipping (relative motion
between ground) is assumed for the contact patch bodies (CPl, CPr).
5) Since a state of slipping is necessary between the contact patch bodies (CPl, CPr) and
ground, the friction components can be determined as functions of the kinematic model
and the corresponding friction model.
6) The three Lagrange multipliers represent the forces required to enforce the constraints.
Here the constraints are the instantaneous, stationary positions of the instant centers of
CPl and CPr. Since no external force is able to be placed on these points, the forces at
these constraints are known and are equal to zero.
Thus, the eight unknowns become the generalized coordinate vector and the three coordinates
locating the instant centers of the left and right contact patch bodies. These are solved in an
iterative fashion according to the flow-chart below.
Local-coordinate Generalized Vector
The generalized coordinate vector, q is expressed in the global frame. Let vector  represent the
generalized coordinate vector expressed in the local frame as,
R I
02 x3 
~
q   R SO2 
  R ,
I 3 x3 
 03 x 2

With q  X P
YP  l
r   x P
y P  l
r .
and
~
~
  R  R
q

With
  sin     cos 
cos   sin   0 0 0
 
 sin   cos  0 0 0
 cos    sin  
I






R
0 2 x3
~
~
R 0
0
1 0 0 , R   R SO2 
0
0

0
0



3x2
3x3 

0
0 1 0
0
0

 0

 0
0
0 0 1
0
0

Then the dynamic relationships can be written as,


~
~
~
~
~
~
~
R T Mq Rη  R T Vq, q   Rη  R T Gq   R T Eτ  R T A T λ
Or
~
~
~
~
~
Mq η  V q, q   G q   Eτ  RA T λ
With
0
2m w d sin 2 
 mc  2m w

0
mc  2m w
 2m w d cos2 
~ 
M  2m w d sin 2   2m w d cos2  2m w d 2  b 2   I *

0
0
0


0
0
0
v x sin 2   v y  2m w d cos2  2 


 v x cos2   2m w d sin 2  

~

V q, q   
2m wv x


0




0


0
0
0
Iw
0
0
0 
0

0
I w 
0 0 0

0 0 0
0 0 0

0 0 0
0 0 0
 sin  mg 
 cos mg 


~

G q   
0


0




0
~
EE
Work on 3/22/11
Now have added all non-conservative forces and have new equations.
Secondly, solved the friction force balance only (assume that M_fric = 0, frx +fry =0). The 1 dof
solution is shown in the plots below:
The final requirement: lambda1, lambda2, lambda3 = 0. These are the forces that constain the
instant center to lie at ylic, yric, xic. If an arbitrary set of ylic, yric, xic were chosen, then the
EOM would be used to find the lamdas (forces) to enforce this constraint. Of course, there
would need to be some magic bar extending out to this point, and reacting against ground to
create these forces. In practice, no such force exists, so that lambda1-lambda3 must be zero and
the EOM are used to solve for the instant center (ylic, yric, xic) that satisfies this case.
and vector q lies in the null space of A. A linear operator S(q) is defined as a matrix function
that spans null space of A, S  N A , ( SqAq  0 ) such that an alternate control vector v that
satisfies the system constraints can be defined as:
q  Sqv .
And one such operator, S defined as,
 A11 A 2 
S

I


with
 r bc  ds  r bc  ds 
2b
 2b

r
r

bs  dc 
 2b bs  dc 
2b

S
r
r


2b
2b


1
0


0
1


And
v  ν1  l ,r


Alternatively, given the generalized coordinate vector, , (coordinates specified in robot frame)
  xP y P   l  r ,   x P y P  l r
Then the kinematic relationships can be written as,


Ar  0
Where
Ar  J1  J 2 
or
1 0  ylic
Ar  1 0  yric
0 1 xic
and  lies in the null space of Ar.
  l rl
0
0
0 
  r rr 
0 