The document discusses Kane's method for modeling multi-body systems. It begins with an introduction to multi-body systems and generalized coordinates. It then covers Kane's method which uses generalized speeds and forces to develop equations of motion in a compact form. The method encapsulates both holonomic and non-holonomic constraints. Kane's method is considered superior to other methods for modeling complex multi-body systems. The document provides details on deriving Kane's equations using virtual work principles and generalized speeds and coordinates.

1. PURPOSE: DYNAMICS OF MULTI BODY SYSTEM BY KANE’S METHOD
FOCUS:
Introduction to multibody system
Dynamic Modeling of Multibody system
Kane’s Method of Multi Body System
Application of Kane’s Method Equation
1
KANE DYNAMICS WITH APPLICATION TO MULTI BODY
SYSTEM LIKE IN MECHANISMS AND MANIPULATORS
preparedbyYohannesR.@JiT

2.  In Systems where all parts are not best described in an inertial
or "world" reference frame are referred to as multi-body
systems (MBSs).
 These are common in:
Robotics
Aerospace
Aviation
Industrial Automation
2
Introduction to multibody system
preparedbyYohannesR.@JiT

3.  To model a given dynamic system S, one has choose variables to
describe the configuration of S, or variables that specify the
location of a reference point and orientation of a reference frame
fixed within each body of S. These variables named as
configuration variables.
 For mechanical scenario, a multibody system is a system that
consists a number of rigid bodies (referred to as links) connected in
succession by kinematic pairs (referred to as joints).
Base
Link0
Joint1
Link2
Link3Joint3
End of Arm
Link1
Joint2
3
preparedbyYohannesR.@JiT

4. 4
 Human hands can be act as MBS in which the arrangement of bones(link) and kinematic
pair of joints provides dexterity (manipulating objects)
 Each joint represent a degree of freedom; there are 27bones, 22joints and thus 22DoF
Example of MBS
preparedbyYohannesR.@JiT

5. 5
The basic laws of dynamics can be formulated (expressed mathematically) in several
ways other that that given by Newton’s Laws. The most important are:
(a) D’Alembert Principle
(b) Lagrange’s Equations
(c) Hamilton’s Equations (f) Kane’s Equations
(d) Gibbs-Appell’s Equations
Popular Methods for Modeling of Multi-body Systems
 According to literatures there are two classes of methods to model Multi-body
Systems
 Vector Methods
Newton-Euler equation, Kane's equation , D’Alembert Principle
 Scalar Methods
Lagrangian Dynamics equation, Gibbs-Appell’s Equations
Kane's method borrows concepts from both, but is classified as a Vector Method.
Can be named optimized equation
(e) Newton-Euler’s Equations
Dynamic Modeling of Multibody system
preparedbyYohannesR.@JiT

6. 6
 Kane’s method (originally called Lagrange form of D'Alambert's principle)
which is a powerful tool for developing dynamical equations for MBS motion
 Applying the Newton-Euler method requires that force and moment balances
be applied for each body taking in consideration every interactive and constraint
force.
 Therefore, the method is inefficient when only a few of the system’s forces need
to be solved for.
 The major disadvantage of Lagrange’s Equations method is the need to
differentiate scalar energy functions (kinetic and potential energy).
 In Kane’s method, With the use of generalized forces the need for examining
interactive and constraint forces between bodies is eliminated
 (Huston 1990) argue that; Kane’s method provides combined means to
develop the dynamics equations for multibody systems that lends itself to
automated numerical computation.
preparedbyYohannesR.@JiT
 Essentially all methods for obtaining equations of motion are equivalent.
 However, some are more suited for multibody dynamics than others.

7.  For example, if a revolute joint connects two bodies, only the joint angle is
needed to describe the configuration of the second body, if the configuration
of the first is known.
 Compared to a set of variables that specifies the location and orientation of
each body relative to a common ground, a set of generalized coordinates is
reduced in number i.e., n<6v, where v is the number of bodies of S and spatial
motion is being considered.
 If M is the number of configuration constraints encapsulated by the choice of
generalized coordinates, then n = 6v-M.
 The generalized coordinates describe only the allowed configurations and thus
encapsulate certain configuration constraints.
7
preparedbyYohannesR.@JiT
 Generalized coordinates are a set of convenient coordinates, usually
independent of one another, used to describe a particular configuration of a
system

8. Kane's method is touted as a superior approach by it proponents
because it:
 Encapsulates holonomic (position) constraints by the use of
generalized coordinates (as in the Lagrangian method).
 Also encapsulates non- holonomic (velocity) constraints through
the use of generalized speeds. (Which requires Lagrange's Method
of Undetermined Multipliers)
 Results in a compact, first order representation of the equations of
motion. (ODE)
 Is more systematic and therefore easier to learn .
 Is becoming the industry standard where complex systems need to
be modeled.
Why Kane's Method?
8
preparedbyYohannesR.@JiT

9. 9
Kane’s Equations
Consider an open-chain multibody system of N interconnected rigid bodies each
subject to external and constraint forces these external forces can be transformed
into an equivalent force and torque Fk and Mk passing through Gk ,which is the
mass center of the body k , (k = 1,2…N).
Similar to the external forces, the constraint forces may be written as and
Using D'Alambert's principle for the force equilibrium of body k, the following is
obtained
0  c
kkk FFF

kkk amF 

where is the inertia force of body k.
The concept of virtual work may be described as follows for a system of N
particles with N degrees of freedom
Professor Thomas R. Kane
1924 -
Stanford University
Kane’s Equations Using the Principle of Virtual Work
c
kF

c
kM

Kane’s Method of Multi Body System
preparedbyYohannesR.@JiT

10. 10
The virtual work is then defined as:


N
i
ii rFW
1


Where is the resultant force acting on the ith particle and is the position vector of the
particle in the inertial reference frame. is the virtual displacement, which is imaginary in
the sense that it is assumed to occur without the passage of time.
iF

ir
ir
Now applying the concept of virtual work to our multibody system considering
only the work due to the forces on the system we obtain:
0)(  
k
c
kkk rFFFW 

(k = 1,2,…,N)
The constraints that are commonly encountered are known as workless constraints so…
0 k
c
k rF 

Which simplifies the virtual work equation to:
0)(  
kkki rFFW 

The positions vector may also be written as:
),( tqrr rkk


t
r
dt
dq
q
r
r kr
r
k
k








t
r
q
q
r k
r
r
k










preparedbyYohannesR.@JiT

11. 11
Taking the partial derivative of with respect to we obtain
kr
rq
r
k
r
k
q
r
q
r








r
k
r
k
q
r
q
v








or
Since the virtual displacement is arbitrary without violating the constraints
we can write * as:
rq
0 
KK ff Kf

Kf
generalized active
generalized inertia forces
r
k
kK
q
v
Ff





r
k
kK
q
v
Ff




 
In a similar fashion it can be shown using
virtual work that the moments can be
written as
0 
KK MM
KM generalized active moments

KM generalized inertia moments
r
k
kK
q
TM






r
k
kkkK
q
IIM





 
 )(
By superposition of the force and moment
equations we arrive at Kane’s equations:
0 
kk FF
KKK MfF 

 KKK MfF
preparedbyYohannesR.@JiT

12. 12
 tri

ir


tvi

 tPi tP i'
 ttP i '
ir


Virtual Path True Path (P)
Newtonian or
Dynamic Path
The Constraint
Space at t
mjra
N
i
i
j
i ,,10
1




   
   
   







0
0
0
21
21
21
tttt
trtr
trtr
ii
ii



1t
2t
Kane’s Equations can be also derived in terms of generalized coordinates we
can write:
Kane’s Equations Using the Generalized Coordinates system
          Niktqzjtqyitqxtqqrtqr iiinii .,1,,,,,,, 1 



 1
  Nikdzjdyidxdt
t
r
dq
q
r
tqrd iii
i
n
j
j
j
i
i .,1,
1









 
2
preparedbyYohannesR.@JiT

13. 13
 
Ni
t
r
q
q
r
td
tqrd
rv i
n
j
j
j
ii
ii .,1
,
1












 

3
Kane and Levinson have shown that with the n generalized coordinates , is useful
to define another n variables , which are linear functions of the n :
jq
iu jq
nrZqYu r
n
j
jrjr .,1
1
 

4
where the matrix is invertible and    nj
nrrjYY
,1
,1


       nj
nrrjWWY
,1
,1
1 



njXuWq j
n
r
rrjj .,1
1
  
5
jrrjrj XandZWY ,, are functions of tandq
are called Generalized Speed (also Nonholonomic Velocities, Quasivelocities.
etc.) and are not unique.
iu
preparedbyYohannesR.@JiT
where the elements of the (nxn) matrix Y and (nx1) matrix Z are functions of
qi (i = 1; …….;n) and possibly the time t. Reciprocal relations express the
generalized coordinate derivatives in terms of the generalized speeds:

14. 14
Kane’s Equations (continue)
Non holonomic constraints are linear relations among either or the ; for m
Non holonomic constraints:
6
7
where k may be n-m or n, depending on whether the non holonomic constraints are
incorporated.
iu jq
nmnsBuAu s
mn
r
rsrs .,1
1
 



If we substitute equations (6) in (5) we obtain a more general expression for : jq
njXuWq j
k
r
rrjj .,1
1
  
Let substitute equation (6) in (3):
 
Ni
t
r
X
q
r
uW
q
r
t
r
XuW
q
r
td
tqrd
rv
i
k
r
n
j
j
j
i
r
n
j
rj
j
i
i
n
j
j
k
r
rrj
j
ii
ii
.,1,
,
1 11
1 1



 






























 
 
 
 

From this equation we can see that
 



 n
j
rj
j
i
r
i
W
q
r
u
v
1

preparedbyYohannesR.@JiT
njXuWq j
n
r
rrjj .,1,
1
  
 Eq.-7 represents for kinematical
differential equations and form the
first part of the state equations
(equations of motion)

15. 15
Kane’s Equations (continue)
8
9
Let use now the equation of differential work:
By defining we obtain:
t
r
X
q
r
v i
n
j
j
j
ii
t









1
Nivu
u
v
v t
i
k
r
r
r
i
i .,1
1







 
 

N
i
iii
N
i
ii rdamrdFdW
11

Equation (9) is now rewritten using (8). On the left side we obtain:
dtvu
u
v
FdtvFrdF
N
i
t
i
k
r
r
r
i
i
N
i
ii
N
i
ii 
















     1 111


10
Similarly, the right side of (9) becomes:
dtvu
u
v
amdtvamrdam
N
i
t
i
k
r
r
r
i
ii
N
i
iii
N
i
iii 
















     1 111



11
Equations (10) and (11) are equated and terms re collected:
    0
11 11


























   
dtamFvdtuam
u
v
F
u
v N
i
iii
t
i
k
r
r
N
i
ii
r
i
N
i
i
r
i 

12
preparedbyYohannesR.@JiT

16. 16
Kane’s Equations (continue)
    0
11 11


























   
dtamFvdtuam
u
v
F
u
v N
i
iii
t
i
k
r
r
N
i
ii
r
i
N
i
i
r
i 

12
The and dt are nonzero and independent and so the coefficients of each
of them must be zero. Also using Newton’s Second Law:
krur ,1
0 iii amF

nrZqYu r
n
j
jrjr ,.,1
1
 

krF
u
v
F
N
i
i
r
i
r ,,1
1







  kram
u
v
F
N
i
ii
r
i
r ,,1'
1








krFF rr ,,10' 
4
13
14
15
Generalized Speeds
Generalized Active Forces
Generalized Inertia
Forces
summary of Kane’s Equations
we have confined Kane's equation of motion:
preparedbyYohannesR.@JiT
0 
kk FF Similar to VRP

17. 17
Application of Kane’s Method Equation
Problem 1
The first problem is a spring-mass-pendulum problem with frictionless sliding. This
was problem is good for showing the general procedure of Kane’s method.
Step 1) Define important points as the center
of mass of A and particle P
general procedure for developing Kane's equations of motion
Solution
Step 2) Select generalized coordinates as shown in
the figure and generate velocity and acceleration
expressions for the important points.
11
ˆnuv AN


APPNANPN
rvv

 
)ˆ()ˆ(ˆ 23211 bLbunu 
1211
ˆˆ bLunu 
11
ˆnuaAN



2
2
21211
ˆˆˆ bLubLunuaPN
 

Step 3) Construct a partial velocity table.
preparedbyYohannesR.@JiT

18. 18
0 
kk FF
Step 4) apply the concept of virtual work
P
r
NA
r
N
k vngMvngMnKqF

 )ˆ()ˆˆ( 222111
11 KqF 
)sin( 222 qLgMF 
P
r
NA
r
NAN
k vngMvaMF


)ˆ()( 221
Generalized Active Forces
Generalized inertia Forces
))sin()cos(( 2
2
22212111 qLuqLuuMuMF 

))cos(( 2
22122 LuqLuMF  
Step 5) Assemble the equation and form matrix for final solution unknown variables
0 
rr FF





 














)sin(
)sin(
)cos(
)cos()(
22
12
2
22
2
1
2
222
2221
qLgM
KqqLuM
u
u
LMqLM
qLMMM


preparedbyYohannesR.@JiT

19. Manipulator dynamic model
19
 As stated by Angeles et al., 1989, Kane’s equations sometimes
referred to as D’Alambert’s equations in Lagrangian form which
are powerful in robot manipulator dynamics
preparedbyYohannesR.@JiT

20. 20
Problem 2
This problem is useful in that it shows how auxiliary generalized speeds can be introduced
to bring constraint forces and torques into evidence. In this case we will introduce u3
to find an expression for Tc (constraint torque about ) The joints at O and P are revolute.
Body A and B are uniform rods with length 4L and 2L respectively. Body A has two times
the mass of body B.
1
ˆa
preparedbyYohannesR.@JiT

21. 21
Step 1) Choose important points: Center of Mass of bodies A and B, and point P.
Step 2) Select generalized coordinates as shown in the figure (plus auxiliary generalized
coordinate u3) and generate velocity and acceleration expressions for the important points.
The prime ( *) in the equations below indicates that the specified quantities contain the auxiliary
generalized coordinate.
Body A
2113
ˆˆ auauAN


)ˆˆ(2 1123 auauLrvv OAANONAN
  

21
ˆauAN



3
2
111
ˆ2ˆ2 aLuauLa AN


 Point P
)ˆˆ(4 1123 auauLrvv OPANONPN



preparedbyYohannesR.@JiT

22. 22
Body B
322113
ˆ)ˆˆ( auauauBAANBN
 

 3212322231221
ˆ)(ˆ)4(ˆ)4( asucuasuuacuuLrvv PBBNPNBN
  

1213221
ˆˆˆ auuauauBAANBAANBN
 
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ˆˆˆ)(ˆ)(ˆˆˆˆ4 acuuacuasusuLasucuuasuacuLauau  
Step 3) Construct a partial velocity table.
preparedbyYohannesR.@JiT

23. 23
0 
kk FF
B
r
N
A
r
N
c
A
r
N
k vnmgaTvnmgF







)ˆ()ˆ()ˆ2( 333 
)8( 1211 cssmgLF 
212 cmgLsF 
cTcmgLcF  213
Step 4) apply the concept of virtual work
A
r
NANA
A
ANA
A
AN
A
r
NAN
k IIvamF






 







)2(
B
r
NBNB
B
BNB
B
AN
B
r
NBN
IIvam






 







)(
22
2
11
22
2
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3
4
ˆˆ
3
4
000
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3
4
0
00
3
4
2 aaLaaLL
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mI A
A

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

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




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
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


33
2
11
2
2
2
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3
1ˆˆ
3
1
3
1
00
000
00
3
1
bbLbbL
L
L
mI B
B





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
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








preparedbyYohannesR.@JiT

24. 24
)(
3
)8(
3
8
2221
2
21
2
1
2
1
2
1 csuusu
mL
umLu
mL
F 













 )2())((4)( 212212
2
2
2
12221
2
sucuusuuscuumL  
 )4())(()(
3
2
222
2
122
2
2
2
12221
22
1222
2
2 usuuscuuscuumLucsu
mL
F 







 )2()4(4)2(
3
212212
2
222
2
1
22
221221
2
3 sucuucusuumLcuucsu
mL
F  























































2
2
2
21
2
2
2
12
2
1
22
221
2
2
2
2
12
2
21
2
2
2
122
2
22
2
1
2
1212122
2
2
2
12
2
2221
2
2
1
2
2
2
2
22
2
2
2
2
2
2
22
2
2
2
2
16216
3
2
)4()(
3
)8()2)(4(
3
14
3
0
3
04
3
4
12
3
8
umLuucmLcmglcumLcuu
mL
uusmLcmglsuucsmLcsu
mL
cssmgluucsuusmLcsuu
mL
T
u
u
smLsmLcs
mL
mL
mL
cmL
cmLs
mL
mL
mL
c


Step 5) Assemble the equation and form matrix for final solution unknown variables
preparedbyYohannesR.@JiT

25. 25
preparedbyYohannesR.@JiT
Velocity propagation to find unknown linear and angular velocities of 2R
Problem 3

26. 26
preparedbyYohannesR.@JiT

27. 27
preparedbyYohannesR.@JiT
or

28. 28
preparedbyYohannesR.@JiT

29. 29
preparedbyYohannesR.@JiT

30. 30
preparedbyYohannesR.@JiT

31. 31
10Q 4 ur attention
preparedbyYohannesR.@JiT