computation of robot inertia matrix using block diagram approach - Fei

COMPUTATION OF ROBOT INERTIA MATRIX
USING BLOCK DIAGRAM APPROACH
Narpat S. Gehlot
Universidade Federal da Paraíba - Departamento de Engenharia Elétrica
58109-970 Campina Grande-PB-Brasil
E-Mail: gehlot@dee.ufpb.br
Pablo J. Alsina
Universidade Estadual da Paraíba - CCT - Departamento de Física
58100 Campina Grande - PB - Brasil
Abstract: In this paper, two new methods for computation of the inertia matrix of robot
manipulators are proposed. The new algorithms are based on a block diagram of
manipulator dynamics, derived from the Newton-Euler formulation. The proposed
methods allows to compute the inertia matrix explicitly, in an efficient recursive manner,
and can be applied for robot dynamics simulation.
r. INTRODUCTION
The inverse dynamics formulation is a c1assic
problem in robotics. The inverse dynamics equation
allows to compute the vector of input generalized
torques as a function of the N joint positions and its
derivatives,
where,
't = Nx I vector ofgeneralized torques.
q = Nx I vector ofjoint positions.
M(q) = NxN inertia matrix.
't m = Nx I vector ofinertial torques.
'to = Nx I vector of coriolis, centrifugai,
gravitational and disturbance torques.
The solution of Eq. (I) is a typical control problem :
finding the adequate vector r needed to impose the
.desired joint trajectory q. In fact, the computed
torque control technique (Yoshikawa , 1990).
utilizes Eq. (I) to obtain the vector of generalized
torques r from the measured values of position and
velocity and computed values of acceleration. The
447
Newton-Euler technique (Luh, 1980) allows to
compute the inverse dynamics Eq. (1) in an efficient
recursive way. The computational efIort of this
method has a linear dependence with the number N
of degrees offreedom (DOF) ofthe manipulator.
The dynamic simulation of robotic manipulators is a
dual of the control problem. The robot simulation
consists on obtaining the arm position as a function
of time, given the vector of generalized torques and
the initial conditions. This is attained by integrating
the direct dynamics, obtained by inverting Eq. (1),
(2)
It can be noted that the inverse of the inertia matrix
is needed in the evaluation of Eq. (2). For this
reason, the Newton-Euler method cannot be used
directly to compute Eq. (2), since this method only
evaluates numerically the vector of generalized
torques, without computing explicitly the inertia
matrix M(q). In spite of this, ao artifice can be used
to compute the inertia matrix through the Newton-
Euler method, as shown-by Walker (Walker & Orin
I982).Several methods were developed to compute
the forward dynamics (and -the inertia matrix) .
Baharin (Baharin & Green(Í991) proposed a

ecursive Lagrangian formulation based on
differential relationships. Also based on the
Lagrangian mechanics, Li (Li, 1988) proposed a
method for the evaluation of the terms of the
Dynamic Equation. Both above-mentioned
have a computational effort of order 0(N3). More
efficient methods (Lilly & Orin, 1991; Featherstone,
1987) , with computational effort of order O(N2),
have been proposed. Balafoutis (Balafoutis & Patel,
1989) described a method, based on · the Newton-
Euler algorithm and using the concept ofgeneralized
and augmented links, for computing the inertia
matrix with a computational effort of order 0(N2).
Furthermore, an efficient method for computing of
the inverse dynamics and the inertia matrix
rearranging the equations for a minimum set of
dynamics parameters, was proposed by Kawasaki
(Kawasaki, et ai., 1992). This method, oriented to
parameter identification, has a computational effort
of order 0(N2). Alsina (Alsina & Gehlot, 1994)
proposed a formulation method of order 0(N2) based
on the Newton-Euler recursion expressed in term of
spatial vectors. This method allows the explicit
computation of the inertia matrix. In order to
improve the real-time performance of the forward
dynamics computation, several paralleI algorithms
(Fijani & Bejczy, 1989) and architectures have been
proposed. Arnim-Javaheri (Amim-Javaheri & Or in,
1988) described several systolic architectures for the
evaluation of the inertia matrix based on dedicated
VLSI robotic processors. Lee (Lee & Chang, 1988)
proposed several parallel algorithms for real-time
simulation of robot forward dynamics based on
single-instruction multiple-data-stream computer,
with as many processor as the number of DOF of the
manipulator. Beside the methods above, several
forward dynamics computation schemes, with no
need of inertia matrix inversion, were proposed. Ko
(Ko, 1992) reformulated the Newton-Euler equations
in a Kalman Filter form for the solution of the
forward dynarnics. Gogoussis (Gogoussis & Donath
1990) proposed a method for the solution of forward
dynamics using the recursive properties of Newton-
Euler formulation associated with algebraic loops
(solved through analog hardware) for the digital
simulation of robot dynamics, A good comparison
between several methods for inverse and forward
dynamics computation is found in (Zomaya, 1991).
In this paper, two novel methods for computing the
inertia matrix of robot manipulator are proposed.
The methods are based on recursive relationships,
obtained from a block diagram representation of the
dynamics equation. This block diagram is obtained
from the Newton-Euler formulation. In Section 2
the Newton-Euler algorithm in a ôx l-vector form is
summarized. The blqck diagram representation of
manipulator dynamics is derived in Section 3. The
448
proposed recursive methods for computation of the
inertia matrix are described in Section 4. The
computational effort of the proposed methods is
discussed in Section 5. Final conclusions are given
in Section 6
2. NEWTON-EULER FORMULATION
The Newton-Euler formulation allows to compute
the manipulator dynamics in a computationally
efficient recursive manner. This formulation is
realized in two stages: Forward and Backward
recursions. In the Forward recursion, the cinematic
variables are transformed frorn the base of the
manipulator to the end-effector. The Backward
recursion updates forces and torques starting from
the tool to the base, obtain ing the driving torque of
each joint. A 6x I spatial vector version of the
Newton-Euler algorithm is summarized below. The
following notation is adopted:
q i Position variable ofjoint i.
v i 3x I Linear velocity vector of link i.
co i 3x I Angular velocity vector of Iink i.
fi 3x I Force vector applied on link i by
link i-I.
n i 3xl Moment vector applied on link i by
linki-I.
3x3 Rotation matrix frorn link i to i+I.
1
P;+I 3xl Position vector oflink i+1 with
respect to link i.
P c; 3xl Center ofmass position vector of
link i.
m i Total mass of link i.
h i 3x3 Inertia tensor of link i in its own
reference frame.
't ; Driving torque/force oflink i.
Defining the 3x3 and 6x6 matrix operators for cross
product of3xl and 6xl vectors respectively as:
I O -pz 1
I P y I
(px) = I pz O -Px I (3)
l-py Px O J
rr V1 1 I (cox) 2(vx)1
= l2(vx) (ox) J
where p, v and 00 are 3x 1 vectors. Thus, the 6x6
spatial rotation matrix, from link i to link i+ 1, is:
o 1
A:+,J
(4)
(5)

The 6x I vectors of spatial veIocity, spatial angular
velocity and spatial effort, and the 6x6 spatial inertia
matrix oflink i, are respectively defined as:
T
V=v [ 1 1
T]T
Cú;
T] T
Cú;
fi.(p .x)Tl
I c, J
h i
The flag of joint i is defined as: for i = Oto N-I
(6)
(7)
(8)
(9)
--------
3. BLOCK DIAGRAM REPRESENTATION
By imposing zero joint accelerations cL ' (i =
I, ....,N), to(q, q) (vector of corioIis, centrifugai,
gravitational and disturbance torques) can be easily
computed through the Newton-Euler algorithm.
Similarly, the inertial torque t m '(t m = M(q)q ),
can be updated through the Newton-Euler algorithrn
by imposing zero joint velocities, initial spatial
acceleration and initial spatial effort:
- Forward Recursionfor inertial torques:
1; =[0 O O O OIr for Rjoint(lO.a) Y · RiT y' 1 00
rn;+1 = i+lo mi + i+lqi+l (17)
1;=[0 O O O Or forPjoint(IO.b)
where R and P stands for rotational and prismatic
joint respectively. Thus, the Newton-Euler algorithm
is given by:
- Forward Recursion:
v. = gravity acceleration
for i = I to N
(lI)
(13)
end i
- Backward Recursion for inertial torques:
FmN+1 = O
for i = N to I
F rn; = R;+IFrni+1 +H;V rni
t m] = JT·F rni
end i
(18)
(19)
According to these recursions, a block diagram
representation of the . manipulator dynamics is
obtained, as shown in Figure lo
.4. COMPUTATION OF INERTIA MATRIX
end i 4 01 The Inverse Path Method
- Backward Recursion:
F N + 1 = Spatial effort on the tool.·
for i = N to I
F = (n.x)H.n .
cj 1 I I
end i
(14)
(15)
(16)
449
In Figure I, the block diagram inside the dashed line
represents the transfer matrix between joint
accelerations and inertial torques, which is the
inertia matrix. Thus, the inertia matrix represents
the relation between an input vector q and an
output vector t rn ' that is, t rn = M(q).q o •The
following notation is adopted: [t rn , Fm]ij is the
transfer matrix of the path from Frnj to t m] and
[t , V ].. is the transfer matrix of the path from
rn rn IJ
V . to 't . o Thus, from the block diagram ofFigure
rnJ rn, .
I, the following recursive relationships are obtained:

Fig. 1 - Block Diagram: Inverse Dynamics and Inertia Matr ix.
Since the inertia matrix is symmetric, it is sufficient
to compute Equation (20) for j varying from i to N
and Equations (21) and (22) for j varying from N to
i. Thus, based on the above reIationships, a recursive
aIgorithm for the computation of the inertia matrix
is proposed, as shown in Figure 2. Since the transfer
matrices are computed recursiveIy in the direction
from torques to joint accelerations, this algorithm
was named: Inverse Path Method .
4.2 The Central Path Méthod
(20)
(22)
In the block diagram of Figure I, a central path
[Frn ,VrnL can be divided in two parallel paths: one
through matrix H. ' and the other through
[Fm >V rn l+l,i+1' R;+l ' On the other hand, the
transfer matrix [Frn ,qh is composed by matrix
R 1+I serially associated with transfer matrix
[Frn ,q]j+l,i' Thus, t?e following recursions are
obtained:
450
The In verse Path Method:
For i = I to N
[-rrn,Frn1 i =f{
r- .....
Inertia Matrix
If(i < N) Then [-rrn ,FrnL+, = [-rrn,Frn1iR;+1
If(i < N-I) Then
ForG=i+2toN)
Endj
[-r rn ' Vrn 1N = [-r rn ,Frn 1N"HN
M iN = [-rrn,Vrn1w l N
If (i < N) Then
For (k = N-I to i)
End k
End i
Fig. 2 - The Inverse Path Method.

(24)
From the recursions above, a new algorithm, named
the Central Path Method, is derived for computation
ofthe inertia matrix, as shown in Figure 3..
The Central Path Method:
[Fm,Vm]NN =HN
[Fm,q]NN = [Fm, Vm]NN·JN
M NN =
For (i = N-I to I)
. i' iT
[Fm ,VmL =Hj + Ri+I·[Fm,VmLI,i+l·Ri+1
[Fm,q);; = [Fm,Vm]ii ·Ji
M ii =JT ·[Fm,q);;
For G= i to 1)
Endj
End i
[Fm,q]j,i+1 = R1+,·[Fm,q]j+l,i+1
Mj,i+1 = Jy.[Fm,q]j ,i+]
Fig. 3 - The Central Path Method.
5. COMPUTATIONALEFFORT
The proposed methods involve several 1x6 vector /
6x6 matrix products and additions. Noting that the
6x6 spatial inertia and spatial rotation matrices
include some zero terms. Thus, the product of a lx6
vector and a 6x6 spatial inertia matrix involves 24
products and 18 additions. The product of a lx õ
vector and a 6x6 spatial rotation matrix involves 27
products and 21 additions. In this way, both
algorithms for the inertia matrix computation
involve a computational effort· of order 0(N2), as
shown in Figure 4. The computational efforts for the
Inverse Path Method and for the Central Path
Method are respectively summarized in Table 1 and
Table2.
Table 1 -Computational Effort: Inverse Path Method.
No.OfProducts
39W - 42N +27
No. of Additions
33W - 36N + 21
45 1
Table 2-Computational Effort: Central Path Method.
No.OfProducts
(27N 2+621 N-648)/2
No. of Additions
(21N2+531N-552)J2
For a sixjoint manipulator, the lnverse Path Method
involves 1179 products and 993 sums, whi1e the
Central Path Method involves 2025 products and
1695 sums for computation of the inertia matrix. As
can be seen in Figure 4, for more than twe1ve joints
(highly redundant manipulators), the Central Path
Method -becomes more efficient than the lnverse
Path Method. This is because, for high number of
DOF, the N 2 term is predominant, and the Central
Path Method has the smallest N 2 coefficient.
10000
' 5000
15000
No . o f P ro d u c ts
o Inverse Path
• C entra I P a th
O O 1O 20
No . o f J o in ts
NO .ofSums
o Inve rsePath
10000 . C entra I P a th
5000
O O 10 20
No . of Joints
Fig. 4 - Computational Effort.
6. CONCLUSIONS
Two novel methods for computation of the inertia .
matrix of robot manipulators were proposed: the
Inverse Path M éthod and the Central Path Method.
The proposed methods were derived from a set of
recursive relationships obtained from a block
diagram representation of the dynamics equation.
The methods update explicit1y the inertia matrix in
an efficient recursive way, thus allowing to app1y
these formulations to the forward dynamics problem
and the dynamics simulation of robot manipulators.
The computational efforts of the proposed methods
have an 0(N2) dependence. The Inverse Path
Method is more efficient for small number of DOF,
while the Central Path Method is more efficient for
highly redundant manipulators.

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