Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 3, pp. 793-801, Warsaw 2014

A CASE STUDY OF INVERSE DYNAMICS CONTROL

OF MANIPULATORS WITH PASSIVE JOINTS

Wojciech Blajer, Krzysztof Kołodziejczyk

University of Technology and Humanities in Radom, Faculty of Mechanical Engineering, Radom, Poland

e-mail: w.blajer@uthrad.pl; k.kolodziejczyk@uthrad.pl

Manipulators with both active and passive joints are examples of underactuated systems,
featured by less control inputs than degrees of freedom. Due to the underactuation, in the
trajectory tracking (servo-constraint) problem, the feed-forward control obtained from an
inverse model is influenced by internal dynamics of the system, leading to a more involved
control design than in the fully actuated case. It is demonstrated that a convenient ap-
proach to the problem solution is to formulate the underactuated system dynamics in the
input-output normal form, with the arising governing equations formulated either as ODEs
(ordinary differential equations) or DAEs (differential-algebraic equations). The interrela-
tionship between the inverse dynamics control and the associated internal dynamics is then
studied and illustrated using a planar manipulator with two active and one passive joint.
Some simulation results for the sample case study are reported.

Keywords: inverse dynamics, underactuatedmanipulators, passive joints, servo-constraints

1. Introduction

Most typically,manipulators aredesignedandmodeled as fullyactuated, inwhich thenumber m
of control inputs equals the number f of degrees of freedom, m= f. Given a (fully) prescribed
motion of the system, the inverse dynamics simulation allows for determination of the desired
control, which, combinedwith feedback linearization, is used in the computed torque controllers
(Paul, 1981).The situation isdifferent formanipulatorswith elastic/passive joints and/orflexible
members (Spong, 1987; De Luca and Oriolo, 2002; Benosman and Le Vey, 2004), where the
relatedunactuateddegrees of freedomresult inunderactuationof the system, m<f (less control
inputs than degrees of freedom). A desired motion of such systems can be partly prescribed by
m outputs,whose number coincideswith the number of inputs.Themotion specifications lead to
servo-constraints on the system (Kirgetov, 1967; Bajodah et al., 2005; Blajer andKołodziejczyk,
2007, 2008) with the system input control forces referred to as reactions of the constraints. The
servo-constraint problem is then a specific inverse dynamics simulation problem in which the
input control strategy that forces an underactuated system to complete the partly specified
motion is to be determined.

The servo-constraint problem analyzed in the previous works (Blajer and Kołodziejczyk,
2007, 2008) addressed mainly the so-called differentially flat problems (Fliess et al., 1995), in
which all system states and control inputs can be expressed algebraically in terms of outputs
and their time derivatives up to a certain order, with no internal/unspecified dynamics left in
the system. The differential flatness for underactuated systems is also consequent to tangen-
tial or mixed orthogonal-tangential realization of the related servo-constraints. By contrast, the
servo-constraint problem analyzed in this paper, for manipulators with both active (actuated)
and passive (unactuated) joints, is a non-flat problem, and orthogonal realization of the related
servo-constraints is observed. The orthogonal realization of servo-constraints means that the
inverse dynamics control can be determined using an input-output normal form, which is then



794 W. Blajer, K. Kołodziejczyk

associated with the internal dynamics. Due to amutual relationship between the inverse control
and internal dynamics, the problem becomes a dynamic problem, and stability of the internal
dynamical is of crucial importance. As shown by Seifried (2012a), the stability can be achie-
ved by a suitable manipulator design or, as motivated hereafter, by appropriate imposition of
motion specifications. All these issues are addressed in the present paper, with a uniform formu-
lation of the governing equations which arise either as ordinary differential equations (ODEs)
or differential-algebraic equations (DAEs). The subject-specific stability problems are then illu-
strated using a planar manipulator with two (first) active and one (last) passive joints, f = 3
and m=2. Some simulation results for the sample case study are also reported.

2. Formulation of the problem

Let us consider an f-degree-of-freedom underactuated mechanical system, described by confi-
guration coordinates q= [q1, . . . ,qf]

T, and controlled by m inputs u= [u1, . . . ,um]
T, m<f.

The nonlinear dynamic equations of the system are

M(q)q̈+k(q, q̇)=g(q, q̇)+B(q)u (2.1)

where M is the f ×f generalized mass matrix, k is the f-vector of dynamic terms, g is the
f-vector of applied forces, and B is the f ×m matrix of maximal column rank that projects
the control inputs u on the directions of q. The m system outputs y = [y1, . . . ,ym]

T can be
expressed in termsof q,with the following relationships at theposition, velocity andacceleration
levels

y=Φ(q) ẏ=H(q)q̇ ÿ=H(q)q̈+h(q, q̇) (2.2)

where Φ = [Φ1, . . . ,Φm]
T are appropriately differentiable functions of q, the m× f Jacobian

matrix H= ∂Φ/∂q is of maximal row rank, and the m-vector h= Ḣq̇ involves accelerations
induced by the output relationship of Eq. (2.2)1.
A desired/specified motion of an underactuated system can be given in time as y = ỹ(t).

The relationships of Eq. (2.2) lead then to servo-constraints on the system with the following
equations at the position, velocity and acceleration levels

ϕ(q, t) =Φ− ỹ=0 γ(q̇,q, t)=Hq̇− ˙̃y=0

η(q̈, q̇,q, t)=Hq̈+h− ¨̃y=0
(2.3)

The servo-constraints introduced in Eq. (2.3)1 aremathematically equivalent to passive (or con-
tact) constraints on the system, ϕ(q, t) = 0. By analogy to the constrained system dynamics,
the generalized actuating force gu = Bu can be viewed as a generalized reaction force of the
servo-constraints. There is, however, a substantial difference between the passive-constraint re-
alization and the servo-constraint realization (Blajer and Kołodziejczyk, 2007, 2008). Whereas
the generalized reaction of (ideal) passive constraints is by principle orthogonal to the passive
constraint manifold, the generalized actuating force can be arbitraily oriented with respect to
the servo-constraint manifold (it may also be tangent to the manifold). As it will be seen fur-
ther, in the case of manipulators with passive joints, there is typically an explicit input-output
relationship (the system outputs y are directly actuated by the inputs u), which indicates an
orthogonal realization of servo-constraints, and is conditioned upon the maximal rank of the
m×m matrix Y = HM−1B, rank(Y) = m and det(Y) 6= 0; see also Blajer and Seifried
(2012), Seifried (2012a,b), and Seifried and Blajer (2013), This is not, however, an ideal ortho-
gonal realization, but a non-ideal orthogonal one, which denotes that the control required for
the servo-constraint realization has an influence also on the internal (unspecified) dynamics of
the system.



A case study of inverse dynamics control of manipulators... 795

3. Governing equations

The servo-constraint problemcanbe formulated in differentways; see e.g. Spong (1998), Fantoni
and Lozano (2002), Bajodah et al. (2005), Blajer and Kołodziejczyk (2007, 2008), and Seifried
(2012b) for some of the possibilities. A convenient setting can be achieved by applying a coor-
dinate transformation, from the configuration coordinates q to a new set of coordinates that
include the system outputs y. Motivated by a similar approach presented by e.g. Spong (1998)
andSeifried (2012a,b), the f generalized coordinates q arefirst partitioned into m actuated and
f −m unactuated coordinates, written symbolically as q= [qTa q

T
u ]
T, conditioned upon the

m×mmatrix Ba, which gathers the rows of B related to qa, is of maximal rank, det(Ba) 6=0.
The partitioned forms of the output relationships introduced in Eqs. (2.2)2,3 are then

ẏ=Haq̇a+Huq̇u ÿ=Haq̈a+Huq̈u+h (3.1)

where H= [Ha
... Hu ], and Ha,Hu are of dimensions, respectively, m×m and m× (f−m).

The new set of coordinates is q′ = [yT qTu ]
T, where, compared to q = [qTa q

T
u ]
T, the

outputs y are introduced instead of qa. At the acceleration level, the transformation formula is

q̈
′ =

[
ÿ

q̈u

]
=

[
Hq̈

q̈u

]
+

[
h

0

]
=

[
Ha Hu
0 I

][
q̈a
q̈u

]
+

[
h

0

]
=H′(q)q̈+h′(q, q̇) (3.2)

where H′ is the f×f matrix of transformation whose inverse is conditioned upon detHa 6=0.
A simple way to express the underactuated system dynamics in the new coordinate set, directly
in the resolved form, is just to substitute q̈=M−1(g−k+Bu), obtained fromEq. (2.1), into
Eq. (3.2), which gives

ÿ=HM−1(g−k)+h+HM−1Bu

q̈u =
[
0
... I
]
M−1(g−k)+

[
0
... I
]
M−1Bu




⇐⇒
{
ÿ= fy(q, q̇)+Y(q)u

q̈u = fq(q, q̇)+Q(q)u
(3.3)

where fy = HM
−1(g−k)+h and fq = [0

... I ]M−1(g−k) are m and (f −m) dimensional
vectors, respectively, Y = HM−1B is the m×m matrix introduced above, and the matrix

Q = [0
... I ]M−1B is of dimension (f −m)×m. For the case of orthogonal realization of

servo-constraints, detY 6=0, the first relationship of Eq. (3.3) explicitly relates u and ÿ. The
feedforwardcontrol u(t) required for exact reproductionof ¨̃y(t) can theneffectively becomputed
as

u(t,q, q̇)=Y−1(¨̃y− fy) (3.4)

A feedback-enhanced controller can be designed by applying ÿstab = ¨̃y+α( ˙̃y− ẏ)+β(ỹ−y)
instead of ¨̃y, where α and β are the diagonal matrices of gain values, and y and ẏ are
respectively the actual output and its time-derivative. A scheme of manipulator control, based
on explicit output-input relationship (3.4), is illustrated in Fig. 1.
The symbolic representation ofEq. (3.3) shows that the servo-constraint problemathandcan

be viewed as an (algebraic) input-output normal form of Eq. (3.3)1, resulting in the feedforward
control law of Eq. (3.4) and anODE representation for the internal dynamics, Eq. (3.3)2. Since
the control required for the orthogonal realization of servo-constraints and the internal dynamics
evolution aremutually dependent, of critical importance for the control strategy is therefore that
the internal dynamics remains bounded. The case of orthogonal realization of servo-constraints
for an underactuated system is therefore not trivial; see also Seifried (2012a) and Seifried and
Blajer (2013), where the problem is given a more detailed consideration.



796 W. Blajer, K. Kołodziejczyk

Fig. 1. Structure of control with inner and outer loops, based on explicit output-input relationship (3.4)

The dynamic equations in the form of Eq. (3.3) can also serve as a basis for a DAE set
of the governing equations. Since the dynamic equations that govern q̈′ = [ÿT q̈Tu ]

T depend
on the states q and q̇, the servo-constraint equations at the position and velocity level must
be appended, and the involvement of the servo-constraint equations at the acceleration level is
equivalent to using ÿ = ¨̃y(t) in Eq. (3.3)2. The arising set of 2f +m DAEs in q, v and u is
the following

q̇u =vu

v̇u = fq(q, q̇)+Q(q)u

0= fy(q, q̇)+Y(q)u− ¨̃y(t)

0=H(q)q̇− ˙̃y(t)

0=Φ(q)− ỹ(t)





⇐⇒





q̇u =vu

v̇u =b(q, q̇,u)

0=η(q, q̇,u, t)

0=γ(q, q̇, t)

0=ϕ(q, t)

(3.5)

The design of control of an underactuated system in the prescribed trajectory motion can be
based on the solution to theDAEs. A simple numerical code for this solution, previously imple-
mented byBlajer andKołodziejczyk (2007), can be based on the Euler backward differentiation
scheme in which the time derivatives of the variables are approximated with their backward
differences with respect to the integration time step ∆t. Given qn and vn at time tn (un are
not required), the values qn+1,vn+1 and un+1 at time tn+1 = tn+∆t are obtained as a solution
to the following nonlinear equations

(qu)n+1− (qu)n−∆t(vu)n+1 =0

(vu)n+1− (vu)n−∆tb(qn+1,vn+1,un+1)=0

η(qn+1,vn+1,un+1, tn+1)=0

γ(qn+1,vn+1, tn+1)=0

ϕ(qn+1, tn+1)=0

(3.6)

In this way, the solutions are advanced from tn to tn+1, and the rough computational scheme
is of acceptable accuracy only for on appropriately small ∆t. The accuracy can be improved
by applying higher-order backward difference approximations or other specialized DAE solvers
(Ascher and Petzold, 1998). A scheme of manipulator control, based on the DAE formulation,
is illustrated in Fig. 2.

Fig. 2. Structure of control based on DAE formulation (3.5)



A case study of inverse dynamics control of manipulators... 797

4. Case study illustration

Consider a 3-degree-of-freedom planarmanipulator seen in Fig. 3a, with the actuated joints O1
and O2, and the passive joint Awith a spring-damper combination. The vanishing torque of the
torsional spring is achieved for θ3 = θ2, and the armmoves in the horizontal plane perpendicular
to the direction of gravity. Defined q = [θ1,θ2,θ3]

T and u = [τ1,τ2]
T, the components of the

arm dynamic equations introduced in Eq. (2.1) are defined by

M1,1 = JC1+m1s
2
1+(m2+m3+m4)l

2
1

M1,2 =M2,1 = [m2s2+(m3+m4)l2]l1cos(θ2−θ1)

M1,3 =M3,1 =(m3s3+m4l3)l1cos(θ3−θ1)

M2,2 = JC2+m2s
2
2+(m3+m4)l

2
2

M2,3 =M3,2 =(m3s3+m4l3)l2cos(θ3−θ2)

M3,3 = JC3+m3s
2
3+m4l

2
3

(4.1)

and

k=



−[m2s2+(m3+m4)l2]l1θ̇

2
2 sin(θ2−θ1)− (m3s3+m4l3)l1θ̇

2
3 sin(θ3−θ1)

[m2s2+(m3+m4)l2]l1θ̇
2
1 sin(θ2−θ1)− (m3s3+m4l3)l2θ̇

2
3 sin(θ3−θ2)

(m3s3+m4l3)l1θ̇
2
1 sin(θ3−θ1)+(m3s3+m4l3)l2θ̇

2
2 sin(θ3−θ2)




g=




0

c(θ3−θ2)+d(θ̇3− θ̇2)

−c(θ3−θ2)−d(θ̇3− θ̇2)


 B=



1 −1
0 1
0 0




(4.2)

where mi, JCi, li and si (i = 1,2,3) are the respective link masses, central mass moments of
inertia, lengths, and locations of the mass center (distances O1C1, O2C2 and AC3), m4 is an
additional mass at the top of link 3, and c and d are the spring and damper constants.

Fig. 3. A planar manipulator armwith two active and one passive joint: (a) the model;
(b) illustration of the assigned task for θ3 = θ2

The desired task is to move the end point E along a circle of radius r, from the point B0
to point Bf on the circle, illustrated in Fig. 3b for the case θ3 = θ2. The maneuver can be
accomplished by imposing the following time-specified function of the angle ϕ (defined in the
figure), used previously in Blajer and Kołodziejczyk (2007)

ϕ̃(t)=ϕ0+
[
126
( t
T

)5
−420

( t
T

)6
+540

( t
T

)7
−315

( t
T

)8
+70
( t
T

)9]
(ϕf −ϕ0) (4.3)

where ϕ0 = ϕ̃(0) and ϕf = ϕ̃(T) are the initial and final values of ϕ, and T is the maneuver
duration. This is a rest-to-rest maneuver assuring the steady state conditions ˙̃ϕ(0)= ˙̃ϕ(T)= 0,



798 W. Blajer, K. Kołodziejczyk

¨̃ϕ(0)= ¨̃ϕ(T)= 0,
...
ϕ̃(0)=

...
ϕ̃(T)= 0 and ϕ̃(4)(0)=

...
ϕ̃
(4)
(T)= 0. In further simulations, we used

then: ϕ0 =0
◦, ϕf =180

◦, l1 =0.8m, l2 = l3 =0.6m, r=1.2m, b=0.2m and h=0.4m.

For the above task, there is a natural tendency to assign the outputs to the coordinates of
the point E, ỹ(t) = [x̃E(t), ỹE(t)]

T, with x̃E(t) = b+ rcos(ϕ̃(t)) and ỹE(t) = h+ r sin(ϕ̃(t)).
This output allocationmay be improper, however, resulting in an unbounded internal dynamics
(Seifried, 2012a,b). The reason for the inappropriateness of the end-effector output assignment
can be illustrated with a little simpler example – a rotational arm with one active and one
passive joint, seen in Fig. 4a. Let us assume that the arm is in rest in the position seen, and
the task is to move the end point E in the y direction, yielding ¨̃yE > 0. Intuitively, one can
expect a positive (counterclockwise) sense of τ required to initiate the upward motion of the
point E. Thismay not be the case, however.More strictly, the effect of τ on link 2 is an upward
reaction R in the joint A (Fig. 4b), whichmoved to the linkmass center C2 is associated with
a couple with a clockwise torque T =Rs2. Assumed link 2 is homogeneous, i.e. s2 = l2/2 and
JC2 =m2l

2
2/12, the acceleration of the point E is then

ÿE =
R

m2
−
Rs2
JC2
(l2−s2)=

R

m2
−
Rl2
2

12

m2l
2
2

l2
2
=−
2R

m2
(4.4)

The action of a “positive” τ has thus the opposite effect from what is expected – a “negati-
ve” ¨̃yE is produced. A “positive”

¨̃yE will thus require a “negative” τ. This “reverse” control is
certainly wrong, and, when continued, will soon lead to collapse of the servo-constraint problem
execution. As shown by Seifried (2012a), the inverse simulation control inappropriateness can
be overcome by modifying (optimizing) the system parameters, resulting specifically in redu-
cing s2 and enlarging JC2 so that the same senses in the input-output relationship are achieved.
Another possibility is reformulation of the desired output ỹ(t), applied in the following for the
present case study.

Fig. 4. A rotational armwith one active and one passive joint: (a) the model; (b) illustration of the
action of control torque τ on link 2

The described inappropriateness in the output-input inverse model, which relates also the
present case study, can be overcome by assigning the outputs to the coordinates of the inner
point P instead of the end point E (Fig. 3a), y = [xP ,yP ]

T, and a reasonable choice used in
the forthcoming simulations is P →C3 (sP = l3/2). The output relationships of Eq. (2.2) are
then defined by

y=

[
xP
yP

]
=

[
l1cosθ1+ l2cosθ2+sP cosθ3
l1 sinθ1+ l2 sinθ2+sP sinθ3

]

H=

[
−l1 sinθ1 −l2 sinθ2 −sP sinθ3
l1cosθ1 l2cosθ2 sP cosθ3

]

h=

[
−l1θ̇

2
1 cosθ1− l2θ̇

2
2 cosθ2−sP θ̇

2
3 cosθ3

−l1θ̇
2
1 sinθ1− l2θ̇

2
2 sinθ2−sP θ̇

2
3 sinθ3

]

(4.5)



A case study of inverse dynamics control of manipulators... 799

The reference trajectories ỹ(t) = [x̃P , ỹP ]
T → ˙̃y(t) = [˙̃xP , ˙̃yP ]

T → ¨̃y(t) = [¨̃xP , ¨̃yP ]
T have been

determined from the manipulator inverse kinematics for θ3 = θ2, illustrated in Fig. 5 together
with the end point E trajectories. The assumedmaneuver duration was T =5s.

Fig. 5. The reference trajectories for points P (black lines) and E (grey lines)

The results of inverse dynamics simulation for the manipulator arm tracking the specified
motion of the point P are illustrated in Fig. 6. In addition to the abovementioned manipulator
geometrical parameters and the task specifications, the inertial parameters used in calculations
were: m1 = 8kg, m2 = m3 = 6kg, m4 = 10kg, JC1 = 0.45kgm

2, JC2 = JC3 = 0.2kgm
2,

and the stiffness and damping coefficients were c = 10Mm/rad and d = 2Mms/rad. The
integration time step was ∆t = 0.001s. As seen from the graphs, the internal dynamics is
bounded (stabilized), and, due to the assumed damping in the passive joint, the manipulator
quickly achieves the final rest position for t > T = 5s. Since the specification of the point
P →C3, ỹ(t) = [x̃P(t), ỹP(t)]

T, the motion of the end point E is affected also by the internal
motion evolution. The resulted trajectory and velocity components of the point E are shown in
Fig. 7 (black lines), compared to those obtained for θ3 = θ2 (grey lines).

Fig. 6. Simulation of the manipulator motion and control in the prescribedmotion



800 W. Blajer, K. Kołodziejczyk

Fig. 7. Motion of the end point E

5. Summary and conclusions

Manipulatorswithpassive joints, studied in this paper, fall into the category of non-flat underac-
tuated systems. A non-ideal orthogonal realization of servo-constraints imposed on the systems
is observed, denoting that the inputs can directly regulate the motion specifications. However,
besides the output-input inverse dynamics model, exemplified in Eqs. (3.3)1, (3.4) and (3.5)3,
an additional internal dynamics arises. The servo-constraint problem becomes thus a dynamical
model, with the internal dynamics which may be bounded (stable) or unbounded (unstable).
Meaningful solutions to the servo-constraint problem require therefore a careful analysis related
both to the design of the underactuatedmanipulator and to the imposedmotion specifications.

For the case study analyzed in the paper, choosing the end-effector position coordinates as
outputs, and then specifying them in time to formulate servo-constraints on the system, is not
a good practice.While the orthogonal realization of the servo-constraints is maintained and the
inverse dynamics control can explicitly be determined, the assisted internal dynamics (affec-
ted by the required control) is unbounded, and the solution leads to collapse. It was shown by
Seifried (2012a) that the situation can be improved by modifying/optimizing the design para-
meters of the underactuated system, usually the geometric dimensions and mass distribution.
The other possibility, motivated in this contribution, is to choose some appropriate outputs.
Regulating some inner point position, instead of the end-effector position, is recommended. The
servo-constraint problemexecution becomes then stablewithout anymodification to the original
system design. The sacrifice is that, whereas the exact tracking of the inner point (specified)
trajectory is assured, the end-effector motion is also affected by the internal dynamics. The
end-effector trajectory tracking is then realized with a limited accuracy. The situation can be
improved by enlarging (or additionally adding) the damping in the system, and, evidently, for
large stiffness in the passive joint the solution becomes similar to that obtained for a “rigid”
manipulator (with the passive joint jammed). Finally, itmay beworth noting that the stabiliza-
tion of the servo-constraint problem execution with limited accuracy was also discussed in e.g.
Seifried (2011,b) and Kovács and Bencsik (2012), where a linearly combined output instead of
the end-effector position was used as the reference trajectory, which is somewhat equivalent to
the regulating of the inner point position used in the present paper.

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Manuscript received January 7, 2014; accepted for print February 27, 2014