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End-Link Dynamics of Redundant Robotic Limbs:

The Reaction Null Space Approach

Naoyuki Hara, Yoichi Handa and Dragomir Nenchev

Abstract It is shown that the Reaction Null Space formula-tion a method developed for motion analysis and reactionlessmotion generation of free-floating and flexible-base robots can be used to fully decouple the end-link dynamics of a kine-matically redundant fixed-base robot. Decoupling is achievedthereby via the pseudoinverse of a coupling-inertia submatrix,in quite a different way from the dynamic decoupling achievedvia the inertia-weighted generalized inverse of the manipulatorJacobian and known from the Operational Space formulation.The properties of the new formulation are clarified with the helpof a comparative study on a representative motion/force control

scenario. The simulation results show that the two formulationsdeliver identical results as far as end-link dynamics are con-cerned. The new method has an advantage with regard to jointspace dynamics, though, which becomes apparent especially inthe neighborhood of kinematic singularities, where the inertia-weighted generalized inverse of the manipulator Jacobian isill-behaved.

I. INTRODUCTION

End-link dynamics play an important role in many appli-

cations, including those when contact is established between

the end link and an object. Special methods of control are

required to deal with contact forces then. Extensive studies

have been performed leading to such well-established inter-

action control methods as impedance control or force/motion

control, for example [1][3].

Since the motion of the end link is of prime interest, anotation in terms of end-link coordinates is an appropriate

form for expressing and controlling the motion dynamics

[4][9]:

Me(X)V + Ce(X, V) + Ge(X) = F. (1)

X, V and F denote end link related six-dimensional spatialquantities for position, velocity and force, respectively, Meis a 6 6 inertia matrix, Ce and Ge are the Coriolis andcentrifugal force and the gravity force vectors, respectively,

in spatial coordinates.

The above dynamics representation has the same structure

as the familiar equation of motion in joint coordinates:

Ml() + cl(, ) + gl() = , (2)

where and are n-dimensional coordinate vectors forthe joint angles and joint torques, respectively, Ml is the

n n manipulator inertia matrix, cl and gl are Coriolis andcentrifugal force and gravity force vectors, respectively, in

joint coordinates.

With regard to the dynamics notation (1), the following

points must be clearly stated. First, although written in end-

link coordinates, the equation does not denote the dynamics

of the real end link, but rather the dynamics of the whole

manipulator represented as an articulated body [10]. Second,

the notation is somewhat misleading in the sense that the

spatial force, F, is not an external force, i.e. it is not due toa contact force. If the end link were to come into contact with

the environment, e.g. under an interaction-task scenario, an

additional contact force component, Fc, should be added tothe l.h.s. Third, note that while joint coordinates can always

be regarded as generalized coordinates, end-link coordinates

cannot be because they describe uniquely the states of a

subclass of manipulators only: those with degree of freedom

(DOF) equal to the number of end-link coordinates (i.e. the

case n = 6). Such type of manipulators constitute the class ofnonredundantmanipulators. Hence, it would be inappropriateto directly apply the dynamics notation (1) for kinematically

redundant manipulators.

Further on, even if we were inclined to limit our attention

to nonredundant manipulators, we have to note the additional

difficulties that will arise when end-link coordinates are

adopted as generalized coordinates. The problem becomes

apparent when relating the two dynamics representations

(1) and (2) via the kinematic equation X = f() and itstime derivatives V = J() and V = J() + J(, ),where f() is the kinematic function and J() = f()/denotes the manipulator Jacobian. We can then represent the

map Me(X) in terms of joint coordinates as:

Me() =

(J()M1l ()JT()

1, (3)

i.e. as the inverse of the mobility tensor [2] JM1l J

T. The

problem arises when the manipulator Jacobian becomes rank-

deficient, i.e. at a kinematic singularity. In this case it would

be inappropriate to make use of the dynamics representation

(1). Possible approaches for dealing with problems due to

redundancy and singularity have been proposed [1], [6] but

nevertheless, redundant manipulators and kinematic singular-

ity have still to be addressed as special cases whenever the

end-link coordinate representation of dynamics is employed.

The main aim of this paper is to highlight the existenceof a full end-link dynamical decoupling notation for a

fixed-base serial link kinematically redundant manipulator,

different from the well known Operational Space method [6],

and to discuss its properties. We use a dynamics formulation

based on an extended set of coordinates that includes both

joint coordinates and end-link coordinates. The approach

we take is as follows. First, we assume that the serial

manipulator under consideration is fully unconstrained. The

aim is to take advantage of a unfixed-base system dynamics

representation extensively studied in the past in relation to

2012 IEEE International Conference on Robotics and AutomationRiverCentre, Saint Paul, Minnesota, USAMay 14-18, 2012

978-1-4673-1404-6/12/$31.00 2012 IEEE 299

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free-floating space robots or flexible-base robots [11] and

more specifically, of the full dynamical decoupling property

for base/manipulator-link motion achieved with the help

of the Reaction Null Space [12][17]. We show how to

reformulate the RNS notation w.r.t. end-link/manipulator-link

dynamical decoupling and how to implement it in interaction

control task scenarios. The results of a comparison (via

simulations) with motion/force control under the Operational

Space formulation [6] will be presented to evaluate the

performance of the new scheme.

I I . BACKGROUND AND NOTATION

A. Serial-link Manipulator Dynamics in Operational Space

It is well known that the dynamics of nonredundant and

redundant serial-link manipulators in end-link coordinates

are expressed in different ways [1], [7]. For a nonredundant

manipulator, the equation is obtained in a straightforward

manner by transforming the joint space dynamics (2) with

the help of the inverse of the transposed Jacobian JT()into:

Me()V + Ce(, ) + Ge() = F. (4)

The inertia matrix Me() is given in (3), the Coriolis andcentrifugal force transform, the gravity force and the force

imposed on the end link are:

Ce(, ) = JT()cl(, ) Me()J(),

Ge() = JT()gl(), (5)

F = JT(),

respectively.

In contrast, when dealing with a redundant manipulator,

the transformation is not as straightforward, as noted in

the Introduction. Khatib [6], [7] suggested to apply the

transpose of the inertia-weighted generalized inverse1 of the

manipulator Jacobian:

J#M() = M

1l ()J

T()Me()

as the underlying transform. The equation of motion is

then of the same form (4), but the terms (5) have to be

rewritten, replacing thereby the inverse transpose JT()

withJ#M()

Teverywhere.

As another alternative, the extended task space approach

can be employed [8], wherein the set of end-link coordinates

is extended by including r = n 6 additional variables toparametrize the self motion:

V

VvN

=

J()P()

J(), (6)

where vN r is the self-motion parameter velocity andP() rn is a map from joint space to redundancyparameter space. This matrix is non-unique. Matrix J()is square and assumed to be invertible. The inverse of its

transpose becomes the underlying transform from joint space

to extended task space. Hence, the dimension of the equation

1Referred to also as the dynamically consistent inverse [7].

of motion (4) is increased from 6 to n, whereby the end-link

spatial acceleration V is replaced with V. In addition, alltransposes and inverses of the Jacobian matrix appearing in

(3) and (5) are replaced with the transposes and inverses of

matrix J().Further on, the dynamical models in end-link coordinates

are applied for control design, as follows. First, based on (4),

we obtain:

F Me()V Ce(, ) Ge() = Fc, (7)

where Fc is the contact force measured by a force/torquesensor. With the help of this equation, a reference end-link

force comprising two components can be designed as [6]:

Fref = Frefm + Frefc , (8)

Frefm = Me()SVref + Ce(, ) + Ge(),

Frefc = SFrefc ,

where Frefm and Frefc are two components referring to end-

link motion and contact force, respectively. S is a selection

matrix suitably defined to specify the unconstrained (motion)directions, while S is its complement.

Next, the control joint torque is to be determined. Usually,

quasistatic relations are employed, e.g. = JT()Fref inthe case of a nonredundant manipulator. In the case of a

redundant manipulator, a torque component that parametrizes

self motion must be added. Under the extended task space

formulation, the quasistatic relation becomes:

= JT

()

Fref

fN

(9)

= JT()Fref + PT()fN,

where fN is the force dual to velocity vN. It must be

emphasized, though, that the use of this relation needs special

attention. As pointed out in [1], if the force fN is chosen as a

nonzero force, self motion will be induced and the quasistatic

relation can cease to hold true.

In order to ensure force decoupling, a proper choice of the

pair J#T and PT is essential such that the product J#TPT

remains always identically zero. Khatib [6], [7] proposed the

following generalized force decomposition:

= JTFref +U JTJ#TM

a, (10)

where the second expression on the r.h.s. is a non-minimum

parameterization of self motion via the arbitrary joint torque

vector a projected onto the null space of the trans-posed inertia-weighted generalized inverse. U stands for

the n n unit matrix. It can be easily confirmed that

J#TM

U JTJ#TM

is identically zero, meaning that a

will not impose an additional force on the end link.

B. Dynamics of a Unfixed-Base Serial-Link Limb

Consider a serial-link limb in mid-air composed of n + 1links (see Fig. 1). The two end links will be denoted as Aand B, respectively. The number of generalized coordinatesis n + 6, including the n joint coordinates and the six spatial

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Fig. 1. Model of a unfixed-base serial-link limb. The system constitutes asingle articulated body, C denoting its CoM.

coordinates of one of the end links. The joint coordinates

are considered actuated, while the end-link coordinates are

not. Hence, we have an underactuatedsystem at hand. In the

following derivation we will refer occasionally to the system

as a single articulated body (AB), as defined in [10].

Without loss of generality, we will pick up end-link A asthe link of reference. The equation of motion can then be

written as:

MA MAlMTAl Ml

VA

+CAcl

+GAgl

=FA

+

ABT

T

JT

FB, (11)

where

Ml nn : link inertia matrix

MA 66 : AB inertia matrix

MAl 6n : coupling inertia matrixcl

n : link Coriolis and centrifugal forces

CA 6

: AB Coriolis and centrifugal forcesgl n : link gravity force vector

GA 6 : AB gravity force vector n : joint coordinate vectorVA

6 : spatial velocity of end-link A n : joint torque vectorF()

6 : spatial forceABT

66 : spatial coordinate transform

Note the presence of the external spatial forces FA and FB .One of these forces will be used later for constraining the

motion of the respective end link to obtain a quasi-fixed-base

limb, which will be the subject of the comparative study in

what follows. The upper part of the above equation denotesthe system AB dynamics. The coordinates are those of end

link A, but the inertial properties are those of the entiresystem, hence the term articulated body. The lower part, on

the other hand, describes the dynamics of the limb. Because

end-link coordinates are used, quantities Ml, cl and J are

those of the respective fixed-base limb. Further on, the whole

equation includes components for intercoupled inertial and

nonlinear generalized forces on the l.h.s., and for external

and/or driving forces on the r.h.s.

We focus henceforth on the AB dynamics which can be

written as:

MAVA + MAl + CA = Fqs, (12)

where

Fqs = FA +ABT

TFB GA

denotes the quasistatic force. The AB dynamic equilibrium

can be then expressed as Fd Fqs = 0, where Fd denotesthe AB dynamical force. This force can be represented as

a function of the generalized coordinates q = (XA, ) andtheir time derivatives q = (VA, ) and q = (VA, ):

Fd = MAVA + Fimp(q, q, ), (13)

where XA denotes the spatial displacement of the end-link.Note that we have separated the inertial force MAVA fromthe imposed force Fimp:

Fimp(q, q, ) MAl + CA. (14)

The imposed force is imposed on end-link A through the

motion of all other links; it results from the inertial andnonlinear couplings between the end link and the rest of

the links.

III. THE REACTION NUL L SPACE: FUL L DECOUPLING OF

END-L IN K DYNAMICS

Assume now that the limb is in contact with the envi-

ronment via its end-links and is motionless, i.e. in static

equilibrium. End-link As spatial velocity and accelerationare zero, and hence:

MAl + CA = 0. (15)

In case of a redundant system, the number of joints is largerthan the DOF of end-link A (n > 6). Therefore, the generalsolution for the joint acceleration is obtained from the above

equation as:

= M#AlCA + (U M#AlMAl)a, (16)

where ()# denotes a generalized inverse and the secondterm on the r.h.s. is a vector from the kernel of the coupling

inertia matrix, a denoting an arbitrary vector with dimen-

sion of joint acceleration. The kernel is referred to as the

RNS w.r.t. end-link A. The last equation shows that there isa set of joint accelerations { : Fimp(t) = 0, t} that wouldguarantee a null imposed forcethroughout the entire motion.

Hence, the initial static equilibrium of end-link A will notbe disturbed, and the end links balance will be maintained

despite the motion in the joints. Any acceleration from the

above set can be therefore characterized as reactionless joint

acceleration w.r.t. the state of end-link A.Three remarks are due. First, the above reactionless joint

acceleration implies full dynamic decoupling of end-link Afrom the rest of the links. This is achieved in a straight-

forward manner via the above RNS decomposition, thus

avoiding the necessity to look for an appropriate pair of

matrices, as under the extended task formulation.

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Second, if the pseudoinverse is adopted as the generalized

inverse in (16), local optimality can be achieved. The opti-

mized quantity is part of the total kinetic energy. It can be

related to the dynamical coupling between end-link A andthe rest of the links, and expressed as 1

2(MAl)TMAl.

Third, at nondegenerate configurations where the coupling

inertia matrix MAl is full rank, limb redundancy implies

the existence of an (n 6)dimensional RNS. We shouldnote, however, that in motion/force control scenarios, it is

usually assumed that end-link A contacts the environment,being thereby constrained along k < 6 directions. Hence,decoupling from the rest 6 k motion-controlled directionscan be achieved with a higher, (nk)dimensional RNS (seee.g. the notation on the selective RNS in [16]). A higher-

dimensional null space may lead to certain advantages, a

detailed discussion on this problem goes however beyond

the scope of the present work. Below we will consider only

the case of an (n 6)dimensional RNS.

IV. APPLICATION TO MOTION/FORCE CONTROL

Here we outline how the proposed formulation could be

applied to motion/force control. The derivations are based on

the theory presented in [3]; the details are omitted thereby

because the problem is somewhat out of the main scope of

the paper and also, due to lack of space.

First, to apply the above floating-chain formulation, we

have to derive a quasi-fixed-base limb by constraining appro-

priately one of the end links. End-linkB is a good candidatesince its dynamics do not appear explicitly in the equation

of motion (11). We attach a very stiff spatial spring to this

end link. The constraint force FB can then be calculated viaHooks law, assuming that a sensor is available to measure

the resulting infinitesimal spatial displacement.

Next, we assume that end-link A contacts the environmentbeing thereby constrained along k < 6 directions. Thiscan be expressed via the equation Jc(q)q = 0, Jc(q) k(6+n) denoting the constraint Jacobian containing partialderivatives related to the environment constraint c(q) = 0.End-link As spatial force is then FA = J

TcA, where

k is the Lagrange multiplier for the forces of constraintand JcA

k6 is a submatrix of the constraint Jacobian

s.t. JcAVA = 0. Further on, we denote VA Sv where 6k is end-link As velocity along the unconstraineddirections and Sv 6(6k) is defined from S

TfSv = 0,

Sf JTcA.

Next, we design the control joint acceleration via the upper

part of (11) and (16) as:

ref

= ref

A + RNS, (17)

where

ref

A = M+Al

Sff MASvv MASv

+ M+Al

ABT

TFB CA GA

,(18)

RNS = (U M+AlMAl)a. (19)

A is the particular component of the general solution; it is

used to realize the motion/force control task described with

control quantities v and f, for motion and force, respec-

tively, within the RNS full dynamic decoupling framework.

Note that the pseudoinverse M+Al has been adopted here

because of its desirable local optimization property. Note also

that the second term on the r.h.s of (18) that compensates

quasi-static and nonlinear forces, contains a component for

base reaction compensation a property we consider to be

quite important in unfixed-base applications. The null space

component RNS, on the other hand, can be used to realize a

desirable motion/force strategy within the RNS via a suitably

chosen a.

It is straightforward to show that the above control ac-

celeration ensures complete decoupling between the force

and velocity controlled subspaces, resulting in = v and = f. In addition, complete decoupling between end-link dynamics and self motion can be achieved as well, due

to the orthogonality of the two control joint acceleration

components.

Finally, the control torque can be obtained by inserting

the control acceleration ref

from (17) into the lower part

of (11):

= A + MlRNS, (20)

where

A = MTAlSvv + Ml

ref

A + cl + gl JTFB (21)

=MTAl MlM

+AlMA

Svv + MlM

+AlSff

+MlM

+Al

ABT

T JT

FB + cl + gl

MlM+Al

CA + GA + MASv

.

As shown in [3], for a decoupled system as above, control

laws for v and f do exist which would ensure asymptotictracking of the desired end-link velocity and acceleration,

with exponential convergence, and in addition, exponential

asymptotic stability with regard to a desired des, both for

rigid and for compliant environments, and even in the pres-

ence of uncertain stiffness/compliance in the environment.

Below we introduce a simple instance of such control laws

to be used in the following comparative study.

V. A COMPARATIVE STUDY

A comparison between the newly introduced formulation

based on the RNS decoupling and the Operational Space

formulation [6], [7] will be performed herein.

A. Models

Two similar planar five-link four R-joint limb models will

be employed, as shown in Fig. 2. The quasi-fixed-base limb

model (Fig. 2a) will be used with the RNS formulation, while

the fixed-base limb model (Fig. 2b) with the Operational

Space formulation. Note that in the former model, end-linkBis the base while end-linkA is the end-effector. To obtain twodynamically equivalent limbs, in the latter model the links

have to be arranged in a reverse order, all link parameters

being identical: link masses, lengths and CoM positions are

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1

2

3

m1

m2m3

a1

l1

a2

l2

a3l3

x

z

: passive joint

A

B

kBzdBz

12

3

m3

m2

m1

a3

l3

a2l2

a1

l1

x

z

: active joint

A

B

kBzdBz

kAzdAz

dBx

kBxfixed basequasi-fixed base

(a) (b)

Fig. 2. Planar five-link four R-joint limb models used in: (a) RNSformulation and (b) Operational Space formulation. The passive jointsappear since the end-link orientation is ignored, thus obtaining equivalent3R redundant limbs.

set as mj = 10 kg, lj = 1.0 m, aj = 0.5 m (j = 1, 2, 3),respectively. Note that the end-links are massless virtual

links. Note also that in each model, there is one passive jointat one of the end-links such that the respective rotational

coordinate can be ignored. Thus, we obtain two dynamically

equivalent 3R redundant limbs, with n = 3 and two end-linkcoordinates (for translational motion only). Further on, for

simplicity and without loss of generality, we will ignore the

presence of gravity.

End-link B in Fig. 2a is the quasi-fixed base. It is con-nected to the environment via two very stiff spring/damper

systems. The two end-effectors (A in Fig. 2a and Bin Fig. 2b) move along a flat horizontal surface. Hence,

the unconstrained motion direction is along the x axis,while the force-controlled direction is along the z axis. A

spring/damper system is attached in the latter direction inorder to model positioning/force errors.

B. Controllers

The performance of the following two motion/force con-

trollers will be examined: the RNS formulation based con-

troller, or RNS-C for short, (21), and the Operational Space

formulation based controller, or OSF-C for short, ((10)

together with (8)). The motion/force feedback laws for these

controllers are:

vrefi = vdesi + kd(v

desi vi) + kp(x

desi xi), (22)

frefi = fdesi + kf(f

desi fi), (23)

where kd, kp and kf are feedback gains used in bothcontrollers. In the case of RNS-C (i = A), vrefA andfrefA are to be used in (21) in place of Svv and Sff,respectively. In the case of OSF-C (i = B), on the otherhand, vrefB and f

refB should replace the reference acceleration

and the reference contact force appearing in (8) as SVref

and SFrefc , respectively.

C. Simulations

Two simulations will be performed. The initial configu-

rations are set as = (75, 60, 60) deg for RNS-C

and = (15, 60, 60) deg for OSF-C, respectively. Theseare shown in Fig. 2. Sampling and simulation times are

set to 1 ms and 10 s, respectively. During the first half ofthe simulation, the null-space components a for RNS-

C and a for OSF-C are set to zero to focus on end-

effector motion/force control only. During the second half,

the RNS-C null-space term is set to the constant a =[0 1 0]T rad/s2, while the respective value for the OSF-Cnull space component is determined by a = Mla.

The desired path in the unconstrained direction along xis defined via a fifth-order spline function with the goal

position set at 1.0 m. The respective feedback gains are setas kd = 10 s1 and kp = 100 s2. The desired force inthe constrained direction along z is designed as a fifth-orderspline function during the first second, with a maximum

value set at 10 N, to be kept constant for the remainingtime. The respective feedback gain is set to kf = 5.

Figures 3 and 4 show the simulation results obtained from

RNS-C and OSF-C, respectively. From Figs. 3a3d and 4a

4d it is apparent that the end-effector position/force data

are almost identical for the two controllers. This means thatthe properties of full motion/force decoupling and full end-

effector/null-space decoupling could be validated with both

controllers.

Further on, Fig. 4e shows that the joint velocity with the

OSF-C undergoes much larger fluctuations than that with the

RNS-C in Fig. 3e. Examination of the determinant of the mo-

bility tensor (3), shown in Fig. 4f, reveals that joint velocity

peaks appear around its minima, that is around 2.2 s, and inthe interval between 6 s and 8.3 s. The fluctuations induceincreased force error around 7.5 s, as seen from Fig. 4d. Inthe case of the RNS-C, on the other hand, the problem could

not be confirmed because the kinematic singularity-sensitive

mobility tensor is not used in the formulation. The RNS-C is sensitive, however, to rank-deficiency of the coupling

inertia matrix. The deficiency property can be related to

the determinant det(MAlMTAl), therefore we included the

respective graph as shown in Fig. 3f. It becomes apparent,

however, that the minima of the graph do not imply large

fluctuations in the joint velocity, as it was the case with the

minima of the mobility tensor determinant.

VI . DISCUSSION AND CONCLUSIONS

A new formulation of end-link dynamics for kinemati-

cally redundant robot limbs was introduced, derived via a

floating-limb notation. It was shown that full dynamic de-

coupling of the end-link is possible within the Reaction NullSpace framework developed earlier for free-floating space

robots and flexible-base robots. The decoupling property was

achieved by minimizing locally only that part of the kinetic

energy that is related to the inertial coupling between the end-

link and the rest of the links, instead of locally minimizing

the total kinetic energy, as in the Operational Space and other

formulations. This lead to improved efficiency in joint space

motion, expressed with smaller peak joint velocities with

less fluctuation, especially in the neighborhood of kinematic

singularities.

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(a) (b)

0

0.5

1

1.5

2

0 2 4 6 8 10

EEposition[m]

Time [s]

xz

xdes

zdes

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10

EEpositionerror[mm]

Time [s]

xz

(c) (d)

-5

0

5

10

15

20

25

0 2 4 6 8 10

EEforce[N]

Time [s]

xz

xdes

zdes

-6

-4

-2

0

2

4

6

0 2 4 6 8 10

EEforceerror[mN]

Time [s]

z

(e) (f)

-8

-6-4-202468

10

0 2 4 6 8 10

Jointvelocity[rad/s]

Time [s]

#1#2#3

0

5

10

15

20

25

0 2 4 6 8 10

Det.ofcouplinginertia

Time [s]

x104

x104

x104

x104

x104

Fig. 3. RNS-C simulation results. Self motion is initialized at 5 s witha = [0 1 0]T rad/s2.

An outline was given how to apply the new formulation

to the design of a motion/force controller. The performance

of this controller was examined by comparing it with that

of a motion/force controller based on the Operational Space

formulation. It was shown that both controllers ensure the

desired decoupling between the motion and force subspaces

and also between the end-link dynamics and the self motion

of the limb. In addition, the efficiency of the new formulationas outlined above was confirmed as well.

The discussion was focused on fixed-base limbs, but it

should be mentioned that the RNS formulation presented

here has an advantage when applied to unfixed-base robots

such as humanoid robots, for example. This is due to the fact

that the base reaction FB appears explicitly in the equationsand can be directly controlled by self motion obtained via the

RNS, without influencing the decoupled motion/force end-

link control loops whatsoever.

Further study is needed on how to better exploit the

advantages of the new formulation in various interactions

task scenarios. Also, problems associated with near-rank-

deficiency of the coupling inertia matrix have to be ad-

dressed.

REFERENCES

[1] R. M. Murray, Z. Li and S. S. Sastry, Hand dynamics and controlin A Mathematical Introduction to Robotic Manipulation, CRC press,1994, ch. 6, pp. 285290.

[2] M. W. Spong, S. Hutchinson and M. Vidyasagar, Force control inRobot Modeling and Control, John Wiley and Sons, Inc., 2005, ch. 9,pp. 319338.

[3] L. Villani and J. De Schutter, Force Control, in Springer Handbookof Robotics, B. Siciliano and O. Khatib, Eds. New York: Springer-Verlag, 2008, ch. 7, pp. 161185.

(a) (b)

1.5

2

2.5

3

3.5

0 2 4 6 8 10

EEposition[m]

Time [s]

xz

xdes

zdes

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10

EEpositionerror[mm]

Time [s]

xz

(c) (d)

-5

0

5

10

15

20

25

0 2 4 6 8 10

EEforce[N]

Time [s]

xz

xdes

zdes

-6

-4

-2

0

2

4

6

0 2 4 6 8 10

EEforceerror[mN]

Time [s]

z

(e) (f)

-8

-6-4-202468

10

0 2 4 6 8 10

Jointvelocity[rad/s]

Time [s]

#1#2#3

1

1.5

2

2.5

0 2 4 6 8 10

Det.ofmobilitytensor

Time [s]

x10-2

x10-2

x10-2

x10-2

Fig. 4. OSF-C simulation results. Self motion is initialized at 5 s witha = Mla, a = [0 1 0]

T rad/s2.

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