An experimental evaluation of a novel minimumjerk cartesian controller for humanoid robots.
ABSTRACT In this paper we describe the design of a Cartesian Controller for a generic robot manipulator. We address some of the challenges that are typically encountered in the field of humanoid robotics. The solution we propose deals with a large number of degrees of freedom, produce smooth, humanlike motion and is able to compute the trajectory online. In this paper we support the idea that to produce significant advancements in the field of robotics it is important to compare different approaches not only at the theoretical level but also at the implementation level. For this reason we test our software on the iCub platform and compare its performance against other available solutions.

Article: Prioritized Optimal Control
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ABSTRACT: This paper presents a new technique to control highly redundant mechanical systems, such as humanoid robots. We take inspiration from two approaches. Prioritized control is a widespread multitask technique in robotics and animation: tasks have strict priorities and they are satisfied only as long as they do not conflict with any higherpriority task. Optimal control instead formulates an optimization problem whose solution is either a feedback control policy or a feedforward trajectory of control inputs. We introduce strict priorities in multitask optimal control problems, as an alternative to weighting task errors proportionally to their importance. This ensures the respect of the specified priorities, while avoiding numerical conditioning issues. We compared our approach with both prioritized control and optimal control with tests on a simulated robot with 11 degrees of freedom.10/2014;  SourceAvailable from: Andrea Del Prete
Article: Prioritized motionforce control of constrained fullyactuated robots:“Task Space Inverse Dynamics”
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ABSTRACT: We present a new framework for prioritized multitask motion/force control of fullyactuated robots. This work is established on a careful review and comparison of the state of the art. Some control frameworks are not optimal, that is they do not find the optimal solution for the secondary tasks. Other frameworks are optimal, but they tackle the control problem at kinematic level, hence they neglect the robot dynamics and they do not allow for force control. Still other frameworks are optimal and consider force control, but they are computationally less efficient than ours. Our final claim is that, for fullyactuated robots, computing the operationalspace inverse dynamics is equivalent to computing the inverse kinematics (at acceleration level) and then the jointspace inverse dynamics. Thanks to this fact, our control framework can efficiently compute the optimal solution by decoupling kinematics and dynamics of the robot. We take into account: motion and force control, soft and rigid contacts, free and constrained robots. Tests in simulation validate our control framework, comparing it with other stateoftheart equivalent frameworks and showing remarkable improvements in optimality and efficiency.Robotics and Autonomous Systems 10/2014; · 1.16 Impact Factor  SourceAvailable from: Sean Ryan Fanello
Conference Paper: 3D Stereo Estimation and Fully Automated Learning of EyeHand Coordination in Humanoid Robots
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ABSTRACT: This paper deals with the problem of 3D stereo estimation and eyehand calibration in humanoid robots. We first show how to implement a complete 3D stereo vision pipeline, enabling online and realtime eye calibration. We then introduce a new formulation for the problem of eyehand coordination. We developed a fully automated procedure that does not require human supervision. The endeffector of the humanoid robot is automatically detected in the stereo images, providing large amounts of training data for learning the visiontokinematics mapping. We report exhaustive experiments using different machine learning techniques; we show that a mixture of linear transformations can achieve the highest accuracy in the shortest amount of time, while guaranteeing realtime performance. We demonstrate the application of the proposed system in two typical robotic scenarios: (1) object grasping and tool use; (2) 3D scene reconstruction. The platform of choice is the iCub humanoid robot.IEERAS International Conference on Humanoid Robots; 11/2014
Page 1
Abstract—In this paper we describe the design of a Cartesian
Controller for a generic robot manipulator. We address some of
the challenges that are typically encountered in the field of
humanoid robotics. The solution we propose deals with a large
number of degrees of freedom, produce smooth, humanlike
motion and is able to compute the trajectory online. In this
paper we support the idea that to produce significant
advancements in the field of robotics it is important to compare
different approaches not only at the theoretical level but also at
the implementation level. For this reason we test our software
on the iCub platform and compare its performance against
other available solutions.
I. INTRODUCTION
S researchers in the field of humanoid robotics we
commonly find ourselves promising our funders that
soon we will see the world populated by robots substituting
humans in everyday tasks. At the same time – and quite sadly
– in our laboratories students spend an enormous amount of
time facing tasks that everyone considers quite easy or at
least “already solved” in the literature. The problem is that, if
on the one hand the scientific literature is full of papers
describing techniques that solve virtually all possible tasks,
on the other hand it is difficult to find the implementation of
those techniques and use them out of the box. This is not to
say that scientific papers are incorrect, do not contain good
work or are not useful. It is true, however, that researchers
put a lot of effort in writing papers and developing new
techniques, but rarely concentrate on writing good
implementations of these techniques and making them
publicly available for comparison purposes or just as
building blocks to work out more sophisticated tasks.
In particular when we decided to implement a Cartesian
Controller for the iCub platform we found that it was not as
easy as we initially expected. In humanoid robotics we often
deal with kinematics structures that have a large (and
variable) number of degrees of freedom. The problem is
further complicated because trajectories have to be computed
quickly in realtime. Finally, humanoid robots are
Manuscript received March 10, 2010. This work was fully supported by
European FP7 ICT project No. 215805 (CHRIS).
U. Pattacini, F. Nori, L. Natale, G. Metta, and G. Sandini are with the
Italian Institute of Technology (IIT), Department of Robotics, Brain and
Cognitive Sciences (RBCS), Via Morego 30, Genova, Italy (email:
ugo.pattacini@iit.it, francesco.nori@iit.it,
giorgio.metta@iit.it, giulio.sandini@iit.it).
lorenzo.natale@iit.it,
programmed to produce smooth movements. All these
aspects rule out the possibility to resort to expensive offline
solutions that are typically employed in industrial settings
[1].
To deal with these issues we designed a novel Cartesian
Controller which extends the MultiReferential dynamical
system approach [2] in two aspects. Firstly we have modified
the trajectory generator to produce trajectories that have
minimumjerk profile. Secondly, we have applied an interior
point optimization technique [3] to solve the inverse
kinematics problem. We have shown that our solution has
some advantages with respect to [2] and standard approaches
in the literature [6], [7]. We also conducted an experimental
comparison between our implementation and publicly
available software, demonstrating the performance gain of
our technique in terms of smoothness, speed, repeatability
and robustness.
Fig. 1. The iCub robot.
II. PLATFORM
The experiments reported in this paper were performed on
the iCub [8]. The iCub is a 53 degrees of freedom humanoid
robot (Fig. 1) shaped as a human child. The iCub was
designed to study manipulation so most of the mechanical
complexity is in the arms and hands, which are actuated by a
total of 16 motors each (9 and 7 in the hand and arm
respectively). The iCub is equipped with cameras, force
sensors, and gyroscopes. All the software running on the
iCub – including the software we describe in this paper – is
released opensource (GPL) and is freely available for
download.
An Experimental Evaluation of a Novel MinimumJerk Cartesian
Controller for Humanoid Robots
Ugo Pattacini, Francesco Nori, Lorenzo Natale, Giorgio Metta, and Giulio Sandini
A
The 2010 IEEE/RSJ International Conference on
Intelligent Robots and Systems
October 1822, 2010, Taipei, Taiwan
9781424466764/10/$25.00 ©2010 IEEE1668
Page 2
III. THE CARTESIAN CONTROLLER
Given the Cartesian position of a target object, reaching is
performed in two separate modules (Fig. 2). The first stage
employs a nonlinear optimization technique to determine the
arm joints configuration that achieves the desired pose (i.e.
endeffector position and orientation). The second stage
consists of a biologically inspired kinematic controller that
computes the velocity of the motors to produce a humanlike
quasistraight trajectory of the endeffector. In the following
these two modules composing the structure of proposed
Cartesian controller are analyzed in depth.
Fig. 2. Diagram of proposed Cartesian controller.
A. The Solver
We consider the general problem of computing the value
of joint encoders
q ∈ in order to reach a given position
3
d x ∈ and orientation
d
α ∈ of the endeffector (where
α is represented in axis/angle notation1). At the same time,
the computed solution has to satisfy a set of given constraints
expressed as inequalities. Formally this problem can be
stated as follows:
( )
(
n
q
∈
where
x
K and Kα are the forward kinematic functions that
respectively compute the position and the orientation of the
endeffector from the joint angles q ;
configuration, W is a diagonal matrix of weighting factors,
β is a positive scalar weighting the influence of
a parameter for tuning the precision of the movement:
typically
1
β <
and
problem (1) has to comply with a set of additional
constraints: for example, we required that the solution lies
between lower and upper bounds (
admissible values.
In our case the joints vector has 10 components (7 joints
for the arm, 3 joints for the torso) and we have chosen the
value of
rest
q
so that the torso of the robot when controlled is
as close as possible to the vertical position. We proposed to
use an interior point optimization technique to solve the
*n
4
d
()()
)
( )
q
2
2
*
arg min
s.t.
,
T
LU
restrest
d
x
d
qKqqqW qq
xK
qqq
α
αβ
ε
=−+⋅−−
−<
<<
(1)
rest
q
is a preferred joint
rest
q
and ε is
54
10
[
10
]
,
ε
−−
∈
. Moreover, the solution to
,
LU
n
q q ∈ ) of physically
1 In axis/angle representation any rotation is described by a unit vector
ω, indicating the direction of rotation, and an angle ϑ that accounts for the
magnitude of rotation around the axis according to the righthand rule;
hence it holds:
[]
,
d
αω θ ω=∈
[]
3
,1,0,
T
ωθπ=∈
.
problem (1), in particular we used IpOpt [3], a public
domain software package designed for largescale nonlinear
optimization. This approach has the following advantages:
1) Quick convergence. IpOpt is reliable and fast enough to
be employed in realtime as demonstrated in the remainder,
especially compared to more traditional iterative methods
such as the Cyclic Coordinate Descent (CCD) [12] adopted
in [2].
2) Scalability. The intrinsic capability of the optimizer to
treat nonlinear problems in any arbitrary number of variables
is here exploited to make the controller’s structure easily
scalable with the dimension n of the joint space. For
example, it is possible to switch at runtime from the control
of the 7DOF iCub arm to the complete 10DOF structure
inclusive of the torso or to any combination of the joints
depending on the task.
3) Automatic handling of singularities and joint limits.
This technique automatically deals with singularities in the
arm Jacobian and joint limits, and can find solutions in
virtually any working conditions.
4) Tasks hierarchy. The task is split in two subtasks: the
control of the orientation and the control of the position of
the endeffector. Different priorities can be assigned to the
subtasks. In our case the control of the position has higher
priority with respect to the orientation subtask (the former is
handled as a nonlinear constraint and thus is evaluated
before the cost) because we deemed that to accomplish a
successful grasp which is the ultimate goal of reaching for a
humanoid it is central to cope with circumstances when the
final object is attainable in position and thus can be touched,
but the orientation cannot be reached perfectly at the same
time.
Fig. 3. MultiReferential scheme. ( )
5) Description of complex constraints. It is easy to add
new constraints as linear and/or nonlinear inequalities either
in task or joint space. In the case of the iCub, for instance,
we added a set of constraints that avoid breaking the tendons
that actuate the three joints of the shoulder: these tendons
indeed are shared among the joints, whose movements are
thus limited by the tendons lengths within a compact subset
of the convex hull described by the physical joints bounds
K
⋅ is the forward kinematic map.
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Page 3
[13]. Thereby, three linear inequalities hold among the
shoulder joints that are conveniently included into (1) taking
the form
sh
lCL
q
≤⋅≤
, being
shoulder joints, l, L the lower and upper limits imposed by
the tendons lengths and C a suitable 3by3 matrix.
sh
q the vector of the three
B. The Controller: design of minimumjerk solution
The goal of the controller’s module is to determine the
smooth velocity profiles in the joint space which steer the
arm from the current posture q to the final configuration
while at the same time ensuring that the joints lie within well
defined limits. This can be obtained by applying the Multi
Referential Dynamical Systems approach [2], in which two
dynamical controllers, one in joint space and one in task
space, evolve concurrently (Fig. 3). The coherence constraint
between the two tasks – providing that x
at each instant with J the Jacobian of the forward kinematic
map – is enforced with the Lagrangian multipliers method
and can be used to modulate the relative influence of each
controller (i.e. to avoid joint angles limits). The advantage of
such a redundant representation of the movement is that a
quasistraight trajectory profile can be generated for the end
effector in the task space reproducing a humanlike behavior
[5], [9], while retaining converge property and robustness
against singularities [2].
In [2] the two controllers are implemented with Vector
IntegrationToEndpoint (VITE)
approximate the neural signal commanding a pair of agonist
antagonist muscles and whose behavior is regulated by a
second order differential equation. According to the
implementation in [2], the angular velocities, output of the
coherence constraint block, are integrated to generate
position references which are then sent to a second cascade
controller that is in charge of yielding the velocity profiles in
closed loop with a proportional law (Fig. 4).
Aside from the connection to biological evidences, a
second significant merit of this approach is the description of
the model given as a compact and autonomous dynamic
equation which makes the controller implementation
straightforward and robust against external perturbation of
the movement creating an attractor landscape towards the
goal, i.e. the target configuration [14]. On the other hand, the
specific choice of a coupled second order dynamic systems
in cascade with a P controller entails a major disadvantage
when applied to the control of a robotic limb: notably, the
generated velocity profiles become less humanlike as the
required execution time becomes shorter. When a fast
response is requested, trajectories approach an exponential
response (typical of a first order dynamical system),
irrespectively of how the controller’s parameters are tuned;
therefore, the corresponding velocities are no longer bell
shaped, having a steep acceleration at the beginning followed
by a slow decay. The reason is twofold: primarily a second
order system cannot reproduce the smoothness typical of
biological motion [9] (for example it does not impose zero
acceleration at starting point) and secondly the presence of
*
q ,
Jq
=
is guaranteed
models [4], which
the proportional controller reduces the performances since it
cannot guarantee the velocities computed by the coherence
constraint block. As result fast movements tend to be jerky
producing unwanted vibrations.
Fig. 4 Schematic of implemented multireferential VITE controllers in the
work of Hersch and Billard [2].
To overcome these limitations, we maintained the multi
referential approach and replaced the VITE with more
complex controllers which reproduce a trajectory that
resemble a minimumjerk profile both in joint and task
space: movements are still represented and controlled in
multiple frames of reference but preserve a smooth (bell
shaped) velocity profile. To this end one might consider to
rely on a trajectory generator which codes for example the
minimumjerk profile over the time interval T and specific
starting and ending points
x x : in literature [10] it is well
known that the desired shape depicted in Fig. 5 (red lines) is
given by the following fifth order polynomial:
The seeming straightforwardness of this formulation hides
a number of somewhat important issues that need to be taken
into account by an effective design: it is required indeed to
generate an internal temporal scale t that has to be
reinitialized any time the feedback is acquired modifying the
coefficients of (2) online; moreover, the feedback turns out
to be mandatory since the coherence block disturbs the true
velocity command causing eventually drifts if not
compensated by feedback. Therefore, a regulator appears to
be a more natural answer for the task and joint space
minimumjerk elements. Possibly the generator (2) can be
applied as the feedforward component of the regulator,
operating merely on the target position and leaving a PID to
take into account the feedback: even so the PID would work
hard to stabilize the response against the drift, delivering
velocities that do not comply with the requisite of human
likeness. This ultimately suggests devising a controller that
intrinsically embeds the desirable property of smoothness.
We took inspiration from the feedback formulation of the
minimumjerk trajectory as the solution of an optimal control
problem as reported in [11], where a third order linear time
varying (LTV) differential equation is derived:
0,
d
( )
x t
()
345
00
10156
.
d
ttt
xxx
TTT
=+−−+
(2)
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()
()
()
323
0100
0010
60 369
−
60
.
d
Tt
TtTtTt
x
x
x
x
x
x
x
−−−
−−−
=+
(3)
The weak point of (3) is that it contains coefficients whose
values become infinite when t approaches the execution time
T. To sidestep this difficulty, we decided to employ a linear
timeinvariant (LTI) system of the same order whose
parameters are tuned to better approximate a minimum jerk
trajectory: in other words we sought the third order
differential system that is the best timeinvariant version of
(3) minimizing the same jerk measure over the interval [0,
T]. Formally, we started from the parametric equation of the
trajectory expressed in the form:
( )()
112
expexp
x tCtC
λ
=⋅+
that is a particular solution of a stable third order differential
system with three independent real negative poles
selection of real negative roots stems from the objective to
identify a stable system and avoid damped resonant terms
since we require a monotonic trend to the target, without
overshoots, as it comes to be relevant for joints limits
avoidance; in addition, oscillating components certainly
introduce jerkiness in the response.
The coefficients
i
C can be determined by imposing null
initial conditions for the position, the velocity and the
acceleration. Other monotonic solutions different from (4)
exist for a generic third order equation depending on the
multiplicity of the roots of the characteristic polynomial, thus
we repeated our treatment exhaustively over all the possible
cases, finding out that the most suitable structure of the
solution simplified to having just one root λ of multiplicity
3. Therefore, defined
( )
M t the measure of the squared jerk
( )
xt
accumulated up to time t as:
t
M tx
=∫
()()
233
exp,
d
tCtx
λλ
⋅+⋅+
(4)
i λ . The
2
( )( )
τ
2
0
,d
τ
(5)
we solved the following minimization problem for the case
T=1 and
(
argminM
λ
∈
d x =1 without loss of generality:
( )
∞
)
( )
1
*
0
s.t..
1x
λ
λ
ε
<
=
≥ −
(6)
The extension of
in the numeric evaluation of the cost and can be performed
since ( )
x t is a monotonic function. Furthermore, the second
constraint in (6) forces the solution to reach the steadystate
with a “rate” specified by the parameter ε. Without this
( )
M T into
( )
M ∞ yields a simplification
lower bound on ( )
function would be permitted, including functions with very
slow time constants. In other words, by setting the parameter
ε we are able to tune the final execution time. It is worth
pointing out that the cost
( )
M ∞ can be easily resolved as
function of the root λand more importantly its gradient has a
close form that enables to carry out the minimization with
reliable gradientbased methods (such as the interior point
algorithm).
Fig. 5 compares the trajectories (position, velocity and the
measure
( )
M t ) of the ideal minimumjerk model against
those obtained with the timeinvariant system derived with
our approach by selecting
0.1
ε =
As expected the LTI system approximates the ideal
minimumjerk position trajectory, having in particular a
slightly faster onset followed by slower convergence to the
steady state. At the same time, however, it provides a very
good compromise between smoothness and simplicity of
implementation.
1x
, any possible monotonically increasing
(we retrieved
*
5.322
λ = −
).
Fig. 5. Comparison between the responses of the 3rd order dynamical time
invariant system found through the minimization (blue) and the responses
of the minimumjerk model (red).
Once the root
λ is known, it is possible to compute the
elements of the dynamic matrix A and input matrix B and
write the controller in the canonical form extended to the
case of generic execution time T (the description of these
steps is out of the scope of this paper):
=
*
( )
3
T
( )
2
T
A
( )
T
( )
3
T
B
0100
0010
.
d
abca
x
x
x
x
x
x
x
λλλλ
+
−
(7)
The system response in t
steadystate value as enforced by the constraint, and the
transient can be considered extinguished for
The first important result achieved by this controller is
visible in Fig. 6: it is clear that minimumjerk controllers can
provide, especially for fast movements, smooth velocity
profiles that are more similar to the desired humanlike
T
=
is equal to the 90% of the
1.5tT
≥⋅
.
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Page 5
prototypes if compared to the profiles generated by the VITE
model.
Fig. 6. Comparison between step responses of VITE’s (blue) and
minimumjerk controller (green), providing the same velocity peak. The 3rd
order system has a faster convergence and an (almost) bellshaped velocity
profile.
C. The Controller: implementation issues
The algorithm follows faithfully the three steps in [2] with
two main exceptions as detailed hereafter.
Fig. 7. A): transfer function of the LTI minimumjerk model. B): an insight
of the joint space minimumjerk controller implemented in closedloop
form.
1) The need to constantly read the feedback
the authors of [2] to introduce a P controller with the purpose
to keep the generation of trajectories and their execution as
two separated functionalities, preventing the evolution of the
feedback from interfering with the internal state of the VITE.
This brought about a series of drawbacks we have analyzed
in section III.B.
In order to adhere to the original diagram of Fig. 3, an
alternative solution has been explored that transforms the
structure of our model as represented in Fig. 7B, where for
sake of clearness only the joint space minimumjerk
controller is presented: from case A that corresponds to the
statespace model in (7) we migrated to the system B that
owns exactly the same transfer function ( )
in A, taking now the actual error between the target and the
feedback as input. The pure integrator plays the role of the
mechanical plant that integrates the received command and
returns the current feedback. All the remaining unmodeled
dynamics and uncertainties are represented by the term D,
whose effects are rejected by the closedloop system. The
disturbance introduced in the minimumjerk controller by the
fb
q motivated
( )
s written
d
q sq
coherence constraint is compensated similarly: as a matter of
fact, the signal computed through the Lagrangian multipliers
does not act like a feedforward component, but rather
perturbs the controller. To conclude, the closedloop
structure realizes exactly the scheme of the multireferential
approach and ensures that the current robot’s position is
correctly fed back in the system.
2) The modulation of weighting matrices that appear in the
coherence reinforcement and serve for the handling of joints
limits avoidance (see section 3.6 of [2]) is achieved by
imposing a cosine shaping relation (see Fig. 8). Nonetheless,
to better exploit the whole arm workspace it is advisable to
assign high priority to the Cartesian controller in a portion of
the joint space that is as large as possible. To this end we
adopted a different weighting policy made of a flat region
connected with hyperbolic tangent functions whose decay
rate is much steeper than the original cosine law, as
illustrated in Fig. 8, showing out its benefits in the execution
of quasistraight Cartesian trajectories for pointtopoint
motion as demonstrated by experiments.
Fig. 8. The implemented shaping policy for the joints limits avoidance
(green) and the original cosine law (blue) used in [2].
IV. EXPERIMENTAL RESULTS
According to the RobotCub opensource philosophy the
whole software developed by the project partners was made
available to the wide community of iCub users; this
facilitates the collaboration and promotes the development of
new algorithms. As result, a collection of libraries and
modules targeting different fields (vision processing, motor
development, machine learning, etc) has been circulating
among partners of the RobotCub Consortium who can easily
reuse code for their research activities without having to be
concerned with the implementation details. In this respect it
has been almost immediate and much valuable comparing on
the same shared platform the performance of our Cartesian
controller2 with the VITEbased system3 whose modules can
2 The standalone application of MinimumJerk Cartesian controller
used to obtain the presented experimental results is named iKinArmCtrl and
relies on the iKin library, a general purpose tool for forward and inverse
1672
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be downloaded from public repositories. Additionally, we
included in the assessment a further controller as an example
of a more conventional strategy making use of the typical
Damped LeastSquares (DLS) rule [15] coupled with a
secondary task that comprises the joints angles limits by
means of the gradient projection method [16]. This solution
employs the thirdparty package Orocos4, a tool for robot
control that implements the DLS approach (used by some
partners of the RobotCub Consortium) and whose public
availability and compliance with realtime constraints
justified its adoption as one of the reference controllers.
Fig. 9. Pointtopoint Cartesian trajectories executed by the three
controllers: the VITEbased method produces on average the blue line, the
minimumjerk controller result is in green, the DLS system using Orocos in
red. Bands containing all the measured paths within a confidential interval
of 95% are drawn in corresponding colors. Controllers settings are: T=2.0 s
for the minimumjerk system, α=0.008, β=0.002, KP=3 for the VITE, and
μ=105 for the damping factor of the DLS algorithm.
In the first experiment we put to the test the three selected
schemes in a pointtopoint motion task wherein the iCub
arm was actuated in 7DOF mode and whose endeffector
was controlled both in position and orientation. It came out
that paths produced by our controller and the DLSbased
system are well restricted in narrow tubes of confidence
intervals and are quite repeatable; conversely the VITE is
affected by a much higher variability.
Fig. 9 highlights what reported for a set of 10 trials of a
typical reaching task where the right hand is moved from the
rest posture towards a location in front of the iCub waist with
the palm turned downward; Tab. 1 summarizes the measured
intarget errors for the three cases: all the controllers behave
satisfactory, but the DLS achieves lower errors because
operates continuously on the current distance from the target
x , being virtually capable of canceling it at infinite time.
On the contrary, strategies based on the interaction with an
d
kinematics of serial link chains. It is available from the repository:
https://robotcub.svn.sourceforge.net/svnroot/robotcub/trunk/iCub.
3 A version of the M. Hersch’s code that accepts target points expressed
in the iCub standard reference frame can be downloaded from the
repository:
https://robotcub.svn.sourceforge.net/svnroot/robotcub/trunk/papers/pattacin
i2010.
4 The kinematic component of Orocos project is reachable here:
http://www.orocos.org/kdl.
external solver bind the controller module to close the loop
on an approximation
d
x of the real target that is determined
by the optimization tolerances as in (1).
TABLE I
Tab. 1. Mean errors along with the confidence levels at 95% computed
when the target is attained. An average measure of the variability of
executed path is also given for the three controllers.
Regarding the analysis of humanlikeness, the new
proposed Cartesian controller outperforms both the
traditional and the VITEbased solution thanks to the
regulator design – so near to the ideal minimumjerk model –
and also as consequence of the wider working region where
the task space module can function due to the replacement of
the shaping policy. It is indeed clear from Fig. 9 how the
trajectory commanded by the minimumjerk controller
(green line) approaches much more a quasistraight path
whereas the red and the blue lines oscillate before reaching
the target. This important aspect becomes evident also once
the velocity of the endeffector in the operational space is
drawn as shown in Fig. 10: the velocity profiles have been
computed in postprocessing from the indirect acquisition of
the endeffector coordinates by reading the joints encoders
values (i.e. ( )( )
()
x t K q t
=
and then have been filtered to
remove the highfrequency components (with a cutting
frequency of 2.5 Hz). The superposition with the curve of the
ideal minimumjerk prototype (sketched black line),
identified by knowing the starting and the ending points of
the motion, underlines the good level of similarity of our
implementation (green line) and, at the same time, the
discrepancy of the other two methods which show a very
sharp onset and a remarkable asymmetry of the response.
Tab. 2 sums up the jerk measures computed in the
Cartesian space: a factor of 43.8% is gained with respect to
the VITE system and even a factor of 69% is achieved
against the DLS.
Fig. 10. Magnitude of the endeffector velocity in the operational space:
blue for VITE, red for DLS and green for minimumjerk controller. The
pointtopoint task begins at t=1s.
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The second evaluation was concerned with the dynamical
characteristics of the DLS and minimumjerk controllers. In
particular, we verified their capabilities to track in position a
quite fast reference trajectory given in the operational space
while keeping the orientation of the endeffector constant;
unfortunately, we did not manage to test the VITE algorithm
as we experienced that the implemented CCD solver was not
fast enough to run in realtime.
TABLE II
Tab. 2. Relative measures of the jerk in the operational space given as
the ratio between the integral of the squared second derivative of the
Cartesian velocity for the three controllers and the same quantity computed
for the exact minimumjerk model.
The target passed to the controllers evolved along a
lemniscateshaped trajectory (Fig. 11), completing one cycle
with two different time periods: TP in {30, 15} seconds. In
the first experiment (TP=30 s), the minimumjerk Controller
ran with the parameter T set to 2.0 s and accomplished the
task considerably well; the DLS method deviated somehow
from the reference and did not perform with the same
accuracy. When the target moved faster (TP=15 s), the
minimumjerk Controller still behaved better, in the sense
that the gap between the executed curve and the reference
was lower compared to the DLS case, as we reduced the
parameter T to 1.5 s in order to get a quicker response; on
the other side, the DLS reactivity remained unchanged
lacking of an analogous builtin tuning.
Fig. 11. Controllers’ responses while tracking a lemniscate shape:
minimumjerk controller in green, DLS in red. The resulting trajectories of
10 trials are shown for the two time periods TP.
Notably, it is crucial to mention for this kind of test that
IpOpt is able to comply with the stringent realtime
constraints and eventually allows to close the loop of Fig. 2:
the average solving time of (1) for the case TP=15 s was
17±28 ms, having the solver running at 33 Hz on a multi
core Intel (R) Xeon with 2.27 GHz of clock frequency.
V. CONCLUSION
We designed a novel and general purpose Cartesian
controller capable of moving the robot endeffector in the
operational space in a way most similar to the behavior that
humans express during reaching tasks, performing motions
that follow quasistraight trajectories with bellshaped
velocity profiles. By inheriting the concept of multi
referential approach from a preexisting VITEbased model,
the controller’s scheme has been improved through an
optimization process that enabled to achieve better results in
terms of similarity to humanlike movements with respect to
the VITE system and traditional control strategies. This was
demonstrated by carrying out an experimental assessment of
the different techniques on the same robotic platform.
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