On Christoffel and standard words and their derivatives

We introduce and study natural derivatives for Christoffel and finite standard words, as well as for characteristic Sturmian words. These derivatives, which are realized as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to $a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are longer than one letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.Comment: 28 pages. Final version, to appear in TC

Efficient computation of inverse dynamics and feedback linearization for VSA-based robots

We develop a recursive numerical algorithm to compute the inverse dynamics of robot manipulators with an arbitrary number of joints, driven by variable stiffness actuation (VSA) of the antagonistic type. The algorithm is based on Newton-Euler dynamic equations, generalized up to the fourth differential order to account for the compliant transmissions, combined with the decentralized nonlinear dynamics of the variable stiffness actuators at each joint. A variant of the algorithm can be used also for implementing a feedback linearization control law for the accurate tracking of desired link and stiffness trajectories. As in its simpler versions, the algorithm does not require dynamicmodeling in symbolic form, does not use numerical approximations, grows linearly in complexity with the number of joints, and is suitable for online feedforward and real-time feedback control. A Matlab/C code is made available

Open and Closed Prefixes of Sturmian Words

A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of reversed standard words.Comment: To appear in WORDS 2013 proceeding

The sequence of open and closed prefixes of a Sturmian word

A finite word is closed if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We are interested in the {\it oc-sequence} of a word, which is the binary sequence whose $n$-th element is $0$ if the prefix of length $n$ of the word is open, or $1$ if it is closed. We exhibit results showing that this sequence is deeply related to the combinatorial and periodic structure of a word. In the case of Sturmian words, we show that these are uniquely determined (up to renaming letters) by their oc-sequence. Moreover, we prove that the class of finite Sturmian words is a maximal element with this property in the class of binary factorial languages. We then discuss several aspects of Sturmian words that can be expressed through this sequence. Finally, we provide a linear-time algorithm that computes the oc-sequence of a finite word, and a linear-time algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of arXiv:1306.225

Real-time computation of distance to dynamic obstacles with multiple depth sensors

We present an efficient method to evaluate distances between dynamic obstacles and a number of points of interests (e.g., placed on the links of a robot) when using multiple depth cameras. A depth-space oriented discretization of the Cartesian space is introduced that represents at best the workspace monitored by a depth camera, including occluded points. A depth grid map can be initialized off line from the arrangement of the multiple depth cameras, and its peculiar search characteristics allows fusing on line the information given by the multiple sensors in a very simple and fast way. The real-time performance of the proposed approach is shown by means of collision avoidance experiments where two Kinect sensors monitor a human-robot coexistence task

Some characterizations of Sturmian words in terms of the lexicographic order

In this paper we present three new characterizations of Sturmian words based on the lexicographic ordering of their factors

Faster Motion on Cartesian Paths Exploiting Robot Redundancy at the Acceleration Level

The problem of minimizing the transfer time along a given Cartesian path for redundant robots can be approached in two steps, by separating the generation of a joint path associated to the Cartesian path from the exact minimization of motion time under kinematic/dynamic bounds along the obtained parameterized joint path. In this framework, multiple suboptimal solutions can be found, depending on how redundancy is locally resolved in the joint space within the first step. We propose a solution method that works at the acceleration level, by using weighted pseudoinversion, optimizing an inertia-related criterion, and including null-space damping. Several numerical results obtained on different robot systems demonstrate consistently good behaviors and definitely faster motion times in comparison with related methods proposed in the literature. The motion time obtained with our method is reasonably close to the global time-optimal solution along same Cartesian path. Experimental results on a KUKA LWR IV are also reported, showing the tracking control performance on the executed motions

A model-based residual approach for human-robot collaboration during manual polishing operations

A fully robotized polishing of metallic surfaces may be insufficient in case of parts with complex geometric shapes, where a manual intervention is still preferable. Within the EU SYMPLEXITY project, we are considering tasks where manual polishing operations are performed in strict physical Human-Robot Collaboration (HRC) between a robot holding the part and a human operator equipped with an abrasive tool. During the polishing task, the robot should firmly keep the workpiece in a prescribed sequence of poses, by monitoring and resisting to the external forces applied by the operator. However, the user may also wish to change the orientation of the part mounted on the robot, simply by pushing or pulling the robot body and changing thus its configuration. We propose a control algorithm that is able to distinguish the external torques acting at the robot joints in two components, one due to the polishing forces being applied at the end-effector level, the other due to the intentional physical interaction engaged by the human. The latter component is used to reconfigure the manipulator arm and, accordingly, its end-effector orientation. The workpiece position is kept instead fixed, by exploiting the intrinsic redundancy of this subtask. The controller uses a F/T sensor mounted at the robot wrist, together with our recently developed model-based technique (the residual method) that is able to estimate online the joint torques due to contact forces/torques applied at any place along the robot structure. In order to obtain a reliable residual, which is necessary to implement the control algorithm, an accurate robot dynamic model (including also friction effects at the joints and drive gains) needs to be identified first. The complete dynamic identification and the proposed control method for the human-robot collaborative polishing task are illustrated on a 6R UR10 lightweight manipulator mounting an ATI 6D sensor

Enumeration and Structure of Trapezoidal Words

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, \emph{semicentral words}, and show that they are characterized by the property that they can be written as $uxyu$, for a central word $u$ and two different letters $x,y$. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.Comment: Accepted for publication in Theoretical Computer Scienc