Abstract. The Euler−Poinsot **rigid** **body** **motion** is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S 2

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∂Ω
x × (V + v ∗ )(V · n)dγ. (3.9)
Indeed the formulation (3.5) and (3.6) might look artificial, but it depends on how to develop the linear theory in the next section (there are actually some other possible ways). In order to define the control space for our problem, we consider six auxiliary adjoint problems, associated with six elementary **rigid** **body** **motion** velocities. For each i ∈ {1, 2, 3}, let (v (i) , q (i) ) be the solution of the generalized Oseen problem

IV. C ONCLUSION
We presented here two 2D models, intrinsically related to the Euler-Poinsot **rigid** **body** **motion**. The new theoretical result of this paper is the explicit construction of the basis of Jacobi vector fields for Darboux-type metrics on 2D surfaces of revolution. The consequences of this computation go beyond the content of this paper. From one hand, it gives an alternative and simplified proof of the conjugate locus equation that we used to describe the structure of the conjugate locus of the Serret-Andoyer metric. On the other hand, even if this result gives no information on the conjugate

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2 Laboratoire Jean Kuntzmann, B.P. 53, 38041 Grenoble Cedex 9, France
February 26, 2014
Abstract
Finding the root mean sum of squared deviations (RMSDs) between two coordinate vec- tors that correspond to the **rigid** **body** **motion** of a macromolecule is an important problem in structural bioinformatics, computational chemistry and molecular modeling. Standard algo- rithms compute the RMSD with time proportional to the number of atoms in the molecule. Here, we present RigidRMSD, a new algorithm that determines a set of RMSDs correspond- ing to a set of **rigid** **body** motions of a macromolecule in constant time with respect to the number of atoms in the molecule. Our algorithm is particularly useful for **rigid** **body** modeling applications such as **rigid** **body** docking, and also for high-throughput analysis of **rigid** **body** modeling and simulation results. We also introduce a constant-time rotation RMSD as a sim- ilarity measure for **rigid** molecules. A C++ implementation of our algorithm is available at http://nano-d.inrialpes.fr/software/RigidRMSD.

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University of Nice, LJAD UMR CNRS 7351, Parc Valrose, 06108 Nice Cedex, France
Abstract
A simple method is developed to couple accurately the **motion** of **rigid** bodies to compressible fluid flows. Solid **rigid** bodies are tracked through a Level-Set function. Numerical diffusion is controlled thanks to a compressive limiter (Overbee) in the frame of MUSCL type scheme, giving an excellent compromise between accuracy and efficiency on unstructured meshes (Chiapolino et al., 2017). The method requires low resolution to preserve solid bodies’ volume. Several coupling methods are then addressed to couple **rigid** **body** **motion** to fluid flow dynamics: a method based on stiff relaxation and two methods based on Ghost cells (Fedkiw et al., 1999) and immersed boundaries. Their accuracy and convergence rates are compared against an immersed piston problem in 1D having exact solution. The second Ghost cell method is shown to be the most efficient. It is then extended to multidimensional computations on unstructured meshes and its accuracy is checked against flow computations around blunt bodies. Reference results are obtained when the flow evolves around a **rigid** **body** at rest. The same **rigid** **body** is then considered with prescribed velocity moving in a flow at rest. Computed results involving wave dynamics match very well. The method is then extended to two-way coupling and illustrated to several examples involving shock wave interaction with solid particles as well as phase transition induced by projectiles **motion** in liquid-gas mixtures.

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6 Evaluation
6.1 Dataset
This section introduces the new dataset we acquired to allow quantitative evalu- ation of human **body** shape estimation from dynamic data. The dataset consists of synchronized acquisitions of dense unstructured geometric **motion** data and sparse **motion** capture (MoCap) data of 6 subjects (3 female and 3 male) cap- tured in 3 different motions and 3 clothing styles each. The geometric **motion** data are sequences of meshes obtained by applying a visual hull reconstruction to a 68-color-camera (4M pixels) system at 30FPS. The basic motions that were captured are walk, rotating the **body**, and pulling the knees up. The captured clothing styles are very tight, layered (long-sleeved layered clothing on upper **body**), and wide (wide pants for men and dress for women). The **body** shapes of 6 subjects vary significantly. Fig. 4 shows some frames of the database.

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The antenna pattern is expanded in terms of spherical harmonics coefficients [ 7 ],[ 8 ]. This approach reduces the amount of data required for describing the full antenna pattern. This has been the chosen manner for describing the antenna pattern in the PyLayers simulation platform.
The ray tracing is used to take into account the interaction on walls of the environment. This is much more questionable to apply the ray tracing for the On-**body** channel. There is current

Fig. 7. **Rigid** **motion** compensation results of a deformable region using
simulated 4D US
At first, we perform a **rigid** **motion** tracking task with an abdominal phantom. The secondary robot repeatedly rotated the abdominal phantom on a turning table in one direction and the opposite direction. In the meantime, the 6-DOF robot holding the 4D US probe is controlled by our method to automatically compensate the **rigid** **motion** of the target region within the phantom. The observed feature error and probe trajectory are shown in Fig. 9 and Fig. 10 (left). To maintain the firm contact between the probe and the phantom, we used a force control along the X axis of the 3D US image.

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The chief technical difﬁculty i n t his c lass o f p roblems i s t hat, o wing t o t he presence of a space-dependent term (the one that was overlooked by Bedeaux & Rubi ( 1987 ) and Pérez-Madrid et al. ( 1990 )), the unsteady disturbance in Fourier space is governed by a set of coupled partial differential equations. This makes it particularly difﬁcult to obtain the solution. For the solid-**body** rotation, this difﬁculty i s e asily overcome by using a rotating reference frame, since the space-dependent term disappears in this frame (Herron et al. 1975 ; Miyazaki 1995 ; Candelier 2008 ). Based on this observation, it seems natural to seek a generic coordinate transformation that removes this term whatever the carrying ﬂow. T his i s t he b ackbone o f t he p resent w ork. M ore precisely, we express the unsteady disturbance problem in a system of moving non-orthogonal coordinates that follow the undisturbed ﬂow. I n F ourier s pace, t he d isturbance i s then determined by a set of ordinary differential equations in these co-moving coordinates, making the problem much easier to solve. Solving these equations and transforming back to the laboratory frame yields the desired inertial corrections irrespective of the nature of the linear carrying ﬂow. T his t echnique i s s imilar i n e ssence t o t he approach used in the rapid distortion theory (RDT), pioneered by Batchelor & Proudman ( 1954 ) to determine how a turbulent velocity ﬂuctuation i s d istorted b y a s trong non-uniform mean ﬂow. I n t he p articular c ontext o f t he ﬂ ow pa st a ri gid bo dy, th is id ea was also used by Miyazaki et al. ( 1995 ), extending a technique developed by Onuki & Kawasaki ( 1980 ) for a scalar ﬁeld, b ut, c ompared t o o ur a pproach, t hey employed it differently, namely by considering time-dependent wavenumbers in the Fourier transform of the disturbance equation. These connections are discussed in more detail at the end of § 3.1 .

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in 3D space, with few or no anatomical consideration, resulting to movements that may not be realized in real surgery.
A new paradigm is then to simulate the procedure itself, instead of the desired result. During surgery, bone fragments are repositioned using clamps, hooks or Schanz screws (figure 2). The fracture is then reduced via the application of forces by the physician. Moreover, the surgeon use the contacts between structures, e.g. lean the ischium on the femoral head, to produce the expected movements. To simulate such a procedure we have chosen to use a mechanical model of the hip joint bony elements, implemented within the non-commercial Artisynth framework [10]. Each bone fragment is considered as an independent **rigid** **body**. One of them is usually considered as fixed, e.g. the anterior or posterior column and/or the femoral head. Collisions are handled to ensure non-penetration between elements, with dry friction (Coulomb) response. The action of a clamp is simulated via a Hill muscle model which extremities are the clamp jaws positions on the bones. The interactive “contraction” of this model apply forces similarly to the real clamp action. In reality, the muscular system apply heavy constraints to the bones during their repositioning. While modeling this accurately is an extremely complex problem, moreover in a patient-specific context, a first approximation is to add a strong global damping to the all system. Even if preferred anatomical directions are not accounted for, this high resistance ensure the response to collisions and numerical instabilities are very low in comparison to the forces directly applied to the bones. When all these elements are set, the dynamic numerical system is solved using traditional methods (Euler implicit, Runge-Kutta…).

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[7] Marchioro C., Pulvirenti M., Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, 1994.
[8] Ortega J., Rosier L., Takahashi T., On the **motion** of a **rigid** **body** immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 24 (2007), no. 1, 139–165.

We introduce theoretical and practical contributions that address these issues. We propose an implicit imaging model for non-**rigid** scenes from which we derive non-**rigid** matching tensors and closure constraints. We give a non- **rigid** Structure-From-**Motion** algorithm based on comput- ing matching tensors over subsequences, from which the im- plicit cameras are extrated. Each non-**rigid** matching tensor is computed, along with the rank of the subsequence, using a robust estimator incorporating a model selection criterion that detects erroneous image points.

The controlled experiments used images of a phantom spine, of a real and a plastic skull, of a real head and of a lumbar plastic spine segment, and the experiments using [r]

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There are few articles in the last decade concerning the controllability results on fluid- structure interaction problem. In a paper of Raymond and Vanninathan [ 37 ], they considered a simplified model in 2D where the fluid equations are replaced by the Helmholtz equations and the **motion** of a solid represented by a harmonic oscillator. In that case, the domain is supposed to be fixed but one of the difficulties comes from the fact that there is no control in the solid part. They established exact controllability results for this model with an internal control only in the fluid part. In the work of Doubova and Fern´ andez-Cara [ 12 ], they proved the local null controllability by boundary controls for a 1D model where point mass is immersed in a fluid which evolves in p´1, 1q. In that case, the domain is not fixed any more and the proof of the result is based on the global null controllability of the linearized system (by Carleman estimates) and on Kakutani’s fixed point theorem. In [ 29 ], the authors established exact con- trollability of a 2D fluid-structure system where the **body** is a ball. In the paper of Boulakia and Osses [ 4 ], the authors dealt with the same problem as in [ 29 ], except that the **body** can have more general shape. In [ 3 ], Boulakia and Guerrero proved the local null controllability of a fluid-solid interaction problem in three dimension. Finally, in [ 34 ], the authors studied the local null controllability problem for the simplified one dimensional model considered in [ 12 ] and they managed to reduce the number of controls.

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Coming back to unbounded linear ﬂows, another series of studies employed the so-called ‘induced-force’ method (hereinafter abbreviated as IF) as an alternative to the MAE approach, based on the formulation developed by Mazur & Bedeaux ( 1974 ) to extend Faxén’s formulae to a sphere undergoing an arbitrary time-dependent **motion** in an inhomogeneous ﬂow. In this method, an extra force is added to the Navier–Stokes equation to ensure that the slip velocity vanishes everywhere within the **body**, rendering the modiﬁed equation valid in the entire domain, both in the ﬂuid and the **body**. This approach was ﬁrst applied by Bedeaux & Rubi ( 1987 ) to ﬁnd the frequency-dependent inertial corrections to the force experienced by a sphere translating in a planar or an axisymmetric purely elongational ﬂow. Pérez-Madrid, Rubi & Bedeaux ( 1990 ) then obtained the quasi-steady form of the friction tensor for the three canonical planar ﬂow conﬁgurations discussed above. While their result agreed with that of Herron et al. ( 1975 ) in a solid-**body** rotation ﬂow, the components of the resistance tensor obtained in the case of a pure shear ﬂow differed from those determined by Harper & Chang ( 1968 ). In particular the component corresponding to the Saffman’s lift force was found to be approximately 2.3 times larger than predicted by the MAE approach (Harper & Chang 1968 ; Saffman 1968 ). This issue was reconsidered by Miyazaki, Bedeaux & Avalos ( 1995 ), who identiﬁed that a non-algebraic term was unduly neglected by Bedeaux & Rubi ( 1987 ) and Pérez-Madrid et al. ( 1990 ), leading to erroneous results in the quasi-steady limit (except in the solid-**body** rotation case where this term does not contribute to the ﬁnal result). Having dealt with this term through a transformation described later, Miyazaki

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considered first. As a consequence, the trajectory tracking and the balance man- agement tasks dealt with already admissible trajectories. The imitation showed good results in terms of all four tasks. Fig. 2 shows the hands and foot simulta- neous trajectories of scaled human (blue) and humanoid (red) movements during on-line tracking. The distances are in [mm] for the left hand (top), the right hand (middle) and the left foot (bottom). The Cartesian values were synchronized in time, which means that the robot **motion** was performed 1) at the same velocity as the human **motion** and 2) the human movement coordination was respected. All the optimized tasks are in the kernel of the last Jacobian, which means they have equivalent priority in the proposed model. The only way to modify the or- der of priority is the tuning of the gains κ ℓ and κ h . Let us also point out that the

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We proposed several simple lemmas to exclude non-**rigid** binary relations which can be turned into computer programs. These lemmas are quite effective not only in a 4- element domain but also in a k-element domain when k ≥ 5. Some techniques used in recent developments in Barto and Stanovský [37] and Jovanovi´c [40] can also help in this aspect. Applying these rules to a 5-element domain, we could obtain a result as in Table 7.1. It would be interesting to find out all strongly **rigid** binary relations out of the potential list on a 5-element domain.

jacques.gangloff@unistra.fr pierre.renaud@insa-strasbourg.fr
The **rigid**-**body** replacement method is often used when de- signing a compliant mechanism. The stiffness of the com- pliant mechanism, one of its main properties, is then highly dependent on the initial choice of a **rigid**-**body** architecture. In this paper, we propose to enhance the efficiency of the syn- thesis method by focusing on the architecture selection. This selection is done by considering the required mobilities and parallel manipulators in singularity to achieve them. Kine- matic singularities of parallel structures are indeed advan- tageously used to propose compliant mechanisms with inter- esting stiffness properties. The approach is first illustrated by an example, the design of a one degree of freedom compliant architecture. Then the method is used to design a medical device where a compliant mechanism with three degrees of freedom is needed. The interest of the approach is outlined after application of the method.

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