Section: New Results
Stabilization of mobile robots and of nonlinear systems
Participants : Claude Samson, Pascal Morin, Minh-Duc Hua, Tarek Hamel [ Univ. of Nice-Sophia Antipolis ] , Masato Ishikawa [ Univ. of Kyoto ] .
Control of snake-like wheeled robots
We are pursuing the development of the Transverse Function (TF) approach for the control of highly nonlinear systems. In relation to this endeavour, the study of snake-like wheeled robots gives us the opportunity to i) apply and adapt this approach to various mechanical systems for which no feedback control solution existed so far, ii) prolong and generalize the control design methodology associated with it, and iii) propose new paradigms for the control of systems whose motion capabilities are based on the generation of oscillatory (or undulatory) shape changes.
The idea of studying biological systems via the study of man-made robotic ersatz is not new. Nor is the mirror concept of bio-observation-inspiration invoked as an effective way to address difficult problems for which no solid theoretical corpus is yet available. For instance, a significant research effort, started many years ago, is devoted to the control of anthropomorphic and animal-like robots in order to better understand legged locomotion. Crawling locomotion, as examplified and perfected in Nature by snakes, is another complex locomotion mode which, despite decades of scrutiny by different scientific communities, still retains many mysteries. Of particular interest to us is the control of snake-like wheeled mechanisms, proposed by various researchers to better understand crawling locomotion (starting with the pioneering works of Hirose et al. [53] [52] ). Indeed, most of the studies devoted to this theme have focused on the generation of open-loop control strategies yielding simple overall displacements along specified (and specific) directions, whereas attempts to synthesize feedback control laws are few, incomplete and (to our point of view) mostly inconclusive due to the non-existence of adequate control design tools. One of our objectives is to show that the TF approach, and its extensions, provide such tools.
The first mechanism of this kind that we have considered (see last year's report) is the so-called trident snake system depicted on Figure 9 , originally proposed by Ishikawa from the University of Kyoto [56] .
This is a mobile robot with a “parallel” mechanical structure, composed of a triangular-shaped body and three rotary articulations. Given a frame attached to the body, we have derived feedback control laws which ensure the “practical” stabilization of any reference trajectory for this frame, with the complementary property of maintaing the articulation angles 
1, 2, 3 away from values for which the system is kinematically singular. We have validated these controls in simulation, and Ishikawa and his students are currently working on their implementation on physical prototypes. The specific structure of the Control Lie Algebra associated with the kinematic equations of this system also gave us the idea to look for new transverse functions which differ from those that we proposed previously in that they are defined on the rotation group SO(3) rather than on the three-dimensional torus T3 . The better performance observed in simulation when using these new functions comes from the fact that they better respect the system's symmetries. We have subsequently generalized the construction of such functions on SO(n) in relation to the case of a control distribution maximally generated by Lie brackets of order less or equal to two (see next subsection for more details). These results have been published in conference articles [26] [29] .
Another snake-like wheeled robot which attracted our attention this year is Hirose's ACM III snake robot, and more specifically the simplified model composed of three segments studied by Ostrowski and Burdick [69] , depicted on Figure 10 . In fact, one can distinguish several options as for the actuation of this system. The one considered by these authors corresponds to the most general case for modeling purposes. It involves five control inputs, namely two articulation angular velocities 
and three “wheel” angular velocities 
. From the control viewpoint, the more the control inputs, the easier the control problem. A significantly more difficult situation is when all three wheel angles are fixed, so that only the two articulation angular velocities can be modified. This latter case has, for instance, been considered by Ishikawa [57] to illustrate the possibility of switching between a set of piecewise sinusoidal inputs to produce a desired net displacement effect. The increased difficulty –in terms of control– also shows up in the way the Control Lie Algebra is generated. In the five-inputs case, Lie brackets of length up to two are sufficient to nominally satisfy the Lie Algebra Rank Condition for local controllability, whereas one has to go to the length three to obtain the same with only two inputs. As in the case of the trident snake, it is also important to pay attention to mechanical singularities and work out control laws which allow the system to move along any direction while ensuring that singularities are never encountered. Now, independently of the control inputs selection, one of the characteristics of this type of mechanism is that the instantaneous pose (position + orientation) velocity of any of the segments is a function of solely the system's “shape” variables (articulations and wheel angles) and their velocities. In this respect they are alike classical nonholonomic car-like vehicles, with or without trailers, and also alike the trident snake mechanism evoked previously, whose motion can be modeled in the form of driftless control systems with velocity inputs as control variables. In the specialized literature, such systems are sometimes described as purely kinematic systems, by contrast to other systems for which complementary dynamic constraints cannot be eliminated when modeling the system's motion. To distinguish between these two types of systems, one can also refer to purely nonholonomic systems on the one hand, and underactuated systems on the other hand. Although the TF approach has been developed for the first set of (driftless) systems originally, we have also worked on extensions of the approach to systems belonging to the second set (such as the underactuated rigid body in both planar and spatial cases).
The simplified snakeboard model (depicted on Figure 11 ) is a third type of wheeled snake-like mechanism whose modeling equations are used in [69] to illustrate the case of an underactuated undulatory mechanical system. Finding out how to apply the TF approach to design feedback control laws for this system gives us the opportunity to prolong our research program devoted to the extension of the TF approach to non-driftless systems and, at the same time, participate in the multidisciplinary research devoted to the comprehension of undulatory locomotion and its control.
The studies evoked above about the control of various wheeled snake-like robots in relation to the development of the TF approach have progressed well, each of them yielding solutions to previously unsolved problems. It remains to finalize them and report them in articles, the first of which will be submitted for publication during next year's first quarter.
Transverse functions on special orthogonal groups
The transverse function (TF) approach relies on a theorem, first proved in [9] , that establishes the equivalence between the two following properties: i) a family of smooth vector fields satisfies the Lie Algebra Rank Condition (LARC) at a point, ii) there exist smooth functions, defined on a torus of adequate dimension, which are transverse to this family of vector fields. The design of such “transverse functions” is an important issue because, when applying the TF approach to the control of a system, the behavior of the controlled system strongly depends on the choice of the transverse function itself. Accordingly, in the last few years we have conducted several studies in order to design transverse functions that allow to achieve some desirable properties (like e.g. the asymptotic stabilization of feasible trajectories [11] , [23] ). In all these studies transverse functions were defined on a torus, as suggested by [9] . In the present study, we show that transverse functions can also be defined on other compact manifolds, like the special orthogonal groups SO(m) . More precisely, given a family of smooth vector fields defined on a m -dimensional manifold M , we show that the two following conditions are equivalent: i) these vector fields satisfy the LARC at the order one at some point p (i.e. the vector fields together with their first-order Lie brackets span the tangent space of M at p ), ii) there exist smooth functions defined on SO(m) , transverse to these vector fields. From a practical point of view, these new functions have been instrumental in controlling several mechanisms like the rolling sphere or snake-like nonholonomic mechanisms (See Section 6.4.1 above). From a theoretical point of view, this result opens the door to new research investigations concerning the characterization of manifolds that can be used as definition domains of transverse functions. The results of this study are reported in the paper [29] , presented at the IEEE Conference on Decision and Control (CDC).
Control and attitude estimation of aerial vehicles
We have continued our work on the control and attitude estimation of aerial vehicles. We are more specifically interested in small VTOL (Vertical Take-Off and Landing) vehicles which raise several difficulties from a control and attitude estimation point of view, due to a combination of factors: high sensitivity to wind gusts (which can provoke important accelerations in both position and orientation), limited actuation power compared to the intensity of aerodynamic effects, use of low cost/low weight sensors which do not provide high quality measurements, low signal-to-noise ratio of (raw) GPS absolute position measurements in the case of (quasi) stationnary flight. From a feedback control point of view, we have complemented our study recently published in the IEEE Transactions on Automatic Control [22] by addressing the combined stabilization of the vehicle's horizontal linear velocity and altitude. This corresponds to a typical teleoperated control mode for aerial vehicles, with the linear horizontal velocity being measured by GPS or pitot-tubes, and the altitude by a barometer. This result is reported in the memoire of Minh-Duc Hua's Ph.D. thesis [21] . We have also continued the work initiated last year on the attitude estimation problem. Attitude estimation is typically obtained by fusing GPS, magnetometers, and IMU (Inertial Measurement Unit) measurements. When the vehicle's linear acceleration is small, it is theoretically possible to reconstruct the vehicle's attitude by using only accelerometers and magnetometers measurements. To build attitude observers/filters, most existing (“classical”) methods rely on this small acceleration assumption. For many VTOL vehicles however, linear accelerations may be important and induce significant errors on the attitude estimation. In a recent paper [65] , Martin and Salaun have proposed a new attitude observer which uses linear velocity measurements (obtained e.g. via a GPS) in order to take into account linear accelerations in the estimation algorithm. The proposed solution shows a significant improvement with respect to classical methods when linear accelerations are not negligible. However, no analysis of the observer's stability and convergence is provided. Motivated by Martin and Salaun's result, we have worked out two new attitude observers based on linear velocity, accelerometers, and magnetometers measurements, for which we have been able to prove stability and convergence properties. The first observer establishes a property of semi-global exponential stability under a high-gain condition. For the second observer, without resorting to high-gain type arguments we show that for any initial condition of the estimator outside a set of zero measure, the estimation errors converge to zero. We also show that the set of “bad” initial conditions (i.e. those for which convergence of the estimation errors cannot be granted) is unstable. Furthermore, in the special case of constant accelerations of the vehicle, almost-global asymptotic stability of the observer is achieved. This is the strongest possible result knowing that there does not exist smooth globally asymptotically stable observers due to the topology of SO(3) . Alike the solution proposed in [65] , simulation results show a net improvement of the attitude estimation with respect to classical solutions when the small linear acceleration assumption does not apply. The results of this study are reported in the paper [29] , presented at the IEEE Conference on Robotics and Automation (ICRA), and in a journal article accepted for publication in Control and Engineering Practice [54] .
