Section: Scientific Foundations
Robot modeling and control
Participants : Claude Samson, Pascal Morin, MinhDuc Hua, Tarek Hamel.
Since robotic, or “robotizable”, mechanisms are structurally nonlinear systems which, in practice, need to be controlled in an efficient and robust manner, the project ARobAS has a natural interest and activities in the domain of Automatic Control related to the theory of control of nonlinear systems. Nonlinear control systems can be classified on the basis of the stabilizability properties of the linear systems which approximate them around equilibrium points. Following [38] , an autonomous controllable nonlinear system is called critical when the corresponding linearized systems are not asymptotically stabilizable (and therefore not controllable either). Whereas local stabilizers for noncritical systems can often be derived from their linear approximations, one has to rely on other –truly nonlinear– methods in the case of critical systems.
For robotic applications, one is concerned in the first place with the design of feedback laws which stabilize statereference trajectories in the sense of ensuring small tracking errors despite adverse phenomena resulting from modeling errors, control discretization, measurement noise,...
The set of critical systems strictly encompasses the one of controllable driftless systems affine in the control input (e.g. kinematic models of nonholonomic wheeled vehicles ). Most of the existing literature on the subject has focused on these latter systems due to their well delimited and understood structural properties. On the other hand, nonlinear controlaffine systems with a drift term which cannot be removed without rendering the system uncontrollable have been much less studied, whereas many locally controllable underactuated mechanical systems (e.g. manipulators with nonactuated degrees of freedom, hovercrafts, blimps, submarines,...) belong to this category of critical systems. However, there exist also underactuated mechanical systems which are not critical in the sense evoked above. Such is the case of flying machines with vertical takeoff capabilities (helicopters, VTOL devices,...) whose linear approximations at an equilibrium are controllable due to the action of an external field of forces (the field of gravity, in the present case). Understandably, the control techniques used for these systems heavily rely on this property eventhough, mathematically, the absence of such a field would not necessarily render the system itself (by opposition to its linear approximation) uncontrollable. This latter observation is important because it means that not all the structural controllability properties of the system have been exploited in the control design. This also implies that general control methods developed for critical systems could be applied to these noncritical systems, with their performance being less critically dependent on the existence and modeling of an external “stabilizing” field. To our knowledge, this research direction has never been explored before.
To summarize, the problem of control of critical nonlinear systems is relevant for most robotic devices other than fullyactuated holonomic manipulators. It is, of course, also relevant for other physical systems presenting similar structural control properties (an example of which are induction electrical motors). We have been advocating for a few years that it needs to be investigated further by developing new control design paradigms and tools. In this respect, our conviction is based on a certain number of elements, a summary of which follows.

Asymptotic stabilization of an equilibrium combining fast convergence (say exponential) and a degree of robustness similar to what can be achieved for linear systems (e.g. stability against structured modeling errors, control discretization, timedelays, and manageable sensitivity w.r.t. noise measurement,...) has never been obtained. Studies that we, and a few other researchers, have conducted towards this goal [41] , [63] , [68] have been rewarded with mitigated success, and we strongly feel now that no solution exists: basically, for these systems, fast convergence rules out robustness.

It is known from [77] that asymptotic stabilization of admissible state trajectories (i.e. trajectories obtainable as solutions to the considered control system) is “generically” solvable by using classical control methods, in the sense that the set of trajectories for which the linear approximation of the associated error system is controllable is dense. Although this is a very interesting result which can (and has been) thoroughly exploited in practice, this is also a delusional result whose limitations have insufficiently been pondered by practitioners. The reason is that it tends to convey the idea that all tracking problems can be solved by applying classical control techniques. The application of Brockett's Theorem [42] to the particular case of a trajectory reduced to a single equilibrium of the system indicates that no smooth purestate feedback can be an asymptotical stabilizer, and thus clearly invalidates this idea. If an asymptotic stabilizer exists, it has to involve a nontrivial dynamic extension of the initial system. Timevarying feedbacks that we have been first to propose [16] to solve this type of problem in the case of nonholonomic systems constitute an example of this. However, solving the problem for fixed equilibria still does not mean that “any” admissible trajectory can be asymptotically stabilized, nor that there exists a “universal” controller, even a complicated one, capable of stabilizing any admissible trajectory –whereas simple solutions to this latter problem are wellknown for linear systems. This lack of completude of the results underlies severe practical implications which have not been sufficiently addressed.

For instance, the nonexistence of a “universal” stabilizer of admissible (feasible) trajectories has been proven in [61] in the case of nonholonomic systems. This result is conceptually important because it definitively ruins the hope of finding a complete solution to the tracking problem (in the usual sense of ensuring asymptotic stabilization), even for the simplest of the critical systems.

To our knowledge, the problem of stabilizing nonadmissible trajectories has never been addressed systematically, even in the case of fullyactuated nonholonomic systems, except by us recently. A decade of active research devoted to the control of these systems (in the 1990's) had left this issue wideopen, eventhough it was known that, for a nonholonomic driftless system, the property of local controllability implies that any continuous nonadmissible trajectory in the state space can be approximated with arbitrary good precision by an admissible trajectory. While several openloop control methods for calculating such an approximation have been proposed by various authors [60] ,[15] , practical stabilization of nonadmissible trajectories –the feedback control version of the problem– seems to have been completely “occulted” by the problem of asymptotic stabilization of admissible trajectories.

The range of feedback control design methods for nonlinear systems, especially those based on geometrical concepts, is limited and needs to be enlarged. Existing methods are often inspired by ideas and techniques borrowed from linear control theory. Whereas this makes good sense when the system is noncritical (including feedback linearizable systems), we contend that critical systems, being structurally different, call for revisiting and adapting the basic concepts and objectives on which control design methods lean. The notion of practical stabilization is an example of such an adaptation.
The objective of practical stabilization is weaker than the classical one of asymptotic stabilization: any asymptotical stabilizer is a practical stabilizer –whereas the converse is not true. However, this objective is not “much” weaker. In particular, instead of ensuring that the error converges to zero, a practical stabilizer ensures that this error is ultimately bounded by some number which can be as small as desired (but different from zero). We assert that this “small” difference in the objective changes everything at the control design level in the sense that none of the obstructions and impossibilities evoked previously holds any more: fast convergence to a set contained in a small neighborhood of the desired state can be achieved in a robust fashion, universal practical stabilizers of state trajectories exist, and, moreover, these trajectories do not have to be admissible. Furthermore, by accepting to weaken the control objective slightly, the set of control solutions is considerably enlarged, so that new control design methods can be elaborated. One of them is the Transverse Function approach that we have initiated a few years ago and that we continue to develop. It is based on a theorem, first published in [9] , which states the equivalence between the satisfaction of the Lie Algebra Rank Condition (LARC) by a set of vector fields and the existence of particular (bounded) periodic functions whose infinitesimal variations are transversal to the directions associated with these vector fields. For control purposes, the timederivatives of the variables on which such transverse functions depend can be used as extra control inputs which facilitate the control of systems whose dynamics are either completely (the case of nonholonomic systems) or partially (the case of underactuated systems) driven by the vector fields with which the transverse function is associated. In the case of mechanical systems, these new control inputs are directly related to the frequency of the “manœuvres” that the system has to perform in order to track a given reference trajectory. With this interpretation in mind, one can say that the approach provides a way of adapting the frequency of the manœuvres automatically.
We have first experimented feedback controllers derived with this approach on our laboratory unicycletype mobile robot with the goal of tracking an omnidirectional vehicle (target) observed by a camera mounted on the robot (visionbased tracking). To our knowledge, this experiment is still unique in its kind. Results that we have obtained show a net improvement with respect to earlier attempts that we had made, based on the use of timevarying feedback techniques [79] . Theoretically, the approach can be applied to any nonholonomic vehicle –carlike vehicles without or with trailers, in particular.
More recently, we have adapted it to the problem of controlling nonholonomic mobile manipulators, i.e. manipulators mounted on nonholonomic mobile platforms, and have derived a general methodology for the coordinated motion of this type of robot [4] . It is based on the concept of omnidirectional companion frame which basically allows to control the mobile platform as if it were omnidirectional. Feedback control laws devised with this methodology have properties never demonstrated before, such as the possibility of ensuring the perfect execution of a manipulation task on a moving object whose motion is not known in advance, with the insurance that the manipulator will never collide into its jointlimits.
Even more recently, we have started to extend the approach to the control of critical underactuated mechanical systems, a problem which is more difficult than the control of fullyactuated nonholonomic systems due to the necessity of including dynamical effects in the modeling equations of the system, yielding a drift term which cannot be treated as a perturbation which can be precompensated. For these systems, the objective is again to practically stabilize any desired trajectory (admissible, or not) defined in the configuration space. To our knowledge, this problem had never been solved before, even for the simplest critical underactuated system (namely, the 3dimensional secondorder chained system). Although we have already much progressed on this subject, and devised a control design method which applies to classical examples of critical underactuated mechanical systems involving a single rigid body [12] , many aspects of the problem have not been explored yet, or need to be studied further. Several are related to the definition and exploitation of criteria to qualify and compare different implementations of the control design method, such as the property of making velocities tend to zero when the reference trajectory is reduced to a fixedpoint. Others concern the applicability and usefulness of the approach when the system is not critical (due to the action of dissipative/friction forces combined with the gravity field, in particular).
Robustness is a central and vast issue for feedback control. Any feedback control design approach has to be justified in terms of the robustness properties which are associated with it. In the case of advanced robotic applications based on the use of exteroceptive sensors, robustness concerns in the first place the capacity of dealing with the imprecise knowledge of the transformations relating the space in which sensor signals live to the Cartesian space in which the robot evolves. A vast literature, including several chapters of [15] and a large part of the publications on visionbased control, has addressed this issue in the case of fully actuated holonomic manipulators. Comparatively, very little has been done on this subject in the case of nonholonomic and underactuated mobile robots. We have thus initiated studies (constituting the core of a PhD work) in order to figure out i) how feedback control schemes based on the use of transverse functions can be adapted to the use of exteroceptive sensors when the above mentioned transformations are not known precisely, and ii) how robust the resulting control laws are. Initial results that we have obtained are encouraging [66] , but the complexity of the analyses also tells us that future research efforts in this direction will have to rely much on simulation and experimentation.