## Section: New Results

### Control of mobile robots

Participants : Claude Samson, Pascal Morin, Minh-Duc Hua, Daniele Pucci, Glauco Scandaroli, Luca Marchetti, Tarek Hamel [Univ. of Nice-Sophia Antipolis] .

#### Control of two-steering-wheels mobile robots with the Transverse Function approach

This study is about the control of ground vehicles with independent front and rear steering wheels. At the kinematical level, this system has three independent control inputs, namely the vehicle's longitudinal velocity along the direction joining the steering wheels axles and the front and rear steering-wheels angular velocities (see figure 7 ).

With respect to classical car-like vehicles with a single steering train, this type of vehicle provides superior maneuvering capabilities and the possibility of orienting the main vehicle's body independently of the translational motion direction. This can be used, for instance, to transport large payloads without changing the payload's orientation with respect to a fixed frame in order to minimize energy consumption. From the control viewpoint, assuming that classical *rolling-without-slipping* nonholonomic constraints are satisfied at the wheel/ground contact level, the kinematical equations of this type of vehicle yield a locally controllable five-dimensional nonholonomic (driftless) system with SE(2)×S^{1}×S^{1} as its configuration space. A complementary constraint, here imposed, is that singular mechanical configurations when either one of the wheel angles _{1} and _{2} is equal to ±/2 must be avoided whatever the desired gross displacement of the vehicle in the plane. This implies that certain reference trajectories in the SE(2) can be stabilized only “practically” by making maneuvers, just as in the case of a car accomplishing sideways lateral displacements. The Transverse Function approach applies to this system whose structure is close to the one of a car with two control inputs in the sense that it is also locally equivalent to a homogeneous (nilpotent) system invariant on a Lie group. However, its Lie Algebra is generated differently due to the third control input. In particular, only Lie brackets of the control vector fields up to the order one are needed to satisfy the Lie Algebra Rank Condition (LARC) –the local controllability condition– at any point, whereas Lie brackets of order two are necessary in the car case. This property reflects the symmetric steering action of the front and rear wheels on the vehicle, and it is of practical importance. In order to respect this symmetry, one is led to consider transverse functions defined on the three-dimensional special orthogonal group SO(3) , rather than on the two-dimensional torus –a solution used in the car case, for instance. Therefore, after the trident snake studied in [52] , and the rolling sphere studied in [63] , this is another example of a mechanical system for which the use of transverse functions defined on SO(3) is natural. As a matter of fact, this example presents the complementary interest, and complication, of involving transverse functions defined on a manifold whose dimension, equal to three, is not minimal. The corresponding extra degree of freedom thus has to be taken into account at the control design level and, if possible, be used effectively. For instance, a desirable feature is to ensure the asymptotic stabilization of *feasible* trajectories for which more classical control solutions, such as those proposed in [62] , apply. In the end one obtains a unique feedback control law which ensures the avoidance of mechanical singularities, the *practical* stabilization of *any* (feasible or non-feasible) trajectory in SE(2) , including fixed points, and the *asymptotic* stabilization of feasible trajectories for which this objective is achievable by using classical feedback control techniques –typically when adequate conditions of *persistent excitation* upon the reference longitudinal velocity are satisfied. The results of this study will soon be submitted for presentation at an international conference. This system will also be used as an illustrating example in a journal paper adressing the construction of transverse functions on special orthogonal groups.

#### Development of an automated shopping cart

An "Action d'Envergure Nationale" (AEN) called "PAL" (for Personnal Assistant Living) has been initiated this year. This AEN regroups INRIA robotics teams with the aim of developing robotic devices that can assist elderly and disabled people in their everyday life. The AROBAS team takes part in this AEN through the development of an automatic shopping cart. The subject involves feedback control of nonholonomic systems, pose reconstruction via vision sensors, and obstacle avoidance in dynamic environments. Luca Marchetti has recently started a post-doc on this topic within the AROBAS team. Experiments will be conducted on the ANG walker developed by the COPRIN team.

#### Time sub-optimal nonlinear PI and PID controllers applied to longitudinal headway control

Proportional integral (PI) and proportional integral derivative (PID) controllers are at the heart of control engineering practice and, owing to their relative simplicity and satisfactory performance for a wide range of processes, have become the standard controllers used by industry. Perhaps only 5-10% of man-implemented control loops cannot be controlled by single input, single output (SISO) PI or PID controllers. However, this widespread usage also goes with numerous problems due to either poor tuning practice or limited capabilities offered by standard PI-PID schemes. These problems have in turn periodically revived the interest from the academic research community in order to work out complementary explanations and solutions. In particular, a well-known source of degradation of performance is the occurrence of control saturation, when the boundedness of the “physical” control that can be applied to the system under consideration is no longer compatible with the application of the (theoretically unbounded) calculated control value. This has the consequence of invalidating the performance index established on the assumption of linearity of the controlled system, and can give rise to various undesired (and unnecessary) effects such as multiple bouncing between minimal and maximal values of the control, and important overshoots of the regulated error variables. The so-called integrator wind-up phenomenon, which worsens the overshoot problem and the reduction of which still motivates various research studies is also commonly presented as a consequence of control saturation combined with the integral action incorporated in the control law in order to compensate for unknown (slowly varying) additive perturbations. Compared to the already huge corpus of studies devoted to PI and PID controllers, the present study has the limited ambition of proposing new nonlinear versions of these controllers that attempt to combine the constraints of control saturation with *i)* the objective of optimizing the control action to reduce the size of initially large tracking errors as fast as possible, and *ii)* the design of integral action terms with limited wind-up effects. The former issue is close to the line of research on “proximate” time-optimal for linear systems admitting closed-form time-optimal solutions. The study is restricted to the simplest first and second order linear systems. In particular, continuous nonlinear proportional (P) and proportional derivative (PD) state feedbacks depending continuously on an extra-parameter whose convergence to infinity yields the discontinuous time-optimal controls for these systems are derived and form the cores of the PI and PID controllers proposed subsequently. As for the latter issue, it is related to the work on anti-windup and “conditional integrators”. This work is also related to the theme of bounded control design based on the use of nested saturation functions with the same concern of proving *global* asymptotic stability of the desired set-point, but with a different way of designing the control solutions. In the second part of the study, the proposed nonlinear PID controller is applied to the longitudinal headway control of a car following a leader. The reason for choosing this application is its good fit with the design constraints and objectives imposed on the control and its performance, namely the existence of different bounds on the car's acceleration and deceleration capabilities, control effectiveness in terms of time of convergence to the desired inter-distance between the two vehicles, absence of bouncing transients –for the comfort of the passengers, fuel economy, and reduced wear-off of mechanical parts–, and very small overshoot in order to avoid collisions with the leader. The results of this study are reported in a paper submitted for presentation at an international conference.

#### Control of aerial vehicles subjected to lift forces

We have continued our work on the development of a general theory for the control of underactuated (ground, marine, and aerial) vehicles whose main propulsion relies on a thrust force exerted in a single (vehicle's related) direction. This is the subject on an ongoing thesis research project. We have previously studied the simplified case when environment forces acting on the vehicle do not depend on the vehicle's attitude (orientation), an idealization of which corresponds to the case when the vehicle's shape is spherical [5] . In the first approximation, this property holds for VTOL vehicles such as helicopters and small drones mostly used for their hovering capabilities at reduced velocities. It may also hold for ellipsoidal or tubular shaped rockets and marine vehicles with small external wing-shaped appendages (such as rudders mostly used to modify the vehicle's attitude). But it does not hold for airplanes the flight's basics of which heavily rely on the existence of important lift forces associated with the presence of large wings attached to the main vehicle's body. The work has this year thus focused on the modelling of such lift forces and their taking into account at the control design level. To simplify the study, we have been concentrating on the 2D case, i.e. motion in the vertical plane exclusively. In the long range, one of our ambitions is to encompass within a unified nonlinear control design framework most of the methodology based on linear control techniques on which current marine and airplane autopilots rely and, from there, improve on this methodology.

#### Vision-based control of helicopter drones

Unmaned Aerial Vehicles (UAVs) can be used for many surveillance and monitoring
applications, both indoors and outdoors. Their effectiveness relies in the first place on the
use of embarked sensors that can provide information on the vehicule's *pose* (i.e.
position and orientation). In teleoperated modes, the human operator can compensate
for the lack of some pose information (like, e.g., the vehicule's position). For
fully autonomous control modes, however, information on both position and orientation is
necessary. This is often a challenging problem for small VTOL UAVs (Vertical
Take-Off and Landing vehicles) due to several reasons. For example, no sensor can provide a
direct measure of the 3D-orientation. Also, GPS sensors that are usually used to retrieve position,
do not provide precise and high-rate measurements, and these sensors are not always operational
(like, e.g., urban canyons). Other sensors should be used to improve UAV's effectiveness, especially those providing information about UAV's local environment. One of the most promising alternatives is the use of vision sensors.

In this work, we address the problem of controlling VTOL UAVs' hover flight, based on measurements provided by a single camera and gyrometers only. The solution relies on the measure of the homography matrix associated with the camera's observation of a planar target. There are several challenges associated with this problem. First, since we do not assume any information on the target (like, e.g., size or inclination), the vehicle's pose cannot be extracted from the homography measure. Then, unlike previous works on the subject, we do not assume that the vehicle's orientation can be reconstructed (using, e.g., information on the target or additional sensors). Finally, we do not have any sensor that provides linear velocity measurements either. An homography-based controller is proposed, together with a complete stability and robustness analysis. A remarkable aspect of this solution is that given any lower bound on the distance between the reference hover flight position and the visual target, control gain that guarantee local asymptotic stability of this position can be designed without explicit knowledge of the associated distance. This work has been submitted for publication at the next ICRA (IEEE Conf. on Robotics and Automation), and is part of Henry de Plinval's Ph.D. work. Henry de Plinval is an engineer at ONERA, co-supervised by P. Morin and P. Mouyon (ONERA).