Global control of mechanics on Riemannian manifolds, and applications to under-actuated aerial vehicles
We consider the problem of designing trajectory tracking feedback control laws for La- grangian mechanical systems in a Riemannian geometric framework. Classical nonlinear control techniques that rely on Euclidean parameterizations of nonlinear confguration manifolds, severely restrict the region of operation of the system due to singularities of local coordinate charts. The primary focus of our study is to develop a generic, con- structive and intrinsic (coordinate independent) procedure for global control design such that the closed loop operational envelop is signifcantly enhanced. An important class of systems where the proposed control design techniques have been applied are underactu- ated unmanned aerial vehicles (UAV). Such systems are physically capable of executing aggressive and global (unrestricted) maneuvers as a result of enhanced mechanical de- sign and actuation technology. However, developing autonomous controllers such that the closed loop system can execute such maneuvers is indeed a formidable problem. Part 1: Global control on Riemannian manifolds (chapter 3 and 4): In the frst part of the thesis, we consider simple Lagrangian mechanical control sys- tems evolving on compact Riemannian manifolds, whose coordinate independent Euler- Lagrange equations of motion are established through the Levi-Civita connection corre- sponding to the kinetic-energy metric tensor. When the system is fully actuated), using the Riemannian connection structure, we develop a generic and constructive trajectory tracking feedback control law based on integrator back-stepping where the confguration error is the gradient of the squared geodesic distance between the confguration of the system and the reference trajectory, and the velocity error is the di erence between the velocity of the system and the parallel translation of the velocity of the reference trajec- tory, along a minimal geodesic connecting the confguration of the system and the point on the reference trajectory. The control law is appended with a feed-forward term which is the covariant derivative of the distance-gradient and the parallel-translation term. We demonstrate that this control law achieves asymptotically stable tracking when the confguration of the system is within injectivity radius of the point on the reference tra- jectory. The primary reason is that the control law does not encounter the cut locus, where it is no longer well defned, and around which it is no longer smooth. We then use our study of the compact Riemannian cut locus (which is the primary topological obstruction in global control design) in chapter 2 where we establish certain structural and dynamical properties, and thereby show that the control gains can be chosen large enough such that the confguration of the system does not intersect the cut locus of the point on the reference trajectory for all positive time, provided it starts away from the cut locus (arbitrarily close to it) initially. We thereby extend the region of stability of the above control law to an arbitrarily large domain of the tangent bundle. We then append the control law with a dynamic feedback in order to achieve globally exponentially stable tracking. We now restrict our attention to compact Lie groups which are naturally equipped with a bi-invariant metric structure, which enables us in constructing an elegant and computationally simple version of the generic control law on Riemannian manifolds. Exploiting the isometry of the group action, we convert the global tracking problem to a local tracking problem within the injectivity radius, and a global stabilization problem. Unlike the generic Riemannian case, we show that the components of the control law can be easily computed using only the Lie group structure (i.e. the group actions, exponential and logarithm map). We fnally study the problem of under-actuated differentially constrained mechanical systems where the velocity is constrained to a regular distribution. The equations of motion are established through a metric-compatible connection called the 'constrained connection', which is not necessarily torsion free if the differential constraint is non- integrable (i.e. non-holonomic). We extend the control design techniques previously established, to achieve global output tracking of such systems. Part 2: Application to under-actuated aerial vehicles on SE(3) (chapter 5 and 6): In the second part of the thesis, we apply the geometric control design techniques to two under-actuated aerial vehicles; multi-rotors and thrust vectored vertical take-o and landing (VTOL) aircraft, whose confguration evolves on the Lie group SE(3). In our study of multi-rotors, we frst design a globally-exponentially stable controller for tracking the position and relative heading angle of a quadrotor, which is considered as a rigid body subjected to a force along the body z axis and three torques about the body axes. The equations of motion can be written as a cascade of two subsystems, one which is a fully actuated rotational subsystem on SO(3), the output of which is the input to a translational subsystem on R3. We use the previously described control design on R3 using the bi-invariant metric (from the cannonical Ad-invariant inner product on the Lie algebra) and cascade this control with a saturated thrust feedback control on R3 in order to achieve global asymptotically stable tracking at an exponential rate. This design is then augmented with a fault tolerant strategy, which ensures that the controller continues to track the position of the center of mass of the quadrotor in spite of a rotor failure. This is achieved by relinquishing control of the heading angle, and designing a reduced-attitude control law which tracks the orientation of the thrust axis on S2. We use the global output tracking control design discussed in part 1 to achieve this. The reduced attitude control law is then cascaded with the saturated thrust feedback control on R3 as in the previous case. We then study the bi-spinner problem which is a rigid body with only two fxed co-axial rotors. Such a vehicle is severely under-actuated and therefore global tracking control is indeed a formidable problem. We propose a multi-scale geometric controller under the assumption that the angular velocity of the bispinner about the thrust-axis is signifcantly higher than the other two angular velocity components. We design a control law which globally tracks position trajectories with only two functioning rotors. In our study of thrust vectored VTOL UAVs, we consider an axis-symmetric aerial vehicle subjected to a terminally applied vectored thrust and torque about the axis of symmetry. Typical examples of such vehicles are thrust-propelled rockets, submersible torpedos, tail-sitter drones etc. The diffculty in control design for such a problem is that we can no longer write the equations of motion as a cascade system as we did in the case of multi rotors. The reason is that the control inputs which produce torques for the rotations in SO(3) also generate forces which result in translations in R3. Further, this coupling results in an unstable inverse input-output system (non-minimumphase), which renders the control design problem formidable. In order to resolve this difficulty, we frst impose a non-integrable differential constraint on the system, such that the constrained system admits a differentially at output i.e. The Huygens center-of-oscillation. We reformulate the translational dynamics with respect to this point to convert the tracking control problem into one which involves a cascade system as in the previous case, and apply the reduced attitude and saturated thrust feedback law to achieve global tracking of the center of mass, when the differential constraint is satisfed. This control law is augmented with another component which ensures that the differential constraint is asymptotically stabilized which ensures global asymptotic tracking for all initial conditions in the tangent bundle. Another important factor that the control design adresses is the constraint on the thrust of the vehicle to be strictly bounded above zero. We provide simulation results which illustrate the effectiveness, robustness and global tracking performance of the proposed controllers.