Many interesting dynamic systems in science and engineering evolve on a nonlinear, or curved, space that cannot be globally identified with a linear vector space. Such nonlinear spaces are referred to as manifolds, and they appear in various mechanical systems such as a simple pendulum, rotational attitude dynamics of a rigid body, or complex multibody systems.
Our research is focused on constructing computational geometric methods for dynamics and control of mechanical systems evolving on a nonlinear manifold. The central idea is constructing nonlinear control systems directly on nonlinear manifolds, and verifying their performances numerically by using geometric numerical integrators that preserve the underlying geometric properties of a dynamic system. All of these are expressed in a coordinate-free form to avoid any singularities, ambiguities, and complexities inherent to local parameterizations. These allow us to study non-local, large, and aggressive motions of complex dynamic systems globally in an accurate and efficient fashion over a long period.
Computational Geometric Control and OptimizationBy explicitly utilizing geometric properties on a nonlinear manifold, this framework yields a systematic method to construct geometric control systems that are particularly useful for autonomous acrobatic maneuvers of complex aerospace systems. Applications include optimal spacecraft formation reconfiguration, robust adaptive control of the attitude dynamics of a rigid satellite, and complex maneuvers of a quadrotor UAV.
Quadrotor UAVs have been envisaged for various applications such as surveillance or mobile sensor networks due to its simple mechanical structures. However, most of the existing controllers exhibit singularities when representing complex rotational maneuvers of a quadrotor UAV, thereby fundamentally restricting their ability to follow nontrivial trajectories.
We developed a geometric control system to stabilize six translational and rotational degrees of freedom of a quadrotor UAV using four thrust inputs, while asymptotically tracking four outputs consisting of its position and heading direction. Almost global exponential attractiveness is guaranteed, and numerical studies illustrate that this can generate unprecedented challenging maneuvers, such as multiple flipping, in a unified way. Currently, a hardware system based on a linux computer module is also under development for an experimental demonstration of extreme maneuverability.
This method has substantial computational advantages. Numerical solutions obtained by discrete mechanics are more faithful to general non-geometric integrators, and consequently more accurate solutions of optimal control problems are obtained. We also find optimal trajectories efficiently as this is free of artificial numerical dissipations introduced by conventional numerical integration schemes.
Computational Geometric MechanicsGeometric numerical integration deals with numerical integration methods that preserve geometric properties of a dynamic system, such as invariants, symmetries, reversibility, or the structure of the configuration manifold. The geometric structure of a dynamic system determines its qualitative dynamical behavior, however, general purpose numerical integrators fail to provide reasonable computational results. For example, numerical simulations based on the conventional implicit Euler method predict that the outer solar system collapses over a time period of several hundred years, but an integration over a hundred million years does not deteriorate with geometric integrators.
We developed geometric numerical integrators, referred to as Lie group variational integrators for mechanical systems evolving on a Lie group, with applications to complex multibody systems. They are developed from the perspective that a geometric numerical integrator for a Lagrangian or Hamiltonian system is a discrete-time Lagrangian or Hamiltonian mechanical system. Therefore, their numerical solutions are consistent with continuous time dynamic characteristics: they are symplectic and momentum preserving, and they exhibit good energy behavior over a long time period. They also preserve the Lie group structures as the update algorithm is based on the corresponding group operation. A recent benchmark studies show that they have substantial advantages in computational accuracy and efficiency: when compared to other geometric integrators, the speedup factor varies from 16 times to 98 times.
Due to these computation advantages, Lie group variational integrators have been applied to various multibody systems. Most importantly, it has been used to study the dynamics of the binary near-Earth asteroid 66391 (1999 KW4) at the Jet Propulsion Laboratory, NASA. Other examples include spacecraft formation flight, rigid bodies connected by ball joints, articulated rigid bodies in a perfect fluid, and tethered spacecraft with a reeling mechanism.
Computational Geometric Uncertainty Propagation and EstimationThis is to develop computational geometric methods to propagate uncertainties through nonlinear Hamiltonian flows with stochastic excitations that evolve on nonlinear Lie group configuration spaces. The main objective is to derive computational methods that preserve the underlying stochastic and geometric properties of a Hamiltonian system to obtain long-term structural stability in the numerical uncertainty propagation. By incorporating mesh-free computational techniques that are globally valid on a Lie group, we obtained numerical algorithms that are well suited to propagating uncertainties along nontrivial trajectories of complex Hamiltonian systems over a long-time period. These represent significant advances over current computational methods that are restricted to moderate trajectories, simple dynamic properties, and short propagation times.
We also developed attitude estimation algorithm directly on the special orthogonal group to avoid complexities, singularities, and ambiguities that appear in other attitude representation such as quaternions and Euler angles.