Feedback synthesis for underactuated systems using sequential second-order needle variations

Abstract

This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actionsthe second-order needle variations of optimal control-as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Lastly, the underactuated dynamic underwater vehicle model demonstrates convergence even in the presence of a velocity field.

Local Koopman Operators for Data-Driven Control of Robotic Systems

“…We use the data-trained Koopman operator to implement linear feedback control (LQR) for tracking. Using the weight matrices Q and R, which penalize the tracking error and control effort, respectively, we define the minimization problem (11) and calculate the infinite-horizon LQR gains. Contrary to work in [49], and in order to illustrate the simplicity and robustness of the proposed scheme, we keep the same weights across all different tasks, such that the same LQR gains, (unless the model is updated) are used in every type of trajectory.…”

“…In fact, several of these methods have already been explored in underwater tasks using robotic fish of different morphologies. Researchers have performed maneuvering, speed and orientation control, collision-avoidance, point-to-point navigation as well as velocity and position tracking using a myriad of control schemes, such as PID [12], [13], LQR [14], SAC [11], [15], fuzzy control [16], [17], geometric control [18], [19], sliding mode control [20], NMPC [21], feedback linearization [22], backstepping control [23] or even a combination of the above [24], [25]. However, the aforementioned methods are either system-specific, apply to dynamics with certain structures, or are computationally prohibitive for real-time identification and control of resource-constrained robots.…”

“…Optimal control theory has reached a level of maturity such that there are a number of available schemes suitable for problems with known dynamics. Examples of such include linear quadratic regulator (LQR) [1], linear model predictive control (LMPC) [2], nonlinear model predictive control (NMPC) [3], feedback linearization [4], differential dynamic programming (DDP) [5], sequential action control (SAC) [6] and variants of the above [7]- [9]. The plethora of available techniques allows one to compute satisfactory solutions for nonlinear and high-dimensional systems.…”

“…These challenges call for feedback policies that can learn or adapt online to unmodeled changes [2] and balance model accuracy and computational efficiency. One can draw from many control schemes, including linear quadratic regulator (LQR) [3], linear model predictive control (LMPC) [4], nonlinear model predictive control (NMPC) [5], feedback linearization [6], differential dynamic programming (DDP) [7], sequential action control (SAC) [8] and variants of the above [9]- [11]. In fact, several of these methods have already been explored in underwater tasks using robotic fish of different morphologies.…”

Iterative Sequential Action Control for Stable, Model-Based Control of Nonlinear Systems

“…(related to controllability-see [24]). Also, we assume that u nom i is not an optimizer of (2) in the current time step (usually selected as constant or zero-see Remark 1), and that J x * i (t) = 0, i.e., system trajectories have not already converged to the desired equilibrium.…”