In this work, the author presents a method called Convex Model Predictive Control (CMPC) to control systems whose states are elements of the rotation matrices SO(n) for n = 2, 3. This is done without charts or any local linearization, and instead is performed by operating over the orbitope of rotation matrices. This results in a novel model predictive control (MPC) scheme without the drawbacks associated with conventional linearization techniques such as slow computation time and local minima. Of particular emphasis is the application to aeronautical and vehicular systems, wherein the method removes many of the trigonometric terms associated with these systems’ state space equations. Furthermore, the method is shown to be compatible with many existing variants of MPC, including obstacle avoidance via Mixed Integer Linear Programming (MILP).
An alternative paradigm lies in optimization, primarily based on a Model Predictive Control (MPC) framework, in which a model of the system is used to optimize a time-discretized trajectory into a finite horizon of the future. References @cite_18 and @cite_13 shares many similarities with our work, adopting a Mixed Integer framework to allow for obstacles, and can be considered an antecedent of this work. Where our work differs is in our use of the convex hull of @math , and @math to constrain the motion of the system, which had previously required an infinite number of Linear Program constraints and had thus only been approximated. Additionally, we develop the method further, incorporating integrator dynamics for our UAV and spacecraft examples. The work of @cite_23 also uses convex programming for model predictive control. In particular, a norm constraint on the control thrust is relaxed, with the relaxation shown to be tight. Going further in examining the Lie group structure of these systems, @cite_6 develops a functional approach to the problem that circumvents the need for time discretization and grapples with the manifold structure of the group directly.
@cite_18 @cite_13 @cite_6 @cite_23
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We investigate the problem of planning under uncertainty, with application to mobile robotics. We propose a probabilistic framework in which the robot bases its decisions on the generalized belief, which is a probabilistic description of its own state and of external variables of interest. The approach naturally leads to a dual-layer architecture: an inner estimation layer, which performs inference to predict the outcome of possible decisions; and an outer decisional layer which is in charge of deciding the best action to undertake. Decision making is entrusted to a model predictive control MPC scheme. The formulation is valid for general cost functions and does not discretize the state or control space, enabling planning in continuous domain. Moreover, it allows to relax the assumption of maximum likelihood observations: predicted measurements are treated as random variables, and binary random variables are used to model the event that a measurement is actually taken by the robot. We successfully apply our approach to the problem of uncertainty-constrained exploration, in which the robot has to perform tasks in an unknown environment, while maintaining localization uncertainty within given bounds. We present an extensive numerical analysis of the proposed approach and compare it against related work. In practice, our planning approach produces smooth and natural trajectories and is able to impose soft upper bounds on the uncertainty. Finally, we exploit the results of this analysis to identify current limitations and show that the proposed framework can accommodate several desirable extensions. There is extensive literature on using convex optimization to derive piece-wise polynomial trajectories for controlling differential flat systems with applications to three-dimensional flight for Micro Aerial Vehicles. In this work, we propose a method to formulate trajectory generation as a quadratic program (QP) using the concept of a Safe Flight Corridor (SFC). The SFC is a collection of convex overlapping polyhedra that models free space and provides a connected path from the robot to the goal position. We derive an efficient convex decomposition method that builds the SFC from a piece-wise linear skeleton obtained using a fast graph search technique. The SFC provides a set of linear inequality constraints in the QP allowing real-time motion planning. Because the range and field of view of the robot's sensors are limited, we develop a framework of Receding Horizon Planning , which plans trajectories within a finite footprint in the local map, continuously updating the trajectory through a re-planning process. The re-planning process takes between 50 to 300 ms for a large and cluttered map. We show the feasibility of our approach, its completeness and performance, with applications to high-speed flight in both simulated and physical experiments using quadrotors. We present a Model-Predictive Control (MPC) system for online synthesis of interactive and physically valid character motion. Our system enables a complex (36-DOF) 3D human character model to balance in a given pose, dodge projectiles, and improvise a get up strategy if forced to lose balance, all in a dynamic and unpredictable environment. Such contact-rich, predictive and reactive motions have previously only been generated offline or using a handcrafted state machine or a dataset of reference motions, which our system does not require. For each animation frame, our system generates trajectories of character control parameters for the near future --- a few seconds --- using Sequential Monte Carlo sampling. Our main technical contribution is a multimodal, tree-based sampler that simultaneously explores multiple different near-term control strategies represented as parameter splines. The strategies represented by each sample are evaluated in parallel using a causal physics engine. The best strategy, as determined by an objective function measuring goal achievement, fluidity of motion, etc., is used as the control signal for the current frame, but maintaining multiple hypotheses is crucial for adapting to dynamically changing environments. More than 15 years after model predictive control (MPC) appeared in industry as an effective means to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge. The issues of feasibility of the on-line optimization, stability and performance are largely understood for systems described by linear models. Much progress has been made on these issues for non-linear systems but for practical applications many questions remain, including the reliability and efficiency of the on-line computation scheme. To deal with model uncertainty ‘rigorously’ an involved dynamic programming problem must be solved. The approximation techniques proposed for this purpose are largely at a conceptual stage. Among the broader research needs the following areas are identified: multivariable system identification, performance monitoring and diagnostics, non-linear state estimation, and batch system control. Many practical problems like control objective prioritization and symptom-aided diagnosis can be integrated systematically and effectively into the MPC framework by expanding the problem formulation to include integer variables yielding a mixed-integer quadratic or linear program. Efficient techniques for solving these problems are becoming available.