705 research outputs found
Constrained LQR Using Online Decomposition Techniques
This paper presents an algorithm to solve the infinite horizon constrained
linear quadratic regulator (CLQR) problem using operator splitting methods.
First, the CLQR problem is reformulated as a (finite-time) model predictive
control (MPC) problem without terminal constraints. Second, the MPC problem is
decomposed into smaller subproblems of fixed dimension independent of the
horizon length. Third, using the fast alternating minimization algorithm to
solve the subproblems, the horizon length is estimated online, by adding or
removing subproblems based on a periodic check on the state of the last
subproblem to determine whether it belongs to a given control invariant set. We
show that the estimated horizon length is bounded and that the control sequence
computed using the proposed algorithm is an optimal solution of the CLQR
problem. Compared to state-of-the-art algorithms proposed to solve the CLQR
problem, our design solves at each iteration only unconstrained least-squares
problems and simple gradient calculations. Furthermore, our technique allows
the horizon length to decrease online (a useful feature if the initial guess on
the horizon is too conservative). Numerical results on a planar system show the
potential of our algorithm.Comment: This technical report is an extended version of the paper titled
"Constrained LQR Using Online Decomposition Techniques" submitted to the 2016
Conference on Decision and Contro
Real-Time Motion Planning of Legged Robots: A Model Predictive Control Approach
We introduce a real-time, constrained, nonlinear Model Predictive Control for
the motion planning of legged robots. The proposed approach uses a constrained
optimal control algorithm known as SLQ. We improve the efficiency of this
algorithm by introducing a multi-processing scheme for estimating value
function in its backward pass. This pass has been often calculated as a single
process. This parallel SLQ algorithm can optimize longer time horizons without
proportional increase in its computation time. Thus, our MPC algorithm can
generate optimized trajectories for the next few phases of the motion within
only a few milliseconds. This outperforms the state of the art by at least one
order of magnitude. The performance of the approach is validated on a quadruped
robot for generating dynamic gaits such as trotting.Comment: 8 page
Imprecise dynamic walking with time-projection control
We present a new walking foot-placement controller based on 3LP, a 3D model
of bipedal walking that is composed of three pendulums to simulate falling,
swing and torso dynamics. Taking advantage of linear equations and closed-form
solutions of the 3LP model, our proposed controller projects intermediate
states of the biped back to the beginning of the phase for which a discrete LQR
controller is designed. After the projection, a proper control policy is
generated by this LQR controller and used at the intermediate time. This
control paradigm reacts to disturbances immediately and includes rules to
account for swing dynamics and leg-retraction. We apply it to a simulated Atlas
robot in position-control, always commanded to perform in-place walking. The
stance hip joint in our robot keeps the torso upright to let the robot
naturally fall, and the swing hip joint tracks the desired footstep location.
Combined with simple Center of Pressure (CoP) damping rules in the low-level
controller, our foot-placement enables the robot to recover from strong pushes
and produce periodic walking gaits when subject to persistent sources of
disturbance, externally or internally. These gaits are imprecise, i.e.,
emergent from asymmetry sources rather than precisely imposing a desired
velocity to the robot. Also in extreme conditions, restricting linearity
assumptions of the 3LP model are often violated, but the system remains robust
in our simulations. An extensive analysis of closed-loop eigenvalues, viable
regions and sensitivity to push timings further demonstrate the strengths of
our simple controller
A Parallel Dual Fast Gradient Method for MPC Applications
We propose a parallel adaptive constraint-tightening approach to solve a
linear model predictive control problem for discrete-time systems, based on
inexact numerical optimization algorithms and operator splitting methods. The
underlying algorithm first splits the original problem in as many independent
subproblems as the length of the prediction horizon. Then, our algorithm
computes a solution for these subproblems in parallel by exploiting auxiliary
tightened subproblems in order to certify the control law in terms of
suboptimality and recursive feasibility, along with closed-loop stability of
the controlled system. Compared to prior approaches based on constraint
tightening, our algorithm computes the tightening parameter for each subproblem
to handle the propagation of errors introduced by the parallelization of the
original problem. Our simulations show the computational benefits of the
parallelization with positive impacts on performance and numerical conditioning
when compared with a recent nonparallel adaptive tightening scheme.Comment: This technical report is an extended version of the paper "A Parallel
Dual Fast Gradient Method for MPC Applications" by the same authors submitted
to the 54th IEEE Conference on Decision and Contro
Smooth Model Predictive Control with Applications to Statistical Learning
Statistical learning theory and high dimensional statistics have had a
tremendous impact on Machine Learning theory and have impacted a variety of
domains including systems and control theory. Over the past few years we have
witnessed a variety of applications of such theoretical tools to help answer
questions such as: how many state-action pairs are needed to learn a static
control policy to a given accuracy? Recent results have shown that continuously
differentiable and stabilizing control policies can be well-approximated using
neural networks with hard guarantees on performance, yet often even the
simplest constrained control problems are not smooth. To address this void, in
this paper we study smooth approximations of linear Model Predictive Control
(MPC) policies, in which hard constraints are replaced by barrier functions,
a.k.a. barrier MPC. In particular, we show that barrier MPC inherits the
exponential stability properties of the original non-smooth MPC policy. Using a
careful analysis of the proposed barrier MPC, we show that its smoothness
constant can be carefully controlled, thereby paving the way for new sample
complexity results for approximating MPC policies from sampled state-action
pairs.Comment: 15 pages, 1 figur
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