Approximation of the Step-to-Step Dynamics Enables Computationally Efficient and Fast Optimal Control of Legged Robots

2021 ◽  
Author(s):  
Pranav Bhounsule ◽  
Myunghee Kim ◽  
Adel Alaeddini
Keyword(s):  
Optimal Control ◽  
Legged Robots ◽  
Computationally Efficient ◽  
Step Dynamics

Approximation of the Step-to-Step Dynamics Enables Computationally Efficient and Fast Optimal Control of Legged Robots

2020 ◽  
Author(s):  
Pranav A. Bhounsule ◽  
Myunghee Kim ◽  
Adel Alaeddini
Keyword(s):  
Optimal Control ◽  
Regression Model ◽  
Closed Form ◽  
Legged Robots ◽  
Input Output ◽  
Output Data ◽  
Region Of Stability ◽  
Step Dynamics ◽  
Step Control ◽  
Closed Form Approximation

Abstract Legged robots with point or small feet are nearly impossible to control instantaneously but are controllable over the time scale of one or more steps, also known as step-to-step control. Previous approaches achieve step-to-step control using optimization by (1) using the exact model obtained by integrating the equations of motion, or (2) using a linear approximation of the step-to-step dynamics. The former provides a large region of stability at the expense of a high computational cost while the latter is computationally cheap but offers limited region of stability. Our method combines the advantages of both. First, we generate input/output data by simulating a single step. Second, the input/output data is curve fitted using a regression model to get a closed-form approximation of the step-to-step dynamics. We do this model identification offline. Next, we use the regression model for online optimal control. Here, using the spring-load inverted pendulum model of hopping, we show that both parametric (polynomial and neural network) and non-parametric (gaussian process regression) approximations can adequately model the step-to-step dynamics. We then show this approach can stabilize a wide range of initial conditions fast enough to enable real-time control. Our results suggest that closed-form approximation of the step-to-step dynamics provides a simple accurate model for fast optimal control of legged robots.

scholarly journals One-Step Deadbeat Control of a 5-Link Biped Using Data-Driven Nonlinear Approximation of the Step-to-Step Dynamics

Robotics ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 90
Author(s):  
Pranav A. Bhounsule ◽  
Ernesto Hernandez-Hinojosa ◽  
Adel Alaeddini
Keyword(s):  
Torque Control ◽  
Deadbeat Control ◽  
Computationally Efficient ◽  
Foot Placement ◽  
Affine Model ◽  
Computed Torque Control ◽  
Process Error ◽  
One Step ◽  
Step Dynamics ◽  
Computed Torque

For bipedal robots to walk over complex and constrained environments (e.g., narrow walkways, stepping stones), they have to meet precise control objectives of speed and foot placement at every single step. This control that achieves the objectives precisely at every step is known as one-step deadbeat control. The high dimensionality of bipedal systems and the under-actuation (number of joint exceeds the actuators) presents a formidable computational challenge to achieve real-time control. In this paper, we present a computationally efficient method for one-step deadbeat control and demonstrate it on a 5-link planar bipedal model with 1 degree of under-actuation. Our method uses computed torque control using the 4 actuated degrees of freedom to decouple and reduce the dimensionality of the stance phase dynamics to a single degree of freedom. This simplification ensures that the step-to-step dynamics are a single equation. Then using Monte Carlo sampling, we generate data for approximating the step-to-step dynamics followed by curve fitting using a control affine model and a Gaussian process error model. We use the control affine model to compute control inputs using feedback linearization and fine tune these using iterative learning control using the Gaussian process error enabling one-step deadbeat control. We demonstrate the approach in simulation in scenarios involving stabilization against perturbations, following a changing velocity reference, and precise foot placement. We conclude that computed torque control-based model reduction and sampling-based approximation of the step-to-step dynamics provides a computationally efficient approach for real-time one-step deadbeat control of complex bipedal systems.

Control of Down-Hole Drilling Process Using a Computationally Efficient Dynamic Programming Method

2018 ◽  
Vol 140 (10) ◽  
Author(s):  
Chong Ke ◽  
Xingyong Song
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Oil And Gas ◽  
Dynamic Programming Method ◽  
Hole Drilling ◽  
Drilling Process ◽  
Computationally Efficient ◽  
Dynamics Model ◽  
Control Optimization ◽  
Drilling System

The unconventional down-hole resources such as shale oil and gas have gradually become a critical form of energy supply thanks to the recent petroleum technology advancement. Its economically viable and reliable production highly depends on the proper operation and control of the down-hole drilling system. The trend of deeper drilling in a complex environment requires a more effective and reliable control optimization scheme, either for predrilling planning or for online optimal control. Given the nonlinear nature of the drilling system, such an optimal control is not trivial. In this paper, we present a method based on dynamic programming (DP) that can lead to a computationally efficient drilling control optimization. A drilling dynamics model that can enable this method is first constructed, and the DP algorithm is customized so that much improved computational efficiency can be achieved compared with using standard DP. A higher-order dynamics model is then used to validate the effectiveness of the optimized control, and the control robustness is also evaluated by adding perturbations to the model. The results verify that the proposed approach is effective and efficient to solve the down-hole drilling control optimization problem.

Unscented MPSP for Optimal Control of a Class of Uncertain Nonlinear Dynamic Systems

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
S. Mathavaraj ◽  
Radhakant Padhi
Keyword(s):  
Optimal Control ◽  
Initial Conditions ◽  
Mean Value ◽  
Control Synthesis ◽  
Benchmark Problems ◽  
Computationally Efficient ◽  
Efficient Manner ◽  
The Mean ◽  
Low Dimensional ◽  
Time Invariant

A new computationally efficient nonlinear optimal control synthesis technique, named as unscented model predictive static programming (U-MPSP), is presented in this paper that is applicable to a class of problems with uncertainties in time-invariant system parameters and/or initial conditions. This new technique is a fusion of two recent ideas, namely MPSP and Riemann–Stieltjes optimal control problems. First, unscented transform is utilized to construct a low-dimensional finite number of deterministic problems. The philosophy of MPSP is utilized next so that the solution can be obtained in a computational efficient manner. The control solution not only ensures that the terminal constraint is met accurately with respect to the mean value, but it also ensures that the associated covariance matrix (i.e., the error ball) is minimized. Significance of U-MPSP has been demonstrated by successfully solving two benchmark problems, namely the Zermelo problem and inverted pendulum problem, which contain parametric and initial condition uncertainties.

A Generalized Time-Optimal Bidirectional Scan Algorithm for Constrained Feed-Rate Optimization

2005 ◽  
Vol 128 (2) ◽  
pp. 379-390 ◽  
Author(s):  
J. Dong ◽  
J. A. Stori
Keyword(s):  
Optimal Control ◽  
Optimization Algorithm ◽  
Feed Rate ◽  
Optimal Solution ◽  
Global Optimality ◽  
Computationally Efficient ◽  
Time Optimal ◽  
Optimal Feed ◽  
Rate Optimization ◽  
Feed Rate Optimization

The problem of generating an optimal feed-rate trajectory has received a significant amount of attention in both the robotics and machining literature. The typical objective is to generate a minimum-time trajectory subject to constraints such as system limitations on actuator torques and accelerations. However, developing a computationally efficient solution to this problem while simultaneously guaranteeing optimality has proven challenging. The common constructive methods and optimal control approaches are computationally intensive. Heuristic methods have been proposed that reduce the computational burden but produce only near-optimal solutions with no guarantees. A two-pass feedrate optimization algorithm has been proposed previously in the literature by multiple researchers. However, no proof of optimality of the resulting solution has been provided. In this paper, the two-pass feed-rate optimization algorithm is generalized and a proof of global optimality is provided. The generalized algorithm maintains computational efficiency, and supports the incorporation of a variety of state-dependent constraints. By carefully arranging the local search steps, a globally optimal solution is achieved. Singularities, or critical points on the trajectory, which are difficult to deal with in optimal control approaches, are treated in a natural way in the generalized algorithm. A detailed proof is provided to show that the algorithm does generate a globally optimal solution under various types of constraints. Several examples are presented to illustrate the application of the algorithm.

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