A Weighted Method for Fast Resolution of Strictly Hierarchical Robot Task Specifications Using Exact Penalty Functions

Abstract

Extensive work has been done on efficiently resolving hierarchical robot task specifications that minimize the -2 norm of linear constraint violations, but not for -1 norm, in which there has recently been growing interest for sparse control. Both approaches require solving a cascade of quadratic programs (QP) or linear programs (LP). In this letter, we introduce alternate and more efficient approaches to hierarchical -1 norm minimization by formulating it as a single LP that can be solved by any off-the-shelf solver. The first approach is a recursive method that transforms the lexicographic LP (LLP) into a single objective problem using Lagrangian duality. The second approach, which forms the main focus of this letter, is a weighted method based on the exact penalty method, that is equivalent to the original LLP for a well chosen set of weights.We propose methods to compute and adapt these weights. The algorithms were applied on an interesting dual arm robot task. We discuss and benchmark the computational efficiency of these methods. Simplicity of the weighted method makes it a promising approach for tackling challenging prioritized robot control problems involving a control horizon or nonlinear constraints. Within this letter, we take the first step towards that goal by demonstrating the efficacy of the weighted method on a simpler instantaneous robot control problem with linear constraints.

A Simple Formulation for Fast Prioritized Optimal Control of Robots using Weighted Exact Penalty Functions

“…With quadratic penalties are used, the weight on the higher priority penalty needs to be infinitely higher than the weight on a lower priority penalty [15] for the weighted method to obtain a lexicographically optimal solution. This is practically infeasible, hence the existing approaches [9]- [12] use only the sequential method.…”

“…The weighted method is promising compared to the previous approaches as only one NLP needs to be solved unlike the sequential method. But it is known that the weighted method is not applicable for quadratic penalties because it requires extensive tuning [11] and also the higher priority weights need to be infinitely higher than the lower priority weights [15] to obtain strict prioritization, which is numerically infeasible. However, for exact penalty functions, such as the 1 norm, correctly chosen finite valued weights are sufficient to enforce strict prioritization.…”

“…Therefore, it is worth exploring whether it is possible to tune weights such that they remain sufficiently high for all the MPC control instances of a task, even if the weights are not optimally small. It was found that a simple combination of manual tuning and weight adaptation worked well for prioritized task-space constraints [15] and compared favourably against the sequential methods in terms of computational performance. In this paper, we extend this idea to control with a predictive horizon.…”

Kinematic Control of Redundant Robots With Online Handling of Variable Generalized Hard Constraints

“…1) Cartesian constraints at the elbow: In the first experiment, the desired EE task is to track three times a 3D circle (m = 3), starting on the path with the initial joint configuration q 0 = (13.50 − 7.76 55. 16 A single control point of dimension d 1 = 2 is considered at the robot elbow (joint 4), which has to satisfy the temporal constraints (21) Figure 8 shows how the robot executes the desired task by complying with the Cartesian constraints. In Fig.…”

“…Otherwise, the relaxation of these constraints in a least square sense leads to a physical violation of the hard limits. As an alternative to the previously mentioned QP approaches that minimize squared ℓ 2 -norms, using the ℓ 1 -norm minimization with suitable penalties offers a computationally efficient solution for hierarchical robot task control [16].…”