Condition-based Design of Variable Impedance Controllers from User Demonstrations

TL;DR

An approach to ensure conditions on Variable Impedance Controllers through the off-line tuning of the parameters involved in its description is presented, and its application to term modulations defined by a Learning from Demonstration technique is proved.

Abstract

This paper presents an approach to ensure conditions on Variable Impedance Controllers through the off-line tuning of the parameters involved in its description. In particular, we prove its application to term modulations defined by a Learning from Demonstration technique. This is performed through the assessment of conditions regarding safety and performance, which encompass heuristics and constraints in the form of Linear Matrix Inequalities. Latter ones allow to define a convex optimisation problem to analyse their fulfilment, and require a polytopic description of the VIC, in this case, obtained from its formulation as a discrete-time Linear Parameter Varying system. With respect to the current state-of-art, this approach only limits the term definition obtained by the Learning from Demonstration technique to be continuous and function of exogenous signals, i.e. external variables to the robot. Therefore, using a solution-search method, the most suitable set of parameters according to assessment criteria can be obtained. Using a 7-DoF Kinova Gen3 manipulator, validation and comparison against solutions with relaxed conditions are performed. The method is applied to generate Variable Impedance Controllers for a pulley belt looping task, inspired by the Assembly Challenge for Industrial Robotics in World Robot Summit 2018, to reduce the exerted force with respect to a standard (constant) Impedance Controller. Additionally, method agility is evaluated on the generation of controllers for one-off modifications of the nominal belt looping task setup without new demonstrations.

Figures (12)

• Figure 1: From human-guided demonstrations, LfD is used to provide the reference trajectory to follow together with the required compliance profile for the task. This is used to define a Variable Impedance Controller, and the proposed approach (Condition-based Design) provides the set of parameters that complete its definition such that safety and performance conditions are fulfilled. The person appearing in this Figure is the first author and gave permission to use his image for this purpose.
• Figure 2: Generation of the polytopic description of the VIC from controller solution and the H-GP model embedding task demonstrations.
• Figure 3: Graphical representation of the User Preference mechanism for three different cases on the same compliance profile in $K\xspace_{max}-K\xspace_{min}$ plane (a) and the corresponding $K$ profiles (b). Case A has both low Scale and Similarity values, which translates into a stiffness modulation $K\xspace(t)$ with a $K\xspace_{min}$ closer to $\underline{K\xspace}$ and very distinct from $K\xspace_{max}$. For B, both $K\xspace_{min}$ and $K\xspace_{max}$ have higher values than in A as the Scale is higher, but also a higher Similarity makes their values closer. Case C has both high Scale and Similarity values and therefore both $K\xspace_{min}$ and $K\xspace_{max}$ are closer to $\overline{K\xspace}$ with similar values.
• Figure 4: Complete scheme of the automated generation of controller solutions for LfD-based VIC description, based on proposed condition assessment.
• Figure 5: Demonstrations of the validation trajectory and generated reference trajectory (a) together with the H-GP mean and confidence intervals for the X axis (b) and the corresponding compliance profile (c). Regions where a constant virtual force of 50[N] is applied in opposite directions are highlighted in the reference trajectory in (a) and represented as shadowed areas in (b) and (c), together with the points Q1 and Q2 where compliancy is evaluated.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• Proposition 3