Keita Sugibayashi*, Eiji Konaka (Meijo Univ. Japan) Design of three-dimensional binary manipulators based on the KS statistic and maximum empty circles

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Serial and Parallel Manipulator Two basic types of manipulators [1]https://robotics.kawasaki.com/ja1/xyz/jp/1804-03/ Links are connected in serial Serial link Parallel link Link Classification based on link type

Binary manipulator: Binary actuator and 3-bit module 3-bit module consists of three binary actuators ଷON/OFF patterns of binary actuators and corresponding shape

Binary manipulator: Serial connection of 3-bit modules • Advantages – Lightweight – Redundant against module failures – Remote control through low- bandwidth communication paths [2]Konaka, SICE Magazine, Vol.56, No.7, pp.503-508, 2017 (in Japanese) modules = ଷ஻ON/OFF patterns

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Design of distribution of reachable end- effecter position • Reachable points: discrete set • Depend on ON/OFF length of each binary actuator Distribution of reachable end-effector position ON/OFF length Distribution Long Wide Short Narrow

Our previous work: ON/OFF length design • ON/OFF length design for one connected workspace – Novel performance index for distribution uniformity – GA-based stochastic optimization [3]Sugibayashi, Konaka. IEICE Tech. Rep. ,Vol.122, no.435, MSS2022-89, pp126-131, 2023. (in Japanese)

Today’s topic: pick-and-place task • Pick-and-place: main task of robot manipulators – Picking and placing: often separated into different areas • Our previous study[3] should be extended into multiple workspace areas

Research objective • Main objective – Design ON/OFF length of each binary actuator – Minimize positioning error – Workspace is separated into two areas • Method – Performance index is modified • Maximum empty circle (MEC) • Kolmogorov-Smirnov (KS) statistic – GA-based stochastic optimization • Numerical experiments

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Maximum empty circle (MEC) large small low high radius density Small MEC radius = small positioning error & high density

Measuring uniformity: KS statistic (non-ideal) distribution Ideal distribution • Ideal distribution: uniform distribution on workspace KS statistic can measure uniformity of distribution

Definition of KS statistic • : Hypothetical CDF(Cumulative distribution function) • CDF of uniform distribution • : Empirical CDF • CDF of reachable points of the manipulator

Definition of KS statistic • : Hypothetical CDF(Cumulative distribution function) • CDF of uniform distribution • : Empirical CDF • CDF of reachable points of the manipulator

KS statistic for separated workspace in 3D space

Performance index Radius of MEC KS-Statistic • Small MEC = high density • Small KS statistic = close to uniform distribution • Design of binary manipulator = minimize

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Proposed method: GA-based stochastic optimization • Genetic Algorithm (GA) – Life-inspired optimization algorithm – Candidate of solution is coded as gene – Genes are evaluated by the performance index • Often called “fitness function” in GA-context – Generate new and potentially good genes by selection and genetic operation – Selection: bad genes are removed from population – Genetic Operation: crossover, mutation • Coding of gene – Length of expansion/contraction for each binary actuator

Proposed algorithm (1,2/8) Population of initial individuals 1. Define the performance index as the fitness function 2. Generate initial individuals with individuals. ・・・individual ・・・gene

Proposed algorithm (3/8) fitness 0.3 0.4 0.6 0.5 0.7 3. Calculate fitness for each individual. ・・・individual ・・・gene

Proposed algorithm (4/8) 4. Selection: Lower 40% of population are removed (“Elite strategy”) 5 4 0.7 0.6 3 2 1 rank individual 0.5 0.4 0.3 fitness selection ・・・individual ・・・gene

Proposed algorithm (5/8) parent 1 parent 2 offspring 2 offspring 1 5. Genetic operation: Two-point crossover between elites

Proposed algorithm (6/8) mutation ! 6. Genetic operation: Mutation of genes at random

Proposed algorithm (7/8) generation 7. Next generation is made. ・・・individual ・・・gene

Proposed algorithm (8/8) generation ・・・ generation generation ・・・individual ・・・gene 8. Repeat G generations. Output the best individual as the solution.

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Numerical experiment setup Problem: Design ON/OFF lengths for a binary manipulator with 4 modules Value Symbol Weight ଵ ଶ Population Mutation ratio Generations Parameters used in GA modules with 12 actuators Workspace

Main result (by generation) Designed distribution Performance index value Workspace

Result in detail (1st gen.) Workspace Workspace

Result in detail (500th gen.) Workspace Workspace

Result in detail (comparison) ௠௘௖ ௠௘௖ 1st gen. 500th gen. Proposed method is useful in binary manipulator design

Outline 1 2 3 4 5 Background Problem setup Proposed method Numerical experiment Summary Summary Work-in-progress Setup and result Performance index and optimization Problem and objective Binary manipulator

Summary • Extend KS statistic to separated workspace • Marginal distribution for x-, y-, and z-axes • The proposed performance index worked better. • GA-based stochastic optimization can find good design for binary manipulator.

Work-in-progress • Trajectory control with obstacle avoidance

Sugibayashi*,Konaka , “Design of three-dimensional binary manipulators based on the KS statistic and maximum empty circles” IEEE IECON2023@Marina Bay Sands Many thanks ! And welcome questions and comments !